Non-Linear Characteristics of a Pendulum System829348/FULLTEXT01.pdf1 Non-Linear Characteristics of a Pendulum System Wei Zongyuan Pir Aamir Ali Shah V. K. C Varma Rudraraju Department

Master's Degree Thesis ISRN: BTH-AMT-EX--2010/D-06--SE

Supervisor: Andreas Josefsson and Ansel Berghuvud, BTH

Department of Mechanical Engineering Blekinge Institute of Technology

Karlskrona, Sweden

2010

Wei Zongyuan Pir Aamir Ali Shah

V. K. C Varma Rudraraju

Non-Linear Characteristics of a Pendulum System

1

Non-Linear Characteristics of a Pendulum System

Wei Zongyuan Pir Aamir Ali Shah

V. K. C Varma Rudraraju

Department of Mechanical Engineering

Blekinge Institue of Technology

Karlskrona, Sweden

2010

Thesis submitted for completion of Master of Science in Mechanical Engineering with emphasis on Structural Mechanics at the Department of Mechanical Engineering, Blekinge Institute of Technology, Karlskrona, Sweden.

Abstract: In this master thesis, the dynamics of a moving point pendulum and coupled pendulum system are analyzed. First the time simulation is performed. The periodic and non-periodic states of the system are analyzed with varying fundamental forcing frequencies and amplitudes. The amount of distortion in the response is studied. The type of non-linearity is detected with varying force levels and is found to be similar to a softening spring. An experimental test is carried out with a coupled beam-pendulum system and the measurement results agree well with simulation results.

Keywords: Nonlinearity, Coupled SDOF-pendulum system, Distortion, Simulation, Frequency Response Function.

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Acknowledgements

This work was carried out at the department of Mechanical Engineering, Blekinge Institute of Technology, Karlskrona, Sweden, under the supervision of Andreas Josefsson and Dr. Ansel Berghuvud.

We would like to thank Andreas Josefsson and Dr. Ansel Berghuvud for guiding us throughout the thesis. Especially we would like to thank Andreas Josefsson for helping us with a lot of valuable discussions and guidance during the whole thesis work.

Karlskrona, August 2010.

Wei Zongyuan

Pir Aamir Ali Shah

V K C Varma Rudraraju

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Table of contents:

Contents:

1. Notations 5 2. Introduction 8

2.1 Purpose 8 2.2 Background 8

3. Theoretical modeling 10 3.1 Introduction 10 3.2 Equations for a simple pendulum 10 3.3 Parametrically excited system 13

3.3.1 Parametric excitation 13 3.3.2 Equations for the simple pendulum suspended by a moving pivot 14

3.4 Equations for a pendulum coupled with SDOF system 17 4. Simulation of Motions 21

4.1 Introduction 21 4.2 Dynamic response of a simple pendulum 21 4.3 Dynamic response of a simple pendulum suspended

by a moving pivot 25 4.4 Dynamic response of a simple pendulum coupled

with SDOF system 29 5. Harmonic Generation 36

5.1 Introduction 36 5.2 Distortion affect 37

5.2.1 Harmonic distortion in the moving pivot pendulum 39 5.2.2 Harmonic distortion in the coupled SDOF system 48

6. Estimation of Frequency Response Function 57 6.1 Introduction 57 6.2 Linearization of differential equations 57 6.3 Changes in the FRF with respect to input force 63

6.3.1 Measurements for Non-Linear systems 63

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6.3.1.1 Transient Excitation 63 6.3.1.2 Harmonic Excitation 76

7. Experimental Work 72 7.1 Experimental setup 72 7.2 Characteristics of beam and pendulum 73 7.3 Changes in the FRF with respect to input force 77 7.4 Higher harmonics 82

7.4.1 Measurement with the sweeping frequency 83 7.4.2 Measurement with sinusoidal input 84

8. Conclusions and Future Work 89 9. References 91

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1. Notations

A Area of whole frequency range

A1 Area of the fundamental forcing frequency

at Tangential acceleration (m/s2)

ax Absolute acceleration in horizontal direction (m/s2)

bh Horizontal damper coefficient

C Damper coefficient

F Force (N)

Fc Force due to damper (N)

Fd Fundamental Frequency of excitation signal (Hz)

Fj Drag force (N)

Fk Force due to spring (N)

Fn Normal force (N)

Fh External force (N)

f frequency (Hz)

g Gravitational acceleration (m/s2)

j Damping coefficient (kg·m2/s)

K Spring constant (N/m)

L Length of pendulum wire (m)

M Mass of the SDOF mass (kg)

m Mass of the pendulum (kg)

P1 Power of Model signal as superposition of Harmonic

P2 Power of the simulated response

s Arc length of the pendulum (m)

T Time period of the pendulum (s)

t time variable (s)

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Δt instantaneous time (s)

x Absolute distance (m)

xp Absolute displacement in horizontal direction (m)

Absolute velocity (m/s)

Absolute acceleration (m/s2)

y output signal

θ Angle swept by the pendulum (rad)

Angular velocity (rad/s)

Angular acceleration (rad/s2)

ω Angular frequency (Hz)

ω0 Undamped natural frequency (Hz)

τ time invariant (s)

τj Drag torque (N.m)

ζ Relative Damping

phase (rad)

Indices: B(s) System Impedance

Crit1 Power Ratio

Crit2 Magnitude Ratio

F(s) Laplace domain representation of forcing function

H Linear system

X(s) Laplace domain representation of system response

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Abbreviations: 2DOF Two degree of freedom

SDOF Single degree of freedom

FRF Frequency response function

ODE Ordinary Differential Equation

RMS Root Mean Square

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2. Introduction

2.1 Purpose

In the design of engineering structures, the structural analysis of the system has a vital role. Because of the flexibility of linearization of the system and the mathematics involved in linear systems, most of the engineers prefer to work with linear systems. However the fact is that most of the systems around us are nonlinear. Therefore, a lot of research is aimed at finding the characteristics of nonlinear dynamics as it cannot be generalized as linear systems. The effect of the non-linear elements complicates the problem to predict the dynamic behavior in real physical systems. During the design stage, it is necessary to do a complete analysis of the system before fabrication. A bad combination of parameters and excitation can lead to an unstable response which may not be desirable.

In this thesis, the rich dynamic behavior of pendulum coupled with a mass-damper-spring system is analyzed. The pendulum oscillations introduce nonlinearities into the system. The time and frequency simulations are compared with the experimental results.

Initially, some theoretical background about the time analysis, frequency analysis, autoparametric resonance and nonlinearity detection is explained after which the simulated results are presented in chapter 4. Distortion effects and estimation of Frequency Response Function (FRF) are done in chapters 5 and 6 respectively. Experimental results are presented in chapter 7. The conclusions are drawn in chapter 8 and future work is proposed.

2.2 Background

Nowadays renewable energies have been highly regarded in our lives. Wave power devices extract energy directly from surface waves or from pressure fluctuations below the surface. In Sweden, there are several projects which have been placed on wave power. The studied system in this project is related to a wave energy device which is currently being developed. This device includes a coupling between horizontal/vertical motion and pendulum motion.

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Based on the master thesis, Dynamic Characteristics of a Pendulum System done by Andres Lorente [3], this thesis work is focusing on the nonlinearity characteristics of coupled pendulum-SDOF system. A simplified model has been taken as shown in Figure 2.1.

Figure 2.1. The coupled pendulum-SDOF system.

In order to better understand dynamic behavior of the coupled pendulum-SDOF system, a simplified model is taken and studied. A theoretical model plays a key role in simulating how the real system behaves. It is good to predict the resonances in reality. The studied system has a very rich dynamic behavior due to its nonlinearity. Thus the time simulation, the detection of type of non-linearity and the generation of higher harmonics of the proposed system will be emphasized in this work.

M Fx(t)

K

m

(t)

C

L

x

Y

10

3. Theoretical Modeling

3.1 Introduction

An important step in the study of the motion of physical systems is to derive the dynamic equations. By using Newton’s second law and equilibrium conditions, the differential equations of the proposed SDOF system can be obtained.

The aim of this chapter is to derive the equations for the coupled pendulum-SDOF system which consisting of mass spring damper system and a pendulum system attached to the mass as shown in Figure 2.1 above.

To derive the equations of the coupled pendulum system, first a simple pendulum is considered, afterwards the simple pendulum suspended by a moving pivot that can move in the horizontal direction on x-axis is considered, in which parametric excitation is also introduced. At last, based on that equation the moving pivot is substituted with a SDOF system. Then the differential equations for SDOF system (mass spring damper system) attached to the simple damped pendulum are derived.

3.2 Equations for A simple pendulum

First the differential equations of motion of a simple pendulum are derived. Generally for deriving the equation of a simple pendulum system the mass of the pendulum wire is neglected and pendulum with single degree of freedom is assumed.

The variables used in deriving the equation are time ‘t’ measured in seconds and angle ‘θ’ made by the pendulum with the equilibrium position which is measured in radians. And also two constants, length of the pendulum ‘L’ measured in meters and mass of the pendulum ‘m’ measured in kilograms are used.

For usual physics based problems the analysis of forces in the system are considered for derivation. The mass of the pendulum experiences two

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different accelerations, normal acceleration and gravitational acceleration [3].

Figure 3.1. A simple pendulum.

Normal acceleration is the acceleration that the angle ‘θ’ experiences with the passage of time. It is found by taking the second derivative of the distance swept by the angle ‘θ’. If θ is measured in radians it is known that:

(3.1)

Here is the length of the pendulum. Now the second derivative of S becomes:

(3.2)

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Here is the tangential acceleration. Also it is commonly known that by Newton’s second law . Therefore the force becomes:

(3.3)

In the tangential acceleration, the weight of the mass due to gravity is considered. Taking the tangential component of gravity, the force using Newton’s second law can be written as:

(3.4)

Negative signs indicate that the mass is moving in the opposite direction of the equilibrium.

Now consider the equilibrium condition that is equating the Equations (3.3) and (3.4).

(3.5)

Dividing both sides by · , the above equation becomes:

(3.6)

So the differential equation governing the motion of simple pendulum is given by:

0 (3.7)

For small angles of θ, Equation (3.7) can be written as:

· 0 (3.8)

If damping is introduced then the Equation (3.8) becomes:

0 (3.9)

A constant proportional to rotational velocity is introduced in order to add damping to the system.

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3.3 Parametrically excited system

3.3.1 Parametric excitation

In the last section, the differential equations for a simple pendulum has been derived and explained. The system which has been studied in section 3.2 is linear and time-invariant when is very small.

In many cases, vibrations can be described as mass-spring-damping systems, the coefficients corresponded to the properties such as mass, stiffness and damping do not vary with time. However, in some cases those coefficients are dependent on time, which leads to parametric excitations [5]. In this section a simple pendulum suspended on a moving point is introduced.

Now the whole system is turned into a nonlinear system which actually is a parametrically excited system. At first the model is created and then the differential equations are derived.

Consider the simple pendulum suspended by a point which can move in horizontal direction. The sketch of the theoretical model is created as shown in Figure 3.2.

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Figure 3.2. Pendulum on a moving pivot.

3.3.2 Equations for a simple pendulum suspended by a moving point

In this case the pivot of the pendulum is moving in horizontal motion. First the differential equation of the above system is derived. To obtain the equations of the motion, forces acting on the mass are studied. In Figure 3.3, all the forces acting on the mass of the pendulum are shown. All these forces can be decomposed into two directions which are normal and tangential respectively. Now it is assumed that the motion of the pivot can be described as with respect to time.

m

θ

x

Y

15

Figure 3.3. All the forces acting on the mass.

The movement of the pendulum along its path of motion is divided into normal acceleration and tangential acceleration which are given as follows.

Normal acceleration:

(3.10)

Tangential acceleration:

(3.11)

By using Newton’s second law and equilibrium conditions the following equations are obtained.

0 (3.12)

0 (3.13)

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Here is a viscous damping force.

Substitute and also substitute in the Equation (3.13)

Then the Equation (3.13) can be rewritten as:

0 (3.14)

Now dividing the whole equation by · , the following differential equation is obtained.

0 (3.15)

As the motion of the pivot is moving in horizontal axis, it is assumed that the displacement of pivot is described by the following equation.

(3.16)

Where angular frequency ω=2πf.

Differentiating the above equation the following equation for velocity is obtained.

(3.17)

Again differentiating the above equation the acceleration of the pivot moving in horizontal direction is obtained.

(3.18)

This acceleration is inserted in Equation (3.15) which can be rewritten as:

0 (3.19)

0 (3.20)

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3.4 Equations for a pendulum coupled with SDOF system

After studying a simple pendulum suspended by a moving point, the final theoretical model which contains a single degree of freedom system (SDOF) coupled with a simple pendulum is considered. The pendulum suspends on a spring damper mass. The schematic figure of the theoretical model has been shown in Figure 2.1.

The motion of the pendulum and the motion of mass damper system is relative to each other. Here the mass of SDOF systems moves horizontally and the pendulum swings freely. As the system has two different motions that are motion due to pendulum and motion due to spring damper system, two equations are obtained [3].

Before deriving the differential equations, following critical assumptions are mentioned in order to establish the theoretical model.

1. The length of the pendulum wire is not variable.

2. The mass of the SDOF system is only moved in one direction i.e., in horizontal direction.

To derive the differential equations, the system is separated into two parts. First the forces acting on the pivot of the pendulum are considered. These forces are shown in the following Figure 3.4.

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Figure3.4. Forces acting on pendulum system.

Form the above Figure 3.4 the total force acting on the horizontal direction is given by the equation:

(3.21)

Here is the absolute acceleration of the pendulum in horizontal direction and T is the tension force by the wire.

The absolute displacement in horizontal direction is given by:

(3.22)

Differentiating the Equation (3.22) twice, the following absolute acceleration in horizontal direction is obtained.

mg

T

Fj

x

θ

Y

m

L

19

(3.23)

Substituting Equation (3.23) into Equation (3.21), the following equation is obtained:

(3.24)

Now, the forces acting on the mass of SDOF are concerned as in Figure 3.5 shown below.

Figure3.5. Coupled spring damper system.

In the above Figure 3.5, Fk and Fc are the forces from the spring and damper respectively in horizontal direction and Fh here is the external force. Fn is the supporting force from the wave. As the system is coupled, the motion of spring damper system is also affected by the motion of the pendulum. Thus tension T is added to the above SDOF system. By using Newton’s second law and equilibrium conditions the following equations are obtained.

In horizontal direction:

(3.25)

where,

M Fc FK Fh

Y

x

T

20

(3.26)

(3.27)

Substituting Equation (3.24) into Equation (3.25), the following equation is obtained.

(3.28)

The differential equation of motion of the pendulum is similar to the equation of motion of simple pendulum mentioned in section 3.32 which can be shown below.

0 (3.29)

Finally by substituting , in Equation (3.28), which is derived in the previous section, two equations for the system can be obtained.

(3.30)

0 (3.31)

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4. Simulation of the motions

4.1 Introduction

To study the nonlinearity of the coupled pendulum system, it is very important to simulate the dynamic motions of the system first. From the previous chapter, the differential equations have been derived for each system. In this chapter, the work is mainly focused on applying these equations in MATLAB. The motions of the system are analyzed in time domain and frequency domain. ODE functions [7] in MATLAB are applied to solve the nonlinear differential equations.

4.2 Dynamic response of a simple pendulum

From the earlier chapter 3.2, the theoretical model for a simple pendulum has been created and also the equations for the motions have been derived. By using Equation (3.9) the dynamic response can be calculated in MATLAB. The parameters for the model are shown in Table 4.1.

Table 4.1. Parameters for the simple pendulum.

Parameter value

Mass, m [kg] 1

Length of the pendulum, L [m] 0.25

damping coefficient due to the friction, j [Nm·s]=[Kg·m2/s] 0.02

Gravity Acceleration, g [m/s2] 9.8

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Figure 4.1. Simulation of the simple pendulum without damping.

0 10 20 30 40 50 60 70 80 90 100-1

-0.5

0

0.5

1

Time [s]

Ang

le o

f the

Pen

dulu

m [r

adia

n] Simple Pendulm without damping Simulation

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

Frequency Domain [Hz]

Mag

nitu

de

Simple Pendulm without damping Simulation

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Figure 4.2. Simulation of the simple pendulum with damping.

Plotted in Figure 4.1 and Figure 4.2 are the time simulations of the simple pendulum with the initial condition 0.5 radians (30°). As seen in Figure 4.1, it shows that when the damping is not considered the motion of the pendulum will never decay.

In both figures the spectra is plotted. As is shown in these two figures, the frequencies of the peaks are all around 1.00 Hz, which means the natural frequency for the pendulum system is 1.00 Hz which fulfills the Equation

(3.11), = · .

..

1.00 Hz. From Equation (3.11), if the

length of the wire is increased, the natural frequency of the pendulum is decreased.

0 10 20 30 40 50 60 70 80 90 100-0.5

0

0.5

1

Time [s]

Ang

le o

f the

Pen

dulu

m [r

adia

n] Simple Damped Pendulm Simulation

0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04


Mag

nitu

de

Simple Damped Pendulm

24

Figure 4.3. Simulation of the simple pendulum after increasing the length of the pendulum.

As seen in Figure 4.3, the length is increased to 0.35 m, which causes the natural frequency decline to 0.84 Hz. Moreover, in Figure 4.3 it can be observed from the movement of the pendulum that the decaying time has been prolonged by increasing the length of the pendulum, since the relative damping is defined as ζ √L

√ g .

0 10 20 30 40 50 60 70 80 90

-0.5

0

0.5

Time [s]

Ang

le o

f the

Pen

dulu

m [r

adia

n] Simple Damped Pendulm SimulationX: 0Y: 0.5236

0 1 2 3 4 5 6 7 8 9 100

0.02

0.04

0.06

0.08

X: 0.8407Y: 0.06355


Mag

nitu

de

Simple Damped Pendulm

25

4.3 Dynamic response of a simple pendulum suspended by a moving point.

Now the simulation for a simple pendulum suspended by a moving point is considered in this section. The interest in this simulation is to observe the motions of pendulum under the parametric excitation with different fundamental frequencies and amplitude.

Equation (3.20) is applied in MATLAB. Before doing this, an input signal acting as a sine function with a certain fundamental frequency and amplitude is assumed to be applied on the pivot.

The parameters for the studied system are shown in Table 4.2

Table 4.2. Parameters for the pendulum system.

Parameter value

Mass, m [kg] 1




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Figure 4.4. A periodic input signal.

Figure 4.4 shows a sinusoidal input force which is described in Equation (3.16), with a fundamental frequency 0.1 Hz and amplitude 0.5 N.

0 20 40 60 80 100 120 140 160 180 200-1

-0.5

0

0.5

1

Time [s]

Dis

plac

emen

t [m

]Input Signal

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8


Mag

nitu

de

27

Figure 4.5. Simulation of the simple pendulum with a moving pivot.

Plotted in Figure 4.5, are the time responses of the pendulum and the corresponding spectra. From time domain, it can be found that the transient response has a small influence on the motion of the pendulum which takes about 20 seconds to make the pendulum turn into steady state. As observed in the frequency domain, it has a peak at 0.1 Hz equal to the fundamental frequency of the input displacement which means the fundamental frequency is dominating the motion.

0 20 40 60 80 100 120 140 160 180 200-0.05

0

0.05

Time [s]

Ang

le o

f the

Pen

dulu

m [r

adia

n] Simple Damped Pendulum with a moving pivot

0 1 2 3 4 5 6 7 8 9 100

0.005

0.01

0.015

0.02


Mag

nitu

de

28

Figure 4.6. Simulation when the fundamental frequency is equal to the natural frequency.

.

Now the fundamental frequency is increased same as the natural frequency i.e., 1 Hz. As it is seen in Figure 4.6, from 0 to 200 seconds the movement of the pendulum continuously turns into big amplitude which is in the unstable state since it is not periodic. Obviously, when increasing the amplitude of the input signal to a certain value and keeping the fundamental frequency same as the natural frequency of the pendulum, the motion of the pendulum will be uneven. The motion can be stable or unstable depending on the amplitude and fundamental frequency of the excitation signal. In next chapter, more details about the distortion of the parametrically excited systems will be studied.

0 20 40 60 80 100 120 140 160 180 200-20

0

20

40

60

Time [s]

Ang

le o

f the

Pen

dulu

m [r

adia

n] Simple Damped Pendulum with a moving pivot

0 1 2 3 4 5 6 7 8 9 100

10

20

30


Mag

nitu

de

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4.4 Dynamic response of a pendulum coupled with SDOF system

At last section, a pendulum suspended by a moving point has been studied. Based on that in this section the simulations for a pendulum coupled with SDOF system will be done. As seen in Figure 3.7, an external force Fh is applied on the mass of SDOF system in horizontal axis. It is important to see how the pendulum and the mass behave under the excitation force.

The parameters which are used in time simulation are shown in Table 4.3.

Table 4.3. Parameters for the coupled system.

Parameter Value

Pendulum Mass, m [kg] 1

SDOF mass, M [kg] 2


Horizontal spring coefficient, Kh [N/m]=[Kg/g2] 500

Horizontal damping coefficient, bh [N·s/m]=[Kg/s] 0.2

Gravity, g [m/s2] 9.8

Angular damping coefficient, j [Nm·s]=[Kg·m2/s] 0.02

Equation (3.30) and (3.31) from chapter 3.4 are used in the simulation in MATLAB. In the first simulation, the external force is neglected. An initial condition is taken in which the pendulum is hold on to 0.5 radians and released.

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Figure 4.7. The time response of the studied system without external force Fh.

In Figure 4.7 (a) and (b), the movements of both the SDOF mass and the pendulum are plotted. The SDOF mass takes a bit more seconds than the pendulum to totally die out. In this case there are two damping coefficients, j and bh which will affect the total damping of the system.

Now one of the damping coefficients is increased and the movement of the system will be observed.

First damping coefficient j is increased to 0.1.

0 50 100-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Time [s]

Hor

izon

tal p

ositi

on o

f the

mas

s [m

](a)

0 50 100-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time [s]

Ang

le o

f the

pen

dulu

m [r

ad]

(b)

31


Figure 4.8 shows the decaying time for both two mass has been shorten after increasing damping j. However the pendulum has been more affected.

Then, j is kept as 0.02 and bh is increased to 2.

0 50 100-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Time [s]

Hor

izon

tal p

ositi

on o

f the

mas

s [m

](a)

0 50 100-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time [s]

Ang

le o

f the

pen

dulu

m [r

ad]

(b)

32


As a result from Figure 4.9 it can be detected that the decaying time for both masses is also decreased. In comparison with Figure 4.8, the SDOF in this case is more affected. It is concluded that if either damping j or bh is increased or decreased the motion of the whole system is affected.

Now the external force is concerned. As in last section, the excitation signal is again defined as sinusoidal signal with a certain fundamental frequency and amplitude. The parameters in Table 4.4 are used.

0 50 100-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Time [s]

Hor

izon

tal p

ositi

on o

f the

mas

s [m

](a)

0 50 100-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time [s]

Ang

le o

f the

pen

dulu

m [r

ad]

(b)

33

Table 4.4. Parameters for the coupled system.

Parameter Value


SDOF mass, M [kg] 2



Horizontal damping coefficient, bh [N·s/m]=[Kg/s] 2



Figure 4.10. The external force Fh.

0 20 40 60 80 100 120 140 160 180 200-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Time [s]

Am

plitu

de [N

]

The excitation force

34

Figure 4.10 shows an external force in time domain with frequency 0.2 Hz and amplitude 0.5 N.

Figure 4.11. The time response of the studied system excited by an external force Fh.

As seen from Figure 4.11, the motions of the SDOF mass and the pendulum are shown. Figure 4.11(a) demonstrates the movement of the mass oscillates between +0.0015 m and -0.0015 m and in Figure 4.11(b) the motion of the pendulum has a transient response in the beginning around 0 to 25 seconds. However, as shown in Figure 4.11(a) and 4.11(b) the motions are quite small because of the small amplitude of the external force.

Here is an example about the unstable state of the pendulum in this system, when the fundamental frequency of the excitation force is increased to 2.5 Hz and the amplitude is increased to 5 N.

0 50 100 150 200-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5x 10

-3

Time [s]

Hor

izon

tal p

ositi

on o

f the

mas

s [m

]

(a)

0 50 100 150 200-8

-6

-4

-2

0

2

4

6

8x 10

-3

Time [s]

Ang

le o

f the

pen

dulu

m [r

ad]

(b)

35

Figure 4.12. The time response of the studied system with an external force Fh.

As seen in Figure 4.12 the motion of the pendulum has big amplitude during 0 to 40 seconds which actually is rotating around the pivot, however after 40 seconds the pendulum turns into static. It manifests that the transient response has a big affect to motion of the pendulum.

In next chapter, more simulation about distortion will be done and also harmonic distortion will be introduced to check the stability by changing the fundamental frequency and amplitude.

0 50 100 150 200-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Time [s]

Hor

izon

tal p

ositi

on o

f the

mas

s [m

](a)

0 50 100 150 200-5

0

5

10

15

20

Time [s]

Ang

le o

f the

pen

dulu

m [r

ad]

(b)

36

5. Harmonic Generation

5.1 Introduction

In the design phase of physical oscillation system, stability analysis is very important. An unstable system leads to many problems, for instance damage to the mechanism or getting out of control. Therefore, for the given models the presence of any higher harmonics [10] is checked.

In this chapter the moving point pendulum and mass-spring-damper system coupled with the pendulum is analyzed for its stable and unstable regions for the detection of the nonlinearities involved in the system. Since there is no unique approach to deal with nonlinear problems either analytically or experimentally several approaches are proposed to experiment in order to ascertain that whether the system can be classified as linear or nonlinear. It can be of extreme help if the modal analysis techniques are employed in order to test for linearity in the nonlinear techniques. But if these principles of nonlinearities are ignored, further analysis of the data will encounter errors.

For a linear system, it must fulfill the criteria [1]: 1) the system shall follow the superposition and 2) the system shall follow homogeneity.

(5.1)

(5.2)

So, the linear systems theory can be broke down to check for the nonlinearities and instabilities in the system [5].

There are various ways to check if a system is linear or not. In this chapter, the amount of distortion in the response is examined.

37

5.2 Distortion Affect

Harmonic or waveform distortion is one of the best suited methods to check for the nonlinearities in the system. It is a consequence of superposition principle in which if the excitation input to a linear system is a mono-harmonic signal, i.e., a sine or cosine wave of frequency ω, the response will also be a mono-harmonic signal having the same frequency in steady-state. When the linear systems are excited by a sinusoidal force, its steady-state response will be sinusoidal with the same frequency as that of input excitation. Whereas, when a nonlinear system is excited by sinusoidal force, the response of the system might contain sub-harmonics or super-harmonics. In extreme conditions, the response will show chaotic behavior, which does not show any periodic behavior. The proof [5] is as follows.

Suppose sin is the input to a linear system. First of all, it is observed that

, implies that , and .

This is because superposition demands that:

(5.3)

And follows in the limit as 0. (Note also that there is an implicit assumption of time invariance here, namely that implies for any τ.) Again, by superposition:

(5.4)

Where is a constant

So taking and gives:

(5.5)

Now, as sin :

0 (5.6)

38

In the steady state, a zero input to a linear system results in a zero output. It therefore allows from Equation (5.3) that:

0 (5.7)

And the general solution of this equation is:

(5.8)

The implication is that such signal will not distort if it is passed through a linear system. Therefore when a signal is passed through a non-linear system additional harmonics are added to the original frequencies.

Figure 5.1. A black box used in simulations. A harmonic input signal gives a distorted output signal containing higher harmonics from an unknown

non-linear system [4].

Two ways to check the distortion percent are used in this work.

The first way is by using the ratios of the power of the response signal.

1 100% (5.9)

In this method, the response is modeled as a sum of orthogonal sinusoids. Where is the power of the simulated response. P1 represents the power of the signal produced by modeling the signal as a superposition of

39

harmonics. 1 is the ratio of and . In this thesis work, this method is defined as ‘power ratio method.

The second way is by using the magnitude ratios instead of the power ratios.

2 100% (5.10)

In the second method, the ratio of the sum of magnitude around the fundamental frequency to the sum of the whole area including the fundamental frequency is taken. The region apart from the fundamental frequency have got energy distributed due to the higher harmonics in the signal showing the nonlinearity in the system, so the sum of that whole area including the fundamental frequency can be taken as ‘A’, where the region around fundamental frequency of the system with some tolerated frequencies about ±1% of the fundamental frequency is taken as ‘A1’. This method here is described as ‘magnitude ratio method’.

In next section both of these distortion methods are simulated, first the moving point pendulum is analyzed and then coupled pendulum-SDOF is analyzed.

5.2.1 Harmonic Distortion in the Moving Pivot Pendulum

In this section the distortion levels of the moving point pendulum are checked as discussed in chapter 3.2. By using the differential Equations (3.16) and (3.20) from previous chapter, the result can be simulated in MATLAB. For the input displacement, amplitude input array and a fundamental frequency array are set as shown in Table 5.1. As studied in the previous chapter 3.2, the natural frequency for the pendulum is around 1.1 Hz.

The given parameters are shown in the Table 5.1.

40

Table 5.1. Parameters for the moving point pendulum.

Parameter value

Mass of the pendulum, m [kg] 1




First a specific fundamental frequency and amplitude are used with the input signal with the form . The motion of the system can be simulated in MATLAB.

Table 5.2. Amplitude and fundamental frequency.

Parameter value

Amplitude of input signal, a1 [m] 0.5

Fundamental frequency, Fd [Hz] 0.5

41

Figure 5.2. Time response of the pendulum.

Figure 5.2(a) and 5.2(b) are the time response for the pendulum and the time response after removing transient response respectively.

0 100 200 300 400 500 600-2

-1

0

1

2

Time [s]

Ang

le o

f the

Pen

dulu

m [r

adia

n] (a)

250 300 350 400 450 500 550 600-1

-0.5

0

0.5

1

Time [s]

Ang

le o

f the

Pen

dulu

m [r

adia

n] (b)

42

Figure 5.3. Motion of the pendulum in frequency domain.

In Figure 5.3, the highest peak is around at 0.5 Hz which is the fundamental frequency. As discussed before about linear and nonlinear system, from Figure 5.2 and 5.3, it can be judged that the pendulum system is regular.

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Mag

nitu

de

43

Figure 5.4. Calculation of Crit2 before cut and after cut.

The given Figure 5.4 shows the comparison of Crit1 and Crit2. In Figure 5.4(a), the red region shows sum of the magnitude of fundamental frequency with a tolerance region of 2% and denoted as A1 and the blue region is the whole region’s magnitude. It can be noticed that Crit1 is 99.69% and Crit2 is 63.8304% which indicates that Crit2 is more sensitive than Crit1. Also in Figure 5.4 (a), there is one green line called ‘cutline’ which is used to remove small ripples. It can be observed that those small ripples contain quite small value which is close to 0. However if all these values are summed it will make the dominator bigger, if these ripples are regarded as 0 value. As shown in Figure 5.4(b), after cutting these small ripples which stay below the green line, Crit2 becomes bigger than the one in Figure 5.4(a). It’s because the sum of the magnitude of the green region which is A as the dominator in function Crit2 is decreased due to removing the small ripples. But still Crit2 is a bit lower than Crit1.

0.45 0.5 0.55 0.6 0.65 0.70

0.2

0.4

0.6

0.8

Crit1=std(xnew)/std(x) =99.9378%Crit2 =A1/A=60.093%


Mag

nitu

de(a)

A,Sum of the magnitude of whole areaA1,Sum of the magnitude around FdCutline,the point below is taken to 0

0.45 0.5 0.55 0.6 0.65 0.70

0.2

0.4

0.6

0.8

Crit1=std(xnew)/std(x) =99.9378%Crit2 =A1/A=81.8167%


Mag

nitu

de

(b)

A,Sum of the magnitude of whole areaA1,Sum of the magnitude around Fd

44

Now an unstable state of the system is checked. The fundamental frequency is set to coincide with the natural frequency of the pendulum. Table 5.3 shows the parameters for the input signal.


Parameter value

Amplitude of input signal, a1 [m] 0.5

Fundamental frequency, Fd [Hz] 1.0

Figure 5.5. Time response of the pendulum.

0 100 200 300 400 500 600-120

-100

-80

-60

-40

-20

0

20

40

60

Time [s]

Ang

le o

f the

Pen

dulu

m [r

adia

n]

45

Figure 5.5 indicates that the system is unstable since a periodic solution is not found. Clearly, Crit1 or Crit2 will be very low for this case since a large amount of the power is at the DC level.

The distortion map can be plotted if the amplitude and fundamental frequencies are applied as an array respectively as shown in Table 5.4.

Table 5.4. Array of Amplitude and Fundamental Frequency.

Parameter Value

Amplitude input array, a1 [m] 0.1:0.1:5

Fundamental frequency, Fd [Hz] 0.1:0.1:5

Figure 5.6. A color map of distortion by using method Crit1to find the periodic and non-periodic solution.

Amp [m]

Fd [H

z]

0.5 1 1.5 2 2.5 3 3.5 4 4.5

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

46

Figure 5.7. A color map of distortion by using method Crit2to find the periodic and non-periodic solution.

Plotted in Figure 5.6 and 5.7 are the color maps which shows the amount of distortion for the pendulum with a moving pivot. The amount of distortion can be very high or very low depending on the input values ‘ ’ and ‘Fd’. That is even at low amplitude input and at the fundamental frequency near the resonance frequency of the pendulum, there is high distortion, but when the amplitude is increased to 1 m, it has a region with very low distortion for almost the whole range of frequency in Crit1 and Crit2. But when ‘ ’ is increased then it has very high distortion above the resonance frequency of the pendulum. When ‘ ’ is reached to 4 m to 4.5 m, a new region with low distortion is found.

Now, when the angular damping coefficient j is increased to 0.1, the following result is shown in Figure 5.8.

Amp [m]

Fd [H

z]

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

47

Figure 5.8. A color map distortion by using method Crit1with damping coefficient j=0.1.

Amp [m]

Fd [H

z]

0.5 1 1.5 2 2.5 3 3.5 4 4.5

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

48

Figure 5.9. A color map of distortion by using method Crit2 with increased damping coefficient j=0.1.

From Figure 5.8 and 5.9, it is found that with increasing damping coefficient, the region with low distortion will be extended to higher frequencies.

5.2 .2 Harmonic distortion in Coupled Pendulum-SDOF

In this section, the Coupled Pendulum-SDOF system with the data given in the Table 5.5 is analyzed similarly as previous section.

Amp [m]

Fd [H

z]

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

49

Table 5.5. Parameters of the system.

Parameter value


SDOF mass, M [kg] 2



Horizontal damping coefficient, bh [N·s/m]=[Kg/s] 2



The harmonic input is given as:

· sin 2 · cos 2 (5.11)

Similarly, a specific fundamental frequency and amplitude for the input force are shown in Table 5.6


Parameter value

Amplitude of input signal, a1 [m] 1

Fundamental frequency, Fd [Hz] 2

50

Figure 5.10. (a) The motion of the SDOF mass in time domain. (b) The motion of the pendulum in time domain.

0 200 400 600-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

Time [s]

Hor

izon

tal p

ositi

on o

f the

mas

s [m

](a)

0 200 400 600-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Time [s]

Ang

le o

f the

pen

dulu

m [r

ad]

(b)

51

Figure 5.11. (a) Time response for the pendulum. (b) Time response in steady state.

0 100 200 300 400 500 600-0.1

-0.05

0

0.05

0.1

Time [s]

Ang

le o

f the

Pen

dulu

m [r

adia

n] (a)

200 250 300 350 400 450 500 550 600-0.04

-0.02

0

0.02

0.04

Time [s]

Ang

le o

f the

Pen

dulu

m [r

adia

n] (b)

52

Figure 5.12. Calculation of Crit2 before cut and after cut.

The simulation results as plotted in above figures show that the coupled system is in the stable condition. Both Crit1 and Crit2 are quite close to 100%.

By using the fundamental frequency in the range of 1 to 5 Hz and amplitude from 1 to 5, the color maps of the stability can be simulated. The parameters are shown as Table 5.7.

1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 2.250

2

4

x 10-3

Crit1=std(xnew)/std(x) =100%Crit2 =A1/A=62.589%


Mag

nitu

de

(a)

A,Sum of the magnitude of whole areaA1,Sum of the magnitude around FdCutline,the point below is taken to 0

1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 2.250

2

4

x 10-3

Crit1=std(xnew)/std(x) =100%Crit2 =A1/A=91.2296%


Mag

nitu

de

(b)

A,Sum of the magnitude of whole areaA1,Sum of the magnitude around Fd

53

Table 5.7. Array of Amplitude and Fundamental Frequency.

Parameter value

Amplitude input array, a1 [m] 0.1:0.5:10

Fundamental frequency, Fd [Hz] 0.1:0.1:5

Figure 5.13. A color map for distortion by using Crit1.

X= 3.6Y= 2.5

Level= 0.80773

Amp [N]

Fd [H

z]

The distortion percent of the SDOF-Pendulum

1 2 3 4 5 6 7 8 9

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

54


Figure 5.13 and Figure 5.14 show the distortion color maps with the increasing force level and frequencies. As seen in both figures, the color maps for both methods, Crit1 and Crit2 are pretty much similar but the average distortion percent in Figure 5.14 is a bit lower than that in Figure 5.14. As seen that the color in Figure 5.14 is a little darker than that in Figure 5.13. It is observed that magnitude ratios method is more sensitive than power ratio method. It is clear that both figures have two narrow cracks located in 2.5 Hz and 1 Hz respectively. The first crack is much bigger than the later one. The frequency of the first is quite close the resonance frequency of the SDOF system however the frequency of the latter is almost the same as the natural frequency of the pendulum. Cracks indicate that the region with high distortion appears when the amplitude of excitation force is increased to a certain amplitude with a certain fundamental frequency. Also when the force amplitude is increased higher

Amp [N]

Fd [H

z]

X= 3.1Y= 2.5

Level= 0.94674

1 2 3 4 5 6 7 8 9

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

55

distortion is observed. Therefore, to avoid the highly nonlinear affect, the input frequency should be avoided from those frequencies.

Now the damping coefficients are increased as shown in Table 5.8 and the results are observed.

Table 5.8. Damping coefficients.

Parameter value

Horizontal damping coefficient, bh [Ns/m]=[Kg/s] 2

Angular damping coefficient, j [Nm·s]=[Kg·m2 /s] 0.1

Figure 5.15. A color map for harmonic distortion by using Crit1.

Amp [N]

Fd [H

z]

The distortion percent of the SDOF-Pendulum

X= 4.1Y= 2.5

Level= 0.74618

1 2 3 4 5 6 7 8 9

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

56


As seen from the above figures the distortion maps haven’t been changed more, whereas the damping coefficient has been increased. However the small crack which is located in 1 Hz has disappeared and the crack in 2.5 Hz gets a bit less narrow than that before increasing the damping. It manifests that changes in damping coefficients will make the regions with high distortion move in the amplitude axis.

X= 3.1Y= 2.5

Level= 0.94847

Amp [N]

Fd [H

z]

1 2 3 4 5 6 7 8 9

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

57

6 Estimation of Frequency Response Function (FRF)

6.1 Introduction

In dynamic systems, the frequency response function or FRF is the most widely used method to check the input-output properties. The FRF summarizes most of the information of the dynamics of a structure resonances, anti-resonances, modal density and phase. An FRF is only defined for a linear system and no such simple input-output relationships exist for a nonlinear system. Despite this, an approximate FRF can still be utilized when studying nonlinear systems. In this case, the FRF depends on the excitation amplitude. Typically, resonance frequencies and damping will change when the excitation force is increased. The shape of FRF constructed at different excitation levels can give information about the type of nonlinearity in the system. In the linear systems theory of FRF’s, the resonance frequencies do not change with varying excitation magnitudes, so the property of linearity is utilized.

In this chapter, the type of nonlinearity involved in the given system is observed. First the linearization of the given system is simulated for the pure FRF and then compared with the approximate nonlinear FRF at different force levels. The latter is referred to as composite FRF and it will be simulated with two types of input excitations: 1) Impulse excitation and 2) Sinusoidal excitation.

6.2 Linearization of the Differential Equations

Linearization of the differential equations [2] is very useful method to obtain a closed-form solution to many of the resulting non-linear differential equations. The differential Equations (3.30) and (3.31) for a pendulum coupled with a SDOF system are recalled and rewritten as:

58

(6.1)

0 (6.2)

In this case, the system is governed by two nonlinear differential equations. In order to get theoretical FRF, linearization has to be done. Some assumptions [2] need to be made before linearization.

The angular oscillation for the pendulum is assumed to be small ( 10°),

1

Equations (6.1) and (6.2) can be linearized and rewritten as follows:

· (6.3)

(6.4)

These equations may be expressed in matrix form as:

1 0

00

(6.5)

From Equation (6.5), the following matrices are identified.

The mass matrix = 1 ,

59

The damping matrix = 0

The stiffness matrix =0

0

The force matrix

The two-degree-of-freedom system may be written as:

(6.6)

Laplace Transform of Equation (6.6) gives:

(6.7)

Here, undamped natural frequencies can be found by assuming that:

C 0 and, F s 0 (6.8)

So Equation (6.7) will become:

0 (6.9)

Now multiplying Equation (6.9) by K and divided by s ), the following equation is obtained.

0 (6.10)

So the Eigen-values can be found from Equation (6.10).

The parameters used in an example are shown in Table 6.1.

60

Table 6.1. Parameters for coupled SDOF-pendulum system.

Parameter value


SDOF mass, M [kg] 3






For the parameters given in Table 6.1, the undamped resonance frequencies using K M are:

0.9133 2.2474 Hz (6.11)

Equation (6.7) can be written as · · . is referred to as the system impedance. Then Equation (6.7) becomes:

· (6.12)

The Laplace domain can be thought of as complex frequency . Therefore, the quantities in Equation (6.7) can be represented as follows:

• - the Laplace domain (complex frequency) representation of the forcing function

• - the Laplace domain (complex frequency) representation of the system response

Equation (6.7) states that the system response is directly related to the system forcing function through the quantity . If the systems force function and response function is known, can be calculated. That is:

61

(6.13)

By defining , Equation (6.13) becomes:

· (6.14)

The quantity is known as the system transfer function. In other words, a transfer function relates the Laplace transform of the system input to the Laplace transform of the system response. From Equations (6.7) and (6.14), the transfer function can be defined as:

// · /

(6.15)

Figure 6.1. Input and output on the coupled pendulum system.

The M,C and K matrices can be obtained from Equation (6.5) , the frequency range is [0 to 10 Hz] and the inno, outno are 1,2 respectively. As seen from Figure 6.1, the estimated Mobility H11 and H21 are shown in the following Figure 6.2 and Figure 6.3.

62

Figure 6.2. Frequency Response Function for H11.

Figure 6.3. Frequency Response Function for H21.

0 1 2 3 4 5 6 7 8 9 1010-5

10-4

10-3

10-2

10-1

100

101Estimated FRF H11

Frequency (Hz)

Mob

ility

(m/N

s)

0 1 2 3 4 5 6 7 8 9 1010

-10

10-8

10-6

10-4

10-2

100

102Estimated FRF H21

Frequency (Hz)

Mob

ility

(rad

/Ns)

63

6.3 Changes in the FRF with respect to the input force

As already stated that for a linear system, the ratios between the input force and the response of a linear system is constant, according to the law of superposition [4]. For nonlinear systems the principle of superposition is not valid, therefore an indication of nonlinear behavior can be found by comparing the FRFs obtained by excitation at different force levels. Therefore, it is important to perform measurements at different force levels and different locations before a system is classified as a linear or non-linear system.

6.3.1 Measurements for Non-linear systems

In dynamic testing the test setup should be properly setup to avoid any non-linearity outside the inherent property of the system. Several different ways are used to excite the systems. The most commonly used methods in modal analysis are transient, random and sinusoidal or harmonic excitation.

6.3.1.1 Transient Excitation

Transient excitation is most commonly performed manually with an impact hammer [1]. It is a broad-band spectra since it excite single frequency with a small energy though. For the simulations in this work, a force pulse with amplitude is defined by the Equation (6.16).

· sin 2 , 0,

(6.16)

Then the impulse, that is the area under this sinusoid is defined by Equation (6.17)

0.345 · (6.17)

The transient spectrum for the given sinusoid pulse of maximum amplitude 45 N and duration is shown in Figure 6.4. According to Equation (6.17),

this pulse has impulse of

, which is around 17.27 Ns. According to Figure 6.4, this value agrees with the transient spectrum at 0 Hz.

64

Figure 6.4. Shape of the force pulse and its corresponding Spectrum.

Now, at various force levels, the composite FRF can be compared with the underlying linear FRF. A simulation result is shown in Figure 6.5 using the parameters shown in Table 6.1.

0 5 10 15 200

5

10

15

20

25

30

35

40

45

50

Time, s

Forc

e, N

0 5 10100

101

102

Frequency, Hz

Tran

sien

t Spe

ctru

m, N

s

65

Figure 6.5. Comparison of FRF H11 with Impulse Excitation.

Figure 6.6. Comparison of FRF H21 with Impulse Excitation.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510-5

10-4

10-3

10-2

10-1

100

101Mobility FRF, H11

Frequency [Hz]

Mag

nitu

de [m

/Ns]

Pure FRFComposite FRF at F=1.725NsComposite FRF at F=10.26NsComposite FRF at F=17.27Ns

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510-10

10-8

10-6

10-4

10-2

100


Frequency [Hz]

Mag

nitu

de [r

ad/N

s]

Pure FRFComposite FRF at F=1.725NsComposite FRF at F=10.26NsComposite FRF at F=17.27Ns

66

In Figure 6.5 and 6.6, there is no affect in the nonlinearity at a small force level. By increasing force level to 10.26 Ns, that is with a maximum amplitude pulse of 28 N and also at a force level of 17.2 Ns having maximum amplitude of 50 N, the changes in the FRF can be noticed compared to the underlying linear FRF.

A zero-memory nonlinear system is detected which is dependent only on the present input [4]. The type of nonlinearity detected in the system is known as softening spring, as the composite FRFs are changing either the beam inherent mass or stiffness, or the inherent property of pendulum’s length is varying. That is either the stiffness of the beam is decreasing or mass is decreasing with increasing force levels, whereas, in the case of pendulum the length is reduced.

6.3.1.2 Harmonic Excitation

By using harmonic excitation only one frequency is excited at a time resulting in all energy in the signal associated with one single frequency. This makes it suitable for exciting all the nonlinearities involved in the structure. There are two different ways to perform harmonic excitation, stepped sine and sweep sine [1]. Using the stepped sine one frequency at a time is excited and measurement is taken when the steady state response is reached. Whereas sweeping-sine is performed slowly and continuously varying the frequency in the given interval. The sweeping sine excitation is used in the detection here. The time and spectrum of the force with input force of 5 N is shown in Figure 6.7.

67

Figure 6.7. Chirp Time domain and Corresponding Spectrum for 5 N Force.

Using the same parameters as from Table 6.1 and Table 6.2, the following plots are shown in Figures 6.8 and 6.9.

Table 6.2. Sweeping frequency and amplitude.

Parameter Value

Amplitude of input signal, a1 [N] 0.1:10

Fundamental frequency, Fd [Hz] 0.001 1 5

0 50 100 150 200-5

-4

-3

-2

-1

0

1

2

3

4

5

time, s

Forc

e, N

0 5 1010-2

10-1

100

Frequency, Hz

Chi

rp S

pect

rum

, N

68

Figure 6.8. Comparison of FRF with sweeping sine.

Figure 6.9. Comparison of FRF with sweeping sine.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510-5

10-4

10-3

10-2

10-1

100


Frequency [Hz]

Mag

nitu

de [m

/Ns]

Pure FRFComposite FRF at F=0.001NComposite FRF at F=1NComposite FRF at F=3N

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

-10

10-8

10-6

10-4

10-2

100


Frequency [Hz]

Mag

nitu

de [r

ad/N

s]

Pure FRFComposite FRF at F=0.001NComposite FRF at F=1NComposite FRF at F=3N

69

So this result from Figure 6.8 and 6.9 indicates the same phenomenon of softening spring is found with impulse excitation. This method is handier compared to impulse excitation in experimentation, since the input of impulse excitation cannot be controlled.

In the step sinusoidal case, a discrete set of frequencies are utilized and checked for various force levels. As shown in Figure 6.10, at a force level of 0.01N the frequency response function is plotted.

Figure 6.10. Comparison of pure FRF and a stepped-sine at 0.01N.

0 1 2 3 4 510

-4

10-3

10-2

10-1

100

101

Mag

nitu

de [m

/Ns]

Mobility FRF, H11

Frequency [Hz]

0 1 2 3 4 510

-8

10-6

10-4

10-2

100

102

Mag

nitu

de [r

ad/N

s]

Mobility FRF, H21

Frequency [Hz]

Pure FRFComposite FRF at 0.01N

Pure FRFComposite FRF at 0.01N

70

Figure 6.11. Comparison of pure FRF and a stepped-sine at 20 N.

Figure 6.11 shows the deflection in the FRF having the same effect as that of softening spring. Similarly it can be noticed as the force is as shown in Figure 6.12.

0 1 2 3 4 510

-4

10-3

10-2

10-1

100

101M

agni

tude

[m/N

s]Mobility FRF, H11

Frequency [Hz]

0 1 2 3 4 510

-8

10-6

10-4

10-2

100

102

Mag

nitu

de [r

ad/N

s]

Mobility FRF, H21

Frequency [Hz]

Pure FRFComposite FRF at 20N


71

Figure 6.12. Comparison of pure FRF and a stepped-sine at 30 N.

0 1 2 3 4 510-4

10-3

10-2

10-1

100

101M

agni

tude

[m/N

s]Mobility FRF, H11

Frequency [Hz]

0 1 2 3 4 510-8

10-6

10-4

10-2

100

102

Mag

nitu

de [r

ad/N

s]

Mobility FRF, H21

Frequency [Hz]



72

7. Experimental work

In this chapter, an experimental model similar to the studied coupled pendulum-SDOF system is built and experiments are conducted on it. The results obtained are compared to the results obtained in the previous chapters which are simulated using MATLAB.

7.1 Experimental Setup

In this setup a cantilever beam is coupled with a pendulum, in which the length of the beam is kept long while the length of the pendulum is kept small so that the pendulum resonance frequency is closer to that of the cantilever beam as shown in Figure7.1.

Figure 7.1. Experimental Setup.

The devices and instruments used in the experimental setup are:

• A Shaker: It was used to excite the system near the free end of the cantilever beam.

73

• A Force Transducer: It was used to measure the force transmitted by the shaker.

• A Signal Generator: It was used to generate a voltage signal to the shaker.

• A Power Amplifier/Gain Master: It was used to amplify the voltage signal to the shaker.

• An Accelerometer: It was used to measure the acceleration at the beam and at the pendulum

• A Data Acquisition Card: It was used to collect and measure the data from the system

7.2 Characterization of the Beam and Pendulum

In order to check the model numerically, the modal parameters are extracted first. The modal mass , static stiffness and damping coefficients of the beam can be extracted by giving an input force to it at its free end and calculating its accelerance. Half-Power Method (3 db-bandwidth method) [1] is applied to get relative damping .

The relative damping can be calculated from the formula:

(7.1)

Where the two points and are identified as the ‘half – power points’.

74

Figure 7.2. Half-Power Bandwidth Method to calculate relative damping.

From Figure 7.2, 0.01437

It is known that for accelerance:

| | / (7.2)

Where the peak value of | | is

| | 2 / (7.3)

Thus, =532.27 N/m

The undamped natural frequency ω0 = (to get m)

= 0.2316kg (the mass of the SDOF system)

0 2 4 6 8 10 12 14 16 18 20-80

-60

-40

-20

0

20

40

60Experimental FRF

Frequency (Hz)

Mag

nitu

de (d

b)

f d 1 = 7 . 6 3 0 2 H z

ζ =1.4377%

75

As is know that ζ = √

Thus, =0.3244 N·s/m

Figure 7.3. To get a better estimate of the parameters from the beam a model is used to fit into that of the measured FRF.

After calculating the modal parameters of the beam, the mass of the pendulum was measured and the length of the wire of the pendulum was calculated from the resonance frequency of the pendulum. The angular damping coefficient j of the pendulum is calculated by comparing measured FRF to the model and adjusting it as seen in Figure7.4.

0 1 2 3 4 5 6 7 8 9 1010

-3

10-2

10-1

100

101

102

103 Dynamic Flexibility H11

Frequency (Hz)

Mag

nitu

de o

f FR

F

Measured (Receptance)Model

76

Figure 7.4. Measured FRF is compare with linear FRF model to get the

value of angular damping coefficient.

Figure7.5. Coherence plot of the measured FRF.

0 1 2 3 4 5 6 7 8 9 1010-5

10-4

10-3

10-2

10-1

100

101

102

103Accelerance H11

Frequency, Hz

Mag

nitu

de o

f Fre

quen

cy R

espo

nse

Measured FRFLinear FRF

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Coherence for Measured FRF H11

Frequency [Hz]

77

The final modal parameters are shown in Table 7.1.

Table 7.1. Parameters for the Beam and Pendulum.

Parameter value

Pendulum Mass, m [kg] 0.139

SDOF mass, M [kg] 0.2316


Horizontal spring coefficient, Kh [N/m]=[Kg/g2] 532.27




7.3 Changes in the FRF with respect to the input force

As is simulated in chapter 6 about the composite FRF of nonlinear system, in this section it will be done in the experiment and compared with that in the theoretical model.

78

Figure 7.6. Five different time Force data with RMS values.

As is seen from Figure 7.6, five different random forces are plotted. The RMS (root mean square) value of them is from smallest to the biggest.

The result from these varying forces is shown in Figure 7.8.

0 20 40 60-0.2

0

0.2Random Force1, RMS= 0.036527(N)

time [s]

Am

plitu

de [N

]

0 20 40 60-1

0

1Random Force2, RMS= 0.20776(N)

time [s]

Am

plitu

de [N

]

0 20 40 60-1

0


time [s]

Am

plitu

de [N

]

0 20 40 60-2

0


time [s]

Am

plitu

de [N

]

0 20 40 60-2

0

2Random Force5, RMS= 0.45909N)

time [s]

Am

plitu

de [N

]

79

Figure 7.7. Comparison at various force level.

0 5 100

0.5

1Coherence for FRF1

Frequency [Hz]0 5 10

0

0.5

1Coherence for FRF2

Frequency [Hz]

0 5 100

0.5

1Coherence for FRF3

Frequency [Hz]0 5 10

0

0.5

1Coherence for FRF4

Frequency [Hz]

0 5 100

0.5

1Coherence for FRF5

Frequency [Hz]

80

Figure 7.8. Comparison at various force level.

From Figure 7.7 and Figure 7.8, it is noticed that with increased force levels, the same effect of softening spring is observed as that of the simulated model.

Using the first, the third and the fifth random forces from the experiment as an input respectively to simulate the composite FRF with parameters as that of Table 7.1 are plotted, the result is as shown in Figure 7.9.

0 1 2 3 4 5 6 7 8 9 1010-3

10-2

10-1

100

101

102

103

Measured FRF H11

Frequency [Hz]

Acc

eler

ance

,Mag

nitu

de o

f Fre

quen

cy R

espo

nse

m2 /N

s

FRF1 Excited by Random Force1FRF2 Excited by Random Force2FRF3 Excited by Random Force3FRF4 Excited by Random Force4FRF5 Excited by Random Force5

81

Figure 7.9. Changes in FRF with increasing force amplitude using the simulation model.

0 1 2 3 4 5 6 7 8 9 1010-4

10-3

10-2

10-1Simulated Composite FRF H11

Frequency [Hz]

Dyn

amic

Fle

xibi

lity,

Mag

nitu

de

Composite FRF1 Excited by Random Force1Composite FRF2 Excited by Random Force2Composite FRF3 Excited by Random Force3

82

Figure 7.10. Changes in FRF with increasing force amplitude using the simulation model. Zooming in on the second peak.

The results shows the changes in the composite FRFs match the result observed from the experiment. It’s quite clear if looking on the second peak as the force level increased the composite FRF is moving down and backward. Thus, it can be concluded that there is a softening spring effect similar to that of the simulated model.

7.4 Higher Harmonics

Three ways have been implemented here to find the presence of higher harmonics in the system. The time data with high resolution is utilised here to illustrate the distortion.

7 7.5 8 8.5 910-3

10-2

10-1Simulated Composite FRF H11

Frequency [Hz]

Composite FRF1 Excited by Random Force1Composite FRF2 Excited by Random Force2Composite FRF3 Excited by Random Force3

83

7.4.1 Measurment with the sweeping frequency In this experiment, the higher harmonics are observed. In order to observe the higher harmonics in the coupled pendulum system, a slowly increasing sweep input is given to it. The pendulum is the removed and the test is repeated for the linear beam system. The output acceleration time history for the coupled system is then compared with the acceleration time history of the linear beam as shown in Figure 7.11.

Figure 7.11. The distortion effect using sweeping sinusoid.

It was observed that the region around the resonance frequency of the pendulum have some distortion affect, so a high resolution data with slowly increasing sweep input is applied to the system. It is noticed from Figure 7.11 that the distortion effect of the coupled beam-pendulum system is more prominent from 3.102Hz to 3.103Hz.

3.102 3.1022 3.1024 3.1026 3.1028 3.103

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

frequency (Hz)

Acc

eler

atio

n (m

/s2 )

Distrotion effect using sweeping sinusoid

Coupled Beam-PendulumBeam

84

In the next two methods, a sinusoidal input with a frequency of 3Hz with increasing force and different measuring position is utilized.

7.4.2 Measurment with the sinusoidal Input Case 1 is shown in Figure 7.12 and Figure 7.13. The system is excited by a sine input at a frequency of 3 Hz while the force level is increased gradually.

Figure 7.12. Distortion effect with sine input at 3Hz with increasing force level.

0 5 10 15 20 25 30 35 40 45 50-4

-2

0

2

4Force input data

Time, s

Forc

e, N

sine input at 3Hz

0 5 10 15 20 25 30 35 40 45 50-1

-0.5

0

0.5Distortion effect observed while measurement at the beam

time, s

Acc

eler

atio

n, m

/s2

85

Figure 7.13. Zooming in on the distorted part.

With the increasing force level, the distortion affect is observed for a certain time interval.

24.5 25 25.5 26 26.5 27 27.5

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Distortion effect observed while measurement at the beam

time, s

Acc

eler

atio

n, m

/s2

86

Case 2 is shown in Figure 7.14, Figure 7.15 and Figure 7.16. In this case the data is observed via accelerometer at the pendulum. The system is excited by a sine input at a frequency of 3 Hz while the force level is increased gradually.

Figure 7.14. Distortion effect with increasing force level while with sine input at 3 Hz.

0 5 10 15 20 25 30 35 40 45 50-0.4

-0.2

0

0.2

0.4

Time ,s

Forc

e, N

Increasing Force level

Input Force at 3 Hz

0 5 10 15 20 25 30 35 40 45 50-0.4

-0.2

0

0.2

0.4Distortion effect while measuring at the pendulum

Time ,s

Acc

eler

atio

n, m

/s2

87

Figure 7.15. Zooming in on the output signal under the low force level.

8.2 8.4 8.6 8.8 9 9.2 9.4 9.6 9.8 10

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01Distortion effect while measuring at the pendulum

Time ,s

Acc

eler

atio

n, m

/s2

88

Figure 7.16. Zooming in on the output signal under the high force level.

It is observed that, the acceleration signal is distorted when the force level is increased, showing the presence of higher harmonics. In summary, the results from the experiments are similar to that of the simulation model.

44.5 45 45.5 46 46.5 47 47.5

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Distortion effect while measuring at the pendulum

Time ,s

Acc

eler

atio

n, m

/s2

89

8. Conclusions and Future Work

The differential equations for the moving point pendulum and coupled system has been shown in this work and its time simulation is implemented in MATLAB verifying the equations for the given systems. Linear and nonlinear terms are identified for both the systems.

The distortion affects are investigated via two proposed methods that is the power ratio method and the magnitude ratio method to check for the amount of distortion in the response of the systems. It is found out that the second method is more sensitive to detect distortion in the system. From Figure 5.8 and Figure 5.9 some uneven response is observed for a moving point pendulum showing the nonlinear behavior of the system. Whereas in the case of coupled system, the distortion is high near the resonance frequency of the pendulum and SDOF system with higher amplitudes of excitation forces.

After checking for the periodicity, the changes in approximate nonlinear FRFs were compared with the linear FRF. At lower force levels, the composite FRFs were fairly linear while with the increase in amplitude of force, the FRF changes its shape. The changes in FRF show a behavior similar to that of a softening spring since the resonance frequency is shifted left side with reduction in magnitude. This implies that either the equivalent stiffness is increased or the equivalent mass is decreasing.

Similar observations were also noticed from experiments for the random force input, verifying the results of simulated model. By increasing the force level of the random input, changes in the FRF were observed as shown in Figure 7.8. The softening spring phenomenon was observed. From the experiments the distortion affects are noticed with a stepped sinusoid near the resonance frequency of pendulum at 3 Hz with increasing force level while taking measurements at the beam and the pendulum. The response observed at the pendulum was found to be highly distorted. In both measurements the distortion in the acceleration time data shows the presence of higher harmonics.

Since the types of non-linearity is known, now a faster algorithm can be developed to find the frequency response functions ‘FRFs’, like harmonic balance method can be applied to find dynamic characteristics of the system.

90

A suggestion for future work is to simulate the color maps (for distortion) using the parameters obtained for the experimental structure and the simulation model. The result can then be verified with experimental testing.

9. References

1. Anders Brandt, (2003). Introductory Noise and Vibration Analysis. Saven Edutech AB.Blekinge Institute of Technology. Sweden.

2. Genta and Giancarlo, second edition. Vibration of Structures And Machines.springer. USA

3. Andres Lorente, (2010). Dynamic Characteristics of a Pendulum System. Department of mechanical engineering, Blekinge Institute of technology, Sweden.

4. Martin Magnevall. Methods For Simulation And Characterization Of Nonlinear Mechanical Structures, Blekinge Institute of Technology, Sweden.

5. Worden, K. and G R Tomlinson, G.R., (2001). Nonlinearity In Structural Dynamics. Institute of Physics Publishing Bristol and Philadelphia, USA.

6. Dr. Randall, J. Allemang,(1994). Analytical and Experimental Modal Analysis. Department of Mechanical, Industrial and Nuclear Engineering, University of Cincinnati, Ohio.

7. Nur Adila Faruk Senan, A brief introduction to using ode45 in MATLAB. Department of Mechanical Engineering, University of California at Berkeley.

8. Armando Enrique Leon Guarena, (2007). Study on vibrations induced by parametric excitations on strings. Department of mechanical engineering, Blekinge Institute of Technology, Sweden.

9. Chen liqun, Liu yanzhu, (2001). Nonlinear Vibration. High Education Publishing, Beijing China.

10. MonsonH.Hayes, (1999). Schaum’s Outlines Digital Signal Processing. The McGraw-Hill Companies,USA.

School of Engineering, Department of Mechanical Engineering Blekinge Institute of Technology SE-371 79 Karlskrona, SWEDEN

Telephone: E-mail:

+46 455-38 50 00 [email protected]

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Non-Linear Characteristics of a Pendulum System829348/FULLTEXT01.pdf1 Non-Linear Characteristics of a Pendulum System Wei Zongyuan Pir Aamir Ali Shah V. K. C Varma Rudraraju Department