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Nonlinear stochastic vibration of a variable cross-section rod with a fractional derivative element
- Source :
- International Journal of Non-Linear Mechanics. 135:103770
- Publication Year :
- 2021
- Publisher :
- Elsevier BV, 2021.
-
Abstract
- The article deals with the problem of computing efficiently the nonlinear response of a rod involving a fractional constitutive model, and exposed to random excitation. The constitutive model is a three-parameter model comprising an instant elasticity modulus, a prolonged elasticity modulus, and a relaxation parameter. The nonlinear term is a linear-plus-cubic force of the Winkler kind. The resulting nonlinear fractional partial differential equation governing the rod displacement has no known exact solution. Thus, the article proposes an approximate analytical solution by relying on the statistical linearization technique. Further, it develops a Boundary Element Method (BEM)-based approach to estimate numerically the rod response statistics. The statistical linearization solution is obtained by representing the rod displacement as the superposition of linear modes of vibration having time-dependent coefficients. In this context, it is shown that the equation governing the time variation of the mode coefficients is a nonlinear fractional ordinary differential equation, whose solution is computed by a surrogate linear system identified by minimizing the response error between the linear system and the nonlinear one in a mean square sense. Relevant Monte Carlo studies pertaining to rods with fixed-fixed, and fixed-free ends show that the proposed analytical solution is in a good agreement with data obtained by the numerical (BEM) approach.
- Subjects :
- Partial differential equation
Applied Mathematics
Mechanical Engineering
Constitutive equation
Mathematical analysis
Linear system
Relaxation (iterative method)
Fractional derivative
02 engineering and technology
021001 nanoscience & nanotechnology
Statistical linearization
Fractional calculus
Nonlinear system
Superposition principle
020303 mechanical engineering & transports
Boundary element method
Longitudinal vibration
Random vibration
Rod
0203 mechanical engineering
Mechanics of Materials
Ordinary differential equation
0210 nano-technology
Mathematics
Subjects
Details
- ISSN :
- 00207462
- Volume :
- 135
- Database :
- OpenAIRE
- Journal :
- International Journal of Non-Linear Mechanics
- Accession number :
- edsair.doi.dedup.....691be7ad56f6a39ab66395934a7995dc