The Concept of Complex Damping

1952 ◽  
Vol 19 (3) ◽  
pp. 284-286
Author(s):  
N. O. Myklestad
Keyword(s):  
Steady State ◽  
Complex Number ◽  
Forced Vibrations ◽  
Damping Factor ◽  
Viscous Damping ◽  
Mass System ◽  
Free And Forced Vibrations ◽  
Phase Angles ◽  
State Case ◽  
Complex Damping

Abstract In this paper it is shown that if the hysteresis loop for a material has a particular shape the damping can be considered adequately by multiplying the modulus of elasticity of the material by the complex number e2bi where 2b is called the complex damping factor. For small values of b it is shown that both for free and forced vibrations of a simple spring-mass system the motion in the case of complex damping is the same as in the case of viscous damping, with b = c/ccr, except that in the steady-state case the phase angles are slightly different. Also, it is shown how complex damping may be applied to cases of forced vibrations of uniform rods and beams. The greatest advantage of using complex damping, however, is in numerical calculations of forced vibrations of engine crankshafts, airplane wings, and other types of structures; and for such calculations it already has been extensively used.

CONTROL OF STEADY-STATE VIBRATIONS OF RECTANGULAR PLATES

2011 ◽  
Vol 11 (03) ◽  
pp. 535-562 ◽  
Author(s):  
K. A. ALSAIF ◽  
M. A. FODA
Keyword(s):  
Steady State ◽  
Function Method ◽  
Forced Vibrations ◽  
Rectangular Plates ◽  
Selected Lines ◽  
Numerical Examples ◽  
Nodal Lines ◽  
Free And Forced Vibrations ◽  
Concentrated Masses ◽  
The Given

The focus of the present research is to eliminate the undesired steady-state vibrations at selected lines or locations in a vibrating plate by means of adding attachments at arbitrary selected locations. These attachments can be either added concentrated masses and/or translational or rotational springs which are connected to the plate at one end and grounded at the other. The case of attachment of translational and/or rotational oscillators systems is examined. In addition, imposing lines of zero displacements (nodal lines) at selected locations are also investigated. The dynamic Green's function method is employed. Several numerical examples are cited to verify the utility of the proposed method. In addition, sample experiments to measure the plate free and forced vibrations for the given boundary conditions are conducted and the experimental measurements are compared with the analytical results.

Closed-Form Exact Solutions for Viscously Damped Free and Forced Vibrations of Longitudinal and Torsional Bars

2017 ◽  
Vol 17 (08) ◽  
pp. 1750093 ◽  
Author(s):  
Jae-Hoon Kang
Keyword(s):  
Free Vibration ◽  
Closed Form ◽  
Closed Form Solution ◽  
Forced Vibrations ◽  
Form Solution ◽  
Forced Response ◽  
Mode Shapes ◽  
Viscous Damping ◽  
Vibration Displacement ◽  
Free And Forced Vibrations

This paper studies the viscously damped free and forced vibrations of longitudinal and torsional bars. The method is exact and yields closed form solution for the vibration displacement in contrast with the well-known eigenfunction superposition (ES) method, which requires expression of the distributed forcing functions and displacement response functions as infinite series sums of free vibration eigenfunctions. The viscously damped natural frequency equation and the critical viscous damping equation are exactly derived for the bars. Then the viscously damped free vibration frequencies and corresponding damped mode shapes are calculated and plotted, aside from the undamped free vibration and corresponding mode shapes typically computed and used in vibration problems. The longitudinal or torsional amplitude versus forcing frequency curves showing the forced response to distributed loadings are plotted for various viscous damping parameters. It is found that the viscous damping affects the natural frequencies and the corresponding mode shapes of longitudinal and torsional bars, especially for the fundamental frequency.

The Vibration of Unsymmetrical Rotating Shafts

1965 ◽  
Vol 32 (1) ◽  
pp. 157-162 ◽  
Author(s):  
S. T. Ariaratnam
Keyword(s):  
Forced Vibrations ◽  
Viscous Damping ◽  
Free And Forced Vibrations ◽  
Rotating Shafts ◽  
Principal Directions

This paper deals with the vibration of heavy unsymmetrical shafts possessing unequal flexural rigidities in the principal directions and running in symmetrical bearings. The existence of several bands of whirling speeds is shown and the effects of stationary and rotary viscous damping on the free and forced vibrations of the shaft are discussed.

On Steady-State Harmonic Vibrations of Nonlinear Systems With Many Degrees of Freedom

1966 ◽  
Vol 33 (2) ◽  
pp. 406-412 ◽  
Author(s):  
W. M. Kinney ◽  
R. M. Rosenberg
Keyword(s):  
Steady State ◽  
Nonlinear Systems ◽  
Degrees Of Freedom ◽  
Forced Vibrations ◽  
Geometrical Method ◽  
Mass System ◽  
Harmonic Vibrations ◽  
Forcing Functions ◽  
Exciting Forces ◽  
The Stability

A nonlinear spring-mass system with many degrees of freedom, and subjected to periodic exciting forces, is examined. The class of admissible systems and forcing functions is defined, and a geometrical method is described for deducing the steady-state forced vibrations having a period equal to that of the forcing functions. The methods used combine the geometrical methods developed earlier in the problem of normal mode vibrations and Rauscher’s method. The stability of these steady-state forced vibrations is examined by Hsu’s method. The results are applied to an example of a system having two degrees of freedom.

Insensitivity of steady-state distributions of generalized semi-Markov processes by speeds

1978 ◽  
Vol 10 (04) ◽  
pp. 836-851 ◽  
Author(s):  
R. Schassberger
Keyword(s):  
Steady State ◽  
General System ◽  
State Distribution ◽  
Lifetime Distributions ◽  
State Dependent ◽  
Semi Markov Process ◽  
Semi Markov Processes ◽  
Infinite State ◽  
State Case ◽  
Residual Lifetimes

A generalized semi-Markov process with speeds describes the fluctuation, in time, of the state of a certain general system involving, at any given time, one or more living components, whose residual lifetimes are being reduced at state-dependent speeds. Conditions are given for the stationary state distribution, when it exists, to depend only on the means of some of the lifetime distributions, not their exact shapes. This generalizes results of König and Jansen, particularly to the infinite-state case.

A Resonance Chart for Damped Vibration

1955 ◽  
Vol 59 (540) ◽  
pp. 850-852 ◽  
Author(s):  
R. E. D. Bishop
Keyword(s):  
Linear System ◽  
Convenient Method ◽  
Harmonic Excitation ◽  
Degree Of Freedom ◽  
Viscous Damping ◽  
Similar Form ◽  
Hysteretic Damping ◽  
Resonance Curves ◽  
Phase Angles

A convenient method is pointed out for calculating the response of a damped linear system with one degree of freedom to harmonic excitation. Results of such calculations are usually represented by the familiar “ resonance curves ”—one curve being plotted for each intensity of damping. These curves are not particularly convenient to use and Yates has overcome several of their defects by throwing them into a nomographic form. Yates' nomogram is based upon the concept of viscous damping and it does not give the information of a conventional set of resonance curves in that it relates to the velocity of vibration. By changing over to hysteretic damping, a nomogram of somewhat similar form may be constructed such that it gives amplitudes and phase angles of displacements while retaining the advantages, over resonance curves, of this form of representation.

Steady-State Dynamic Response of a Limited Slip System

1968 ◽  
Vol 35 (2) ◽  
pp. 322-326 ◽  
Author(s):  
W. D. Iwan
Keyword(s):  
Steady State ◽  
Dynamic Response ◽  
Slip System ◽  
External Load ◽  
Initial Conditions ◽  
Viscous Damping ◽  
Steady State Response ◽  
Response Curves ◽  
State Response ◽  
System Nonlinearity

The steady-state response of a system constrained by a limited slip joint and excited by a trigonometrically varying external load is discussed. It is shown that the system may possess such features as disconnected response curves and jumps in response depending on the strength of the system nonlinearity, the level of excitation, the amount of viscous damping, and the initial conditions of the system.

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