窄带外激作用下多自由度非线性动力系统响应的研究
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摘要
本文用解析和数值方法研究了窄带随机噪声外激下非线性系统的响应问题。论文的主要内容如下：
第一章阐明了窄带随机噪声激励下非线性系统响应问题的研究现状，简述了论文的主要内容和结构安排。
第二章和第三章研究了系统(1．1)在窄带随机外部激励下的共振响应问题。主要讨论了w_2≈3w_1，Ω≈3w_1时的情况。对该系统的研究分为两个部分：首先在第二章集中讨论了相应的确定性系统，即v=0时的情况。用多尺度法分离了系统的快变项，并详细推导出以系统振幅为未知数的代数方程组；对得到的代数方程组，由于其复杂性，不能直接进行讨论，而只能借助计算机编程实现对方程的求解；对系统响应的稳定性，也进行了讨论。在第二章基础上，第三章将多尺度法引入到相应的随机系统的研究中；严格推导了系统的约简方程，用矩方法求出稳态解应满足的方程，获得一些结果；并且数值模拟结果与理论推导的结果是一致的；并注意到，与其对应的确定性系统相比较，系统响应从周期解变为近似周期解，系统的相轨线从极限环变为扩大的近似极限环；随着激励带宽的增大，此扩大的近似极限环的宽度将增大。
第四章，给出了全文的总结，并对需要进一步进行的工作做了介绍。
In this dissertation, the response in two-degree-of-freedom nonlinear system under random external excitation is investigated by using analytical and numerical method. Some conclusions are obtained. The main contents of the dissertation are as follows:
Chapter one summarizes the situation and application of response in nonlinear dynamical system under narrow-band random excitation and presents the main contents and arrangement of the dissertation.
In chapter two and chapter three, the response of two-degree-of freedom nonlinear system (1.1) is studied.In chapter two, the case when v = 0 is discussed. The method of multiple scales is used to determine the equations of modulation of amplitude and phase .The steady state response can be obtained by solving a couple of algebraic equations, which have been achieved by careful deduction under some conditions. And because of the complexity of the equations, programs are necessary to solve the equations mentioned above, and certain graphs are presented. Based on chapter two, In chapter three, the method of multiple scales is introduced to the study of the multiple-dimensional nonlinear stochastic systems under random external excitation. Using method of multiple scales, we strictly deduce the equation of modulation of amplitude and phase. The effects of random excitations are analyzed; numerical simulations verify the results. Theoretical and numerical simulations show that when the intensity of the random ex
citation increases, the nontrivial steady state solution may be changed from a limit cycle to a diffused limit cycle.
In chapter four we concludes the work and points out some aspects to be further studied.
第一章阐明了窄带随机噪声激励下非线性系统响应问题的研究现状，简述了论文的主要内容和结构安排。
第二章和第三章研究了系统(1．1)在窄带随机外部激励下的共振响应问题。主要讨论了w_2≈3w_1，Ω≈3w_1时的情况。对该系统的研究分为两个部分：首先在第二章集中讨论了相应的确定性系统，即v=0时的情况。用多尺度法分离了系统的快变项，并详细推导出以系统振幅为未知数的代数方程组；对得到的代数方程组，由于其复杂性，不能直接进行讨论，而只能借助计算机编程实现对方程的求解；对系统响应的稳定性，也进行了讨论。在第二章基础上，第三章将多尺度法引入到相应的随机系统的研究中；严格推导了系统的约简方程，用矩方法求出稳态解应满足的方程，获得一些结果；并且数值模拟结果与理论推导的结果是一致的；并注意到，与其对应的确定性系统相比较，系统响应从周期解变为近似周期解，系统的相轨线从极限环变为扩大的近似极限环；随着激励带宽的增大，此扩大的近似极限环的宽度将增大。
第四章，给出了全文的总结，并对需要进一步进行的工作做了介绍。
In this dissertation, the response in two-degree-of-freedom nonlinear system under random external excitation is investigated by using analytical and numerical method. Some conclusions are obtained. The main contents of the dissertation are as follows:
Chapter one summarizes the situation and application of response in nonlinear dynamical system under narrow-band random excitation and presents the main contents and arrangement of the dissertation.
In chapter two and chapter three, the response of two-degree-of freedom nonlinear system (1.1) is studied.In chapter two, the case when v = 0 is discussed. The method of multiple scales is used to determine the equations of modulation of amplitude and phase .The steady state response can be obtained by solving a couple of algebraic equations, which have been achieved by careful deduction under some conditions. And because of the complexity of the equations, programs are necessary to solve the equations mentioned above, and certain graphs are presented. Based on chapter two, In chapter three, the method of multiple scales is introduced to the study of the multiple-dimensional nonlinear stochastic systems under random external excitation. Using method of multiple scales, we strictly deduce the equation of modulation of amplitude and phase. The effects of random excitations are analyzed; numerical simulations verify the results. Theoretical and numerical simulations show that when the intensity of the random ex
citation increases, the nontrivial steady state solution may be changed from a limit cycle to a diffused limit cycle.
In chapter four we concludes the work and points out some aspects to be further studied.
引文
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1．周纪卿，朱因远，非线性振动．西安交通大学出版社：西安，1998；
2. Tso W K, Asmis K G., Multiple parametric resonance in a nonlinear two-degree-Of-freedom system. Nonlinear Mech., 1974, 9:269～458;
3. Tezak E G, Mook D T, Neyfeh A H., Nonlinear analysis of the lateral response of columns to periodic loads. J. Mechanical Design, 1978, 100, 651～659;
4. Feng Z C, Sethna P R., Global bifurcation and chaos in parametrically forced systems with one-one resonance. Dynamics and Stability of Systems, 1990, 5(4):201～225;
5. Banerjee B, Bajaj A K., Chaotic responses in two degree-of-freedom systems with1:2 internal resonances. Nonlinear Dynamics and Stochastic Mechanics, 1996, 9:1～21;
6. HaQuang N., The response of multi-degree-of-freedom systems with cubic nonlinearities subjected to parametric and external excitation. In: Ph, D. Dissertation. Institute and State University, 1986;
7. Burton T.D.and Rahman Z., On the multi-scale analysis of strongly nonlinear forced oscillators. Nonlinear Mechanics, 1986,21 (2): 135～146;
8．李骊．强非线性振动系统的定性理论与定量方法．科学出版社：北京，1997；
9. Lin Y.K. and Cai G.Q., Probability Structural Dynamics: Advanced Theory and Applications. Singapore, McGraw-Hill, 1995;
10．朱位秋．随机振动．科学出版社：北京，1998；
11. Atalik,T.S., and Utku,S., Stochastic Linearization of multi-degree-of-freedom systems. Earthquake Engineering and Structural Dynamics. 1976,4:411～420;
12. Roberts,J.B. and Spanos,P.D., Random Vibration and Statistical Linearization. New York, Wiley, 1990;
13. Spanos,P.D., Stochastic Linearization in Structural Dynamics. Applied Mechanics Review, 1981,34(1): 1～8;
14．朱位秋．随机平均法及其应用．力学进展，1987，4：342～352；
15. Spanos,P. D., Approximate Analysis of Random Vibration through Stochastic Averaging. Nonlinear Dynamics, 1997,4:357～372;
16. Xu Z, Cheung Y K., Averaging method using generalized Harmonic functions for strongly non-linear oscillators. Journal of Sound and Vibration, 1994, 174 (4):563～576;
17．胡岗，随机力与非线性系统．上海科技教育出版社：上海，1994；
18. Risken H., The Fokker-Planck Equation. New York, Springer, 1983;
19. Utz V.W. and Wedig W.V., On the Calculation of Stationary Solutions of Multi-Dimensional Fokker-Planck Equations by Orthogonal Functions. Nonlinear Dynamics,2000,21,289～306;
20. Mcwilliam S., Knappett D.J., Numerical Solution of the Stationary FPK Equation using Shannon Wavelets. Journal of Sound and Vibration, 2000, 232(2):405～430;
21. Naess.A, Moe V., Generalized cell mapping versus path integration.
Computational Stochastic Mechanics, Spans, (eds.). Rotterdam: Balkma, 1998,385～390;
22．张柄根、赵玉芝，科学与工程中的随机微分方程．海洋出版社，1980；
23. Jahedi A. and Ahmadi G., Application of Wiener-hermite expansion to nonstationary random vibration of a Duffing oscillator 1983, ASME, Journal of Applied Mechanics, 50: 436～442;
24. Jahedi A. and Ahmadi G., Application of Wiener-hermite expansion to nonstationary random vibration of a Duffing oscillator.1983,ASME,Journal of Applied Mechanics, 50: 436～442;
25. Orabi I.I, and Ahmadi G., Nonstationary response analysis of a Duffing oscilator by the Wiener-hermite expansion method, 1987,54:434～440;
26. Orabi I.I, and Ahmadi G., A functional series expansion method excitations. International Journal of Nonlinear Mechanics, 1987,22(6):451～465;
27. Hartog J.E, Mechanical Vibrations, New York: McGraw-Hill,1956,forth edition;
28. Lyon R.H., Heckl M. and Hazelgrove C.B., Response of hard-spring oscillator to narrow band excitation. Journal of the Acoustical Society of America, 1961,33:1404～1411;
29. Dementberg M.F., Oscilations of a system with a nonlinear cubic characteristic under narrow band excitation. Mekhaniea Tveidogo Tela, 1970, 6:150～154;
30. Iyengar R.N., Random vibration of a second order nonlinear elastic system. Journal of Sound and Vibration, 1975,40:155～165;
31. Davies H.G. and Nandall D., Phase plane for narrow band random excitation of a Duffing oscillator. Journal of Sound and Vibration, 1886, 104:277～283;
32. Richard K. and Anand G. V., Nonlinear resonance in strings under narrow band random excitation, part Ⅰ:planar response and stability. Journal of Sound and Vibration, 1983,86:85～98;
33. Iyengar R.N., Stochastic response and stability of the Duffing oscillator under narrow band excitation. Journal of Sound and Vibration, 1988, 126:255～263;
34. Iyengar R.N., Response of nonlinear systems to narrow band excitation. Strctural Safety, 1989,6:177～185;
35. Lennox W. C. and Kuak Y. C., Narrow band excitation of a nonlinear oscillator Journal of Applied Mechanics, 1976, 43:340～344;
36. Grigoriu M., Probabilistic analysis of response of Duffing oscillator to narrow band stationary Gaussian excitation Proceedings of First Pan-American Congress of Applied Mechanics, Rio de Janeiro, Brazil, 1989;
37. Davies H.G. and Liu Q., The response probability density function of a Duffing oscillator with random narrow band excitation. Journal of Sound and Vibration, 1990,139:1～8;
38. Kapitaniak T., Stochastic response with bifurcations to non-linear Duffing oscillators. Journal of Sound and Vibration, 1985,102:440～441;
39. Fang T. and Dowell E. H., Numerical simulations of jump phenomena in stable Duffing systems. International Journal of Nonlinear Mechanics,
1987, 22:267～274;
40. Zhu W. Q., Lu M. Q.and Wu Q. T., Stochastic jump and bifurcation of a Duffing oscillator under narrow band excitation. Journal of Sound and Vibration, 1983,165(2):285～304;
41. Wedig W. V., Invariant measures and Lyapunov exponents for generalized parameter fluctuations, Structural Safety, 1990, 8:13～25;
42. Dementberg M., Statistical dynamics of nonlinear time-varying system. England, Research Studies Press, 1988;
43. Wedig W., Analysis and simulation of nonlinear stochastic system, Nonlinear Dynamics in Engineering Systems. Schiehlen W. (ed.), Berlin, Germany, Springer Verlag, 1989:337～344;
44. Ariaratnam S T., Stochastic stability of viscoelastic systems under bounded noise excitation. IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics, 1996,11～18;
45. Hou Z, Zhou Y, Dimentberg M, Noori M A. Non-stationary stochastic model for periodic excitation with random phase modulation. Probabilistic Engineering Mechanics, 1995,10:73～81;
46. Hou Z, Zhou Y, Dimentberg M, Noori M A. A stationary model for periodic excitation with uncorrelated random disturbances. Probabilistic Engineering Mechanics, 1996,11:191～203;
47. Hou Z, Zhou Y, Dimentberg M, Noori M A., Random response to periodic excitation with correlated disturbances. ASCE, Journal of Engineering Mechanics, 1996,122(11): 1101～1108;
48. Rajah S. andDavies H. G., Multiple time scaling of the response of a Duffing oscillator to narrow-band excitations. Journal of Sound and Vibration, 1988, 123:497～506;
49. Nayfeh A.H. and Serhan S.J., Response statistics of nonlinear systems to combined deterministic and random excitations. International Journal of Nonlinear Mechanics, 1990,25(5):493～509;
50. Rong Hai Wu, Xu Wei and Fang Tong, Principal response of Duffing oscillator to combined deterministic and narrow-band random parametric excitation. Journal of Sound and Vibration, 1998,210(4): 483～515;
51．戎海武，徐伟，方同，谐和与窄带随机噪声联合作用下Duffng系统的参数主共振．力学学报，1998，30(2)：178～185；
52. Shinozuka M., Simulation of multivariate and multidimensional random processes. Journal of Sound and Vibration, 1971,19:357～367;
53. Shinozuka M., Digital simulation of random processes and its applications. Journal of Sound and Vibration, 1972,25:111～128;
54. Natsiavas,S.and Tratskas, P.,On vibration isolation of mechanical systems with nonlinear foundations. Journal of Sound and Vibration, 194,1996,1.73～185;
55. Ho, C.H., Scott,R.A.,and Eisley, J.G., Non-planar, nonlinear oscillations of a beams: Ⅰ.Forced motions. International Journal of Non-Linear Mechanics 10,1975,113～127;
56. Chin, C.-M. and Nayfeh, A.H., Three-to-one internal resonances in hinged-clamped beams. Nonlinear Dynamics 12,1997,129～154;
57. Williams,C.J.H., The stability of nodal patterns in disk vibration. International Journal of Mechanical Science 8,1966,421～432;
58. Yang, X.L. and Sethna, P, R., Non-linear phenomena in forced vibrations of a nearly square plate: Antisymetric case. Journal of Sound and Vibration 155, 1992,413～441;
59. Chang, S.I., Bajaj,A.K.,and Krousgrill,C.M., Non-linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance. Nonlinear Dynamics 4, 1993,433～460;
60. Nayfeh, T.A. and Vakakis, A., Subharmonic traveling waves in a geometrically nonlinear circular plate. International Journal of Non-Linear Mechanics;
61. Nayfeh, A.H. and Mook, D.T., Nonlinear Oscillations, Wiley-Interscienee, New York, 1979;
62. Nayfeh,A.H. and Balachandran, B., Modal interactions in dynamical and structural systems Applied Mechanics Review 42,1989,S 175～S201;
63. Bajaj, A.K., and Tousi, S., Torus doublings and chaotic amplitude modulations in a two degree-of-freedom resonantly forced mechanical system. International Journal of Non-Linear Mechanics 25,1990,625～642;
64. Roth, B., Computations in kinematics, J. Angeles, G. Hommel, and P. Kovacs(eds.), Kluwer, Dordreeht, 1993, 3～14.
1．周纪卿，朱因远，非线性振动．西安交通大学出版社：西安，1998；
2. Tso W K, Asmis K G., Multiple parametric resonance in a nonlinear two-degree-Of-freedom system. Nonlinear Mech., 1974, 9:269～458;
3. Tezak E G, Mook D T, Neyfeh A H., Nonlinear analysis of the lateral response of columns to periodic loads. J. Mechanical Design, 1978, 100, 651～659;
4. Feng Z C, Sethna P R., Global bifurcation and chaos in parametrically forced systems with one-one resonance. Dynamics and Stability of Systems, 1990, 5(4):201～225;
5. Banerjee B, Bajaj A K., Chaotic responses in two degree-of-freedom systems with1:2 internal resonances. Nonlinear Dynamics and Stochastic Mechanics, 1996, 9:1～21;
6. HaQuang N., The response of multi-degree-of-freedom systems with cubic nonlinearities subjected to parametric and external excitation. In: Ph, D. Dissertation. Institute and State University, 1986;
7. Burton T.D.and Rahman Z., On the multi-scale analysis of strongly nonlinear forced oscillators. Nonlinear Mechanics, 1986,21 (2): 135～146;
8．李骊．强非线性振动系统的定性理论与定量方法．科学出版社：北京，1997；
9. Lin Y.K. and Cai G.Q., Probability Structural Dynamics: Advanced Theory and Applications. Singapore, McGraw-Hill, 1995;
10．朱位秋．随机振动．科学出版社：北京，1998；
11. Atalik,T.S., and Utku,S., Stochastic Linearization of multi-degree-of-freedom systems. Earthquake Engineering and Structural Dynamics. 1976,4:411～420;
12. Roberts,J.B. and Spanos,P.D., Random Vibration and Statistical Linearization. New York, Wiley, 1990;
13. Spanos,P.D., Stochastic Linearization in Structural Dynamics. Applied Mechanics Review, 1981,34(1): 1～8;
14．朱位秋．随机平均法及其应用．力学进展，1987，4：342～352；
15. Spanos,P. D., Approximate Analysis of Random Vibration through Stochastic Averaging. Nonlinear Dynamics, 1997,4:357～372;
16. Xu Z, Cheung Y K., Averaging method using generalized Harmonic functions for strongly non-linear oscillators. Journal of Sound and Vibration, 1994, 174 (4):563～576;
17．胡岗，随机力与非线性系统．上海科技教育出版社：上海，1994；
18. Risken H., The Fokker-Planck Equation. New York, Springer, 1983;
19. Utz V.W. and Wedig W.V., On the Calculation of Stationary Solutions of Multi-Dimensional Fokker-Planck Equations by Orthogonal Functions. Nonlinear Dynamics,2000,21,289～306;
20. Mcwilliam S., Knappett D.J., Numerical Solution of the Stationary FPK Equation using Shannon Wavelets. Journal of Sound and Vibration, 2000, 232(2):405～430;
21. Naess.A, Moe V., Generalized cell mapping versus path integration.
Computational Stochastic Mechanics, Spans, (eds.). Rotterdam: Balkma, 1998,385～390;
22．张柄根、赵玉芝，科学与工程中的随机微分方程．海洋出版社，1980；
23. Jahedi A. and Ahmadi G., Application of Wiener-hermite expansion to nonstationary random vibration of a Duffing oscillator 1983, ASME, Journal of Applied Mechanics, 50: 436～442;
24. Jahedi A. and Ahmadi G., Application of Wiener-hermite expansion to nonstationary random vibration of a Duffing oscillator.1983,ASME,Journal of Applied Mechanics, 50: 436～442;
25. Orabi I.I, and Ahmadi G., Nonstationary response analysis of a Duffing oscilator by the Wiener-hermite expansion method, 1987,54:434～440;
26. Orabi I.I, and Ahmadi G., A functional series expansion method excitations. International Journal of Nonlinear Mechanics, 1987,22(6):451～465;
27. Hartog J.E, Mechanical Vibrations, New York: McGraw-Hill,1956,forth edition;
28. Lyon R.H., Heckl M. and Hazelgrove C.B., Response of hard-spring oscillator to narrow band excitation. Journal of the Acoustical Society of America, 1961,33:1404～1411;
29. Dementberg M.F., Oscilations of a system with a nonlinear cubic characteristic under narrow band excitation. Mekhaniea Tveidogo Tela, 1970, 6:150～154;
30. Iyengar R.N., Random vibration of a second order nonlinear elastic system. Journal of Sound and Vibration, 1975,40:155～165;
31. Davies H.G. and Nandall D., Phase plane for narrow band random excitation of a Duffing oscillator. Journal of Sound and Vibration, 1886, 104:277～283;
32. Richard K. and Anand G. V., Nonlinear resonance in strings under narrow band random excitation, part Ⅰ:planar response and stability. Journal of Sound and Vibration, 1983,86:85～98;
33. Iyengar R.N., Stochastic response and stability of the Duffing oscillator under narrow band excitation. Journal of Sound and Vibration, 1988, 126:255～263;
34. Iyengar R.N., Response of nonlinear systems to narrow band excitation. Strctural Safety, 1989,6:177～185;
35. Lennox W. C. and Kuak Y. C., Narrow band excitation of a nonlinear oscillator Journal of Applied Mechanics, 1976, 43:340～344;
36. Grigoriu M., Probabilistic analysis of response of Duffing oscillator to narrow band stationary Gaussian excitation Proceedings of First Pan-American Congress of Applied Mechanics, Rio de Janeiro, Brazil, 1989;
37. Davies H.G. and Liu Q., The response probability density function of a Duffing oscillator with random narrow band excitation. Journal of Sound and Vibration, 1990,139:1～8;
38. Kapitaniak T., Stochastic response with bifurcations to non-linear Duffing oscillators. Journal of Sound and Vibration, 1985,102:440～441;
39. Fang T. and Dowell E. H., Numerical simulations of jump phenomena in stable Duffing systems. International Journal of Nonlinear Mechanics,
1987, 22:267～274;
40. Zhu W. Q., Lu M. Q.and Wu Q. T., Stochastic jump and bifurcation of a Duffing oscillator under narrow band excitation. Journal of Sound and Vibration, 1983,165(2):285～304;
41. Wedig W. V., Invariant measures and Lyapunov exponents for generalized parameter fluctuations, Structural Safety, 1990, 8:13～25;
42. Dementberg M., Statistical dynamics of nonlinear time-varying system. England, Research Studies Press, 1988;
43. Wedig W., Analysis and simulation of nonlinear stochastic system, Nonlinear Dynamics in Engineering Systems. Schiehlen W. (ed.), Berlin, Germany, Springer Verlag, 1989:337～344;
44. Ariaratnam S T., Stochastic stability of viscoelastic systems under bounded noise excitation. IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics, 1996,11～18;
45. Hou Z, Zhou Y, Dimentberg M, Noori M A. Non-stationary stochastic model for periodic excitation with random phase modulation. Probabilistic Engineering Mechanics, 1995,10:73～81;
46. Hou Z, Zhou Y, Dimentberg M, Noori M A. A stationary model for periodic excitation with uncorrelated random disturbances. Probabilistic Engineering Mechanics, 1996,11:191～203;
47. Hou Z, Zhou Y, Dimentberg M, Noori M A., Random response to periodic excitation with correlated disturbances. ASCE, Journal of Engineering Mechanics, 1996,122(11): 1101～1108;
48. Rajah S. andDavies H. G., Multiple time scaling of the response of a Duffing oscillator to narrow-band excitations. Journal of Sound and Vibration, 1988, 123:497～506;
49. Nayfeh A.H. and Serhan S.J., Response statistics of nonlinear systems to combined deterministic and random excitations. International Journal of Nonlinear Mechanics, 1990,25(5):493～509;
50. Rong Hai Wu, Xu Wei and Fang Tong, Principal response of Duffing oscillator to combined deterministic and narrow-band random parametric excitation. Journal of Sound and Vibration, 1998,210(4): 483～515;
51．戎海武，徐伟，方同，谐和与窄带随机噪声联合作用下Duffng系统的参数主共振．力学学报，1998，30(2)：178～185；
52. Shinozuka M., Simulation of multivariate and multidimensional random processes. Journal of Sound and Vibration, 1971,19:357～367;
53. Shinozuka M., Digital simulation of random processes and its applications. Journal of Sound and Vibration, 1972,25:111～128;
54. Natsiavas,S.and Tratskas, P.,On vibration isolation of mechanical systems with nonlinear foundations. Journal of Sound and Vibration, 194,1996,1.73～185;
55. Ho, C.H., Scott,R.A.,and Eisley, J.G., Non-planar, nonlinear oscillations of a beams: Ⅰ.Forced motions. International Journal of Non-Linear Mechanics 10,1975,113～127;
56. Chin, C.-M. and Nayfeh, A.H., Three-to-one internal resonances in hinged-clamped beams. Nonlinear Dynamics 12,1997,129～154;
57. Williams,C.J.H., The stability of nodal patterns in disk vibration. International Journal of Mechanical Science 8,1966,421～432;
58. Yang, X.L. and Sethna, P, R., Non-linear phenomena in forced vibrations of a nearly square plate: Antisymetric case. Journal of Sound and Vibration 155, 1992,413～441;
59. Chang, S.I., Bajaj,A.K.,and Krousgrill,C.M., Non-linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance. Nonlinear Dynamics 4, 1993,433～460;
60. Nayfeh, T.A. and Vakakis, A., Subharmonic traveling waves in a geometrically nonlinear circular plate. International Journal of Non-Linear Mechanics;
61. Nayfeh, A.H. and Mook, D.T., Nonlinear Oscillations, Wiley-Interscienee, New York, 1979;
62. Nayfeh,A.H. and Balachandran, B., Modal interactions in dynamical and structural systems Applied Mechanics Review 42,1989,S 175～S201;
63. Bajaj, A.K., and Tousi, S., Torus doublings and chaotic amplitude modulations in a two degree-of-freedom resonantly forced mechanical system. International Journal of Non-Linear Mechanics 25,1990,625～642;
64. Roth, B., Computations in kinematics, J. Angeles, G. Hommel, and P. Kovacs(eds.), Kluwer, Dordreeht, 1993, 3～14.