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.

Experimental Periodic Localized Motions in Coupled Beams With Active Nonlinearities

1995 ◽  
Author(s):  
Melvin E. King ◽  
Johannes Aubrecht ◽  
Alexander F. Vakakis
Keyword(s):  
Steady State ◽  
Vibrational Energy ◽  
Normal Modes ◽  
Coupled System ◽  
Nonlinear Normal Modes ◽  
Nonlinear Frequency ◽  
Nonlinear Phenomenon ◽  
Response Curves ◽  
Nonlinear Motion ◽  
Spatially Extended

Abstract Steady-state nonlinear motion confinement is experimentally studied in a system of weakly coupled cantilever beams with active stiffness nonlinearities. Quasi-static swept-sine tests are performed by periodically forcing one of the beams at frequencies close to the first two closely-spaced modes of the coupled system, and experimental nonlinear frequency response curves for certain nonlinearity levels are generated. Of particular interest is the detection of strongly localized steady-state motions, wherein vibrational energy becomes spatially confined mainly to the directly excited beam. Such motions exist in neighborhoods of strongly localized anti-phase nonlinear normal modes (NNMs) which bifurcate from a spatially extended NNMs of the system. Steady-state nonlinear motion confinement is an essentially nonlinear phenomenon with no counterpart in linear theory, and can be implemented in vibration and shock isolation designs of mechanical systems.

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.

Successive Synthesis of Substructure Modes

1991 ◽  
Vol 58 (3) ◽  
pp. 759-765 ◽  
Author(s):  
Luis E. Suarez ◽  
Mahendra P. Singh
Keyword(s):  
Closed Form ◽  
Characteristic Equation ◽  
Degrees Of Freedom ◽  
Coupled System ◽  
Closed Form Expression ◽  
Element Discretization ◽  
Root Finding ◽  
Complete Set ◽  
Conventional Solution ◽  
Second Eigenvalue

A mode synthesis approach is presented to calculate the eigenproperties of a structure from the eigenproperties of its substructures. The approach consists of synthesizing the substructures sequentially, one degree-of-freedom at a time. At each coupling stage, the eigenvalue is obtained as the solution of a characteristic equation, defined in closed form in terms of the eigenproperties obtained in the preceding coupling stage. The roots of the characteristic equation can be obtained by a simple Newton-Raphson root finding scheme. For each calculated eigenvalue, the eigenvector is defined by a simple closed-form expression. The eigenproperties obtained in the final coupling stage provide the desired eigenproperties of the coupled system. Thus, the approach avoids a conventional solution of the second eigenvalue problem. The approach can be implemented with the complete set or a truncated number of substructure modes; if the complete set of modes is used, the calculated eigenproperties would be exact. The approach can be used with any finite element discretization of structures. It requires only the free interface eigenproperties of the substructures. Successful application of the approach to a moderate size problem (255 degrees-of-freedom) on a microcomputer is also demonstrated.

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.

Closed-Form Estimation of Normal Modes From a Partially Sampled Water Column

2020 ◽  
Vol 45 (4) ◽  
pp. 1574-1582
Author(s):  
Houcem Gazzah ◽  
Sergio M. Jesus
Keyword(s):  
Closed Form ◽  
Water Column ◽  
Normal Modes
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