Nonlineardynamicbehaviorofa conicalspringwithtopmass · Nonlineardynamicbehaviorofa conicalspringwithtopmass ing.L.J ... 4 Dynamic modeling and pre-design of the ... 1.1 Nonlinear

Nonlinear dynamic behavior of a

conical spring with top mass

ing. L.J.A. den Boer

DCT 2009.007

Master’s thesis

Coach: dr.ir. R.H.B. Fey

Supervisor: prof.dr. H. Nijmeijer

Eindhoven University of TechnologyDepartment Mechanical Engineering

Dynamics and Control group

Eindhoven, January, 2009

II

Contents

Notations VIII

Samenvatting IX

Abstract XI

1 Introduction 1

1.1 Nonlinear coil springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Literature study 5

2.1 Conical springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Rodriguez’ derivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Static analysis of the conical spring 13

3.1 Considerations for obtaining a strong nonlinear spring characteristic. . . . . . . . . 13

3.2 Parameter studies for the nonlinear load-deflection characteristic . . . . . . . . . . . . 16

3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

III

4 Dynamic modeling and pre-design of the experimental setup 21

4.1 Dynamic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 Linear dynamic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3 Additional constraints on design parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.4 A set of parameter values satisfying the constraints . . . . . . . . . . . . . . . . . . . . . . . 26

4.5 Parameter study with respect to setup steady-state-dynamics. . . . . . . . . . . . . . . . 27

4.6 Pre-design of the experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Design and parameter identification of the experimental setup 47

5.1 Design of the experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2 Identification shaker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.3 Identification top mass-conical spring system . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6 Numerical and experimental results 59

6.1 Static spring characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.2 Linear dynamic analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.3 Frequency amplitude plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.4 Domains of attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.5 Detailed steady-state analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7 Conclusions and recommendations 75

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Bibliography 78

A Conical spring model of Wu and Hsu 79

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

IV

A.2 Linear general helical spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

A.3 Linear conical spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

A.4 Nonlinear conical spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

B Flexible connection between shaker and lower guiding 85

C Parameter identification using least square fit method 87

V

VI

Notations

B frequency range where frequency hysteresis occurs [Hz]

d coil diameter [m]d1 damping constant of conical spring [Ns/m]

d1,l damping constant of conical spring A [Ns/m]

d1,f extra damping of conical spring A, chosen to get a good fit [Ns/m]

d2 damping constant of shaker [Ns/m]

D mean coil diameter [m]

D1 mean spring diameter of smallest initially active coil [m]

D2 mean spring diameter of biggest initially active coil [m]

fi eigenfrequency of mode i [Hz]

f1 eigenfrequency where the shaker table shows dominant deformation [Hz]

f2 eigenfrequency where the conical spring shows dominant deformation [Hz]

F compression force [N]Fcs(x) nonlinear conical spring force [N]

FC maximum compression force [N]

FT transition compression force (change between linear and nonlinear regime) [N]

G shear modulus [N/m2]

H initial spring height [m]

H1 height of biggest coil of spring A [m]

H2 height of smallest coil of spring A [m]

k1 linear stiffness of spring [N/m]

k2 linear stiffness of shaker [N/m]

m1 top mass of spring (50 % of mspring is included) [kg]

m2 mass of shaker (50 % of mspring is included) [kg]mspring mass of spring [kg]

n initial number of coils [coils]

nD number of coils, as a continuous variable running from 0 to n [coils]

nf number of free coils (variable during nonlinear regime) [coils]

PSD Power Spectral Density [m2/Hz]

ui eigencolumn of mode i [-]

U input voltage amplifier [V]

V matrix with eigencolumns [-]

W amplifier gain [N/V]

VII

x total spring deflection [m]

xf total deflection of free coils [m]

xs total deflection of solid coils [m]

x0 coils compressed at ground for x>x0 [m]

y absolute displacement of shaker [m]

β total angular deflection of one end of the coil with respect to the other end [o]

δf elementary deflection of solid coils [m]δs deflection of solid coils [m]

λi eigenvalue of mode i [-]

ζ1 dimensionless damping coefficient of spring [-]

ζ1,l dimensionless damping coefficient of spring A [-]

ζ2 dimensionless damping coefficient of shaker [-]

τmax maximum shear stress [N/m2]

ρ mass density [kg/m3]

ωi eigenmode of mode i [rad/s]

viii

Samenvatting

Wanneer conische veren in dynamische systemen worden toegepast is het belangrijk om het

effect van conische veren op het dynamisch gedrag te kennen. Dit omdat conische veren niet-

linear gedrag vertonen wat ontstaat doordat de actieve windingen bij compressie geleidelijk gaanaanliggen. In de literatuur is weinig te vinden over de dynamica van conische veren. In deze

thesis is het niet-lineaire dynamische gedrag van conische veren met topmassa onderzocht. Deze

conische veer heeft een constante spoed en is telescopisch, wat inhoudt dat elke actieve winding

in de volgende winding valt tijdens het samendrukken.

Uit de literatuur is een veerkracht-indrukking relatie met een constante spoed gekozen en geïm-plementeerd in een statisch model. Daarna is een dynamisch systeem met twee vrijheidsgraden

gemodelleerd. De eerste graad van vrijheid is de verplaatsing van de top massa van het conische

veer systeem. De tweede vrijheidsgraad is de verplaatsing van de shaker tafel. Vervolgens is een

numerieke niet-lineaire dynamische analyse gedaan om een frequentie amplitude plot te krijgen.

Het gelineariseerde systeem heeft twee eigenmodes. Omdat we hoofdzakelijk geïnteresseerd zijn

in het niet-lineaire gedrag van het top massa-conische veer systeem, concentreren we ons op de

mode die de grootste compressie van de conische veer geeft. Een test opstelling is gerealiseerd

welke bestaat uit een elektromagnetische shaker met stroomversterker, een conische veer, een

top massa, lucht lagers, een PC en een kracht- en verplaatsingssensor. De conische veer is op de

shaker tafel geplaatst. The shaker tafel oefent een harmonische kracht met een voorgeschreven

frequentie uit op de onderzijde van de conische veer. De top massa is bovenop de conische veerbevestigd. Vervolgens is de statische veerkracht-verplaatsing relatie experimenteel bepaald in de

test setup. De massa, stijfheid en lineaire demping van de shaker en de lineaire demping van

het top massa-conische veer systeem zijn experimenteel geïdentificeerd. Als laatste zijn de the-

oretische en experimentele frequentie amplitude plots bepaald en vergeleken, net als de Power

Spectral Density plots van verschillende werkpunten.

Tussen de theoretische en experimentele statische veerkarakteristieken is geen goede kwanti-

tatieve overeenkomst gevonden. Dit verschil is veroorzaakt doordat de experimentele veer geen

constante spoed heeft. Dit blijkt fabricagetechnisch moeilijk realiseerbaar te zijn. Daarom is in

het model niet de theoretische statische veerkarakteristiek gebruikt, maar een fit van de experi-

mentele statische veerkarakteristiek. De frequentie amplitude plots van het model en de exper-

imenten komen kwalitatief en kwantitatief overeen wanneer extra demping wordt toegevoegd.

IX

x

Abstract

When conical springs are used in dynamic systems it is important to know the effect of conical

springs on the dynamic behavior. This because conical springs show nonlinear behavior, which

occurs when the active coils gradually are compressed to the ground. In literature, research onthe dynamics of conical springs seems limited. In this thesis the nonlinear dynamic behavior of

conical springs carrying a top mass is investigated. The conical springs in this thesis have a con-

stant pitch and are telescoping, which means that every active coil fits in the following coil when

coils are compressed. This research exists of a theoretical/numerical part and an experimental

part.

From the literature a load-deflection relation with a constant pitch is chosen and implemented in a

static model. Then a dynamic two degree of freedom system is modeled. One degree of freedom,

the displacement of the top mass of the conical spring system, is free, whereas the other is the

displacement of the shaker table. Subsequently, a numerical nonlinear dynamic analysis is made

to get the frequency amplitude plot. The linearized system shows two eigenmodes. Since we are

mainly interested in the nonlinear dynamic behavior of the top mass-conical spring system, we

focus on that mode, which gives largest deflections of the conical spring. A dynamic test setup

is designed and realized, which consists of an electromagnetic shaker with power amplifier, a

conical spring, a top mass, air bearings, a PC and a force and a displacement sensor. The conical

spring is placed on a shaker table. The shaker table exerts a harmonic force with a prescribed

frequency to the bottom of the conical spring. On top of the conical spring a top mass is placed.Next, the static load defection relation of the conical spring is experimentally determined in the

test setup. Then, the mass, stiffness and linear damping of the shaker and the linear damping

of the top mass-conical spring system are experimentally determined and identified. Finally the

theoretical and experimental frequency amplitude plots are obtained and compared, as well as

Power Spectral Density plots for several operation points.

From the static theoretical and experimental spring characteristic it can be concluded that they do

not match quantitatively. The mismatch is caused by the fact that the experimental spring does

not have a constant pitch. For manufacturing reasons, this seems difficult to realize. In themodel

therefore, the theoretical static spring characteristic is not used, but a fit of the experimental static

spring characteristic. The frequency amplitude plots of the model and the experiments match

qualitatively as well as quantitatively, when extra damping is added.

XI

xii

Chapter 1

Introduction

1.1 Nonlinear coil springs

Helical springs are often used in mechanical systems. They can be designed in such a way that

they show nonlinear behavior. This means that the spring stiffness is not constant but dependson the compression. This nonlinear behavior occurs when the number of active coils decreases

or increases with varying compression. The nonlinear behavior of a spring can be achieved by

- varying the coil diameter

- varying the pitch- varying the mean spring diameter in axial direction

It is evident that also combinations of these three options can be used.

The research of this thesis focusses on conical springs, so the nonlinear behavior will be achieved

by varying themean spring diameter in axial direction. Conical springs can have some advantages

compared to cylindrical springs. In nontelescoping springs, the coils stack one above the other

during compression (figure 1.1, right). Telescoping springs can be designed in such a way (figure

1.1, left) that every active coil fits in the following coil when coils are compressed. The advantage of

this telescoping spring is that the spring height, when fully compressed, is only the coil diameter.

Another advantage is that conical springs can have a higher sideways stability, so they will betterresist buckling.

For this research, a conical spring with a constant pitch and a constant coil diameter is used. A

telescoping spring is used, because it has a stronger nonlinearity than the nontelescoping spring.

Also, a telescoping spring is investigated because, as stated before, this type has the advantagethat it has a lower installation height. The conical spring parameters used in this thesis are (see

figures 1.1 and 1.2)

1

1. introduction

d : coil diameter [m]

D1 : mean spring diameter of smallest initially active coil [m]

D2 : mean spring diameter of biggest initially active coil [m]

n : initial number of coils [coils]

H : initial spring height [m]

x : axial spring deflection [m]

F : axial compression force (load) [N]

D1

D2d

Figure 1.1: Left: telescoping conical spring. Right: nontelescoping conical spring.

F

x

H

Figure 1.2: Compression of a telescoping conical spring.

1.2 Motivation

Conical springs can be used in many different mechanisms like engine valves, railway and au-

tomotive (figure 1.3) suspension systems or as a buffer for an elevator. Conical springs also can

be used on a smaller scale, for example as a microactuator in microelectromechanical systems

(MEMS) (figure 1.4). This application is called the conical spring actuator which is capable of

stepwise displacements vertical to the substrate [Hata et al., 2003]. Conical springs are often cho-

sen for one special characteristic, for example their ability to telescope, meaning they use very

little space at maximum compression while storing as much energy as cylindrical springs. This

can be useful to lower a car (the center of mass is lowered) which improves its performance.

Another advantage is that a specific spring characteristic can be prescribed (by varying the pitchand the mean spring diameters). This can be desirable for example for a sports car which needs

a higher stiffness when it passes a corner.

2

1. introduction

Figure 1.3: Conical springs used for automotive suspension. Left: old application in a T-Ford[Triddle, 2007]. Right: modern application in racing equipment [Chevy, 2008].

Figure 1.4: Conical spring actuator (MEMS).

1.3 Objective

When conical springs are used in dynamic systems it is important to know the effect of conical

springs on the dynamic behavior. This because of the nonlinear behavior of the conical spring.

In literature, research on the dynamics of conical springs seems limited. The main objective of

this thesis is to investigate the nonlinear dynamic behavior of conical springs carrying a top mass.

The top mass will be heavy enough to justify neglection of inertia properties of the conical springitself.

1.4 Outline

The outline of this thesis is as follows. In chapter 2, a literature study about theoretical deriva-

tions of the load-deflection relation of conical springs will be presented. In chapter 3, a suitableload-deflection relation will be chosen and static analysis of the conical spring will be performed.

In chapter 4, a dynamic two degree of freedom system will be modeled. One degree of freedom,the displacement of the top mass of the conical spring system, is free, whereas the other is the

displacement of the shaker table. Subsequently, a numerical nonlinear dynamic analysis is made

3

1. introduction

to get the frequency amplitude plot. Chapter 5, starts with the design and realization of a dy-namic test setup, which contains a shaker which will be driven by a harmonic excitation signal.

First the static load defection relation is experimentally determined in the test setup. Next, the

mass, stiffness and linear damping of the shaker and the linear damping of the top mass-conical

spring system are experimentally determined and identified. In chapter 6, first the theoretical andexperimental static load-deflection relations are compared and a fitted load-deflection relation is

derived. Next, the results of the numerical model and of the experiments will be presented and

compared. Finally in chapter 7, conclusions and recommendations will be given.

4

Chapter 2

Literature study

2.1 Conical springs

Many books are written about springs, but only a few discuss conical springs. In these books,

often only the linear behavior of conical springs is treated. Wahl [1963] has derived the load-

deflection relation of the helical cylindrical spring. Timoshenko [1966] extended this relation to

a conical spring, but only its linear behavior. Only two papers are found, in which the nonlinear

behavior of the conical springs is discussed. The first paper is "Analytical behavior law for a

constant pitch conical compression spring" by Rodriguez et al. [2006] and the second paper

is "Modeling the static and dynamic behavior of a conical spring by considering the coil close

and damping effect" by Wu and Hsu [1998]. This nonlinear behavior occurs when the numberof active coils decreases or increases with varying compression. During compression, first the

biggest coil starts gradually to bottom which causes a gradually stiffening of the spring. A paper

was found about "Optimal design of conical springs" [Paredes and Rodriguez, 2008]. This paper

is not discussed because is was published when this thesis was almost finished.

In literature, very little about the dynamic research of conical springs can be found. Only Wu andHsu propose a basic equation of motion, but do not give experimental verification of this model.

In this chapter, the two papers of Rodriguez et al. and Wu and Hsu will be discussed. The

derivation of the load-deflection relation according to Rodriguez et al. is shown in section 2.2.

The derivation according toWu andHsu is shown in appendix A, as the static and dynamicmodelin this thesis are not based on this derivation. In table 2.1, the papers are compared globally.

The spring dimensions used in this chapter are depicted in figure 1.1 and 1.2.

2.2 Rodriguez’ derivation

First the load-deflection relation for a linear helical cylindrical spring will be derived (section

2.2.1) according to Wahl [1963]. This derivation will be extended for a linear conical spring (sec-

5

2. literature study

Table 2.1: Global comparison of nonlinear conical spring models of Rodriguez et al. [2006] andWu and Hsu [1998].

Rodriguez et al. [2006] Wu and Hsu [1998]

conical spring type telescoping and nontelescoping nontelescoping

coil type constant pitch constant pitch angle

load-deflection relation continuous relation discrete relation

strain energy terms - torsion - torsion- bending- tension and compression- direct shear

static experimental verification very good 4.6 % error

tion 2.2.2), according to Timoshenko [1966]. Finally, a continuous relation will be derived for anonlinear conical spring (section 2.2.3), according to Rodriguez et al. [2006].

2.2.1 Linear cylindrical spring

According to Wahl’s assumptions, the derivation is accurate for cases where deflections per coil

in axial direction of the spring are not too large (not more than half the mean spring radius) and

pitch angles are less than 10◦. The reason for this is partly that the pitch angle is assumed zero

and partly because the coil radius changes with deflection.

xF

d

β

Figure 2.1: Spring coil behaves essentially as a straight bar in pure torsion.

In this elementary theory, curvature and direct shear effects are neglected. This theory is based on

the assumption that an element of an axially loaded helical cylindrical spring behaves essentially

as a straight bar in pure torsion (figure 2.1). Each element of the spring coil is assumed to be

subjected to a torque M = FD/2 acting about the spring center, where D is the mean spring

diameter. A linear shear (τ ) distribution along the bar radius is assumed, as shown in figure 2.2.

At a distance ρ from the center O the shearing stress will be τ = 2ρτmax/d. The moment dMtaken from a ring with width dρ at a radius ρ will be dM = 4πτmaxρ3dρ/d. The total moment

FD/2 then becomes

FD

2=

∫ d/2

0dM =

∫ d/2

0

4πτmaxρ3dρ

d=

πd3τmax

16(2.1)

6

2. literature study

Figure 2.2: Cross-sectional element of spring under torsion. [Wahl, 1963]

and gives a maximum shear stress of

τmax =8FD

πd3(2.2)

An element ab (figure 2.2) on the surface of the bar and parallel to the coil axis is considered.

After deformation, this element will rotate through a small angle γ (figure 2.2) to the position ac.From elastic theory, this angle γ will be

γ =τmax

G=

8FD

πd3G(2.3)

with G being the shear modulus. Since the distance bc = γdxab for small angles such as consid-

ered here, the elementary angle dβ through which one cross section rotates with respect to the

other will be equal to 2γdxab/d. Again assuming that the spring may be considered as a straight

bar of length L = πnD, the total angular deflection β (figure 2.1) of one end of the coil with

respect to the other end becomes

β =

∫ πnD

0

2γdxab

d=

∫ πnD

0

16FDdxab

πd4G=

16FD2n

Gd4(2.4)

The effective moment arm of the load F is equal to D/2, so the total axial deflection of the spring

at load F will be (figure 2.1)

x =βD

2=

8FD3n

Gd4(2.5)

This is the commonly used formula for cylindrical spring deflections.

2.2.2 Linear conical spring

The formula for cylindrical helical springs will now be extended to a formula for conical helical

springs without coil close (implying that the spring will behave linearly). Now the variable diam-

7

2. literature study

eter of the spring as function of coil number nD is (nD being the continuous variable, runningfrom 0 to n)

D(nD) = D1 +(D2 − D1)nD

n(2.6)

The total axial deflection of the spring will be obtained from (2.5) and (2.6) and gives

x =8F

Gd4

∫ n

0

[

D1 +(D2 − D1)nD

n

]3

dnD =2Fn(D2

1 + D22)(D1 + D2)

Gd4(2.7)

2.2.3 Nonlinear conical spring

In this subsection, telescoping conical springs with a constant pitch are considered. In compres-

sion, these springs show a two regime load deflection relation (see figure 2.3), where the first

regime is linear and the second regime is nonlinear. In extension, the load-deflection relation for

these springs is linear.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

500

1000

1500

2000

2500

OT

C

x [m]

F[N

]

Figure 2.3: Telescoping conical spring characteristic. Point O: no compression. Transition pointT: start of active coil-ground contact; start of nonlinear behavior. Point C: maximal compression(all active coils in contact with the ground).

Linear regimeIn the linear regime of the deflection curve (from point O to T in figure 2.3), the largest coil is freeso deflects, as all other coils of the spring. Thus the load-deflection relation is linear according to

(2.7).

8

2. literature study

H

n nf

Figure 2.4: Telescoping spring. Left: linear behavior, n free coils. Middle: nonlinear behavior,nf < n free coils. Right: maximum deflection, nf = 0. [Rodriguez et al., 2006]

Nonlinear regimeAlong the nonlinear regime (from point T to C in figure 2.3), the active coils are gradually com-

pressed to the ground. During this regime, nf coils are free and n − nf coils are compressed

to the ground (these coils are called solid coils), see figure 2.4. The coil is considered as a in-

finitely number of elementary angular parts. When the first elementary part of the largest coil

has reached its maximum physical deflection, it starts to be a nonactive element of the spring.

This defines the transition point T. The first regime of compression then stops and the second

one begins. During the second regime of compression, nf continuously decreases from n to 0and leads to a gradual increase of the spring stiffness. This explains why this second regimeshows a nonlinear load-deflection relation.

The total conical spring deflection (x) at a certain axial load (F ), is an addition of the total ax-

ial deflection of the free coils (xf ) and the total axial deflection of the solid coils (xs). This is

approximated by an addition of all the elementary axial deflections of the free coils (δf ) and theelementary axial deflections of the solid coils (δs). This gives a total deflection of

x = xf + xs =

∫ nf

0δf (nD) +

∫ n

nf

δs (2.8)

Other algorithms use discretization of the coil into several angular parts. The deflection of the

spring for a given load is determined by adding the individual deflections of each part of a cylindri-

cal spring. Each individual deflection of each part is considered to be part of a cylindrical spring.

Each individual deflection is limited to its maximum geometrical value. The method introduced

in this chapter is based on the same principle as the other algorithms, but here discretization is

replaced by an integral approach (see (2.8)).

Every single elementary axial deflection of the free coils (δf ) can be written as (taking the load-

deflection relation of the cylindrical spring (2.5), where mean spring diameter D is replaced with

the variable diameter of the conical spring D(nD) (2.6))

δf (nD) =8F [D(nD)]3

Gd4dnD (2.9)

This derivation is based on a constant pitch, which implies that the axial distance between thecoils is constant. Therefore can be stated that, for every angular part, the elementary deflection of

the solid coils (δs) corresponds to the maximum geometrical elementary deflection. This can be

9

2. literature study

calculated as follows

δs =H

ndnD (2.10)

FT is the load for which the largest active coil (with local spring diameter D2) reaches its maxi-

mum deflection δs. So at the transition point T can be written

xf (n) = xs (2.11)

Using (2.9) and (2.10), this can be written as

8FT (D2)3

Gd4=

H

n(2.12)

so

FT =Gd4H

8D32n

(2.13)

On the conical spring load-deflection curve, the maximum point C defines the ultimate com-

pression state of the spring. FC is the load for which the smallest active coil (with local spring

diameter D1) reaches its maximum deflection H . So, like the transition point, this can be written

as

FC =Gd4H

8D31n

(2.14)

The elementary axial deflections of the free coils (δf ) have a variable number of coils nD, running

from 0 to n. Every single element reaches its maximum deflection at nD = nf coils (with nf is

the number of free coils). This corresponds with the elementary axial deflection of the solid coils

(δs), so

δf (nf ) = δs (2.15)

Using (2.6), (2.9) and (2.10), this can be written as

8F (D(nf ))3

Gd4=

H

n(2.16)

So that

nf =n

D2 − D1

[

(

HGd4

8Fn

)1/3

− D1

]

(2.17)

10

2. literature study

As nf is defined, the continues load-deflection relation can be written (using (2.8)),

x(F ) =2FD4

1n

Gd4(D2 − D1)

[

[

1 +

(

D2

D1− 1

)

nf

n

]4

− 1

]

+ H(

1 − nf

n

)

(2.18)

In this chapter the load-deflection relation for a conical spring with a constant pitch is derived,

see (2.17) and (2.18). This relation will be used for the analysis in this thesis.

11

2. literature study

12

Chapter 3

Static analysis of the conical spring

In this chapter, the static analysis is based on Rodriguez’ load-deflection relation for a conical

spring, as derived in chapter 2. The research of this thesis concerns an introductory study of the

steady-state behavior of a nonlinear conical spring. In order to increase the choice to see non-linear dynamic effects later on, preferably a spring with a strong nonlinear static load-deflection

characteristic has to be used. The objective of this chapter is to insight in the influence of conical

spring design parameters (d, D1, D2, n andH) on the nonlinear static load-deflection characteris-

tic. In section 3.1, considerations will be made to obtain a strong nonlinear spring characteristic.

In section 3.2, a parameter study will be performed for the nonlinear load-deflection characteristic

of a telescoping spring.

3.1 Considerations for obtaining a strong nonlinear spring characteristic

As stated before, for steady-state experiments to be carried out later on, a strong nonlinear load-

deflection characteristic is desired. First, the characteristic is only nonlinear from point T to C, see

figure 3.1. So, it would be desirable if the operating deflection range of the spring has a relatively

large nonlinear range, in other words xC −xT should be large with respect to xT −x0. As another

indicator of strong nonlinear behavior, the ratio between the first derivative, i.e. the stiffness, at

FC/2 (force at point C divided by 2) and the first derivative at FT (force at point T) could be used.Obviously, the choice of taking the stiffness at FC/2 is rather arbitrary, but can be motivated by

the observation that this point also lies within the working range of the spring, which is less likely

for point FC for example, because there the spring bottoms. A high value of this stiffness ratio is

wanted, because it indicates a relatively strong nonlinear spring characteristic. This relation will

be derived now.

According to Rodriguez’ approximation (see (2.17) and (2.18)) the load-deflection relation for the

13

3. static analysis of the conical spring

0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

500

1000

1500

2000

2500

OT

C

x [m]

F[N

]

Figure 3.1: Telescoping conical spring characteristic. Point O: no compression. Transition pointT: start of active coil-ground contact; start of nonlinear behavior. Point C: maximal compression(all active coils in contact with the ground)

nonlinear part of a telescoping conical spring is

x(F ) =2FD4

1n

Gd4(D2 − D1)

[

[

1 +

(

D2

D1− 1

)

nf

n

]4

− 1

]

+ H(

1 − nf

n

)

(3.1)

with

nf =n

D2 − D1

[

(

HGd4

8Fn

)1/3

− D1

]

(3.2)

Because it is a telescoping spring, it is allowed to use the initial spring height H in formula (3.1)

and (3.2)). The flexibility, i.e. the derivative of x(F ) (see (3.1) and (3.2)) with respect to F is

dx(F )

dF=

n/8(HGd4/F/n)4/3 − 2nD41

Gd4(D2 − D1)(3.3)

A numeric example of x(F ) and its first derivative, i.e. the flexibility, is shown in figure 3.2 (with

parameters values given in table 3.1). In the upper diagram of figure 3.2 it can be seen that, directlyafter point T, x(F ) changes significant (which indicates a strong nonlinear behavior) which is not

the case when point C is approached. In the lower diagram, the opposite phenomenon can be

14


0 500 1000 1500 2000 25000

0.005

0.01

0.015

T

C

0 500 1000 1500 2000 25000

0.5

1

1.5x 10

−4

T

C

x(F

)[m

]

F [N]

dx(F

)dF

[m/N

]

Figure 3.2: Up: spring characteristic (nonlinear part). Below: derivative of spring characteristicwith respect to F (nonlinear part).

Table 3.1: Spring parameters

parameter value

d 0.0025 mD1 0.010 mD2 0.036 mn 2.0 coilsH 0.0120 mG 82.0 GPa (spring steel)

seen. The forces FC/2 and FT are ((2.13) and (2.14))

FT =Gd4H

8D32n

(3.4)

FC/2 =Gd4H

16D31n

(3.5)

The flexibility at FT is (using (3.3) and (3.4))

dx(F )

dF

∣

∣

∣

∣

FT

=2n(D3

2 + D1D22 + D2

1D2 + D31)

Gd4(3.6)

15


The flexibility at FC/2 is (using (3.3) and (3.5))

dx(F )

dF

∣

∣

∣

∣

FC/2

=2nD4

1(24/3 − 1)

Gd4(D2 − D1)(3.7)

The flexibility ratio between the flexibilities at FC/2 and FT is

dx(F )dF

∣

∣

∣

FC/2

dx(F )dF

∣

∣

∣

FT

=D4

124/3 − D4

1

D42 − D4

1

(3.8)

In order to obtain a high (for strong nonlinear behavior) stiffness ratio, a low flexibility ratio is

desired. Therefore it can be concluded that for D1 a low value should be chosen and for D2

a high value. This is also physically easy to understand since a cylindrical spring with a large

spring diameter obviously has a lower stiffness than a cylindrical spring with a small spring

diameter. Apparently, the parameters d, n and H have no influence on (3.8). Obviously, they do

have influence on the nonlinear behavior, see (3.1) and (3.2). Moreover, there are constraints on

the choice of d, D1, D2, n and H for the spring to be of telescoping type. These constraints will

be given in chapter 4. For now, it suffices to mention that these constraints are fulfilled in all

telescopic springs considered in this chapter.

The flexibility at point T (see figure 3.1) can be calculated with the linear equation (2.7) and with

the nonlinear equations (3.1) and (3.2) at FT . To check if this is correct, the linear equation (2.7)

and the nonlinear equation (3.6) must correspond and this is indeed the case.

3.2 Parameter studies for the nonlinear load-deflection characteristic

In this section, a number of spring characteristics will be presented to get insight in the influence

of several telescoping spring parameters. The spring parameter values initially are chosen as in

table 3.1. The values of the parameters d, D1, D2, n and H are varied one at a time, as shown in

table 3.2. The varied parameter value has a value which is 20 % lower and 20 % higher than its

initial value.

In figure 3.3 it can be seen that when parameter d is increased, the total characteristic decreases

with F . Also it can be seen that when d is increased, xC − xT does not change with respect to

xT − x0. In figure 3.4 it can be seen that when parameter D1 is increased, point C increases with

F where point T remains unchanged. Also it can be seen that when D1 is increased, xC − xT

decreases with respect to xT −x0. In figure 3.5 it can be seen that when parameterD2 is increased,

point T increases with F where point C remains unchanged. Also it can be seen that when D2

is increased, xC − xT increases with respect to xT − x0. In figure 3.6 it can be seen that when

parameter n is increased, the total characteristic decreases with F . Also it can be seen that whenn is increased, xC − xT does not change with respect to xT − x0. In figure 3.7 it can be seen

that when parameter H is increased, the total characteristic increases with F . Also it can be seen

16


that when H is increased, xC − xT does not change with respect to xT − x0. From figure 3.3 till3.7 it can be concluded that a strong nonlinearity of the spring characteristic can be achieved by

choosing a low value for D1 and a high value for D2.

Table 3.2: Parameter influences on spring characteristic.

parameter initial varied (80 %) varied (120 %) figure

d 0.0025 m 0.0020 m 0.0030 m 3.3D1 0.010 m 0.008 m 0.012 m 3.4D2 0.036 m 0.0288 m 0.0432 m 3.5n 2.0 coils 1.6 coils 2.4 coils 3.6H 0.0120 m 0.0096 m 0.0144 m 3.7

In table 3.3, the parameter influences on the flexibility at FC/2 and FT , as well as their flexibility

ratio are given. When the parameter variations d at 80 % and 120 % are compared it can be seen

that the flexibilities vary, but the flexibility ratio remains for both variations 0.0091. Also when

the parameters n and H are varied, the flexibilities vary but the flexibility ratio remains 0.0091.When parameter D1 is varied, the flexibility changes as well as the flexibility ratio. The same

phenomenon is shown when D2 varied. That the variations of d, D1, D2, n and H influence the

flexibility is obvious. But remarkable is that the flexibility ratio does not change with d, n and H ,

but does change with D1 and D2. As a low flexibility ratio indicates a strong nonlinear behavior,

it can be concluded that a low value for parameter D1 should be chosen and a high value for D2.

Table 3.3: Parameter influences on the flexibility (at FC/2 and FT ) and the flexibility ratio.

parameter dx(F )dF

∣

∣

∣

FC/2

dx(F )dF

∣

∣

∣

FT

dx(F )dF

∣

∣

∣

FC/2

dx(F )dF

∣

∣

∣

FT

d (80 %) 1.782 · 10−6 m/N 1.958 · 10−4 m/N 0.0091D1 (80 %) 2.777 · 10−7 m/N 7.473 · 10−5 m/N 0.0037D2 (80 %) 1.010 · 10−6 m/N 4.503 · 10−5 m/N 0.0224n (80 %) 5.840 · 10−7 m/N 6.415 · 10−5 m/N 0.0091H (80 %) 7.300 · 10−7 m/N 8.019 · 10−5 m/N 0.0091

d (120 %) 3.520 · 10−7 m/N 3.867 · 10−5 m/N 0.0091D1 (120 %) 1.640 · 10−6 m/N 8.632 · 10−5 m/N 0.0190D2 (120 %) 5.717 · 10−7 m/N 1.306 · 10−4 m/N 0.0044n (120 %) 8.760 · 10−7 m/N 9.623 · 10−5 m/N 0.0091H (120 %) 7.300 · 10−7 m/N 8.019 · 10−5 m/N 0.0091

3.3 Conclusion

The conclusion of the analytical approach is that a strong nonlinearity of the spring characteristiccan be achieved by choosing a low value for D1 and a high value for D2.

17


0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

x [m]

F[N

]

Figure 3.3: Spring characteristic. Dashdot: d = 0.0020 m. Solid: d = 0.0025 m. Dashed:d = 0.0030 m.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

x [m]

F[N

]

Figure 3.4: Spring characteristic. Dashdot: D1 = 0.008 m. Solid: D1 = 0.010 m. Dashed:D1 = 0.012 m.

18


0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

500

1000

1500

2000

2500

x [m]

F[N

]

Figure 3.5: Spring characteristic. Dashdot: D2 = 0.0288 m. Solid: D2 = 0.036 m. Dashed:D2 = 0.0432 m.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

500

1000

1500

2000

2500

3000

3500

x [m]

F[N

]

Figure 3.6: Spring characteristic. Dashdot: n = 1.6 coils. Solid: n = 2.0 coils. Dashed: n = 2.4coils.

19


0 0.005 0.01 0.0150

500

1000

1500

2000

2500

3000

x [m]

F[N

]

Figure 3.7: Spring characteristic. Dashdot: H = 0.0096 m. Solid: H = 0.0120 m. Dashed:H = 0.0144 m.

20

Chapter 4

Dynamic modeling and pre-design of

the experimental setup

In this chapter, the dynamic model of the experimental setup will be derived. First the eigenfre-

quencies and eigenmodes of the top mass-conical spring system with shaker will be investigated

(the system shows linear behavior as long as no coils are compressed to the ground). Due to thepresence of shaker mass, damping and stiffness the system has two dof, so two modes will be

considered. Since we are mainly interested in the nonlinear dynamic behavior of the top mass-

conical spring system, we will focus on that mode, which gives largest deflections of the conical

spring. The two modes should not influence each other, which constrains the choice of the pa-

rameters. Next, the additional constraints on design parameters are discussed and a parameter

study of the setup and the conical spring is made. Finally a pre-design of the experimental setup

is provided by determining the parameters m1, m2, d, D1, D2, n, H and G.

As stated before, in this research the dynamic behavior of a telescoping spring with strong non-

linear behavior carrying a top mass will be examined. As a dynamic measure for the nonlinearity,

the frequency range for which frequency hysteresis occurs will be taken. In the experimental

setup the conical spring-top mass will be excited by a shaker.

4.1 Dynamic model

For the dynamic research, the conical spring is placed on a shaker. The shaker table will exert

a harmonic force with a prescribed frequency (f ) to the bottom of the conical spring. On top of

the conical spring a top mass (m1) is placed. The total system can therefore be modeled as a top

mass-conical spring system in combination with a shaker. This two degrees of freedom system ismodeled as shown in figure 4.1. The equations of motion are

m1(x + y) + d1x + Fcs(x) − m1g = 0 (4.1)

21

4. dynamic modeling and pre-design of the experimental setup

UW cos(2πft)

x

yk2

d1

d2

m1

m2

g

Figure 4.1: Top mass-conical spring system with shaker. (Two degrees of freedom).

m2y + d2y − d1x + k2y − Fcs(x) − m2g = UW cos(2πft) (4.2)

with

Fcs(x) =

k1x if x ≥ x0

2FD41n

Gd4(D2−D1)

[

[

1 +(

D2D1

− 1)

nf

n

]4− 1

]

+ H(

1 − nf

n

)

if x < x0(4.3)

the nonlinear restoring force of the conical telescoping spring and with

nf =n

D2 − D1

[

(

HGd4

8Fn

)1/3

− D1

]

(4.4)

the number of free coils (a continuous variable!) The linear stiffness of the conical spring is

k1 =2Fn(D2

1 + D22)(D1 + D2)

Gd4(4.5)

In this model, x is the displacement of the top mass relative to the shaker and y is the absolute

displacement of the shaker. x and y are 0 m for the unloaded situation, so with gravity taken as

g = 0 m/s2 (instead of g = 9.81 m/s2). The definitions of the other parameters are given in table

4.1. The dimensionless damping factor for the conical spring is defined by ζ1 = d1/(2√

m1k1).The dimensionless damping factor for the shaker is defined by ζ2 = d2/(2

√m2k2).

4.2 Linear dynamic model

In this section, generic expressions for the eigenfrequencies and eigenmodes of the undamped,

linear system, i.e. for x ≥ x0, will be derived. Now, (4.1) and (4.2) can be rearranged and

simplified to

Mq + Kq = 0 (4.6)

22


Table 4.1: System parameters

x0 m start of coil to ground compression for x < x0

m1 kg top massd1 Ns/m damping constant of springζ1 - dimensionless damping coefficient of spring

m2 kg mass of shakerd2 Ns/m damping constant of shakerζ2 - dimensionless damping coefficient of shakerk2 N/m stiffness of shaker

U V input voltage amplifierW N/V amplifier gain

where the column of generalized coordinates q is defined as

q =

[

x + yy

]

(4.7)

and the mass and stiffness matrix are given by

M =

[

m1 00 m2

]

, K =

[

k1 −k1

−k1 k1 + k2

]

(4.8)

The natural angular eigenfrequencies ωi = 2πfi and corresponding eigenmodes ui of the system

[de Kraker and van Campen, 2001] can be determined by solving the corresponding eigenvalueproblem

[

M − ω2K]

u = 0 (4.9)

Nontrivial solutions exist if and only if the determinant of the matrix[

M − ω2i K

]

vanishes

det[

M − ω2i K

]

= 0 (4.10)

with (4.10) being known as the characteristic equation or frequency equation.

The two angular eigenfrequencies of the system are

ω1,2 =1

[2m1m2]1/2

· (4.11)

[

k1m2 + k1m1 + k2m1 ±[

k21m

22 + 2k

21m1m2 − 2k1k2m1m2 + k

21m

21 + 2k1k2m

21 + k

22m

21

]1/2]1/2

and the two corresponding eigenmodes are

u1,2 = (4.12)

23


[

ω2

1,2m2

2−

k2

2−

ω2

1,2m1

2∓ 1

2

[

k22 + 2ω2

1,2k2m1 − 2ω21,2k2m2 + ω4

1,2m21 − 2ω4

1,2m1m2 + ω41,2m

22 + 4k2

1

]1/2

1

]

Keeping in mind the objectives of this project, the following observations are made:

1. Basically we are interested in the dynamic behavior of the telescoping spring carrying its

top mass. So we are interested in that frequency range which includes the eigenfrequency

for which the corresponding eigenmode shows dominant deformation of the telescoping

spring.

2. We are not interested in the frequency range including the eigenfrequency which corre-

sponding eigenmode shows dominant deformation of the shaker suspension.

Obviously, these observations put constraints on the design variables, because

1. The two modes described above should be uncoupled as much as possible. Their eigenfre-

quencies should not be too close to each other.

2. The eigenfrequency of the interesting eigenmode must lie in the excitation frequency range

of the shaker (0 Hz - 9000 Hz).

4.3 Additional constraints on design parameters

In this section, additional constraints on the design parameters of the experimental setup will be

discussed.

4.3.1 Minimum diameter D1

For manufacturing reasons of the spring, the minimal mean spring diameter (D1) must be at

least three times greater than the coil diameter (d).

D1 > 3d (4.13)

4.3.2 Minimum distance between coilsAs stated before, in this project the telescoping conical spring is used. This means that during

compression to the ground, every coil must fit in the following coil. When the spring is fully

compressed to the ground, the distance from one coil to the next (heart to heart) is D2−D12n . The

distance from one coil to the next (e) when the spring is fully compressed to the ground is e =D2−D1

2n − d. A spring is telescoping when e > 0 m, otherwise the spring is nontelescoping. So

the constraint can be written as

D2 − D1

2n− d > 0 m (4.14)

4.3.3 Minimum top massThe top mass m1 and the shaker table mass m2 are assumed to be constant and the spring mass

24


is neglected. Note further that in equation (4.1) and (4.2), the mass of the spring in motion is notconstant during compression because the coils reach the ground, i.e. the shaker table, gradually.

For this reason, the top mass m1 is taken at least as twenty times greater than the spring mass,

assuming that m2 > m1. If these constraints are fulfilled it is assumed that the influence of the

spring mass variation on the dynamics may be neglected. So

m1 > 20mspring (4.15)

with

mspring = π2d2(D1 + D2)nρ/8 (4.16)

where ρ = 7800 kg/m3 is the mass density of the (steel) spring.

4.3.4 Minimum eigenfrequencyWhen piezoelectric accelerometers sensors are used in the setup, important system frequencies

should not be lower that, say 5 Hz. Below this frequency, piezoelectric accelerometers are not

reliable. So the lowest eigenfrequency is chosen to be

f1 > 5 Hz (4.17)

These piezoelectric accelerometers were initially used for the identification of the setup (see ap-

pendix C). As the results of the identification are not reliable, a LVDT sensor and a force sensor

are used.

4.3.5 Maximum voltageThe shaker delivers a maximum force of 98 N. The current amplifier and the shaker (see chapter

5) have a amplifier gain of W = 28.54 N/V. This results in a maximal input voltage (U ) of the

current amplifier of 3.4 V. So

U < 3.4 V (4.18)

4.3.6 Influence of subharmonic resonance peaksThis subsection is only relevant if the interesting mode is the second one. In nonlinear systems

[Thompson and Stuart, 1986], subharmonic resonance peaks can occur near fi·j, for j = 2, 3, 4...etc. This means that subharmonic resonance peaks of the first mode can influence the resonance

peak of the second mode. An example of this can be seen in the figures 4.6 and 4.7 (the corre-

sponding parameter values can be found in table 4.2), where the resonance peak at f1 = 28 Hzcauses the subharmonic peak near f2 ≈ f1 · 2 = 56 Hz. This influence is not desirable, because

in fact only the dynamic behavior of the top mass-conical spring is of interest. So the frequency

peak of the top mass-conical spring system (at f2) should be taken in such a way that it does not

coincide with a subharmonic resonance peak caused by f1.

4.3.7 Influence of superharmonic resonance peaksThis subsection is only relevant if the interesting mode is the first one. In nonlinear systems,

25


superharmonic resonances may occur at fi/j, for j = 2, 3, 4... etc. So the situation where f1 ≈f2/j should be avoided to prevent the second mode influencing the first mode.

4.3.8 Constraints to shaker mass displacementThe maximum physical displacements of the shaker mass (y) is ±0.0088 m.

4.3.9 Constraints to compression of telescoping springThe compression of telescoping springs have a maximum of spring height H .

4.3.10 Frequency hysteresisAn important indicator for strong nonlinear dynamic behavior is the width of the frequency in-

terval in which frequency hysteresis occurs, indicated by B. B should be as large as possible and

will be further introduces in section 4.5.

Table 4.2: System parameter values.

parameter value

d 0.0025 mD1 0.010 mD2 0.036 mn 2.0 coilsH 0.0120 mG 82.0 GPa (spring steel)

x0 −0.0036 mm1 0.2 kgd1 2.99 Ns/mζ1 3.0 %

m2 1.0 kgd2 4.85 Ns/mζ2 1.77 %k2 18930 N/m

U 2.7 VW 28.54 N/V

4.4 A set of parameter values satisfying the constraints

Table 4.2 shows a set of parameter values satisfying the design constraints mentioned before.Note that the shaker related parameters b2 and k2 have fixed values. The identification of these

parameters will be discussed in chapter 5. The parameter values from table 4.2 result in the

following eigenfrequencies and eigenmodes

f1 =ω1

2π= 20.8 Hz, f2 =

ω2

2π= 48.3 Hz (4.19)

26


Table 4.3: Verification of the constraints for a set of parameters as in table 4.2.

section verification of the constraint

(4.3.1) fulfilled, because D1 = 0.010 > 3d = 0.0075 m

(4.3.2) fulfilled, because D2−D12n − d = 0.004 m > 0 m

(4.3.3) fulfilled, because m2 > m1 and m1 = 0.2 > 20mspring = 0.11 kg(4.3.4) fulfilled, because f1 ≈ 20.8 > 5 Hz(4.3.5) fulfilled, because U = 2.7 < 3.4 V(4.3.6) fulfilled, f2 ≈ 48.3 Hz does not coincide with f1 · 2 ≈ 41.6 and f1 · 3 ≈ 62.4 Hz(4.3.7) Not relevant because the second mode is of interest.(4.3.8) fulfilled, because ymax ≈ 0.003 < 0.0088 m (see figure 4.3)(4.3.9) fulfilled, because xmax ≈ 0.0115 < 0.012 m(4.3.10) this will be discussed in the next sections.

u1 =

[

1.31

]

, u2 =

[

−41

]

(4.20)

From the eigenmodes it can be concluded that the second mode is the mode of interest because

here the deformation of the telescoping spring is dominant. Table 4.3 shows to which extent

the constraints are fulfilled for the parameter values from table 4.2. It is important to note that

the damping parameter d1 (which can be translated to ζ1) is estimated at this moment since an

experimental set-up is not available yet to identify the value of the damping parameter d1 (or ζ1).

As we will see in coming chapters, in the experiments ζ1 will be identified to be much lower than3 %. As a consequence, the needed excitation voltage (U ) will also appear to be much lower than

the values used in this chapter.

4.5 Parameter study with respect to setup steady-state-dynamics

In this section, the influence of different parameters on the steady-state dynamics is investigated.

This is done by solving the equations of motion for a certain excitation frequency range (equations

(4.1) and (4.2)) using the ordinary differential equation solver ODE45 in Matlab [Matlab, 2007].

Starting with zero initial conditions and a low excitation frequency, first the excitation frequency is

slowly increased using small equidistant frequency steps (sweep up). Subsequently the excitation

frequency is slowly decreased again using small discrete frequency steps (sweep down). The

end conditions of the previous frequency step (x, x, y and y) are each time taken as the initial

conditions for the next frequency step. The maximum and minimum amplitudes of the steady-

state solutions are plotted in the frequency amplitude plot (for example figure 4.2 and 4.3). In

these plots it can be seen that in some frequency ranges different solutions occur during thesweep up and sweep down for the same excitation frequency.

In the parameter studies presented in the following subsections, the width of the frequency in-

terval where hysteresis occurs, i.e. where different solutions are found, is indicated by B (see for

example figure 4.3). A high value for B indicates a high amount of nonlinearity, so we will try tomaximize B. Furthermore, in all analyses the excitation voltage U will be adjusted so that at one

frequency maximum compression of the telescoping spring occurs, whereas simultaneously the

27


Table 4.4: Different configurations.

configuration 1 2 3 4 5 6 7

figure 4.2,4.3 4.4,4.5 4.6,4.7 4.8, 4.9 4.10, 4.11 4.12, 4.13 4.14, 4.15

m1 [kg] 0.2 0.24 − − − − −m2 [kg] 1.0 − 0.3 − − − −d [m] 0.0025 − − 0.0023 − − −

D1 [m] 0.010 − − − − − −D2 [m] 0.036 − − − 0.0432 − −n [coils] 2.0 − − − − 2.4 −H [m] 0.012 − − − − − 0.0144

U [V] 2.7 2.3 1.8 2.2 2.0 2.6 3.4B [Hz] 6.7 5.8 5.2 6.9 7.0 7.5 7.1

constraint on the shaker mass displacement is fulfilled. Maximum compression of the telescop-

ing spring is desired to appeal to the nonlinear behavior of the spring as much as possible. In

the configurations which will be discussed in the following subsections all parameter values will

be according to configuration 1 in table 4.4 except for changed parameter values which will be

indicated. Parameters m1, d, D2, n and H are increased by a factor 1.2. As parameter d seems to

be a sensitive parameter, d could not be increased or decreased by factor 1.2 because the second

mode (f2) would then be influenced by the first (f1 ·2) and second (f1 ·3) subharmonic resonance

peak of the first mode. Therefore parameter d is decreased by a factor 1.09. Parameter m2 is

decreased from 1.0 kg to 0.3 kg to see the influence when the shaker has its lowest possible mass.

The influence of parameter D1 is not discussed separately, as its influence is opposite to D2. Asmay be clear now, a parameter variation is usually accompanied by a change in excitation voltage.

Damping constant d1 is increased till B has almost disappeared.

4.5.1 Influence of top mass m1

The influence of the top mass on the steady-state dynamics is investigated. The top mass m1 is

increased from 0.2 kg to 0.24 kg, see configurations 1 and 2 in table 4.4. When the top mass

is increased, the first resonance frequency does not change much, only the second resonancefrequency decreases, compare figures 4.2 and 4.4. As the resonance peaks move away from each

other, they influence each other less. This is preferable because only the second resonance peak

is of interest. B decreases from 6.7 Hz to 5.8 Hz, compare figure 4.3 and 4.5. The conclusion is

that a lower top mass is preferred.

4.5.2 Influence of shaker mass m2

The influence of the shaker mass on the steady-state dynamics is investigated. The shaker mass

m2 is decreased from 1.0 kg to 0.3 kg, see configurations 1 and 3 in table 4.4. When the shakermass is decreased, both resonance frequencies increase, compare figures 4.2 and 4.6. B de-

creases from 6.7 Hz to 5.2 Hz, compare figures 4.3 and 4.7, so a higher shaker mass is preferred.

28


4.5.3 Influence of d

The influence of the coil diameter on the steady-state dynamics is investigated. The coil diameter

d is decreased from 0.0025 m to 0.0023 m, see configurations 1 and 4 in table 4.4. Doing so, the

first resonance frequency does not changemuch, only the second resonance frequency decreases,

compare figures 4.2 and 4.8. B increases form 6.7 Hz to 6.9 Hz, compare figures 4.3 and 4.9.

The conclusion is that a lower coil diameter is preferred.

4.5.4 Influence of D2

The influence of mean diameter on the steady-state dynamics is investigated. The mean coil

diameter D2 is increased from 0.036 to 0.0432 m, see configurations 1 and 5 in table 4.4. Doing

so, the first resonance frequency does not change much, only the second resonance frequency

decreases, compare figures 4.2 and 4.10. B increases form 6.7 Hz to 7.0 Hz, compare figures

4.3 and 4.11. The conclusion is that a higher mean coil diameter D2 is preferred. As D1 has the

opposite influence as D2, a lower mean coil diameter D1 is preferred.

4.5.5 Influence of n

The influence of the number of coils on the steady-state dynamics is investigated. The number

of coils n is increased from 2.0 coils to 2.4 coils, see configurations 1 and 6 in table 4.4. Doing

so, the first resonance frequency does not change much, only the second resonance frequency

decreases, compare figures 4.2 and 4.12. B increases form 6.7 Hz to 7.5 Hz, compare figures 4.3

and 4.13. The conclusion is that a higher number of coils is preferred.

4.5.6 Influence of H

The influence of the spring height on the steady-state dynamics is investigated. The spring height

H is increased from 0.012 m to 0.0144 m, see configurations 1 and 7 in table 4.4. Doing so, both

resonance frequencies do not change much, compare figures 4.2 and 4.14. B increases form

6.7 Hz to 7.1 Hz, compare figures 4.3 and 4.15. The conclusion is that a higher spring height ispreferred.

4.5.7 Influence of d1

Finally, the influence of the damping constant of the spring on the steady-state dynamics is inves-

tigated. As reference, configuration 1 is taken (table 4.4). The damping constant d1 is increased

from 2.99 Ns/m to 15.00 Ns/m and the input voltage U from 2.7 V to 12.5 V, see figure 4.16.Doing so, B has decreased from 6.7 Hz to almost 0 Hz. It is obvious that a low value of the

damping constant d1 is preferred.

29


4.6 Pre-design of the experimental setup

For the pre-design of the experimental setup, the parameters m1, m2, d, D1, D2, n, H and Gmust be chosen. When a system with high value for B is aimed, the system parameters must be

chosen with keeping the following points in mind.

• As the second mode is of interest, the influence of the fist mode must be minimized.Therefore a big frequency difference is desirable between first and second mode. This can

be achieved by a stiff spring, which is a result of the choice of the parameters d, D1, D2,

n and G (chapter 3). These parameters must be chosen with keeping in mind that a lower

coil d, a lower D1, a higher D2, a higher n and a higher H are preferred.

• From this chapter can be concluded that a lower top mass and a higher shaker mass are

preferred.

The resonance peak of the second mode, which is of interest, is chosen in between the first (at

f1 · 2) and the second (at f1 · 3) subharmonic resonance peak of the first mode. The resonance

peak of the second mode is not chosen in between the resonance peak of the first mode (at f1)and the first subharmonic resonance peak (at f1 ·2) of the first mode, because the resonance peak

of the first mode would influence the dynamics of the second mode too much.

The mass of the shaker (m2) is taken 1.0 kg. A shaker mass of 0.3 kg is not chosen, because the

maximum amplitude of the shaker is then about 0.0070 m (see figure 4.7) and approaches themaximum physical deflection of the shaker of 0.0088 m.

The top mass (m1) is taken 0.2 kg. A top mass lower than 0.2 kg is not preferable, because it does

not provide enough material to construct a guiding of.

As the top mass is already chosen, this eigenfrequency of the top mass-conical spring system

must be obtained by taking the correct stiffness. This is achieved with a combination of the

parameters d, D1, D2, n and G taking the constraints on the design parameters into account.

This results in a d of 0.0025 m, a D1 of 0.010 m, a D2 of 0.036 m and a n of 2.0 coils. As material

of the spring, the common used spring steel is chosen which has a shear modulus of G = 82.0GPa.

The higher the spring height (H), the higher the input voltage which is needed to compress the

spring solid. The spring height is chosen as H = 0.012 m. This results in a input voltage of 2.7V, which is well below the maximum input of 3.5 V.

30


10 20 30 40 50 60 70

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

10 20 30 40 50 60 70

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

f [Hz]

x[m

]

f [Hz]

y[m

]

Figure 4.2: Frequency amplitude plot (configuration 1). Black: sweep up. Gray: sweep down.Horizontal solid lines: physical limits. Horizontal dotted line: change linear/nonlinear behavior.

31


40 45 50 55 60

-0.01

-0.005

0

0.005

0.01

0.015

40 45 50 55 60-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

f [Hz]

x[m

]

f [Hz]

y[m

]B

Figure 4.3: Frequency amplitude plot (configuration 1), zoomed on second eigenmode of figure 4.2.Black: sweep up. Gray: sweep down. Horizontal solid line: physical limits. Horizontal dotted line:change linear/nonlinear behavior.

32


10 20 30 40 50 60 70

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

10 20 30 40 50 60 70−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

f [Hz]

x[m

]

f [Hz]

y[m

]


33


35 40 45 50 55

−0.01

−0.005

0

0.005

0.01

35 40 45 50 55−0.01

−0.008

−0.006

−0.004

−0.002

0

0.002

0.004

0.006

0.008

0.01

f [Hz]

x[m

]

f [Hz]

y[m

]

Figure 4.5: Frequency amplitude plot (configuration 2), zoomed on second eigenmode of figure 4.4.Black: sweep up. Gray: sweep down. Horizontal solid lines: physical limits. Horizontal dottedline: change linear/nonlinear behavior.

34


20 30 40 50 60 70

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

20 30 40 50 60 70

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

f [Hz]

x[m

]

f [Hz]

y[m

]

subharmonic resonance


35


52 54 56 58 60 62 64 66 68 70 72

-0.01

-0.005

0

0.005

0.01

0.015

52 54 56 58 60 62 64 66 68 70 72-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

f [Hz]

x[m

]

subharmonic resonance

f [Hz]

y[m

]


36


10 20 30 40 50 60 70

−0.01

0

0.01

0.02

0.03

0.04

10 20 30 40 50 60 70−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

f [Hz]

x[m

]

f [Hz]

y[m

]


37


35 40 45 50 55

−0.01

−0.005

0

0.005

0.01

0.015

35 40 45 50 55−0.01

−0.008

−0.006

−0.004

−0.002

0

0.002

0.004

0.006

0.008

0.01

f [Hz]

x[m

]

f [Hz]

y[m

]


38


10 20 30 40 50 60 70

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

10 20 30 40 50 60 70

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

f [Hz]

x[m

]

f [Hz]

y[m

]


39


35 40 45 50 55

−0.01

−0.005

0

0.005

0.01

0.015

35 40 45 50 55

−10

−8

−6

−4

−2

0

2

4

6

8

x 10−3

f [Hz]

x[m

]

f [Hz]

y[m

]

Figure 4.11: Frequency amplitude plot (configuration 5), zoomed on second eigenmode of figure4.10. Black: sweep up. Gray: sweep down. Horizontal solid lines: physical limits. Horizontaldotted line: change linear/nonlinear behavior.

40


10 20 30 40 50 60 70

−0.01

0

0.01

0.02

0.03

0.04

0.05

10 20 30 40 50 60 70

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

f [Hz]

x[m

]

f [Hz]

y[m

]


41


35 40 45 50 55

−0.01

−0.005

0

0.005

0.01

0.015

35 40 45 50 55−0.01

−0.008

−0.006

−0.004

−0.002

0

0.002

0.004

0.006

0.008

0.01

f [Hz]

x[m

]

f [Hz]

y[m

]


42


10 20 30 40 50 60 70

−0.01

0

0.01

0.02

0.03

0.04

10 20 30 40 50 60 70

−0.1

−0.05

0

0.05

0.1

f [Hz]

x[m

]

f [Hz]

y[m

]


43


40 45 50 55 60

−0.01

−0.005

0

0.005

0.01

0.015

40 45 50 55 60−0.01

−0.008

−0.006

−0.004

−0.002

0

0.002

0.004

0.006

0.008

0.01

f [Hz]

x[m

]

f [Hz]

y[m

]


44


40 45 50 55 60

-0.01

-0.005

0

0.005

0.01

0.015d1 = 2.99 Ns/m

d1 = 12.00 Ns/m

f [Hz]

x[m

]

Figure 4.16: Frequency amplitude plot with d1 = 2.99 Ns/m and d1 = 12.00 Ns/m. Black: sweepup. Gray: sweep down. Horizontal solid line: physical limits. Horizontal dotted line: changelinear/nonlinear behavior.

45


46

Chapter 5

Design and parameter identification

of the experimental setup

In this chapter, the design of the experimental setup is discussed. Next, the unknown parame-

ters of the model in figure 4.1 will be experimentally determined. Mass m1 and m2 are simply

weighted. The conical spring force Fcs(x), the stiffness k2, the linear damping constants d1 andd2 and the gain W will be experimentally determined. The stiffnesses will be determined by mea-

suring the displacement and the force, while the load is slowly increased. The linear damping

constants will be determined by a dynamic test using the time response after an initial puls force.

The gain will be determined by a quasi static test, where the relation between the input voltage of

the amplifier and the displacement of the shaker is measured. Initially it was tried to determine

all these parameters with a least square fit procedure of a frequency response function. There-

fore only the frequency response functions of the shaker table and the top mass-conical spring

system would have been needed. As the parameters according to this procedure are considered

to be linear, the nonlinear conical spring force Fcs(x) should be determined separately for large

displacements. This procedure will not be used because some uncertainties occur as will be

explained in appendix C.

5.1 Design of the experimental setup

The new test setup is depicted in figure 5.1. The setup is designed to be compact and relatively

light, enabling transport to demonstration sites. The setup has been designed in such a way, that

other types of springs also can be experimentally investigated with this setup in the future. Belowthe specific components of the current setup will be discussed.

Conical springThe spring with dimensions as specified in table 3.1 is ordered from the Amsterdam Technische

Verenfabriek [ATV, 2008]. The delivered spring is depicted in figure 5.2 and has dimensionsas spring A in table 6.1 (the height H of the delivered spring differs from the ordered spring).

The spring consists of two active coils. A spring with a constant pitch height is ordered, which

47

5. design and parameter identification of the experimental setup

1 upper air bearing

2 lower air bearing

3 position (LVDT) sensor

4 flexible connection

5 force sensor

6 shaker

7 mattress

8 upper guiding

9 conical spring

10 lower guiding

1

2

3

4

5

6

7

8

9

10

Figure 5.1: Experimental setup.

48


H = 0.0093 m

n = 2.0 coils

H2

H1

d = 0.0025 m

D1 = 0.010 m

D2 = 0.036 m

Figure 5.2: Geometry of conical spring A.

49


means that H1 and H2 should be equal. This is obviously not the case because the first coil hasheight H1 = 0.0032 m and the second coil has height H2 = 0.0061 m (H = H1 + H2). This will

obviously cause a mismatch between the theoretical and experimental analysis.

Table 5.1: Shaker and amplifier specifications.

shaker LDS type 403 (naturally cooled)maximum force: ± 98 Nmaximum displacement: ± 0.0088 mfrequency range: 9 kHzeffective armature mass: 0.2 kg

current amplifier TU/e 35-1276maximum input: ± 2.5 Vmaximum output: ± 10 A

ShakerThe electromechanical shaker that will be used to apply the force on the setup is manufactured by

Ling Dynamic Systems. The force generated by the shaker is proportional to the current through

the coil which is powered by a current amplifier. The main property of a current amplifier is

that the output current is proportional to the input control signal (U [V]). The principle on which

the shaker design is based can be modeled as a single degree of freedom mass-spring-damper

system. The specifications of the shaker and the amplifier are shown in table 5.1.

Flexible connection between shaker and lower guidingAmisalignment between the shaker and the lower guiding may damage the lower air bearing. To

prevent this, a flexible connection is designed. For information about the design of the flexible

connection, see appendix B.

Upper and lower guidingThe top mass and the shaker table need to be guided. The shaker table is already guided by its

built in leaf springs (figure 5.3 left), but for large axial deflections these may not be stiff enough.

For the guiding, a very low friction must be realized, so air bearings are chosen (figure 5.3 right).

The chosen type of air bearing (air bushing) is placed around a shaft. The smaller the gap between

the air bushing and the shaft to be guided is, the stiffer the guiding is. The specifications of the

air bushings and shafts are shown in table 5.2.

For a good performance of the air bushing, a shaft is needed with a tolerance of plus 0 µm and

minus 7.62 µm. The shaft has a tolerance of plus 0 µm and minus 13 µm, so the gap between

the air bushing and the shaft might be too big. Nevertheless, this shaft is chosen and when the

guiding will not perform good enough, it is possible to thicken the shaft, but this seemed not

necessary. The shafts are chosen of hardened steel.

SensorsA position sensor (i.e. a linear variable differential transformer LVDT) and a force sensor are

used in the setup and are part of the shaker table mass (see figures 5.1 numbers 3 and 5). Themeasured signals of the force and position sensors are amplified with separate amplifiers. Withthese sensors, the force as a function of the displacement is measured and this way stiffnesses k1

and k2 (see figure 4.1) are determined. For specifications of the sensors see table 5.3.

50


Table 5.2: Air bushings and shafts with diameters respectively 0.020 m and 0.025 m.

air bushing New Way type S302001

bore diameter: 0.0200203 m +5.08−0 µm

shaft diameter: 0.020 m +0−7.62 µm

length: 0.0508 mmass: 0.054 kg

air bushing New Way type S302502

bore diameter: 0.0250215 m +5.08−0 µm

shaft diameter: 0.025 m +0−7.62 µm

length: 0.05715 mmass: 0.083 kg

shaft diameter 0.020 m +0−13 µm

shaft diameter 0.025 m +0−13 µm

Figure 5.3: Left: Arrows indicate leaf springs of shaker. Right: Air bushings.

Furthermore, an acceleration sensor and an excitation hammer are used in combination with the

data acquisition system Siglab. With this equipment, damping d1 and d2 (see figure 4.1) and

the frequency response functions (used to identify the system parameters, see appendix C) aredetermined. For specifications of the acceleration sensor and excitation hammer see table 5.3.

TUeDACS MicrogiantA TUeDACS Microgiant system is used to convert the digital signal of the PC to an analog signal

for the current amplifier of the shaker. Also analog signals from the force and LVDT amplifiersare converted to digital signals for the PC. The signals of the PC are real time sent and read

by Matlab Simulink using the operating system Linux Knoppix. The TUeDACS sample fre-

51


quency is set to 2000 Hz.

Table 5.3: Sensors.

force sensor Kistler type 9311B (± 5 kN)frequency range: up to 70 kHz

charge amplifier Kistler type 5007

LVDT sensor Schaevitz type 1000DC-Elinear range: ± 0.025 mlinearity: 0.25 % of full rangeacceleration range: 100 m/s2

frequency range: up to 2 kHzamplifier TU/e 35-1327

acceleration sensor Kistler type 8732Aacceleration range: 5000 m/s2

frequency range: 2-10000 Hz

excitation hammer PCB type 086B03frequency range: up to 22 kHz

5.2 Identification shaker

During the experiments for the identification, the shaker is connected to the current amplifier

which power is switched on. This is done because the electric circuit is part of the shaker’s

dynamics. Also during these experiments, the shaker is decoupled from the conical spring with

top mass. Then the effective mass of the shaker table is 0.2 kg.

5.2.1 Static experiment to find k2

To determine the stiffness of the shaker (k2), a force sensor is placed on the setup, see number 5in figure 5.1. For this experiment, the LVDT sensor is mounted in such a way that it measures the

absolute displacement of the shaker y. Subsequently a mass is slowly added by hand, in about 10seconds. The experimental force-displacement relation is shown in figure 5.4. For −0.004 ≤ y ≤0.004 m (negative range for y not shown in figure 5.4), the relation is approximately linear with a

stiffness of k2 = 75.72 N / 0.004 m = 18930 N/m.

5.2.2 Linear dynamic experiment to find d2

To determine the linear damping constant of the shaker (d2), an acceleration sensor is attached

to the shaker table and a pulse force is applied with an excitation hammer. The measured time

response of the acceleration is shown in the lower diagram of figure 5.5 (solid line).

To determine damping d2, the Hilbert transform (yHilbert) [Iglesias, 2000] is used to determine

52


−2 0 2 4 6 8 10

x 10−3

−20

0

20

40

60

80

100

120

140

160

180

y [m]

F[N

]

Figure 5.4: Static experiment of shaker: load - shaker displacement relation.

a new signal

¯y(t) = y(t) + iyHilbert(t) (5.1)

The time signal is replaced with the impuls response function of a single degree of freedom

system

y(t) = Ae−ζωtsin(ω(√

1 − ζ2)t) (5.2)

The oscillatory component is eliminated by taking the magnitude of the analytical signal y

|¯y(t)| = Ae−ζωt (5.3)

By taking the natural logarithm on both sides this is depending linearly on time

ln(|¯y(t)|) =ln(A) − ζωt (5.4)

In this function, ω [rad/s] represents the natural angular eigenfrequency and t [s] the time. When

this function is fitted to the experimental data, the dimensionless damping factor ζ can be deter-

mined. In the lower diagram of figure 5.5, the time signal y and the fitted Hilbert transform isshown. Taking ω = 337.8 rad/s (based on the resonance frequency of f = ω

2π = 53.8 Hz, see

lower diagram of figure 5.5) gives ζ2 = 3.94 %.

53


Now, the damping parameter d2 can be determined by (with m2 = 0.2 kg and k2 = 18930 N/m)

d2 = ζ22√

m2k2 = 4.845 Ns/m (5.5)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−200

−100

0

100

200

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

2

4

6

t [s]

y[m

/s2]

ln(|y|)

[m/s

2]

Figure 5.5: Above: Hilbert transform of the experimental signal (solid) and fit (dashed). Be-low: acceleration signal y(t) of the experiment of the shaker (solid) and fitted Hilbert transform(dashed).

Note that the damped resonance frequency visible in figure 5.5 is equal to f = ω2π = 53.8 Hz.

For the identification the shaker table is disconnected from the setup. The resulting mass of the

shaker table is 0.2 kg. In the experiments which will be presented in the next chapter, the setup

is connected to the shaker table.

5.2.3 Quasi static experiment to find amplifier gain W

The force generated by the shaker is proportional to the current through the coil which is pow-

ered by a current amplifier. The main property of a current amplifier is that the output current

is proportional to the input control signal U [V]. The principle on which the shaker design is

based can be modeled as a single degree of freedom mass-spring-damper system. The voltage-

displacement relation is measured by giving the current amplifier an input of Asin(2πft), withA = 2.5 V and f = 0.01 Hz. (There is assumed that this also yields for higher frequencies, but

this is not checked.) In other words, the system is loaded quasi statically. Again, the LVDT sensor

is mounted in such a way that it measures the absolute displacement of the shaker y. In figure5.6 the measured voltage-displacement relation is shown, which is approximately linear. This

gives an amplifier gain of W = k20.00377 m

2.5 V = 28.54 N/V.

54


0 0.5 1 1.5 2 2.5−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

−3

U [V]

y[m

]

Figure 5.6: Measured relation: input voltage U - displacement of shaker y.

5.3 Identification top mass-conical spring system

5.3.1 Static experiment to find Fcs(x)

The (nonlinear) conical spring force (Fcs(x)) is determined in the same way as the stiffness of

the shaker (k2). The LVDT sensor and the force sensor are positioned as shown in figure 5.1. Themeasured spring load-deflection relation is shown in figure 5.7. For −0.0036 < x < 0 m the

spring is linear with k1 = 11513 N/m. At x = x0 = −0.0036 m the coils start to be compressed

to the ground till x = −0.00932 m. In this range the spring stiffness increases for increasing x.Then the spring is fully compressed to the ground. In the range −0.00965 < x < −0.0093 m

the spring stiffness increases due to a geometrical imperfection of the connection between the

spring and the top mass. Therefore this part of the spring characteristic is not taken into account

any further.

5.3.2 Linear dynamic experiment to find d1

The linear damping constant of the conical spring (d1) is identified using the same procedure as

used for the identification of the linear damping constant of the shaker. For this experiment, the

shaker table is fixed so the top mass-conical spring system is isolated. An acceleration sensor is

placed on the top mass and a force is applied on the top mass with an excitation hammer. Theresulting time response for the acceleration of the top mass is shown in the lower diagram of

figure 5.8 (solid line). Taking ω = 267.7 rad/s (based on the resonance frequency of f = ω2π =

55


-9 -8 -7 -6 -5 -4 -3 -2 -1 0

x 10-3

-200

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

x [m]

F[N

]

x0

Figure 5.7: Static experiment of conical spring: load - displacement relation. O: transition pointfrom linear to nonlinear behavior.

42.6 Hz, see lower diagram of figure 5.8) gives ζ1 = 0.28 %. The damping parameter d1 can then

be determined by (using m1 = 0.170 kg)

d1 = ζ12√

m1k1(x > x0) = 0.2477 Ns/m (5.6)

In the response range of x(t), the spring stiffness is linear (because x > x0), so k1(x > x0) =11513 N/m (with x0 = −0.0036 m).

56


0 0.5 1 1.5−40

−20

0

20

40

0 0.5 1 1.52

2.5

3

3.5

t [s]

x[m

/s2]

ln(|x|)

[m/s

2]

Figure 5.8: Above: Hilbert transform of the experimental signal (solid) and fit (dashed). Below:acceleration signal x(t) of the experiment of the top mass-conical spring system (solid) and fittedHilbert transform (dashed).

57


58

Chapter 6

Numerical and experimental results

In this chapter first the numerical and experimental dynamic response for the shaker-conical

spring-top mass system will be compared. First, the static spring characteristics of two experi-

mental springs A and B will be discussed and spring A will be applied in the dynamic analysis.For the dynamic model a fitted, a linear approximated and a theoretical spring characteristic will

be used and the frequency amplitude plots will be compared. A numerical stability study will be

made to study which initial conditions lead to which stable solution. Subsequently, for the model

a linear analysis will be made for the damped system and time plots and Poincaré maps will be

used to study the periodicity at different frequencies. Finally Power Spectral Density plots of the

model and the experiment will be compared.

6.1 Static spring characteristics

6.1.1 Experimental spring A and B

Spring A, with parameter values as shown in table 6.1, is manufactured by the company Am-

sterdam Technische Verenfabriek in Almere [ATV, 2008]. The spring is depicted in figure 5.2.

The theoretical and the experimental static spring characteristics of spring A are plotted in figure

6.1. Up to a compression of x = 0.0055 m, the match between the theoretical and experimental

characteristics is reasonably good. For x > 0.0055 m, the mismatch becomes more than 10 %.

The transition points from linear to nonlinear behavior, indicated in figure 6.1 by circles, match

well. As the match of spring A was not satisfactory, a new spring B was ordered (with parametervalues as shown in table 6.1). The theoretical and experimental static spring characteristics are

shown in figure 6.2. The linear parts correspond very well but the mismatch between and after

the transition points is very large. For x > 0.006 m the mismatch becomes more than 10 %.

Spring A is chosen for the dynamic experiment, because of spring A 60 % of the total springcompression ratio shows an error of less than 10 % compared to 52 % for spring B. Another

reason is that during compression, spring B has a relatively small nonlinear regime. This is

59

6. numerical and experimental results

undesirable because in this investigation the focus is on the nonlinear behavior.

Table 6.1: Parameter values of experimental springs A and B.

spring A B

d [m] 0.0025 0.0025D1 [m] 0.010 0.010D2 [m] 0.036 0.036n [coils] 2.0 2.0H [m] 0.0093 0.0112

G [GPa] (spring steel) 82.0 82.0

−9 −8 −7 −6 −5 −4 −3 −2 −1 0

x 10−3

−200

−150

−100

−50

0

x [m]

Fcs

[N]

Figure 6.1: Static spring characteristic of spring A. Dashed: theoretical curve. Solid: experimentalcurve.

The mismatch between the theoretical and experimental spring characteristic of both spring A

and B is caused by the fact that the pitch of the spring is not manufactured well. The used theoryof Rodriguez [2006] is based on a constant pitch. Also using the theory of Wu and Hsu [1998],

which assumes a constant pitch angle, no good match will be found. Spring A and B have not a

constant pitch nor a constant pitch angle. The expressions for the pitch of spring A and B are not

investigated, because this is not within the scope of this research.

6.1.2 Fitting of the theoretical spring characteristics

The linear regime of (experimental) spring B is very large compared to the nonlinear part (figure

6.2). This differs much from the theoretical characteristic where the linear regime is more than

60


−10 −8 −6 −4 −2 0

x 10−3

−350

−300

−250

−200

−150

−100

−50

0

x [m]

Fcs

[N]

Figure 6.2: Static spring characteristic of spring B. Dashed: theoretical curve. Solid: experimentalcurve.

half as small. The reason for this is that when the spring is compressed in the experiment, thefirst coil reaches the ground at once (at the transition point) instead of gradually as is the case

for a constant pitch. After the transition point, the characteristic is approximately linear, which

means that the second coil also reaches the ground approximately at once and not gradually.

Spring A shows more or less comparable behavior (figure 6.1). When spring A is compressed,

the experimental curve changes with a kink after the transition point, whereas the theoretical

curve changes gradually.

As the theoretical spring characteristic does not match well with the experimental characteristic,

a fit will be made for the linear and the nonlinear regimes of spring A. The fitted conical spring

force Fcs(x) becomes

Fcs(x) =

{

11513 if x > x0

1.5290 · 106x2 + 8.7885 · 103x − 8.9590 if x ≤ x0

(6.1)

with x0 = −0.0036 m. The fitted characteristic and the experimental characteristic of spring A

both are shown in figure 6.1. In this figure it is clearly shown that they almost coincide.

The static equilibrium point of the top mass is x = xs = − g m1

k1= −1.45 10−4 m and of the

shaker table y = ys = − g (m1+m2)k2

= −5.42 10−4 m. The parameter values are shown in table

6.2.

61


Table 6.2: System parameters for the linear analysis.

x m relative displacement between top mass and shakerm1 0.170 kg top mass of springd1 0.367 Ns/m damping of springk1 11513 N/m linear stiffness of springy m absolute displacement of shaker

m2 0.875 kg mass of shakerd2 4.85 Ns/m damping of shakerk2 18930 N/m stiffness of shakerg 9.81 m/s2 gravity

6.2 Linear dynamic analysis

In this section, the eigenfrequencies and eigenmodes of the damped, linear dynamic system,

i.e. for x > x0 = −0.0036 m, will be derived. The parameter values are shown in table 6.2.

A damping constant of d1 = d1,l + d1,f = 0.367 Ns/m is chosen, because this gives the best

fit with experimental results. This will be explained later. Now, equations (4.1) and (4.2) can be

rearranged and simplified to

Mq + Bq + Kq = 0 (6.2)

where the column of generalized coordinates q is defined as

q =

[

x + yy

]

(6.3)

and the mass, damping and stiffness matrix are given by

M =

[

m1 00 m2

]

, B =

[

d1 −d1

−d1 d1 + d2

]

, K =

[

k1 −k1

−k1 k1 + k2

]

(6.4)

The natural angular eigenvalues (λi) and corresponding eigencolumns (vi) of the system can be

determined by solving the corresponding linear eigenvalue problem [de Kraker, 2004]

[λiAm]vi = 0 i = 1, 2, 3, 4 (6.5)

with

Am =

[

O I−M−1K −M−1B

]

(6.6)

62


where O is a (2, 2) null matrix and I the (2, 2) identity matrix. This gives the eigenvalues

λi = [−2.066 ± 130.5j − 1.982 ± 292.0j] (6.7)

and matrix V with the corresponding complex eigencolumns (first element of each eigencolumn

normalized to 1)

V =

1.0000 1.00000.7484 ∓ 0.0069j −0.2596 ∓ 0.0054j

−2.0732 ± 130.5542j −1.98 ± 292.0612j−0.6481 ± 97.7123j 2.08 ∓ 75.8088j

(6.8)

The eigenvalues in (6.7) occur in complex conjugate pairs, so the modes are undercritically

damped. The real parts are obviously negative which means that free vibrations will show a

decreasing amplitude. The imaginary parts of the complex eigenvalues can be interpreted as the

damped angular frequencies in rad/s. This leads to the damped eigenfrequencies of

f1,2 =Im[λ1,2]

2π= 20.778 Hz, f3,4 =

Im[λ3,4]

2π= 46.482Hz (6.9)

Accordingly the eigencolumns also occur in complex conjugate pairs, see (6.8). Only the first

two rows of each eigencolumn contains the essential vibrational information, so the interesting

(displacement) parts are [de Kraker, 2004]

[

1.0000 1.00000.7484 ∓ 0.0069j −0.2596 ∓ 0.0054j

]

(6.10)

In terms of the generalized coordinates

[

xy

]

, the eigencolumns can be written as

[

1.0000 1.00002.9712 ∓ 0.1092j −0.2061 ∓ 0.0034j

]

(6.11)

As these eigencolumns here have essentially complex values, the system is generally viscously

damped. The imaginary part of the eigencolumns indicates a phase difference between mass m1

and m2. For this system, the imaginary parts are very small compared to the real parts, which

means that the phase difference is very small. This means that at the damped eigenfrequency of

20.778 Hz, m1 and m2 are almost in phase and at the eigenfrequency of 46.482 Hz m1 and m2

are almost in anti-phase. The first eigenmodes mainly show deformation of the shaker spring

and is therefore of less interest. The second eigenmode shows dominant deformation of the

conical spring. Therefore, for studying the dynamic nonlinear behavior of the conical spring, aswill be carried out in the next section, attention should be focused on the frequency range near

the second eigenfrequency (46.5 Hz).

63


6.3 Frequency amplitude plot

6.3.1 Numerical results

The numerical frequency amplitude plots will be obtained as in section 4.5. As theoretical model,

the model of figure 4.1 will be used with the parameters as in table 6.3. For conical spring

force Fcs the fitted equation (6.1) and the theoretical equation (4.3) are used and for the damping

constant d1 different values will be used.

6.3.2 Experimental results

For the experiments, simulation program Matlab Simulink is used to send and receive data of

the setup. The excitation frequency (f ) is varied from 40 to 57 Hz. During the sweep up, f is

incrementally increased with 0.1 Hz after every 50 excitation periods with a constant excitation

frequency, during the sweep up as well as the sweep down. After these 50 periods, the system is

considered to be in a excitation steady-state. During this sweep up and sweep down, the relative

displacement x is measured with the LVDT sensor, with a sample frequency of 2000 Hz. During

every frequency step, the maximum and minimum values are taken of the last periods. These

values are plotted in the frequency amplitude plot in figure 6.3 (solid line).

Table 6.3: System parameters with experimental values.

x m relative displacement between top mass and shakerx0 -0.0036 m coils compressed at ground for x>x0,m1 0.170 kg top mass of spring (50 % of mspring is included)

mspring 0.007 kg mass of springd1,l 0.247 Ns/m damping constant of springd1,f 0.120 Ns/m extra damping of spring, chosen to get a good fitζ1,l 0.28 % dimensionless damping coefficient of spring

Fcs(x) N/m conical spring force (equation (4.3) and (6.1))

y m absolute displacement of shakerm2 0.875 kg mass of shaker (50 % of mspring is included)d2 4.85 Ns/m damping constant of shakerζ2 1.89 % dimensionless damping coefficient of shakerk2 18930 N/m stiffness of shaker for -0.004≤y≤0.004 m

U 0.35 V input voltage amplifierW 28.54 N/V amplifier gain

6.3.3 Comparing numerical and experimental results

In this section the experimental results will be compared with numerical results of the model.For the model, first the fitted Fcs(x) of equation (6.1) will be used. Then an extra damping will

be added to improve the match. Next will be used the theoretical Fcs(x).

64


Model with fitted Fcs(x)

The conical spring force Fcs(x) is taken as equation (6.1). In figure 6.3, the numerical and ex-

perimental results are shown. The match is very good, apart from the fact that the maximum

amplitude of the numerical curve is at 52.5 Hz and of the experimental curve is at 53.8 Hz. As

damping d1, the linear damping of the spring d1,l (= 0.247 Ns/m) as defined in section 5.3.2, is

used. In figure 6.3, frequency hysteresis can be seen from 47.5 Hz up to 52.5 Hz which means

that in this situation at least two stable solutions coexist.

Model with fitted Fcs(x) and extra damping

To improve the match between the numerical and experimental results, the linear damping con-

stant d1 is increased by means of an extra damping d1,f (= 0.120 Ns/m). This results in a new

linear damping d1 = d1,l +d1,f = 0.367 Ns/m. The corresponding steady-state results are shown

in figure 6.4. The match is indeed improved and a very good quantitative match is obtained.

The damping d1 is taken linear. In fact, the damping can be considered to be linear until the

coils start to be compressed to the ground. In fact, then the coils start to clash with an impact to

the ground [Wu and Hsu, 1998]. The more coils are compressed to the ground, the higher the

impact is and the higher the damping becomes, so actually a nonlinear dampingmodel should be

considered. However, for now no further attempts are done to improve the theoretical damping

model, because a nonlinear damping will hardly improve the match of the steady-state behavior

between model and the experiment (see figure 6.4). So a linear damping model is maintained.

That the extra damping is needed could also have other reasons, for example when the parameters

d2 (damping of the shaker) and W [N/V] (gain of the amplifier) are not estimated well. But thisinaccuracy is not expected to have that much influence.

Model with theoretical Fcs(x)

Now for Fcs(x) the theoretical expressions from equations (4.3) and (4.4) are used with the pa-

rameter values of spring A as shown in table 6.1. The resulting theoretical frequency amplitude

plot is compared with the experimental results in figure 6.5.

6.3.4 Conclusion

The conclusion is that the numerical and experimental match of figure 6.4 is very good. So forthe following analysis, the fitted Fcs(x) of equation (6.1) will be used and the damping will be

taken as d1 = d1,l + d1,f .

65


40 42 44 46 48 50 52 54 56

−0.01

−0.005

0

0.005

0.01

x[m

]

f [Hz]

Figure 6.3: Frequency amplitude plot. Solid: experiment. Dashed: model, with fitted Fcs(x) andd1 taken as d1,l. Black: sweep up. Gray: sweep down.

40 42 44 46 48 50 52 54 56

−8

−6

−4

−2

0

2

4

6

8

10

x 10−3

x[m

]

f [Hz]

Figure 6.4: Frequency amplitude plot. Solid: experiment. Dashed: model, with fitted Fcs(x) andd1 is taken as d1,l + d1,f . Black: sweep up. Gray: sweep down.

66


40 42 44 46 48 50 52 54 56

−8

−6

−4

−2

0

2

4

6

8

10

x 10−3

x[m

]

f [Hz]

Figure 6.5: Frequency amplitude plot. Solid: experiment. Dashed: model, using theoretical Fcs(x)as in equation (4.3). Black: sweep up. Gray: sweep down.

6.4 Domains of attraction

As can be seen in figure 6.4 in the frequency range 47.5 - 52.5 Hz, two stable solutions co-

exist. Each of these solutions will have a domain of attraction, i.e. a set of initial conditions

converging to the solution. A numerical global stability study is done for a limited range of initial

conditions. The study is carried out at 51 Hz. The considered range of initial conditions of the

nonautonomous system is defined as follows.

q(t) =

x(t)x(t)y(t)y(t)

, q(0) =

−0.016 < x(0) < 0.012−0.009 < x(0) < 0.009

00

(6.12)

Figure 6.6 shows that certain initial conditions result in the low amplitude solution and other

initial conditions result in the high amplitude solution. The maximum steady-state amplitude

xsteady(= max(x)) in figure 6.6 indicates the low or the high amplitude solution.

6.5 Detailed steady-state analysis

For some excitation frequencies, i.e. at 10 Hz, 21 Hz, 41 Hz and 51 Hz the steady state behavior

is investigated in more detail by means of time histories, phase portraits and Poincaré sections

67


−0.02

−0.01

0

0.01

−0.01

−0.005

0

0.005

0.010

2

4

6

8

x 10−3

xst

eady

[m]

initial x [m] initial x [m]

Figure 6.6: Global stability research at 51 Hz, with initial conditions as in equation (6.12).

in figure 6.8 and 6.9. Note that for f = 51 Hz two stable periodic solutions coexist. Global

information about the steady-state solutions can be seen in figure 6.7. To get a better comparison

between the displacements of the top mass and the shaker table, the static equilibrium point (xs

and ys, see section 6.1.2) are not taken into account.

Situation 1: f = 10 Hz. (before first eigenfrequency f1)

The numerical results are shown in figure 6.8 a. Masses m1, m2 and excitation force F (t) =UW cos(2πft) are in phase.

Situation 2: f = 21 Hz. (≈ first eigenfrequency f1)

The numerical results are shown in 6.8 b. Massesm1 andm2 are in phase. This corresponds with

the linear theory from section 6.2, which results in an eigenmode u1 of

[

xy

]

=

[

1.00002.9712 ∓ 0.1092j

]

(6.11), which is indeed an in phase motion.

Situation 3: f = 41 Hz.

The numerical results are shown in figure 6.8 c and the experimental results in figure 6.8 d. Aseigenfrequency f2 = 46.5 Hz is approached, m1 and m2 interchange energy.

Situation 4: f = 51 Hz (high amplitude solution).

The numerical results are shown in figure 6.9 a and the experimental results in figure 6.9 b.Eigenfrequency f2. Masses m1 and m2 are in anti-phase. This corresponds more or less with the

linear theory from section 6.2, which gives a f2 of 46.482 Hz (equation 6.9) and an eigenmode

u2 of

[

xy

]

=

[

1.0000−0.2061 ∓ 0.0034j

]

(6.11), which is indeed an anti-phase motion.

Situation 5: f = 51 Hz (low amplitude solution).

68


10 20 30 40 50 60 70

−5

0

5

10

x 10−3

10 20 30 40 50 60 70

−0.01

0

0.01

x[m

]

f [Hz]

y[m

]

Figure 6.7: Frequency amplitude plot (numerical analysis). Black: sweep up. Gray: sweep down.Horizontal solid lines: physical limits. Horizontal dotted line: change linear/nonlinear behavior.

The numerical results are shown in figure 6.9 c and the experimental results in figure 6.9 d.Masses m1 and m2 again are more or less in anti-phase.

The phase portraits (right figures in figure 6.8 c and figure 6.9 a and c), consist of the responseresulting from five excitation periods. In each plot the five Poincaré points (indicated by a circle)

coincide. The conclusion is that for each excitation frequency a harmonic solution is found:

q(t) = q(t + T ) with T = 1/f .

Note that nonlinear behavior is present only at f = 21 Hz and at the high amplitude solution at

f = 51 Hz, since in these cases x becomes lower than x0 = −0.0036 m.

In addition, the Power Spectral Density (PSD) is studied for the two coexisting stable solutions at

f = 51 Hz. The PSD is defined as follows [de Kraker, 2004]

Sxx(f) =1

TmeasX(f)∗X(f) (6.13)

Where X(f) is the discrete Fourier transformation of the time signal x measured at f = 51 Hz.

For both the numerical (model) and the experimental time signals the following parameter values

are used:

f = 51 Hz (UW cos(2πft))T = 1/f = 0.0196 s

69


fsample = 2000 Hz (sample frequency)Tsample = 1/fsample = 0.0005 s

NS = 30 periods (number of periods)

Tmeas = NS · T = 0.5882 s

NFFT = Tmeas/Tsample + 1 = 1177 points (number of points)

ffold = fsample/2 = 1000 Hz (maximum frequency)

In figure 6.10, the PSD of the model (dashed line) and of the experiment (solid line) are plotted

for the low amplitude stable solution at 51 Hz. The system is linear for this stable solution,

because x > x0 (x0 = −0.0036 m). This explains why only at f = 51 Hz a resonance peak is

found. The maximum value of the PSD at f for the model is 0.37 m2/Hz and for the experiment

0.69 m2/Hz.

The PSD’s of the model (dashed line) and of the experiment (solid line) for the high amplitude

stable solution at 51 Hz are plotted in figure 6.11. In this stable solution the nonlinearity in

the system is addressed. [Thompson and Stuart, 1986] So superharmonic peaks can be found at

f·2, f·3, f·4. The maximum value of the PSD at f = 51 Hz is respectively 44.55 m2/Hz for the

model and 47.45 m2/Hz for the experiment, which is a good match. Mind that the figure has a

logarithmic scale, and that the maximum value at f · 2 of the experiment is 0.10 m2/Hz, which

is only 0.2 % of the maximum value at f .

The conclusion is that between the model and the experiments, a qualitative and a quantitative

match can be found although the quantitative match for the high amplitude stable solution is

better.

6.6 Summary

In this chapter the numerical and experimental dynamic response for the shaker-conical spring-

top mass system are compared. The static experimental spring characteristics of spring A and B

are compared and no good quantitative match can be found. The conclusion is that that the mis-

match between the theoretical and experimental spring characteristic is caused by the fact that the

experimental spring does not have a constant pitch. For the model a linear analysis is made for

the damped system. Spring A is applied in the dynamic analysis. For the dynamic model a fitted,a linear approximated and a theoretical spring characteristic is used and the frequency amplitude

plots are compared. The frequency amplitude plots of the model with the fitted characteristic and

the experiments match qualitatively as well as quantitatively, when extra damping is added. The

steady-state behavior of the system has been investigated for the frequency range 0 < f < 70 Hz.

Five different steady-state situations for specific excitation frequencies have been examined in de-

tail using time histories, phase portraits, Poincaré maps and Power Spectral Densities. All steady-

state solutions found appeared to be harmonic solutions. A brief numerical global stability study

is done to investigate which initial conditions lead to which of the two coexisting stable solutions

at the resonance frequency of the conical spring top mass system. The Power Spectral Density

(PSD) plots of the model and the experiment at f = 51 Hz match qualitatively and quantitatively

for harmonic solutions. At the low amplitude solution, no superharmonic frequency compo-nents occur because the system is linear. At the high amplitude solution, these superharmonic

frequency components are present due to nonlinear behavior. [Thompson and Stuart, 1986].

70


0 0.02 0.04 0.06 0.08 0.1

−5

0

5

x 10−4

0 0.02 0.04 0.06 0.08 0.1−0.04

−0.02

0

0.02

0.04

x(-

),y(−

−)

[m]

x(-

),y(−

−)

[m]

t [s]

(a) f = 10 Hz, numerical

0 0.02 0.04 0.06 0.08 0.1

−0.01

0

0.01

0 0.02 0.04 0.06 0.08 0.1−2

−1

0

1

2

x(-

),y(−

−)

[m]

x(-

),y(−

−)

[m]

t [s]

(b) f = 21 Hz, numerical

0 0.02 0.04 0.06 0.08 0.1

−5

0

5

x 10−4

0 0.02 0.04 0.06 0.08 0.1

−0.1

0

0.1

x(-

),y(−

−)

[m]

x(-

),y(−

−)

[m]

t [s]−5 0 5

x 10−4

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

x [m]

x[m

/s]

(c) f = 41 Hz, numerical

0 0.02 0.04 0.06 0.08 0.1

−5

0

5

x 10−4

x(-

),y(−

−)

[m]

(d) f = 41 Hz, experimental

Figure 6.8: Numerical and experimental time plots, phase planes and Poincaré maps.

71


0 0.02 0.04 0.06 0.08 0.1

−5

0

5

x 10−3

0 0.02 0.04 0.06 0.08 0.1−2

−1

0

1

2

x(-

),y(−

−)

[m]

x(-

),y(−

−)

[m]

t [s]−6 −4 −2 0 2 4 6

x 10−3

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x [m]

x[m

/s]

(a) f = 51 Hz, high amplitude solution, numerical

0 0.02 0.04 0.06 0.08 0.1

−5

0

5

x 10−3

x(-

),y(−

−)

[m]

(b) f = 51 Hz, experimental

0 0.02 0.04 0.06 0.08 0.1

−1

−0.5

0

0.5

1

x 10−3

0 0.02 0.04 0.06 0.08 0.1−0.4

−0.2

0

0.2

0.4

x(-

),y(−

−)

[m]

x(-

),y(−

−)

[m]

t [s]−1 −0.5 0 0.5 1

x 10−3

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

x [m]

x[m

/s]

(c) f = 51 Hz, low amplitude solution, numerical

0 0.02 0.04 0.06 0.08 0.1−1

−0.5

0

0.5

1

x 10−3

x(-

),y(−

−)

[m]

(d) f = 51 Hz, experimental

Figure 6.9: Numerical and experimental time plots, phase planes and Poincaré maps.

72


50 100 150 200 25010

−12

10−10

10−8

10−6

10−4

10−2

100

PSD

[m2/H

z]

f [Hz]

Figure 6.10: PSD at 51 Hz, low amplitude solution. Dashed: model. Solid: experiment.

50 100 150 200 25010

−10

10−8

10−6

10−4

10−2

100

102

PSD

[m2/H

z]

f [Hz]

Figure 6.11: PSD at 51 Hz, high amplitude solution. Dashed: model. Solid: experiment.

73


74

Chapter 7

Conclusions and recommendations

7.1 Conclusions

The objective of this thesis is to investigate the nonlinear dynamic behavior of conical springs

carrying a top mass. The top mass is heavy enough to justify neglection of inertia properties of

the conical spring itself. This thesis is started with a literature study about theoretical derivations

of the load-deflection relation of conical springs. A suitable load-deflection relation is chosen andstatic analysis of the conical spring are performed. Then a dynamic two degree of freedom system

is modeled. One degree of freedom, the displacement of the top mass of the conical spring sys-

tem, is free, whereas the other is the displacement of the shaker table. Subsequently, a numerical

nonlinear dynamic analysis is made to get the frequency amplitude plot. The experimental part

starts with the design and realization of a dynamic test setup, which contains a shaker which will

be driven by a harmonic excitation signal. First the static load defection relation is experimen-

tally determined in the test setup. The theoretical and experimental static spring characteristic

are compared. For the model a fitted spring characteristic is used. Next, the mass, stiffness and

linear damping of the shaker and the linear damping of the top mass-conical spring system are

experimentally determined and identified. Finally the theoretical and experimental frequency

amplitude plots are obtained and compared.

From the theoretical and experimental spring characteristic it can be concluded that they do not

match quantitatively. The theoretical characteristic is based on Wahl’s assumptions, which is

accurate for cases where deflections per coil are not more than half the mean spring radius and

pitch angles are less than 10◦. These two conditions are fulfilled, so this could not be the cause of

the mismatch. At the theoretical spring characteristic, a constant pitch is assumed. A spring witha constant pitch has been ordered, but it appeared that this spring does not have a constant pitch

at all. The conclusion is that that the mismatch between the theoretical and experimental spring

characteristic is caused by the fact that the experimental spring does not have a constant pitch.

In the theoretical model, the theoretical spring characteristic based on a constant pitch is not used.Instead a fit of the experimental static load-deflection curve is used. The frequency amplitude

plots of the model and the experiments match qualitatively as well as quantitatively, when extra

75

7. conclusions and recommendations

damping is added. In the model extra damping is added to the experimentally identified lineardamping to get a good fit between the frequency amplitude plots. In fact, the damping can be

considered linear till the coils start being compressed to the ground. Then the coils clash to the

ground which causes an increasing nonlinear damping. However, no further attempts are made

to improve the damping model, because the match between the model and the experiment is

already very good. So a linear damping model is maintained.

The steady-state behavior of the system has been investigated for the frequency range 0 < f < 70Hz. Five different steady-state situations for specific excitation frequencies have been examined

in detail using time histories, phase portraits, Poincaré maps and Power Spectral Densities. All

steady-state solutions found appeared to be harmonic solutions.

A brief numerical global stability study is made to investigate which initial conditions lead to

which of the two coexisting stable solutions at the resonance frequency of the conical spring top

mass system.

The Power Spectral Density (PSD) plots of the model and the experiment at f = 51 Hz matchqualitatively and quantitatively for harmonic solutions. At the low amplitude solution, no su-

perharmonic frequency components occur because the system is linear. At the high amplitude

solution, these superharmonic frequency components are present due to nonlinear behavior.

[Thompson and Stuart, 1986].

7.2 Recommendations

Although the frequency amplitude plots of the model and the experiments match very good sev-

eral recommendations can be made. Firstly, the analytical linear damping model can be extended

taking the damping effect due to coil clash into account. A start is made by Wu and Hsu [1998].

Secondly, in this thesis the top mass is heavy enough to justify neglection of the inertia properties

of the conical spring. The analytical model can be improved by taking the inertia properties of

the conical spring into account.

For the experiments conical springs were ordered with a constant pitch. The delivered springs

however did not have a constant pitch. Therefore it is recommended to be well informed about

possibilities to manufacture these types of springs before ordering them.

This report is a first step in the research of the nonlinear dynamic behavior of conical springs. It

is important to think about possible exploitation of the conical spring and the consequences of

this type of behavior in applications. An interesting application of the conical spring is a valve

systems which is exerted by a cam shaft [Nagaya et al., 2008].

In this thesis the nonlinear behavior of conical springs is investigated. It is recommended also

to investigate different types of springs showing comparable nonlinear behavior. For example,

cylindrical springs with a varying pitch or a varying coil diameter could be investigated.

76

Bibliography

[ATV, 2008] ATV (2008). Amsterdam Technische Verenfabriek. see http://www.atv.nl.

[Chevy, 2008] Chevy (2008). QA1 Motorsports’ Coilover Front Shock Conversion. Avail-

able at http://www.chevyhiperformance.com/techarticles/148-0208-coilover-front-shock-

conversion/index.html.

[de Kraker, 2004] de Kraker, B. (2004). A numerical-experimental approach in structural dynamics.Shaker Publishing B.V.

[de Kraker and van Campen, 2001] de Kraker, B. and van Campen, D. (2001). Mechanical Vibra-tions. Shaker Publishing B.V.

[Hata et al., 2003] Hata et al., S. (2003). Integrated conical spring linear actuator. Microelectronic

Engineering 67/68 page 574/581, Elsevier Science B.V.

[Iglesias, 2000] Iglesias, A. (2000). Investigating various modal analysis extraxtion techniques toestimate damping ratio. Masters thesis, Virginia polytechnic Institute and State University.

[Matlab, 2007] Matlab (2007). Version R2007a. The MathWorks, Inc.

[Nagaya et al., 2008] Nagaya, K., Kobayashi, Nenoi, Hosokawa, and Murakami (2008). Low driv-ing energy engine valve mechanism using permanent-electromagnet and conical spring. InternationalJournal of Applied Electromagnetics and Mechanics 28 267/273, IOS Press.

[Paredes and Rodriguez, 2008] Paredes, M. and Rodriguez, E. (2008). Optimal design of conicalsprings. Engineering with Computers, DOI 10.1007/s00366-008-0112-3.

[Rodriguez et al., 2006] Rodriguez, E., Paredes, M., and Sartor, M. (2006). Analytical behaviorlaw for a constant pitch conical compression spring. Journal of mechanical design, Vol. 128, ASME.

[Rosielle and Keker, 1996] Rosielle, P. and Keker, E. (1996). Dictaat: Constructieprincipes 1. Be-doeld voor het nauwkeurik bewegen en positioneren. Lecture note-book, Eindhoven University of

Technology.

[Thompson and Stuart, 1986] Thompson, J. and Stuart, H. (1986). Nonlinear dynamics and chaos.John Wiley and sons Ltd.

[Triddle, 2007] Triddle (2007). The suspension components of a Ford Model T. The coil-spring device is an aftermarket accessory, the ’Hassler shock absorber.’. Available at

http://en.wikipedia.org/wiki/Ford-Model-T.

77

BIBLIOGRAPHY

[Wahl, 1963] Wahl, A. (1963). Mechanical springs. McGraw=Hill book company, Inc.

[Wu and Hsu, 1998] Wu, M. and Hsu, W. (1998). Modelling the static and dynamic behavior of aconical spring by considering the coil close and damping effects. Journal of sound and vibration,

214.

78

Appendix A

Conical spring model of Wu and Hsu

In the text below the following parameters will be used:

F : load

d : coil diameterD : mean spring diameter

r : mean spring radius

r1 : mean spring radius of the smallest coil

r2 : mean spring radius of the biggest coil

n : original number of coils

H : original spring height

A : area of the cross-section

G : shear modulus

E : elastic modulus

IB : moment of inertia of the wire cross-section about b-axis

IN : moment of inertia of the wire cross-section about n-axisJ : polar momentum

L : coil length

p : pitch angle

x : axial spring deflection

A.1 Introduction

In [Wu and Hsu, 1998], first the spring constant for a general helical spring is derived. Then sim-plifications are made to achieve the load-deflection relation for a nontelescoping conical spring

with constant pitch angle.

79

a. conical spring model of wu and hsu

A.2 Linear general helical spring

To derive the spring constant of the general helical spring with variable helix radius and variable

pitch angle, a coordinate system is defined, as shown in figure A.1. A general helix, is parameter-

ized over the arc length and can be expressed as

X(s) = r(s) cos θ(s)i + r(s) sin θ(s)j + h(s)k (A.1)

where mean radius r, polar angle θ, and local helix height h, are all functions of helical length s.Then the tangent of the parametric curve is expressed as the derivative with respect to the helical

length s, as

T = X ′(s) = (r′ cos θ − θ′r sin θ)i + (r′ sin θ + θ′r cos θ)j + h′k (A.2)

Figure A.1: Coordinate system of general helical spring

The unit tangent is

t = T (s)/|T (s)| = (1/W )[(r′ cos θ − θ′r sin θ)i + (r′ sin θ + θ′r cos θ)j + h′k] (A.3)

where

W =√

r′2 + r2θ′2 + h′2

The unit vector n of the helical curve is perpendicular to the unit tangent and is

n = (1/√

r′2 + r2θ′2)[−(r′ sin θ + θ′r cos θ)i + (r′ cos θ − θ′r sin θ)j]. (A.4)

The unit vector b is perpendicular to t and n, so

b = t · n =1

W ×√

r′2 + r2θ′2

×[−h′(r′ cos θ − θ′r sin θ)i − h′(r′ sin θ + θ′ cos θ)j + (r′2 + r2θ′2)k]

(A.5)

80


The three unit vectors t, n and b form a right-handed orthonormal co-ordinate system. A staticload F is acting on the spring along the center line, so

F = F k (A.6)

and then the moment at the spring wire becomes

M = Fr sin θi − Fr cos θj (A.7)

Three components of force F and moment M in t, n and b directions can be expressed as

FT = F · t = Fh′/W, FB = F · b = F√

r′2 + r2θ′2/W, FN = F · n = 0,

MT = M · t = −Fr2θ′/W, MB = M · b = Fr2h′θ′/W√

r′2 + r2θ′2,

MN = M · n = −2Frr′ sin2 θ/√

r′2 + r2θ′2.

(A.8)

The total strain energy Utotal is the sum of the strain energy terms from the above components.

Utotal = U1 + U2 + U3 + U4 + U5 (A.9)

where

U1 =1

2GA

∫ L

0F 2

Bds, U2 =1

2EA

∫ L

0F 2

T ds, U3 =1

2EIB

∫ L

0M2

Bds,

U4 =1

2GJ

∫ L

0M2

T ds, U5 =1

2EIN

∫ L

0M2

Nds.

(A.10)

U1 is the strain energy for direct shear in b-direction, U2 for tension and compression in the

t-direction, U3 for bending about the b-axis, U4 for torsion about the t-axis and U5 for bending

about the n-axis.

According to Castigliano’s theorem, the axial deflection x is

x = δUtotal/δF = F/k (A.11)

where the spring rate k of a general helical spring is equivalent to

k = 1/(C1 + C2 + C3 + C4 + C5) (A.12)

81


where

C1 =1

GA

∫ L

0

(

r′2 + r2θ′2)

W 2ds, C2 =

1

EA

∫ L

0

h′2

W 2ds,

C3 =1

EIB

∫ L

0

r4h2θ′2

W 2 (r′2 + r2θ′2)ds,

C4 =1

GJ

∫ L

0

r4θ′2

W 2ds, C5 =

1

EIN

∫ L

0

4r2r′2 sin2 θ

W 2 (r′2 + r2θ′2)ds.

(A.13)

A.3 Linear conical spring

Spring rate for a conical spring with constant pitch, p(s, t) is a constant,

h′ = sin p, r(s) = r1 + ([r2 − r1]/L)s, r′(s) = (r2 − r1)/L,

θ(s) =

∫ s

0cos p/r(ζ)dζ, θ′(s) = cos p/r(s), L = π(r1 + r2)n.

(A.14)

where n is the original number of active coils, r1 is the radius of the smallest coil, and r2 is the

radius of the largest coil. Then

C1 =L

(

r′2 + cos2 p)

GA (1 + r′2), C2 =

sin2 pL

EA (1 + r′2), C3 =

L(

r22 + r1r2 + r2

1

)

cos2 p

3GJ (1 + r′2),

C4 =4r′2L

(

r22 + r1r2 + r2

1

)

sin2 p

3EIN (r′2 + cos2 p), C5 =

L(

r22 + r1r2 + r2

1

)

sin2 p cos2 p

EIB (1 + r′2) (r′2 + cos2 p).

(A.15)

A.4 Nonlinear conical spring

This nonlinear regime can only be determined by a discretizing algorithm. Let H be the total

deflection to compress the spring solid

H =

∫ L

0sin p(ζ)dζ − nd (A.16)

The coil is discretized over the length in a number of pieces (steps indicates the number of

pieces). Dy is the variable mean spring diameter, and the mean spring diameter of the next piece

is

Dyn = Dy + Dy/steps (A.17)

82


The height, the number of coils and the length of one piece is

Hc = H/steps

nc = n/steps

Lc = π(Dy + Dyn)/2

(A.18)

According to equations (A.12) and (A.15), the spring rate kc can be calculated for a piece of the

spring. F , needed for the piece with diameter Dy starting to contact with the next piece for the

height Hc, can be calculated with

Fy = Hc · kc (A.19)

Also, the spring rate k2 for the remaining coil length can be determined. Therefore, the deflection

under load F can be expressed as two parts. The total load-deflection relationship becomes

x = Hclose + Hopen = F/k2 + (nf/n)H (A.20)

83


84

Appendix B

Flexible connection between shaker

and lower guiding

Wire spring with stiffened middle part

The connection between the shaker and the lower guiding of the setup must be flexible to correct

the misalignment. This flexible connection must be as stiff as possible in the axial direction

and as weak as possible in the radial direction. Namely, the force in radial direction will exert a

moment on the air bearing. The maximum moment on the air bearing (Mair) is specified by the

supplier as 1.92 Nm.

As a flexible connection a wire spring with stiffened middle part is chosen. The length of

the stiffened middle part is equal to 56 of the length between the hinges (l). The thin parts

both have a length of 16 l, because the pole is half in between the length of the elastic part.

[Rosielle and Keker, 1996]

Figure B.1: Wire spring with stiffened middle part. [Rosielle and Keker, 1996]

The maximum axial force (Fax) in the flexible connection is caused by the shaker and is assumed

to be 265 N. Fax is approximated by a maximum dynamic electrostatically force of 98 N (specified

in the shaker specification) and a static force of 0.0088 m18930 N/m = 167 N (where 0.0088 m is the

maximum displacement of the shaker and is 18930 N/m the shaker stiffness).

l = 0.025 m (length between the poles of the flexible connection)

d = 0.0015 m (diameter of the thin part of the flexible connection)

85

b. flexible connection between shaker and lower guiding

E = 210 GPa (elastic modulus)τmax = 500 · 108 Pa (maximum shear stress)

z = 0.0004 m (approximated maximummisalignment between shaker and air bearing)

Fax = 265 N (approximated maximum axial force of the shaker)

Mair = 1.92 Nm (maximum moment on the air bearing)

Axial stiffnessThe axial stiffness of the flexible connection is

cax =3Eπd2

4l= 3.7 · 107 N/m (B.1)

This leads to a maximal axial compression of Faxcax

= 7.1 · 10−6 m, which is an acceptable value.

Radial stiffnessThe radial stiffness of the flexible connection is

crad =0.7Ed4

l3= 2.7 · 104 N/m (B.2)

The force in radial direction is Frad = z · crad = 11.0 N. The force acts at a axial distance of 0.150m on the air bearing (the axial distance between the shaker and the air bearing), which results ina moment on the air bearing of 1.6 Nm. This is acceptable, because this is below the maximum

Mmax.

Shear stressThe shear stress which occurs due to bending of the flexible connection is

τ =3Edz

l2= 4.2 · 108 Pa (B.3)

The maximum shear stress (τmax) is much higher than the acting shear stress τ so it is satisfac-

tory.

BucklingThe flexible connection has a axial buckling limit of

Fz =36π2EI

l2= 21000 N withI =

πd4

64(B.4)

The acting axial force Fax is much lower than the maximum axial force Fz so it is satisfactory.

Conclusion

The flexible connection with d = 0.0015 m and l = 0.025 m is chosen and satisfies the limits of

the axial and radial stiffness, the shear stress and the buckling.

86

Appendix C

Parameter identification using least

square fit method

The identification of the parameters of the shaker (mass m2, damping d2, stiffness k2 and gain

W ) and the top mass-conical spring system (mass m1, damping d1 and stiffness k1) can be done

by a least squares fit of the frequency response data. This is a linear approach, so all parametersare assumed to be linear. As k1 is not linear, but depends on the compression x, k1 should be

determined separately. In this chapter, first the identification of the shaker is treated and the

occurring problems.

The LDS shaker consist of a permanent magnetic with in the center a core which is supported by

leaf springs. Around the core, a coil is winded. When a current flows through the coil a force willbe generated by the magnetic field between the coil and the permanent magnet. The shaker can

be modeled as a 1 DOF system

m2y + d2y + k2y = UW (C.1)

where m2 represents the mass, d2 the damping and k2 the stiffness of the shaker. W [N/V] is the

gain. U [V] represents the amplitude of the input control. The parameters m2, d2, k2 and W need

to be determined.

The frequency response function (FRF) is measured from input control U (noise signal, delivered

by Siglab), to the acceleration of the core y (using an acceleration sensor). To get the desired

displacement y as shown in figure C.1, the acceleration y is divided by (2πf)2.

The FRF with the peak at the left (see figure C.1 with the eigenfrequency at fm+mex = 44.5 Hz)

is performed with an extra mass mex. The FRF with the peak at the right is performed without

the extra mass (see figure C.1 with the eigenfrequency at fm = 52.0 Hz).

Knowing that

mex = 0.084 kg

87

c. parameter identification using least square fit method

fm = 52.0 Hzfm+mex = 44.5 Hz

fm2 =1

2π

√

k2

m2; fm2+mex =

1

2π

√

k2

m2 + mex(C.2)

the mass of the shaker m2 can be determined and is 0.23 kg. Next a least squares fit is made of

the FRFs and are also plotted in figure C.1 (the fits are so well fitted that they coincide almost com-

pletely with the measurement). The least squares fit is made with the Matlab function Invfreqsand fits the FRFs as

|H| =W

m2s2 + d2s + k2(C.3)

Results:m2 = 0.23 kg

d2 = 15.3 Ns/m (based on the FRF with extra mass)

ζ2 = 4.5 %

k2 = 2.46 · 104 N/m

W = 17.2 N/V

Af [Hz]

|H|[

m/V

]

Figure C.1: Transfer function |H| [m/V] from input voltage U to shaker displacement y

ProblemTill so far the parameter identification seems successful. But a strange phenomenon occurs when

the root mean square (RMS) of the input voltage (U ) is changed. When the RMS is changed, the

88


eigenfrequency of the FRF changes, which is not expected. In figure C.2 the FRFs are plotted forthe RMS values (from left resonance peak to the right) 0.300, 0.200, 0.150, 0.100, 0.050, 0.020 and

0.005 V. In figure C.3 (solid line) the eigenfrequencies at the different RMS values are plotted,

with RMS on a logarithmic scale. A almost straight line is visible, which means a exponential

relation between the RMS and eigenfrequency, where no dependency is expected. In figure C.4

the maximal amplitudes at the different RMS values are plotted. A discontinue relation is found,

where a continue relation should be expected.

20 30 40 50 60 70 80

10−3

10−2

20 30 40 50 60 70 80

−150

−100

−50

0

|H|[

m/V

]phase

[◦]

f [Hz]

Figure C.2: FRFs of the shaker for the RMS values (U) of (from left resonance peak to the right)0.300, 0.200, 0.150, 0.100, 0.050, 0.020 and 0.005 V.

Conclusion

Some uncertainties occur during the parameter identification using the least square fit method.

Namely, during the FRF measurements, the eigenfrequency changes with the RMS input volt-

age (U ), which should not be the case. Therefore this method is not used for the parameters

identification.

89


10−2

10−1

51.5

52

52.5

53

53.5

54

54.5

55

55.5

RMS [mV]

eige

nfr

equen

cyf

[Hz]

Figure C.3: Plot of the eigenfrequencies at different RMS values.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.351.75

1.8

1.85

1.9

1.95

2

2.05

2.1

2.15

RMS [mV]

max

(|H

|)[m

/V]

Figure C.4: Plot of the maximal amplitudes at different RMS values.

90

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