On the forced vibrations of an elastoplastic oscillator with a geometrically nonlinear behaviour

The scope of this study is to present a contribution to the geometrically nonlinear free and forced vibration of multiple-stepped beams, based on the theories of Euler–Bernoulli and von Karman, in order to calculate their corresponding amplitude-dependent modes and frequencies. Discrete expressions of the strain energy and kinetic energies are derived, and Hamilton’s principle is applied to reduce the problem to a solution of a nonlinear algebraic system and then solved by an approximate method. The forced vibration is then studied based on a multimode approach. The effect of nonlinearity on the dynamic behaviour of multistepped beams in the free and forced vibration is demonstrated and discussed. The effect of varying some geometrical parameters of the stepped beams in the free and forced cases is investigated and illustrated, among which is the variation in the level of excitation.

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Analysis of Nonlinear Forced Vibrations of Shallow Shells With Cut-Outs by Using the R-Function Method

Volume 7: Dynamic Systems and Control; Mechatronics and Intelligent Machines, Parts A and B ◽

10.1115/imece2011-64349 ◽

2011 ◽

Author(s):

Galyna Pilgun ◽

Marco Amabili

Keyword(s):

Equations Of Motion ◽

Continuation Method ◽

Forced Vibrations ◽

Second Step ◽

Geometrically Nonlinear ◽

Shallow Shells ◽

Natural Modes ◽

Mesh Free ◽

Nonlinear Equations Of Motion ◽

Admissible Functions

Geometrically nonlinear forced vibrations of shells based on the domains with cut-outs are investigated. Classical nonlinear shallow-shell theories retaining in-plane inertia is used to calculate the strain energy; the shear deformation is neglected. A mesh-free technique based on classic approximate functions and the R-function theory is used to build the discrete model of the nonlinear vibrations. This allowed for constructing the sequences of admissible functions that satisfy given boundary conditions in domains with complex geometries. Shell displacements are expanded by using Chebyshev orthogonal polynomials. A two-step approach is implemented to solve the problem: first a linear analysis is conducted to identify natural frequencies and corresponding natural modes to be used in the second step as a basis for nonlinear displacements. The system of ordinary differential equations is obtained by using Lagrange approach on both steps. The convergence of the solution is studied by using different multimodal expansions. The pseudo-arclength continuation method and bifurcation analysis are used to study the nonlinear equations of motion. Numerical responses are obtained in the spectral neighbourhood of the lowest natural frequency. When possible, obtained results are compared to those available in the literature.

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Topology Optimization of Large-Displacement Flexure Hinges

Volume 5A: 39th Mechanisms and Robotics Conference ◽

10.1115/detc2015-46444 ◽

2015 ◽

Cited By ~ 2

Author(s):

Min Liu ◽

Xianmin Zhang ◽

Sergej Fatikow

Keyword(s):

Topology Optimization ◽

Compliant Mechanisms ◽

Optimization Method ◽

Large Displacement ◽

Element Formulation ◽

Nonlinear Behaviour ◽

Geometrically Nonlinear ◽

Flexure Hinges ◽

Lagrangian Finite Element ◽

Moving Asymptotes

Research on topology optimization of compliant mechanisms is extensive but the design of flexure hinges using topology optimization method is comparatively rare. This paper deals with topology optimization of flexure hinges undergoing large-displacement. The basic optimization model is developed for topology optimization of the revolute hinge. The objective function for the synthesis of large-displacement flexure hinges are proposed together with constraints function. The geometrically nonlinear behaviour of flexure hinge is modelled using the total Lagrangian finite element formulation. The equilibrium is found by using a Newton-Raphson iterative scheme. The sensitivity analysis of the objective functions are calculated by the adjoint method and the optimization problem is solved using the method of moving asymptotes (MMA). Numerical examples are used to show the validity of the proposed method and the differences between the results obtained by linear and nonlinear modelling are large.

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