Optimization of secondary suspension of piecewise linear vibration isolation systems

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0020-7403/$ - se

doi:10.1016/j.ijm

�CorrespondE-mail addr

Sudhir.Mehta@

(G.N. Jazar).

International Journal of Mechanical Sciences 48 (2006) 341–377

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Optimization of secondary suspension of piecewise linear vibrationisolation systems

Sagar Deshpande, Sudhir Mehta, G. Nakhaie Jazar�

Department of Mechanical Engineering and Applied Mechanics, North Dakota State University, Fargo, ND 58105-5285, USA

Received 20 April 2005; received in revised form 25 October 2005; accepted 10 November 2005

Abstract

A comprehensive optimal design solution is presented for piecewise-linear vibration isolation systems. First, primary suspension

optimum parameters are established, followed by an investigation of jump-avoidance conditions for the secondary suspension. Within

the no-jump zones, an optimal design solution is then obtained for the secondary system and overall results are discussed.

Averaging method is employed to obtain an implicit function for frequency response of a bilinear system under steady-state

conditions. This function is examined for jump-avoidance and a condition is derived which when met ensures that the undesirable

phenomenon of ‘jump’ does not occur and the system response is functional and unique. Optimal stiffness and damping parameters for

the primary suspension are extracted from a recently established work for passive linear vibration systems. For each point of the primary

suspension optimal curve, jump-free zones are identified. Iterating this process, a boundary surface between no-jump (unique response)

and jump (multiple-response) areas is established. Keeping optimal parameters for the primary suspension system fixed, the secondary

suspension stiffness and damping parameters are varied inside the no-jump zones to explore optimum solutions for the secondary.

The root mean square (RMS) of the absolute acceleration is minimized against the RMS of the relative displacement (Z). It is observedthat there is a certain band of parameters defined by primary damping, within which a valid frequency response can be obtained. An

optimum numerical solution is sought within this band of parameters. Optimal solution curves are achieved for the secondary

suspension. These can be used in conjunction with the optimal curve for the primary suspension to select design parameter values for the

best possible vibration isolation performance in a given application.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Vibration isolators; Piecewise linear system; Non-linear system; Jump avoidance

1. Introduction

A linear vibration system is defined as one in which thequantities of mass (or inertia), stiffness and damping arelinear in behavior and do not vary with time [1]. Althoughmathematical models employing a linear ordinary differ-ential equation with constant coefficients portray a simpleand manageable system for analytical scrutiny, in mostcases they are an incomplete representation simplified forthe sake of study. Most real physical vibration systems aremore accurately depicted by non-linear governing equa-

e front matter r 2005 Elsevier Ltd. All rights reserved.

ecsci.2005.11.006

ing author. Tel.:+1701 231 8303; fax: +1701 231 8913.

esses: [email protected] (S. Deshpande),

ndsu.edu (S. Mehta), [email protected]

tions, in which the non-linearity may stem from structuralconstraints causing a sudden change in stiffness anddamping characteristics, or from inherent non-linearbehavior of internal springs and dampers. The former aremostly more difficult to analyze as sudden changes insystem parameters display unprecedented complex beha-vior. This research focuses on a general form of such a non-linear system. This study of piecewise-linear systems willallow hazardous system behavior over operating frequencyranges to be gauged and controlled in order to avoidpremature fatigue damage, and prolong the life of thesystem. Piecewise linear vibration isolation systems areones in which the stiffness and damping are linear over aparticular range of amplitude and change drastically toanother set of linear values, once a threshold is reached.They have been used to represent automotive suspension

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Nomenclature

a relative displacement amplitude of frequencyresponse (m)

A ¼ a/Y dimensionless relative displacement ampli-tude of frequency response

c linear damping coefficient (N s/m)f frequency (Hz)g3, g4 piecewise linear functionsk linear stiffness coefficient (N/m)m mass of sprung mass (kg)r ¼ o=o1 excitation frequency ratiot time (s)u ¼ x/Y dimensionless mass displacementv ¼ y/Y dimensionless base excitationw ¼ u�v dimensionless relative displacementx mass displacement coordinate (m)X amplitude of displacement (m)y base displacement coordinate (m)Y amplitude of base harmonic displacement

excitation (m)z ¼ x�y mass-base relative displacement coordinate

(m)Zi parametric function of dynamical character-

istics and amplitude (m)b Phase of relative displacementd ¼ D/Y dimensionless clearance

t ¼ o1t dimensionless timer ¼ o2=o1 secondary/primary stiffness ratioD clearance (m)x damping ratiox1 ¼ c1=ð2mo1Þ primary suspension damping charac-

teristicx2 ¼ c2=ð2mo2Þ secondary suspension damping char-

acteristicx3 ¼ c3=ð2mo3Þ combined suspension damping charac-

teristico excitation frequency (rad/s) or (Hz)o1 ¼

ffiffiffiffiffiffiffiffiffiffiffik1=m

pprimary suspension frequency character-

istico2 ¼

ffiffiffiffiffiffiffiffiffiffiffik2=m

pprimary suspension frequency character-

isticf ¼ otþ b angle of mass-base relative displacement

(rad)f0 ¼ sin�1(d/A) angle of secondary system activation

(rad)ðÞ0 dðÞ=dt

over dot dðÞ=t

Subscripts

1 primary2 secondary3 ¼ 1+2 primary+secondary

S. Deshpande et al. / International Journal of Mechanical Sciences 48 (2006) 341–377342

systems, engine mounts, gear-pair systems, and isolationsystems for various kinds of externally excited vibrationsystems, and have rightfully received the interest of severalscientists over the years.

2. Background on piecewise linear isolators

Den Hartog and Mikina [2] were the first scientists toinvestigate an approximate solution to a piecewise linearsystem in 1932, in which they analyzed the behavior of asystem with bilinear stiffness under steady state. Hartogand Heiles [3] then presented in 1936 a closed form solutionfor evaluating the dynamic amplitude of a mass attached toa symmetric bilinear spring under harmonic excitation.Gurtin [4] proposed the use of clearance in viscous dampersto limit high-frequency force transmission, in which heexamined a system with amplitude-dependent damping.Maezawa [5] developed in 1961 a Fourier series solution forforced vibration of a piecewise linear system under steadystate, and a year later [6] applied the Fourier series to studysub-harmonic vibration of piecewise linear systems. Masri[7] in 1965 presented analytical and experimental studies ofimpact dampers, and in 1978 extended his work toinvestigate dynamic piecewise systems with a physicalgap. Rosenberg [8] proved the existence of a steady-statesolution for a general form of a non-linear system in 1966.

Loud [9] in 1968 analyzed the branching of periodicsolutions of non-autonomous piecewise linear systems.Masri and Stott [10] used experimental methods to studythe behavior of a non-linear vibration system underrandom excitation. In 1982, Badrakhan [11] devisedLagrangian formulations to study the existence of firstintegrals in a bilinear hysteresis oscillator according toNoether’s generalized theorem, and found that firstintegrals existed only if the restoring force was linear.Shaw and Holmes [12] in 1983 explored an oscillator with apiecewise linear restoring force, and found that the solutionbranches into harmonic, sub-harmonic and chaotic mo-tions. In 1986, Nyugen et al. [13] investigated time behaviorof a piecewise linear system using numerical integrationmethods. Natsiavas [14] presented exact solutions andstability analysis for a system with a symmetric tri-linearrestoring force subjected to harmonic excitation, in 1989.He derived a set of six transcendental equations definingthe system behavior, using periodicity conditions, andfound that both stable and unstable time solutions exist fordifferent combinations of system parameters. He later[15,16] published analytical studies on steady-state beha-vior and stability of non-linear Duffing-type systems, andagain inferred that parameter values play an important rolein improving system performance. Ji [17–19] utilized thesame approach and analyzed piecewise linear systems

ARTICLE IN PRESS

c2 c1k1k2

Sprung mass mx

y

Primary Suspension

Secondary Suspension

Relative displacement

k1

Stif

fnes

s ra

te

c1 + c2

k1 + k2

Relative displacement

c1

Dam

ping

rat

e

�

−� �

−� �

Fig. 1. Mechanical model of the bilinear system and representation of the

stiffness and damping coefficients.

S. Deshpande et al. / International Journal of Mechanical Sciences 48 (2006) 341–377 343

subject to saturation constraints and showed possiblechaotic behavior. Mahfouz and Badrakhan [20] in 1990investigated chaotic behavior in piecewise linear systems.Dai and Singh [21] examined the oscillatory motion of aspring–mass system in time domain subjected to apiecewise excitation. Padmanabhan and Singh [22] utilizedthe technique of parametric continuation to study steady-state response and global dynamics of a two d.o.f.piecewise non-linear system. In 1995, Narayanan andSekar [23] published a study on response of a single d.o.f.vibrating system with asymmetrical piecewise linearstiffness subjected to combined harmonic and flowinduced excitations. Wang [24] presented his findings ona similar system using a finite element method. Chatterjeeet al. [25] in 1995 extended the method of equivalentlinearization to obtain periodic responses of harmonicallyexcited piecewise non-linear oscillators. In 1998, Natsiavas[26] presented a stability analysis for periodic motions of aclass of harmonically excited single d.o.f oscillators withpiecewise linear characteristics. In 2001, Chicurel-Uziel [27]proposed a method to obtain, in a single equation, theexact closed form displacement response of free undampedand forced damped vibrations of systems with piecewiselinear springs. Jazar and Golnaraghi [28] studied steady-state vibrations of piecewise linear systems using averagingmethod to find the frequency response and found thatdamping had a more pronounced effect on amplitudereduction and jump avoidance in such a system. Xu et al.[29] in 2002 used an incremental harmonic balancemethod to perform stability studies on a system withasymmetric piecewise linearities and found the occurrenceof period doubling bifurcation leading to chaotic behavior.Nakai et al. [30] put forth their analysis of suchsystems with boundaries in space and time using thecircumferential vibration of a gear pair structure as anexample. More recently, comprehensive studies on largeamplitude non-linear normal modes and mapping dy-namics of piecewise linear systems have been presented byJiang et al. [31] in 2003 and Luo [32] in 2004, respectively.Till date, exact solutions proposed by Natsiavas [26] havebeen used as a benchmark to compare newly developedapproximate solutions; and comparative studies usingnumerical analysis methods and direct integration havebeen employed in several later studies. However, the exactsolution method suffers from a lack of closed form solutionin the frequency domain and requires excessive numericalanalysis. In 2003, Jazar et al. [33] presented studies ontransient and steady-state behavior of passive linearvibration isolation systems and introduced an optimaldesign procedure for these systems using a cost functionRMS of absolute acceleration minimized with respect toRMS of relative displacement. Narimani et al. [34] in 2004presented a frequency response equation for symmetricpiecewise linear systems with dual damping and stiffnessbehavior, using a modified averaging method. Thisresearch commences where Narimani et al. concluded,and employs graphical and numerical analysis to study the

behavior of piecewise linear systems and arrive at acomprehensive optimal design solution.

3. Mathematical model and analysis

An ideal example for this study is a bilinear system withtwo sets of stiffness and damping coefficients as shown inFig. 1. This system will provide us with a high degree of non-linearity along with simplicity of numerical calculation.The sprung mass m acts as the body of the system

supported by two stages of vibration isolation. The firststage is called ‘primary suspension’ and the second stage,which is effective beyond the clearance amplitude D, is called‘secondary suspension’. It is known that a softer suspensionprovides better force transmissibility at the cost of relativedisplacement. But there is a physical constraint on themaximum displacement possible, and this physical spaceusually is a fixed application-specific value. The secondarysuspension system prevents excessive relative displacementof the mass m from the base by effecting a step-increase inthe stiffness and damping characteristics after the clearanceamplitude D is exceeded. This represents a sudden change inthe system properties which accounts for the inherent hardnon-linearity of the piecewise linear system.The governing differential equation of the system after

non-dimensionalization is

€wþ 2x1 _wþ w ¼ �€vþ g3ðw; _wÞ, (1)

where

g3ðw; _wÞ ¼

�2rx2 _w� r2wþ r2d w4d;

0 jwjod;

�2rx2 _w� r2w� r2d wo� d

8><>:

and

w ¼ u� v ¼z

Y; z ¼ x� y; u ¼

x

Y; v ¼

y

Y; d ¼

DY

,

ARTICLE IN PRESSS. Deshpande et al. / International Journal of Mechanical Sciences 48 (2006) 341–377344

2x1o1 ¼c1

m; 2x2o2 ¼

c2

m; o2

1 ¼k1

m; o2

2 ¼k2

m,

2x3o3 ¼ðc1 þ c2Þ

m¼

c3

m¼ 2x1o1 þ 2x2o2,

o23 ¼ðk1 þ k2Þ

m¼

k3

m¼ o2

1 þ o22,

_w ¼dw

dt; w0 ¼

dw

dt; t ¼ o1t; r ¼

oo1; r ¼

o2

o1.

g4 ¼

0 0ofof0;

�2rx2rA cos f� r2A sin fþ r2A sin f0 f0ofop� f0;

0 p� f0ofopþ f0;

�2rx2rA cos f� r2A sin f� r2A sin f0 pþ f0ofo2p� f0;

0 2p� f0ofo2p:

8>>>>>><>>>>>>:

(12)

The system response depends on six-dimensionlessparameters, x1, x2, r ¼ o2Xo1, r ¼ oXo1, t, and d. Theequation of motion under a harmonic base excitation

v ¼y

Y¼ sinðrtÞ (2)

becomes

€wþ 2x1 _wþ w ¼ r2 sinðrtÞ þ g3ðw; _wÞ. (3)

Assuming a solution of the form

wðtÞ ¼ AðtÞ sinðrtþ bðtÞÞ, (4)

where

_wðtÞ ¼ AðtÞr cosðfðtÞÞ, (5)

fðtÞ ¼ rtþ bðtÞ (6)

provide

_A sin fþ A _b cos f ¼ 0, (7)

€w ¼ _Ar cos f� Ar2 sin f� Ar _b sin f. (8)

Using Eqs. (4), (5), and (8) the equation of motionbecomes

_Ar cos f� Ar2 sin f� Ar _b sin fþ 2x1Ar cos f

þ A sin f� r2 sinðrtÞ � g3 ¼ 0. ð9Þ

By solving Eq. (7) and (9) the problem transforms to aset of two first-order ode.

_A ¼ A sin f cos f r�1

r

� �þ r cos f sinðf� bÞ

þg4 cos f

r� 2x1A cos2 f, ð10Þ

_b ¼ sin2 f1

r� r

� �þ 2x1 sin f cos f�

g4 sin fAr

þr sin f sinðb� fÞ

A. ð11Þ

The variables _A and _b are slowly varying with time, andit can be assumed that their averages over a period ofoscillation remain constant [35]. A constant j0 is intro-duced such that d ¼ A sin f0, f0 being the value of j(t)when the system touches the secondary suspension. So g4becomes

Integrating this expression over the intervals of j(t) thefollowing expressions are obtained:

_A ¼1

2pð�2x1Ap� rp sin bþ 2rx2Aðsin 2f0 � pþ 2f0ÞÞ,

(13)

_b ¼ �1

2prp cos b

Aþ p r�

1þ r2

r

� ��

þ2r2f0 þ r2 sin 2f0

r

�. ð14Þ

For a steady-state response of the system, _A ¼ 0 and_b ¼ 0. By eliminating b between Eqs. (13) and (14), thefollowing expression for frequency response is obtained[34] which is an implicit function of the amplitude andfrequency ratio.

ðA2 � 1Þp2r4 þ ðZ21 � 2AZ2pÞr2 þ Z2

2 ¼ 0, (15)

where

Z1 ¼ �2x1Apþ 4rx2d cos f0 � 2rx2Aðp� 2f0Þ,

Z2 ¼ Apð1þ r2Þ � 2Af0r2 � 2r2d cos f0.

Also, phase of relative displacement

b ¼ tan�1Z1r

�Apr2 þ Z2

� �. (16)

The Eq. (1) and solution (15) have been analyzed indetail, and experimental and numerical validations areprovided by Narimani et al. [34]. A numerical plot of thefrequency response is observed to exhibit single unique andmultiple response behavior for different values of para-meters x1, x2 and r. Figs. 2 and 3 depict the frequencyresponse and phase of motion for a piecewise linearvibration isolator compared to a linear isolator, to

ARTICLE IN PRESS

2.5

2

2

1.5

1.5

1

1

0.5

0.50

0

A

r

δ = 2

Linearξ1=0.2ξ2=0ρ = 0δ = infinity

Non-linearξ1=0.2ξ2=1ρ = 7δ = 2

Fig. 2. Relative displacement frequency response of piecewise linear

system compared with linear system.

160

140

120

100

80

60

40

20

0.5 1 1.5 2 2.5 3r

β

Linear

δ = 2

ξ1=0.2

ξ2=0

ρ = 0

δ = infinity

Non-linearξ1=0.2ξ2=1ρ = 7

Fig. 3. Phase lag response of piecewise linear system compared with linear

system.

2.16

2.14

2.12

2.1

2.08

2.06

2.04

2.02

21 1.2 1.4 1.6

A

r

Bacward JumpResponse path 2

Forward JumpResponse path 1

δ = 2

ξ1=0.2ξ2=1ρ = 7

Fig. 4. Jump and multiple response behavior.


illustrate the effect of introducing the secondary suspen-sion.

4. The phenomenon of ‘jump’ and condition for jump

avoidance

When the excitation frequency is gradually increased atconstant excitation amplitude, there comes a point where asudden drop in amplitude is observed. This phenomenon isknown as ‘jump’, which has been observed in the piecewiselinear vibration system when studied experimentally byJazar et al. [34], and discussed in detail in non-lineartextbooks [35]. Jump phenomenon is shown in Fig. 4.

A good vibration isolator design should eliminate thepossibility of jump. For this to happen, the conditionsunder which jump occurs must be identified and isolated.On these lines, if a scientific method can be developed toensure that the system functions in no-jump zones, such astep would assume immense significance in optimizingisolator performance, and would be a major deciding

factor in successful experimental corroboration of specu-lated designs. Jump can be avoided if it can be ensured thatthe frequency response of the system is unique. It ispossible that a single unique response exists within aparticular range of the parameters x1, x2, and rU Toinvestigate if such a jump-avoidance parameter range canbe established, the frequency response equation is analyzedas follows.In our case the frequency response (15) can be solved for

r2 to be written as

r2 ¼�Z2

1 þ 2AZ2p�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ4

1 � 4Z21Z2Apþ 4Z2

2p2q2p2ðA2 � 1Þ

. (17)

Substituting Z1 and Z2 and differentiating r2 with respectto amplitude A, provides an equation

dr

dA¼�ð�Z2

1 þ 2AZ2p� Z5ÞA

2rp2ðA2 � 1Þ2

þ�Z1Z3 þ Z2pþ AZ4p� Z6=4Z5

2rp2ðA2 � 1Þð18Þ

to determine the position of having vertical slope,where Zi are functions in x1, x2, A, d, and r and listed inAppendix A.It can be said that a multiple behavior frequency

response curve will have exactly two points where dr/

dA ¼ 0. A unique response curve will have just one or nopoint C where dr/dA ¼ 0. The boundary case whereunique response behavior ends and multiple-responsebehavior begins, will have exactly one point on the curvewhere dr/dA ¼ 0. It can be inferred from Fig. 5 that r ¼3:6324 is the highest value of r to avoid jump, this beingtrue for x1 ¼ 0:2 and x2 ¼ 1. So there is a limiting value ofr for every set of x1 and x2, for which there is a singleunique response and for values higher than the limitingvalue, a multiple-behavior frequency response is obtained.

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2.12

2.1

2.08

2.06

2.04

2.02

2

3 3.5 4 4.5 5 5.5 6ρ

A

D

JumpAvoidanceRegion

Jump

RegionOccurance

E

F

ξ1 = 0.2

ξ2 = 1

δ = 2

Fig. 6. Plot of A vs. r for jump avoidance region.

2.14

2.12

2.1

2.08

2.06

2.04

2.02

A

20.9 1 1.1 1.2 1.3 1.4 1.5 1.6

r

Cdr/dA=0

ξ1=0.2ξ2=1

δ = 2ρ = 3.6324

Fig. 5. No jump—single response.


To find this limiting value of r, a plot of Eq. (18) withdr/dA ¼ 0 is examined in Fig. 6.

From the Fig. 6, it can be seen that when r ¼ 3:6324there is just one point (point D) at which dr/dA ¼ 0, whichimplies the case as in Fig. 5, when the value of r is highestpossible without the occurrence of jump; the limiting valuefor jump avoidance. When r ¼ 4:5, there are two points(E and F) at which dr/dA ¼ 0, which suggests a situa-tion similar to Fig. 4, i.e. a multiple-behavior responseexhibiting jump.

Malatkar and Nayfeh [36] have developed an analyticalmethod to calculate the jump frequencies for single-degree-of-freedom non-linear systems, using two methods: simpleanalytical isolation and the Grobner-basis. Similarly, inthis work, it was attempted to obtain rlimiting analyticallyby setting dr/dA ¼ 0 in Eq. (18). But this could not be doneas the Eq. (19) so obtained is implicit in r and cannot besolved analytically. Therefore the graphical criterion forjump avoidance is employed for further analysis. This hassignificant advantages also, as each A vs. r curve can be

visualized for post-verification.

�ð�Z21 þ 2AZ2p� Z5ÞA

ðA2 � 1Þ� Z1Z3 þ Z2pþ AZ4p

�Z6

4Z5¼ 0. ð19Þ

5. Parameter boundary for jump avoidance

Keeping d constant, numerical values for x1 and x2 aresubstituted in the condition for jump avoidance (19) andrlimiting for each set is investigated. A range of parametersin which jump cannot occur is obtained. The bounds onthis range are the parameter boundaries for furtheroptimization of the secondary system within the no-jumpzone. First, using the condition for jump-avoidance, anumerical plot of A vs. r is constructed for one value of x1and one value of x2. The limiting value for r is retrievedfrom the plot data. An array consisting of [x1, x2, rlimiting] iscreated. This procedure is performed for different values ofx1 and x2 to yield a matrix with [x1, x2, rlimiting] as rows. Asurface plot is constructed using this matrix. This surfaceplot defines the boundary between jump avoidance andjump occurrence regions. The region below the surfaceshows a single unique frequency response without thepossibility of jump. The values on the surface also showsingle response. Once parameter values increase from thesurface outward, a multiple behavior frequency response isseen. The parameter boundary surface plot obtained ford ¼ 2 is as shown in Fig. 7.The parameter boundary is observed to be at a minimum

for rlimiting at [x1 ¼ 0, x2 ¼ 0]. As x1 is increased, rlimiting

moves upward and has a maximum at x1 ¼ 0.25. Theupward shift in the boundary is gradual for lower values ofx2. But for higher values of x2 the transition is much fasterindicating that the boundary is more sensitive to primarydamping as the secondary damping x2 is increased. Assecondary damping x2 is increased, rlimiting moves upwardand has a maximum at x2 ¼ 5. The upward shift in theboundary is gradual for lower values of x1. Again, forhigher values of x1 the transition is much faster indicatingthat the boundary is more sensitive to secondary dampingas the primary damping x1 is increased.Fig. 8 depicts a two-dimensional plot of x1 vs. x2 for

different values of r and facilitates the selection ofparameters for jump avoidance. This criterion would behelpful in the design of piecewise linear vibration isolators,in that the possibility of jump avoidance can be eliminatedby proper selection of system parameters, for a givenclearance d. Similar plots can be constructed in anyapplication with predefined structural constraints andknown system configuration.It can be inferred from Fig. 8 that in general, there is a

reduction in the jump free zones with increasing stiffnessratio. A unit stiffness ratio provides the maximum area for

ARTICLE IN PRESS

0.25

0.2

0.15

0.1

0.05

0

ξ 1

ξ2

4 2 0

ρ=8

ρ=5

ρ=3 ρ=2.5 ρ=2 ρ=1.5 ρ=1

Fig. 8. 2D design curves for selecting the dynamic parameters of the

piecewise vibration isolator in order to avoid jumps.

8

6

4

2

0

4

2

00 0.05 0.1 0.15 0.2 0.25

ρ

ρ

ξ2

ξ1ξ1

ξ2

Jump avoidanceboundary

Jump avoidanceboundary

8

6

4

2

00.25 0.2 0.15 0.1 0.05

0

4

2

0

δ = 2

δ = 2

Fig. 7. Parameter boundary surface between jump avoidance and jump occurrence regions of the frequency response.


avoiding jump. This indicates that the jump avoidanceperformance is best when the primary and secondarysuspension stiffness values are equal. As the secondarystiffness is increased over the primary, a gradual decreasein the parameter space for jump-prevention is observed. Itis also notable that for higher values of stiffness ratio, bothprimary and secondary damping need to be increased forjump-free operation of the isolator. The jump avoidanceparameter boundaries are obtained in a matrix form withrows [x1,o1, x2, rlimiting].

6. Parameter study

Before exploring an optimum design, a parameter studyis needed to predict the effect of change in system

parameters x1, x2, r and d on the frequency response (19)and the associated change in peak relative displacementamplitude. It is equally important to examine variation ofjump behavior with system parameters.Effect of changing primary damping coefficient x1 on

frequency response is shown in Fig. 9a. An increase inprimary damping x1, with other three parameters constant,leads to a reduction in peak amplitude and a decreasingtendency towards jump occurrence.Effect of changing secondary damping coefficient x2 on

frequency response is illustrated in Fig. 9b. An increase insecondary system damping x2 causes expected reduction inpeak amplitude. Also, increasing x2 shows a gradual shiftfrom multiple response behavior to single unique response.The peak amplitude change is less sensitive to changes in x2as x2 is increased.Effect of changing stiffness ratio r on frequency

response is plotted in Fig. 9c in which the systemshows a tendency to shift to multiple response behaviorexhibiting jump when the value of r is increased beyond acertain threshold. This implies again that there is a limitingvalue of r beyond which the piecewise linear systemshows occurrence of jump. Also observed is thereduction in peak amplitude with increase in rU Effect ofchanging clearance amplitude d on frequency response isshown in Fig. 9d. There is an increase in peak amplitudewith increase in d. Also observed is a reduction inthe operating frequency range with increasing d. Asthe clearance d is reduced it is observed that frequencyresponse plots can be obtained for x140:25. It remainsto be seen whether definite regions of optimumperformance can be found within this limited band ofparameter x1.It can be observed that the peak amplitude of relative

displacement is sensitive to changes in system parametersx1, x2, r and d. The phenomenon of jump occurrence also issensitive to changes in all four system parameters.

ARTICLE IN PRESS

2.35

2.3

2.25

2.2

2.15

2.1

2.05

2

A A

A A

ξ2=1ξ1=0.05 ξ2=1

ξ2=2

ξ2=10

ξ1=0.1

ξ1=0.15

ξ1=0.2

ξ1=0.25

ρ = 4

δ = 2

ξ2=0.15

ρ = 4

δ = 2

ξ2=0.1

ξ2=1

ρ = 4

ξ2=0.15

ρ = 4

δ = 2

1 1.2 1.4 1.6r r

2.2

2.15

2.1

2.05

2

2.2

2.15

2.1

2.05

2

0.9 1 1.1 1.2 1.3 1.4

4

3.5

3

2.5

2

1.50.8 1 1.2 1.4 1.6 1.8

r0.8 1 1.2 1.4 1.6 1.8

r

ρ = 1

ρ = 2

δ = 4

δ = 3.5

δ = 3

δ = 2.5

δ = 2δ = 1.5

(a) (b)

(c) (d)

Fig. 9. Effect of changing parameters on the frequency response: (a) effect of changing x1; (b) effect of changing x2; (c) effect of changing r; (d) effect ofchanging d.

10

8

6

4

2

03 2 1 0

0

0.1

0.2

ρ ρ

δ=2

δ=1.5

δ=1.25

δ=1.05

No-jumpszones

No-jumpszones

ξ2

ξ1

ξ1

ξ2

Jump zones

Jump zones10

8

6

4

2

0

0

2

1

0

0.250.2 0.15

0.10.05

δ=2

δ=1.05

δ=2

Fig. 10. Jump avoidance boundary variation with clearance d.


A significant step in optimal design of the piecewiselinear system is ensuring effective jump avoidance. It istherefore beneficial to study the effect of varying dynamicalparameters on the jump avoidance boundary. The resultinginformation will be instrumental in devising a bettervibration isolator. Fig. 10 shows jump avoidance boundaryvariation with change in clearance dU The parameterboundary shifts downward with increasing clearance delta,

reducing the no-jump zone. However the reduction in theno-jump area is not uniform over the entire range ofprimary damping. An increasing tendency of the boundarysurface to flex upward is visible for increasing values of d.Increasing d reduces the no-jump zone for x1o0:15,whereas beyond x1 ¼ 0:15, the boundary flexes to increasethe no-jump zone for higher values of d (d ¼ 1:5 onwards).This indicates a rising dominance of primary damping on

ARTICLE IN PRESS

0.4

0.3

0.2

0

0.1

ξ 1 ξ 1ξ 1ξ 1

3 2 1 0ξ2

3 2 1 0ξ2

3 2 1 0ξ2

3 2 1 0ξ2

ρ = 10

ρ = 7

ρ = 6

ρ = 5

ρ = 4

ρ = 3 ρ = 2

ρ = 9 ρ = 8 ρ = 7 ρ = 6 ρ = 5ρ = 4 ρ = 2

ρ = 3 ρ = 9

ρ = 8

ρ = 7

ρ = 6

ρ = 5

ρ = 10

ρ = 9

ρ = 8

ρ = 7

ρ = 6

ρ = 5

ρ = 4

ρ = 3

ρ = 2 ρ = 1

ρ = 4 ρ = 3 ρ = 2

δ = 1.05δ = 1.25

Jump free zonesJump free zones

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

0.25

0.2

0.15

0.1

0.05

0

0.25

0.2

0.15

0.1

0.05

0

δ = 1.5Jump free zonesδ = 2Jump free zones

(a) (b)

(d)(c)

Fig. 11. 2D design curves for jump avoidance parameter-selection.


the system behavior as x1 is increased, which is expected,because when primary damping is amplified, the piecewiselinear system should show a reducing tendency to engagethe secondary suspension. Jump avoidance behavior fordifferent values of d can be seen clearly from anotherviewpoint in Fig. 10.

The above figures depict the changing behavior of thepiecewise linear system with variation in primary andsecondary damping, stiffness ratio and clearance gapwithin normal operating parameter ranges. It is notablethat for higher values of clearance d, the jump avoidancecondition is more sensitive to changes in x1 and x2.

More investigation on numerical values tabulated inAppendix B shows that the jump avoidance conditionrange drops from x1 ¼ 0:45 to x1 ¼ 0:25 when clearance dis increased from 1.05 to 2. There is a critical value ofprimary damping x1 beyond which the primary damper isso strong that it does not allow the secondary system toengage, i.e. the system vibrations are damped to anamplitude lower than the clearance d. The jump avoidancecondition is not valid after this critical value of primarydamping. Expectedly, the critical primary damping goesdown with increase in clearance d. The variation of the

jump avoidance condition is relatively uniform withchanging secondary damping x2. The jump-free zones alsoincrease at a steady slope.The 2D design criterion plots for jump avoidance

are as shown in Fig. 11. From these plots it can beobserved that there is a progressive change in thejump avoidance condition when the clearance d isincreased from 1.05 to 2. For d ¼ 1:05 (Fig. 11a), thejump avoidance curve is less sensitive to changes inprimary damping coefficient x1 and shifts uniformly withincrease in stiffness ratio r. The no-jump zones arerelatively large and distinct. As the clearance amplitudeis increased, the system becomes more and moresensitive to primary damping, and the effect of a dominantprimary damping coefficient is visible for d ¼ 2 in Fig. 11d.It is obvious that the jump avoidance zones appearto be shrinking for increasing clearance amplitudes,when other parameters are held constant. This suggeststhat widening of the clearance gap increases susceptibilityof the system to jump occurrence, which is quitecontrary to the seemingly logical notion that moreclearance amplitude will amount to better isolator designand increased system life.


7. Optimal design solution

Effective optimization of vibration isolator performancehas been the focus of attention in the research efforts ofseveral scientists and engineers over the years. One of theearliest studies was conducted by Sevin and Pilkey in 1968[37], in which a concise mathematical statement for theoptimization problem of shock isolation systems waspresented, and two computational approaches for itssolution were introduced. In 1985, Nissen et al. [38] cameforth with notable work on optimization of a non-lineardynamic vibration absorber, in which the maximumsuppression bandwidth was used as performance index.The result was a method which predicted system para-meters to maximize the operating frequency ranges for ageneral form of non-linear absorber. In 1986, Chalasani[39] employed RMS functions of vertical acceleration and

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

A

r

δ = 2r1 r2

Linearξ1=0.2ξ2=0ρ=0δ=infinity

Non-linearξ1=0.2ξ2=1ρ=7δ=2

Fig. 12. Limits of integration.

Fig. 13. R�Z plot fo

suspension travel as objectives to explore optimal para-meter ranges for passive linear suspensions and comparedthe results with optimal active suspensions. Benefits ofusing an active suspension were pronounced near reso-nance conditions, and a 20% overall improvement wasobserved in RMS acceleration transmission. Jordanov andCheshankov [40] in 1987 introduced a numerical method toobtain optimal solutions for generic forms of linear andnon-linear vibration absorbers, where the objective func-tions were based on relative displacement amplitude andmass ratio. In 1989, Lin and Zhang [41] developed analgorithm for frequency domain equivalent optimal controland applied it to a four-degree-of-freedom vehicle passivesuspension system. The optimization was performed forminimum body response acceleration under allowablesuspension displacements. More recently in 1996, Roystonand Singh [42] proposed an optimization method based onvibratory power transmission. In this method optimizationwas performed in the frequency domain and active andpassive vibratory systems were employed in conjunction tominimize the objective function of transmitted power.A drawback of this method is that it can converge to localminima. Foumani et al. [43] in 2003 introduced anexperimental method for frequency domain optimizationof engine mount characteristics. However, this work didnot cover non-linear possibilities and jump occurrence.Most recently, Alkhatib et al. [44] published their work onoptimization of passive linear suspension using a geneticalgorithm technique, wherein root mean square (RMS)functions of absolute acceleration and relative displace-ment were employed as objectives. This method however,can be computationally expensive. Till date there has beenno research performed on optimization of piecewise linearvibration isolation systems.

r linear system.

ARTICLE IN PRESSS. Deshpande et al. / International Journal of Mechanical Sciences 48 (2006) 341–377 351

Now an optimal solution is sought for the piecewiselinear system within the parameter boundaries definedby the plot in Fig. 7. Absolute displacement, relativedisplacement, Transmissibility have been all been em-ployed to the task, but a good result have been obtainedusing RMS cost functions by Jazar et al. [33] and Chalasani[39] for passive linear vibration systems. RMS of absoluteacceleration [33] is chosen as the cost function, andit is minimized with respect to the RMS of relativedisplacement.

The expressions for relative displacement, absoluteacceleration and absolute displacement are derived asfollows.

From (4) and (15) the steady-state periodic timeresponse for relative displacement of the system is

Fig. 14. R� Z plot for optimum linear


wðtÞ ¼ AðtÞ sinðrtþ bðtÞÞ. An expression for the absolutedisplacement at steady-state condition can be derived:

u ¼ A sinðrtþ bÞ þ sinðrtÞ ¼ B sinðrtþ gÞ, (20)

where

B ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2 þ 2A cos bþ 1

qg ¼ tan�1

A sin bA cos bþ 1

� �.

(21)

Similarly, expressions are derived for amplitude andphase of absolute acceleration:

€u ¼ �Ar2 sinðrtÞ cos b� Ar2 cosðrtÞ sin b� r2 sinðrtÞ

¼ �C sinðrtþ aÞ, ð22Þ

point [x1 ¼ 0:01, o1 ¼ 1:884956].

point [x1 ¼ 0:05, o1 ¼ 9:424778].

d � 1


where

C ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2r4 þ r4 þ 2Ar4 cos b;

q

a ¼ tan�1A sin b

A cos bþ 1

� �¼ g. ð23Þ

Plots of absolute displacement and absolute acceleration canbe constructed using (21) and (23). The cost function RMSabsolute acceleration for the secondary suspension is defined as

R ¼1

ðr2 � r1Þ

Z r2

r1

C2 dr

¼1

ðr2 � r1Þ

Z r2

r1

ðA2r4 þ r4 þ 2Ar4 cos bÞdr ð24Þ



and RMS relative displacement for the secondary suspension isgiven by

Z ¼1

ðr2 � r1Þ

Z r2

r1

A2 dr. (25)

The limits of integration r1 and r2 are the extremities of thefrequency response curve (4). They are a function of d and canbe obtained from the linear frequency response expression [33].

r1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2 � 2x21d

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�4x21d

4þ 4x41d

4þ d2

qr

2, (26)

point [x1 ¼ 0:1025, o1 ¼ 20:10619].

point [x1 ¼ 0:15, o1 ¼ 30:15929].


r2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2 � 2x21d

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�4x21d

4þ 4x41d

4þ d2

qr

d2 � 1. (27)

Fig. 12 illustrates r1 and r2.Now the optimal design curve for a linear system as

obtained in [33] is reproduced. The optimal parameters x1and o1 for a linear system are extracted from the plotin Fig. 13 [33] and are used as primary suspensionparameters to further optimize the secondary system. Foroptimizing the secondary, for each point on the linearoptimal curve [x1, o1], values of secondary dampingcoefficient x2 are chosen and rlimiting for every such stepis found using the jump avoidance condition. A matrix



with rows [x1, o1, x2, rlimiting] is stored. Numerical values ofR and Z are calculated by using a looped code. The coderuns for a fixed value of d. The following procedure isfollowed for a fixed d.The stored matrix of [x1, o1, x2, rlimiting] is accessed. For

every set of values of x1 and x2, r is increased in steps from0 to rlimiting, and a frequency response is plotted. The valueof peak relative displacement amplitude is extracted fromthis plot, which is used in Eqs. (24) and (25). Using x1, o1,x2, r, and A, the RMS absolute acceleration ‘R’ and RMSrelative displacement ‘Z’ are calculated.‘R vs. Z’ plots are constructed for each set of [x1, o1,

x2, r]. There is a unique optimal design plot for each set of[x1, o1, x2, r]. Lines of minima are the objective in ‘R vs. Z’

point [x1 ¼ 0:20125, o1 ¼ 40:8407].

point [x1 ¼ 0:25, o1 ¼ 49:63716].

ARTICLE IN PRESS

4.5

4

3.5

3

2.5

2

A

0.9 1 1.1 1.2 1.3r

Linearξ2=0, ρ = 0

ξ1=0.1025

ω1=20.10619

δ=2

ξ2=0.5, ρ = 0

ξ2=0.5, ρ = 0.7

ξ2=0.5, ρ = 1.746148

Non-Optimal

Optimal

Jump-avoidanceboudary

Fig. 20. Relative displacement comparison for optimal, non-optimal and

linear system.

5

4

3

2

1

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2r

C

Linearξ2 = 0, � = 0

ξ1 = 0.1025ω1 = 20.10619

δ = 2

ξ2 = 0.5, � = 0.1

ξ2 = 0.5, � = 0.7

ξ2 = 0.5, � = 1.746148

Non-Optimal

Optimal

Jump-avoidance boundary

Fig. 21. Absolute acceleration comparison for optimal, non-optimal and

linear system.

Fig. 22. Relation between optimal par


plots. The minimum transmitted acceleration and mini-mum displacement for a given set of parameters will lie onthe lines of minima. These lines will express the parametervalues for optimal system performance.Sample optimal plots at regular intervals on the linear

optimal design curve are presented (Figs. 14–19). Each plothas a line of minima as the optimal design curve for thesecondary suspension.Optimal plots show similar behavior over the interval of

interest [x1 ¼ 0 to 0.25, x2 ¼ 0 to 5, r ¼ 0 to rlimiting]. It canbe observed from the optimal plots that a minimizationtradeoff for absolute acceleration and relative displacementcan be obtained along the line of minima. The line ofminima runs close to the constant r � 1 line for most cases,indicating that optimal conditions exist close to r � 1 ork1 � k2. Keeping stiffness values constant, increase insecondary damping x2 further reduces transmitted accel-eration and relative displacement. But this advantage isseen to be obtained only up to a certain threshold value ofx2 beyond which the x2 ¼ const. lines curve downward.

8. Results and discussion

The behavior of design parameters x1, x2, and r in thefrequency response is investigated and a boundary surfaceFig. 7 is found which specifies the domain in which thephenomenon of jump cannot occur. The limiting values ofr are successfully obtained for range of values of dampingcoefficients x1 and x2, and a matrix is constructedconsisting of [x1, o1, x2, rlimiting]. Then, optimization ofthe system performance is carried out by minimizing thenon-dimensionalized transmitted acceleration with respectto relative displacement, within the jump avoidancedomain specified by the parameter plot Fig. 7. RMS valuesof absolute acceleration and relative displacement are used

ameters for x1 from 0.01 to 0.075.

ARTICLE IN PRESS

Fig. 23. Relation between optimal parameters for x1 from 0.1025 to 0.15.

Fig. 24. Relation between optimal parameters for x1 from 0.15 to 0.25.


to generate a numerical R vs. Z plot. These RMS values areobtained using the numerical values of peak relativedisplacement amplitude A extracted from the frequencyresponse plot. It is observed that the frequency responsecan be obtained within a limited band of the parameter x1,beyond which the response function shows complex rootsand cannot generate a valid numerical plot. This limitingvalue of x1 signifies the minimum value of primarydamping which prevents the limiter from reaching thesecondary system.

R vs. Z plots are generated within the functional domainof x1. Each plot displays lines of minima for x2 ¼ constant

curves. The system parameters can be chosen along theselines to obtain minimal force transmission under allowablerelative displacement.Comparative plots showing absolute acceleration

and relative displacement for optimal and non-optimalparameter values are shown in Figs. 20 and 21. It canbe verified from Fig. 21 that both, the absolute accelera-tion and its integral over frequency ratio are minimumfor optimal parameter values, albeit at the expense ofslightly increased relative displacement (Fig. 20). Optimalparameter relation plots are shown in Figs. 22–24.These plots are trend plots for optimization and can

ARTICLE IN PRESS

Table 1

Optimal parameters for jump avoidance


be used to select parameter values for optimal systemperformance.

x1 o1 x2 rlimiting

0.00375 0.62832 0 0.975989843

0.00375 0.62832 0.25 1.261207889

0.00375 0.62832 0.5 1.343185247

0.00375 0.62832 0.75 1.495016779

0.00375 0.62832 1 1.634304491

0.00375 0.62832 1.5 1.862443374

0.00375 0.62832 2 2.049416052

0.00375 0.62832 3 2.352381534

0.00375 0.62832 4 2.599308828

0.00375 0.62832 5 2.811762817

0.00625 1.25664 0 0.976066255

0.00625 1.25664 0.25 1.26461027

0.00625 1.25664 0.5 1.35090379

0.00625 1.25664 0.75 1.507807204

0.00625 1.25664 1 1.649594818

0.00625 1.25664 1.5 1.884633645

0.00625 1.25664 2 2.078081674

0.00625 1.25664 3 2.394675248

0.00625 1.25664 4 2.655111529

0.00625 1.25664 5 2.881188446

0.01 1.88496 0 0.976252928

0.01 1.88496 0.25 1.269789048

0.01 1.88496 0.5 1.362768769

0.01 1.88496 0.75 1.525970816

0.01 1.88496 1 1.673099963

0.01 1.88496 1.5 1.918580383

0.01 1.88496 2 2.122090152

0.01 1.88496 3 2.458553798

0.01 1.88496 4 2.740177732

0.01 1.88496 5 2.987113556

9. Conclusion

Piecewise linear vibration isolation systems are highlynon-linear systems which cannot be designed and opti-mized analytically. In this research a numerical analysisapproach to optimizing the performance of a piecewiselinear vibration isolation system is introduced. Implicitfunctions for the jump-avoidance condition and frequencyresponse are plotted numerically in multiple iterativeloops, and values of minimum stiffness ratio and maximumamplitude are extracted from these plots, to constitute arange of cost functions, which are in turn examinedfor minima conditions. The optimum solution for thispiecewise linear system is found to be dependant uponprimary and secondary damping coefficients x1 and x2,stiffness ratio r, and clearance d. Optimal parameter plotsare presented for one value of clearance d to help theselection of optimal parameter values. This optimiza-tion process can be used to design all symmetric bilinearsystems.

The result of this research is a novel numerical procedurefor the optimal design of a piecewise linear vibrationisolation system, which can be further adapted and appliedto several highly non-linear systems hitherto considered asvery difficult to examine.

0.01375 2.51327 0 0.976526831

0.01375 2.51327 0.25 1.275062453

0.01375 2.51327 0.5 1.374997894

0.01375 2.51327 0.75 1.54438326

0.01375 2.51327 1 1.697194283

0.01375 2.51327 1.5 1.951676153

0.01375 2.51327 2 2.165517088

0.01375 2.51327 3 2.523971103

0.01375 2.51327 4 2.826632254

0.01375 2.51327 5 3.094944482

0.01625 3.14159 0 0.976758481

0.01625 3.14159 0.25 1.27863321

0.01625 3.14159 0.5 1.38336462

0.01625 3.14159 0.75 1.556888072

0.01625 3.14159 1 1.713486695

0.01625 3.14159 1.5 1.974351433

0.01625 3.14159 2 2.195270978

0.01625 3.14159 3 2.567269595

0.01625 3.14159 4 2.884290334

0.01625 3.14159 5 3.167924732

0.02 3.76991 0 0.977180678

0.02 3.76991 0.25 1.284076214

0.02 3.76991 0.5 1.396254645

0.02 3.76991 0.75 1.576028284

0.02 3.76991 1 1.736905108

0.02 3.76991 1.5 2.009141879

Appendix A. List of abbreviations

Z1 ¼ �2x1Apþ 4rx2d cos f0 � 2rx2A ðp� 2f0Þ,

Z2 ¼ Apð1þ r2Þ � 2r2f0A� 2r2d cos f0,

Z3 ¼ �2x1pþ4rx2d

3

A3 cos f0

� 2rx2ðp� 2f0Þ �4rx2d

A cos f0

,

Z4 ¼ pð1þ r2Þ þ2r2d

A cos f0

� 2r2f0 �2r2d3

A3 cos f0

,

Z5 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ4

1 � 4Z21Z2Apþ 4Z2

2p2q

,

Z6 ¼ 4Z31Z3 � 8Z1Z2Z3Ap� 4Z2

1Z2p� 4Z21Z4Ap

þ 8p2Z2Z4.

0.02 3.76991 2 2.240103659

0.02 3.76991 3 2.633880181

0.02 3.76991 4 2.972710483

0.02 3.76991 5 3.27865404

0.02375 4.39823 0 0.977694269

0.02375 4.39823 0.25 1.289628774

0.02375 4.39823 0.5 1.40947233

Appendix B. Optimal design tables

Optimal parameters for jump avoidance are given inTable 1 and optimal parameters for linear system are givenin Table 2.

ARTICLE IN PRESS

Table 1 (continued )

x1 o1 x2 rlimiting

0.02375 4.39823 0.75 1.595631282

0.02375 4.39823 1 1.760878592

0.02375 4.39823 1.5 2.043378358

0.02375 4.39823 2 2.285331976

0.02375 4.39823 3 2.700885952

0.02375 4.39823 4 3.062507404

0.02375 4.39823 5 3.391222195

0.02625 5.02655 0 0.978088572

0.02625 5.02655 0.25 1.293394486

0.02625 5.02655 0.5 1.417755913

0.02625 5.02655 0.75 1.608992222

0.02625 5.02655 1 1.777209048

0.02625 5.02655 1.5 2.066463133

0.02625 5.02655 2 2.316205463

0.02625 5.02655 3 2.746292115

0.02625 5.02655 4 3.123257562

0.02625 5.02655 5 3.467458519

0.03 5.65487 0 0.97875991

0.03 5.65487 0.25 1.299144105

0.03 5.65487 0.5 1.430543541

0.03 5.65487 0.75 1.627989101

0.03 5.65487 1 1.802345981

0.03 5.65487 1.5 2.102056501

0.03 5.65487 2 2.361936066

0.03 5.65487 3 2.815062918

0.03 5.65487 4 3.215019334

0.03 5.65487 5 3.583203626

0.0325 6.28319 0 0.979262074

0.0325 6.28319 0.25 1.30304769

0.0325 6.28319 0.5 1.439323588

0.0325 6.28319 0.75 1.640763215

0.0325 6.28319 1 1.819006868

0.0325 6.28319 1.5 2.125985968

0.0325 6.28319 2 2.393058516

0.0325 6.28319 3 2.861239177

0.0325 6.28319 4 3.276940122

0.0325 6.28319 5 3.661060377

0.03625 6.91150 0 0.980099853

0.03625 6.91150 0.25 1.308817721

0.03625 6.91150 0.5 1.452901129

0.03625 6.91150 0.75 1.660366169

0.03625 6.91150 1 1.843514064

0.03625 6.91150 1.5 2.161318581

0.03625 6.91150 2 2.440352455

0.03625 6.91150 3 2.931814344

0.03625 6.91150 4 3.371140716

0.03625 6.91150 5 3.779599063

0.04 7.53982 0 0.981042677

0.04 7.53982 0.25 1.356391087

0.04 7.53982 0.5 1.481916745

0.04 7.53982 0.75 1.680479733

0.04 7.53982 1 1.868578676

0.04 7.53982 1.5 2.197831704

0.04 7.53982 2 2.487743639

0.04 7.53982 3 3.002626728

0.04 7.53982 4 3.466620701

0.04 7.53982 5 3.900096996

0.0425 8.16814 0 0.981731825

0.0425 8.16814 0.25 1.360035695

0.0425 8.16814 0.5 1.490585565

0.0425 8.16814 0.75 1.69420178

0.0425 8.16814 1 1.885758518

0.0425 8.16814 1.5 2.222644377

0.0425 8.16814 2 2.520432157


x1 o1 x2 rlimiting

0.0425 8.16814 3 3.050718005

0.0425 8.16814 4 3.53113238

0.0425 8.16814 5 3.981016618

0.04625 8.79646 0 0.98286024

0.04625 8.79646 0.25 1.365603775

0.04625 8.79646 0.5 1.503961341

0.04625 8.79646 0.75 1.715007137

0.04625 8.79646 1 1.912059332

0.04625 8.79646 1.5 2.259038987

0.04625 8.79646 2 2.568484879

0.04625 8.79646 3 3.123655317

0.04625 8.79646 4 3.628802576

0.04625 8.79646 5 4.104652818

0.05 9.42478 0 0.984107399

0.05 9.42478 0.25 1.37130007

0.05 9.42478 0.5 1.51711179

0.05 9.42478 0.75 1.734932579

0.05 9.42478 1 1.937584571

0.05 9.42478 1.5 2.296610433

0.05 9.42478 2 2.618161171

0.05 9.42478 3 3.197047525

0.05 9.42478 4 3.727939208

0.05 9.42478 5 4.229518608

0.0525 10.05310 0 0.985007959

0.0525 10.05310 0.25 1.375172875

0.0525 10.05310 0.5 1.525785204

0.0525 10.05310 0.75 1.748526775

0.0525 10.05310 1 1.95490713

0.0525 10.05310 1.5 2.322315825

0.0525 10.05310 2 2.651018134

0.0525 10.05310 3 3.247040533

0.0525 10.05310 4 3.79442587

0.0525 10.05310 5 4.314100899

0.05625 10.68142 0 0.986467767

0.05625 10.68142 0.25 1.381101526

0.05625 10.68142 0.5 1.539198404

0.05625 10.68142 0.75 1.769385184

0.05625 10.68142 1 1.981424181

0.05625 10.68142 1.5 2.359869341

0.05625 10.68142 2 2.701372339

0.05625 10.68142 3 3.322498656

0.05625 10.68142 4 3.895835089

0.05625 10.68142 5 4.442088814

0.05875 11.30973 0 0.987517028

0.05875 11.30973 0.25 1.38513767

0.05875 11.30973 0.5 1.548425303

0.05875 11.30973 0.75 1.783588915

0.05875 11.30973 1 1.999624843

0.05875 11.30973 1.5 2.385607066

0.05875 11.30973 2 2.735506163

0.05875 11.30973 3 3.37302792

0.05875 11.30973 4 3.964587845

0.05875 11.30973 5 4.528815194

0.0625 11.93805 0 0.989211467

0.0625 11.93805 0.25 1.391324946

0.0625 11.93805 0.5 1.56272279

0.0625 11.93805 0.75 1.805442986

0.0625 11.93805 1 2.026855289

0.0625 11.93805 1.5 2.425224786

0.0625 11.93805 2 2.786514419

0.0625 11.93805 3 3.450452666

0.0625 11.93805 4 4.068236632

0.0625 11.93805 5 4.659646059

0.065 12.56637 0 0.990425625


ARTICLE IN PRESS


x1 o1 x2 rlimiting

0.065 12.56637 0.25 1.395543199

0.065 12.56637 0.5 1.57257831

0.065 12.56637 0.75 1.819906364

0.065 12.56637 1 2.044693613

0.065 12.56637 1.5 2.450997438

0.065 12.56637 2 2.821758872

0.065 12.56637 3 3.502292611

0.065 12.56637 4 4.138651506

0.065 12.56637 5 4.748553055

0.06875 13.19469 0 0.992381562

0.06875 13.19469 0.25 1.402019328

0.06875 13.19469 0.5 1.587883231

0.06875 13.19469 0.75 1.841151388

0.06875 13.19469 1 2.072272055

0.06875 13.19469 1.5 2.490857267

0.06875 13.19469 2 2.873831459

0.06875 13.19469 3 3.58129108

0.06875 13.19469 4 4.244983109

0.06875 13.19469 5 4.882299908

0.07125 13.82301 0 0.993780416

0.07125 13.82301 0.25 1.406441343

0.07125 13.82301 0.5 1.598457211

0.07125 13.82301 0.75 1.855694444

0.07125 13.82301 1 2.090997007

0.07125 13.82301 1.5 2.518209361

0.07125 13.82301 2 2.909526179

0.07125 13.82301 3 3.635069298

0.07125 13.82301 4 4.316431894

0.07125 13.82301 5 4.973209892

0.075 14.45133 0 0.996030722

0.075 14.45133 0.25 1.412793094

0.075 14.45133 0.5 1.614600554

0.075 14.45133 0.75 1.878021451

0.075 14.45133 1 2.119938922

0.075 14.45133 1.5 2.558577152

0.075 14.45133 2 2.963351509

0.075 14.45133 3 3.715711031

0.075 14.45133 4 4.425509027

0.075 14.45133 5 5.110714951

0.0775 15.07964 0 0.997638532

0.0775 15.07964 0.25 1.417007117

0.0775 15.07964 0.5 1.625007616

0.0775 15.07964 0.75 1.893261191

0.0775 15.07964 1 2.138505548

0.0775 15.07964 1.5 2.586119837

0.0775 15.07964 2 2.999610719

0.0775 15.07964 3 3.770269145

0.0775 15.07964 4 4.498873614

0.0775 15.07964 5 5.203417842

0.08125 15.70796 0 1.000223569

0.08125 15.70796 0.25 1.423495168

0.08125 15.70796 0.5 1.641239764

0.08125 15.70796 0.75 1.916684135

0.08125 15.70796 1 2.167217259

0.08125 15.70796 1.5 2.628529262

0.08125 15.70796 2 3.054959585

0.08125 15.70796 3 3.853503726

0.08125 15.70796 4 4.61074242

0.08125 15.70796 5 5.345448301

0.08375 16.33628 0 1.002070202

0.08375 16.33628 0.25 1.427938366

0.08375 16.33628 0.5 1.652505744

0.08375 16.33628 0.75 1.931654575

0.08375 16.33628 1 2.186888091


x1 o1 x2 rlimiting

0.08375 16.33628 1.5 2.656332034

0.08375 16.33628 2 3.092102661

0.08375 16.33628 3 3.909368213

0.08375 16.33628 4 4.686112601

0.08375 16.33628 5 5.440064808

0.0875 16.96460 0 1.004665881

0.0875 16.96460 0.25 1.434791961

0.0875 16.96460 0.5 1.669883981

0.0875 16.96460 0.75 1.95470022

0.0875 16.96460 1 2.217239725

0.0875 16.96460 1.5 2.699321945

0.0875 16.96460 2 3.148995562

0.0875 16.96460 3 3.994452208

0.0875 16.96460 4 4.800795684

0.0875 16.96460 5 5.584827125

0.09 17.59292 0 1.006428499

0.09 17.59292 0.25 1.439494519

0.09 17.59292 0.5 1.681779411

0.09 17.59292 0.75 1.970458216

0.09 17.59292 1 2.236988827

0.09 17.59292 1.5 2.728759448

0.09 17.59292 2 3.187120615

0.09 17.59292 3 4.052356463

0.09 17.59292 4 4.878313216

0.09 17.59292 5 5.683309276

0.09375 18.22124 0 1.009259787

0.09375 18.22124 0.25 1.446762825

0.09375 18.22124 0.5 1.700120546

0.09375 18.22124 0.75 1.994802262

0.09375 18.22124 1 2.267153531

0.09375 18.22124 1.5 2.772283081

0.09375 18.22124 2 3.245597875

0.09375 18.22124 3 4.13996244

0.09375 18.22124 4 4.995937231

0.09375 18.22124 5 5.830985597

0.09625 18.84956 0 1.011281456

0.09625 18.84956 0.25 1.451760382

0.09625 18.84956 0.5 1.712679278

0.09625 18.84956 0.75 2.011517706

0.09625 18.84956 1 2.287983769

0.09625 18.84956 1.5 2.802290435

0.09625 18.84956 2 3.284815079

0.09625 18.84956 3 4.19865281

0.09625 18.84956 4 5.075806308

0.09625 18.84956 5 5.93141673

0.09875 19.47787 0 1.013418619

0.09875 19.47787 0.25 1.456887377

0.09875 19.47787 0.5 1.724438149

0.09875 19.47787 0.75 2.027821229

0.09875 19.47787 1 2.309195367

0.09875 19.47787 1.5 2.832865596

0.09875 19.47787 2 3.324822257

0.09875 19.47787 3 4.258858552

0.09875 19.47787 4 5.156150026

0.09875 19.47787 5 6.033518044

0.1025 20.10619 0 1.01685814

0.1025 20.10619 0.25 1.51624105

0.1025 20.10619 0.5 1.746147555

0.1025 20.10619 0.75 2.052418988

0.1025 20.10619 1 2.340850179

0.1025 20.10619 1.5 2.878276078

0.1025 20.10619 2 3.385230286

0.1025 20.10619 3 4.350090603

0.1025 20.10619 4 5.278897259


ARTICLE IN PRESS


x1 o1 x2 rlimiting

0.1025 20.10619 5 6.187662361

0.105 20.73451 0 1.019319787

0.105 20.73451 0.25 1.520951073

0.105 20.73451 0.5 1.757696799

0.105 20.73451 0.75 2.06928727

0.105 20.73451 1 2.362141559

0.105 20.73451 1.5 2.909587883

0.105 20.73451 2 3.426485542

0.105 20.73451 3 4.411181538

0.105 20.73451 4 5.361474849

0.105 20.73451 5 6.291822078

0.10875 21.36283 0 1.023289979

0.10875 21.36283 0.25 1.528242459

0.10875 21.36283 0.5 1.775788725

0.10875 21.36283 0.75 2.095441717

0.10875 21.36283 1 2.39508888

0.10875 21.36283 1.5 2.956890115

0.10875 21.36283 2 3.488755556

0.10875 21.36283 3 4.505103257

0.10875 21.36283 4 5.487502425

0.10875 21.36283 5 6.451377202

0.11125 21.99115 0 1.026138302

0.11125 21.99115 0.25 1.533264259

0.11125 21.99115 0.5 1.788403387

0.11125 21.99115 0.75 2.113406404

0.11125 21.99115 1 2.417891865

0.11125 21.99115 1.5 2.988957458

0.11125 21.99115 2 3.531414862

0.11125 21.99115 3 4.568742907

0.11125 21.99115 4 5.573101269

0.11125 21.99115 5 6.559158117

0.11375 22.61947 0 1.02916257

0.11375 22.61947 0.25 1.538423424

0.11375 22.61947 0.5 1.80149744

0.11375 22.61947 0.75 2.130956157

0.11375 22.61947 1 2.440062169

0.11375 22.61947 1.5 3.021823926

0.11375 22.61947 2 3.57417282

0.11375 22.61947 3 4.633115971

0.11375 22.61947 4 5.659265058

0.11375 22.61947 5 6.667735073

0.1175 23.24779 0 1.034060425

0.1175 23.24779 0.25 1.546437565

0.1175 23.24779 0.5 1.82146507

0.1175 23.24779 0.75 2.157641832

0.1175 23.24779 1 2.474180535

0.1175 23.24779 1.5 3.071085516

0.1175 23.24779 2 3.639895467

0.1175 23.24779 3 4.731193417

0.1175 23.24779 4 5.79079414

0.1175 23.24779 5 6.833007224

0.12 23.87610 0 1.037590455

0.12 23.87610 0.25 1.551976879

0.12 23.87610 0.5 1.834741266

0.12 23.87610 0.75 2.176068041

0.12 23.87610 1 2.497761431

0.12 23.87610 1.5 3.104955185

0.12 23.87610 2 3.684024089

0.12 23.87610 3 4.797647712

0.12 23.87610 4 5.880361269

0.12 23.87610 5 6.945811223

0.1225 24.50442 0 1.04135398

0.1225 24.50442 0.25 1.557684759

0.1225 24.50442 0.5 1.848391551


x1 o1 x2 rlimiting

0.1225 24.50442 0.75 2.194983923

0.1225 24.50442 1 2.521955202

0.1225 24.50442 1.5 3.139390391

0.1225 24.50442 2 3.729115993

0.1225 24.50442 3 4.86533391

0.1225 24.50442 4 5.970799386

0.1225 24.50442 5 7.059480159

0.12625 25.13274 0 1.047484151

0.12625 25.13274 0.25 1.566586117

0.12625 25.13274 0.5 1.86958996

0.12625 25.13274 0.75 2.224087021

0.12625 25.13274 1 2.557693162

0.12625 25.13274 1.5 3.191182864

0.12625 25.13274 2 3.797880672

0.12625 25.13274 3 4.968354003

0.12625 25.13274 4 6.108664872

0.12625 25.13274 5 7.233391695

0.12875 25.76106 0 1.051930215

0.12875 25.76106 0.25 1.572763888

0.12875 25.76106 0.5 1.884230755

0.12875 25.76106 0.75 2.243036346

0.12875 25.76106 1 2.582185036

0.12875 25.76106 1.5 3.227189118

0.12875 25.76106 2 3.844815373

0.12875 25.76106 3 5.038137261

0.12875 25.76106 4 6.202517659

0.12875 25.76106 5 7.351345171

0.1325 26.38938 0 1.059210692

0.1325 26.38938 0.25 1.582428602

0.1325 26.38938 0.5 1.906983193

0.1325 26.38938 0.75 2.27259316

0.1325 26.38938 1 2.620533764

0.1325 26.38938 1.5 3.281096369

0.1325 26.38938 2 3.916206091

0.1325 26.38938 3 5.145127182

0.1325 26.38938 4 6.345608174

0.1325 26.38938 5 7.531220097

0.135 27.01770 0 1.064521784

0.135 27.01770 0.25 1.589158424

0.135 27.01770 0.5 1.922173784

0.135 27.01770 0.75 2.293049733

0.135 27.01770 1 2.645907535

0.135 27.01770 1.5 3.318213549

0.135 27.01770 2 3.964663743

0.135 27.01770 3 5.217605486

0.135 27.01770 4 6.443210599

0.135 27.01770 5 7.654112981

0.1375 27.64602 0 1.070245403

0.1375 27.64602 0.25 1.596136451

0.1375 27.64602 0.5 1.937137446

0.1375 27.64602 0.75 2.314203216

0.1375 27.64602 1 2.67180423

0.1375 27.64602 1.5 3.355775428

0.1375 27.64602 2 4.013984565

0.1375 27.64602 3 5.291579879

0.1375 27.64602 4 6.541764158

0.1375 27.64602 5 7.777839792

0.14125 28.27433 0 1.072695757

0.14125 28.27433 0.25 1.603842669

0.14125 28.27433 0.5 1.958989065

0.14125 28.27433 0.75 2.343752061

0.14125 28.27433 1 2.710804563

0.14125 28.27433 1.5 3.413367416

0.14125 28.27433 2 4.08877844


ARTICLE IN PRESS


x1 o1 x2 rlimiting

0.14125 28.27433 3 5.40406352

0.14125 28.27433 4 6.692659582

0.14125 28.27433 5 7.967394749

0.14375 28.90265 0 1.079187252

0.14375 28.90265 0.25 1.610994252

0.14375 28.90265 0.5 1.974661066

0.14375 28.90265 0.75 2.365044856

0.14375 28.90265 1 2.73764665

0.14375 28.90265 1.5 3.451611855

0.14375 28.90265 2 4.141652115

0.14375 28.90265 3 5.481331801

0.14375 28.90265 4 6.795636531

0.14375 28.90265 5 8.097088406

0.14625 29.53097 0 1.086191426

0.14625 29.53097 0.25 1.618438603

0.14625 29.53097 0.5 1.990862937

0.14625 29.53097 0.75 2.38705823

0.14625 29.53097 1 2.765685461

0.14625 29.53097 1.5 3.491255741

0.14625 29.53097 2 4.193223233

0.14625 29.53097 3 5.559201728

0.14625 29.53097 4 6.900892647

0.14625 29.53097 5 8.228729437

0.15 30.15929 0 1.097257632

0.15 30.15929 0.25 1.696888158

0.15 30.15929 0.5 2.01777462

0.15 30.15929 0.75 2.421593695

0.15 30.15929 1 2.809353358

0.15 30.15929 1.5 3.553523148

0.15 30.15929 2 4.273835578

0.15 30.15929 3 5.679846262

0.15 30.15929 4 7.061119256

0.15 30.15929 5 8.430892435

0.1525 30.78761 0 1.104996298

0.1525 30.78761 0.25 1.704420874

0.1525 30.78761 0.5 2.034642227

0.1525 30.78761 0.75 2.445649326

0.1525 30.78761 1 2.839928845

0.1525 30.78761 1.5 3.596456011

0.1525 30.78761 2 4.330409827

0.1525 30.78761 3 5.761882379

0.1525 30.78761 4 7.171053144

0.1525 30.78761 5 8.567756351

0.155 31.41593 0 1.113048114

0.155 31.41593 0.25 1.712266246

0.155 31.41593 0.5 2.052399181

0.155 31.41593 0.75 2.470609105

0.155 31.41593 1 2.869163438

0.155 31.41593 1.5 3.638271734

0.155 31.41593 2 4.385473652

0.155 31.41593 3 5.846261335

0.155 31.41593 4 7.282997527

0.155 31.41593 5 8.707307628

0.15875 32.04425 0 1.125765289

0.15875 32.04425 0.25 1.724675586

0.15875 32.04425 0.5 2.080896678

0.15875 32.04425 0.75 2.505917972

0.15875 32.04425 1 2.914321687

0.15875 32.04425 1.5 3.70416833

0.15875 32.04425 2 4.472380265

0.15875 32.04425 3 5.975134748

0.15875 32.04425 4 7.454935351

0.15875 32.04425 5 8.923663367

0.16125 32.67256 0 1.134688094


x1 o1 x2 rlimiting

0.16125 32.67256 0.25 1.732875775

0.16125 32.67256 0.5 2.101005846

0.16125 32.67256 0.75 2.530376187

0.16125 32.67256 1 2.945918213

0.16125 32.67256 1.5 3.750450365

0.16125 32.67256 2 4.53246919

0.16125 32.67256 3 6.062887762

0.16125 32.67256 4 7.572092609

0.16125 32.67256 5 9.070413514

0.16375 33.30088 0 1.144006566

0.16375 33.30088 0.25 1.740730693

0.16375 33.30088 0.5 2.119254551

0.16375 33.30088 0.75 2.555835523

0.16375 33.30088 1 2.978983629

0.16375 33.30088 1.5 3.796957463

0.16375 33.30088 2 4.5922578

0.16375 33.30088 3 6.153358695

0.16375 33.30088 4 7.693378589

0.16375 33.30088 5 9.220669154

0.1675 33.92920 0 1.158811052

0.1675 33.92920 0.25 1.753221932

0.1675 33.92920 0.5 2.147988816

0.1675 33.92920 0.75 2.596102509

0.1675 33.92920 1 3.031122768

0.1675 33.92920 1.5 3.867748408

0.1675 33.92920 2 4.687127815

0.1675 33.92920 3 6.292226313

0.1675 33.92920 4 7.878297495

0.1675 33.92920 5 9.453316471

0.17 34.55752 0 1.16929041

0.17 34.55752 0.25 1.762071716

0.17 34.55752 0.5 2.168162281

0.17 34.55752 0.75 2.624427764

0.17 34.55752 1 3.064349289

0.17 34.55752 1.5 3.917666792

0.17 34.55752 2 4.751168743

0.17 34.55752 3 6.388824121

0.17 34.55752 4 8.005526919

0.17 34.55752 5 9.61201176

0.17375 35.18584 0 1.181640444

0.17375 35.18584 0.25 1.776228864

0.17375 35.18584 0.5 2.200165881

0.17375 35.18584 0.75 2.668208277

0.17375 35.18584 1 3.117259033

0.17375 35.18584 1.5 3.99450168

0.17375 35.18584 2 4.85132172

0.17375 35.18584 3 6.536477783

0.17375 35.18584 4 8.201973561

0.17375 35.18584 5 9.857739952

0.17625 35.81416 0 1.189874343

0.17625 35.81416 0.25 1.786322765

0.17625 35.81416 0.5 2.222777849

0.17625 35.81416 0.75 2.696625186

0.17625 35.81416 1 3.154744373

0.17625 35.81416 1.5 4.046647465

0.17625 35.81416 2 4.920419356

0.17625 35.81416 3 6.637753788

0.17625 35.81416 4 8.337552314

0.17625 35.81416 5 9.548455657

0.17875 36.44247 0 1.198557225

0.17875 36.44247 0.25 1.797003804

0.17875 36.44247 0.5 2.246446083

0.17875 36.44247 0.75 2.726434985

0.17875 36.44247 1 3.19394181


ARTICLE IN PRESS


x1 o1 x2 rlimiting

0.17875 36.44247 1.5 4.101457271

0.17875 36.44247 2 4.990624913

0.17875 36.44247 3 6.743529372

0.17875 36.44247 4 8.476174218

0.17875 36.44247 5 9.196552828

0.1825 37.07079 0 1.21250297

0.1825 37.07079 0.25 1.814265214

0.1825 37.07079 0.5 2.28426138

0.1825 37.07079 0.75 2.773925108

0.1825 37.07079 1 3.253147591

0.1825 37.07079 1.5 4.186200077

0.1825 37.07079 2 5.101689894

0.1825 37.07079 3 6.905886184

0.1825 37.07079 4 8.692594673

0.1825 37.07079 5 8.658381759

0.185 37.69911 0 1.222503144

0.185 37.69911 0.25 1.826708235

0.185 37.69911 0.5 2.308190419

0.185 37.69911 0.75 2.807728523

0.185 37.69911 1 3.293863902

0.185 37.69911 1.5 4.24436468

0.185 37.69911 2 5.177439223

0.185 37.69911 3 7.017798246

0.185 37.69911 4 8.841406107

0.185 37.69911 5 8.343838062

0.18875 38.32743 0 1.238684361

0.18875 38.32743 0.25 1.847001251

0.18875 38.32743 0.5 2.345956381

0.18875 38.32743 0.75 2.860309247

0.18875 38.32743 1 3.359257801

0.18875 38.32743 1.5 4.33754577

0.18875 38.32743 2 5.297297226

0.18875 38.32743 3 7.193457796

0.18875 38.32743 4 9.074602947

0.18875 38.32743 5 10.94630383

0.19125 38.95575 0 1.250325454

0.19125 38.95575 0.25 1.861775754

0.19125 38.95575 0.5 2.372971882

0.19125 38.95575 0.75 2.895231882

0.19125 38.95575 1 3.40577414

0.19125 38.95575 1.5 4.400075016

0.19125 38.95575 2 5.380027919

0.19125 38.95575 3 7.316352582

0.19125 38.95575 4 9.235595402

0.19125 38.95575 5 11.14813343

0.195 39.58407 0 1.269333362

0.195 39.58407 0.25 1.886145534

0.195 39.58407 0.5 2.416626018

0.195 39.58407 0.75 2.951733286

0.195 39.58407 1 3.474961997

0.195 39.58407 1.5 4.50140362

0.195 39.58407 2 5.51033633

0.195 39.58407 3 7.507506986

0.195 39.58407 4 9.48905606

0.195 39.58407 5 11.46280257

0.1975 40.21239 0 1.283185182

0.1975 40.21239 0.25 1.904111364

0.1975 40.21239 0.5 2.448011095

0.1975 40.21239 0.75 2.992335956

0.1975 40.21239 1 3.524821568

0.1975 40.21239 1.5 4.570776625

0.1975 40.21239 2 5.601551048

0.1975 40.21239 3 7.640408126

0.1975 40.21239 4 9.375080549


x1 o1 x2 rlimiting

0.1975 40.21239 5 11.682194

0.20125 40.84070 0 1.305970031

0.20125 40.84070 0.25 2.011085128

0.20125 40.84070 0.5 2.496114009

0.20125 40.84070 0.75 3.055856817

0.20125 40.84070 1 3.604632926

0.20125 40.84070 1.5 4.682067094

0.20125 40.84070 2 5.744593621

0.20125 40.84070 3 7.850925029

0.20125 40.84070 4 9.942652163

0.20125 40.84070 5 12.02801062

0.205 41.46902 0 1.331577221

0.205 41.46902 0.25 2.035643725

0.205 41.46902 0.5 2.546181494

0.205 41.46902 0.75 3.122242959

0.205 41.46902 1 3.687599087

0.205 41.46902 1.5 4.799461039

0.205 41.46902 2 5.897654303

0.205 41.46902 3 8.0744865

0.205 41.46902 4 10.23799904

0.205 41.46902 5 12.39513475

0.2075 42.09734 0 1.350479485

0.2075 42.09734 0.25 2.053814342

0.2075 42.09734 0.5 2.58277902

0.2075 42.09734 0.75 3.170699031

0.2075 42.09734 1 3.748126026

0.2075 42.09734 1.5 4.884311509

0.2075 42.09734 2 6.006801278

0.2075 42.09734 3 8.232525956

0.2075 42.09734 4 10.44591019

0.2075 42.09734 5 12.65324289

0.21125 42.72566 0 1.375424018

0.21125 42.72566 0.25 2.084355283

0.21125 42.72566 0.5 2.643310773

0.21125 42.72566 0.75 3.247812594

0.21125 42.72566 1 3.842507171

0.21125 42.72566 1.5 5.01670523

0.21125 42.72566 2 6.178819032

0.21125 42.72566 3 8.483209222

0.21125 42.72566 4 10.77639779

0.21125 42.72566 5 13.06514227

0.215 43.35398 0 1.401859014

0.215 43.35398 0.25 2.117413572

0.215 43.35398 0.5 2.706048323

0.215 43.35398 0.75 3.330417866

0.215 43.35398 1 3.947276662

0.215 43.35398 1.5 5.161088133

0.215 43.35398 2 6.365368242

0.215 43.35398 3 8.754761998

0.215 43.35398 4 11.13364575

0.215 43.35398 5 13.50836416

0.2175 43.98230 0 1.421829971

0.2175 43.98230 0.25 2.140880257

0.2175 43.98230 0.5 2.7515371

0.2175 43.98230 0.75 3.392066331

0.2175 43.98230 1 4.02047914

0.2175 43.98230 1.5 5.266059877

0.2175 43.98230 2 6.483371796

0.2175 43.98230 3 8.949005611

0.2175 43.98230 4 11.39000853

0.2175 43.98230 5 13.82444512

0.22125 44.61062 0 1.456329383

0.22125 44.61062 0.25 2.181599634

0.22125 44.61062 0.5 2.828605269


ARTICLE IN PRESS


x1 o1 x2 rlimiting

0.22125 44.61062 0.75 3.488206819

0.22125 44.61062 1 4.142875062

0.22125 44.61062 1.5 5.43419575

0.22125 44.61062 2 6.710180499

0.22125 44.61062 3 9.262870388

0.22125 44.61062 4 11.439624

0.22125 44.61062 5 Infinity

0.225 45.23893 0 1.497741923

0.225 45.23893 0.25 2.427262811

0.225 45.23893 0.5 2.944380029

0.225 45.23893 0.75 3.598678805

0.225 45.23893 1 4.275910699

0.225 45.23893 1.5 5.620872094

0.225 45.23893 2 6.955209481

0.225 45.23893 3 9.610354211

0.225 45.23893 4 12.25755616

0.225 45.23893 5 Infinity

0.22875 45.86725 0 1.547269255

0.22875 45.86725 0.25 2.472214237

0.22875 45.86725 0.5 3.027782772

0.22875 45.86725 0.75 3.720135362

0.22875 45.86725 1 4.426944268

0.22875 45.86725 1.5 5.830529073

0.22875 45.86725 2 7.223414192

0.22875 45.86725 3 9.998901035

0.22875 45.86725 4 12.76661092

0.22875 45.86725 5 Infinity

0.2325 46.49557 0 1.594689905

0.2325 46.49557 0.25 2.521675795

0.2325 46.49557 0.5 3.122947911

0.2325 46.49557 0.75 3.860311897

0.2325 46.49557 1 4.59975209

0.2325 46.49557 1.5 6.068393518

0.2325 46.49557 2 7.529619877

0.2325 46.49557 3 10.44025585

0.2325 46.49557 4 13.34357666

0.2325 46.49557 5 Infinity

0.23625 47.12389 0 1.656101133

0.23625 47.12389 0.25 2.583762143

0.23625 47.12389 0.5 3.244313389

0.23625 47.12389 0.75 4.024260854

0.23625 47.12389 1 4.800794254

0.23625 47.12389 1.5 6.345887181

0.23625 47.12389 2 7.883838522

0.23625 47.12389 3 10.94956352

0.23625 47.12389 4 Infinity

0.23625 47.12389 5 Infinity

0.24 47.75221 0 1.736803654

0.24 47.75221 0.25 2.664766893


x1 o1 x2 rlimiting

0.24 47.75221 0.5 3.397306798

0.24 47.75221 0.75 4.219956026

0.24 47.75221 1 5.041270606

0.24 47.75221 1.5 6.67475488

0.24 47.75221 2 8.303476702

0.24 47.75221 3 11.5532324

0.24 47.75221 4 Infinity

0.24 47.75221 5 Infinity

0.24375 48.38053 0 1.828188929

0.24375 48.38053 0.25 2.76680756

0.24375 48.38053 0.5 3.587264421

0.24375 48.38053 0.75 4.462359488

0.24375 48.38053 1 5.335996648

0.24375 48.38053 1.5 7.080102141

0.24375 48.38053 2 8.819575647

0.24375 48.38053 3 12.29180227

0.24375 48.38053 4 Infinity

0.24375 48.38053 5 Infinity

0.2475 49.00885 0 1.958789961

0.2475 49.00885 0.25 2.91529925

0.2475 49.00885 0.5 3.833330433

0.2475 49.00885 0.75 4.777225147

0.2475 49.00885 1 5.720157356

0.2475 49.00885 1.5 7.603063681

0.2475 49.00885 2 9.483720333

0.2475 49.00885 3 13.24174418

0.2475 49.00885 4 Infinity

0.2475 49.00885 5 Infinity

0.25 49.63716 0 2.081178396

0.25 49.63716 0.25 3.274975319

0.25 49.63716 0.5 4.07484342

0.25 49.63716 0.75 5.054326208

0.25 49.63716 1 6.055629094

0.25 49.63716 1.5 8.059575945

0.25 49.63716 2 10.06146234

0.25 49.63716 3 Infinity

0.25 49.63716 4 Infinity

0.25 49.63716 5 Infinity

0.25 50.26548 0 2.081178399

0.25 50.26548 0.25 3.274975319

0.25 50.26548 0.5 4.07484342

0.25 50.26548 0.75 5.054326208

0.25 50.26548 1 6.055629094

0.25 50.26548 1.5 8.059575949

0.25 50.26548 2 10.06146235

0.25 50.26548 3 Infinity

0.25 50.26548 4 Infinity

0.25 50.26548 5 Infinity

Table 2

Overall optimal parameters for the piecewise linear system for d ¼ 2

Clearance d ¼ 2

Optimal parameters Jump avoidance

limit rlimiting

Peak relative

displacement

amplitude A

RMS relative

displacement

RMS absolute

acceleration

Primary

damping

coeff. x1

Primary

natural

frequency o1

Secondary

damping

coeff. x2

Stiffness

ratio r

0.0038 0.628319 0 0 0.97599 6 6 5.451198

0.0038 0.628319 0.25 0 1.261208 6 6 5.451198


ARTICLE IN PRESS


Clearance d ¼ 2


limit rlimiting

Peak relative

displacement

amplitude A

RMS relative

displacement

RMS absolute

acceleration

Primary

damping

coeff. x1

Primary

natural

frequency o1

Secondary

damping

coeff. x2

Stiffness

ratio r

0.0038 0.628319 0.5 0.8 1.343185 3.785369 3.785369 4.615775

0.0038 0.628319 0.75 0.8 1.495017 3.270343 3.270343 3.85156

0.0038 0.628319 1 0.9 1.634304 2.927001 2.927001 3.466817

0.0038 0.628319 1.5 0.9 1.862443 2.675241 2.675241 3.080201

0.0038 0.628319 2 0.9 2.049416 2.540955 2.540955 2.881464

0.0038 0.628319 3 1 2.352382 2.371594 2.371594 2.679276

0.0038 0.628319 4 1 2.599309 2.300892 2.300892 2.575105

0.0038 0.628319 5 1 2.811761 2.255494 2.255494 2.5106

0.0063 1.256637 0 0 0.976066 6 6 5.45146

0.0063 1.256637 0.25 0 1.26461 6 6 5.45146

0.0063 1.256637 0.5 0.8 1.350904 3.75918 3.75918 4.576593

0.0063 1.256637 0.75 0.8 1.507807 3.25464 3.25464 3.828663

0.0063 1.256637 1 0.9 1.649595 2.917143 2.917143 3.451478

0.0063 1.256637 1.5 0.9 1.884634 2.66807 2.66807 3.069488

0.0063 1.256637 2 0.9 2.078082 2.535926 2.535926 2.874208

0.0063 1.256637 3 1 2.394675 2.368388 2.368388 2.674515

0.0063 1.256637 4 1 2.655112 2.298622 2.298622 2.571879

0.0063 1.256637 5 1 2.881187 2.253444 2.253444 2.507784

0.01 1.884956 0 0 0.976253 6 6 5.452099

0.01 1.884956 0.25 0 1.269789 6 6 5.452099

0.01 1.884956 0.5 0.8 1.362769 3.719983 3.719983 4.518085

0.01 1.884956 0.75 0.8 1.525971 3.231307 3.231307 3.794754

0.01 1.884956 1 0.8 1.6731 2.977529 2.977529 3.428259

0.01 1.884956 1.5 0.9 1.91858 2.657657 2.657657 3.054018

0.01 1.884956 2 0.9 2.12209 2.528583 2.528583 2.863689

0.01 1.884956 3 1 2.458554 2.363689 2.363689 2.667598

0.01 1.884956 4 1 2.740178 2.29528 2.29528 2.567186

0.01 1.884956 5 1 2.987114 2.250428 2.250428 2.503694

0.0138 2.513274 0 0 0.976527 6 6 5.453031

0.0138 2.513274 0.25 0 1.275062 6 6 5.453031

0.0138 2.513274 0.5 0.8 1.374998 3.681971 3.681971 4.461518

0.0138 2.513274 0.75 0.8 1.544383 3.207405 3.207405 3.760153

0.0138 2.513274 1 0.9 1.697194 2.887218 2.887218 3.405201

0.0138 2.513274 1.5 0.9 1.951676 2.647636 2.647636 3.039231

0.0138 2.513274 2 0.9 2.165517 2.521471 2.521471 2.853587

0.0138 2.513274 3 1 2.523971 2.359115 2.359115 2.660938

0.0138 2.513274 4 1 2.826632 2.292013 2.292013 2.562663

0.0138 2.513274 5 1 3.094943 2.24748 2.24748 2.499761

0.0163 3.141593 0 0 0.976758 6 6 5.453816

0.0163 3.141593 0.25 0 1.278633 6 6 5.453816

0.0163 3.141593 0.5 0.8 1.383365 3.656305 3.656305 4.423412

0.0163 3.141593 0.75 0.8 1.556888 3.19235 3.19235 3.738444

0.0163 3.141593 1 0.9 1.713487 2.876987 2.876987 3.389475

0.0163 3.141593 1.5 0.9 1.974351 2.64116 2.64116 3.029731

0.0163 3.141593 2 0.9 2.195271 2.516851 2.516851 2.847074

0.0163 3.141593 3 1 2.56727 2.356149 2.356149 2.656662

0.0163 3.141593 4 1 2.88429 2.289874 2.289874 2.559738

0.0163 3.141593 5 1 3.167923 2.245551 2.245551 2.497223

0.02 3.769911 0 0 0.977181 6 6 5.455237

0.02 3.769911 0.25 0 1.284076 6 6 5.455237

0.02 3.769911 0.5 0.8 1.396255 3.618982 3.618982 4.368147

0.02 3.769911 0.75 0.8 1.576028 3.169178 3.169178 3.705135

0.02 3.769911 1 0.8 1.736905 2.932875 2.932875 3.365527

0.02 3.769911 1.5 0.9 2.009142 2.631827 2.631827 3.016127

0.02 3.769911 2 1 2.240104 2.473809 2.473809 2.837059

0.02 3.769911 3 1 2.63388 2.351802 2.351802 2.650457

0.02 3.769911 4 1.1 2.97271 2.267667 2.267667 2.554281

0.02 3.769911 5 1 3.278654 2.242709 2.242709 2.493538


ARTICLE IN PRESS


Clearance d ¼ 2


limit rlimiting

Peak relative

displacement

amplitude A

RMS relative

displacement

RMS absolute

acceleration

Primary

damping

coeff. x1

Primary

natural

frequency o1

Secondary

damping

coeff. x2

Stiffness

ratio r

0.0238 4.39823 0 0 0.977694 6 6 5.456952

0.0238 4.39823 0.25 0 1.289629 6 6 5.456952

0.0238 4.39823 0.5 0.8 1.409472 3.582809 3.582809 4.314761

0.0238 4.39823 0.75 0.8 1.595631 3.146364 3.146364 3.672482

0.0238 4.39823 1 0.8 1.760879 2.91663 2.91663 3.342931

0.0238 4.39823 1.5 0.9 2.043378 2.622874 2.622874 3.003178

0.0238 4.39823 2 1 2.285332 2.467316 2.467316 2.827227

0.0238 4.39823 3 0.9 2.700886 2.372475 2.372475 2.644423

0.0238 4.39823 4 1.1 3.062507 2.263893 2.263893 2.548758

0.0238 4.39823 5 1 3.391222 2.239933 2.239933 2.490004

0.0263 5.026548 0 0 0.978089 6 6 5.458257

0.0263 5.026548 0.25 0 1.293394 6 6 5.458257

0.0263 5.026548 0.5 0.8 1.417756 3.558418 3.558418 4.27885

0.0263 5.026548 0.75 0.8 1.608992 3.132001 3.132001 3.652015

0.0263 5.026548 1 0.8 1.777209 2.906283 2.906283 3.328614

0.0263 5.026548 1.5 0.9 2.066463 2.616432 2.616432 2.993893

0.0263 5.026548 2 0.9 2.316205 2.497308 2.497308 2.819852

0.0263 5.026548 3 0.9 2.746292 2.368967 2.368967 2.639887

0.0263 5.026548 4 1.1 3.123258 2.261433 2.261433 2.545198

0.0263 5.026548 5 1 3.467459 2.238118 2.238118 2.48773

0.03 5.654867 0 0 0.97876 6 6 5.460457

0.03 5.654867 0.25 0 1.299144 6 6 5.460457

0.03 5.654867 0.5 0.7 1.430544 3.656882 3.656882 4.225623

0.03 5.654867 0.75 0.8 1.627989 3.109248 3.109248 3.619677

0.03 5.654867 1 0.8 1.802346 2.889615 2.889615 3.305614

0.03 5.654867 1.5 0.9 2.102057 2.605942 2.605942 2.978818

0.03 5.654867 2 0.9 2.361936 2.489429 2.489429 2.809008

0.03 5.654867 3 0.9 2.815063 2.363828 2.363828 2.633311

0.03 5.654867 4 1.1 3.215019 2.257821 2.257821 2.540034

0.03 5.654867 5 1.2 3.583204 2.207097 2.207097 2.48354

0.0325 6.283185 0 0 0.979262 6 6 5.462086

0.0325 6.283185 0.25 0 1.303048 6 6 5.462086

0.0325 6.283185 0.5 0.7 1.439324 3.630498 3.630498 4.190424

0.0325 6.283185 0.75 0.8 1.640763 3.094399 3.094399 3.598655

0.0325 6.283185 1 0.8 1.819007 2.878449 2.878449 3.290271

0.0325 6.283185 1.5 0.9 2.125986 2.599185 2.599185 2.969171

0.0325 6.283185 2 0.9 2.393059 2.484325 2.484325 2.802039

0.0325 6.283185 3 0.9 2.861239 2.36048 2.36048 2.629073

0.0325 6.283185 4 1.1 3.27694 2.255469 2.255469 2.536714

0.0325 6.283185 5 1.2 3.66106 2.205135 2.205135 2.480678

0.0363 6.911504 0 0 0.9801 6 6 5.46477

0.0363 6.911504 0.25 0 1.308818 6 6 5.46477

0.0363 6.911504 0.5 0.7 1.452901 3.590273 3.590273 4.136895

0.0363 6.911504 0.75 0.8 1.660366 3.073308 3.073308 3.56894

0.0363 6.911504 1 0.8 1.843514 2.862464 2.862464 3.268429

0.0363 6.911504 1.5 0.9 2.161319 2.589425 2.589425 2.955336

0.0363 6.911504 2 0.9 2.440352 2.47688 2.47688 2.791957

0.0363 6.911504 3 0.9 2.931814 2.355572 2.355572 2.622928

0.0363 6.911504 4 1.1 3.371141 2.252025 2.252025 2.531914

0.0363 6.911504 5 1.2 3.779597 2.202252 2.202252 2.476533

0.04 7.539822 0 0 0.981043 6 6 5.467742

0.04 7.539822 0.25 0 1.356391 6 6 5.467742

0.04 7.539822 0.5 0.7 1.481917 3.550882 3.550882 4.084673

0.04 7.539822 0.75 0.8 1.68048 3.050884 3.050884 3.537436

0.04 7.539822 1 0.8 1.868579 2.847324 2.847324 3.247889

0.04 7.539822 1.5 0.9 2.197832 2.580103 2.580103 2.942238

0.04 7.539822 2 0.9 2.487744 2.469701 2.469701 2.782337

0.04 7.539822 3 1 3.002627 2.327198 2.327198 2.616429


ARTICLE IN PRESS


Clearance d ¼ 2


limit rlimiting

Peak relative

displacement

amplitude A

RMS relative

displacement

RMS absolute

acceleration

Primary

damping

coeff. x1

Primary

natural

frequency o1

Secondary

damping

coeff. x2

Stiffness

ratio r

0.04 7.539822 4 1.1 3.466621 2.248662 2.248662 2.527301

0.04 7.539822 5 1.2 3.900097 2.199448 2.199448 2.472574

0.0425 8.168141 0 0 0.981732 6 6 5.469883

0.0425 8.168141 0.25 0.7 1.360036 4.539493 4.539493 5.44281

0.0425 8.168141 0.5 0.7 1.490586 3.525681 3.525681 4.051387

0.0425 8.168141 0.75 0.8 1.694202 3.036461 3.036461 3.517266

0.0425 8.168141 1 0.8 1.885759 2.836283 2.836283 3.232929

0.0425 8.168141 1.5 0.9 2.222644 2.574076 2.574076 2.933832

0.0425 8.168141 2 0.9 2.520432 2.465078 2.465078 2.7762

0.0425 8.168141 3 1 3.050718 2.323721 2.323721 2.611727

0.0425 8.168141 4 1.1 3.531132 2.246463 2.246463 2.524327

0.0425 8.168141 5 1.2 3.981017 2.197615 2.197615 2.470027

0.0463 8.796459 0 0 0.98286 6 6 5.473333

0.0463 8.796459 0.25 0.5 1.365604 4.916657 4.916657 5.321455

0.0463 8.796459 0.5 0.7 1.503961 3.487323 3.487323 4.000852

0.0463 8.796459 0.75 0.8 1.715007 3.015967 3.015967 3.488756

0.0463 8.796459 1 0.8 1.912059 2.819883 2.819883 3.210812

0.0463 8.796459 1.5 0.9 2.259039 2.565295 2.565295 2.921676

0.0463 8.796459 2 0.9 2.568485 2.458319 2.458319 2.767308

0.0463 8.796459 3 1 3.123655 2.318634 2.318634 2.604921

0.0463 8.796459 4 1 3.628803 2.259222 2.259222 2.519773

0.0463 8.796459 5 1.2 4.104653 2.194918 2.194918 2.466336

0.05 9.424778 0 0 0.984107 6 6 5.477067

0.05 9.424778 0.25 0.5 1.3713 4.806438 4.806438 5.196472

0.05 9.424778 0.5 0.7 1.517112 3.45044 3.45044 3.952473

0.05 9.424778 0.75 0.8 1.734933 2.99397 2.99397 3.458222

0.05 9.424778 1 0.8 1.937585 2.80437 2.80437 3.190052

0.05 9.424778 1.5 0.9 2.29661 2.555176 2.555176 2.90765

0.05 9.424778 2 1 2.618161 2.420069 2.420069 2.75784

0.05 9.424778 3 1 3.197048 2.313713 2.313713 2.598428

0.05 9.424778 4 1 3.727939 2.255378 2.255378 2.515052

0.05 9.424778 5 1.1 4.229519 2.204312 2.204312 2.462683

0.0525 10.0531 0 0 0.985008 6 6 5.479714

0.0525 10.0531 0.25 0.5 1.375173 4.735214 4.735214 5.115912

0.0525 10.0531 0.5 0.7 1.525785 3.425981 3.425981 3.920491

0.0525 10.0531 0.75 0.8 1.748527 2.979909 2.979909 3.438802

0.0525 10.0531 1 0.8 1.954907 2.794496 2.794496 3.176926

0.0525 10.0531 1.5 0.9 2.322316 2.548365 2.548365 2.898257

0.0525 10.0531 2 1 2.651018 2.415702 2.415702 2.75164

0.0525 10.0531 3 1 3.247041 2.310518 2.310518 2.594262

0.0525 10.0531 4 1 3.794426 2.25287 2.25287 2.512015

0.0525 10.0531 5 1.1 4.314101 2.202254 2.202254 2.460126

0.0563 10.68142 0 0 0.986468 6 6 5.483919

0.0563 10.68142 0.25 0.5 1.381102 4.632277 4.632277 4.999796

0.0563 10.68142 0.5 0.7 1.539198 3.389904 3.389904 3.873478

0.0563 10.68142 0.75 0.7 1.769385 3.040192 3.040192 3.410997

0.0563 10.68142 1 0.8 1.981424 2.778681 2.778681 3.155929

0.0563 10.68142 1.5 0.9 2.359869 2.538587 2.538587 2.884886

0.0563 10.68142 2 0.9 2.701372 2.438503 2.438503 2.741448

0.0563 10.68142 3 1 3.322499 2.305832 2.305832 2.588226

0.0563 10.68142 4 1 3.895835 2.249185 2.249185 2.507619

0.0563 10.68142 5 1.1 4.442089 2.199229 2.199229 2.45643

0.0588 11.30973 0 0 0.987517 6 6 5.486877

0.0588 11.30973 0.25 0.5 1.385138 4.565742 4.565742 4.924943

0.0588 11.30973 0.5 0.7 1.548425 3.366277 3.366277 3.842796

0.0588 11.30973 0.75 0.8 1.783589 2.945566 2.945566 3.391666

0.0588 11.30973 1 0.8 1.999625 2.767926 2.767926 3.141702

0.0588 11.30973 1.5 0.9 2.385607 2.53227 2.53227 2.876316


ARTICLE IN PRESS


Clearance d ¼ 2


limit rlimiting

Peak relative

displacement

amplitude A

RMS relative

displacement

RMS absolute

acceleration

Primary

damping

coeff. x1

Primary

natural

frequency o1

Secondary

damping

coeff. x2

Stiffness

ratio r

0.0588 11.30973 2 0.9 2.735506 2.433296 2.433296 2.734717

0.0588 11.30973 3 1 3.373028 2.302777 2.302777 2.58434

0.0588 11.30973 4 1 3.964588 2.246777 2.246777 2.504791

0.0588 11.30973 5 1.1 4.528818 2.197252 2.197252 2.454055

0.0625 11.93805 0 0 0.989211 6 6 5.491547

0.0625 11.93805 0.25 0.5 1.391325 4.469519 4.469519 4.816999

0.0625 11.93805 0.5 0.7 1.562723 3.331022 3.331022 3.797161

0.0625 11.93805 0.75 0.8 1.805443 2.924777 2.924777 3.363294

0.0625 11.93805 1 0.8 2.026855 2.752597 2.752597 3.121569

0.0625 11.93805 1.5 0.9 2.425225 2.52308 2.52308 2.86395

0.0625 11.93805 2 0.9 2.786514 2.425718 2.425718 2.725016

0.0625 11.93805 3 1 3.450453 2.298293 2.298293 2.578708

0.0625 11.93805 4 1 4.068237 2.243238 2.243238 2.500699

0.0625 11.93805 5 1.1 4.659646 2.194343 2.194343 2.450625

0.065 12.56637 0 0 0.990426 6 6 5.494814

0.065 12.56637 0.25 0.5 1.395543 4.407146 4.407146 4.747219

0.065 12.56637 0.5 0.7 1.572578 3.308121 3.308121 3.767633

0.065 12.56637 0.75 0.7 1.819906 2.986291 2.986291 3.344061

0.065 12.56637 1 0.8 2.044694 2.742772 2.742772 3.108754

0.065 12.56637 1.5 0.9 2.450997 2.517136 2.517136 2.856018

0.065 12.56637 2 0.9 2.821759 2.420851 2.420851 2.718852

0.065 12.56637 3 1 3.502293 2.295368 2.295368 2.575082

0.065 12.56637 4 1 4.138652 2.240924 2.240924 2.498068

0.065 12.56637 5 1.1 4.748553 2.19244 2.19244 2.448422

0.0688 13.19469 0 0 0.992382 6 6 5.499943

0.0688 13.19469 0.25 0.5 1.402019 4.31688 4.31688 4.646535

0.0688 13.19469 0.5 0.7 1.587883 3.273711 3.273711 3.723399

0.0688 13.19469 0.75 0.7 1.841151 2.964195 2.964195 3.316893

0.0688 13.19469 1 0.8 2.072272 2.727924 2.727924 3.089466

0.0688 13.19469 1.5 0.8 2.490857 2.546396 2.546396 2.843582

0.0688 13.19469 2 0.9 2.873831 2.413745 2.413745 2.709942

0.0688 13.19469 3 0.9 3.581291 2.310822 2.310822 2.569587

0.0688 13.19469 4 1 4.244983 2.237519 2.237519 2.49426

0.0688 13.19469 5 1.1 4.8823 2.189637 2.189637 2.44524

0.0713 13.82301 0 0 0.99378 6 6 5.503514

0.0713 13.82301 0.25 0.5 1.406441 4.258513 4.258513 4.581621

0.0713 13.82301 0.5 0.7 1.598457 3.251441 3.251441 3.69489

0.0713 13.82301 0.75 0.7 1.855694 2.9491 2.9491 3.298381

0.0713 13.82301 1 0.8 2.090997 2.717282 2.717282 3.075644

0.0713 13.82301 1.5 0.8 2.518209 2.539294 2.539294 2.83497

0.0713 13.82301 2 0.9 2.909526 2.409132 2.409132 2.704218

0.0713 13.82301 3 0.9 3.635069 2.307377 2.307377 2.565691

0.0713 13.82301 4 1 4.316432 2.235291 2.235291 2.491813

0.0713 13.82301 5 1.1 4.97321 2.187803 2.187803 2.443198

0.075 14.45133 0 0 0.996031 6 6 5.509097

0.075 14.45133 0.25 0.5 1.412793 4.173186 4.173186 4.486995

0.075 14.45133 0.5 0.7 1.614601 3.217908 3.217908 3.652091

0.075 14.45133 0.75 0.7 1.878021 2.926306 2.926306 3.270524

0.075 14.45133 1 0.8 2.119939 2.701954 2.701954 3.055877

0.075 14.45133 1.5 0.9 2.558577 2.491802 2.491802 2.822432

0.075 14.45133 2 0.9 2.963352 2.402389 2.402389 2.69594

0.075 14.45133 3 0.9 3.715711 2.302329 2.302329 2.560058

0.075 14.45133 4 1.2 4.425509 2.204329 2.204329 2.48717

0.075 14.45133 5 1.1 5.110715 2.1851 2.1851 2.44025

0.0775 15.07964 0 0 0.997639 6 6 5.512967

0.0775 15.07964 0.25 0.5 1.417007 4.118252 4.118252 4.426269

0.0775 15.07964 0.5 0.7 1.625008 3.196175 3.196175 3.624474

0.0775 15.07964 0.75 0.7 1.893261 2.911949 2.911949 3.253106


ARTICLE IN PRESS


Clearance d ¼ 2


limit rlimiting

Peak relative

displacement

amplitude A

RMS relative

displacement

RMS absolute

acceleration

Primary

damping

coeff. x1

Primary

natural

frequency o1

Secondary

damping

coeff. x2

Stiffness

ratio r

0.0775 15.07964 1 0.8 2.138506 2.692133 2.692133 3.043305

0.0775 15.07964 1.5 0.9 2.58612 2.485298 2.485298 2.81389

0.0775 15.07964 2 1 2.999611 2.371052 2.371052 2.690057

0.0775 15.07964 3 1.1 3.770269 2.262433 2.262433 2.55624

0.0775 15.07964 4 1.2 4.498874 2.201991 2.201991 2.484083

0.0775 15.07964 5 1.1 5.203418 2.183329 2.183329 2.438359

0.0813 15.70796 0 0 1.000224 6 6 5.518995

0.0813 15.70796 0.25 0.5 1.423495 4.038481 4.038481 4.338367

0.0813 15.70796 0.5 0.7 1.64124 3.163562 3.163562 3.583159

0.0813 15.70796 0.75 0.7 1.916684 2.890009 2.890009 3.22656

0.0813 15.70796 1 0.8 2.167217 2.678118 2.678118 3.025516

0.0813 15.70796 1.5 0.9 2.628529 2.475849 2.475849 2.801589

0.0813 15.70796 2 1 3.05496 2.364284 2.364284 2.681012

0.0813 15.70796 3 1.1 3.853504 2.257759 2.257759 2.550087

0.0813 15.70796 4 1.2 4.610742 2.198558 2.198558 2.479621

0.0813 15.70796 5 1 5.34545 2.191603 2.191603 2.435323

0.0838 16.33628 0 0 1.00207 5.986139 5.986139 5.511124

0.0838 16.33628 0.25 0.5 1.427938 3.986632 3.986632 4.281406

0.0838 16.33628 0.5 0.7 1.652506 3.142283 3.142283 3.556316

0.0838 16.33628 0.75 0.7 1.931655 2.875141 2.875141 3.208621

0.0838 16.33628 1 0.8 2.186888 2.667823 2.667823 3.012412

0.0838 16.33628 1.5 0.9 2.656332 2.469743 2.469743 2.793715

0.0838 16.33628 2 0.9 3.092103 2.38535 2.38535 2.675102

0.0838 16.33628 3 1.1 3.909368 2.254715 2.254715 2.546133

0.0838 16.33628 4 1.1 4.686113 2.208574 2.208574 2.476742

0.0838 16.33628 5 1 5.440067 2.18957 2.18957 2.43334

0.0875 16.9646 0 0 1.004666 5.730892 5.730892 5.295865

0.0875 16.9646 0.25 0.5 1.434792 3.911554 3.911554 4.199209

0.0875 16.9646 0.5 0.7 1.669884 3.110727 3.110727 3.516657

0.0875 16.9646 0.75 0.7 1.9547 2.854099 2.854099 3.183432

0.0875 16.9646 1 0.8 2.21724 2.652426 2.652426 2.99291

0.0875 16.9646 1.5 0.9 2.699322 2.46086 2.46086 2.782366

0.0875 16.9646 2 0.9 3.148996 2.377493 2.377493 2.665531

0.0875 16.9646 3 1.1 3.994452 2.250251 2.250251 2.540415

0.0875 16.9646 4 1.1 4.800796 2.204721 2.204721 2.472175

0.0875 16.9646 5 1 5.584827 2.186584 2.186584 2.430493

0.09 17.59292 0 0 1.006428 5.573799 5.573799 5.163792

0.09 17.59292 0.25 0.5 1.439495 3.862746 3.862746 4.145941

0.09 17.59292 0.5 0.7 1.681779 3.089896 3.089896 3.490575

0.09 17.59292 0.75 0.7 1.970458 2.840047 2.840047 3.166673

0.09 17.59292 1 0.8 2.236989 2.642615 2.642615 2.980585

0.09 17.59292 1.5 0.8 2.728759 2.488005 2.488005 2.774204

0.09 17.59292 2 0.9 3.187121 2.372441 2.372441 2.659451

0.09 17.59292 3 1 4.052356 2.263291 2.263291 2.536357

0.09 17.59292 4 1.1 4.878313 2.202208 2.202208 2.469245

0.09 17.59292 5 1 5.683309 2.18463 2.18463 2.428671

0.0938 18.22124 0 0 1.00926 5.354085 5.354085 4.979602

0.0938 18.22124 0.25 0.5 1.446763 3.791336 3.791336 4.068255

0.0938 18.22124 0.5 0.7 1.700121 3.059495 3.059495 3.452688

0.0938 18.22124 0.75 0.7 1.994802 2.818236 2.818236 3.140702

0.0938 18.22124 1 0.8 2.267154 2.628554 2.628554 2.963075

0.0938 18.22124 1.5 0.8 2.772283 2.477619 2.477619 2.762137

0.0938 18.22124 2 0.9 3.245598 2.365072 2.365072 2.650682

0.0938 18.22124 3 1 4.139962 2.258236 2.258236 2.530453

0.0938 18.22124 4 1.1 4.995937 2.198519 2.198519 2.465017

0.0938 18.22124 5 1.3 5.830986 2.152411 2.152411 2.425451

0.0963 18.84956 0 0 1.011281 5.21703 5.21703 4.86504

0.0963 18.84956 0.25 0.5 1.45176 3.74569 3.74569 4.018794


ARTICLE IN PRESS


Clearance d ¼ 2


limit rlimiting

Peak relative

displacement

amplitude A

RMS relative

displacement

RMS absolute

acceleration

Primary

damping

coeff. x1

Primary

natural

frequency o1

Secondary

damping

coeff. x2

Stiffness

ratio r

0.0963 18.84956 0.5 0.7 1.712679 3.038959 3.038959 3.427162

0.0963 18.84956 0.75 0.7 2.011518 2.804406 2.804406 3.124359

0.0963 18.84956 1 0.8 2.287984 2.619549 2.619549 2.951956

0.0963 18.84956 1.5 0.8 2.80229 2.470917 2.470917 2.754428

0.0963 18.84956 2 0.9 3.284815 2.360294 2.360294 2.64506

0.0963 18.84956 3 1 4.198653 2.254946 2.254946 2.526664

0.0963 18.84956 4 1.1 5.075806 2.19611 2.19611 2.462305

0.0963 18.84956 5 1.3 5.931417 2.150395 2.150395 2.422932

0.0988 19.47787 0 0 1.013419 5.086755 5.086755 4.756396

0.0988 19.47787 0.25 0.5 1.456887 3.700463 3.700463 3.969894

0.0988 19.47787 0.5 0.7 1.724438 3.01953 3.01953 3.403151

0.0988 19.47787 0.75 0.7 2.027821 2.791137 2.791137 3.10878

0.0988 19.47787 1 0.8 2.309195 2.609105 2.609105 2.938935

0.0988 19.47787 1.5 0.9 2.832866 2.432615 2.432615 2.746555

0.0988 19.47787 2 0.9 3.324822 2.355617 2.355617 2.63961

0.0988 19.47787 3 1 4.258859 2.251716 2.251716 2.522988

0.0988 19.47787 4 1.1 5.15615 2.19374 2.19374 2.459675

0.0988 19.47787 5 1.3 6.033518 2.148412 2.148412 2.420494

0.1025 20.10619 0 0 1.016858 4.903119 4.903119 4.603695

0.1025 20.10619 0.25 0.5 1.516241 3.635229 3.635229 3.899644

0.1025 20.10619 0.5 0.7 1.746148 2.989298 2.989298 3.365836

0.1025 20.10619 0.75 0.7 2.052419 2.76975 2.76975 3.083626

0.1025 20.10619 1 0.8 2.34085 2.594125 2.594125 2.92042

0.1025 20.10619 1.5 0.9 2.878276 2.423087 2.423087 2.734629

0.1025 20.10619 2 0.9 3.38523 2.348787 2.348787 2.631743

0.1025 20.10619 3 1 4.350091 2.24698 2.24698 2.51768

0.1025 20.10619 4 1.1 5.278897 2.190257 2.190257 2.455881

0.1025 20.10619 5 1.3 6.187662 2.145499 2.145499 2.416983

0.105 20.73451 0 0 1.01932 4.787917 4.787917 4.508184

0.105 20.73451 0.25 0.5 1.520951 3.592243 3.592243 3.853484

0.105 20.73451 0.5 0.7 1.757697 2.970029 2.970029 3.342196

0.105 20.73451 0.75 0.7 2.069287 2.755932 2.755932 3.067478

0.105 20.73451 1 0.8 2.362142 2.584586 2.584586 2.908735

0.105 20.73451 1.5 0.9 2.909588 2.416935 2.416935 2.727007

0.105 20.73451 2 1 3.426486 2.322194 2.322194 2.62664

0.105 20.73451 3 1 4.411182 2.243894 2.243894 2.514273

0.105 20.73451 4 1.1 5.361475 2.187981 2.187981 2.453448

0.105 20.73451 5 1.3 6.291822 2.143595 2.143595 2.414737

0.1088 21.36283 0 0 1.02329 4.624951 4.624951 4.373472

0.1088 21.36283 0.25 0.5 1.528242 3.530204 3.530204 3.787138

0.1088 21.36283 0.5 0.7 1.775789 2.940932 2.940932 3.306605

0.1088 21.36283 0.75 0.7 2.095442 2.736324 2.736324 3.044765

0.1088 21.36283 1 0.8 2.395089 2.570889 2.570889 2.892115

0.1088 21.36283 1.5 0.9 2.95689 2.408039 2.408039 2.716109

0.1088 21.36283 2 1 3.488756 2.315118 2.315118 2.617696

0.1088 21.36283 3 1 4.505103 2.239365 2.239365 2.509351

0.1088 21.36283 4 1.1 5.487502 2.184632 2.184632 2.449937

0.1088 21.36283 5 1.3 6.451377 2.140796 2.140796 2.411502

0.1113 21.99115 0 0 1.026138 4.522377 4.522377 4.288935

0.1113 21.99115 0.25 0.5 1.533264 3.489515 3.489515 3.743761

0.1113 21.99115 0.5 0.5 1.788403 3.072052 3.072052 3.283189

0.1113 21.99115 0.75 0.7 2.113406 2.72255 2.72255 3.028799

0.1113 21.99115 1 0.7 2.417892 2.603514 2.603514 2.880738

0.1113 21.99115 1.5 0.8 2.988957 2.430374 2.430374 2.708487

0.1113 21.99115 2 1 3.531415 2.310531 2.310531 2.611965

0.1113 21.99115 3 1 4.568743 2.23641 2.23641 2.506193

0.1113 21.99115 4 1 5.573101 2.193331 2.193331 2.447461

0.1113 21.99115 5 1.3 6.559158 2.138965 2.138965 2.409433


ARTICLE IN PRESS


Clearance d ¼ 2


limit rlimiting

Peak relative

displacement

amplitude A

RMS relative

displacement

RMS absolute

acceleration

Primary

damping

coeff. x1

Primary

natural

frequency o1

Secondary

damping

coeff. x2

Stiffness

ratio r

0.1138 22.61947 0 0 1.029163 4.424307 4.424307 4.208303

0.1138 22.61947 0.25 0.5 1.538423 3.450076 3.450076 3.701861

0.1138 22.61947 0.5 0.5 1.801497 3.047452 3.047452 3.258344

0.1138 22.61947 0.75 0.7 2.130956 2.708688 2.708688 3.012774

0.1138 22.61947 1 0.8 2.440062 2.551651 2.551651 2.868798

0.1138 22.61947 1.5 0.8 3.021824 2.423499 2.423499 2.700801

0.1138 22.61947 2 1 3.574173 2.306042 2.306042 2.606413

0.1138 22.61947 3 0.9 4.633116 2.24832 2.24832 2.503077

0.1138 22.61947 4 1 5.659265 2.190802 2.190802 2.445025

0.1138 22.61947 5 1.2 6.667735 2.144855 2.144855 2.407293

0.1175 23.24779 0 0 1.03406 4.28482 4.28482 4.093952

0.1175 23.24779 0.25 0.5 1.546438 3.392144 3.392144 3.640528

0.1175 23.24779 0.5 0.5 1.821465 3.012893 3.012893 3.223712

0.1175 23.24779 0.75 0.7 2.157642 2.689005 2.689005 2.990221

0.1175 23.24779 1 0.8 2.474181 2.536996 2.536996 2.851126

0.1175 23.24779 1.5 0.8 3.071086 2.413519 2.413519 2.689764

0.1175 23.24779 2 0.9 3.639895 2.319125 2.319125 2.5978

0.1175 23.24779 3 1.1 4.731193 2.215297 2.215297 2.498083

0.1175 23.24779 4 1 5.790794 2.187086 2.187086 2.441516

0.1175 23.24779 5 1.2 6.833007 2.141831 2.141831 2.404166

0.12 23.8761 0 0 1.03759 4.196782 4.196782 4.022

0.12 23.8761 0.25 0.5 1.551977 3.354924 3.354924 3.601295

0.12 23.8761 0.5 0.5 1.834741 2.988971 2.988971 3.199733

0.12 23.8761 0.75 0.7 2.176068 2.676576 2.676576 2.976113

0.12 23.8761 1 0.8 2.497761 2.527656 2.527656 2.839972

0.12 23.8761 1.5 0.9 3.104955 2.380374 2.380374 2.68253

0.12 23.8761 2 0.9 3.684024 2.314081 2.314081 2.592117

0.12 23.8761 3 1.1 4.797648 2.211851 2.211851 2.49394

0.12 23.8761 4 1 5.880361 2.184657 2.184657 2.439271

0.12 23.8761 5 1.2 6.945811 2.139854 2.139854 2.402169

0.1225 24.50442 0 0 1.041354 4.112141 4.112141 3.95298

0.1225 24.50442 0.25 0.5 1.557685 3.317494 3.317494 3.561905

0.1225 24.50442 0.5 0.5 1.848392 2.966476 2.966476 3.177345

0.1225 24.50442 0.75 0.7 2.194984 2.662698 2.662698 2.9602420.1225 24.50442 1 0.8 2.521955 2.518655 2.518655 2.829312

0.1225 24.50442 1.5 0.9 3.13939 2.37389 2.37389 2.674703

0.1225 24.50442 2 0.9 3.729116 2.309149 2.309149 2.586617

0.1225 24.50442 3 1.1 4.865334 2.208474 2.208474 2.489928

0.1225 24.50442 4 1 5.970799 2.182267 2.182267 2.437097

0.1225 24.50442 5 1.2 7.059481 2.137908 2.137908 2.400238

0.1263 25.13274 0 0 1.047484 3.991577 3.991577 3.854961

0.1263 25.13274 0.25 0.5 1.566586 3.263437 3.263437 3.505276

0.1263 25.13274 0.5 0.5 1.86959 2.932285 2.932285 3.143371

0.1263 25.13274 0.75 0.7 2.224087 2.642826 2.642826 2.937708

0.1263 25.13274 1 0.7 2.557693 2.540958 2.540958 2.81245

0.1263 25.13274 1.5 0.9 3.191183 2.364477 2.364477 2.663463

0.1263 25.13274 2 0.9 3.797881 2.301955 2.301955 2.578695

0.1263 25.13274 3 1.1 4.968354 2.20353 2.20353 2.484147

0.1263 25.13274 4 1.3 6.108665 2.150281 2.150281 2.43307

0.1263 25.13274 5 1.2 7.233392 2.135043 2.135043 2.397465

0.1288 25.76106 0 0 1.05193 3.915177 3.915177 3.793035

0.1288 25.76106 0.25 0.5 1.572764 3.228528 3.228528 3.468863

0.1288 25.76106 0.5 0.5 1.884231 2.91047 2.91047 3.121834

0.1288 25.76106 0.75 0.7 2.243036 2.630284 2.630284 2.923622

0.1288 25.76106 1 0.7 2.582185 2.53067 2.53067 2.801376

0.1288 25.76106 1.5 0.9 3.227189 2.35844 2.35844 2.656343

0.1288 25.76106 2 0.9 3.844815 2.297289 2.297289 2.573623

0.1288 25.76106 3 1.1 5.038137 2.200312 2.200312 2.480445


ARTICLE IN PRESS


Clearance d ¼ 2


limit rlimiting

Peak relative

displacement

amplitude A

RMS relative

displacement

RMS absolute

acceleration

Primary

damping

coeff. x1

Primary

natural

frequency o1

Secondary

damping

coeff. x2

Stiffness

ratio r

0.1288 25.76106 4 1.3 6.202518 2.147872 2.147872 2.430131

0.1288 25.76106 5 1.2 7.351345 2.133169 2.133169 2.395695

0.1325 26.38938 0 0 1.059211 3.805931 3.805931 3.704737

0.1325 26.38938 0.25 0.5 1.582429 3.177199 3.177199 3.415518

0.1325 26.38938 0.5 0.5 1.906983 2.877295 2.877295 3.089128

0.1325 26.38938 0.75 0.7 2.272593 2.61114 2.61114 2.902167

0.1325 26.38938 1 0.8 2.620534 2.480877 2.480877 2.784746

0.1325 26.38938 1.5 0.8 3.281096 2.37325 2.37325 2.645918

0.1325 26.38938 2 0.9 3.916206 2.290476 2.290476 2.566314

0.1325 26.38938 3 1.1 5.145127 2.195598 2.195598 2.475108

0.1325 26.38938 4 1.3 6.345608 2.144336 2.144336 2.425899

0.1325 26.38938 5 1.2 7.53122 2.130407 2.130407 2.393153

0.135 27.0177 0 0 1.064522 3.736991 3.736991 3.649202

0.135 27.0177 0.25 0.5 1.589158 3.143322 3.143322 3.380419

0.135 27.0177 0.5 0.5 1.922174 2.856047 2.856047 3.068311

0.135 27.0177 0.75 0.7 2.29305 2.597841 2.597841 2.887259

0.135 27.0177 1 0.8 2.645908 2.471719 2.471719 2.774087

0.135 27.0177 1.5 0.8 3.318214 2.366429 2.366429 2.638586

0.135 27.0177 2 1 3.964664 2.268161 2.268161 2.560895

0.135 27.0177 3 1.1 5.217605 2.192528 2.192528 2.471688

0.135 27.0177 4 1.2 6.443211 2.15002 2.15002 2.423157

0.135 27.0177 5 1.1 7.654113 2.135664 2.135664 2.391504

0.1375 27.64602 0 0 1.070245 3.670582 3.670582 3.595829

0.1375 27.64602 0.25 0.5 1.596136 3.11041 3.11041 3.346451

0.1375 27.64602 0.5 0.5 1.937137 2.834791 2.834791 3.047529

0.1375 27.64602 0.75 0.7 2.314203 2.585119 2.585119 2.873111

0.1375 27.64602 1 0.8 2.671804 2.462842 2.462842 2.763839

0.1375 27.64602 1.5 0.8 3.355775 2.359786 2.359786 2.631512

0.1375 27.64602 2 1 4.013985 2.263273 2.263273 2.555074

0.1375 27.64602 3 1.1 5.29158 2.18952 2.18952 2.468385

0.1375 27.64602 4 1.2 6.541764 2.147453 2.147453 2.420346

0.1375 27.64602 5 1.1 7.777839 2.133626 2.133626 2.389779

0.1413 28.27433 0 0 1.072696 3.575432 3.575432 3.519573

0.1413 28.27433 0.25 0.5 1.603843 3.062392 3.062392 3.297105

0.1413 28.27433 0.5 0.5 1.958989 2.803138 2.803138 3.016681

0.1413 28.27433 0.75 0.7 2.343752 2.567016 2.567016 2.853178

0.1413 28.27433 1 0.7 2.710805 2.479848 2.479848 2.747323

0.1413 28.27433 1.5 0.8 3.413367 2.350142 2.350142 2.621361

0.1413 28.27433 2 1 4.088778 2.256148 2.256148 2.546701

0.1413 28.27433 3 1 5.404064 2.195888 2.195888 2.463309

0.1413 28.27433 4 1.2 6.69266 2.143687 2.143687 2.416301

0.1413 28.27433 5 1.1 7.967387 2.130627 2.130627 2.387305

0.1438 28.90265 0 0 1.079187 3.514627 3.514627 3.470968

0.1438 28.90265 0.25 0.5 1.610994 3.03067 3.03067 3.264607

0.1438 28.90265 0.5 0.5 1.974661 2.782802 2.782802 2.996985

0.1438 28.90265 0.75 0.7 2.365045 2.554133 2.554133 2.838946

0.1438 28.90265 1 0.7 2.737647 2.469944 2.469944 2.736936

0.1438 28.90265 1.5 0.9 3.451612 2.321719 2.321719 2.61367

0.1438 28.90265 2 1 4.141652 2.251531 2.251531 2.541347

0.1438 28.90265 3 1 5.481332 2.19257 2.19257 2.45998

0.1438 28.90265 4 1.2 6.795637 2.14123 2.14123 2.413716

0.1438 28.90265 5 1.1 8.097079 2.128664 2.128664 2.385729

0.1463 29.53097 0 0 1.086191 3.455882 3.455882 3.42411

0.1463 29.53097 0.25 0.5 1.618439 2.999839 2.999839 3.233147

0.1463 29.53097 0.5 0.5 1.990863 2.761762 2.761762 2.976579

0.1463 29.53097 0.75 0.7 2.387058 2.541143 2.541143 2.82463

0.1463 29.53097 1 0.8 2.765685 2.430308 2.430308 2.726382

0.1463 29.53097 1.5 0.9 3.491256 2.315273 2.315273 2.60624


ARTICLE IN PRESS


Clearance d ¼ 2


limit rlimiting

Peak relative

displacement

amplitude A

RMS relative

displacement

RMS absolute

acceleration

Primary

damping

coeff. x1

Primary

natural

frequency o1

Secondary

damping

coeff. x2

Stiffness

ratio r

0.1463 29.53097 2 1 4.193223 2.247015 2.247015 2.536166

0.1463 29.53097 3 1 5.559202 2.189312 2.189312 2.456757

0.1463 29.53097 4 1.2 6.900892 2.138814 2.138814 2.411216

0.1463 29.53097 5 1.1 8.228729 2.126729 2.126729 2.384211

0.15 30.15929 0 0 1.097258 3.371364 3.371364 3.356866

0.15 30.15929 0.25 0.5 1.696888 2.954941 2.954941 3.187545

0.15 30.15929 0.5 0.5 2.017775 2.732402 2.732402 2.948382

0.15 30.15929 0.75 0.7 2.421594 2.522688 2.522688 2.804499

0.15 30.15929 1 0.8 2.809353 2.41657 2.41657 2.710753

0.15 30.15929 1.5 0.9 3.553523 2.305952 2.305952 2.595631

0.15 30.15929 2 0.9 4.273836 2.25507 2.25507 2.528298

0.15 30.15929 3 1 5.679846 2.184537 2.184537 2.452115

0.15 30.15929 4 1.2 7.061119 2.135264 2.135264 2.40762

0.15 30.15929 5 1.1 8.430892 2.123878 2.123878 2.382035

0.1525 30.78761 0 0 1.104996 3.317321 3.317321 3.313978

0.1525 30.78761 0.25 0.5 1.704421 2.924762 2.924762 3.156951

0.1525 30.78761 0.5 0.5 2.034642 2.711629 2.711629 2.92836

0.1525 30.78761 0.75 0.7 2.445649 2.511012 2.511012 2.791896

0.1525 30.78761 1 0.7 2.839929 2.435072 2.435072 2.700616

0.1525 30.78761 1.5 0.9 3.596456 2.299958 2.299958 2.588895

0.1525 30.78761 2 0.9 4.33041 2.250113 2.250113 2.523115

0.1525 30.78761 3 1 5.761882 2.181424 2.181424 2.449142

0.1525 30.78761 4 1.2 7.171053 2.132946 2.132946 2.405322

0.1525 30.78761 5 1.5 8.567756 2.099435 2.099435 2.380403

0.155 31.41593 0 0 1.113048 3.265002 3.265002 3.27254

0.155 31.41593 0.25 0.5 1.712266 2.896207 2.896207 3.128183

0.155 31.41593 0.5 0.5 2.052399 2.692026 2.692026 2.909615

0.155 31.41593 0.75 0.7 2.470609 2.498453 2.498453 2.778263

0.155 31.41593 1 0.7 2.869163 2.424875 2.424875 2.690043

0.155 31.41593 1.5 0.8 3.638272 2.312876 2.312876 2.582351

0.155 31.41593 2 0.9 4.385474 2.245266 2.245266 2.518104

0.155 31.41593 3 1 5.846261 2.178366 2.178366 2.446264

0.155 31.41593 4 1.1 7.282997 2.137773 2.137773 2.403098

0.155 31.41593 5 1.5 8.707309 2.097287 2.097287 2.377792

0.1588 32.04425 0 0 1.125765 3.189667 3.189667 3.213019

0.1588 32.04425 0.25 0.5 1.724676 2.853396 2.853396 3.085162

0.1588 32.04425 0.5 0.5 2.080897 2.662973 2.662973 2.881937

0.1588 32.04425 0.75 0.7 2.505918 2.479511 2.479511 2.757775

0.1588 32.04425 1 0.7 2.914322 2.410289 2.410289 2.675088

0.1588 32.04425 1.5 0.8 3.704168 2.30286 2.30286 2.572058

0.1588 32.04425 2 0.9 4.47238 2.238196 2.238196 2.510895

0.1588 32.04425 3 1.2 5.975135 2.154782 2.154782 2.440725

0.1588 32.04425 4 1.1 7.454936 2.133981 2.133981 2.39965

0.1588 32.04425 5 1.5 8.923665 2.094133 2.094133 2.374048

0.1613 32.67256 0 0 1.134688 3.141467 3.141467 3.17503

0.1613 32.67256 0.25 0.5 1.732876 2.825222 2.825222 3.056952

0.1613 32.67256 0.5 0.5 2.101006 2.64352 2.64352 2.86344

0.1613 32.67256 0.75 0.7 2.530376 2.467546 2.467546 2.744973

0.1613 32.67256 1 0.8 2.945918 2.375724 2.375724 2.6649

0.1613 32.67256 1.5 0.8 3.75045 2.296401 2.296401 2.565501

0.1613 32.67256 2 0.9 4.532469 2.233609 2.233609 2.506285

0.1613 32.67256 3 1.2 6.062888 2.1514 2.1514 2.436773

0.1613 32.67256 4 1.1 7.572093 2.131505 2.131505 2.397448

0.1613 32.67256 5 1.5 9.070413 2.092075 2.092075 2.371663

0.1638 33.30088 0 0 1.144007 3.094913 3.094913 3.138416

0.1638 33.30088 0.25 0.5 1.740731 2.798109 2.798109 3.029944

0.1638 33.30088 0.5 0.5 2.119255 2.625132 2.625132 2.846098

0.1638 33.30088 0.75 0.7 2.555836 2.456075 2.456075 2.732807


ARTICLE IN PRESS


Clearance d ¼ 2


limit rlimiting

Peak relative

displacement

amplitude A

RMS relative

displacement

RMS absolute

acceleration

Primary

damping

coeff. x1

Primary

natural

frequency o1

Secondary

damping

coeff. x2

Stiffness

ratio r

0.1638 33.30088 1 0.8 2.978984 2.366513 2.366513 2.654653

0.1638 33.30088 1.5 0.9 3.796957 2.27273 2.27273 2.558659

0.1638 33.30088 2 1 4.592258 2.215439 2.215439 2.500806

0.1638 33.30088 3 1.2 6.153359 2.148084 2.148084 2.432952

0.1638 33.30088 4 1.1 7.693379 2.129069 2.129069 2.39532

0.1638 33.30088 5 1.5 9.220672 2.090051 2.090051 2.369364

0.1675 33.9292 0 0 1.158811 3.027654 3.027654 3.085629

0.1675 33.9292 0.25 0.5 1.753222 2.757501 2.757501 2.989602

0.1675 33.9292 0.5 0.5 2.147989 2.596405 2.596405 2.818948

0.1675 33.9292 0.75 0.7 2.596103 2.437592 2.437592 2.713105

0.1675 33.9292 1 0.7 3.031123 2.375693 2.375693 2.639835

0.1675 33.9292 1.5 0.9 3.867748 2.262714 2.262714 2.547493

0.1675 33.9292 2 1 4.687128 2.207766 2.207766 2.492176

0.1675 33.9292 3 1.2 6.292226 2.14323 2.14323 2.427456

0.1675 33.9292 4 1.1 7.878297 2.125487 2.125487 2.392262

0.1675 33.9292 5 1.5 9.453316 2.087076 2.087076 2.366068

0.17 34.55752 0 0 1.16929 2.984437 2.984437 3.051779

0.17 34.55752 0.25 0.5 1.762072 2.732011 2.732011 2.964472

0.17 34.55752 0.5 0.5 2.168162 2.578067 2.578067 2.801741

0.17 34.55752 0.75 0.7 2.624428 2.425232 2.425232 2.699982

0.17 34.55752 1 0.7 3.064349 2.365727 2.365727 2.629745

0.17 34.55752 1.5 0.9 3.917667 2.256261 2.256261 2.540391

0.17 34.55752 2 1 4.751169 2.202801 2.202801 2.486671

0.17 34.55752 3 1.2 6.388824 2.14007 2.14007 2.423944

0.17 34.55752 4 1.1 8.005527 2.123145 2.123145 2.390309

0.17 34.55752 5 1.5 9.612012 2.085132 2.085132 2.36397

0.1738 35.18584 0 0 1.18164 2.922018 2.922018 3.002989

0.1738 35.18584 0.25 0.5 1.776229 2.692666 2.692666 2.92569

0.1738 35.18584 0.5 0.5 2.200166 2.550759 2.550759 2.776199

0.1738 35.18584 0.75 0.7 2.668208 2.407679 2.407679 2.681557

0.1738 35.18584 1 0.7 3.117259 2.351442 2.351442 2.615448

0.1738 35.18584 1.5 0.9 3.994502 2.24694 2.24694 2.530274

0.1738 35.18584 2 1 4.851322 2.195565 2.195565 2.478766

0.1738 35.18584 3 1.1 6.536478 2.142691 2.142691 2.418811

0.1738 35.18584 4 1.5 8.201973 2.097679 2.097679 2.38676

0.1738 35.18584 5 9.86 9.857743 2.010791 2.010791 2.355388

0.1763 35.81416 0 0 1.189874 2.881852 2.881852 2.971649

0.1763 35.81416 0.25 0.5 1.786323 2.667686 2.667686 2.901237

0.1763 35.81416 0.5 0.5 2.222778 2.532334 2.532334 2.758982

0.1763 35.81416 0.75 0.7 2.696625 2.396556 2.396556 2.670011

0.1763 35.81416 1 0.8 3.154744 2.321592 2.321592 2.605477

0.1763 35.81416 1.5 0.9 4.046647 2.240929 2.240929 2.523837

0.1763 35.81416 2 1 4.920419 2.190878 2.190878 2.473721

0.1763 35.81416 3 1.1 6.637754 2.139314 2.139314 2.415387

0.1763 35.81416 4 1.5 8.337552 2.095109 2.095109 2.383534

0.1763 35.81416 5 9.55 9.548317 2.011116 2.011116 2.354737

0.1788 36.44247 0 0 1.198557 2.842916 2.842916 2.941318

0.1788 36.44247 0.25 0.5 1.797004 2.642403 2.642403 2.876516

0.1788 36.44247 0.5 0.5 2.246446 2.514931 2.514931 2.742859

0.1788 36.44247 0.75 0.7 2.726435 2.384195 2.384195 2.657043

0.1788 36.44247 1 0.8 3.193942 2.312342 2.312342 2.595435

0.1788 36.44247 1.5 0.8 4.101457 2.248864 2.248864 2.517584

0.1788 36.44247 2 1 4.990625 2.186295 2.186295 2.468848

0.1788 36.44247 3 1.1 6.743529 2.136 2.136 2.412077

0.1788 36.44247 4 1.5 8.476175 2.092585 2.092585 2.380423

0.1788 36.44247 5 9.19 9.187191 2.011563 2.011563 2.35434

0.1825 37.07079 0 0 1.212503 2.786326 2.786326 2.897294

0.1825 37.07079 0.25 0.5 1.814265 2.606011 2.606011 2.841156


ARTICLE IN PRESS


Clearance d ¼ 2


limit rlimiting

Peak relative

displacement

amplitude A

RMS relative

displacement

RMS absolute

acceleration

Primary

damping

coeff. x1

Primary

natural

frequency o1

Secondary

damping

coeff. x2

Stiffness

ratio r

0.1825 37.07079 0.5 0.5 2.284261 2.488063 2.488063 2.71794

0.1825 37.07079 0.75 0.7 2.773925 2.36594 2.36594 2.638017

0.1825 37.07079 1 0.7 3.253148 2.317011 2.317011 2.581107

0.1825 37.07079 1.5 0.8 4.1862 2.239054 2.239054 2.507957

0.1825 37.07079 2 0.9 5.10169 2.18951 2.18951 2.461663

0.1825 37.07079 3 1.1 6.905886 2.131146 2.131146 2.407319

0.1825 37.07079 4 1.5 8.692595 2.088882 2.088882 2.375961

0.1825 37.07079 5 1.5 8.664668 2.075848 2.075848 2.354575

0.185 37.69911 0 0 1.222503 2.750051 2.750051 2.869127

0.185 37.69911 0.25 0.5 1.826708 2.581913 2.581913 2.817822

0.185 37.69911 0.5 0.5 2.30819 2.470496 2.470496 2.701723

0.185 37.69911 0.75 0.7 2.807729 2.354424 2.354424 2.626158

0.185 37.69911 1 0.7 3.293864 2.307171 2.307171 2.571371

0.185 37.69911 1.5 1 4.244365 2.207407 2.207407 2.500966

0.185 37.69911 2 0.9 5.177439 2.184595 2.184595 2.456888

0.185 37.69911 3 1.1 7.017798 2.127984 2.127984 2.404278

0.185 37.69911 4 1.5 8.841407 2.086468 2.086468 2.373119

0.185 37.69911 5 1.3 8.292474 2.081076 2.081076 2.352787

0.1888 38.32743 0 0 1.238684 2.697384 2.697384 2.828285

0.1888 38.32743 0.25 0.5 1.847001 2.546761 2.546761 2.783964

0.1888 38.32743 0.5 0.5 2.345956 2.445119 2.445119 2.678456

0.1888 38.32743 0.75 0.7 2.860309 2.337976 2.337976 2.60941

0.1888 38.32743 1 0.7 3.359258 2.293092 2.293092 2.55761

0.1888 38.32743 1.5 0.9 4.337546 2.209301 2.209301 2.490131

0.1888 38.32743 2 1.1 5.297297 2.159308 2.159308 2.449754

0.1888 38.32743 3 1.1 7.193458 2.123349 2.123349 2.399906

0.1888 38.32743 4 1.5 9.074603 2.082924 2.082924 2.369045

0.1888 38.32743 5 10 10.9463 2.008363 2.008363 2.330665

0.1913 38.95575 0 0 1.250325 2.663427 2.663427 2.801985

0.1913 38.95575 0.25 0.5 1.861776 2.523631 2.523631 2.761776

0.1913 38.95575 0.5 0.5 2.372972 2.427438 2.427438 2.662199

0.1913 38.95575 0.75 0.7 2.895232 2.325443 2.325443 2.596476

0.1913 38.95575 1 0.8 3.405774 2.267606 2.267606 2.547751

0.1913 38.95575 1.5 0.9 4.400075 2.202594 2.202594 2.483013

0.1913 38.95575 2 1.1 5.380028 2.154175 2.154175 2.443806

0.1913 38.95575 3 1 7.316353 2.12634 2.12634 2.397009

0.1913 38.95575 4 1.5 9.235595 2.08061 2.08061 2.36645

0.1913 38.95575 5 11 11.14813 2.006348 2.006348 2.318059

0.195 39.58407 0 0 1.269333 2.614267 2.614267 2.76396

0.195 39.58407 0.25 0.5 1.886146 2.489724 2.489724 2.729414

0.195 39.58407 0.5 0.5 2.416626 2.402461 2.402461 2.639451

0.195 39.58407 0.75 0.7 2.951733 2.30755 2.30755 2.578217

0.195 39.58407 1 0.8 3.47496 2.25368 2.25368 2.533042

0.195 39.58407 1.5 0.9 4.501404 2.192873 2.192873 2.472841

0.195 39.58407 2 1.1 5.510336 2.146705 2.146705 2.435291

0.195 39.58407 3 1.5 7.507507 2.094804 2.094804 2.391926

0.195 39.58407 4 1.3 9.489056 2.084497 2.084497 2.362585

0.195 39.58407 5 11 11.4628 2.005593 2.005593 2.312027

0.1975 40.21239 0 0 1.283185 2.582327 2.582327 2.739264

0.1975 40.21239 0.25 0.5 1.904111 2.467429 2.467429 2.708226

0.1975 40.21239 0.5 0.5 2.448011 2.38568 2.38568 2.624189

0.1975 40.21239 0.75 0.7 2.992336 2.29623 2.29623 2.566806

0.1975 40.21239 1 0.8 3.524804 2.244773 2.244773 2.523751

0.1975 40.21239 1.5 0.9 4.570777 2.186631 2.186631 2.466406

0.1975 40.21239 2 1.1 5.601551 2.14187 2.14187 2.42987

0.1975 40.21239 3 1.5 7.640408 2.091573 2.091573 2.387681

0.1975 40.21239 4 1.3 9.383137 2.081863 2.081863 2.360032

0.1975 40.21239 5 12 11.68219 2.005122 2.005122 2.308348


ARTICLE IN PRESS


Clearance d ¼ 2


limit rlimiting

Peak relative

displacement

amplitude A

RMS relative

displacement

RMS absolute

acceleration

Primary

damping

coeff. x1

Primary

natural

frequency o1

Secondary

damping

coeff. x2

Stiffness

ratio r

0.2013 40.8407 0 0 1.30597 2.536319 2.536319 2.703747

0.2013 40.8407 0.25 0.5 2.011085 2.434734 2.434734 2.677315

0.2013 40.8407 0.5 0.5 2.496114 2.360402 2.360402 2.601245

0.2013 40.8407 0.75 0.7 3.055857 2.279754 2.279754 2.550349

0.2013 40.8407 1 0.7 3.604633 2.24408 2.24408 2.509925

0.2013 40.8407 1.5 0.9 4.682067 2.17757 2.17757 2.457198

0.2013 40.8407 2 1 5.744594 2.141747 2.141747 2.422043

0.2013 40.8407 3 1.3 7.850925 2.09523 2.09523 2.381428

0.2013 40.8407 4 9.9 9.942652 2.0092 2.0092 2.34581

0.2013 40.8407 5 12 12.02813 2.004491 2.004491 2.303526

0.205 41.46902 0 0 1.331577 2.491781 2.491781 2.669375

0.205 41.46902 0.25 0.5 2.035644 2.402857 2.402857 2.647362

0.205 41.46902 0.5 0.5 2.546181 2.336675 2.336675 2.579938

0.205 41.46902 0.75 0.7 3.122243 2.261368 2.261368 2.531818

0.205 41.46902 1 0.8 3.687599 2.218285 2.218285 2.496428

0.205 41.46902 1.5 1 4.799461 2.159521 2.159521 2.44735

0.205 41.46902 2 1 5.897654 2.134207 2.134207 2.414365

0.205 41.46902 3 1.3 8.074487 2.089847 2.089847 2.37539

0.205 41.46902 4 10 10.238 2.008402 2.008402 2.337816

0.205 41.46902 5 12 12.39513 2.004178 2.004178 2.300959

0.2075 42.09734 0 0 1.350479 2.463215 2.463215 2.647355

0.2075 42.09734 0.25 0.5 2.053814 2.381663 2.381663 2.627516

0.2075 42.09734 0.5 0.5 2.582779 2.319572 2.319572 2.564474

0.2075 42.09734 0.75 0.7 3.170699 2.249705 2.249705 2.520206

0.2075 42.09734 1 0.8 3.748126 2.208502 2.208502 2.486279

0.2075 42.09734 1.5 1 4.884312 2.15273 2.15273 2.43979

0.2075 42.09734 2 1 6.006801 2.129326 2.129326 2.409477

0.2075 42.09734 3 1.3 8.232526 2.086348 2.086348 2.371551

0.2075 42.09734 4 10 10.44591 2.007954 2.007954 2.333209

0.2075 42.09734 5 12 12.65325 2.003972 2.003972 2.299387

0.2113 42.72566 0 0 1.375424 2.421426 2.421426 2.615136

0.2113 42.72566 0.25 0.5 2.084355 2.350668 2.350668 2.598661

0.2113 42.72566 0.5 0.5 2.643311 2.295604 2.295604 2.543036

0.2113 42.72566 0.75 0.7 3.247813 2.233059 2.233059 2.503839

0.2113 42.72566 1 0.8 3.842507 2.194505 2.194505 2.471962

0.2113 42.72566 1.5 1 5.016705 2.142893 2.142893 2.429017

0.2113 42.72566 2 1 6.178819 2.122213 2.122213 2.402475

0.2113 42.72566 3 1.3 8.483209 2.08123 2.08123 2.366059

0.2113 42.72566 4 10 10.7764 2.007295 2.007295 2.326738

0.2113 42.72566 5 13.1 13.06515 2.002977 2.002977 2.292901

0.215 43.35398 0 0 1.401859 2.381228 2.381228 2.584156

0.215 43.35398 0.25 0.5 2.117414 2.320023 2.320023 2.570283

0.215 43.35398 0.5 0.5 2.706048 2.271893 2.271893 2.521922

0.215 43.35398 0.75 0.7 3.330418 2.215666 2.215666 2.486746

0.215 43.35398 1 0.7 3.947277 2.190182 2.190182 2.458568

0.215 43.35398 1.5 0.9 5.161088 2.14002 2.14002 2.418857

0.215 43.35398 2 1.3 6.365368 2.09963 2.09963 2.395054

0.215 43.35398 3 1.3 8.754762 2.076261 2.076261 2.360871

0.215 43.35398 4 11.1 11.13365 2.005219 2.005219 2.313101

0.215 43.35398 5 13.5 13.50836 2.002509 2.002509 2.289979

0.2175 43.9823 0 0 1.42183 2.355212 2.355212 2.564105

0.2175 43.9823 0.25 0.5 2.14088 2.300102 2.300102 2.551948

0.2175 43.9823 0.5 0.5 2.751537 2.255591 2.255591 2.507396

0.2175 43.9823 0.75 0.7 3.392066 2.203568 2.203568 2.474865

0.2175 43.9823 1 0.7 4.020479 2.180728 2.180728 2.449738

0.2175 43.9823 1.5 0.9 5.26606 2.133327 2.133327 2.412198

0.2175 43.9823 2 1.2 6.493516 2.098951 2.098951 2.388653

0.2175 43.9823 3 1.3 8.949006 2.07303 2.07303 2.357577


ARTICLE IN PRESS


Clearance d ¼ 2


limit rlimiting

Peak relative

displacement

amplitude A

RMS relative

displacement

RMS absolute

acceleration

Primary

damping

coeff. x1

Primary

natural

frequency o1

Secondary

damping

coeff. x2

Stiffness

ratio r

0.2175 43.9823 4 11.4 11.39001 2.004632 2.004632 2.308626

0.2175 43.9823 5 13.8 13.82445 2.002222 2.002222 2.288321

0.2213 44.61062 0 0 1.456329 2.317246 2.317246 2.53483

0.2213 44.61062 0.25 0.5 2.1816 2.270893 2.270893 2.525226

0.2213 44.61062 0.5 0.5 2.828605 2.232612 2.232612 2.487131

0.2213 44.61062 0.75 0.7 3.488207 2.186304 2.186304 2.458133

0.2213 44.61062 1 0.8 4.142922 2.15808 2.15808 2.435438

0.2213 44.61062 1.5 0.9 5.434196 2.123632 2.123632 2.402706

0.2213 44.61062 2 1.2 6.712894 2.090916 2.090916 2.379453

0.2213 44.61062 3 9.26 9.26287 2.009521 2.009521 2.347863

0.2213 44.61062 4 11.4 11.40874 2.004161 2.004161 2.304703

0.225 45.23893 0 0 1.497742 2.280672 2.280672 2.506623

0.225 45.23893 0.25 0.5 2.427263 2.241694 2.241694 2.498644

0.225 45.23893 0.5 0.5 2.94438 2.208863 2.208863 2.466184

0.225 45.23893 0.75 0.7 3.598679 2.170026 2.170026 2.442609

0.225 45.23893 1 0.8 4.275911 2.143275 2.143275 2.420669

0.225 45.23893 1.5 1.1 5.620872 2.103788 2.103788 2.392891

0.225 45.23893 2 1.2 6.955209 2.083176 2.083176 2.370824

0.225 45.23893 3 9.61 9.610354 2.007761 2.007761 2.333794

0.225 45.23893 4 12 12.25755 2.00328 2.00328 2.298795

0.2288 45.86725 0 0 1.547269 2.245298 2.245298 2.479317

0.2288 45.86725 0.25 0.5 2.472214 2.213481 2.213481 2.473179

0.2288 45.86725 0.5 0.5 3.027783 2.185633 2.185633 2.445834

0.2288 45.86725 0.75 0.7 3.720135 2.15134 2.15134 2.424613

0.2288 45.86725 1 0.8 4.426944 2.129247 2.129247 2.406942

0.2288 45.86725 1.5 1.1 5.830529 2.093112 2.093112 2.380887

0.2288 45.86725 2 1.1 7.223414 2.078965 2.078965 2.362711

0.2288 45.86725 3 10 9.998901 2.006194 2.006194 2.321728

0.2288 45.86725 4 12 12.76661 2.002895 2.002895 2.296168

0.2325 46.49557 0 0 1.59469 2.2111 2.2111 2.452891

0.2325 46.49557 0.25 0.5 2.521676 2.185361 2.185361 2.447951

0.2325 46.49557 0.5 0.5 3.122948 2.163101 2.163101 2.426255

0.2325 46.49557 0.75 0.7 3.860312 2.133569 2.133569 2.407781

0.2325 46.49557 1 0.7 4.599752 2.120664 2.120664 2.394097

0.2325 46.49557 1.5 1.1 6.068394 2.082897 2.082897 2.369713

0.2325 46.49557 2 1.1 7.52962 2.071212 2.071212 2.354949

0.2325 46.49557 3 10 10.44026 2.005336 2.005336 2.31428

0.2325 46.49557 4 13 13.34358 2.001904 2.001904 2.290436

0.2363 47.12389 0 0 1.656101 2.17802 2.17802 2.427296

0.2363 47.12389 0.25 0.5 2.583762 2.158054 2.158054 2.423675

0.2363 47.12389 0.5 0.5 3.244313 2.139342 2.139342 2.405589

0.2363 47.12389 0.75 0.7 4.024261 2.116846 2.116846 2.392233

0.2363 47.12389 1 0.9 4.800794 2.096152 2.096152 2.378774

0.2363 47.12389 1.5 1 6.345887 2.076005 2.076005 2.359189

0.2363 47.12389 2 1.1 7.883839 2.063738 2.063738 2.347672

0.2363 47.12389 3 10 10.94956 2.004506 2.004506 2.30776

0.24 47.75221 0 0 1.736804 2.145987 2.145987 2.402469

0.24 47.75221 0.25 0.5 2.664767 2.130694 2.130694 2.399519

0.24 47.75221 0.5 0.5 3.397307 2.117304 2.117304 2.386694

0.24 47.75221 0.75 0.7 4.219956 2.09798 2.09798 2.374629

0.24 47.75221 1 0.9 5.041271 2.081398 2.081398 2.363902

0.24 47.75221 1.5 1 6.674755 2.065964 2.065964 2.349347

0.24 47.75221 2 2 8.303477 2.038149 2.038149 2.337098

0.24 47.75221 3 12 11.55323 2.002582 2.002582 2.296889

0.2438 48.38053 0 0 1.828189 2.115051 2.115051 2.378453

0.2438 48.38053 0.25 0.5 2.766808 2.1044 2.1044 2.376579

0.2438 48.38053 0.5 0.5 3.587264 2.093589 2.093589 2.366334

0.2438 48.38053 0.75 0.7 4.462359 2.079248 2.079248 2.35742


ARTICLE IN PRESS


Clearance d ¼ 2


limit rlimiting

Peak relative

displacement

amplitude A

RMS relative

displacement

RMS absolute

acceleration

Primary

damping

coeff. x1

Primary

natural

frequency o1

Secondary

damping

coeff. x2

Stiffness

ratio r

0.2438 48.38053 1 0.8 5.336231 2.069697 2.069697 2.350078

0.2438 48.38053 1.5 1.5 7.080102 2.045417 2.045417 2.338189

0.2438 48.38053 2 8 8.819576 2.007485 2.007485 2.323289

0.2438 48.38053 3 12 12.2918 2.001842 2.001842 2.292736

0.2475 49.00885 0 0 1.95879 2.084988 2.084988 2.355057

0.2475 49.00885 0.25 0.5 2.915299 2.077517 2.077517 2.353303

0.2475 49.00885 0.5 0.5 3.83333 2.070742 2.070742 2.346963

0.2475 49.00885 0.75 0.7 4.777205 2.06169 2.06169 2.34168

0.2475 49.00885 1 1.1 5.720399 2.049374 2.049374 2.336184

0.2475 49.00885 1.5 1.5 7.603064 2.033263 2.033263 2.322905

0.2475 49.00885 2 9.5 9.48372 2.003694 2.003694 2.303652

0.2475 49.00885 3 13 13.24175 2.001032 2.001032 2.289026

0.25 49.63716 0 0 2.081178 2.06554 2.06554 2.339896

0.25 49.63716 0.25 0.5 3.274975 2.060829 2.060829 2.339088

0.25 49.63716 0.5 0.5 4.074843 2.056449 2.056449 2.335025

0.25 49.63716 0.75 0.9 5.054326 2.045488 2.045488 2.329569

0.25 49.63716 1 1.2 6.055629 2.036454 2.036454 2.323765

0.25 49.63716 1.5 8 8.059576 2.005754 2.005754 2.313179

0.25 49.63716 2 10 10.06146 2.002458 2.002458 2.296911

0.25 50.26548 0 0 2.081178 2.06554 2.06554 2.339896

0.25 50.26548 0.25 0.5 3.274975 2.060829 2.060829 2.339088

0.25 50.26548 0.5 0.5 4.074843 2.056449 2.056449 2.335025

0.25 50.26548 0.75 0.9 5.054326 2.045488 2.045488 2.329569

0.25 50.26548 1 1.2 6.055629 2.036454 2.036454 2.323765

0.25 50.26548 1.5 8 8.059576 2.005754 2.005754 2.31318

0.25 50.26548 2 10 10.06146 2.002458 2.002458 2.296911


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Documents

Optimization of secondary suspension of piecewise linear vibration isolation systems