Download Non-Linear Characteristics of a Pendulum System

Master's Degree Thesis
ISRN: BTH-AMT-EX--2010/D-06--SE
Non-Linear Characteristics of a
Pendulum System
Wei Zongyuan
Pir Aamir Ali Shah
V. K. C Varma Rudraraju
Department of Mechanical Engineering
Blekinge Institute of Technology
Karlskrona, Sweden
2010
Supervisor:
Andreas Josefsson and Ansel Berghuvud, BTH
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 nonlinearity 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.
1 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
2 Table of contents:
Contents:
1. Notations
2. Introduction
2.1 Purpose
2.2 Background
3. Theoretical modeling
3.1 Introduction
3.2 Equations for a simple pendulum
3.3 Parametrically excited system
3.3.1 Parametric excitation
3.3.2 Equations for the simple pendulum
suspended by a moving pivot
3.4 Equations for a pendulum coupled with SDOF system
4. Simulation of Motions
4.1 Introduction
4.2 Dynamic response of a simple pendulum
4.3 Dynamic response of a simple pendulum suspended
by a moving pivot
4.4 Dynamic response of a simple pendulum coupled
with SDOF system
5. Harmonic Generation
5.1 Introduction
5.2 Distortion affect
5.2.1 Harmonic distortion in the moving pivot pendulum
5.2.2 Harmonic distortion in the coupled SDOF system
6. Estimation of Frequency Response Function
6.1 Introduction
6.2 Linearization of differential equations
6.3 Changes in the FRF with respect to input force
6.3.1 Measurements for Non-Linear systems
3 5
8
8
8
10
10
10
13
13
14
17
21
21
21
25
29
36
36
37
39
48
57
57
57
63
63
6.3.1.1 Transient Excitation
6.3.1.2 Harmonic Excitation
7. Experimental Work
7.1 Experimental setup
7.2 Characteristics of beam and pendulum
7.3 Changes in the FRF with respect to input force
7.4 Higher harmonics
7.4.1 Measurement with the sweeping frequency
7.4.2 Measurement with sinusoidal input
8. Conclusions and Future Work
9. References
4 63
76
72
72
73
77
82
83
84
89
91
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)
5 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
6 Abbreviations:
2DOF
Two degree of freedom
SDOF
Single degree of freedom
FRF
Frequency response function
ODE
Ordinary Differential Equation
RMS
Root Mean Square
7 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 massdamper-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.
8 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.
x
Y
C
Fx(t)
M
K
(t) L
m
Figure 2.1. The coupled pendulum-SDOF system.
In order to better understand dynamic behavior of the coupled pendulumSDOF 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.
9 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 pendulumSDOF 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
10 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)
11 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.
12 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.
13 x
Y θ
m
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.
14 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
0
15 (3.12)
(3.13)
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)
· , the following differential
Now dividing the whole equation by 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
0
16 (3.19)
(3.20)
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.
17 x
Y L
θ
T
m
Fj
mg
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.
18 (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.
x
Y FK
Fc
M
Fh
T
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,
19 (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)
20 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
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
21 value
Angle of the Pendulum [radian]
Simple Pendulm without damping Simulation
1
0.5
0
-0.5
-1
0
10
20
30
0
1
2
3
40
50
60
70
80
Time [s]
Simple Pendulm without damping Simulation
90
100
9
10
Magnitude
0.8
0.6
0.4
0.2
0
4
5
6
Frequency Domain [Hz]
7
8
Figure 4.1. Simulation of the simple pendulum without damping.
22 Angle of the Pendulum [radian]
Simple Damped Pendulm Simulation
1
0.5
0
-0.5
0
10
20
30
0
1
2
3
40
50
60
70
Time [s]
Simple Damped Pendulm
80
90
100
8
9
10
Magnitude
0.04
0.03
0.02
0.01
0
4
5
6
Frequency Domain [Hz]
7
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.
23 Angle of the Pendulum [radian]
Simple Damped Pendulm Simulation
X: 0
Y: 0.5236
0.5
0
-0.5
0
10
20
30
40
50
60
70
Time [s]
Simple Damped Pendulm
80
90
8
9
Magnitude
0.08
0.06
X: 0.8407
Y: 0.06355
0.04
0.02
0
0
1
2
3
4
5
6
Frequency Domain [Hz]
7
10
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
.
24 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
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
25 value
Input Signal
Displacement [m]
1
0.5
0
-0.5
-1
0
20
40
60
0
1
2
3
80
100
120
Time [s]
140
160
180
200
7
8
9
10
Magnitude
0.8
0.6
0.4
0.2
0
4
5
6
Frequency Domain [Hz]
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.
26 Angle of the Pendulum [radian]
Simple Damped Pendulum with a moving pivot
0.05
0
-0.05
0
20
40
60
0
1
2
3
80
100
120
Time [s]
140
160
180
200
7
8
9
10
Magnitude
0.02
0.015
0.01
0.005
0
4
5
6
Frequency Domain [Hz]
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.
27 Angle of the Pendulum [radian]
Simple Damped Pendulum with a moving pivot
60
40
20
0
-20
0
20
40
60
0
1
2
3
80
100
120
Time [s]
140
160
180
200
7
8
9
10
Magnitude
30
20
10
0
4
5
6
Frequency Domain [Hz]
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.
28 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
Length of the pendulum, L [m]
0.25
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.
29 (a)
(b)
0.02
0.5
0.4
0.3
0.01
Angle of the pendulum [rad]
Horizontal position of the mass [m]
0.015
0.005
0
-0.005
-0.01
0.2
0.1
0
-0.1
-0.2
-0.3
-0.015
-0.02
-0.4
0
50
Time [s]
-0.5
100
0
50
Time [s]
100
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.
30 (a)
(b)
0.015
0.5
0.4
0.3
Angle of the pendulum [rad]
Horizontal position of the mass [m]
0.01
0.005
0
-0.005
0.2
0.1
0
-0.1
-0.2
-0.01
-0.3
-0.015
0
50
Time [s]
-0.4
100
0
50
Time [s]
100
Figure 4.8. The time response of the studied system without external force
Fh.
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.
31 (a)
(b)
0.02
0.5
0.4
0.3
0.01
Angle of the pendulum [rad]
Horizontal position of the mass [m]
0.015
0.005
0
-0.005
-0.01
0.2
0.1
0
-0.1
-0.2
-0.3
-0.015
-0.02
-0.4
0
50
Time [s]
-0.5
100
0
50
Time [s]
100
Figure 4.9. The time response of the studied system without external force
Fh.
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.
32 Table 4.4. Parameters for the coupled system.
Parameter
Value
Pendulum Mass, m [kg]
1
SDOF mass, M [kg]
2
Length of the pendulum, L [m]
0.25
Horizontal spring coefficient, Kh [N/m]=[Kg/g2]
500
Horizontal damping coefficient, bh [N·s/m]=[Kg/s]
2
Gravity, g [m/s2]
9.85
Angular damping coefficient, j [Nm·s]=[Kg·m2/s]
0.02
The excitation force
0.8
0.6
Amplitude [N]
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0
20
40
60
80
100
120
Time [s]
140
160
Figure 4.10. The external force Fh.
33 180
200
Figure 4.10 shows an external force in time domain with frequency 0.2 Hz
and amplitude 0.5 N.
-3
2.5
-3
(a)
x 10
8
6
1.5
4
Angle of the pendulum [rad]
Horizontal position of the mass [m]
2
1
0.5
0
-0.5
-1
2
0
-2
-4
-6
-1.5
-2
(b)
x 10
0
50
100
Time [s]
150
-8
200
0
50
100
Time [s]
150
200
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.
34 (a)
(b)
0.2
20
15
0.1
Angle of the pendulum [rad]
Horizontal position of the mass [m]
0.15
0.05
0
-0.05
-0.1
10
5
0
-0.15
-0.2
0
50
100
Time [s]
150
-5
200
0
50
100
Time [s]
150
200
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.
35 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.
36 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 monoharmonic 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 steadystate. When the linear systems are excited by a sinusoidal force, its steadystate 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 superharmonics. In extreme conditions, the response will show chaotic behavior,
which does not show any periodic behavior. The proof [5] is as follows.
sin Suppose
observed that
is the input to a linear system. First of all, it is
, 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:
Where
(5.4)
is a constant
So taking
and
gives:
(5.5)
sin Now, as :
0
(5.6)
37 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
38 harmonics.
1 is the ratio of and
is defined as ‘power ratio method.
. In this thesis work, this 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.
39 Table 5.1. Parameters for the moving point pendulum.
Parameter
value
Mass of the pendulum, 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
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
40 Angle of the Pendulum [radian]
Angle of the Pendulum [radian]
(a)
2
1
0
-1
-2
0
100
200
300
Time [s]
(b)
400
500
600
1
0.5
0
-0.5
-1
250
300
350
400
450
Time [s]
500
550
600
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.
41 0.9
0.8
0.7
Magnitude
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
Frequency Domain [Hz]
15
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.
42 (a)
Magnitude
0.8
A,Sum of the magnitude of whole area
A1,Sum of the magnitude around Fd
Cutline,the point below is taken to 0
0.6
Crit1=std(xnew)/std(x) =99.9378%
Crit2 =A1/A=60.093%
0.4
0.2
0
0.45
0.5
0.55
0.6
Frequency Domain [Hz]
(b)
0.65
0.7
Magnitude
0.8
A,Sum of the magnitude of whole area
A1,Sum of the magnitude around Fd
0.6
Crit1=std(xnew)/std(x) =99.9378%
Crit2 =A1/A=81.8167%
0.4
0.2
0
0.45
0.5
0.55
0.6
Frequency Domain [Hz]
0.65
0.7
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.
43 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.
Table 5.3. Amplitude and fundamental frequency.
Parameter
value
Amplitude of input signal, a1 [m]
0.5
Fundamental frequency, Fd [Hz]
1.0
60
40
Angle of the Pendulum [radian]
20
0
-20
-40
-60
-80
-100
-120
0
100
200
300
Time [s]
400
500
Figure 5.5. Time response of the pendulum.
44 600
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
Fd [Hz]
5
1
4.5
0.9
4
0.8
3.5
0.7
3
0.6
2.5
0.5
2
0.4
1.5
0.3
1
0.2
0.5
0.1
0.5
1
1.5
2
2.5
Amp [m]
3
3.5
4
4.5
Figure 5.6. A color map of distortion by using method Crit1to find the
periodic and non-periodic solution.
45 5
0.9
4.5
0.8
4
0.7
3.5
0.6
Fd [Hz]
3
0.5
2.5
0.4
2
0.3
1.5
0.2
1
0.1
0.5
0.5
1
1.5
2
2.5
3
Amp [m]
3.5
4
4.5
5
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.
46 Fd [Hz]
5
1
4.5
0.9
4
0.8
3.5
0.7
3
0.6
2.5
0.5
2
0.4
1.5
0.3
1
0.2
0.5
0.1
0.5
1
1.5
2
2.5
Amp [m]
3
3.5
4
4.5
Figure 5.8. A color map distortion by using method Crit1with damping
coefficient j=0.1.
47 5
0.9
4.5
0.8
4
0.7
3.5
0.6
Fd [Hz]
3
0.5
2.5
0.4
2
0.3
1.5
0.2
1
0.1
0.5
0.5
1
1.5
2
2.5
3
Amp [m]
3.5
4
4.5
5
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.
48 Table 5.5. Parameters of the system.
Parameter
value
Pendulum Mass, m [kg]
1
SDOF mass, M [kg]
2
Length of the pendulum, L [m]
0.25
Horizontal spring coefficient, Kh [N/m]=[Kg/g2]
500
Horizontal damping coefficient, bh [N·s/m]=[Kg/s]
2
Gravity, g [m/s2]
9.81
Angular damping coefficient, j [Nm·s]=[Kg·m2/s]
0.02
The harmonic input is given as:
· sin 2
· cos 2
Similarly, a specific fundamental frequency
input force
are shown in Table 5.6
(5.11)
and amplitude
Table 5.6. Amplitude and fundamental frequency.
Parameter
value
Amplitude of input signal, a1 [m]
1
Fundamental frequency, Fd [Hz]
2
49 for the
(a)
(b)
0.01
0.06
0.04
0.006
Angle of the pendulum [rad]
Horizontal position of the mass [m]
0.008
0.004
0.002
0
-0.002
-0.004
0.02
0
-0.02
-0.006
-0.04
-0.008
-0.01
0
200
400
Time [s]
-0.06
600
0
200
400
Time [s]
600
Figure 5.10. (a) The motion of the SDOF mass in time domain. (b) The
motion of the pendulum in time domain.
50 Angle of the Pendulum [radian]
Angle of the Pendulum [radian]
(a)
0.1
0.05
0
-0.05
-0.1
0
100
200
300
Time [s]
(b)
400
500
600
0.04
0.02
0
-0.02
-0.04
200
250
300
350
400
Time [s]
450
500
550
600
Figure 5.11. (a) Time response for the pendulum. (b) Time response in
steady state.
51 -3
(a)
Magnitude
x 10
A,Sum of the magnitude of whole area
A1,Sum of the magnitude around Fd
Cutline,the point below is taken to 0
4
Crit1=std(xnew)/std(x) =100%
Crit2 =A1/A=62.589%
2
0
1.85
1.9
1.95
-3
Magnitude
x 10
2
2.05
2.1
Frequency Domain [Hz]
(b)
2.2
2.25
A,Sum of the magnitude of whole area
A1,Sum of the magnitude around Fd
4
Crit1=std(xnew)/std(x) =100%
Crit2 =A1/A=91.2296%
2
0
1.85
2.15
1.9
1.95
2
2.05
2.1
Frequency Domain [Hz]
2.15
2.2
2.25
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.
52 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
The distortion percent of the SDOF-Pendulum
1
5
4.5
0.9
4
0.8
3.5
0.7
X= 3.6
Y= 2.5
Level= 0.80773
Fd [Hz]
3
0.6
2.5
0.5
2
0.4
1.5
0.3
1
0.2
0.1
0.5
1
2
3
4
5
Amp [N]
6
7
8
9
Figure 5.13. A color map for distortion by using Crit1.
53 5
0.9
4.5
0.8
4
0.7
3.5
Fd [Hz]
3
0.6
X= 3.1
Y= 2.5
Level= 0.94674
0.5
2.5
0.4
2
0.3
1.5
0.2
1
0.1
0.5
1
2
3
4
5
Amp [N]
6
7
8
9
Figure 5.14. A color map for distortion by using Crit2.
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
54 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
The distortion percent of the SDOF-Pendulum
5
1
4.5
0.9
4
0.8
3.5
0.7
X= 4.1
Y= 2.5
Level= 0.74618
Fd [Hz]
3
0.6
2.5
0.5
2
0.4
1.5
0.3
1
0.2
0.5
0.1
1
2
3
4
5
Amp [N]
6
7
8
9
Figure 5.15. A color map for harmonic distortion by using Crit1.
55 5
0.9
4.5
0.8
4
0.7
3.5
Fd [Hz]
3
0.6
X= 3.1
Y= 2.5
Level= 0.94847
0.5
2.5
0.4
2
0.3
1.5
0.2
1
0.1
0.5
1
2
3
4
5
Amp [N]
6
7
8
9
Figure 5.16. A color map for distortion by using Crit1.
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.
56 6 Estimation of
Function (FRF)
Frequency
Response
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:
57 (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 (
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:
0
1
0
0
(6.5)
From Equation (6.5), the following matrices are identified.
The mass matrix
=
1 ,
58 10°),
The damping matrix
=
The stiffness matrix
= 0
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
following equation is obtained.
and divided by s ), the
(6.10)
0
So the Eigen-values can be found from Equation (6.10).
The parameters used in an example are shown in Table 6.1.
59 Table 6.1. Parameters for coupled SDOF-pendulum system.
Parameter
value
Pendulum Mass, m [kg]
2
SDOF mass, M [kg]
3
Length of the pendulum, L [m]
0.25
Horizontal spring coefficient, Kh [N/m]=[Kg/g2]
500
Horizontal damping coefficient, bh [N·s/m]=[Kg/s]
0.4
Gravity, g [m/s2]
9.81
Angular damping coefficient, j [Nm·s]=[Kg·m2/s]
0.05
For the parameters given in Table 6.1, the undamped resonance frequencies
using K
M are:
0.9133
Hz
2.2474
(6.11)
Equation (6.7) can be written as ·
·
. 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:
•
•
is
.
- 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:
60 (6.13)
, Equation (6.13) becomes:
By defining ·
(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.
61 Estimated FRF H11
1
10
0
10
Mobility (m/Ns)
-1
10
-2
10
-3
10
-4
10
-5
10
0
1
2
3
4
5
6
7
8
9
10
Frequency (Hz)
Figure 6.2. Frequency Response Function for H11.
Estimated FRF H21
2
10
0
Mobility (rad/Ns)
10
-2
10
-4
10
-6
10
-8
10
-10
10
0
1
2
3
4
5
6
7
8
9
Frequency (Hz)
Figure 6.3. Frequency Response Function for H21.
62 10
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 nonlinearity 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
,
(6.16)
0,
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.
63 2
50
10
45
40
Transient Spectrum, Ns
35
Force, N
30
25
20
15
1
10
10
5
0
0
0
5
10
Time, s
15
20
10
0
5
Frequency, Hz
10
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.
64 Mobility FRF, H11
1
10
0
10
-1
Magnitude [m/Ns]
10
-2
10
-3
10
Pure FRF
Composite FRF at F=1.725Ns
Composite FRF at F=10.26Ns
Composite FRF at F=17.27Ns
-4
10
-5
10
0
0.5
1
1.5
2
2.5
3
Frequency [Hz]
3.5
4
4.5
5
Figure 6.5. Comparison of FRF H11 with Impulse Excitation.
Mobility FRF, H21
2
10
0
10
Magnitude [rad/Ns]
-2
10
-4
10
-6
10
Pure FRF
Composite FRF at F=1.725Ns
Composite FRF at F=10.26Ns
Composite FRF at F=17.27Ns
-8
10
-10
10
0
0.5
1
1.5
2
2.5
3
Frequency [Hz]
3.5
4
4.5
5
Figure 6.6. Comparison of FRF H21 with Impulse Excitation.
65 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.
66 0
5
10
4
3
Chirp Spectrum, N
2
Force, N
1
0
-1
-1
10
-2
-3
-4
-5
-2
0
50
100
time, s
150
10
200
0
5
Frequency, Hz
10
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
67 Mobility FRF, H11
1
10
0
10
-1
Magnitude [m/Ns]
10
-2
10
-3
10
Pure FRF
Composite FRF at F=0.001N
Composite FRF at F=1N
Composite FRF at F=3N
-4
10
-5
10
0
0.5
1
1.5
2
2.5
3
Frequency [Hz]
3.5
4
4.5
5
Figure 6.8. Comparison of FRF with sweeping sine.
Mobility FRF, H21
2
10
0
10
Magnitude [rad/Ns]
-2
10
-4
10
-6
10
Pure FRF
Composite FRF at F=0.001N
Composite FRF at F=1N
Composite FRF at F=3N
-8
10
-10
10
0
0.5
1
1.5
2
2.5
3
Frequency [Hz]
3.5
4
4.5
Figure 6.9. Comparison of FRF with sweeping sine.
68 5
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.
Mobility FRF, H11
1
10
0
0
10
Magnitude [rad/Ns]
Magnitude [m/Ns]
10
-1
10
-2
10
-3
-2
10
-4
10
-6
10
10
Pure FRF
Composite FRF at 0.01N
-4
10
Mobility FRF, H21
2
10
0
1
2
3
4
Frequency [Hz]
Pure FRF
Composite FRF at 0.01N
-8
10
5
0
1
2
3
4
Frequency [Hz]
5
Figure 6.10. Comparison of pure FRF and a stepped-sine at 0.01N.
69 Mobility FRF, H11
1
10
0
0
10
Magnitude [rad/Ns]
Magnitude [m/Ns]
10
-1
10
-2
10
-3
-2
10
-4
10
-6
10
10
Pure FRF
Composite FRF at 20N
-4
10
Mobility FRF, H21
2
10
0
1
2
3
4
Frequency [Hz]
Pure FRF
Composite FRF at 20N
-8
10
5
0
1
2
3
4
Frequency [Hz]
5
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.
70 Mobility FRF, H11
1
10
0
0
10
Magnitude [rad/Ns]
Magnitude [m/Ns]
10
-1
10
-2
10
-3
-2
10
-4
10
-6
10
10
Pure FRF
Composite FRF at 30N
-4
10
Mobility FRF, H21
2
10
0
1
2
3
4
Frequency [Hz]
Pure FRF
Composite FRF at 30N
-8
10
5
0
1
2
3
4
Frequency [Hz]
5
Figure 6.12. Comparison of pure FRF and a stepped-sine at 30 N.
71 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.
72 •
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 dbbandwidth method) [1] is applied to get relative damping .
The relative damping can be calculated from the formula:
(7.1)
Where the two points
points’.
and
are identified as the ‘half – power
73 Experimental FRF
60
40
Magnitude (db)
20
0
-20
fd1
=
7.6302 Hz
-40
ζ =1.4377%
-60
-80
0
2
4
6
8
10
12
14
16
18
20
Frequency (Hz)
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 |
|
Thus,
|
2
| is
/
(7.3)
=532.27 N/m
The undamped natural frequency ω0 =
(to get m)
= 0.2316kg (the mass of the SDOF system)
74 As is know that ζ =
Thus,
√
=0.3244 N·s/m
Dynamic Flexibility H11
3
10
Measured (Receptance)
Model
2
Magnitude of FRF
10
1
10
0
10
-1
10
-2
10
-3
10
0
1
2
3
4
5
6
7
8
9
10
Frequency (Hz)
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.
75 Accelerance H11
3
10
Measured FRF
Linear FRF
2
Magnitude of Frequency Response
10
1
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
0
1
2
3
4
5
6
Frequency, Hz
7
8
9
10
Figure 7.4. Measured FRF is compare with linear FRF model to get the
value of angular damping coefficient.
Coherence for Measured FRF H11
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
Frequency [Hz]
7
8
9
Figure7.5. Coherence plot of the measured FRF.
76 10
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
Length of the pendulum, L [m]
0.025
Horizontal spring coefficient, Kh [N/m]=[Kg/g2]
532.27
Horizontal damping coefficient, bh [N·s/m]=[Kg/s]
0.3245
Gravity, g [m/s2]
9.81
Angular damping coefficient, j [Nm·s]=[Kg·m2/s]
0.00006
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.
77 Amplitude [N]
0
20
40
60
time [s]
Random Force3, RMS= 0.30659(N)
1
0
-1
0
20
40
60
time [s]
Random Force5, RMS= 0.45909N)
2
Amplitude [N]
-0.2
Amplitude [N]
0
Amplitude [N]
Amplitude [N]
Random Force1, RMS= 0.036527(N)
0.2
Random Force2, RMS= 0.20776(N)
1
0
-1
0
20
40
60
time [s]
Random Force4, RMS= 0.34847(N)
2
0
20
0
-2
40
60
time [s]
0
-2
0
20
40
60
time [s]
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.
78 Coherence for FRF1
Coherence for FRF2
1
1
0.5
0.5
0
0
5
Frequency [Hz]
Coherence for FRF3
10
0
1
1
0.5
0.5
0
0
5
Frequency [Hz]
Coherence for FRF5
10
0
5
Frequency [Hz]
10
0
0
5
Frequency [Hz]
Coherence for FRF4
10
0
5
Frequency [Hz]
10
1
0.5
0
Figure 7.7. Comparison at various force level.
79 Measured FRF H11
3
Accelerance,Magnitude of Frequency Response m2/Ns
10
2
10
1
10
0
10
-1
10
FRF1 Excited by
FRF2 Excited by
FRF3 Excited by
FRF4 Excited by
FRF5 Excited by
-2
10
-3
10
0
1
2
3
4
5
6
Frequency [Hz]
7
Random
Random
Random
Random
Random
8
Force1
Force2
Force3
Force4
Force5
9
10
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.
80 Simulated Composite FRF H11
-1
Dynamic Flexibility,Magnitude
10
-2
10
-3
10
Composite FRF1 Excited by Random Force1
Composite FRF2 Excited by Random Force2
Composite FRF3 Excited by Random Force3
-4
10
0
1
2
3
4
5
6
Frequency [Hz]
7
8
9
10
Figure 7.9. Changes in FRF with increasing force amplitude using the
simulation model.
81 Simulated Composite FRF H11
-1
10
-2
10
Composite FRF1 Excited by Random Force1
Composite FRF2 Excited by Random Force2
Composite FRF3 Excited by Random Force3
-3
10
7
7.5
8
Frequency [Hz]
8.5
9
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.
82 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.
Distrotion effect using sweeping sinusoid
0.2
Coupled Beam-Pendulum
Beam
0.15
Acceleration (m/s 2)
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
3.102
3.1022
3.1024
3.1026
frequency (Hz)
3.1028
3.103
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.
83 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.
Force input data
4
sine input at 3Hz
Force, N
2
0
-2
-4
0
5
0
5
10
15
20
25
30
35
40
45
Time, s
Distortion effect observed while measurement at the beam
10
15
20
50
Acceleration, m/s 2
0.5
0
-0.5
-1
25
time, s
30
35
40
45
50
Figure 7.12. Distortion effect with sine input at 3Hz with increasing force
level.
84 Distortion effect observed while measurement at the beam
0.3
0.2
Acceleration, m/s 2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
24.5
25
25.5
26
time, s
26.5
27
27.5
Figure 7.13. Zooming in on the distorted part.
With the increasing force level, the distortion affect is observed for a
certain time interval.
85 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.
Increasing Force level
0.4
Input Force at 3 Hz
Force, N
0.2
0
-0.2
-0.4
0
5
0
5
10
15
20
25
30
35
40
Time ,s
Distortion effect while measuring at the pendulum
10
15
20
45
50
45
50
Acceleration, m/s 2
0.4
0.2
0
-0.2
-0.4
25
Time ,s
30
35
40
Figure 7.14. Distortion effect with increasing force level while with sine
input at 3 Hz.
86 Distortion effect while measuring at the pendulum
0.01
0.005
Acceleration, m/s 2
0
-0.005
-0.01
-0.015
-0.02
-0.025
-0.03
8.2
8.4
8.6
8.8
9
9.2
Time ,s
9.4
9.6
9.8
10
Figure 7.15. Zooming in on the output signal under the low force level.
87 Distortion effect while measuring at the pendulum
0.3
0.2
Acceleration, m/s 2
0.1
0
-0.1
-0.2
-0.3
44.5
45
45.5
46
Time ,s
46.5
47
47.5
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.
88 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.
89 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
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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
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MATLAB. Department of Mechanical Engineering, University of
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8. Armando Enrique Leon Guarena, (2007). Study on vibrations induced
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The McGraw-Hill Companies,USA. 90 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]