1. Robotics - IJRRD - The Resonance and Stability - Ashraf Almandouh (1)

8/10/2019 1. Robotics - IJRRD - The Resonance and Stability - Ashraf Almandouh (1)

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www.tjprc.org [email protected]

International Journal of Robotics

Research and Development (IJRRD)

ISSN(P): 2250-1592 ISSN(E): 22789421

Vol. 4, Issue 5, Oct 2014, 1-16

TJPRC Pvt. Ltd.

THE RESONANCE AND STABILITY OF A ROBOT ARM SIMULATED BY A

CANTILEVER BEAM

M. EISSA1, M. KAMEL

2& A. ALMANDOUH

3

1,2Department of Engineering Mathematics, Faculty of Electronic Engineering,

Menoufia University, Menouf, Egypt

3Faculty of Science in Tureif, Northen Border University, Kingdom of Saudi Arabia

ABSTRACT

In this paper, the oscillations of a nonplanar system of a cantilever beam, simulating a manipulator and an

industrial robot arm subjected to harmonic axial and transverse excitations at the free end are considered and investigated.

The method of multiple time scale perturbation technique (MSPT) is applied throughout to solve the nonlinear differential

equations describing this system up to the end including the second order approximation. All possible resonance cases are

deduced at that approximation order. The steady state solution is obtained around the worst resonance case, which was

confirmed numerically. The stability is studied applying both the frequency response equations and the phase-plane

method. The effects of the different parameters on the response and stability of the vibrating system are investigated.

Some recommendations are given regarding the design of the system. A comparison is made with the available published

work.

KEYWORDS:Cantilever Beam, Multiple Time Scale, Resonance Cases, Stability

INTRODUCTION

In the past decade, the researchers conducted a number of investigations regarding nonlinear oscillations,

bifurcation and chaos of thin plates, cantilever beams and thin shallow arch structures. Zhang [1,2] studied the dynamics of

parametrically excited, simply supported rectangular thin plates. Furthermore, the global bifurcations and chaotic dynamics

of this system were analyzed. Kamel and Amer [3] studied the behavior of the axial vibration of a cantilever beam under

multi-parametric excitation forces with different quadratic damping and cubic stiffness nonlinearities. Eissa and Amer [4]

studied vibration control of a cantilever beam subject to both external and parametric excitations. They obtained a simple

nonlinear feedback law, which was devised to suppress the vibration of the first mode of the cantilever beam. Eissa et. al.

[5], studied the vibrations of a cantilever beam subjected to multi-parametric excitation forces. They showed the

occurrence of saturation phenomena at different parameters values. Alhazza et. al. [6], studied nonlinear vibration of

parametrically excited cantilever beam, subjected to nonlinear delayed-feedback control and during their study they

examined three nonlinear cubic delayed-feedback control (position, velocity and acceleration). Mahmoodi et. al. [7]

studied experimentally, the nonlinear vibration and frequency response analysis of cantilever viscoelastic beams.

The analytically derived frequency response was verified experimentally through harmonic force excitation of samples of

carbon beams.

The nonlinear dynamics of beams have received considerable attention because of their importance in many

engineering applications as spacecraft station, satellite antenna and flexible manipulator. Therefore, research on the
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2 M. Eissa , M. Kamel & A. Almandouh

Impact Factor (JCC): 1.5422 Index Copernicus Value (ICV): 3.0

nonlinear dynamic response of cantilever beam subjected to narrow-band parametric excitation has seldom been dealt with

and investigated. Feng, Lan and Zhu [8-10] studied the nonlinear integro-differential equations of motion for a slender

cantilever beams subjected to axial narrow-band random with parametric excitation of its base. The method of multiple

scale is used to analyze this system of differential equations and the stochastic jump and bifurcation of the first and secondmodal parametric principal resonance, which are also investigated and examined numerically.

Haider, Nayfeh and Chin [11] derived a set of integro-partial-differential equations that describes the nonlinear

nonplanar response of a cantilever metallic beam, to a principle parametric excitation. They applied the method of multiple

scales to derive the modulation equations governing the amplitudes and the phases of the two modes. The effects of both

cross-section and forcing frequency detuning on the static and dynamic bifurcations were investigated and codimension-2

bifurcations were found. Zhang, Wang and Yao [12] studied global bifurcations and chaotic dynamics for the nonlinear

nonplanar oscillations of a cantilever beam subjected to a harmonic axial and transverse excitations at the free end. This is

focused on the 2:1internal resonance, principal parametric resonance, 1/2 sub-harmonic resonance for the in-plane mode

and fundamental parametric-primary resonance for the out-of-plane mode. Also, Yao and Zhang [13] showed that the

multipulse Shilnikov type chaotic motions can occur for the nonlinear nonplanar oscillations of this cantilever beam.

The analysis of the global dynamics indicated that there exists the multipulse jumping orbit in the perturbed phase space of

this cantilever beams. Finally, Zhang [14] presented an analysis of the chaotic motion and its control for the nonlinear

nonplanar oscillations of this cantilever beam subjected to both harmonic axial and transverse excitations at the free end.

A new method of controlling chaotic motion was proposed and studied. The numerical results indicated that the transverse

excitation at the free end controlled the chaotic motion of the cantilever beam.

Lee and Pak [15] studied stability analysis of nonplanar free vibrations of a cantilever beam which is made using

both the nonlinear normal mode concept, and Synges stability concept, they examined the stability of each mode.

Also, the stability criterion that obtained analytically is confirmed numerically by plotting Poincare map of the motions

neighboring on each mode. Finally, Aghababaei et. al. [16 -17] derived the nonlinear differential equations and boundary

conditions of nonplanar vibrations of geometrically imperfect in extensional beams, using the extended Hamilton s

principle. The validity of the model was investigated using the existing experimental data. Also, they applied the

combination of the multiple scales method and the Galerkin procedure to derive two nonlinear integro-differential

equations. Variation of the steady state amplitude curves and their stability are shown in bifurcation diagrams for both the

imperfect and perfect cases.

In this paper, the vibration of a nonlinear oscillations of a nonplanar cantilever beam subjected to harmonic axial

excitation and two transverse excitations at the free end are studied and solved. The nonlinear oscillations of the considered

system are described by a two-degrees-of-freedom system subjected to both multi external and parametric excitations.

The method of multiple time scale perturbation technique (MSPT) [18,19] is applied to analyze the response of the system

near sub-harmonic resonance to obtain semi-closed form solution to the second order approximations. The steady state

solution near the selected worst resonance case is studied using frequency response equations. Different resonance cases

are reported and studied numerically. The effects of different parameters on the vibrating system are studied numerically

and the Runge-Kuttas fourth order method [20,21] is applied to explore the nonlinear dynamic behavior of the system.

Some recommendations are reported regarding the design of such systems. A comparison is made with the available

published work.


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The Resonance and Stability of a Robot Arm Simulated by a Cantilever Beam 3


MATHEMATICAL ANALYSIS

A system of nonlinear nonplanar cantilever beam with length ,L mass m per unit length subjected to harmonicaxial excitation and two transverse excitations at the free end is considered, as shown in Figure 1(a). where oxyz is

A Cartesian coordinate system, , and are the principal axes of the cross section for the cantilever beam at position

,s as shown in Figure 1(b). The symbols ( , )v s t and ( , )w s t denote the displacements of a point in the mid-line of the

cantilever beam in the y and zdirections, respectively.

Figure 1: The Model of a Cantilever Beam [14]. With length L, Mass m per Unit Length and Subjected to aHarmonic Axial Excitation and Transverse Excitations at the Free End: (a) The Model (b) A Segment

The modified model of the nonlinear nonplanar cantilever beam under combined multi external and parametric

excitations of the following form [14] is considered as:

3

2 2 2 31 1 i i 0 2 3 y y 3

i=1

y + cy + y 2 y F cos( T ) + y(yy + y + zz + z ) + y + (

3 3

2 2 2 3 2y 4 y 5 i i 0 i i 0

i=1 i=1y

1+(1 ) (1 ) )yz 2 (y + yz ) Fcos( T ) = f cos( T )

(1)

3

2 2 2 32 1 i i 0 2 3 y y 3

i=1

z + cz + z 2 z F cos( T ) + z(yy + y + zz + z ) + z + (

3 3

2 2 2 3 2y 4 y 5 i i 0 i i 0

i=1 i=1y

1(1 ) (1 ) )zy 2 (z + zy ) Fcos( T ) = g cos( T )

(2)

Where y and zdenote the displacements of a point in the middle line of the cantilever beam, 1 and 2 are

the lowest linear natural frequencies, is a small perturbation parameter, c is the linear damping coefficient, 1 is the

coefficient of linearity, 2 3 4 5, , , are the coefficients of the cubic nonlinearity, if and ig are the excitation


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(10)

(11)

The general solution of Eqs. (5) and (6) can be written in the form

1 0

0 0 1 2( , ) ,i T

y A T T e cc

(12)

2 0

0 0 1 2( , ) ,

i T

z B T T e cc

(13)

Where cc stands for the conjugate of the preceding terms. 0A and 0B are complex functions in 1T and 2T at this

stage of the analysis. They will be determined by eliminating the secular terms and small-divisor terms at the next

approximation order. Inserting Eqs. (12) and (13) into Eqs. (8) and (9), we get

1 02 2 2 2 2

0 1 1 1 1 0 0 2 1 3 y 0 0 0 0 0 y 3 y 4 5

y

1(D + )y = [-i (2D A + cA ) + (2 -3 )A A -2A B B ( + (1- ) - (1 ) )]

i T

y e

1 0 1 1 0 1 1 0 1 2 0 1 2 03i T i( + )T i( - )T i( + )T i( - )T3 2

0 2 1 3 y 1 1 0 1 2 0+A (2 - )e + FA (e + e ) + F A (e + e )

1 3 0 1 3 0 1 2 0i(

+ )T i( - )T i( +2 )T2 2 21 3 0 0 0 2 2 y 3 y 4 y 5

y

1+ F A (e + e ) + A B (2 - ( + (1- ) - (1- ) ))e

1 2 0 1 0 2 0 3 0i( -2 )T i T i T i T2 2 2 31 20 0 2 2 y 3 y 4 y 5

y

ff f1+A B (2 - ( + (1- ) - (1- ) ))e + e + e + e + cc

2 2 2

(14)

2 0i T2 2 2 2 2

0 2 1 2 1 0 0 2 2 3 0 0 0 0 0 y 3 y 4 y 5

y

1(D + )z = (-i (2D B +cB )+(2 -3 )B B -2A A B ( -( 1- ) - (1 - ) ))e

2 0 2 1 0 2 1 0 2 2 0 2 2 03i T i( + )T i( - )T i( + )T i( - )T3 2

0 2 2 3 1 1 0 1 2 0+B (2 - )e + F B (e + e ) + F B (e + e )

2 3 0 2 3 0 1 2 0i( + )T i( - )T i(2 + )T2 2 2

1 3 0 0 0 2 1 y 3 y 4 y 5y

1+ F B (e + e ) + A B (2 - ( - (1- ) - (1- ) ))e


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1 2 0 1 0 2 0 3 0i(2 - )T i T i T i T2 2 2 31 22 1 0 0 y 3 y 4 y 5

y

gg g1+(2 - A B ( - (1- ) - (1- ) ))e + e + e + e + cc

2 2 2

(15)

NUMERICAL RESULTS

The numerical solution for the second order nonlinear differential equations are obtained applying fourth order

Rung-Kutta method using MATLAP 7. The steady state response of the vibrating system is obtained and its stability is

investigated applying phase-plane, trajectories.

For the non-resonant case it can be seen from Figure 2, that the amplitude of the oscillation is a monotonic

decreasing function in time up to a steady state value of about (y=0.28) for the first mode and, (z=1.2) for the second mode.

y 1 2 3

1 2 1 2 3 4 5 1 2 3

1 2 3 1 2 3 1

Figure 2 : Non - resonant system behavior (basic case), c = 0.02, = 2.56, F = 15, F = 20, F = 25,

=1.6, = 2.8, = 0.004, = 0.002, = 0.1, = 0.03, = 0.4, = 4, = 2.9, =1.2,

= 7, = 6, = 5, F = 0.02, F = 0.03, F = 0.04, f = 2

2 3 1 2 3.5, f = 3, f = 3.5, g = 8, g = 10, g = 12 :

The deduced resonance cases at this approximation order are:

Trivial Resonance: 1 2 0

Primary Resonance: 1 2 1 2, , , , , ( 1, 2, 3)j k s s j k j k

Sub-Harmonic Resonances:1 2 2 1 2 1

1 2 1 2

2 , 2 , 2 , 3 ,

4 , 4 , 3 , 3

j k

j k s s

Super-Harmonic Resonances:1

2 12

Internal Resonance: 1 2

Combined Resonance: 1 1

1 2 1 2 1 22 22( ), ( ), ( ), 2( )j j k j k s

Simultaneous Resonance: 12 2 1 1 1 2 3 1 1 3 1 2 1 32

( , ), ( 2 , 2 ) , ( )and

Any combination of the above resonance cases is considered as simultaneous resonance.

Table 1, illustrates the results of some worst resonance cases. It is clear that the simultaneous sub-harmonic


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resonance case is the more worst one. The reported resonance case in table should be avoided in the design of such system.

Table 1: Summary of the Worst Resonance Cases

No Resonance Cases Resonance Condition Amplitude Ratio

1 simultaneous Primary 2 2 1 1, 500%, 280%

2 simultaneous sub-harmonic 1 2 3 12 , 2 465%, 185%

3 Simultaneous Combined 11 3 1 2 1 32

, ( ) 320%, 265%

4 Combined 1 1 22 570%, 175%

Figure 3: System Behavior at Simultaneous Resonance Case,2 2 1 1&

Figure 4: System Behavior at Simultaneous Resonance Case,1 2 3 12 & 2

Figure 5: System Behavior at Simultaneous Resonance Case, 11 3 1 2 1 32

, ( )


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Figure 6: System Behavior at Combined Resonance Case1 1 22

STABILITY ANALYSIS

To investigate the stability around sub-harmonic simultaneous resonance case, we introduce detuning parameters

1 2, , such that

1 2 2 3 1 1 1 2 2 3 1 1 = 2 + & = 2 + or 2 = & 2 = (16)

This case represents one of the main system worst cases as shown previously in table 1. Substituting Eqs. (12),

(13) and (16) into Eqs. (8) and (9). and eliminating the secular terms, leads to the solvability conditions for the second

order approximation noting that both 0A and 0B are functions in 1 2,T T we get

1 1

2

y i T2 2

1 1 0 0 3 y 2 1 0 0 0 0 0 y 3 y 4 5 1 3 0

y

(1- )i (2D A + cA ) + (3 2 )A A + 2A B B [ + (1 ) ] f A e = 0

(17)

2 1

2

y i T2 22 1 0 0 3 2 2 0 0 0 0 0 y 3 y 4 5 1 1 0

y

(1- )i (2D B + cB ) + (3 2 )B B + 2A A B [ (1 ) ] f B e =0

(18)

We consider 0A and 0B in the polar form as

1 1 2 1i (T ) i (T )1 10 02 2

a(T ) b(T )A = e , B = e (19)

Where ,a b are the steady state amplitudes and 1 2, are the phases of the motion. Substituting Eq. (19) into

Eqs. (17) and (18) and separating real and imaginary parts yields

1 31

1

f acaa + sin = 0,

2 2 (20)

23 2y2 1 3

1 2 1 3 y y 3 y 4 5 1

1 1 y 1

(1- ) f aa aba + (2 3 ) [ + (1 ) ] + cos = 0,

8 4 2 (21)

1 12

2

f bcbb + sin = 0

2 2 (22)


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23 2y2 1 1

2 2 2 3 y 3 y 4 5 2

2 2 y 2

(1- ) f bb a bb + (2 3 ) [ (1 ) ]+ cos = 0,

8 4 2 (23)

Where 1 1 1 1 = T 2 and 2 2 1 2 = T 2

Stability of the Fixed Points

The steady state solution of our dynamical system corresponding to the fixed points of Eqs. (20 to 23) are

obtained when 0a b and 1 2 0 , then we get the frequency response equations (FRE) for practical case

(where 0a and 0b ) as follows:

2

1 1 1 1 + + = 0 (24)

22 2 2 2 + + = 0 (25)

where 1 2 1 , , and 2 are defined in the Appendix.

To determine the stability of the steady state solution, one lets

10 11 1 1 10 11a = a + a (T ), = + (26)

10 11 1 2 20 21b = b + b (T ), = + (27)

Where 10 10 10, ,a b and 20 are the solutions of Eqs. (20 to 23) and 11 11 11, ,a b and 21 are perturbations,

which are assumed to be small compared to 10 10 10, ,a b and 20 . Substituting Eqs. (26) and (27) into Eqs. (20 to 23) and

keeping only the linear terms in 11 11 11, ,a b and 21 , we obtain

1 3 1 3 1011 10 11 10 11

1 1

f f aca = (- + sin )a + ( cos )

2 2 2 (28)

2 2 2210 2 1 3 y y10 1 3 1 3

11 1 y 3 y 4 5 10 11 10 11

10 1 1 y 1 1

3a (2 - 3 ) (1- )b f f 1 = ( [ + [ + (1 ) ]+ cos )])a ( sin )

a 4 2 (29)

1 1 1 111 20 11 10 20 21

2 2

f f cb = (- + sin )b + ( b cos ) ,

2 2 2 (30)

22 2 2y10 2 2 3 10 1 1 1 1

21 2 y 3 y 4 5 20 11 20 21

10 2 2 y 2 2

(1- )3b (2 - 3 ) a f f 1 = ( [ + [ (1 ) ] + cos ])b ( sin )

b 4 2 (31)

The Eigen Values of the above system of equations are given by

4 3 2

1 2 3 4 + r + r + r + r = 0 (32)

Where 1 2 3 4( , , , )r r r r are functions of the parameters 1 2 1 2 1( , , , , , , , ,a b c


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2 3 4 5 1 3 1 2, , , , , f , f , , )y . If and only if the real parts of all the eigenvalues are negative, then the

periodic solution near the worst resonance case is stable; otherwise, it is unstable. According to the Routh-Huriwitz

criterion, the necessary and sufficient conditions for all the roots of Eq. (32) to have negative real parts if and only if the

determinant D and all its principle minors are positive:

D=

1 2

3 4

5 6

7 8

+ 0 0

+ 0 0

0 0 +

0 0 +

(33)

The expressions for , 1,2, ...,8i i are given in appendix A.

RESULTS AND DISCUSSIONS

Results are presented in graphical form as steady state amplitudes a and b against detuning parameters 1 and

2 respectively for both vibrating modes. The solutions of the frequency response equations regarding the stability of the

system are shown in Figure 7 and 8. Figure 7a shows the effect of the detuning parameter 1 (while other parameters are

held constant), on the steady state amplitude a for the stability of the practical case, where 0a and 0b . It can be seen

that for a narrow range around 1 0.015 , the system is unstable. From Figure 7b, 7c and 7f we have the steady state

amplitude is a monotonic decreasing function to the parameters 1,c and 3 . It is clear from this figure that the stability

zone is decreased as c and 1 are increased. From Figure 7d, 7e and 7i we have the steady state amplitude is increasing

when the parameters 1 2, and 3f are increased. The stability zone is increased as 1 and 3f are increased also.

The nonlinear parameters 4 and 5 has no effect on stability zone and there exist a shift of the frequency response curve

to right for 4 and shift to the left for 5 as the values of both them are increasing as shown in Figure 7g and 7h.

Figure 8a shows the effect of the detuning parameter 2 on the steady state amplitude b for the same discussed stability

practical case. It can be seen that there is a narrow range around 2 0.036 , where the system is unstable.

From Figure 8b, 8c and 8f we have the steady state amplitude is decreasing when as the parameters 2 3,c and are

increased. The stability zone of these parameters is decreased also. From Figure 8d, 8e and 8i we have the steady state

amplitude is increasing when the values of the parameters 1 2, and 1f are increase. The stability zone of these

parameters is increased also. For the nonlinear parameters 4 and 5 , we have the steady state amplitude is increased and

shifted to the right when 4 is increased. The stability zone of this parameters is increased. The steady state amplitude is

decreased and shifted to left when 5 is increased. Also the stability zone of 5 is increased as illustrated in Figure 8g

and 8h.


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CONCLUSIONS

The system of nonlinear nonplanar oscillations of the cantilever beam subjected to a harmonic axial and

transverse excitations is analyzed using (MSPT). The steady state response and its stability near the simultaneous

sub-harmonic solution is obtained and studied. Effects of different parameters on the steady state response and their

stability are investigated.

The numerical solution and its stability of the vibrating system is obtained and studied applying Runge-Kutta

numerical technique. From this study, the following may be concluded:

The resonance cases should be avoided in the design of such a system as it may lead to system damage. The most

worst resonance case of the system is sub-harmonic simultaneous and

The steady state amplitude of the system are monotonic increasing function in the coefficients .

The steady state amplitude of the system are monotonic decreasing functions to .

The analytical solutions are in excellent agreement with the numerical integrations as demonstrated in Figure (7 a)

and (8 a).

Reference [14] studied the chaotic motion and its control for the nonlinear nonplanar oscillations of a cantilever

beam subjected to a harmonic axial and transverse excitations at the free end for the first time by using numerical

approaches. The case of 2:1 internal resonance, principal parametric resonance 1/2 sub-harmonic resonance was

considered. In the present paper we studied the stability at simultaneous sub-harmonic resonance case (the most

worst resonance case) using both frequency response equations and phase-plane method. The effects of the

different parameters on the steady state response of the vibrating system are studied

Figure 7a: Effect of the Detuning Parameter 1 Figure 7b: Effect of the Damping Coefficient c

Figure 7c: Effect of the Natural Frequency 1 Figure 7d: Effect of the Nonlinear Parameter 1


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Figure 7e: Effect of the Nonlinear Parameter 2 Figure 7f: Effect of the Nonlinear Parameter 3

Figure 7g: Effect of the Nonlinear Parameter 4 Figure 7h: Effect of the Nonlinear Parameter 5

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.10

1

2

3

4

5

6

a

7if3=25

f3=21

f3=20

Figure 7i: Effect of the Excitation Amplitude 3f

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.020

0.2

0.4

0.6

0.8

1

1.2

1.4

b

8a

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.020

0.2

0.4

0.6

0.8

1

1.2

1.4

b

8bc=0.002c=0.02

c=0.04

Figure 8a: Effect of the Detuning Parameter 2 Figure 8b: Effect of the Damping Coefficient c

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.020

0.2

0.4

0.6

0.8

1

1.2

1.4

b

8c

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

b

8d

Figure 8c: Effect of the Natural Frequency 2 Figure 8d: Effect of the NonlinearParameter 1


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-0.1 -0.08 -0.06 -0.04 -0.02 0 0.020

0.2

0.4

0.6

0.8

1

1.2

1.4

b

8e

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.020

0.2

0.4

0.6

0.8

1

1.2

1.4

b

8f

Figure 8e: Effect of the Nonlinear Parameter 2 Figure 8f: Effect of the Nonlinear Parameter 3

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.080

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

b

8g

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.040

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

b

8h

Figure 8g: Effect of the Nonlinear Parameter 4 Figure 8h: Effect of the Nonlinear Parameter 5

-0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.060

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

b

8if1=25

f1=20

f1=15

Figure 8i: Effect of the Excitation Amplitude 1f

Appendix A

22 2y2

1 2 1 3 y y 3 y 4 5

1 1 y

(1 )a b = (2 3 ) [ + (1 ) ],

2

22 2 2 2y2 2 21 3

1 2 1 3 y y 3 y 4 52

1 1 1 y

(1 ) F a b = c + ( (2 3 ) [ + (1 ) ]) , 4 2

22 2y2

2 2 2 3 y 3 y 4 5

2 2 y

(1 )b a = (2 3 ) [ (1 ) ]

2

22 2 2 2y2 2 21 1

2 2 2 3 y 3 y 4 52

2 2 2 y

(1 ) F b a = c + ( (2 3 ) [ (1 ) ])

4 2

1 3

1 10

1

Fc = + sin ,

2 2

1 3 102 10

1

F a = cos ,

2


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2 2 2210 2 1 3 y y10 1 3

3 1 y 3 y 4 5 10

10 1 1 y 1

3a (2 3 ) (1 )b F1 = [ + [ + (1 ) ] + cos )]

a 4 2

1 3 1 1 1 14 10 5 20 6 10 20

1 2 2

F F Fc = sin , = + sin , = b cos , 2 2 2 ,

22 2 2y10 2 2 3 10 1 1 1 1

7 2 y 3 y 4 5 20 8 20

10 2 2 y 2 2

(1 )3b (2 3 ) a F F1 = [ + [ (1 ) ] + cos ], = sin

b 4 2

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