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Principal parametric resonances of non-linear mechanical system with two-frequency and self-excitations
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 32 (2005) 337–350 www.elsevier.com/locate/mechrescom
Principal parametric resonances of non-linear mechanical system with two-frequency and self-excitations A.F. EL-Bassiouny Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt Available online 11 November 2004
Abstract The principal parametric resonance of a single-degree-of-freedom system with non-linear two-frequency parametric and self-excitations is investigated. In particular, the case in which the parametric excitation terms with close frequencies is examined. The method of multiple scales is used to determine the equations that describe to first-order the modulation of the amplitude and phase. Qualitative analysis and asymptotic expansion techniques are employed to predict the existence of steady state responses. Stability is investigated. The effect of damping, magnitudes of non-linear excitation and self-excitation are analyzed. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Non-linear system; Parametric excitation; Self-excitation; Principal resonance; Multiple scales method; Stability.
1. Introduction There are many phenomena in which parametric and self-excited vibrations interact with one another. Examples are flow-induced vibrations and vibrations in rotor systems. Moreover, some parametric excitation may contain two or more periodic components of one another. In particular, complex dynamic behavior has been observed when some of the excitation frequencies are close to one another. The response of single- or multi-degree-of-freedom systems to parametric excitations with single frequency have been investigated in (Kotera and Yano, 1985; Kojima et al., 1985; Yano, 1987; Lu and To, 1991; Schmidt and Tondl, 1986; Dosch et al., 1992; Oueini and Nayfeh, 1999; Sanchez and Nayfeh, 1997; Nayfeh and Zavodney, 1986; Zavodney and Nayfeh, 1988; Zavodney et al., 1989; Pratt, 1998; Arafat
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et al., 1998). Kotera and Yano (1985) dealt with a Van der Pol-Mathieu type equation with cubic non-linearities in the restoring force, which described a beam subjected to a periodic axial force and simultaneously to a flow-induced vibration. Periodic solution in the regions of principal and fundamental parametric resonances were approximated by the sum of two frequency components and a stability criterion for periodic solutions was established. Kojima et al. (1985) considered second-order superharmonic and one-half order subharmonic resonances in the vibration of a beam with a many subjected to vibration of a beam with a many subjected to an alternating electromagnetic force. Yano (1987) investigated some non-linear models describes by typical excitation mechanisms, in which the self-excitation was idealized by a non-linear resonance of Van der Pol Type and parametric excitation contains quadratic and cubic non-linearities. Both principal and fundamental parametric resonances were discussed in comparison to the analysis of linear modeling. Lu and To (1991) investigated the response of a single-degree-of-freedom non-linear system containing two-frequency parametric and self-excitations undergoing a principal parametric resonance. In particular, the case in which the parametric terms which have close frequencies is examined. In the monograph by Schmidt and Tondl (1986) there are some discussions on the interactions between parametric and self-excitations. Dosch et al. (1992) investigated a self-sensing piezoelectric actuator for collocated control. Oueini and Nayfeh (1999) studied single-mode control of a cantilever beam under principal parameter excitation. Sanchez and Nayfeh (1997) investigated the global behavior of a biased-non-linear oscillator under external and parametric excitations. Nayfeh and Zavodney (1986) studied the response of twodegree-of-freedom system with quadratic non-linearities undergoing a combination parametric resonance. In Zavodney and Nayfeh (1988), Zavodney et al. (1989) analyzed the responses of fundamental and principal resonance for a one-degree-of-freedom system with quadratic and cubic non-linearities. Complicated dynamic behavior, such as phenomena of jump, quenching, modal saturation, Hop bifurcation, periodmultiplying and demultiplying bifurcation s and chaos, etc. were exhibited through numerical computations. Moreover, multiple steady state responses and other periodic responses which depend on initial conditions were also found by numerical integration. Pratt (1998) investigated a non-linear vibration absorber for flexible structures. Arafat et al. (1998) studied non-linear non-planar dynamics of parametrically excited cantilever beam. El-Bassiouny and Abdelhafez (2001) investigated the predication of bifurcations for external and parametric excited one-degree-of-freedom system with quadratic, cubic, and quartic non-linearities. Elnaggar and El-Bassiouny (2003) studied the harmonic resonance of non-linear system of rods to harmonic excitation. El-Bassiouny (1999) analyzed the primary resonance of three-degree-offreedom system with cubic non-linearities in which the third mode is subjected to harmonic excitations. The response of linear and non-linear system subjected to multifrequency parametric excitation have been investigated in Nayfeh, 1983a,b; Elnaggar and El-Bassiouny, 1993, 1992; Chin et al., 1994; Barr, 1980. Nayfeh (1983a,b) investigated the response of combinations and principal parametric resonances of a two-degree-of-freedom system to multifrequency parametric excitations. The steady state responses and their stability were integration. Elnaggar and El-Bassiouny (1993, 1992) studied the response of selfexcited two-degree- and three-degree-of-freedom systems to multi-frequency excitations. They investigated harmonic, subharmonic, superharmonic, simultaneous sub/super-harmonic and combination resonances. Chin et al. (1994) studied the response of a non-linear system with a non-semisimple one-to-one resonance to a combination parametric resonance. Barr (1980) gave a review on the stability of non-linear parametric vibrations, including some discussion on multifrequency excitations. Chin and Nayfeh (1999) studied threeto-one internal resonance in parametrically excited hinged-clamped beams. El-Bassiouny et al. (2003) investigated Two-to-one internal resonance in non-linear two-degree-of-freedom system with parametric and external excitations. Davis and Rosenblat (1980) studied the transition curves between stable and unstable regions in the parameter plane for a generalized linear Mathieu-Hill equation with two-frequency parametric excitations. However, very limited attention has been paid to the study of the responses of systems with excitations of nearly equal frequencies possess beating phenomena, it is extremely important to seek a better understanding of the mechanics involved.
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The present work is therefore concerned with the response of single -degree-of-freedom non-linear system containing non-linear two-frequency and self-excitations undergoing principal parametric resonance of order 12. The system is governed by the following equation: 3 V€ þ 2e lV_ þ cV_ þ x2 V þ 2e½a1 cos U1 t þ a2 cos U2 tV 3 ¼ 0 ð1Þ where the dots indicate differentiation with respect to time t, e is a small dimensionless parameter, and l, x, c, a1, a2, U1, and U2 are constants. The method of multiple scales is used to, obtain the equations that govern the amplitude and phase and construct a first-order uniform expansion of the solution of Eq. (1). The fixed points of these equations are obtained and their stability is determined. Numerical solutions are presented.
2. General analysis To determine a first-order uniform expansion of the solutions of Eq. (1), one can use the method of multiple scales (Nayfeh, 1973, 1981; Guckenheimer and Holmes, 1983; Nayfeh and Balachandran, 1994) and let V ðt; eÞ ¼ V 0 ðT ; sÞ þ eV 1 ðT ; sÞ where T = t, s = et. Denote D0 ¼ powers of e on both sides yields Order of e0:
o , ot
ð2Þ D1 ¼
o . os
Substituting Eq. (2) into (1) and equating coefficients of like
D20 V 0 þ x2 V 0 ¼ 0
ð3Þ
Order of e:
h i 3 D20 V 1 þ 2D0 D1 V 0 þ 2 lðD0 V 0 Þ þ cðD0 V 0 Þ þ x2 V 1 þ 2½a1 cos U1 t þ a2 cos U2 tV 30 ¼ 0
ð4Þ
The solution of Eq. (3) can be expressed in the form V 0 ¼ AðsÞ expðixtÞ þ cc
ð5Þ
where cc represents the complex conjugate of the preceding terms, A(s) is the amplitude of the response and is a function of s. Hence, Eq. (4) becomes a1 h 3 D20 V 1 þ x2 V 1 ¼ 2ix A0 þ lA þ cx2 A2 A expðixtÞ 2ix3 cA3 expð3ixT Þ A exp fið/1 þ 3xÞT g 2 i 2 3 þ3A2 A expfið/1 þ xÞT g þ 3AA expfið/1 xÞT g þ A expfið/1 3xÞT g a2 h 3 2 A expfið/2 þ 3xÞT g þ 3A2 A expfið/2 þ xÞT g þ 3AA expfið/2 xÞT g 2 i 3 þA expfið/2 3xÞT g þ cc ð6Þ where the prime stands for the derivative with respect to s and the overbar stands for the complex conjugate. Any particular solution of Eq. (6) contains secular terms, which are generated by the first term on the right-hand side of Eq. (6). Moreover, it may contain small-divisor terms depending on the resonance condition. From Eq. (6), it can be seen that resonances occur when /1 ffi 2x or /2 ffi 2x or both hold. In what follows we shall investigate the principal parametric resonance of the system (1), in which /1 and /2 are either way from or close to each other.
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3. Principal parametric resonance (I) Here we consider the case of principal parametric resonance of order 12 when /1 ffi 2x but /2, is away from /1. The case of /2 ffi 2x but /1 being away from /2, can be treated in the same way. Introduce the detuning parameter r as follows: /1 ¼ 2x þ er
ð7Þ
and writes ð/1 xÞT ¼ xT þ rs
ð8Þ
Using (8), the small-divisor term arising from exp{i(/1 2x)} in Eq. (6) can be transformed into a secular term. Then, eliminating the secular terms yields 3 2 2ix A0 þ lA þ cx2 A2 A þ a1 AA expðirsÞ ¼ 0 ð9Þ 2 one can take 1 AðsÞ ¼ aðsÞ expfibðsÞg 2 in Eq. (9), where a and b are real, separate real and imaginary parts, and obtain 1 3a1 3 a0 ¼ la cx2 a3 a sinðrs 2bÞ 4 16x ab0 ¼
3a1 3 a cosðrs 2bÞ 16x
ð10Þ
ð11Þ ð12Þ
let f ¼ a2 ;
g ¼ rs 2b
ð13Þ
Then Eqs. (11) and (12) become 1 3a1 2 f sin g f0 ¼ 2lf cx2 f2 2 8x
ð14Þ
1 3a1 2 f ð r g0 Þ ¼ f cos g 2 8x
ð15Þ
which are the first-order non-linear equations governing the modulation of the amplitude and phase. Then, the first-order uniform expansion of the solution of Eq. (1) is given by 1 1 2 U ¼ ðfðetÞÞ cos / t gðetÞ þ cc þ OðeÞ ð16Þ 2 1 It is obvious that Eqs. (14) and (15) have a trivial solution f = 0 which corresponds to the trivial steady state response. Non-trivial steady state response correspond to the non-trivial fixed points of Eqs. (14) and (15). That is, they satisfy f 0 = 0, g 0 = 0 and are given by 1 3a1 2l þ cx2 f þ f sin g ¼ 0 2 8x
ð17Þ
1 3a1 r f cos g ¼ 0 2 8x
ð18Þ
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Eliminating sin g and cos g from Eqs. (17) and (18) yields the frequency response relation 2 1 2 1 9a21 2 f ¼0 2l þ cx f þ r2 2 4 64x2 ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9a21 1 2 4 1 2 2 2 4 2 lcx l c x 4 c x 16x2 4l þ 4 r n¼ 9a21 1 2 4 c x 2 4 64x
341
ð19Þ
2
i:e:
ð20Þ
To determine the stability of the trivial steady state response, it is convenient to rewrite A in the form 1 irs ð21Þ A ¼ ðx þ iyÞ exp 2 into linearized form of Eq. (9), one obtains 1 x0 ¼ lx þ ry 2 1 y 0 ¼ rx ly 2 The eigenvalues of the coefficient matrix of Eqs. (22) and (23) are 1 k2 þ 2lk þ l2 þ r2 ¼ 0 4
ð22Þ ð23Þ
ð24Þ
Then, the trivial solution is asymptotically stable if the real parts of kn(n = 1, 2) are less than zero, and it is unstable if at least one of the real parts of kn is larger than zero. The stable (unstable) solutions are represented by solid (dashed) lines on the r-axis. To determine the stability of the non-trivial steady state responses given by Eqs. (14) and (15), we use Eqs. (17) and (18). Let f ¼ f0 þ f1 ;
g ¼ g0 þ g1
ð25Þ
where (f0, g0) is given by Eqs. (17) and (18). Inserting Eq. (25) into Eqs. (14) and (15), using Eqs. (17) and (18) and keeping only the linear terms in f1 and g1, one obtains 3a1 3a1 2 0 2 f0 sin g0 f1 f cos g0 g1 f1 ¼ 2l cx f0 ð26Þ 4x 4x 0 r 3a1 3a1 0 g1 ¼ f0 cos g f1 þ f0 sin g0 g1 ð27Þ f0 4x 8x Eqs. (26) and (27) admit solution of the form (f1, g1)a exp(ks) provided that the solution is stable if and only if the real part of each of the eigenvalues of the coefficient of the matrix is less than zero. The stable solutions are represented by solid lines and the unstable by broken lines on the frequency response curves.
4. Principal parametric resonance (II) Here we consider the case of principal parametric resonance when /1 and /2 are close to 2x1. Assuming that /1 ffi /2. Since it reduces to the case of single -frequency parametric excitation if /1 5 /2. Introducing the detuning parameters r and r1, as follows:
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/1 ¼ 2x þ er;
/2 ¼ /1 þ er1
ð28Þ
Thus we have ð/1 xÞT ¼ xT þ rs;
ð/2 xÞT ¼ xT þ ðr þ r1 Þs
ð29Þ
Using Eq. (29), the small-divisor terms arising from exp{i(/n 2x)} in Eq. (6) can be transformed into secular terms. Then, eliminating the secular terms yields 3 3 2 2 2ix A0 þ lA þ cx2 A2 A þ a1 AA expðirsÞ þ a2 AA expfiðr þ r1 Þsg ¼ 0 ð30Þ 2 2 Inserting the polar form Eq. (10) of A and separating the real and imaginary parts of Eq. (30) yields 1 3a1 3 3a2 3 a sin g a sin g1 a0 ¼ la cx2 a3 4 16x 16x ab0 ¼
3a1 3 3a1 3 a cos g þ a cos g1 16x 16x
ð31Þ ð32Þ
where g1 ¼ ðr þ r1 Þs 2b ¼ r1 s þ g
ð33Þ
Using Eq. (13) into Eqs. (31) and (32), we get 1 3a1 2 3a2 2 f sin g f sin g1 f0 ¼ 2lf cx2 f2 2 8x 8x
ð34Þ
1 3a1 2 3a2 2 fðr g0 Þ ¼ f cos g þ f cos g1 2 8x 8x
ð35Þ
which are the first-order equations governing the modulation of the amplitude and phase in the case of /1 and /2 being close to 2x. We remark that the main difference between the modulation Eqs. (14), (15) and (34), (35) lies in the latter is a non-autonomous system. The corresponding perturbation solution V(t) is still given by (16).
Fig. 1. The amplitude f as a function of the detuning parameter r for increasing the coefficient of linear damping l.
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Fig. 2. The amplitude f as a function of the detuning parameter r for decreasing the coefficients of linear damping l and cubic damping c.
Fig. 3. The amplitude f as a function of the detuning parameter r for increasing the coefficient cubic damping c.
It is obvious that Eqs. (34) and (35) have a trivial solution f = 0. However, they have no non-trivial steady state solution at all. Indeed, if there were a non-trivial steady state solutions of Eqs. (34) and (35), then it should satisfy the following relations: 1 3a1 3a2 2l þ cx2 f þ f sin g þ f sin g1 ¼ 0 2 8x 8x
ð36Þ
1 3a1 3a2 r f cos g f cos g1 ¼ 0 2 8x 8x
ð37Þ
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Fig. 4. The amplitude f as a function of the detuning parameter r for increasing the natural frequency x.
Fig. 5. The amplitude f as a function of the detuning parameter r for decreasing the natural frequency x.
From Eqs. (36) and (37), we can get the frequency response equation
1 2l þ cx2 f 2
i:e: n ¼
2
C2
3a1 1 9 2 2 1 2 3ar 2 cx f cos g ¼ 0 2l þ þ r2 þ f a a f f sin g þ 1 2 4 64x2 2 8x 4x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C22 4C1 C3 2C1
ð38Þ
ð39Þ
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Fig. 6. The amplitude f as a function of the detuning parameter r for increasing the coefficient of the first non-linear excitation a1.
Fig. 7. The amplitude f as a function of the detuning parameter r for decreasing the coefficient of the first non-linear excitation a1.
where 3a1 cx 1 9 2 sin g C1 ¼ c2 x4 þ a a22 þ 4 64x2 1 8 3a1 l 3a1 r sin g cos g C2 ¼ 2lcx2 þ 2x 8x 1 C3 ¼ 4l2 þ r2 4
ð40Þ
The stability of the trivial solution is given also by Eq. (24) and the stability of the non-trivial solution can be obtained by the same method as Section 3.
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Fig. 8. The amplitude f as a function of the detuning parameter r for increasing the natural frequency x.
Fig. 9. The amplitude f as a function of the detuning parameter r for increasing the coefficient of cubic damping c.
5. Numerical results and discussion The analytical analysis is represented graphically by the numerical method. The frequency response Eqs. (19) and (38) are non-linear algebraic equations in the amplitude a. These equations are solved numerically by using bisection method. We focus our attention on the positive real roots of these equations. Figs. 1–7 represent the frequency response curves of the case of principal parametric resonance (I) for the parameters l = 0.3, c = 0.9, x = 1 and a1 = 10. From the geometry of the Figs. 1–7 we note that there exist two solutions, the trivial solution corresponding to a = 0 which has stable solution and the non-trivial solution which correspond to a 5 0 which has single valued and symmetric stable solution about the origin. In Fig. 1, the frequency-response curve has single valued and symmetric stable solution about the origin. When l is increased further, the symmetric branch moves to the upper with increasing magnitudes and when l = 10, we observe that the regions of definition and stability are decreasing. As l is decreased up to 0.01, we are not strongly affected and when c is decreased up to 0.01, we get the same variation. Also,
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Fig. 10. The amplitude f as a function of the detuning parameter r for decreasing the coefficient of cubic damping c.
Fig. 11. The amplitude f as a function of the detuning parameter r for increasing the coefficient of linear damping l.
the variation of decreasing the coefficient of cubic damping c is similar to the variation of l, as shown in Fig. 2. Figs. 3 and 4 show the effect of the coefficient of cubic damping c and the natural frequency x. For increasing c and x respectively, we note that the amplitude a has increasing stable solutions and for further increasing c and x respectively, the regions of definition and stability are decreasing with increasing in the amplitude a. Figs. 5 and 6 show the effect of the natural frequency x and the coefficient of the first parametric excitations a1. As x and a1 are decreasing respectively, the amplitude a has decreasing magnitudes respectively. When a1 is decreased respectively, we get the same variation as Fig. 4 when x is increased, Fig. 7. Figs. 8–14 represent the frequency response curves of the case of principal parametric resonance (II) for the parameters l = 1.5, c = 2, x = 1, a1 = 14 and a2 = 0. From the geometry of these Figs. 8–14 we note that there exist two solutions, the trivial solution corresponding to a = 0 has stable and unstable solutions which represented on the r-axis by dashed and solid lines respectively and the non-trivial solution a 5 0,
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Fig. 12. The amplitude f as a function of the detuning parameter r for decreasing the coefficient of the first non-linear excitation a1.
Fig. 13. The amplitude f as a function of the detuning parameter r for increasing the coefficient of the second non-linear excitation a2.
the response amplitude has stable and unstable solution which represented by solid and broken lines on the response curves. Fig. 8 shows the influence of the natural frequency x on the response curve. As x decreases, the amplitude a has decreasing magnitudes which given increasing in the zone of definition and the solution become unstable. Fig. 9 shows the effect of c on the frequency response curves. As c increases, the continuous curve has increasing magnitudes with decreasing in the zones of definition and stability. When c is decreased up to 0.01, we note that the frequency response curves are not strongly affected by this variation, Fig. 10. Fig. 11 shows the effect of l on the frequency response curves. As l increases, the continuous curve moves to the upper with increasing magnitudes and the region of stability is increased respectively. Fig. 12 shows the influence of the amplitudes of the first excitation a1 for a given r. Here also one observes that the amplitude a has decreased magnitudes through the interval (20 3) and after this interval it has increasing magnitudes with decreasing in the region of definition and the region of stability is increased. Fig. 13 shows the influence of the amplitude of the second excitation a2 for a given r. As a2
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Fig. 14. The amplitude f as a function of the detuning parameter r for decreasing the coefficient of the second non-linear excitation a2.
increases, the continuous curve has decreasing magnitudes which give increasing and decreasing in the zones of definition and stability respectively. As a2 is decreased further, we get the same variation as Fig. 9 with increasing in the region of stability, Fig. 14. 6. Conclusions The method of multiple scales has used to investigate the response of a non-linear mechanical system with two-frequency parametric and self-excitations. The analysis is focuses on principal parametric resonance of order 12. Two first-order ordinary differential equations are derived for the modulation of the amplitude and phase. Qualitative analysis and asymptotic expansion techniques are employed to predict existence of the steady state responses and stability. The following conclusions can be drawn from the analysis: (1) In the first case, we note that the amplitude a has symmetric solution about the origin. (2) From the two cases we note that when the damping coefficient l is increased, the continuous curve moves to the upper and given decreasing in the region of definition. (3) In the first case, we observe that all the non-trivial solutions have stable solutions. (4) In the second case, the trivial and non-trivial solutions have stable and unstable solutions. (5) The solution is unstable for decreasing the natural frequency x in the second case. (6) When the coefficient of cubic damping c is decreased for the two cases, we note that the frequency response curves are not strongly affected and we also show the same phenomena in the first case when l is decreased. (7) When l = 12 in the second case, we note that all the solutions are stable. References Arafat, N., Nayfeh, A.H., Chin, C.-M., 1998. Non-linear nonplanar dynamics of parametrically excited cantilever beam. Nonlinear Dynamics 15, 31–61. Barr, A.D.S., 1980. Some developments in parametric stability and nonlinear vibration. Recent Advances in Structural Dynamics 2, 545–567.
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Chin, C.-M., Nayfeh, A.H., 1999. Three-to-one internal resonance in parametrically excited hinged clamped beams. Nonlinear Dynamics 20, 131–158. Chin, C., Nayfeh, A.H., Mook, D.T., 1994. The response of a nonlinear system with a nonsemisimple one-to-one resonance to a combination parametric resonance. International Journal of Bifurcation and Chaos 5 (4), 971–982. Davis, S.H., Rosenblat, S., 1980. A quasiperiodic Mathieu–Hill equation. Siam Journal of Applied Mathematics 34, 139–155. Dosch, J.J., Inman, D.J., Garcia, E., 1992. A self-sensing piezoelectric actuator for collocated control. Journal of Intelligent Materials Systems and Structures 3, 166–185. El-Bassiouny, A.F., 1999. Response of a three-degree-of-freedom system with cubic non-linearities to harmonic excitation. Applied Mathematics and Computation 104, 65–84. El-Bassiouny, A.F., Abdelhafez, H.M., 2001. Prediction of bifurcation for external and parametric excited-degree-of-freedom system with quadratic, cubic, and quartic non-linearities. Mathematics and Computers in Simulation 57, 61–80. Elnaggar, A.M., El-Bassiouny, A.F., 1992. Response of self-excited three-degree-of-freedoms to multifrequency excitations. International Journal of Theoretical Physics 31 (8), 1531–1548. Elnaggar, A.M., El-Bassiouny, A.F., 1993. Harmonic, subharmonic, super-harmonic, simultaneous sub/super harmonic and combination resonances of self-excited two coupled second order systems to multifrequency excitations. Mechanica Sinica 8 (1), 61–71. Elnaggar, A.M., El-Bassiouny, A.F., 2003. Periodic and non-periodic combination resonance in kinematically excited system of rods. Applied Mathematics and Computation 139, 179–199. El-Bassiouny, A.F., Kamel, M.M., Abdel-Kalik, A., 2003. Two-to-one internal resonance in nonlinear two-degree-of-freedom system with parametric and external excitations. Mathematics and Computers in Simulation 57, 61–80. Guckenheimer, J., Holmes, P.J., 1983. Non-linear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springerverlager-Verlag, New York. Kojima, H., Nagaya, K., Shiraishi, H., Yamashita, A., 1985. Non-linear vibration of a beam with a mass subjected to alternating electromagentic force. Bulletin of the Japan Society of Mechanical Engineers 28, 468–475. Kotera, S., Yano, S., 1985. Periodic solutions and the stability in a non-linear parametric excitation system. Bulletin of the Japan Society of Mechanical Engineers 28, 1473–1480. Lu, Q.S., To, C.W.S., 1991. Principal resonance of a non-linear system with two-frequency parametric and self-excitations. Nonlinear Dynamics 2, 419–444. Nayfeh, A.H., 1973. Perturbation Methods. Wiley Interscience, New York. Nayfeh, A.H., 1981. Introduction of Perturbation Techniques. Wiley Interscience, New York. Nayfeh, A.H., 1983a. Response of two-degree-of-freedom systems to multifrequency parametric excitations. Journal of Sound and Vibration 88, 1–10. Nayfeh, A.H., 1983b. Parametric excited multi-degree-of-freedom systems with repeated frequencies. Journal of Sound and Vibration 88, 145–150. Nayfeh, A.H., Balachandran, B., 1994. Applied Non-linear Dynamics. Wiley Interscience, New York. Nayfeh, A.M., Zavodney, L.D., 1986. The response of two-degree-of-freedom systems with quadratic non-linearities to a combination parametric resonance. Journal of Sound and Vibration 107, 329–350. Oueini, S.S., Nayfeh, A.H., 1999. Single-mode control of a cantilever beam under principal parametric excitation 224 (1), 33–47. Pratt, J.R., 1998. A non-linear vibration absorber for flexible structures. Nonlinear Dynamics 15, 259–282. Sanchez, N.E., Nayfeh, A.H., 1997. Global behavior of a biased-non-linear oscillator under external and parametric excitations. Journal of Sound and Vibration 207 (2), 137–149. Schmidt, G., Tondl, A., 1986. Non-linear Vibrations. Akademic-Verlag, Berlin. Yano, S., 1987. Analytic research on dynamic phenomena of parametrically and self-excited mechanical systems. Ingenieur-Archiv 57, 51–60. Zavodney, L.D., Nayfeh, A.H., 1988. The response of a single-degree-of-freedom system with quadratic and cubic non-linearities to a fundamental parametric resonance. Journal of Sound and Vibration 120, 63–93. Zavodney, L.D., Nayfeh, A.H., Sanchez, E.N., 1989. The response of a single-degree-of-freedom system with quadratic and cubic nonlinearities to a principal parametric resonance. Journal of Sound and Vibration 129, 417–442.
Mechanics Research Communications 32 (2005) 337–350 www.elsevier.com/locate/mechrescom
Principal parametric resonances of non-linear mechanical system with two-frequency and self-excitations A.F. EL-Bassiouny Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt Available online 11 November 2004
Abstract The principal parametric resonance of a single-degree-of-freedom system with non-linear two-frequency parametric and self-excitations is investigated. In particular, the case in which the parametric excitation terms with close frequencies is examined. The method of multiple scales is used to determine the equations that describe to first-order the modulation of the amplitude and phase. Qualitative analysis and asymptotic expansion techniques are employed to predict the existence of steady state responses. Stability is investigated. The effect of damping, magnitudes of non-linear excitation and self-excitation are analyzed. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Non-linear system; Parametric excitation; Self-excitation; Principal resonance; Multiple scales method; Stability.
1. Introduction There are many phenomena in which parametric and self-excited vibrations interact with one another. Examples are flow-induced vibrations and vibrations in rotor systems. Moreover, some parametric excitation may contain two or more periodic components of one another. In particular, complex dynamic behavior has been observed when some of the excitation frequencies are close to one another. The response of single- or multi-degree-of-freedom systems to parametric excitations with single frequency have been investigated in (Kotera and Yano, 1985; Kojima et al., 1985; Yano, 1987; Lu and To, 1991; Schmidt and Tondl, 1986; Dosch et al., 1992; Oueini and Nayfeh, 1999; Sanchez and Nayfeh, 1997; Nayfeh and Zavodney, 1986; Zavodney and Nayfeh, 1988; Zavodney et al., 1989; Pratt, 1998; Arafat
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et al., 1998). Kotera and Yano (1985) dealt with a Van der Pol-Mathieu type equation with cubic non-linearities in the restoring force, which described a beam subjected to a periodic axial force and simultaneously to a flow-induced vibration. Periodic solution in the regions of principal and fundamental parametric resonances were approximated by the sum of two frequency components and a stability criterion for periodic solutions was established. Kojima et al. (1985) considered second-order superharmonic and one-half order subharmonic resonances in the vibration of a beam with a many subjected to vibration of a beam with a many subjected to an alternating electromagnetic force. Yano (1987) investigated some non-linear models describes by typical excitation mechanisms, in which the self-excitation was idealized by a non-linear resonance of Van der Pol Type and parametric excitation contains quadratic and cubic non-linearities. Both principal and fundamental parametric resonances were discussed in comparison to the analysis of linear modeling. Lu and To (1991) investigated the response of a single-degree-of-freedom non-linear system containing two-frequency parametric and self-excitations undergoing a principal parametric resonance. In particular, the case in which the parametric terms which have close frequencies is examined. In the monograph by Schmidt and Tondl (1986) there are some discussions on the interactions between parametric and self-excitations. Dosch et al. (1992) investigated a self-sensing piezoelectric actuator for collocated control. Oueini and Nayfeh (1999) studied single-mode control of a cantilever beam under principal parameter excitation. Sanchez and Nayfeh (1997) investigated the global behavior of a biased-non-linear oscillator under external and parametric excitations. Nayfeh and Zavodney (1986) studied the response of twodegree-of-freedom system with quadratic non-linearities undergoing a combination parametric resonance. In Zavodney and Nayfeh (1988), Zavodney et al. (1989) analyzed the responses of fundamental and principal resonance for a one-degree-of-freedom system with quadratic and cubic non-linearities. Complicated dynamic behavior, such as phenomena of jump, quenching, modal saturation, Hop bifurcation, periodmultiplying and demultiplying bifurcation s and chaos, etc. were exhibited through numerical computations. Moreover, multiple steady state responses and other periodic responses which depend on initial conditions were also found by numerical integration. Pratt (1998) investigated a non-linear vibration absorber for flexible structures. Arafat et al. (1998) studied non-linear non-planar dynamics of parametrically excited cantilever beam. El-Bassiouny and Abdelhafez (2001) investigated the predication of bifurcations for external and parametric excited one-degree-of-freedom system with quadratic, cubic, and quartic non-linearities. Elnaggar and El-Bassiouny (2003) studied the harmonic resonance of non-linear system of rods to harmonic excitation. El-Bassiouny (1999) analyzed the primary resonance of three-degree-offreedom system with cubic non-linearities in which the third mode is subjected to harmonic excitations. The response of linear and non-linear system subjected to multifrequency parametric excitation have been investigated in Nayfeh, 1983a,b; Elnaggar and El-Bassiouny, 1993, 1992; Chin et al., 1994; Barr, 1980. Nayfeh (1983a,b) investigated the response of combinations and principal parametric resonances of a two-degree-of-freedom system to multifrequency parametric excitations. The steady state responses and their stability were integration. Elnaggar and El-Bassiouny (1993, 1992) studied the response of selfexcited two-degree- and three-degree-of-freedom systems to multi-frequency excitations. They investigated harmonic, subharmonic, superharmonic, simultaneous sub/super-harmonic and combination resonances. Chin et al. (1994) studied the response of a non-linear system with a non-semisimple one-to-one resonance to a combination parametric resonance. Barr (1980) gave a review on the stability of non-linear parametric vibrations, including some discussion on multifrequency excitations. Chin and Nayfeh (1999) studied threeto-one internal resonance in parametrically excited hinged-clamped beams. El-Bassiouny et al. (2003) investigated Two-to-one internal resonance in non-linear two-degree-of-freedom system with parametric and external excitations. Davis and Rosenblat (1980) studied the transition curves between stable and unstable regions in the parameter plane for a generalized linear Mathieu-Hill equation with two-frequency parametric excitations. However, very limited attention has been paid to the study of the responses of systems with excitations of nearly equal frequencies possess beating phenomena, it is extremely important to seek a better understanding of the mechanics involved.
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The present work is therefore concerned with the response of single -degree-of-freedom non-linear system containing non-linear two-frequency and self-excitations undergoing principal parametric resonance of order 12. The system is governed by the following equation: 3 V€ þ 2e lV_ þ cV_ þ x2 V þ 2e½a1 cos U1 t þ a2 cos U2 tV 3 ¼ 0 ð1Þ where the dots indicate differentiation with respect to time t, e is a small dimensionless parameter, and l, x, c, a1, a2, U1, and U2 are constants. The method of multiple scales is used to, obtain the equations that govern the amplitude and phase and construct a first-order uniform expansion of the solution of Eq. (1). The fixed points of these equations are obtained and their stability is determined. Numerical solutions are presented.
2. General analysis To determine a first-order uniform expansion of the solutions of Eq. (1), one can use the method of multiple scales (Nayfeh, 1973, 1981; Guckenheimer and Holmes, 1983; Nayfeh and Balachandran, 1994) and let V ðt; eÞ ¼ V 0 ðT ; sÞ þ eV 1 ðT ; sÞ where T = t, s = et. Denote D0 ¼ powers of e on both sides yields Order of e0:
o , ot
ð2Þ D1 ¼
o . os
Substituting Eq. (2) into (1) and equating coefficients of like
D20 V 0 þ x2 V 0 ¼ 0
ð3Þ
Order of e:
h i 3 D20 V 1 þ 2D0 D1 V 0 þ 2 lðD0 V 0 Þ þ cðD0 V 0 Þ þ x2 V 1 þ 2½a1 cos U1 t þ a2 cos U2 tV 30 ¼ 0
ð4Þ
The solution of Eq. (3) can be expressed in the form V 0 ¼ AðsÞ expðixtÞ þ cc
ð5Þ
where cc represents the complex conjugate of the preceding terms, A(s) is the amplitude of the response and is a function of s. Hence, Eq. (4) becomes a1 h 3 D20 V 1 þ x2 V 1 ¼ 2ix A0 þ lA þ cx2 A2 A expðixtÞ 2ix3 cA3 expð3ixT Þ A exp fið/1 þ 3xÞT g 2 i 2 3 þ3A2 A expfið/1 þ xÞT g þ 3AA expfið/1 xÞT g þ A expfið/1 3xÞT g a2 h 3 2 A expfið/2 þ 3xÞT g þ 3A2 A expfið/2 þ xÞT g þ 3AA expfið/2 xÞT g 2 i 3 þA expfið/2 3xÞT g þ cc ð6Þ where the prime stands for the derivative with respect to s and the overbar stands for the complex conjugate. Any particular solution of Eq. (6) contains secular terms, which are generated by the first term on the right-hand side of Eq. (6). Moreover, it may contain small-divisor terms depending on the resonance condition. From Eq. (6), it can be seen that resonances occur when /1 ffi 2x or /2 ffi 2x or both hold. In what follows we shall investigate the principal parametric resonance of the system (1), in which /1 and /2 are either way from or close to each other.
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3. Principal parametric resonance (I) Here we consider the case of principal parametric resonance of order 12 when /1 ffi 2x but /2, is away from /1. The case of /2 ffi 2x but /1 being away from /2, can be treated in the same way. Introduce the detuning parameter r as follows: /1 ¼ 2x þ er
ð7Þ
and writes ð/1 xÞT ¼ xT þ rs
ð8Þ
Using (8), the small-divisor term arising from exp{i(/1 2x)} in Eq. (6) can be transformed into a secular term. Then, eliminating the secular terms yields 3 2 2ix A0 þ lA þ cx2 A2 A þ a1 AA expðirsÞ ¼ 0 ð9Þ 2 one can take 1 AðsÞ ¼ aðsÞ expfibðsÞg 2 in Eq. (9), where a and b are real, separate real and imaginary parts, and obtain 1 3a1 3 a0 ¼ la cx2 a3 a sinðrs 2bÞ 4 16x ab0 ¼
3a1 3 a cosðrs 2bÞ 16x
ð10Þ
ð11Þ ð12Þ
let f ¼ a2 ;
g ¼ rs 2b
ð13Þ
Then Eqs. (11) and (12) become 1 3a1 2 f sin g f0 ¼ 2lf cx2 f2 2 8x
ð14Þ
1 3a1 2 f ð r g0 Þ ¼ f cos g 2 8x
ð15Þ
which are the first-order non-linear equations governing the modulation of the amplitude and phase. Then, the first-order uniform expansion of the solution of Eq. (1) is given by 1 1 2 U ¼ ðfðetÞÞ cos / t gðetÞ þ cc þ OðeÞ ð16Þ 2 1 It is obvious that Eqs. (14) and (15) have a trivial solution f = 0 which corresponds to the trivial steady state response. Non-trivial steady state response correspond to the non-trivial fixed points of Eqs. (14) and (15). That is, they satisfy f 0 = 0, g 0 = 0 and are given by 1 3a1 2l þ cx2 f þ f sin g ¼ 0 2 8x
ð17Þ
1 3a1 r f cos g ¼ 0 2 8x
ð18Þ
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Eliminating sin g and cos g from Eqs. (17) and (18) yields the frequency response relation 2 1 2 1 9a21 2 f ¼0 2l þ cx f þ r2 2 4 64x2 ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9a21 1 2 4 1 2 2 2 4 2 lcx l c x 4 c x 16x2 4l þ 4 r n¼ 9a21 1 2 4 c x 2 4 64x
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ð19Þ
2
i:e:
ð20Þ
To determine the stability of the trivial steady state response, it is convenient to rewrite A in the form 1 irs ð21Þ A ¼ ðx þ iyÞ exp 2 into linearized form of Eq. (9), one obtains 1 x0 ¼ lx þ ry 2 1 y 0 ¼ rx ly 2 The eigenvalues of the coefficient matrix of Eqs. (22) and (23) are 1 k2 þ 2lk þ l2 þ r2 ¼ 0 4
ð22Þ ð23Þ
ð24Þ
Then, the trivial solution is asymptotically stable if the real parts of kn(n = 1, 2) are less than zero, and it is unstable if at least one of the real parts of kn is larger than zero. The stable (unstable) solutions are represented by solid (dashed) lines on the r-axis. To determine the stability of the non-trivial steady state responses given by Eqs. (14) and (15), we use Eqs. (17) and (18). Let f ¼ f0 þ f1 ;
g ¼ g0 þ g1
ð25Þ
where (f0, g0) is given by Eqs. (17) and (18). Inserting Eq. (25) into Eqs. (14) and (15), using Eqs. (17) and (18) and keeping only the linear terms in f1 and g1, one obtains 3a1 3a1 2 0 2 f0 sin g0 f1 f cos g0 g1 f1 ¼ 2l cx f0 ð26Þ 4x 4x 0 r 3a1 3a1 0 g1 ¼ f0 cos g f1 þ f0 sin g0 g1 ð27Þ f0 4x 8x Eqs. (26) and (27) admit solution of the form (f1, g1)a exp(ks) provided that the solution is stable if and only if the real part of each of the eigenvalues of the coefficient of the matrix is less than zero. The stable solutions are represented by solid lines and the unstable by broken lines on the frequency response curves.
4. Principal parametric resonance (II) Here we consider the case of principal parametric resonance when /1 and /2 are close to 2x1. Assuming that /1 ffi /2. Since it reduces to the case of single -frequency parametric excitation if /1 5 /2. Introducing the detuning parameters r and r1, as follows:
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/1 ¼ 2x þ er;
/2 ¼ /1 þ er1
ð28Þ
Thus we have ð/1 xÞT ¼ xT þ rs;
ð/2 xÞT ¼ xT þ ðr þ r1 Þs
ð29Þ
Using Eq. (29), the small-divisor terms arising from exp{i(/n 2x)} in Eq. (6) can be transformed into secular terms. Then, eliminating the secular terms yields 3 3 2 2 2ix A0 þ lA þ cx2 A2 A þ a1 AA expðirsÞ þ a2 AA expfiðr þ r1 Þsg ¼ 0 ð30Þ 2 2 Inserting the polar form Eq. (10) of A and separating the real and imaginary parts of Eq. (30) yields 1 3a1 3 3a2 3 a sin g a sin g1 a0 ¼ la cx2 a3 4 16x 16x ab0 ¼
3a1 3 3a1 3 a cos g þ a cos g1 16x 16x
ð31Þ ð32Þ
where g1 ¼ ðr þ r1 Þs 2b ¼ r1 s þ g
ð33Þ
Using Eq. (13) into Eqs. (31) and (32), we get 1 3a1 2 3a2 2 f sin g f sin g1 f0 ¼ 2lf cx2 f2 2 8x 8x
ð34Þ
1 3a1 2 3a2 2 fðr g0 Þ ¼ f cos g þ f cos g1 2 8x 8x
ð35Þ
which are the first-order equations governing the modulation of the amplitude and phase in the case of /1 and /2 being close to 2x. We remark that the main difference between the modulation Eqs. (14), (15) and (34), (35) lies in the latter is a non-autonomous system. The corresponding perturbation solution V(t) is still given by (16).
Fig. 1. The amplitude f as a function of the detuning parameter r for increasing the coefficient of linear damping l.
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Fig. 2. The amplitude f as a function of the detuning parameter r for decreasing the coefficients of linear damping l and cubic damping c.
Fig. 3. The amplitude f as a function of the detuning parameter r for increasing the coefficient cubic damping c.
It is obvious that Eqs. (34) and (35) have a trivial solution f = 0. However, they have no non-trivial steady state solution at all. Indeed, if there were a non-trivial steady state solutions of Eqs. (34) and (35), then it should satisfy the following relations: 1 3a1 3a2 2l þ cx2 f þ f sin g þ f sin g1 ¼ 0 2 8x 8x
ð36Þ
1 3a1 3a2 r f cos g f cos g1 ¼ 0 2 8x 8x
ð37Þ
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Fig. 4. The amplitude f as a function of the detuning parameter r for increasing the natural frequency x.
Fig. 5. The amplitude f as a function of the detuning parameter r for decreasing the natural frequency x.
From Eqs. (36) and (37), we can get the frequency response equation
1 2l þ cx2 f 2
i:e: n ¼
2
C2
3a1 1 9 2 2 1 2 3ar 2 cx f cos g ¼ 0 2l þ þ r2 þ f a a f f sin g þ 1 2 4 64x2 2 8x 4x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C22 4C1 C3 2C1
ð38Þ
ð39Þ
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Fig. 6. The amplitude f as a function of the detuning parameter r for increasing the coefficient of the first non-linear excitation a1.
Fig. 7. The amplitude f as a function of the detuning parameter r for decreasing the coefficient of the first non-linear excitation a1.
where 3a1 cx 1 9 2 sin g C1 ¼ c2 x4 þ a a22 þ 4 64x2 1 8 3a1 l 3a1 r sin g cos g C2 ¼ 2lcx2 þ 2x 8x 1 C3 ¼ 4l2 þ r2 4
ð40Þ
The stability of the trivial solution is given also by Eq. (24) and the stability of the non-trivial solution can be obtained by the same method as Section 3.
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Fig. 8. The amplitude f as a function of the detuning parameter r for increasing the natural frequency x.
Fig. 9. The amplitude f as a function of the detuning parameter r for increasing the coefficient of cubic damping c.
5. Numerical results and discussion The analytical analysis is represented graphically by the numerical method. The frequency response Eqs. (19) and (38) are non-linear algebraic equations in the amplitude a. These equations are solved numerically by using bisection method. We focus our attention on the positive real roots of these equations. Figs. 1–7 represent the frequency response curves of the case of principal parametric resonance (I) for the parameters l = 0.3, c = 0.9, x = 1 and a1 = 10. From the geometry of the Figs. 1–7 we note that there exist two solutions, the trivial solution corresponding to a = 0 which has stable solution and the non-trivial solution which correspond to a 5 0 which has single valued and symmetric stable solution about the origin. In Fig. 1, the frequency-response curve has single valued and symmetric stable solution about the origin. When l is increased further, the symmetric branch moves to the upper with increasing magnitudes and when l = 10, we observe that the regions of definition and stability are decreasing. As l is decreased up to 0.01, we are not strongly affected and when c is decreased up to 0.01, we get the same variation. Also,
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Fig. 10. The amplitude f as a function of the detuning parameter r for decreasing the coefficient of cubic damping c.
Fig. 11. The amplitude f as a function of the detuning parameter r for increasing the coefficient of linear damping l.
the variation of decreasing the coefficient of cubic damping c is similar to the variation of l, as shown in Fig. 2. Figs. 3 and 4 show the effect of the coefficient of cubic damping c and the natural frequency x. For increasing c and x respectively, we note that the amplitude a has increasing stable solutions and for further increasing c and x respectively, the regions of definition and stability are decreasing with increasing in the amplitude a. Figs. 5 and 6 show the effect of the natural frequency x and the coefficient of the first parametric excitations a1. As x and a1 are decreasing respectively, the amplitude a has decreasing magnitudes respectively. When a1 is decreased respectively, we get the same variation as Fig. 4 when x is increased, Fig. 7. Figs. 8–14 represent the frequency response curves of the case of principal parametric resonance (II) for the parameters l = 1.5, c = 2, x = 1, a1 = 14 and a2 = 0. From the geometry of these Figs. 8–14 we note that there exist two solutions, the trivial solution corresponding to a = 0 has stable and unstable solutions which represented on the r-axis by dashed and solid lines respectively and the non-trivial solution a 5 0,
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Fig. 12. The amplitude f as a function of the detuning parameter r for decreasing the coefficient of the first non-linear excitation a1.
Fig. 13. The amplitude f as a function of the detuning parameter r for increasing the coefficient of the second non-linear excitation a2.
the response amplitude has stable and unstable solution which represented by solid and broken lines on the response curves. Fig. 8 shows the influence of the natural frequency x on the response curve. As x decreases, the amplitude a has decreasing magnitudes which given increasing in the zone of definition and the solution become unstable. Fig. 9 shows the effect of c on the frequency response curves. As c increases, the continuous curve has increasing magnitudes with decreasing in the zones of definition and stability. When c is decreased up to 0.01, we note that the frequency response curves are not strongly affected by this variation, Fig. 10. Fig. 11 shows the effect of l on the frequency response curves. As l increases, the continuous curve moves to the upper with increasing magnitudes and the region of stability is increased respectively. Fig. 12 shows the influence of the amplitudes of the first excitation a1 for a given r. Here also one observes that the amplitude a has decreased magnitudes through the interval (20 3) and after this interval it has increasing magnitudes with decreasing in the region of definition and the region of stability is increased. Fig. 13 shows the influence of the amplitude of the second excitation a2 for a given r. As a2
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Fig. 14. The amplitude f as a function of the detuning parameter r for decreasing the coefficient of the second non-linear excitation a2.
increases, the continuous curve has decreasing magnitudes which give increasing and decreasing in the zones of definition and stability respectively. As a2 is decreased further, we get the same variation as Fig. 9 with increasing in the region of stability, Fig. 14. 6. Conclusions The method of multiple scales has used to investigate the response of a non-linear mechanical system with two-frequency parametric and self-excitations. The analysis is focuses on principal parametric resonance of order 12. Two first-order ordinary differential equations are derived for the modulation of the amplitude and phase. Qualitative analysis and asymptotic expansion techniques are employed to predict existence of the steady state responses and stability. The following conclusions can be drawn from the analysis: (1) In the first case, we note that the amplitude a has symmetric solution about the origin. (2) From the two cases we note that when the damping coefficient l is increased, the continuous curve moves to the upper and given decreasing in the region of definition. (3) In the first case, we observe that all the non-trivial solutions have stable solutions. (4) In the second case, the trivial and non-trivial solutions have stable and unstable solutions. (5) The solution is unstable for decreasing the natural frequency x in the second case. (6) When the coefficient of cubic damping c is decreased for the two cases, we note that the frequency response curves are not strongly affected and we also show the same phenomena in the first case when l is decreased. (7) When l = 12 in the second case, we note that all the solutions are stable. References Arafat, N., Nayfeh, A.H., Chin, C.-M., 1998. Non-linear nonplanar dynamics of parametrically excited cantilever beam. Nonlinear Dynamics 15, 31–61. Barr, A.D.S., 1980. Some developments in parametric stability and nonlinear vibration. Recent Advances in Structural Dynamics 2, 545–567.
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Chin, C.-M., Nayfeh, A.H., 1999. Three-to-one internal resonance in parametrically excited hinged clamped beams. Nonlinear Dynamics 20, 131–158. Chin, C., Nayfeh, A.H., Mook, D.T., 1994. The response of a nonlinear system with a nonsemisimple one-to-one resonance to a combination parametric resonance. International Journal of Bifurcation and Chaos 5 (4), 971–982. Davis, S.H., Rosenblat, S., 1980. A quasiperiodic Mathieu–Hill equation. Siam Journal of Applied Mathematics 34, 139–155. Dosch, J.J., Inman, D.J., Garcia, E., 1992. A self-sensing piezoelectric actuator for collocated control. Journal of Intelligent Materials Systems and Structures 3, 166–185. El-Bassiouny, A.F., 1999. Response of a three-degree-of-freedom system with cubic non-linearities to harmonic excitation. Applied Mathematics and Computation 104, 65–84. El-Bassiouny, A.F., Abdelhafez, H.M., 2001. Prediction of bifurcation for external and parametric excited-degree-of-freedom system with quadratic, cubic, and quartic non-linearities. Mathematics and Computers in Simulation 57, 61–80. Elnaggar, A.M., El-Bassiouny, A.F., 1992. Response of self-excited three-degree-of-freedoms to multifrequency excitations. International Journal of Theoretical Physics 31 (8), 1531–1548. Elnaggar, A.M., El-Bassiouny, A.F., 1993. Harmonic, subharmonic, super-harmonic, simultaneous sub/super harmonic and combination resonances of self-excited two coupled second order systems to multifrequency excitations. Mechanica Sinica 8 (1), 61–71. Elnaggar, A.M., El-Bassiouny, A.F., 2003. Periodic and non-periodic combination resonance in kinematically excited system of rods. Applied Mathematics and Computation 139, 179–199. El-Bassiouny, A.F., Kamel, M.M., Abdel-Kalik, A., 2003. Two-to-one internal resonance in nonlinear two-degree-of-freedom system with parametric and external excitations. Mathematics and Computers in Simulation 57, 61–80. Guckenheimer, J., Holmes, P.J., 1983. Non-linear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springerverlager-Verlag, New York. Kojima, H., Nagaya, K., Shiraishi, H., Yamashita, A., 1985. Non-linear vibration of a beam with a mass subjected to alternating electromagentic force. Bulletin of the Japan Society of Mechanical Engineers 28, 468–475. Kotera, S., Yano, S., 1985. Periodic solutions and the stability in a non-linear parametric excitation system. Bulletin of the Japan Society of Mechanical Engineers 28, 1473–1480. Lu, Q.S., To, C.W.S., 1991. Principal resonance of a non-linear system with two-frequency parametric and self-excitations. Nonlinear Dynamics 2, 419–444. Nayfeh, A.H., 1973. Perturbation Methods. Wiley Interscience, New York. Nayfeh, A.H., 1981. Introduction of Perturbation Techniques. Wiley Interscience, New York. Nayfeh, A.H., 1983a. Response of two-degree-of-freedom systems to multifrequency parametric excitations. Journal of Sound and Vibration 88, 1–10. Nayfeh, A.H., 1983b. Parametric excited multi-degree-of-freedom systems with repeated frequencies. Journal of Sound and Vibration 88, 145–150. Nayfeh, A.H., Balachandran, B., 1994. Applied Non-linear Dynamics. Wiley Interscience, New York. Nayfeh, A.M., Zavodney, L.D., 1986. The response of two-degree-of-freedom systems with quadratic non-linearities to a combination parametric resonance. Journal of Sound and Vibration 107, 329–350. Oueini, S.S., Nayfeh, A.H., 1999. Single-mode control of a cantilever beam under principal parametric excitation 224 (1), 33–47. Pratt, J.R., 1998. A non-linear vibration absorber for flexible structures. Nonlinear Dynamics 15, 259–282. Sanchez, N.E., Nayfeh, A.H., 1997. Global behavior of a biased-non-linear oscillator under external and parametric excitations. Journal of Sound and Vibration 207 (2), 137–149. Schmidt, G., Tondl, A., 1986. Non-linear Vibrations. Akademic-Verlag, Berlin. Yano, S., 1987. Analytic research on dynamic phenomena of parametrically and self-excited mechanical systems. Ingenieur-Archiv 57, 51–60. Zavodney, L.D., Nayfeh, A.H., 1988. The response of a single-degree-of-freedom system with quadratic and cubic non-linearities to a fundamental parametric resonance. Journal of Sound and Vibration 120, 63–93. Zavodney, L.D., Nayfeh, A.H., Sanchez, E.N., 1989. The response of a single-degree-of-freedom system with quadratic and cubic nonlinearities to a principal parametric resonance. Journal of Sound and Vibration 129, 417–442.