轴向运动体系的非线性振动研究
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摘要
本文应用多元Lindstedt-Poincaré(L-P)法及增量谐波平衡法(the incremental harmonic balance method, IHB)研究轴向运动体系的横向非线性振动。
首先,研究无轴向运动的梁的强非线性振动,应用改进的L-P法分析了梁的自由振动和在外激励力作用下的基谐波响应、次谐波响应、超谐波响应,并与IHB法结果比较,证明改进的L-P法的有效性和正确性。
其次,研究轴向运动梁的横向非线性振动,根据哈密顿原理建立轴向运动梁的横向运动微分方程,利用Galerkin方法离散方程,再应用多元L-P法分别研究当外激励力频率Ω接近于系统第一阶固有频率ω_(10)、第二阶固有频率ω_(20)、第一阶固有频率的1/3 (即1/3ω_(10))和第一、第二阶固有频率的平均值1/2 (ω_(10) +ω_(20))时伴随内部共振的基谐波响应、次谐波响应、超谐波响应和组合谐波响应,同时也分析有和没有模态阻尼的差别,从中揭示了丰富多彩的非线性振动现象。
接着,应用IHB法计算上述各种非线性振动响应问题,并与多元L-P法的计算结果进行比较,说明两种方法的有效性和正确性。
最后,给出本文工作的结论和对今后工作的展望。
本论文的部分研究成果已在国内外力学类核心期刊上发表或录用待发表。
The multiple dimension Lindstedt-Poincaré(L-P) method and the incremental harmonic balance (IHB) method are employed for nonlinear vibration analysis of axially moving systems.
Firstly, the modified Lindstedt-Poincarémethod is used to study the strongly nonlinear vibrations of beams without axially moving. The free vibration, the fundamental resonance, subharmonic resonance, supharmonic resonance are studied respectively. All the results are compared with those obtained by the IHB method to show the effectiveness and accuracy.
Secondly, the multiple dimension Lindstedt-Poincarémethod is used to study the nonlinear vibration of axially moving beams. The equation of motion of axially moving beams are derived by using Hamilton principle and discretized by using Galerkin method. Various resonance such as fundamental harmonic resonance, subharmonic resonance, supharmonic resonance, supharmonic resonance, combination harmonic resonance, are studied in detail, which occurred when the excitation frequencyΩis near to the first natural frequencyω_(10), the second frequencyω_(20), the one third of first frequency 1/3ω_(10), the one half ofω_(10) +ω_(20), respectively. From which, many varied and interesting nonlinear phenomenon are revealed. The differences between the results with and without the model damping are analyzed at the same time.
Thirdly, the IHB method is employed to study every resonance mentioned above. The results obtained by the IHB method and the multiple dimension L-P method are compared one by one to show the accuracy and effectiveness of both methods. Finally, the conclusion and recommendation are given.
Parts of this work have been or will be published in several international and national journals.
首先,研究无轴向运动的梁的强非线性振动,应用改进的L-P法分析了梁的自由振动和在外激励力作用下的基谐波响应、次谐波响应、超谐波响应,并与IHB法结果比较,证明改进的L-P法的有效性和正确性。
其次,研究轴向运动梁的横向非线性振动,根据哈密顿原理建立轴向运动梁的横向运动微分方程,利用Galerkin方法离散方程,再应用多元L-P法分别研究当外激励力频率Ω接近于系统第一阶固有频率ω_(10)、第二阶固有频率ω_(20)、第一阶固有频率的1/3 (即1/3ω_(10))和第一、第二阶固有频率的平均值1/2 (ω_(10) +ω_(20))时伴随内部共振的基谐波响应、次谐波响应、超谐波响应和组合谐波响应,同时也分析有和没有模态阻尼的差别,从中揭示了丰富多彩的非线性振动现象。
接着,应用IHB法计算上述各种非线性振动响应问题,并与多元L-P法的计算结果进行比较,说明两种方法的有效性和正确性。
最后,给出本文工作的结论和对今后工作的展望。
本论文的部分研究成果已在国内外力学类核心期刊上发表或录用待发表。
The multiple dimension Lindstedt-Poincaré(L-P) method and the incremental harmonic balance (IHB) method are employed for nonlinear vibration analysis of axially moving systems.
Firstly, the modified Lindstedt-Poincarémethod is used to study the strongly nonlinear vibrations of beams without axially moving. The free vibration, the fundamental resonance, subharmonic resonance, supharmonic resonance are studied respectively. All the results are compared with those obtained by the IHB method to show the effectiveness and accuracy.
Secondly, the multiple dimension Lindstedt-Poincarémethod is used to study the nonlinear vibration of axially moving beams. The equation of motion of axially moving beams are derived by using Hamilton principle and discretized by using Galerkin method. Various resonance such as fundamental harmonic resonance, subharmonic resonance, supharmonic resonance, supharmonic resonance, combination harmonic resonance, are studied in detail, which occurred when the excitation frequencyΩis near to the first natural frequencyω_(10), the second frequencyω_(20), the one third of first frequency 1/3ω_(10), the one half ofω_(10) +ω_(20), respectively. From which, many varied and interesting nonlinear phenomenon are revealed. The differences between the results with and without the model damping are analyzed at the same time.
Thirdly, the IHB method is employed to study every resonance mentioned above. The results obtained by the IHB method and the multiple dimension L-P method are compared one by one to show the accuracy and effectiveness of both methods. Finally, the conclusion and recommendation are given.
Parts of this work have been or will be published in several international and national journals.
引文
1. Mote Jr C D. Dynamic stability of axially moving materials. Shock Vib Dig, 1972, 4(4) : 2-11.
2. Wickert J A and Mote C D Jr. Current research on the vibration and stability of axially-moving materials. Shock and Vibration, 1988, 29(5): 3-13.
3. Wang K W, Liu S P. On the noise and vibration of chain drive systems. Shock Vib Dig, 1991, 23(4): 8-13.
4. Abrate A S. Vibration of belts and belt drivers. Mech Mach Theory, 1992, 27(6): 645-659.
5.陈立群, Jean W Z.轴向运动弦线的纵向振动及其控制.力学进展, 2001, 31(4): 535-546.
6.王建军,邹西凤,李其汉.轴向移动系统的参数振动问题研究进展.应用力学学报, 2003, 20(4): 33-36.
7. Skutch R. Uber die bewegung eines gespannten fadens, Weicher gezwungen ist, durch zwei feste punkte, mit einer constanten geschwindigkeit zu gehen, und zwischen denselben in transversal-schwingungen von gerlinger amplitude versetzt wird. Annalen der Physik und Chemie, 1897, 61: 190-195.
8. Sack R A. Transverse oscillations in traveling strings. British J. Appl. Phys., 1954, 5: 224-226.
9. Wickert J A and Mote C D Jr. Classical vibration analysis of axially moving continua. Journal of Applied Mechanics, 1990, 57: 738-744.
10. Meirovitch L. A new method of solution of the eigenvalue problem for gyroscopic systems. AIAA Journal, 1974, 12(10): 1337-1342.
11. Meirovitch L. A modal analysis for the response of linear gyroscopic systems. ASME Journal of Applied Mechanics, 1975, 42: 446-450.
12. D’Eleuterio G M T, Huges P C. Dynamics of gyroclastic continua. ASME Journal of Applied Mechanics, 1984, 51: 415-422.
13. Mote Jr C D. A study of bandsaw vibrations. J. Franklin Inst., 1965, 279(6): 430-444.
14. Perkins N C, Mote Jr C D. Comments on curve veering in eigenvalue problems. Journal Sound and Vibration, 1986, 106(3): 451-463.
15. Perkins N C. Linear dynamics of a translating string on an elastic foundation. J Vib Acoust, 1990, 112(1): 2-7.
16. Wickert J A. Response solutions for the vibration of a traveling string on an elastic foundation. J Vib Acoust, 1994, 116(1): 137-139.
17. Zhu W D, Mote Jr C D. Propagation of boundary disturbances in an axially moving strip in contact with rigid and flexible constrains. J Appl Mech, 1995, 62(4): 873-879.
18. Chen J S. Natural frequencies and stability of an axially traveling string in contact with a stationary load system. J Vib Acoust, 1997, 152(2): 152-157.
19. Ames W F, Lee S Y, Zaiser J N. Non-linear vibration of a traveling threadline. Intl. J. Nonlin. Mech., 1968, 3: 449-469.
20. Shih L Y. Three-dimensional non-linear vibration of a traveling string. Intl. J. Nonlin. Mech., 1971, 6(4): 427-434.
21. Shih L Y. Motion of elliptic ballooning for a traveling string. Intl. J. Nonlin. Mech., 1975, 10(3/4): 183-191.
22. Mack C. Theory of the spinning balloon. Quart. J. Mech. Appl. Math., 1958, 11(2): 196-207.
23. Mote Jr C D. On the nonlinear oscillation of an axially moving string. ASME J. Appl. Mech., 1966, 33: 463-464.
24. Bapat V A, Srinivasan P. Nonlinear transverse oscillations in traveling strings by the method of harmonic balance. J. Appl. Mech., Trans, ASME, 1967, 34(3): 775-777.
25. Thurman A L, Mote Jr C D. Free, periodic, nonlinear oscillation of an axially moving strip. J. Appl. Mech., Trans. ASME, 1969, 36: 83-94.
26. Simpson A. Transverse modes and frequencies of beams translating between fixed end supports. J. Mech. Engrg. Sci., 1973, 15(3): 159-164.
27. Kim Y, Tabarrok B. On the nonlinear vibration of traveling strings. J. Franklin Inst., 1972, 293: 381-399.
28. Schajer G S. The vibration of a rotating circular strings subject to a fixed elastic restraint. Journal Sound and Vibration, 1984, 92(1): 11-19.
29. Bhat R B, Xistris G D, Sankar T S. Dynamics behavior of a moving belt supported on an elastic foundation. J Mech Design, 1982, 104(1): 143-147.
30. Ulsoy A G, Whitesell J E, Hooven M D. Design of belt-tensioner systems for dynamics stability. J Vib Acoust Stress, Reliability in Desi, 1985, 107: 285-290.
31. Zhu W D, Mote Jr C D. Free and forced response of an axially moving string transporting a damped linear oscillator. Journal Sound and Vibration, 1994, 177(5): 591-610.
32. Swope R D, Ames W F. Vibrations of a moving threadline. J.Franklin Inst., 1963, 275(1): 36-55.
33. Ames W F, Vicario A A Jr. On the longitudinal wave propagation on a traveling threadline. Developments in Mechanics, Proc. 11th Midwest. Mech. Conf., 1969, 5: 733-746.
34. Ames W F, Lee S Y, Vicario A A Jr. Longitudinal wave propagation on a traveling threadline --Ⅱ, Intl. J. Nonlin. Mech., 1970, 5(3): 413-426.
35. Mote C D Jr. Stability of systems transporting accelerating axially moving materials. J. Dynam. Syst. Meas. Control, Trans. ASME, 1975, 97: 96-98.
36. Mahalingam S. Transverse oscillations of power transmission chains. Brit. J. Appl. Phys., 1957, 8: 145-148.
37. Ulsoy A G, Whitesell J E, Hooven M D. Design of belt-tensioner systems for dynamics stability. J Vib Acoust Stress, Reliability in Desi, 1985, 107: 285-290.
38. Wickert J A and Mote C D Jr.. On the energetics of axially continua. J Acoust Soc Am, 1989, 85(3): 1365-1368.
39. Lee S Y, Mote C D Jr. A generalized treatment of the energetics of translating continua, part 1: strings and second order tensioned pipes. Journal Sound and Vibration, 1997, 204(5): 717-734.
40. Sack R A. Transverse oscillations in traveling strings. British J. Appl. Phys., 1954, 5: 224-226.
41. Mahalingam S. Transverse oscillations of power transmission chains. Brit. J. Appl. Phys., 1957, 8: 145-148.
42. Swope R D, Ames W F. Vibrations of a moving threadline. J.Franklin Inst., 1963, 275(1): 36-55.
43. Nelson H M. Transverse vibration of a moving strip. J. Sound Vib., 1979, 65(3): 381-389.
44. Alspaugh D W. Torsional vibration of a moving band. J. Franklin Inst., 1967, 283(4): 328-338.
45. Soler A I. Vibration and stability of a moving band. J. Franklin Inst., 1968, 286(4): 295-307.
46. Roth W. Torsionsschwingungen des gezogenes bandes. ZAMM, 1977, 57(12): 722-725.
47. Mote Jr C D. Divergence buckling of an edge-loaded axially moving band. Intl. J. Mech. Sci., 1968, 10(4): 281-295.
48. Naguleswaran S, Williams C J H. Lateral vibration of band-saw, pulley belts and the like. Int J Mech Sci, 1968, 10: 239-250.
49. Doyle E, Hornung K G. Lateral vibration of V-belts. ASME Paper, 1969, 69-vibr-29.
50. Lai J L, Chen C H. Vibration and dynamic stability of an axially moving belt. ASME Paper, 1971, 71-vibr-31.
51. Kozin F, Milstead R M. The stability of a moving elastic strip subjected to random parametric excitation. J. Appl. Mech., Trans. ASME, 1979, 46(2): 404-410.
52. Kozic P, Paviovic R. Stability of a thin moving elastic strip subjected to random parametric excitations. Journal of Sound and Vibration, 1997, 206(2): 280-285.
53. Wu W Z, Mote C D Jr. Parametric excitation of an axially moving band by periodic edge loading. J. Sound Vib., 1986, 110(1): 27-39.
54. Barakat R. Transverse vibrations of a moving thin rod. J. Acoust. Soc. Amer., 1967, 43(3): 533-539.
55. Taleb I A, Misra A K. Dynamic of an axially moving beam submerged in a fluid. AIAA J. Hydronau., 1981, 15(1-4): 62-66.
56. Mote Jr C D. and Naguleswaran S. Theoretical and experimental band saw vibrations. J. Engrg. Indus., Trans. ASME, 1966, 88(2): 151-156.
57. Cherchas D B, Gossain D M. Dynamics of a flexible solar array during deployment from a spinning spacecraft. CASI Trans, 1974, 7(1): 10-17.
58. Weeks G E. Dynamic analysis of a deployable space structure. J. Spacecraft Rockets, 1986, 23(1): 102-107.
59. Mansfield L, Simmonds J G. The reverse spaghetti problem: drooping motion of an elastica issuing from a horizontal guide. J. Appl. Mech., Trans. ASME, 1987, 54: 147-150.
60. Lips K W, M di V J. Transient attitude dynamics of satellites with deploying flexible appendages, ACTA Astronau, 1978, 5(10): 797-815.
61. Tabarrok B, Leech C M, Kim Y I. On the dynamics of an axially moving beam. J. Franklin Inst., 1974, 297(3): 201-220.
62. Stylianou M, Tabarrok B. Finite element analysis of an axially moving beam part I: time integration. Journal of Sound and Vibration, 1994, 178(4): 433-453.
63. Stylianou M, Tabarrok B. Finite element analysis of an axially moving beam part II: stability analysis. Journal of Sound and Vibration, 1994, 178(4): 455-481.
64. Fung R F, Lu P Y. Non-linearly dynamic modeling of an axially beam with a tap mass. Journal of Sound and Vibration, 1998, 218(4): 559-571.
65. AI-Bedoor B O, Kbulief Y A. An approximate analytical solution of beam vibration during axial motion. Journal of Sound and Vibration, 1996, 192(1): 159-171.
66.李山虎,杨靖波,黄清华,陈德成.轴向运动悬臂梁的独立模态振动控制--Ⅰ近似理论分析.应用力学学报, 2002, 19(1): 35-38.
67. Hayashi C. Non-linear oscillation in physical systems. McGraw-Hill, New York.1964.
68. Jordan D W, Smith P. Non-linear ordinary differential equation. Oxford University Press, Oxford, 1977.
69. Nayfeh A H, Mook D T. Nonlinear oscillations. John Wiley and Sons, New York, 1979.
70. Dooren R V. Combination tones of summed type in a non-linear damped vibratory system with two degrees of freedom. Int. J. Non-linear Mechanics, 1971, 6: 237-254.
71. Chia C Y, Prabhakara M K. Large deflection of unsymmetric cross-ply and angle-ply plates. J. Mech. Eng. Sci., 1976, 18: 179-183
72. Tongue B H. Limit cycle oscillations of a nonlinear rotorcraft model. AIAA Journal, 1984, 22: 967-974.
73. Cheung Y K, Iu V P. An implicit implementation of harmonic balance method for nonlinear dynamic systems. Engineering Computations, 1988, 5: 134-140.
74. Leung A Y T, Fung T C. Phase increment analysis of damped doffing oscillators. InternationalJournal for Numerical Methods in Engineering, 1989, 28: 193-210.
75. Leung A Y T, Fung T C. Construction of chaotic regions. Journal of Sound and Vibration, 1989, 131: 445-455.
76. Fung T C. Steady state solutions of nonlinear dynamic systems. Ph. D. Thesis, University of Hong Kong, 1989.
77. Dunn P, Dugundji J. Nonlinear stall flutter and divergence analysis of cantilevered graphite/epoxy wings. AIAA Journal, 1992, 30(1): 153-162.
78. Kim T, Dugundji J. Nonlinear large amplitude aeroelastic behavior of composite rotor blades. AIAA Journal, 1993, 31(8): 1489-1497.
79. Blankenship G W, Kahraman A. Steady state forced response of a mechanical oscillator with combined parametric excitation and clearance type non-linearity. Journal of Sound and Vibration, 1995, 185(5): 743-765.
80.郑小平,姚振汉,杜庆华.利用谐波平衡法计算局部非线性运动系统的稳态响应.清华大学学报, 1997, 37(8): 36-39.
81. Lau S L and Cheung Y K. Amplitude incremental variational principle for non-linear vibration of elastic systems. ASME, Journal of Applied Mechanics, 1981, 48: 959-964.
82. Cheung Y K, Lau S L. Incremental time-space finite strip method for nonlinear structural vibrations. Earthquake Engineering and Structural Dynamics, 1982, 10: 239-253.
83. Lau S L, Cheung Y K and Wu S Y. Nonlinear vibration of thin elastic plates, partⅠ: generalized incremental hamilton’s principle and element formulation. ASME Journal of Applied Mechanics, 1984, 51: 837-844.
84. Lau S L, Cheung Y K. Incremental hamilton’s principle with multiple time scales for nonlinear aperiodic vibrations of shells. ASME Journal of Applied Mechanics, 1986, 53: 465-466.
85. Iu V P, Cheung Y K and Lau S L. Nonlinear vibration analysis of multiplayer beams by incremental finite elements, partⅠ: theory and numerical formulation. Journal of Sound and Vibration, 1985, 100: 373-382.
86. Iu V P, Cheung Y K. Nonlinear vibration analysis of multilayer sandwich plates by incremental finite elements, partⅠ: theoretical development. Engineering Computation, 1986, 3: 35-42.
87. Lau S L, Cheung Y K and Wu S Y. A variable parameter incrementation method for dynamic instability of linear and nonlinear elastic systems. ASME Journal of Applied Mechanics, 1982, 49: 849-853.
88. Lau S L, Cheung Y K and Wu S Y. Incremental harmonic balance method with multiple time scales for aperiodic vibration of nonlinear systems. ASME Journal of Applied Mechanics, 1983, 50: 871-876.
89. Pierre C, Dowell E H. A study of dynamic instability of plates by an extended incrementalharmonic balance method. ASME Journal of Applied Mechanics, 1985, 52: 693-697.
90. Pierre C. Ferri A A and Dowell E H. Multi-harmonic analysis of dry friction damped systems using an incremental harmonic balance method. ASME Journal of Applied Mechanics, 1985, 52: 958-964.
91. Ferri A A, Dowell E H. Frequency domain solutions to multi-degree-of-freedom dry friction damped systems. Journal of Sound and Vibration, 1988, 124: 207-224.
92. Cheung Y K, Chen S H and Lau S L. Application of the incremental harmonic balance method to cubic non-linearity systems. Journal of Sound and Vibration, 1990, 140: 273-286.
93. Lau S L, Yuen S W. Hopf bifurcation and limit cycle by the incremental harmonic balance method. Computer Methods in Applied Mechanics and Engineering, 1991, 91(1-3): 1109-1121.
94. Lau S L, Zhang W S. Nonlinear vibrations of piecewise-linear systems by incremental harmonic balance method, 1992, 59(1): 153-160.
95. Leung A Y T, Chui S K. Non-linear vibration of coupled duffing oscillators by an improved incremental harmonic balance method. Journal of Sound and Vibration, 1995, 181(4): 619-633.
96. Leamy M J, Perkins N C. Nonlinear periodic response of engine accessory drives with dry friction tensioners. Journal of Vibration and Acoustics, Transactions of the ASME, 1998, 120(4): 909-916.
97. Raghothama A, Narayanan S. Periodic response and chaos in nonlinear systems with parametric excitation and time delay. Nonlinear Dynamics, 2002, 27(4): 341-365.
98. Minorsky N. Non-linear oscillations. Robert E. Krieger, Huntington, New York, 1962.
99. Nayfeh A H. Perturbation methods. John Wiley, New York, 1973.
100. Lindstedt A. Ueder die integration einer fur die storungstheorie wichtigen differentialgleichung. Astron, Nach, 1882, 103: 211-220.
101. PoincaréH. Sur les integrals des equations linéaires. Acta, Math, 1886, 8: 295-344.
102. Lighthill M J. A technique for rendering approximate solutions to physical problems uniformly valid. Phil. Mag., 1949, 40: 1179-1201.
103. Kuo Y H. On the flow of an incompressible viscous fluid past a flat plate at moderate Reynolds numbers. J. Math. and Phys, 1953, 32: 83-101.
104. Kuo Y H. Viscous flow along a flat plate moving at high supersonic speeds. J. Aeron. Sci., 1956, 23: 125-136.
105. Tsien H S. The Poincaré-Lighthill-Kuo method. Advan. Appl. Mech., 1956, 4: 281-349.
106. Chen S H. Generalization of the lindstedt-poincarémethod for analysis of non-linear vibrations. Ph. D. Thesis, University of Hong Kong, 1990.
107. Cheung Y K, Chen S H and Lau S L. A modified Lindstedt-Poincare method for certain strongly nonlinear oscillators. Int. J. Non-linear Mechanics, 1991, 26(3/4): 367-378.
108. Chen S H, Cheung Y K. A modified Lindstedt-Poincare method for a strongly non-linear two degree-of-freedom system. Journal of Sound and Vibration, 1996, 193(4): 751-762.
109. Chen S H, Cheung Y K. A modified Lindstedt-Poincarémethod for a strongly non-linear system with quadratic and cubic nonlinearities. Shock and Vibration, 1996, 3(4): 279-285.
110.唐驾时,尹小波.一类强非线性振动的分叉.力学学报, 1996, 28(3): 363-369.
111.唐驾时.求强非线性系统次谐共振解的MLP方法.应用数学和力学, 2000, 21(10): 1039-1045.
112. Lau S L, Cheung Y K, Chen S H. An alternative perturbation procedure of multiple scale for nonlinear dynamics systems. ASME J. Appl. Mech., 1989, 56(3): 667-675.
113. Chen S H, Cheung Y K, Lau S L. On the internal resonance of multi-degree-of-freedom systems with cubic nonlinearity. Journal Sound and Vibration, 1989, 128(1): 13-24.
114. Sturrock P A. Nonlinear effects in electron plasmas. Proc. Roy. Soc. 1957, A242: 277-299.
115. Sturrock P A. Nonlinear theory of electromagnetic waves in plasmas. Stanford University Microwave Laboratory Repl. 1963, 1004.
116. Frieman E A. On a new method in the theory of irreversible processes. J. Math. Physis, 1963, 4: 410-418.
117. Nayfeh A H. A perturbation method for treating nonlinear oscillation problems. J. Math. and Phys, 1965, 44: 368-374.
118. Sandri G. A new method of expansion in mathematical physics. Nuovo Cimento, 1965, B36: 67-93.
119. Cole J D, Kevorkian J. Uniformly valid asymptotic approximations for certain nonlinear differential equations. Nonlinear Differential Equation and Nonlinear Mechanics, Academic, New York, 1963, 113-120.
120. Morrison J A. Comparison of the modified method of averaging and the two variable expansion procedure. SIAM Rev., 1966, 8: 66-85.
121. Nayfeh A H. The Van Der Pol oscillator with delayed amplitude limiting. Proc. IEEE, 1967, 55: 111-112.
122. Kuzmak G E. Asymptotic solutions of nonlinear second order differential equations with variable coefficients. J. Appl. Math. Mech., 1959, 23: 730-744.
123. Mahony J J. An expansion method for singular perturbation problems. J. Australian Mat. Soc., 1962, 2: 440-463.
124. Nayfeh A H. A perturbation method for treating nonlinear oscillation problems. J. Math. and Phys, 1965, 44: 368-374.
125. Abdelhafez H M. Resonance of multiple frequency excitated systems with quadratic, cubic and quartic non-linearity. Mathematics and Computers in Simulation, 2002, 61: 17-34.
126. Zhao Y Y, Wang L H, Chen D L, Jiang L Z. Non-linear dynamic analysis of the two-dimensional simplified model of an elastic cable. Journal of Sound and Vibration, 2002, 255(1): 43-59.
127. El- Bassiouny. Three-mode interaction in harmonically excited system with cubic nonlinearities. Applied Mathematics and Computation, 2003, 139: 201-230.
128. Arafat H N, Nayfeh A H. Non-linear response of suspended cables to primary resonance excitations. Journal of Sound and Vibration, 2003, 266: 325-354.
129. Li X Y, Chen Y S, Wu Z Q. Non-linear normal modes and their bifurcation of a class of systems with three double of pure imaginary roots and dual. International Journal of Non-linear Mechanics, 2004, 39: 189-199.
130. Krylov N, Bogoliubov N N. Introduction to nonlinear mechanics. Princeton University Press, Princeton. N. J., 1947.
131. Bogoliubov N N, Mitropolski Y A. Asymptotic methods in the theory of nonlinear oscillations. Gordon and Breach, New York, 1961.
132. Yang Y R. KBM method of analyzing limit cycle flutter of a wing with an external store and comparison with a wind-tunnel test. Journal Sound and Vibration, 1995, 187(2), 271-280.
133. Zhou Y H, Tzou H S. Active control of nonlinear piezoelectric circular shallow spherical shells. International Journal of Solids and Structures, 2000, 37: 1663-1677.
134.孟光.延时反馈振荡器在弱正弦外激下的强制振动.西北工业大学学报, 20(1): 1-5.
135.褚亦清.非线性振动分析.北京理工大学出版社, 1996.
136. Pellicano F and Vestroni F. Nonlinear dynamics and bifurcations of an axially moving beam. Journal of Vibration and Acoustics, 2000, 122: 21-30.
137. Wickert J A. Non-linear vibration of a traveling tensioned beam. International Journal of Non-Linear Mechanics, 1992, 27(3): 503-517.
138. Chakrabporty G, Mallik A K, Hatal H. Non-linear vibration of traveling beam. International Journal of Non-Linear Mechanics, 1999, 34: 655-670.
139. .Chakrabporty G, Mallik A K. Non-linear vibration of a traveling beam having an intermediate guide. Nonlinear Dynamics, 1999, 20: 247-265.
140. Al-jawi A A N, Pierre C, Ulsoy A G. Vibration localization in dual-span axially moving beams, PartΙ: formulation and results. Journal of Sound and Vibration, 1995, 179(2): 243-266.
141. Chen J S. Natural frequencies and stability of an axially-traveling string in contract with a stationary load system. ASME Journal of Vibration and Acoustics, 1997, 119: 152-156.
142. Riedel C H and Tan C A. Coupled, forced response of an axially moving strip with internalresonance. International Journal of Non-Linear Mechanics, 2002, 37: 101-116.
143. Pakdemirli M, Ulsoy A G, Ceranoglu A. Transverse vibration of an axially accelerating string. Journal of Sound and Vibration, 1994, 169: 179-196.
144. .Pakdemirli M, Ulsoy A G. Stability analysis of an axially accelerating string. Journal of Sound and Vibration, 1997, 203: 815-832.
145. Mockensturm E M, Perkins N C, Galip Ulsoy A. Stability and limit cycles of parametrically excited, axially moving strings. Journal of Vibration and Acoustics, 1996, 118, 346-351.
146. Zhang L, Zu J W. Nonlinear Vibration of Parametrically Excited Moving Belts, Part I: Dynamic Response. Journal of Applied Mechanics, 1999, 66: 396-402.
147. Zhang L, and Zu J W. Nonlinear Vibration of Parametrically Excited Moving Belts, Part II: Stability Analysis. Journal of Applied Mechanics, 1999, 66: 403-409.
148. Suweken G, Horssen Van W T. On the weakly nonlinear, transversal vibrations of a conveyor belt with a low and time-varying velocity. Nonlinear Dynamics, 2003, 31(2): 197-223.
149. Oz H R, Pakdemirli M. Vibration of an axilly moving beam with time dependent velocity. Journal of Sound and Vibration, 1999, 227: 239-257.
150. Oz H R. On the vibration of an axially traveling beam on fixed supports with variable velocity. Journal of Sound and Vibration, 2001, 239(3): 556-564.
151. Fung R F, Chang H C. Dynamic and energetic analyses of a string/slider non-linear coupling system by variable grid finite difference. Journal of Sound and Vibration, 2001, 239(3): 505-514.
152. Ravindra B, Zhu W D. Low–dimensional chaotic response of axially accelerating continuum in the supercritical regime. Archive of Applied Mechanics, 1998, 68: 195-205.
153. Moon J, Wickert J A. Non-linear vibration of power transmission belts. Journal of Sound and Vibration, 1997, 200: 419-431.
154. Lakshmikumaran A V, Wickert J A. On the vibration of coupled traveling string and air bearing system. ASME Journal of Vibration and Acoustics, 1996, 118: 398-405.
155. Bajkowdki J, Szemplinska-Stupnicka W. Internal resonance effects—simulation versus analytical methods results. Journal of Sound and Vibration, 1986,104: 259-275.
156. Lee C L, Perkins N C. Nonlinear oscillations of suspended cables containing a two-to-one internal resonance. Nonlinear Dyn, 1992, 3: 465-490.
157.胡海昌.弹性力学的变分原理及其应用.科学出版社, 1981.
2. Wickert J A and Mote C D Jr. Current research on the vibration and stability of axially-moving materials. Shock and Vibration, 1988, 29(5): 3-13.
3. Wang K W, Liu S P. On the noise and vibration of chain drive systems. Shock Vib Dig, 1991, 23(4): 8-13.
4. Abrate A S. Vibration of belts and belt drivers. Mech Mach Theory, 1992, 27(6): 645-659.
5.陈立群, Jean W Z.轴向运动弦线的纵向振动及其控制.力学进展, 2001, 31(4): 535-546.
6.王建军,邹西凤,李其汉.轴向移动系统的参数振动问题研究进展.应用力学学报, 2003, 20(4): 33-36.
7. Skutch R. Uber die bewegung eines gespannten fadens, Weicher gezwungen ist, durch zwei feste punkte, mit einer constanten geschwindigkeit zu gehen, und zwischen denselben in transversal-schwingungen von gerlinger amplitude versetzt wird. Annalen der Physik und Chemie, 1897, 61: 190-195.
8. Sack R A. Transverse oscillations in traveling strings. British J. Appl. Phys., 1954, 5: 224-226.
9. Wickert J A and Mote C D Jr. Classical vibration analysis of axially moving continua. Journal of Applied Mechanics, 1990, 57: 738-744.
10. Meirovitch L. A new method of solution of the eigenvalue problem for gyroscopic systems. AIAA Journal, 1974, 12(10): 1337-1342.
11. Meirovitch L. A modal analysis for the response of linear gyroscopic systems. ASME Journal of Applied Mechanics, 1975, 42: 446-450.
12. D’Eleuterio G M T, Huges P C. Dynamics of gyroclastic continua. ASME Journal of Applied Mechanics, 1984, 51: 415-422.
13. Mote Jr C D. A study of bandsaw vibrations. J. Franklin Inst., 1965, 279(6): 430-444.
14. Perkins N C, Mote Jr C D. Comments on curve veering in eigenvalue problems. Journal Sound and Vibration, 1986, 106(3): 451-463.
15. Perkins N C. Linear dynamics of a translating string on an elastic foundation. J Vib Acoust, 1990, 112(1): 2-7.
16. Wickert J A. Response solutions for the vibration of a traveling string on an elastic foundation. J Vib Acoust, 1994, 116(1): 137-139.
17. Zhu W D, Mote Jr C D. Propagation of boundary disturbances in an axially moving strip in contact with rigid and flexible constrains. J Appl Mech, 1995, 62(4): 873-879.
18. Chen J S. Natural frequencies and stability of an axially traveling string in contact with a stationary load system. J Vib Acoust, 1997, 152(2): 152-157.
19. Ames W F, Lee S Y, Zaiser J N. Non-linear vibration of a traveling threadline. Intl. J. Nonlin. Mech., 1968, 3: 449-469.
20. Shih L Y. Three-dimensional non-linear vibration of a traveling string. Intl. J. Nonlin. Mech., 1971, 6(4): 427-434.
21. Shih L Y. Motion of elliptic ballooning for a traveling string. Intl. J. Nonlin. Mech., 1975, 10(3/4): 183-191.
22. Mack C. Theory of the spinning balloon. Quart. J. Mech. Appl. Math., 1958, 11(2): 196-207.
23. Mote Jr C D. On the nonlinear oscillation of an axially moving string. ASME J. Appl. Mech., 1966, 33: 463-464.
24. Bapat V A, Srinivasan P. Nonlinear transverse oscillations in traveling strings by the method of harmonic balance. J. Appl. Mech., Trans, ASME, 1967, 34(3): 775-777.
25. Thurman A L, Mote Jr C D. Free, periodic, nonlinear oscillation of an axially moving strip. J. Appl. Mech., Trans. ASME, 1969, 36: 83-94.
26. Simpson A. Transverse modes and frequencies of beams translating between fixed end supports. J. Mech. Engrg. Sci., 1973, 15(3): 159-164.
27. Kim Y, Tabarrok B. On the nonlinear vibration of traveling strings. J. Franklin Inst., 1972, 293: 381-399.
28. Schajer G S. The vibration of a rotating circular strings subject to a fixed elastic restraint. Journal Sound and Vibration, 1984, 92(1): 11-19.
29. Bhat R B, Xistris G D, Sankar T S. Dynamics behavior of a moving belt supported on an elastic foundation. J Mech Design, 1982, 104(1): 143-147.
30. Ulsoy A G, Whitesell J E, Hooven M D. Design of belt-tensioner systems for dynamics stability. J Vib Acoust Stress, Reliability in Desi, 1985, 107: 285-290.
31. Zhu W D, Mote Jr C D. Free and forced response of an axially moving string transporting a damped linear oscillator. Journal Sound and Vibration, 1994, 177(5): 591-610.
32. Swope R D, Ames W F. Vibrations of a moving threadline. J.Franklin Inst., 1963, 275(1): 36-55.
33. Ames W F, Vicario A A Jr. On the longitudinal wave propagation on a traveling threadline. Developments in Mechanics, Proc. 11th Midwest. Mech. Conf., 1969, 5: 733-746.
34. Ames W F, Lee S Y, Vicario A A Jr. Longitudinal wave propagation on a traveling threadline --Ⅱ, Intl. J. Nonlin. Mech., 1970, 5(3): 413-426.
35. Mote C D Jr. Stability of systems transporting accelerating axially moving materials. J. Dynam. Syst. Meas. Control, Trans. ASME, 1975, 97: 96-98.
36. Mahalingam S. Transverse oscillations of power transmission chains. Brit. J. Appl. Phys., 1957, 8: 145-148.
37. Ulsoy A G, Whitesell J E, Hooven M D. Design of belt-tensioner systems for dynamics stability. J Vib Acoust Stress, Reliability in Desi, 1985, 107: 285-290.
38. Wickert J A and Mote C D Jr.. On the energetics of axially continua. J Acoust Soc Am, 1989, 85(3): 1365-1368.
39. Lee S Y, Mote C D Jr. A generalized treatment of the energetics of translating continua, part 1: strings and second order tensioned pipes. Journal Sound and Vibration, 1997, 204(5): 717-734.
40. Sack R A. Transverse oscillations in traveling strings. British J. Appl. Phys., 1954, 5: 224-226.
41. Mahalingam S. Transverse oscillations of power transmission chains. Brit. J. Appl. Phys., 1957, 8: 145-148.
42. Swope R D, Ames W F. Vibrations of a moving threadline. J.Franklin Inst., 1963, 275(1): 36-55.
43. Nelson H M. Transverse vibration of a moving strip. J. Sound Vib., 1979, 65(3): 381-389.
44. Alspaugh D W. Torsional vibration of a moving band. J. Franklin Inst., 1967, 283(4): 328-338.
45. Soler A I. Vibration and stability of a moving band. J. Franklin Inst., 1968, 286(4): 295-307.
46. Roth W. Torsionsschwingungen des gezogenes bandes. ZAMM, 1977, 57(12): 722-725.
47. Mote Jr C D. Divergence buckling of an edge-loaded axially moving band. Intl. J. Mech. Sci., 1968, 10(4): 281-295.
48. Naguleswaran S, Williams C J H. Lateral vibration of band-saw, pulley belts and the like. Int J Mech Sci, 1968, 10: 239-250.
49. Doyle E, Hornung K G. Lateral vibration of V-belts. ASME Paper, 1969, 69-vibr-29.
50. Lai J L, Chen C H. Vibration and dynamic stability of an axially moving belt. ASME Paper, 1971, 71-vibr-31.
51. Kozin F, Milstead R M. The stability of a moving elastic strip subjected to random parametric excitation. J. Appl. Mech., Trans. ASME, 1979, 46(2): 404-410.
52. Kozic P, Paviovic R. Stability of a thin moving elastic strip subjected to random parametric excitations. Journal of Sound and Vibration, 1997, 206(2): 280-285.
53. Wu W Z, Mote C D Jr. Parametric excitation of an axially moving band by periodic edge loading. J. Sound Vib., 1986, 110(1): 27-39.
54. Barakat R. Transverse vibrations of a moving thin rod. J. Acoust. Soc. Amer., 1967, 43(3): 533-539.
55. Taleb I A, Misra A K. Dynamic of an axially moving beam submerged in a fluid. AIAA J. Hydronau., 1981, 15(1-4): 62-66.
56. Mote Jr C D. and Naguleswaran S. Theoretical and experimental band saw vibrations. J. Engrg. Indus., Trans. ASME, 1966, 88(2): 151-156.
57. Cherchas D B, Gossain D M. Dynamics of a flexible solar array during deployment from a spinning spacecraft. CASI Trans, 1974, 7(1): 10-17.
58. Weeks G E. Dynamic analysis of a deployable space structure. J. Spacecraft Rockets, 1986, 23(1): 102-107.
59. Mansfield L, Simmonds J G. The reverse spaghetti problem: drooping motion of an elastica issuing from a horizontal guide. J. Appl. Mech., Trans. ASME, 1987, 54: 147-150.
60. Lips K W, M di V J. Transient attitude dynamics of satellites with deploying flexible appendages, ACTA Astronau, 1978, 5(10): 797-815.
61. Tabarrok B, Leech C M, Kim Y I. On the dynamics of an axially moving beam. J. Franklin Inst., 1974, 297(3): 201-220.
62. Stylianou M, Tabarrok B. Finite element analysis of an axially moving beam part I: time integration. Journal of Sound and Vibration, 1994, 178(4): 433-453.
63. Stylianou M, Tabarrok B. Finite element analysis of an axially moving beam part II: stability analysis. Journal of Sound and Vibration, 1994, 178(4): 455-481.
64. Fung R F, Lu P Y. Non-linearly dynamic modeling of an axially beam with a tap mass. Journal of Sound and Vibration, 1998, 218(4): 559-571.
65. AI-Bedoor B O, Kbulief Y A. An approximate analytical solution of beam vibration during axial motion. Journal of Sound and Vibration, 1996, 192(1): 159-171.
66.李山虎,杨靖波,黄清华,陈德成.轴向运动悬臂梁的独立模态振动控制--Ⅰ近似理论分析.应用力学学报, 2002, 19(1): 35-38.
67. Hayashi C. Non-linear oscillation in physical systems. McGraw-Hill, New York.1964.
68. Jordan D W, Smith P. Non-linear ordinary differential equation. Oxford University Press, Oxford, 1977.
69. Nayfeh A H, Mook D T. Nonlinear oscillations. John Wiley and Sons, New York, 1979.
70. Dooren R V. Combination tones of summed type in a non-linear damped vibratory system with two degrees of freedom. Int. J. Non-linear Mechanics, 1971, 6: 237-254.
71. Chia C Y, Prabhakara M K. Large deflection of unsymmetric cross-ply and angle-ply plates. J. Mech. Eng. Sci., 1976, 18: 179-183
72. Tongue B H. Limit cycle oscillations of a nonlinear rotorcraft model. AIAA Journal, 1984, 22: 967-974.
73. Cheung Y K, Iu V P. An implicit implementation of harmonic balance method for nonlinear dynamic systems. Engineering Computations, 1988, 5: 134-140.
74. Leung A Y T, Fung T C. Phase increment analysis of damped doffing oscillators. InternationalJournal for Numerical Methods in Engineering, 1989, 28: 193-210.
75. Leung A Y T, Fung T C. Construction of chaotic regions. Journal of Sound and Vibration, 1989, 131: 445-455.
76. Fung T C. Steady state solutions of nonlinear dynamic systems. Ph. D. Thesis, University of Hong Kong, 1989.
77. Dunn P, Dugundji J. Nonlinear stall flutter and divergence analysis of cantilevered graphite/epoxy wings. AIAA Journal, 1992, 30(1): 153-162.
78. Kim T, Dugundji J. Nonlinear large amplitude aeroelastic behavior of composite rotor blades. AIAA Journal, 1993, 31(8): 1489-1497.
79. Blankenship G W, Kahraman A. Steady state forced response of a mechanical oscillator with combined parametric excitation and clearance type non-linearity. Journal of Sound and Vibration, 1995, 185(5): 743-765.
80.郑小平,姚振汉,杜庆华.利用谐波平衡法计算局部非线性运动系统的稳态响应.清华大学学报, 1997, 37(8): 36-39.
81. Lau S L and Cheung Y K. Amplitude incremental variational principle for non-linear vibration of elastic systems. ASME, Journal of Applied Mechanics, 1981, 48: 959-964.
82. Cheung Y K, Lau S L. Incremental time-space finite strip method for nonlinear structural vibrations. Earthquake Engineering and Structural Dynamics, 1982, 10: 239-253.
83. Lau S L, Cheung Y K and Wu S Y. Nonlinear vibration of thin elastic plates, partⅠ: generalized incremental hamilton’s principle and element formulation. ASME Journal of Applied Mechanics, 1984, 51: 837-844.
84. Lau S L, Cheung Y K. Incremental hamilton’s principle with multiple time scales for nonlinear aperiodic vibrations of shells. ASME Journal of Applied Mechanics, 1986, 53: 465-466.
85. Iu V P, Cheung Y K and Lau S L. Nonlinear vibration analysis of multiplayer beams by incremental finite elements, partⅠ: theory and numerical formulation. Journal of Sound and Vibration, 1985, 100: 373-382.
86. Iu V P, Cheung Y K. Nonlinear vibration analysis of multilayer sandwich plates by incremental finite elements, partⅠ: theoretical development. Engineering Computation, 1986, 3: 35-42.
87. Lau S L, Cheung Y K and Wu S Y. A variable parameter incrementation method for dynamic instability of linear and nonlinear elastic systems. ASME Journal of Applied Mechanics, 1982, 49: 849-853.
88. Lau S L, Cheung Y K and Wu S Y. Incremental harmonic balance method with multiple time scales for aperiodic vibration of nonlinear systems. ASME Journal of Applied Mechanics, 1983, 50: 871-876.
89. Pierre C, Dowell E H. A study of dynamic instability of plates by an extended incrementalharmonic balance method. ASME Journal of Applied Mechanics, 1985, 52: 693-697.
90. Pierre C. Ferri A A and Dowell E H. Multi-harmonic analysis of dry friction damped systems using an incremental harmonic balance method. ASME Journal of Applied Mechanics, 1985, 52: 958-964.
91. Ferri A A, Dowell E H. Frequency domain solutions to multi-degree-of-freedom dry friction damped systems. Journal of Sound and Vibration, 1988, 124: 207-224.
92. Cheung Y K, Chen S H and Lau S L. Application of the incremental harmonic balance method to cubic non-linearity systems. Journal of Sound and Vibration, 1990, 140: 273-286.
93. Lau S L, Yuen S W. Hopf bifurcation and limit cycle by the incremental harmonic balance method. Computer Methods in Applied Mechanics and Engineering, 1991, 91(1-3): 1109-1121.
94. Lau S L, Zhang W S. Nonlinear vibrations of piecewise-linear systems by incremental harmonic balance method, 1992, 59(1): 153-160.
95. Leung A Y T, Chui S K. Non-linear vibration of coupled duffing oscillators by an improved incremental harmonic balance method. Journal of Sound and Vibration, 1995, 181(4): 619-633.
96. Leamy M J, Perkins N C. Nonlinear periodic response of engine accessory drives with dry friction tensioners. Journal of Vibration and Acoustics, Transactions of the ASME, 1998, 120(4): 909-916.
97. Raghothama A, Narayanan S. Periodic response and chaos in nonlinear systems with parametric excitation and time delay. Nonlinear Dynamics, 2002, 27(4): 341-365.
98. Minorsky N. Non-linear oscillations. Robert E. Krieger, Huntington, New York, 1962.
99. Nayfeh A H. Perturbation methods. John Wiley, New York, 1973.
100. Lindstedt A. Ueder die integration einer fur die storungstheorie wichtigen differentialgleichung. Astron, Nach, 1882, 103: 211-220.
101. PoincaréH. Sur les integrals des equations linéaires. Acta, Math, 1886, 8: 295-344.
102. Lighthill M J. A technique for rendering approximate solutions to physical problems uniformly valid. Phil. Mag., 1949, 40: 1179-1201.
103. Kuo Y H. On the flow of an incompressible viscous fluid past a flat plate at moderate Reynolds numbers. J. Math. and Phys, 1953, 32: 83-101.
104. Kuo Y H. Viscous flow along a flat plate moving at high supersonic speeds. J. Aeron. Sci., 1956, 23: 125-136.
105. Tsien H S. The Poincaré-Lighthill-Kuo method. Advan. Appl. Mech., 1956, 4: 281-349.
106. Chen S H. Generalization of the lindstedt-poincarémethod for analysis of non-linear vibrations. Ph. D. Thesis, University of Hong Kong, 1990.
107. Cheung Y K, Chen S H and Lau S L. A modified Lindstedt-Poincare method for certain strongly nonlinear oscillators. Int. J. Non-linear Mechanics, 1991, 26(3/4): 367-378.
108. Chen S H, Cheung Y K. A modified Lindstedt-Poincare method for a strongly non-linear two degree-of-freedom system. Journal of Sound and Vibration, 1996, 193(4): 751-762.
109. Chen S H, Cheung Y K. A modified Lindstedt-Poincarémethod for a strongly non-linear system with quadratic and cubic nonlinearities. Shock and Vibration, 1996, 3(4): 279-285.
110.唐驾时,尹小波.一类强非线性振动的分叉.力学学报, 1996, 28(3): 363-369.
111.唐驾时.求强非线性系统次谐共振解的MLP方法.应用数学和力学, 2000, 21(10): 1039-1045.
112. Lau S L, Cheung Y K, Chen S H. An alternative perturbation procedure of multiple scale for nonlinear dynamics systems. ASME J. Appl. Mech., 1989, 56(3): 667-675.
113. Chen S H, Cheung Y K, Lau S L. On the internal resonance of multi-degree-of-freedom systems with cubic nonlinearity. Journal Sound and Vibration, 1989, 128(1): 13-24.
114. Sturrock P A. Nonlinear effects in electron plasmas. Proc. Roy. Soc. 1957, A242: 277-299.
115. Sturrock P A. Nonlinear theory of electromagnetic waves in plasmas. Stanford University Microwave Laboratory Repl. 1963, 1004.
116. Frieman E A. On a new method in the theory of irreversible processes. J. Math. Physis, 1963, 4: 410-418.
117. Nayfeh A H. A perturbation method for treating nonlinear oscillation problems. J. Math. and Phys, 1965, 44: 368-374.
118. Sandri G. A new method of expansion in mathematical physics. Nuovo Cimento, 1965, B36: 67-93.
119. Cole J D, Kevorkian J. Uniformly valid asymptotic approximations for certain nonlinear differential equations. Nonlinear Differential Equation and Nonlinear Mechanics, Academic, New York, 1963, 113-120.
120. Morrison J A. Comparison of the modified method of averaging and the two variable expansion procedure. SIAM Rev., 1966, 8: 66-85.
121. Nayfeh A H. The Van Der Pol oscillator with delayed amplitude limiting. Proc. IEEE, 1967, 55: 111-112.
122. Kuzmak G E. Asymptotic solutions of nonlinear second order differential equations with variable coefficients. J. Appl. Math. Mech., 1959, 23: 730-744.
123. Mahony J J. An expansion method for singular perturbation problems. J. Australian Mat. Soc., 1962, 2: 440-463.
124. Nayfeh A H. A perturbation method for treating nonlinear oscillation problems. J. Math. and Phys, 1965, 44: 368-374.
125. Abdelhafez H M. Resonance of multiple frequency excitated systems with quadratic, cubic and quartic non-linearity. Mathematics and Computers in Simulation, 2002, 61: 17-34.
126. Zhao Y Y, Wang L H, Chen D L, Jiang L Z. Non-linear dynamic analysis of the two-dimensional simplified model of an elastic cable. Journal of Sound and Vibration, 2002, 255(1): 43-59.
127. El- Bassiouny. Three-mode interaction in harmonically excited system with cubic nonlinearities. Applied Mathematics and Computation, 2003, 139: 201-230.
128. Arafat H N, Nayfeh A H. Non-linear response of suspended cables to primary resonance excitations. Journal of Sound and Vibration, 2003, 266: 325-354.
129. Li X Y, Chen Y S, Wu Z Q. Non-linear normal modes and their bifurcation of a class of systems with three double of pure imaginary roots and dual. International Journal of Non-linear Mechanics, 2004, 39: 189-199.
130. Krylov N, Bogoliubov N N. Introduction to nonlinear mechanics. Princeton University Press, Princeton. N. J., 1947.
131. Bogoliubov N N, Mitropolski Y A. Asymptotic methods in the theory of nonlinear oscillations. Gordon and Breach, New York, 1961.
132. Yang Y R. KBM method of analyzing limit cycle flutter of a wing with an external store and comparison with a wind-tunnel test. Journal Sound and Vibration, 1995, 187(2), 271-280.
133. Zhou Y H, Tzou H S. Active control of nonlinear piezoelectric circular shallow spherical shells. International Journal of Solids and Structures, 2000, 37: 1663-1677.
134.孟光.延时反馈振荡器在弱正弦外激下的强制振动.西北工业大学学报, 20(1): 1-5.
135.褚亦清.非线性振动分析.北京理工大学出版社, 1996.
136. Pellicano F and Vestroni F. Nonlinear dynamics and bifurcations of an axially moving beam. Journal of Vibration and Acoustics, 2000, 122: 21-30.
137. Wickert J A. Non-linear vibration of a traveling tensioned beam. International Journal of Non-Linear Mechanics, 1992, 27(3): 503-517.
138. Chakrabporty G, Mallik A K, Hatal H. Non-linear vibration of traveling beam. International Journal of Non-Linear Mechanics, 1999, 34: 655-670.
139. .Chakrabporty G, Mallik A K. Non-linear vibration of a traveling beam having an intermediate guide. Nonlinear Dynamics, 1999, 20: 247-265.
140. Al-jawi A A N, Pierre C, Ulsoy A G. Vibration localization in dual-span axially moving beams, PartΙ: formulation and results. Journal of Sound and Vibration, 1995, 179(2): 243-266.
141. Chen J S. Natural frequencies and stability of an axially-traveling string in contract with a stationary load system. ASME Journal of Vibration and Acoustics, 1997, 119: 152-156.
142. Riedel C H and Tan C A. Coupled, forced response of an axially moving strip with internalresonance. International Journal of Non-Linear Mechanics, 2002, 37: 101-116.
143. Pakdemirli M, Ulsoy A G, Ceranoglu A. Transverse vibration of an axially accelerating string. Journal of Sound and Vibration, 1994, 169: 179-196.
144. .Pakdemirli M, Ulsoy A G. Stability analysis of an axially accelerating string. Journal of Sound and Vibration, 1997, 203: 815-832.
145. Mockensturm E M, Perkins N C, Galip Ulsoy A. Stability and limit cycles of parametrically excited, axially moving strings. Journal of Vibration and Acoustics, 1996, 118, 346-351.
146. Zhang L, Zu J W. Nonlinear Vibration of Parametrically Excited Moving Belts, Part I: Dynamic Response. Journal of Applied Mechanics, 1999, 66: 396-402.
147. Zhang L, and Zu J W. Nonlinear Vibration of Parametrically Excited Moving Belts, Part II: Stability Analysis. Journal of Applied Mechanics, 1999, 66: 403-409.
148. Suweken G, Horssen Van W T. On the weakly nonlinear, transversal vibrations of a conveyor belt with a low and time-varying velocity. Nonlinear Dynamics, 2003, 31(2): 197-223.
149. Oz H R, Pakdemirli M. Vibration of an axilly moving beam with time dependent velocity. Journal of Sound and Vibration, 1999, 227: 239-257.
150. Oz H R. On the vibration of an axially traveling beam on fixed supports with variable velocity. Journal of Sound and Vibration, 2001, 239(3): 556-564.
151. Fung R F, Chang H C. Dynamic and energetic analyses of a string/slider non-linear coupling system by variable grid finite difference. Journal of Sound and Vibration, 2001, 239(3): 505-514.
152. Ravindra B, Zhu W D. Low–dimensional chaotic response of axially accelerating continuum in the supercritical regime. Archive of Applied Mechanics, 1998, 68: 195-205.
153. Moon J, Wickert J A. Non-linear vibration of power transmission belts. Journal of Sound and Vibration, 1997, 200: 419-431.
154. Lakshmikumaran A V, Wickert J A. On the vibration of coupled traveling string and air bearing system. ASME Journal of Vibration and Acoustics, 1996, 118: 398-405.
155. Bajkowdki J, Szemplinska-Stupnicka W. Internal resonance effects—simulation versus analytical methods results. Journal of Sound and Vibration, 1986,104: 259-275.
156. Lee C L, Perkins N C. Nonlinear oscillations of suspended cables containing a two-to-one internal resonance. Nonlinear Dyn, 1992, 3: 465-490.
157.胡海昌.弹性力学的变分原理及其应用.科学出版社, 1981.
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