Abstract
In this paper, a new methodology for the determination of the boundaries between oscillatory and non-oscillatory motion for nonviscously damped nonproportional systems is proposed. It is assumed that the damping forces are expressed as convolution integrals of the velocities via hereditary exponential kernels. Oscillatory motion is directly related to the complex nature of eigensolutions in a frequency domain and, in turn, on the value of the damping parameters. New theoretical results are derived on critical eigenmodes for viscoelastic systems with multiple degrees of freedom, with no restrictions on the number of hereditary kernels. Furthermore, these outcomes enable the construction of a numerical approach to draw the critical curves as solutions of certain parameter-dependent eigenvalue problems. The method is illustrated and validated through two numerical examples, covering discrete and continuous systems.
Highlights
Nonviscous damping is characterized by dissipative forces which depend on the past history of the velocity response via convolution integrals over hereditary kernel functions
The most important contributions, from a theoretical point of view, have been presented: (i) the new characterization of the critical modes given in Equation (36); (ii) the derivation of the modal critical Equation (45), and (iii) the development of a numerical model—summarized in both Equations (52) and (53)—which enables reducing the computation of any critical curve to solve an eigenvalue problem repeatedly along the interval α ∈
The dissipative model is represented by damping forces with linear dependency of the velocities via hereditary kernel functions
Summary
There are many applications, from micro-scale systems to large structures, that require the control of vibration, sound, and wave propagation for proper operation. We investigate criticality in nonviscously damped multiple-degrees-of-freedom (dof) systems, considering any number of exponential hereditary kernels. The first attempts to determine the conditions of overcritical damping were proposed by Muravyov and Hutton [12] and Adhikari [13] These works were developed with just one hereditary kernel, addressing the problem by carrying out an exhaustive analysis of the nature of the root of the resulting third-order characteristic polynomial. The critical oscillatory motion of nonviscous beams has been studied by Pierro [15], solving the eigenvalues for one and two exponential kernels and discussing their nature (real or complex). The problem of determining the critical manifolds of a single degree-offreedom oscillator for any number of hereditary kernels has been analytically solved in exact form by Lázaro [17], by transforming Equation (4) into parametric closed-form expressions. The proposed approach is validated by means of two numerical examples: a four-dof discrete lumped-mass system, and a continuous beam finite element model with viscoelastic supports
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