On Classical Normal Modes of a Damped Linear System

1961 ◽  
Vol 28 (3) ◽  
pp. 458-458 ◽  
Author(s):  
Morris Morduchow
Keyword(s):  
Linear System ◽  
Normal Modes

scholarly journals The Normal Modes of Nonlinear n-Degree-of-Freedom Systems

1962 ◽  
Vol 29 (1) ◽  
pp. 7-14 ◽  
Author(s):  
R. M. Rosenberg
Keyword(s):  
Linear System ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Minimum Problem ◽  
Infinite Class ◽  
Coupled Equations ◽  
Geometrical Problem ◽  
Maximum Minimum

A system of n masses, equal or not, interconnected by nonlinear “symmetric” springs, and having n degrees of freedom is examined. The concept of normal modes is rigorously defined and the problem of finding them is reduced to a geometrical maximum-minimum problem in an n-space of known metric. The solution of the geometrical problem reduces the coupled equations of motion to n uncoupled equations whose natural frequencies can always be found by a single quadrature. An infinite class of systems, of which the linear system is a member, has been isolated for which the frequency amplitude can be found in closed form.

Generalized Cumulants Representing a Transient Random Process in a Linear System

1977 ◽  
Vol 99 (2) ◽  
pp. 103-108 ◽  
Author(s):  
Arthur Mayer
Keyword(s):  
Linear System ◽  
Generating Function ◽  
State Vector ◽  
Tracking System ◽  
Normal Modes ◽  
Moment Generating Function ◽  
Initial Distribution ◽  
The State ◽  
Symmetric Tensors ◽  
The Moment

The state vector of a linear system responding to gaussian noise satisfies a Langevin equation, and the moment-generating function of the probability distribution of the state vector satisfies a partial differential equation. The logarithm of the moment-generating function is expanded in a power series, whose coefficients are organized into a sequence of symmetric tensors. These are the generalized cumulants of the time-dependent distribution of the state vector. They separately satisfy an infinite sequence of uncoupled ordinary differential tensor equations. The normal modes of each of the generalized cumulants are given by an easy formula. This specifies transient response and proves that all cumulants of a stable system are stable. Also, all cumulants of an unstable system are unstable. As an example, a particular non-gaussian initial distribution is assumed for the state vector of a second-order tracking system, and the transient fourth cumulant is calculated.

Vibration Response of Linear Damped Complex Systems

1963 ◽  
Vol 30 (1) ◽  
pp. 70-74 ◽  
Author(s):  
Robert Plunkett
Keyword(s):  
Linear System ◽  
Complex Systems ◽  
Normal Modes ◽  
Continuous System ◽  
Vibration Response ◽  
Degree Of Freedom ◽  
The Other ◽  
Maximum Response ◽  
Approximate Expressions ◽  
Two Degree Of Freedom

Hahnkamm has found the changes in the amplitudes of each of the two maxima of the unit vibration response of a two-degree-of-freedom linear system as the strength of the single linear dashpot is changed. This paper develops two approximate expressions for the change in all of the response maxima of a multidegree or continuous system as the dashpot constant of the single linear damper is changed. One of these approximations is derived from a perturbation solution around the minimax values, and the other is derived from an expansion in normal modes. These expressions are useful in determining the sensitivity of the maximum response value to small changes in the damping constant.

Normal Modes of Nonlinear Dual-Mode Systems

1960 ◽  
Vol 27 (2) ◽  
pp. 263-268 ◽  
Author(s):  
R. M. Rosenberg
Keyword(s):  
Linear System ◽  
Closed Form ◽  
Normal Modes ◽  
Coupled System ◽  
Dual Mode ◽  
Great Simplicity

A system consisting of two unequal masses, interconnected by a coupling spring, and each connected to an anchor spring, is examined. The springs may all be unequal and nonlinear, but each resists being compressed to the same degree as being stretched. The concept of normal modes is rigorously defined, and methods of finding them are given. A knowledge of these modes reduces the coupled system to two uncoupled ones which can always be integrated in quadrature. There exists an infinity of systems, of which the linear is one, which can be integrated in closed form. This approach yields, even for the linear system, new results of great simplicity.

Nonlinear Modal Analysis of a Cracked Beam

1997 ◽  
Author(s):  
M. Chati ◽  
R. H. Rand ◽  
S. Mukherjee
Keyword(s):  
Linear System ◽  
Modal Analysis ◽  
Natural Frequencies ◽  
Piecewise Linear ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Degree Of Freedom ◽  
Mode Shapes ◽  
Cracked Beam ◽  
Piecewise Linear System

Abstract This paper addresses the problem of vibrations of a cracked beam. In general, the motion of such a beam can be very complex. This phenomenon can be attributed to the presence of the nonlinearity due to the opening and closing of cracks. The focus of this paper is the modal analysis of a cantilever beam with a transverse edge crack. The nonlinearity mentioned above has been modelled as a piecewise-linear system. In an attempt to define effective natural frequencies for this piecewise-linear system, the idea of a “bilinear frequency” is utilized. The bilinear frequency is obtained by computing the associated frequencies of each of the linear pieces of the piecewise-linear system. The finite element method is used to obtain the natural frequencies in each linear region. In order to better understand the essential nonlinear dynamics of the cracked beam, a piecewise-linear two degree of freedom model is studied. Perturbation methods are used to obtain the nonlinear normal modes of vibration and the associated period of the motion. Results of this piecewise-linear model problem are shown to justify the definition of the bilinear frequency as the effective natural frequency. It is therefore expected that calculating piecewise mode shapes and bilinear frequencies is useful for understanding the dynamics of the infinite degree of freedom cracked beam.

Normal modes of a defected linear system of beaded springs

2017 ◽  
Vol 85 (3) ◽  
pp. 193-201
Author(s):  
Amir Aghamohammadi ◽  
M. Ebrahim Foulaadvand ◽  
Mohammad Hassan Yaghoubi ◽  
Amir Hossein Mousavi
Keyword(s):  
Linear System ◽  
Normal Modes

Chaos in a manifold piecewise linear system

1981 ◽  
Vol 64 (10) ◽  
pp. 9-17 ◽  
Author(s):  
Toshimichi Saito ◽  
Hiroichi Fujita
Keyword(s):  
Linear System ◽  
Piecewise Linear ◽  
Piecewise Linear System
Sign in / Sign up

Export Citation Format

Share Document