Further Results on Parametric Excitation of a Dynamic System

1965 ◽  
Vol 32 (2) ◽  
pp. 373-377 ◽  
Author(s):  
C. S. Hsu
Keyword(s):  
Dynamic System ◽  
Parametric Excitation ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Excitation Frequency ◽  
Constant Matrix ◽  
System Parameters ◽  
Multiple Degrees Of Freedom ◽  
The Stability ◽  
The Given

A dynamic system having multiple degrees of freedom and being under parametric excitation has been studied in an earlier paper [2]. However, the analysis given there necessitates certain restrictions on the distribution of the natural frequencies of the system. In this paper those restrictions are removed. The analysis presented here shows how to obtain a constant matrix whose eigenvalues determine the stability or instability of a system of ordinary differential equations with periodic coefficients at a given excitation frequency. The constant matrix is expressed entirely in terms of the given system parameters and the excitation frequency.

On the Parametric Excitation of a Dynamic System Having Multiple Degrees of Freedom

1963 ◽  
Vol 30 (3) ◽  
pp. 367-372 ◽  
Author(s):  
C. S. Hsu
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Dynamic System ◽  
Ordinary Differential Equations ◽  
Parametric Excitation ◽  
Degrees Of Freedom ◽  
Periodic Coefficients ◽  
Multiple Degrees Of Freedom ◽  
First Approximation ◽  
Approximation Analysis

A dynamical system having multiple degrees of freedom and under parametric excitation is governed by a system of ordinary differential equations with periodic coefficients. In this paper a first-approximation analysis is carried out and criteria for instability are derived explicitly.

The Natural Frequencies of Free and Constrained Non-Uniform Beams

1960 ◽  
Vol 64 (599) ◽  
pp. 697-699 ◽  
Author(s):  
R. P. N. Jones ◽  
S. Mahalingam
Keyword(s):  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Normal Modes ◽  
Iteration Process ◽  
Conservative System ◽  
Basic System ◽  
Orthogonal Functions ◽  
Added Masses ◽  
The Given

The Rayleigh-Ritz method is well known as an approximate method of determining the natural frequencies of a conservative system, using a constrained deflection form. On the other hand, if a general deflection form (i.e. an unconstrained form) is used, the method provides a theoretically exact solution. An unconstrained form may be obtained by expressing the deflection as an expansion in terms of a suitable set of orthogonal functions, and in selecting such a set, it is convenient to use the known normal modes of a suitably chosen “ basic system.” The given system, whose vibration properties are to be determined, can then be regarded as a “ modified system,” which is derived from the basic system by a variation of mass and elasticity. A similar procedure has been applied to systems with a finite number of degrees of freedom. In the present note the method is applied to simple non-uniform beams, and to beams with added masses and constraints. A concise general solution is obtained, and an iteration process of obtaining a numerical solution is described.

scholarly journals On Cauchy’s Interlacing Theorem and the Stability of a Class of Linear Discrete Aggregation Models Under Eventual Linear Output Feedback Controls

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 712 ◽  
Author(s):  
Manuel De la Sen
Keyword(s):  
Dynamic System ◽  
Output Feedback ◽  
Iteration Step ◽  
Stability And Convergence ◽  
Feedback Controls ◽  
Time Aggregation ◽  
The Stability ◽  
The Given ◽  
Stability Results

This paper links the celebrated Cauchy’s interlacing theorem of eigenvalues for partitioned updated sequences of Hermitian matrices with stability and convergence problems and results of related sequences of matrices. The results are also applied to sequences of factorizations of semidefinite matrices with their complex conjugates ones to obtain sufficiency-type stability results for the factors in those factorizations. Some extensions are given for parallel characterizations of convergent sequences of matrices. In both cases, the updated information has a Hermitian structure, in particular, a symmetric structure occurs if the involved vector and matrices are complex. These results rely on the relation of stable matrices and convergent matrices (those ones being intuitively stable in a discrete context). An epidemic model involving a clustering structure is discussed in light of the given results. Finally, an application is given for a discrete-time aggregation dynamic system where an aggregated subsystem is incorporated into the whole system at each iteration step. The whole aggregation system and the sequence of aggregated subsystems are assumed to be controlled via linear-output feedback. The characterization of the aggregation dynamic system linked to the updating dynamics through the iteration procedure implies that such a system is, generally, time-varying.

scholarly journals On the Role of Nonsynchronous Rotating Damping in Rotordynamics

2000 ◽  
Vol 6 (6) ◽  
pp. 467-475 ◽  
Author(s):  
Giancarlo Genta ◽  
Eugenio Brusa
Keyword(s):  
General Model ◽  
Degrees Of Freedom ◽  
Dynamic Behaviour ◽  
System Parameters ◽  
Spin Speed ◽  
Rotating Systems ◽  
Jeffcott Rotor ◽  
The Stability ◽  
Rotor Model

Nonsynchronous rotating damping, i.e. energy dissipations occurring in elements rotating at a speed different from the spin speed of a rotor, can have substantial effects on the dynamic behaviour and above all on the stability of rotating systems.The free whirling and unbalance response for systems with nonsynchronous damping are studied using Jeffcott rotor model. The system parameters affecting stability are identified and the threshold of instability is computed. A general model for a multi-degrees of freedom model for a general isotropic machine is then presented. The possibility of synthesizing nonsynchronous rotating and nonrotating damping using rotor- and stator-fixed active dampers is then discussed for the general case of rotors with many degrees of freedom.

On the Stability and Nonlinear Dynamics of a Horizontally Base-Excited Shaker/Mould Structure With Unsymmetric End Stiffnesses

1991 ◽  
Author(s):  
John K. H. Cheng ◽  
K. W. Wang
Keyword(s):  
Rotational Motion ◽  
Excitation Frequency ◽  
Mechanical Systems ◽  
Excitation Amplitude ◽  
Governing Equations ◽  
System Parameters ◽  
The Past ◽  
Transverse Vibrations ◽  
Time Varying Parameter ◽  
The Stability

Abstract This paper presents a dynamic analysis of a horizontally base -excited shaker/ mould structure with unsymmetric gripper stiffnesses. The study explains the large rotational and transverse vibrations of the mould at specific operating frequencies observed in the experiments. The governing equations consist of a time-dependent coefficient which indicates the existence of parametric excitation effects. It is concluded that differences between the gripper stiffnesses are responsible for this phenomenon and could destabilize the system. The value of the time-varying parameter is related to the horizontal vibration amplitude of the mould and hence is a function of the system parameters and excitation frequency. The mould’s rotational motion is directly parametrically excited while its transverse vibration is excited indirectly through coupling with the rotational motion. A thorough analysis of this class of mechanical systems has not been performed in the past. In this research, studies are conducted to identify the contributions of various system parameters, such as gripper stiffness, damping, mould inertia, and excitation amplitude to the system dynamic characteristics. The results provide new insight and guidelines toward optimizing such mechanical systems.

Condition Monitoring of Rotor Using Active Magnetic Actuator

2008 ◽  
Author(s):  
Jerzy T. Sawicki ◽  
Michael I. Friswell ◽  
Alex H. Pesch ◽  
Adam Wroblewski
Keyword(s):  
Parametric Excitation ◽  
Excitation Frequency ◽  
Harmonic Excitation ◽  
Normal Operation ◽  
Robust Techniques ◽  
Induced Changes ◽  
Vibration Signatures ◽  
The Given ◽  
Rotor Dynamic ◽  
Cracked Rotor System

It has been widely recognized that the changes in the dynamic response of a rotor could be utilized for general fault detection and monitoring. Current methods rely on the monitoring of synchronous response of the machine during its transient or normal operation. Very little progress has been made in developing robust techniques to detect subtle changes in machine condition caused by rotor cracks. It has been demonstrated that the crack-induced changes in the rotor dynamic behavior produce unique vibration signatures. When the harmonic excitation force is applied to the cracked rotor system, nonlinear resonances occur due to the nonlinear parametric excitation characteristics of the crack. These resonances are the result of the coexistence of a parametric excitation term and different frequencies present in the system, namely critical speed, the synchronous frequency, and excitation frequency from the externally applied perturbation signals. This paper presents the application of this approach on an experimental test rig. The simulation and experimental study for the given rig configuration, along with the application of active magnetic bearings as a force actuator, are presented.

scholarly journals Analysis of the nodal beam isolation system based on application of helical spring

2021 ◽  
Author(s):  
Krzysztof Michalczyk ◽  
Wojciech Sikora
Keyword(s):  
Excitation Frequency ◽  
Elastic Beam ◽  
Experimental Tests ◽  
Operating Conditions ◽  
Helical Spring ◽  
Static Stiffness ◽  
Isolation System ◽  
System Parameters ◽  
Nodal Points ◽  
The Given

AbstractA nodal beam isolation system allows the transmission of vibration from the source to an isolated element to be limited using nodal points on the elastic beam connecting them. These points are selected in such a way that their position during vibration is constant. The application of a helical spring as an elastic beam reduces the dimensions of the system and increases its applications. An effective computational model of the nodal beam isolation system based on a helical spring application as an elastic beam is presented in the paper. The model allows the position of nodal points to be determined for a given excitation frequency. It also allows the influence of system parameters on spring vibration amplitudes and static stiffness of the connection between the source and isolated element to be analysed. The analysis makes it possible to formulate conclusions facilitating the designer to select the proper system parameters for the given operating conditions. The results of numerical and experimental tests exhibit high compliance with the results of the presented model.

scholarly journals Parametrically Excited Nonlinear Two-Degree-of-Freedom Systems With Repeated Natural Frequencies

1993 ◽  
Author(s):  
A. H. Nayfeh ◽  
C. Chin ◽  
D. T. Mook
Keyword(s):  
Parametric Excitation ◽  
Nonlinear Response ◽  
Normal Forms ◽  
Natural Frequencies ◽  
Linear Part ◽  
Degree Of Freedom ◽  
Cubic Nonlinearity ◽  
Simply Supported ◽  
The Stability ◽  
Two Degree Of Freedom

Abstract The method of normal forms is used to study the nonlinear response of two-degree-of-freedom systems with repeated natural frequencies and cubic nonlinearity to a principal parametric excitation. The linear part of the system has a nonsemisimple one-to-one resonance. The character of the stability and various types of bifurcation are analyzed. The results are applied to the flutter of a simply-supported panel in a supersonic airstream.

scholarly journals ANALYTICAL MODELING OF PLANETARY GEAR AND SENSITIVITY OF NATURAL FREQUENCIES

2013 ◽  
pp. 35-40
Author(s):  
MAJID MEHRABI ◽  
DR. V.P. SINGH
Keyword(s):  
Natural Frequency ◽  
Degrees Of Freedom ◽  
Analytical Modeling ◽  
Natural Frequencies ◽  
Vibration Mode ◽  
Planetary Gear ◽  
Vibration Modes ◽  
Planetary Gears ◽  
System Parameters ◽  
Inertia Parameters

This work develops an analytical model of planetary gears and uses it to investigate their natural frequencies and vibration modes. The model admits three planar degrees of freedom for each of the sun, ring, carrier and planets. Vibration modes are classified into rotational, translational and planet modes. The natural frequency sensitivities to system parameters are investigated for tuned (cyclically symmetric) planetary gears. Parameters under consideration include support and mesh stiffnesses, component masses, and moments of inertia. Using the well-defined vibration mode properties of tuned planetary gears, the eigen sensitivities are calculated and expressed in simple exact formulae. These formulae connect natural frequency sensitivity with the modal strain or kinetic energy and provide efficient means to determine the sensitivity to all stiffness and inertia parameters by inspection of the modal energy distribution.

Study on Dynamic Responses of Human-Structure Interaction System

2013 ◽  
Vol 438-439 ◽  
pp. 775-778
Author(s):  
Yi He Wang ◽  
Na Yang
Keyword(s):  
Dynamic System ◽  
Fundamental Frequency ◽  
Dynamic Characteristics ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Coupled System ◽  
Dynamic Responses ◽  
Major Topic ◽  
Integer Multiple ◽  
Structure Interaction

t has previously been shown that human-structure dynamic system received much attention as a major topic in the serviceability performance and safety problems. In this study, the structure occupied by human are considered as a two degrees-of-freedom system. The dynamic characteristics of human-structure system are investigated by deriving the eigenvalue equation of the system. The response of structure to a person walking across it at various rates of walking is also researched. The results show that the pair natural frequencies of coupled system have a contrary trend when the crowd densities increase. It also demonstrates that the resonant situation occurs when structural fundamental frequency is equal to or an integer multiple of the pacing frequency.

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