Damped Vibration Systems Research Articles

Modified multiple time scale method for solving strongly nonlinear damped forced vibration systems

Abstract In this paper, modified multiple time scale (MTS) method is employed to solve strongly nonlinear forced vibration systems. The first-order approximation is only considered in order to avoid complexicity. The formulations and the determination of the solution procedure are very easy and straightforward. The classical multiple time scale (MS) and multiple scales Lindstedt-Poincare method (MSLP) do not give desire result for the strongly damped forced vibration systems with strong damping effects. The main aim of this paper is to remove these limitations. Two examples are considered to illustrate the effectiveness and convenience of the present procedure. The approximate external frequencies and the corresponding approximate solutions are determined by the present method. The results give good coincidence with corresponding numerical solution (considered to be exact) and also provide better result than other existing results. For weak nonlinearities with weak damping effect, the absolute relative error measures (first-order approximate external frequency) in this paper is only 0.07% when amplitude A = 1.5 , while the relative error gives MSLP method is surprisingly 28.81%. Furthermore, for strong nonlinearities with strong damping effect, the absolute relative error found in this article is only 0.02%, whereas the relative error obtained by MSLP method is 24.18%. Therefore, the present method is not only valid for weakly nonlinear damped forced systems, but also gives better result for strongly nonlinear systems with both small and strong damping effect.

Existence Results for Impulsive Damped Vibration Systems

Employing a critical point theorem (local minimum result) for differentiable functionals, we prove the existence of at least one non-trivial weak solution for a class of impulsive damped vibration systems under an asymptotical behavior of the nonlinear datum at zero. Two examples are presented to illustrate the feasibility and effectiveness of the results.

Homoclinic orbits for damped vibration systems with asymptotically quadratic or subquadratic potentials

In this paper, we study a class of damped vibration systems, $$ \ddot{u}(t)+B\dot{u}(t)-L(t)u(t)+\nabla W\bigl(t,u(t)\bigr)=0, \quad \forall t \in \mathbb{R}, $$ where $W(t,u)$ is of indefinite sign. By using a critical point theorem of Ding, we establish a new criterion to guarantee that the above system has infinitely many nontrivial homoclinic orbits under the assumption that $W(t,u)$ is asymptotically quadratic or subquadratic as $|u|\rightarrow\infty$ . Recent results in the literature are generalized and significantly improved.

Ground state homoclinic orbits of superquadratic damped vibration systems

In the case where nonlinearities are superquadratic at infinity, we study the existence of ground state homoclinic orbits for damped vibration systems without periodic conditions by using variational methods. Here the (local) Ambrosetti-Rabinowitz superquadratic condition is replaced by a general superquadratic condition.

Non-periodic damped vibration systems with sublinear terms at infinity: Infinitely many homoclinic orbits

In this paper, we study a class of non-periodic damped vibration systems with subquadratic terms. We obtain infinitely many nontrivial homoclinic orbits by a variant fountain theorem developed recently by Zou. To the best of our knowledge, there is no result published concerning the existence of infinitely many homoclinic orbits for this class of non-periodic damped vibration systems with subquadratic terms at infinity.

Nonperiodic Damped Vibration Systems with Asymptotically Quadratic Terms at Infinity: Infinitely Many Homoclinic Orbits

We study a class of nonperiodic damped vibration systems with asymptotically quadratic terms at infinity. We obtain infinitely many nontrivial homoclinic orbits by a variant fountain theorem developed recently by Zou. To the best of our knowledge, there is no result published concerning the existence (or multiplicity) of nontrivial homoclinic orbits for this class of non-periodic damped vibration systems with asymptotically quadratic terms at infinity.

Infinitely many solutions for a class of second-order damped vibration systems

In this paper, by using the variational approach, we study the existence of infinitely many solutions for a class of second-order damped vibration systems under super- quadratic and sub-quadratic conditions. Some new results are established and some recent results in the literature are generalized and significantly improved.

On homoclinic orbits for a class of damped vibration systems

In this article, we establish the new result on homoclinic orbits for a class of damped vibration systems. Some recent results in the literature are generalized and significantly improved.

On eigenproblem solution of damped vibrations associated with gyroscopic moments

A new efficient approach is presented for solving the quadratic eigenvalue problem of weakly, nonproportionally damped vibration systems. In the analysis of these systems, gyroscopic moments and external damping are both considered. Traditional restriction of symmetry of inertia, damping and stiffness matrices is slightly relaxed. A second-order perturbation theory is developed such that the perturbed solution is based on the eigensolution of an unperturbed subproblem. This subproblem considers the unperturbed system in two different forms: (i) a conservative, gyroscopic part of an original problem, or (ii) a nonconservative, gyroscopic part of an original problem that is proportionally damped. To cope with asymmetry of the system matrices, a Duncan's like state formulation is used to bring these matrices into a suitable form for perturbations. Two numerical examples are introduced for explaining the detailed implementation of the presented approach. Additionally, a practical problem of rotor supported by two tilting pad-bearings is investigated. The eigensolutions obtained by the current approach match, to a great extent, other solutions obtained by time-consuming exact methods. The investigation procedure given here gives a framework to handle vibration problems of weakly nonproportional damping and/or weakly asymmetric inertia, damping and stiffness matrices.

Model Order Reduction of Viscously Damped Vibration Systems Using Accelerated Iterative Dynamic Condensation

An accelerated iterative dynamic condensation method for the model order reduction of vibration systems with viscous damping is proposed. A group of governing equations for dynamic condensation matrix are derived from the eigenvalue equations defined in the state space. Two iterative schemes for solving these governing equations are provided. Based on different state space formulations, two more groups of governing equations are developed. A comparison of the present method with three iterative approaches proposed recently is provided. The present approach is implemented into two practical vibration systems, a tall building with one tuned mass damper and a floating raft isolation system. The results show that the proposed method has much higher accuracy than the other three approaches while the computational effort is almost the same.

Dynamic condensation method of nonclassically damped vibration systems

Dynamic condensation scheme, which has been widely applied to reduce the number of degrees of freedem of finite element modeling, is presented in this paper. Most of them, however, are valid for undamped systems. An efficient iterative approach for the dynamic condensation of nonclassically damped systems is proposed. The classical subspace iteration method for undamped models is extended to nonclassically damped models. Then, a governing equation for the dynamic condensation matrix in state space is derived from the extended subspace iteration. Two iterative schemes are proposed to slove the governing equation. Because the dynamic condensation matrix is independent of the system matrices and the eigenpairs (eigenvalues and eigenvectos) of the reduced model, it is unnecessary to compute them in every iteration. This makes the proposed method much more computationally efficient than those approaches proposed in the past. The convergence of the proposed approach is also proven. Two numerical examples, one discrete mass-damper-spring system and one floating raft isolation system, are included to demonstrate the convergence of the proposed method. The results show that the convergence of the proposed method is much faster than the previous approaches, especially when the dynamic characteristics of the reduced model are very close to the full model.

Efficient Method for Dynamic Condensation of Nonclassically Damped Vibration Systems

Dynamic condensation methods have been widely used to reduce the number of degrees of freedom of finite element models. Most of them, however, are valid for undamped systems. An efficient iterative approach for the dynamic condensation of nonclassically damped systems is proposed. The classical subspace iteration method for undamped models is extended to nonclassically damped models. Then, a governing equation for the dynamic condensation matrix in state space is derived from the extended subspace iteration. Two iterative schemes are proposed to solve the governing equation. Because the dynamic condensation matrix is independent of the system matrices and the eigenpairs (eigenvalues and eigenvectors) of the reduced model, it is unnecessary to compute them in every iteration. This makes the present method much more computationally efficient than those approaches proposed in the past. The convergence of the proposed approach is also proven. Two numerical examples, one discrete mass-damper-spring system and one floating raft isolation system, are included to demonstrate the convergence of the present method. The results show that the convergence of the present method is much faster than the previous approaches, especially when the dynamic characteristics of the reduced model are very close to the full model.

Efficient method for dynamic condensation of nonclassically damped vibration systems

Efficient method for dynamic condensation of nonclassically damped vibration systems

Dynamic condensation method for viscously damped vibration systems in engineering

An iterative dynamic condensation method for the model reduction of viscously damped vibration systems is presented in this paper. The dynamic condensation matrix, which relates the displacements on kept and deleted degrees of freedom, is defined in state space. A governing equation for the dynamic condensation matrix is derived from the dynamic equations of the full model in state space. Since the dynamic condensation matrix is independent of the eigenpairs of the reduced model, it is unnecessary to calculate them in every iteration. This makes the iterative scheme very efficient especially when the number of kept degrees of freedom is large. It is proven that the dynamic condensation matrix is a natural characteristic of the system and is not affected by the external forces. Two engineering systems, a tall building with tuned mass damper and a floating raft isolation system, are also included in this paper. The effect of the selection of kept degrees of freedom on the accuracy of the reduced model is examined by using the first example. The results show that the proposed iterative method is efficient for the model reduction of viscously damped systems.

An accurate modal truncation method for eigenvector derivatives

This paper deals with obtaining accurate eigenvector derivatives in damped vibration systems with modal truncation method only by system matrices and a few available lower modes. A numerical example shows that the method of this paper is correct and efficient.

A new matrix perturbation method for analytical solution of the complex modal eigenvalue problem of viscously damped linear vibration systems

A new matrix perturbation analysis method is presented for efficient approximate solution of the complex modal quadratic generalized eigenvalue problem of viscously damped linear vibration systems. First, the damping matrix is decomposed into the sum of a proportional- and a nonproportional-damping parts, and the solutions of the real modal eigenproblem with the proportional dampings are determined, which are a set of initial approximate solutions of the complex modal eigenproblem. Second, by taking the nonproportional-damping part as a small modification to the proportional one and using the matrix perturbation analysis method, a set of approximate solutions of the complex modal eigenvalue problem can be obtained analytically. The result is quite simple. The new method is applicable to the systems with viscous dampings which do not deviate far away from the proportional- damping case. It is particularly important that the solution technique be also effective to the systems with heavy, but not over, dampings. The solution formulas of complex modal eigenvlaues and eigenvectors are derived up to second- order perturbation terms. The effectiveness of the perturbation algorithm is illustrated by an exemplar numerical problem with heavy dampings. In addition, the practicability of approximately estimating the complex modal eigenvalues, under the proportional-damping hypothesis, of damped vibration systems is discussed by several numerical examples.

A Complex Modal Analysis Method for Damped Vibration Systems (The Representation in the Second Order Differential Form of a Modal Equation and its Use for Practical Application)

Second order uncoupled differential equations for the general damped vibration systems are derived theoretically. The equations are written in a form similar to the classical real modal equations by using the natural frequency, the modal damping ratio, and the newly defined complex modal mass. Introducing supplementary variables, the response analysis is carried out in a similar manner to the real modal analysis. By comparing these equations to the classical ones, physical meanings of the derived equations are clarified. For the vibration problems near the resonant point, approximate complex modal equations are derived which have almost the same form as the classical one. Some applications of the proposed method to vibration problems are discussed.

非線型減衰振動系における過渡現象(第3報)

非線型減衰振動系における過渡現象(第3報)

非線型減衰振動系における過渡現象(第1.2報)

Abstract The object of the non-linear damped vibration systems treated in this paper is chiefly the Oleo-damper systems. The non-linear problems are analyzed by applying Laplace Transformation. It is to be desired that reasonable damping is essentially the acceleration proportional type (or the inertia control type) in the shock absorbing point.