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Nonlinear oscillation analysis of a pendulum wrapping on a cylinder
Scientia Iranica B (2012) 19 (2), 335–340
Sharif University of Technology Scientia Iranica Transactions B: Mechanical Engineering www.sciencedirect.com
Research note
Nonlinear oscillation analysis of a pendulum wrapping on a cylinder H. Mazaheri a , A. Hosseinzadeh b , M.T. Ahmadian a,∗ a b
School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran School of Mechanical Engineering, Amirkabir University of Technology, Tehran, P.O. Box 15875-4413, Iran
Received 26 July 2011; revised 12 October 2011; accepted 21 December 2011
KEYWORDS Nonlinear oscillation; Multiple scales method; Large amplitude; Simple pendulum.
Abstract In this paper, the nonlinear oscillation of a pendulum wrapping and unwrapping on two cylindrical bases is studied, and an analytical solution is obtained using the multiple scales method. The equation of motion is derived based on an energy conservation technique. By applying the perturbation method to the differential equation, the nonlinear natural frequency of the system is calculated, along with its time response. Analytical results are compared with numerical findings and good agreement is found. The effect of large amplitude and radius of cylinders on system frequency is evaluated. The results indicate that as the radius of the cylinder increases, the system frequency is increased. Also, it is illustrated that initial amplitude plays a dual role in the frequency. As the initial amplitude increases up to a certain point, the frequency is increased, while by increasing it to higher values, the system frequency decreases. © 2012 Sharif University of Technology. Production and hosting by Elsevier B.V. All rights reserved.
1. Introduction A simple pendulum has been a basic and practical example for the category of vibration problems. From its discovery, the most popular applications of the pendulum have been timekeeping, seismometers and, sometimes, accelerometers. It has been the basis for well-known mechanisms, such as inverted pendulums, variable length pendulums, etc. Many researchers have paid attention to obtaining the governing equation of pendulums with additional conditions along with their dynamic behavior. Belato et al. [1] investigated a nonideal system, consisting of a pendulum whose support point is vibrated along a horizontal guide. The motion is produced by means of a mechanism that converts rotational motion into longitudinal motion. They studied oscillations of the pendulum through variations of the supplied voltage of the electromotor.
∗
Corresponding author. E-mail address: [email protected] (M.T. Ahmadian).
1026-3098 © 2012 Sharif University of Technology. Production and hosting by Elsevier B.V. All rights reserved. Peer review under responsibility of Sharif University of Technology. doi:10.1016/j.scient.2012.02.014
Cai et al. [2] obtained the solution to a class of nonlinear problems with slowly varying parameters, with multiple scales and the KBM (Krylov-Bogoliubov-Mitropolsky) method. They solved the equation of motion of a varying length pendulum and compared the results of two methods. They also showed that the multiple scales method is equivalent to the KBM method for the first order approximation. Eissa and Sayed [3] studied the vibration reduction of a three-degree-of-freedom spring-pendulum system, subjected to harmonic excitation. They considered the effects of a transversely tuned absorber and active control on the behavior of the system near primary resonances. Amore and Aranda [4] applied Linear Delta Expansion to the Linstedt–Poincaré method and found improved approximate solutions for nonlinear problems. They examined their method for nonlinear pendulum problems and for general unharmonic excitations. Some researchers have studied chaos in the problem. For example, Idowu et al. [5] studied chaotic solutions of a parametrically undamped pendulum. They applied a shooting method to the problem and concluded that this method is more intuitive. Also, it gives more information about system behavior when chaos occurs. Amer and Bek [6] analyzed the chaotic response of a harmonically excited spring pendulum moving in a circular path. They used the multiple scales method in their solution, and showed that the system has bifurcation leading to chaos. Also, the dynamic control of these systems and especially inverted pendulums, is mostly studied in the literature. Anh et al. [7] investigated the vibration reduction for a stable
336
H. Mazaheri et al. / Scientia Iranica, Transactions B: Mechanical Engineering 19 (2012) 335–340
in which the absolute value is used, in order for the equation to be valid for negative values of θ . Let r /L = ε , then: sin θ
g
θ¨ +
L (1 − ε |θ |)
−
θ˙ 2 sgn (θ ) = 0, L (1 − ε |θ |)
r
(5)
in which ε is a small parameter. Expanding terms involving θ , results in: 1 sin θ = θ − θ 3 + · · · , 6 1 = 1 + ε |θ| + εθ 2 + · · · , ε |θ | ≪ 1. (6) (1 − ε |θ |) Substitution of Eq. (6) into Eq. (5) results in: g
θ¨ +
L
θ − θ + · · · 1 + ε |θ| + εθ 2 + · · · 1
3
6
r − θ˙ 2 1 + ε |θ| + εθ 2 + · · · sgn (θ ) = 0.
Figure 1: Simple pendulum wrapping around the cylinders.
L
inverted pendulum with a passive mass-spring-pendulum-type Dynamic Vibration Absorber (DVA). They found that the DVA, on an inverted pendulum, is more effective than the DVA on a normal pendulum. Ovseyevich [8] investigated the stability of the upper equilibrium position of a pendulum in cases where the suspension point makes rapid random oscillation with small amplitude. In this paper, the nonlinear vibration of a simple pendulum bounded by two cylinders at the point of suspension is studied (Figure 1). As the pendulum oscillates, its length varies due to wrapping around the cylinders. It is highly important to note that the cylinders could be considered as a magnified form of surface waviness in the pendulum support.
In this part, the equation of motion of a simple pendulum wrapping and unwrapping around cylinders at the point of suspension is presented. As shown in Figure 1, this system has one degree of freedom, and can be described by the generalized coordinates, θ . As a result of the wrapping and unwrapping of a pendulum around the cylinders, the string length is changed, causing a change in the natural frequency, even at small amplitude of vibration. The system shown in Figure 1 is a conservative system, thus we use the concept of conservation of energy to derive the equation of motion. The kinetic energy of the system is: T =
(1)
U = mg {L − (L − r |θ |) cos(θ ) − r sin(|θ|)} .
(2)
Using the energy conservation, we have: d(T + U )
= m (L − r |θ |)2 θ˙ θ¨ − r (L − r |θ |) θ˙ 3 + mg (L − r |θ|) θ˙ sin(θ ) + r θ˙ sgn(θ ) cos(θ ) − r θ˙ sgn(θ ) cos(|θ |) = 0. (3)
Finally, after some simplifications and considering that cos(|θ |) = cos(θ ), we have:
θ¨ +
g sin θ L − r |θ|
−
r θ˙ 2 L − r |θ |
sgn (θ ) = 0,
1
1
6
6
θ¨ + ω02 θ − ω02 θ 3 + εω02 θ |θ| − εω02 θ 3 |θ | − ε θ˙ 2 sgn (θ ) = 0, (8) √ g g where ω0 = g /L. Introducing L = ε r , the equation of motion
is:
1 g
1
6 r
6
θ¨ + ω02 θ − ε θ 3 + εω02 θ |θ| − εω02 θ 3 |θ | − ε θ˙ 2 sgn (θ ) = 0,
(9)
2.1. Solution method The multiple scales method is used to solve Eq. (9). First, the solution is assumed as [9]:
θ = θ0 + εθ1 + ε 2 θ2 + · · ·
(4)
(10)
Also, the differential operator is defined as below [9]: d dt d2
m(L − r |θ |) θ˙ . 2 2
2 Also, the potential energy of the system, with respect to the equilibrium point, is:
dt
Expanding the above equation, and neglecting θ 5 and higher powers of θ , the equation of motion may be written as:
where the nonlinear effects are expressed in terms of the perturbation parameter, ε , which is less than unity.
2. Theory and formulation
1
(7)
dt 2
= D0 + ε D1 + ε 2 D2 + · · · , = D20 + 2ε D0 D1 + ε 2 2D0 D2 + D22 + · · · ,
(11)
in which: Di =
d dTi
,
Ti = ε i t ,
i = 0, 1, 2, . . .
(12)
Substituting Eqs. (11) and (12) into Eq. (9) and expanding all terms up to the first order of ε , the coefficients of ε 0 and ε 1 on both sides of the equation should be equal:
ε 0 : D20 θ0 + ω02 θ0 = 0, ε 1 : D20 θ1 + ω02 θ1 = −2D0 D1 θ0 + 1
(13) 1L 6r
ω02 θ03
− ω02 θ0 |θ0 | + ω02 θ03 |θ0 | + θ˙02 sgn (θ0 ) .
6 The solution of Eq. (13) is:
θ0 = Aeiω0 T0 + Ae−iω0 T0 ,
(14)
(15)
H. Mazaheri et al. / Scientia Iranica, Transactions B: Mechanical Engineering 19 (2012) 335–340
in which A is a function of T1 . Substituting θ0 on the right hand side of Eq. (14), a nonhomogeneous differential equation in θ1 , with respect to T0 , is obtained. Because the solution must be periodic and bounded as t increases, coefficients of eiω0 T0 on the right hand side of Eq. (14), called secular terms, should be zero [9]. Thus: 1L
− 2iω0 D1 A +
1
ω02 A2 A − ω02 e1 + ω02 f1 + g1 = 0,
2r
6
(16)
where e1 , f1 and g1 are the coefficients of eiω0 T0 in the Fourier expansion of the last three terms in Eq. (8), and can be defined as [10]:
ω en = 2π
ω0
fn =
θ0 |θ0 | e−inωT0 dT0 ,
− ωπ
(17)
0
2π
π ω0
θ03 |θ | e−inω0 T0 dT0 ,
π ω0
ω0 2π
gn =
π ω0
π ω0
(18)
θ˙02 .sgn(θ )e−inω0 T0 dT0 .
− ωπ
(19)
0
A=
2
iβ
ae ,
a = a(T1 ),
ω0 T0 + β = φ,
dT0 =
ω0
.
(20)
The values of integrals in Eq. (17) to (19) for n = 1 can be calculated as follows: e1 =
= f1 =
a2 8π 4 3π
e
iβ
π+β
eiφ + e−iφ eiφ + e−iφ e−iφ .dT
−π +β 2 iβ
a e ,
a3 1
(21) π
eiβ
eiφ + e−iφ
3 iφ e + e−iφ e−iφ dφ
16 2π −π 16a4 iβ = e , 15π 2 a2 iβ π g1 = e −ω02 eiφ − e−iφ sgn eiφ + e−iφ 8π −π e−iφ dφ =
2a ω 2
3π
2 0 iβ
e .
(22)
− iω0 a + iaβ + ′
1L
(23)
2r
ω
2a 0
3
8
−ω
2 0
4 3π
a
2
β = −ω0 a
1 L 16 r
2
a−
3π
+
8 45π
a2
T0 + β0 .
(26)
In order to calculate the particular solution of Eq. (14), Fourier expansion of all nonhomogeneous terms should be evaluated first:
θ0 |θ0 | = e1 eiω0 T0 + e2 e2iω0 T0 + e3 e3iω0 T0 + e4 e4iω0 T0 + e5 e5iω0 T0 + · · · + C .C . θ03 |θ0 | = f1 eiω0 T0 + f2 e2iω0 T0 + f3 e3iω0 T0 + f4 e4iω0 T0 + f5 e5iω0 T0 + · · · + C .C . θ˙ 2 sgn(θ ) = g1 eiω0 T0 + g2 e2iω0 T0 + g3 e3iω0 T0 + g4 e4iω0 T0 + g5 e5iω0 T0 + · · · + C .C .,
(27)
in which C .C . is the complex conjugate of all previous terms, and en , fn and gn are obtained for n = 2, 3, 4, 5 from Eq. (17) to (19) as follows: e2 = f2 = g2 = 0,
g3 =
4 a2 15 π 14 a2 15 π
e3iβ , e3iβ ,
16 a4
16 a4
e3iβ , 35 π 4 a2 5iβ e5 = − e , 105 π
f3 =
e5iβ ,
14 a2
e5iβ . (28) 315 π 15 π It is highly important to note that since Fourier coefficients for n > 5 are very small, they can be neglected. Using the Fourier expansion obtained above, the solution of Eq. (14) becomes:
f5 =
g5 =
e3 + f3 + g3
ei3ω0 T0 × e3iβ ω02 − 9ω02 e5 + f5 + g5 i5ω0 T0 + 2 e × e5iβ + C .C . ω0 − 25ω02 e3 + f3 + g3 i(3ω0 T0 +3β) e =− 8ω02
θ1 =
−
Substituting Eqs. (21)–(23) into Eq. (16), the governing differential equation of a and β becomes:
a = a0 ,
e3 =
β = β(T1 ), dφ
By solving the two differential equations above, a and β could be obtained as:
e4 = f4 = g4 = 0
In order to determine A in terms of T1 , we define: 1
337
e5 + f5 + g5 i(5ω0 T0 +5β) e + C .C . 24ω02
(29)
Now, we can obtain θ by the first approximation:
θ = a cos(ωt + β0 ) − ε −ε
e5 + f5 + g5 24ω02
e3 + f3 + g3 8ω02
cos(3ωt + 3β0 )
cos(5ωt + 5β0 ),
1 L
2
8
Separating the real and imaginary parts of Eq. (24) and equating each part to zero results in:
ω = ω0 1 − ε a a− + a 16 r 3π 45π 1 2 2 8 2 = ω0 1 − a − ε a − + a . (30) 16 3π 45π It should be noted that the values of a and β0 should be calcu-
a′ = 0,
lated using the initial conditions of the problem. By imposing the initial condition on Eq. (30), one obtains:
1
+ ω02 6
β′ = − = −
16a
4
15π
1 L
16 r 1 L 16 r
+
ω 0 a2 − ω0 a20 −
2 3π
2 3π 2 3π
a2 ω02 = 0.
ω0 a +
8 45π 8
ω 0 a0 +
(24)
a2
45π
a20
θ |t =0 = a −
.
(25)
θ˙ |t =0 = 0.
1 8ω02
(e3 + f3 + g3 ) −
1 24ω02
2
(e5 + f5 + g5 ) , (31)
338
H. Mazaheri et al. / Scientia Iranica, Transactions B: Mechanical Engineering 19 (2012) 335–340
Figure 3: Time response of the system.
follows:
ω = ω0 1 − ε a
= ω0 1 −
1 16
1 L 16 r
a−
2 3π
a − εa − 2
+
2 3π
8 45π
+
a
8 45π
2
a
2
.
(32)
It is evident that when ε approaches zero, the nonlinear frequency is:
1 ω = ω 0 1 − a2 , 16
Figure 2: Time response of the system. (Analytic and numerical solution for different values of θ0 and r /L).
From the second condition of Eq. (31), β0 is obtained equal to zero. Then, a numerical technique, based on the Newton–Raphson method, is implemented to obtain a from Eq. (31) above. 3. Results and discussion In this section, we present and compare the analytical and numerical results obtained by the multiple scales and Runge–Kutta methods, respectively. Also, in order to validate the nonlinear frequency evaluated in Eq. (30), we rewrite as
(33)
which is equal to the nonlinear frequency of the large amplitude oscillation of a simple pendulum, as reported in [9]. As in any other vibration problem, it is necessary to obtain the time response of the system to validate the approximated solution. In Figure 2, the time response of the system, obtained by analytical and Runge–Kutta methods, is shown. It can be seen that for low values of θ0 or r /L, the agreement between the approximated analytical solution and the exact numerical solution is very good, and error is less than 1%. It should be noted that the larger the initial amplitude or r /L, the larger the error. In Figure 3 the time response of the wrapping pendulum is compared with the simple pendulum, in the case where no cylinder exists. As the system is nonlinear, we are to analyze its nonlinearity and specifically its effect on the frequency of vibration. There are two sources of nonlinearity in the system: the large amplitude of motion and wrapping around the fillet corner of the pendulum support. In Figure 4, the dependency of system frequency on initial amplitude is shown and compared for several values of cylinder radius. It can be easily seen that by increasing the initial amplitude, system frequency is increased, while, after a certain θ0 value, the frequency tends to decrease with respect to θ0 . In Figure 5, the effect of cylinder radius on the frequency of the system is shown. It can be recognized that the frequency varies, with respect to the cylinder radius, linearly. Variation of frequency versus θ and r /L in three-dimensional form is presented in Figure 6. The frequency of vibration is presented versus the pendulum length in Figure 7; it can be seen that increasing the length of the pendulum makes it oscillate with the lower frequency.
H. Mazaheri et al. / Scientia Iranica, Transactions B: Mechanical Engineering 19 (2012) 335–340
339
Figure 4: Dependency of nonlinear frequency on initial amplitude. Figure 7: Nonlinear frequency vs. pendulum length.
parameters on the vibration frequency is also investigated. It is shown that the nonideal suspension of the pendulum affects system behavior, especially its frequency. Nonlinear vibration of the pendulum is due to radius of the cylinder and large amplitude of oscillation. As the radius of cylinder increases, the nonlinear frequency is increased. But the effect of large initial amplitude is different. Results show that the nonlinear frequency increases by increasing the amplitude of pendulum for smaller values of the initial amplitude. But the nonlinear frequency of the pendulum decreases by increasing the amplitude for the higher values of the initial amplitude. References
Figure 5: Dependency of nonlinear frequency on cylinder radius (L = 1 m).
[1] Belato, D., Weber, H.I., Balthazar, J.M. and Mook, D.T. ‘‘Chaotic vibrations of a non ideal electro-mechanical system’’, Internat. J. Solids Structures, 38(10–13), pp. 1699–1706 (2001). [2] Cai, J., Wu, X. and Li, Y.P. ‘‘Comparison of multiple scales and KBM methods for strongly nonlinear oscillators with slowly varying parameters’’, Mech. Res. Comm., 31(5), pp. 519–524 (2004). [3] Eissa, M. and Sayed, M. ‘‘Vibration reduction of a three DOF nonlinear spring pendulum’’, Commun. Nonlinear Sci. Numer. Simul., 13(2), pp. 465–488 (2008). [4] Amore, P. and Aranda, A. ‘‘Improved Lindstedt-Poincaré method for the solution of nonlinear problems’’, J. Sound Vib., 283(3–5), pp. 1115–1136 (2005). [5] Idowu, B.A., Vincent, U.E. and Njah, A.N. ‘‘Synchronization of chaos in nonidentical parametrically excited systems’’, Chaos Solitons Fractals, 39(5), pp. 2322–2331 (2009). [6] Amera, T.S. and Bek, M.A. ‘‘Chaotic responses of a harmonically excited spring pendulum moving in circular path’’, Nonlinear Anal. RWA, 10(5), pp. 3196–3202 (2009). [7] Anh, N.D., Matsuhisa, H., Viet, L.D. and Yasuda, M. ‘‘Vibration control of an inverted pendulum type structure by passive mass-spring-pendulum dynamic vibration absorber’’, J. Sound Vib., 307(1–2), pp. 187–201 (2007). [8] Ovseyevich, A.I. ‘‘The stability of an inverted pendulum when there are rapid random oscillations of the suspension point’’, J. Appl. Math. Mech., 70(5), pp. 762–768 (2006). [9] Nayfeh, A.H. and Mook, D.T. ‘‘Conservative single-degree-of-freedom systems’’, In Nonlinear Oscillations, John Wiley & Sons, Inc. (1979). [10] Myint, U.T. and Debnath, L. ‘‘Fourier series and integrals with applications’’, In Linear Partial Differential Equations for Scientists and Engineers, fourth ed., pp. 167–220, Birkhäuser, Boston (2006).
Figure 6: Nonlinear frequency vs. cylinder radius and initial amplitude.
4. Conclusion The dynamic response of a large amplitude pendulum wrapping around a cylinder under the effect of gravitational force is studied. The multiple scales method is implemented to solve the nonlinear differential equation of the system and obtain the natural frequency. The effect of geometric
Hashem Mazaheri received his B.S. and M.S. degrees in Mechanical Engineering from Sharif University of Technology, Tehran, Iran, in 2007 and 2010, respectively, where he is now a Ph.D. degree candidate. His interests and research areas include vibration analysis in continuous environments, impact mechanics and solid mechanic. Ali Hosseinzadeh received his B.S. and M.S. degrees in Mechanical Engineering from Sharif University of Technology, Tehran, Iran, in 2007 and 2010, respectively, and is now a Ph.D. degree student in the Mechanical Engineering
340
H. Mazaheri et al. / Scientia Iranica, Transactions B: Mechanical Engineering 19 (2012) 335–340
Department of Amirkabir University of Technology. His interests and research areas include vibration and dynamic analysis, especially in micromechanical and piezoelectric systems. Mohammad Taghi Ahmadian received B.S. and M.S. degrees in Mechanical Engineering, and obtained Ph.D. degrees in both Space and Plasma Physics in 1980, and Mechanical Engineering in 1986 from the University of Kansas
at Lawrence, USA. From 1984 to 1985, he was Assistant Professor at the University of Missouri, Spring Field, and from 1985 to 1988, he was Assistant Professor in the Department of Mechanical Engineering at the University of Kansas at Lawrence, USA. Currently, he is Professor in the Department of Mechanical Engineering at Sharif University of Technology, Tehran, Iran. His research interests include: vibration of dynamic systems, design and analysis of micromechanical systems, structural analysis using finite element method, and dynamic analysis of laminated composite structures.
Sharif University of Technology Scientia Iranica Transactions B: Mechanical Engineering www.sciencedirect.com
Research note
Nonlinear oscillation analysis of a pendulum wrapping on a cylinder H. Mazaheri a , A. Hosseinzadeh b , M.T. Ahmadian a,∗ a b
School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran School of Mechanical Engineering, Amirkabir University of Technology, Tehran, P.O. Box 15875-4413, Iran
Received 26 July 2011; revised 12 October 2011; accepted 21 December 2011
KEYWORDS Nonlinear oscillation; Multiple scales method; Large amplitude; Simple pendulum.
Abstract In this paper, the nonlinear oscillation of a pendulum wrapping and unwrapping on two cylindrical bases is studied, and an analytical solution is obtained using the multiple scales method. The equation of motion is derived based on an energy conservation technique. By applying the perturbation method to the differential equation, the nonlinear natural frequency of the system is calculated, along with its time response. Analytical results are compared with numerical findings and good agreement is found. The effect of large amplitude and radius of cylinders on system frequency is evaluated. The results indicate that as the radius of the cylinder increases, the system frequency is increased. Also, it is illustrated that initial amplitude plays a dual role in the frequency. As the initial amplitude increases up to a certain point, the frequency is increased, while by increasing it to higher values, the system frequency decreases. © 2012 Sharif University of Technology. Production and hosting by Elsevier B.V. All rights reserved.
1. Introduction A simple pendulum has been a basic and practical example for the category of vibration problems. From its discovery, the most popular applications of the pendulum have been timekeeping, seismometers and, sometimes, accelerometers. It has been the basis for well-known mechanisms, such as inverted pendulums, variable length pendulums, etc. Many researchers have paid attention to obtaining the governing equation of pendulums with additional conditions along with their dynamic behavior. Belato et al. [1] investigated a nonideal system, consisting of a pendulum whose support point is vibrated along a horizontal guide. The motion is produced by means of a mechanism that converts rotational motion into longitudinal motion. They studied oscillations of the pendulum through variations of the supplied voltage of the electromotor.
∗
Corresponding author. E-mail address: [email protected] (M.T. Ahmadian).
1026-3098 © 2012 Sharif University of Technology. Production and hosting by Elsevier B.V. All rights reserved. Peer review under responsibility of Sharif University of Technology. doi:10.1016/j.scient.2012.02.014
Cai et al. [2] obtained the solution to a class of nonlinear problems with slowly varying parameters, with multiple scales and the KBM (Krylov-Bogoliubov-Mitropolsky) method. They solved the equation of motion of a varying length pendulum and compared the results of two methods. They also showed that the multiple scales method is equivalent to the KBM method for the first order approximation. Eissa and Sayed [3] studied the vibration reduction of a three-degree-of-freedom spring-pendulum system, subjected to harmonic excitation. They considered the effects of a transversely tuned absorber and active control on the behavior of the system near primary resonances. Amore and Aranda [4] applied Linear Delta Expansion to the Linstedt–Poincaré method and found improved approximate solutions for nonlinear problems. They examined their method for nonlinear pendulum problems and for general unharmonic excitations. Some researchers have studied chaos in the problem. For example, Idowu et al. [5] studied chaotic solutions of a parametrically undamped pendulum. They applied a shooting method to the problem and concluded that this method is more intuitive. Also, it gives more information about system behavior when chaos occurs. Amer and Bek [6] analyzed the chaotic response of a harmonically excited spring pendulum moving in a circular path. They used the multiple scales method in their solution, and showed that the system has bifurcation leading to chaos. Also, the dynamic control of these systems and especially inverted pendulums, is mostly studied in the literature. Anh et al. [7] investigated the vibration reduction for a stable
336
H. Mazaheri et al. / Scientia Iranica, Transactions B: Mechanical Engineering 19 (2012) 335–340
in which the absolute value is used, in order for the equation to be valid for negative values of θ . Let r /L = ε , then: sin θ
g
θ¨ +
L (1 − ε |θ |)
−
θ˙ 2 sgn (θ ) = 0, L (1 − ε |θ |)
r
(5)
in which ε is a small parameter. Expanding terms involving θ , results in: 1 sin θ = θ − θ 3 + · · · , 6 1 = 1 + ε |θ| + εθ 2 + · · · , ε |θ | ≪ 1. (6) (1 − ε |θ |) Substitution of Eq. (6) into Eq. (5) results in: g
θ¨ +
L
θ − θ + · · · 1 + ε |θ| + εθ 2 + · · · 1
3
6
r − θ˙ 2 1 + ε |θ| + εθ 2 + · · · sgn (θ ) = 0.
Figure 1: Simple pendulum wrapping around the cylinders.
L
inverted pendulum with a passive mass-spring-pendulum-type Dynamic Vibration Absorber (DVA). They found that the DVA, on an inverted pendulum, is more effective than the DVA on a normal pendulum. Ovseyevich [8] investigated the stability of the upper equilibrium position of a pendulum in cases where the suspension point makes rapid random oscillation with small amplitude. In this paper, the nonlinear vibration of a simple pendulum bounded by two cylinders at the point of suspension is studied (Figure 1). As the pendulum oscillates, its length varies due to wrapping around the cylinders. It is highly important to note that the cylinders could be considered as a magnified form of surface waviness in the pendulum support.
In this part, the equation of motion of a simple pendulum wrapping and unwrapping around cylinders at the point of suspension is presented. As shown in Figure 1, this system has one degree of freedom, and can be described by the generalized coordinates, θ . As a result of the wrapping and unwrapping of a pendulum around the cylinders, the string length is changed, causing a change in the natural frequency, even at small amplitude of vibration. The system shown in Figure 1 is a conservative system, thus we use the concept of conservation of energy to derive the equation of motion. The kinetic energy of the system is: T =
(1)
U = mg {L − (L − r |θ |) cos(θ ) − r sin(|θ|)} .
(2)
Using the energy conservation, we have: d(T + U )
= m (L − r |θ |)2 θ˙ θ¨ − r (L − r |θ |) θ˙ 3 + mg (L − r |θ|) θ˙ sin(θ ) + r θ˙ sgn(θ ) cos(θ ) − r θ˙ sgn(θ ) cos(|θ |) = 0. (3)
Finally, after some simplifications and considering that cos(|θ |) = cos(θ ), we have:
θ¨ +
g sin θ L − r |θ|
−
r θ˙ 2 L − r |θ |
sgn (θ ) = 0,
1
1
6
6
θ¨ + ω02 θ − ω02 θ 3 + εω02 θ |θ| − εω02 θ 3 |θ | − ε θ˙ 2 sgn (θ ) = 0, (8) √ g g where ω0 = g /L. Introducing L = ε r , the equation of motion
is:
1 g
1
6 r
6
θ¨ + ω02 θ − ε θ 3 + εω02 θ |θ| − εω02 θ 3 |θ | − ε θ˙ 2 sgn (θ ) = 0,
(9)
2.1. Solution method The multiple scales method is used to solve Eq. (9). First, the solution is assumed as [9]:
θ = θ0 + εθ1 + ε 2 θ2 + · · ·
(4)
(10)
Also, the differential operator is defined as below [9]: d dt d2
m(L − r |θ |) θ˙ . 2 2
2 Also, the potential energy of the system, with respect to the equilibrium point, is:
dt
Expanding the above equation, and neglecting θ 5 and higher powers of θ , the equation of motion may be written as:
where the nonlinear effects are expressed in terms of the perturbation parameter, ε , which is less than unity.
2. Theory and formulation
1
(7)
dt 2
= D0 + ε D1 + ε 2 D2 + · · · , = D20 + 2ε D0 D1 + ε 2 2D0 D2 + D22 + · · · ,
(11)
in which: Di =
d dTi
,
Ti = ε i t ,
i = 0, 1, 2, . . .
(12)
Substituting Eqs. (11) and (12) into Eq. (9) and expanding all terms up to the first order of ε , the coefficients of ε 0 and ε 1 on both sides of the equation should be equal:
ε 0 : D20 θ0 + ω02 θ0 = 0, ε 1 : D20 θ1 + ω02 θ1 = −2D0 D1 θ0 + 1
(13) 1L 6r
ω02 θ03
− ω02 θ0 |θ0 | + ω02 θ03 |θ0 | + θ˙02 sgn (θ0 ) .
6 The solution of Eq. (13) is:
θ0 = Aeiω0 T0 + Ae−iω0 T0 ,
(14)
(15)
H. Mazaheri et al. / Scientia Iranica, Transactions B: Mechanical Engineering 19 (2012) 335–340
in which A is a function of T1 . Substituting θ0 on the right hand side of Eq. (14), a nonhomogeneous differential equation in θ1 , with respect to T0 , is obtained. Because the solution must be periodic and bounded as t increases, coefficients of eiω0 T0 on the right hand side of Eq. (14), called secular terms, should be zero [9]. Thus: 1L
− 2iω0 D1 A +
1
ω02 A2 A − ω02 e1 + ω02 f1 + g1 = 0,
2r
6
(16)
where e1 , f1 and g1 are the coefficients of eiω0 T0 in the Fourier expansion of the last three terms in Eq. (8), and can be defined as [10]:
ω en = 2π
ω0
fn =
θ0 |θ0 | e−inωT0 dT0 ,
− ωπ
(17)
0
2π
π ω0
θ03 |θ | e−inω0 T0 dT0 ,
π ω0
ω0 2π
gn =
π ω0
π ω0
(18)
θ˙02 .sgn(θ )e−inω0 T0 dT0 .
− ωπ
(19)
0
A=
2
iβ
ae ,
a = a(T1 ),
ω0 T0 + β = φ,
dT0 =
ω0
.
(20)
The values of integrals in Eq. (17) to (19) for n = 1 can be calculated as follows: e1 =
= f1 =
a2 8π 4 3π
e
iβ
π+β
eiφ + e−iφ eiφ + e−iφ e−iφ .dT
−π +β 2 iβ
a e ,
a3 1
(21) π
eiβ
eiφ + e−iφ
3 iφ e + e−iφ e−iφ dφ
16 2π −π 16a4 iβ = e , 15π 2 a2 iβ π g1 = e −ω02 eiφ − e−iφ sgn eiφ + e−iφ 8π −π e−iφ dφ =
2a ω 2
3π
2 0 iβ
e .
(22)
− iω0 a + iaβ + ′
1L
(23)
2r
ω
2a 0
3
8
−ω
2 0
4 3π
a
2
β = −ω0 a
1 L 16 r
2
a−
3π
+
8 45π
a2
T0 + β0 .
(26)
In order to calculate the particular solution of Eq. (14), Fourier expansion of all nonhomogeneous terms should be evaluated first:
θ0 |θ0 | = e1 eiω0 T0 + e2 e2iω0 T0 + e3 e3iω0 T0 + e4 e4iω0 T0 + e5 e5iω0 T0 + · · · + C .C . θ03 |θ0 | = f1 eiω0 T0 + f2 e2iω0 T0 + f3 e3iω0 T0 + f4 e4iω0 T0 + f5 e5iω0 T0 + · · · + C .C . θ˙ 2 sgn(θ ) = g1 eiω0 T0 + g2 e2iω0 T0 + g3 e3iω0 T0 + g4 e4iω0 T0 + g5 e5iω0 T0 + · · · + C .C .,
(27)
in which C .C . is the complex conjugate of all previous terms, and en , fn and gn are obtained for n = 2, 3, 4, 5 from Eq. (17) to (19) as follows: e2 = f2 = g2 = 0,
g3 =
4 a2 15 π 14 a2 15 π
e3iβ , e3iβ ,
16 a4
16 a4
e3iβ , 35 π 4 a2 5iβ e5 = − e , 105 π
f3 =
e5iβ ,
14 a2
e5iβ . (28) 315 π 15 π It is highly important to note that since Fourier coefficients for n > 5 are very small, they can be neglected. Using the Fourier expansion obtained above, the solution of Eq. (14) becomes:
f5 =
g5 =
e3 + f3 + g3
ei3ω0 T0 × e3iβ ω02 − 9ω02 e5 + f5 + g5 i5ω0 T0 + 2 e × e5iβ + C .C . ω0 − 25ω02 e3 + f3 + g3 i(3ω0 T0 +3β) e =− 8ω02
θ1 =
−
Substituting Eqs. (21)–(23) into Eq. (16), the governing differential equation of a and β becomes:
a = a0 ,
e3 =
β = β(T1 ), dφ
By solving the two differential equations above, a and β could be obtained as:
e4 = f4 = g4 = 0
In order to determine A in terms of T1 , we define: 1
337
e5 + f5 + g5 i(5ω0 T0 +5β) e + C .C . 24ω02
(29)
Now, we can obtain θ by the first approximation:
θ = a cos(ωt + β0 ) − ε −ε
e5 + f5 + g5 24ω02
e3 + f3 + g3 8ω02
cos(3ωt + 3β0 )
cos(5ωt + 5β0 ),
1 L
2
8
Separating the real and imaginary parts of Eq. (24) and equating each part to zero results in:
ω = ω0 1 − ε a a− + a 16 r 3π 45π 1 2 2 8 2 = ω0 1 − a − ε a − + a . (30) 16 3π 45π It should be noted that the values of a and β0 should be calcu-
a′ = 0,
lated using the initial conditions of the problem. By imposing the initial condition on Eq. (30), one obtains:
1
+ ω02 6
β′ = − = −
16a
4
15π
1 L
16 r 1 L 16 r
+
ω 0 a2 − ω0 a20 −
2 3π
2 3π 2 3π
a2 ω02 = 0.
ω0 a +
8 45π 8
ω 0 a0 +
(24)
a2
45π
a20
θ |t =0 = a −
.
(25)
θ˙ |t =0 = 0.
1 8ω02
(e3 + f3 + g3 ) −
1 24ω02
2
(e5 + f5 + g5 ) , (31)
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H. Mazaheri et al. / Scientia Iranica, Transactions B: Mechanical Engineering 19 (2012) 335–340
Figure 3: Time response of the system.
follows:
ω = ω0 1 − ε a
= ω0 1 −
1 16
1 L 16 r
a−
2 3π
a − εa − 2
+
2 3π
8 45π
+
a
8 45π
2
a
2
.
(32)
It is evident that when ε approaches zero, the nonlinear frequency is:
1 ω = ω 0 1 − a2 , 16
Figure 2: Time response of the system. (Analytic and numerical solution for different values of θ0 and r /L).
From the second condition of Eq. (31), β0 is obtained equal to zero. Then, a numerical technique, based on the Newton–Raphson method, is implemented to obtain a from Eq. (31) above. 3. Results and discussion In this section, we present and compare the analytical and numerical results obtained by the multiple scales and Runge–Kutta methods, respectively. Also, in order to validate the nonlinear frequency evaluated in Eq. (30), we rewrite as
(33)
which is equal to the nonlinear frequency of the large amplitude oscillation of a simple pendulum, as reported in [9]. As in any other vibration problem, it is necessary to obtain the time response of the system to validate the approximated solution. In Figure 2, the time response of the system, obtained by analytical and Runge–Kutta methods, is shown. It can be seen that for low values of θ0 or r /L, the agreement between the approximated analytical solution and the exact numerical solution is very good, and error is less than 1%. It should be noted that the larger the initial amplitude or r /L, the larger the error. In Figure 3 the time response of the wrapping pendulum is compared with the simple pendulum, in the case where no cylinder exists. As the system is nonlinear, we are to analyze its nonlinearity and specifically its effect on the frequency of vibration. There are two sources of nonlinearity in the system: the large amplitude of motion and wrapping around the fillet corner of the pendulum support. In Figure 4, the dependency of system frequency on initial amplitude is shown and compared for several values of cylinder radius. It can be easily seen that by increasing the initial amplitude, system frequency is increased, while, after a certain θ0 value, the frequency tends to decrease with respect to θ0 . In Figure 5, the effect of cylinder radius on the frequency of the system is shown. It can be recognized that the frequency varies, with respect to the cylinder radius, linearly. Variation of frequency versus θ and r /L in three-dimensional form is presented in Figure 6. The frequency of vibration is presented versus the pendulum length in Figure 7; it can be seen that increasing the length of the pendulum makes it oscillate with the lower frequency.
H. Mazaheri et al. / Scientia Iranica, Transactions B: Mechanical Engineering 19 (2012) 335–340
339
Figure 4: Dependency of nonlinear frequency on initial amplitude. Figure 7: Nonlinear frequency vs. pendulum length.
parameters on the vibration frequency is also investigated. It is shown that the nonideal suspension of the pendulum affects system behavior, especially its frequency. Nonlinear vibration of the pendulum is due to radius of the cylinder and large amplitude of oscillation. As the radius of cylinder increases, the nonlinear frequency is increased. But the effect of large initial amplitude is different. Results show that the nonlinear frequency increases by increasing the amplitude of pendulum for smaller values of the initial amplitude. But the nonlinear frequency of the pendulum decreases by increasing the amplitude for the higher values of the initial amplitude. References
Figure 5: Dependency of nonlinear frequency on cylinder radius (L = 1 m).
[1] Belato, D., Weber, H.I., Balthazar, J.M. and Mook, D.T. ‘‘Chaotic vibrations of a non ideal electro-mechanical system’’, Internat. J. Solids Structures, 38(10–13), pp. 1699–1706 (2001). [2] Cai, J., Wu, X. and Li, Y.P. ‘‘Comparison of multiple scales and KBM methods for strongly nonlinear oscillators with slowly varying parameters’’, Mech. Res. Comm., 31(5), pp. 519–524 (2004). [3] Eissa, M. and Sayed, M. ‘‘Vibration reduction of a three DOF nonlinear spring pendulum’’, Commun. Nonlinear Sci. Numer. Simul., 13(2), pp. 465–488 (2008). [4] Amore, P. and Aranda, A. ‘‘Improved Lindstedt-Poincaré method for the solution of nonlinear problems’’, J. Sound Vib., 283(3–5), pp. 1115–1136 (2005). [5] Idowu, B.A., Vincent, U.E. and Njah, A.N. ‘‘Synchronization of chaos in nonidentical parametrically excited systems’’, Chaos Solitons Fractals, 39(5), pp. 2322–2331 (2009). [6] Amera, T.S. and Bek, M.A. ‘‘Chaotic responses of a harmonically excited spring pendulum moving in circular path’’, Nonlinear Anal. RWA, 10(5), pp. 3196–3202 (2009). [7] Anh, N.D., Matsuhisa, H., Viet, L.D. and Yasuda, M. ‘‘Vibration control of an inverted pendulum type structure by passive mass-spring-pendulum dynamic vibration absorber’’, J. Sound Vib., 307(1–2), pp. 187–201 (2007). [8] Ovseyevich, A.I. ‘‘The stability of an inverted pendulum when there are rapid random oscillations of the suspension point’’, J. Appl. Math. Mech., 70(5), pp. 762–768 (2006). [9] Nayfeh, A.H. and Mook, D.T. ‘‘Conservative single-degree-of-freedom systems’’, In Nonlinear Oscillations, John Wiley & Sons, Inc. (1979). [10] Myint, U.T. and Debnath, L. ‘‘Fourier series and integrals with applications’’, In Linear Partial Differential Equations for Scientists and Engineers, fourth ed., pp. 167–220, Birkhäuser, Boston (2006).
Figure 6: Nonlinear frequency vs. cylinder radius and initial amplitude.
4. Conclusion The dynamic response of a large amplitude pendulum wrapping around a cylinder under the effect of gravitational force is studied. The multiple scales method is implemented to solve the nonlinear differential equation of the system and obtain the natural frequency. The effect of geometric
Hashem Mazaheri received his B.S. and M.S. degrees in Mechanical Engineering from Sharif University of Technology, Tehran, Iran, in 2007 and 2010, respectively, where he is now a Ph.D. degree candidate. His interests and research areas include vibration analysis in continuous environments, impact mechanics and solid mechanic. Ali Hosseinzadeh received his B.S. and M.S. degrees in Mechanical Engineering from Sharif University of Technology, Tehran, Iran, in 2007 and 2010, respectively, and is now a Ph.D. degree student in the Mechanical Engineering
340
H. Mazaheri et al. / Scientia Iranica, Transactions B: Mechanical Engineering 19 (2012) 335–340
Department of Amirkabir University of Technology. His interests and research areas include vibration and dynamic analysis, especially in micromechanical and piezoelectric systems. Mohammad Taghi Ahmadian received B.S. and M.S. degrees in Mechanical Engineering, and obtained Ph.D. degrees in both Space and Plasma Physics in 1980, and Mechanical Engineering in 1986 from the University of Kansas
at Lawrence, USA. From 1984 to 1985, he was Assistant Professor at the University of Missouri, Spring Field, and from 1985 to 1988, he was Assistant Professor in the Department of Mechanical Engineering at the University of Kansas at Lawrence, USA. Currently, he is Professor in the Department of Mechanical Engineering at Sharif University of Technology, Tehran, Iran. His research interests include: vibration of dynamic systems, design and analysis of micromechanical systems, structural analysis using finite element method, and dynamic analysis of laminated composite structures.