On normal mode vibrations

1964 ◽  
Vol 60 (3) ◽  
pp. 595-611 ◽  
Author(s):  
R. M. Rosenberg
Keyword(s):  
Finite Number ◽  
Linear Systems ◽  
Normal Mode ◽  
Degrees Of Freedom ◽  
Normal Modes ◽  
Free Vibrations ◽  
Non Linear ◽  
Non Linear Systems

1. Introduction. In linear systems, the concept of ‘free vibrations in normal modes’ is well defined and fully understood. The meaning of this phrase is far less clear when it is applied to non-linear systems. It is the purpose here to define and examine the free vibrations in normal modes (and their stability) in certain non-linear systems composed of masses and springs and having a finite number of degrees of freedom. Of necessity, such a paper is in some degree conceptual in nature.

Effects of Suspension Design on the Attitudes of a Car during Braking and Acceleration

1973 ◽  
Vol 187 (1) ◽  
pp. 787-794
Author(s):  
J. R. Ellis
Keyword(s):  
Steady State ◽  
Linear Systems ◽  
Degrees Of Freedom ◽  
Coupled Systems ◽  
Two Degrees Of Freedom ◽  
Simulation Techniques ◽  
Non Linear ◽  
Non Linear Systems ◽  
Effort Distribution

Two degrees of freedom models of a car are employed to demonstrate the effects of the suspension derivative ∂ x/∂ z on the pitch and bounce attitudes during braking or accelerating. The work equation is employed to show that brake effort distribution between the axles has a significant effect on the attitudes when anti-dive suspension characteristics are utilized. The steady-state positions in both pitch and bounce are developed for linear systems of typical suspensions that may be either standard or coupled systems. Non-linear systems are considered using simulation techniques. A description of some simulation circuits is contained in an appendix.

scholarly journals Non-Linear Normal Modes, Invariance, and Modal Dynamics Approximations of Non-Linear Systems

1993 ◽  
Author(s):  
Nicolas Boivin ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Linear Systems ◽  
Invariant Manifolds ◽  
Computational Cost ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
High Order Accuracy ◽  
Continuous Systems ◽  
Continuous Approach ◽  
Non Linear ◽  
Non Linear Systems

Abstract Non-linear systems are here tackled in a manner directly inherited from linear ones, i.e., by denning proper normal modes of motion. These are defined in terms of invariant manifolds in the system’s phase space, on which the uncoupled system dynamics can be studied. Two different methodologies which were previously developed to derive the non-linear normal modes of continuous systems — one based on a purely continuous approach, and one based on a discretized approach where the theory developed for discrete systems can be applied — are simultaneously applied to the same study case — an Euler-Bernoulli beam constrained by a non-linear spring —, and compared as regards accuracy and reliability, resulting in the abandonment of the continuous approach for lack of reliability. Numerical simulations of purely non-linear modal motions are performed using these approaches, and compared to simulations of equations obtained by a classical projection onto the linear modes. The invariance properties of the nonlinear normal modes are demonstrated, and it is also found that, for a purely non-linear modal motion, the invariant manifold approach achieves the same accuracy as that obtained using several linear normal modes, but with significantly reduced computational cost. This is mainly due to the possibility of obtaining high-order accuracy in the dynamics by solving only one non-linear ordinary differentia] equation.

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