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A generalized mathematical model for a class of mechanical systems with lumped and distributed parameters

1 Chair of Applied Mathematics, East Siberian State University of Technology and Management, Ulan-Ude, 670013, Russia
2 Institute of Mathematics and Computer Science, Department of Applied Mathematics and differential equations, Buryat State University, Ulan-Ude, 670000, Russia

Topical Section: Mathematical modeling

A hybrid system of differential equations, which represents a generalized mathematical model for a system of rigid bodies mounted on an Euler-Bernoulli beam with the aid of springs, is described in the general form. A hybrid system of differential equations is understood as a system of differential equations composed of ordinary differential equations and partial differential equations. Hybrid systems of differential equations of such type are normally constructed in the process of inference of dynamic equations for a given class of mechanical systems with the use of the Hamiltonian variation principle. The paper considers the analytical-numerical method proposed by the author, which is based on the mathematical apparatus of generalized functions. The comparative analysis of results of numerical computations obtained by the author’s method to the computational results obtained by the techniques known from the literature has shown the plausibility and universality of the author’s approach.
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Keywords mechanical systems; system of rigid bodies; generalized mathematical model; Euler-Bernoulli beam; hybrid systems of differential equations; Hamiltonian variation principle

Citation: Arsalan D. Mizhidon. A generalized mathematical model for a class of mechanical systems with lumped and distributed parameters. AIMS Mathematics, 2019, 4(3): 751-762. doi: 10.3934/math.2019.3.751

References

  • 1.A. D. Mizhidon, K. A. Mizhidon, Eigenvalues of boundary value problem for a hybrid system of differential equations, Siberian Electronic Mathematical Reports, 13 (2016), 911-922.
  • 2.D. W. Chen, The exact solution for free vibration of uniform beams carrying multiple two-degree-of-freedom spring-mass systems, J. Sound Vib., 295 (2006), 342-361.    
  • 3.T. P. Chang, C. Y. Chang, Vibration analysis of beams with a two-degree-of-freedom spring-mass system, Int. J. Solids Struct., 35 (1998), 383-401.    
  • 4.P. D. Cha, Free vibrations of a uniform beam with multiple elastically mounted two-degree-of-freedom systems, J. Sound Vib., 307 (2007), 386-392.    
  • 5.J.-J. Wu, A. R. Whittaker, The natural frequencies and mode shapes of a uniform cantilever beam with multiple two-DOF spring-mass systems, J. Sound Vib., 227 (1999), 361-381.    
  • 6.S. Naguleswaran, Transverse vibrations of an Euler-Bernoulli uniform beam carrying several particles, Int. J. Mech. Sci., 44 (2002), 2463-2478.    
  • 7.S. Naguleswaran, Transverse vibration of an Euler-Bernoulli uniform beam on up a five resilient supports including ends, J. Sound Vib., 261 (2003), 372-384.    
  • 8.H. Su, J. R. Banerjee, Exact natural frequencies of structures consisting of two part beam-mass systems, Struct. Eng. Mech., 19 (2005), 551-566.    
  • 9.J.-S. Wu, Alternative approach for free vibration of beams carrying a number of two-degree of freedom spring-mass systems, J. Struct.Eng., 128 (2002), 1604-1616.    
  • 10.J.-S. Wu, H.-M. Chou, A new approach for determining the natural frequencies and mode shapes of a uniform beam carrying any number of spring masses, J. Sound Vib., 220 (1999), 451-468.    
  • 11.J.-S. Wu, D.-W. Chen, Dynamic analysis of a uniform cantilever Beam carrying a number of elastically mounted point masses with dampers, J. Sound Vib., 229 (2000), 549-578.    
  • 12.S. Kukla, B. Posiadala, Free vibrations of beams with elastically mounted masses, J. Sound Vib., 175 (1994), 557-564.    
  • 13.S. Kukla, The green function method in frequency analysis of a beam with intermediate elastic supports, J. Sound Vib., 149 (1991), 154-159.    
  • 14.A. D. Mizhidon, Theoretical foundations of the study of a certain class of hybrid systems of differential equations, Itogi Nauki i Tekhniki. Seriya "Sovrem. Mat. Pril. Temat. Obz.", 155 (2018), 38-64.

 

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