Forced Harmonic Response of a Continuous System Displaying Eigenvalue Veering Phenomena

1993 ◽  
Author(s):  
Jerry H. Ginsberg ◽  
Hoang Pham
Keyword(s):  
Natural Frequencies ◽  
Continuous System ◽  
Harmonic Excitation ◽  
Forced Response ◽  
Steady State Response ◽  
Concentrated Force ◽  
Peak Displacement ◽  
Harmonic Response ◽  
State Response ◽  
Exact Eigenvalue

Abstract Prior studies of self-adjoint linear vibratory systems have extensively explored the phenomenon of veering of eigenvalue loci that depict the dependence of natural frequencies on a system parameter. The present work is an exploration of the effect of such phenomena on the response of a continuum to harmonic excitation. The focus of the analysis is the prototypical system of a two-span beam with a strong torsional spring at the intermediate pin support. The results of an exact eigenvalue analysis, not previously disclosed, are used to perform a modal analysis of the steady-state response of the beam to a harmonic concentrated force applied to the middle of one span. The analysis is used to identify situations in which the forced response is localized to one span, as well as the degree to which the location and magnitude of the peak displacement displays parameter sensitivity.

Forced Harmonic Response of a Continuous System Displaying Eigenvalue Veering Phenomena

1995 ◽  
Vol 117 (4) ◽  
pp. 439-444 ◽  
Author(s):  
J. H. Ginsberg ◽  
Hoang Pham
Keyword(s):  
Natural Frequencies ◽  
Continuous System ◽  
Harmonic Excitation ◽  
Force Response ◽  
Concentrated Force ◽  
Peak Displacement ◽  
Harmonic Response ◽  
State Response ◽  
Exact Eigenvalue ◽  
Hysteretic Loss

Prior studies of self-adjoint linear vibratory systems have extensively explored the free vibration phenomena associated with veering of eigenvalue loci that depict the dependence of natural frequencies on a system parameter. The present work is an exploration of the effect of such phenomena on the response of a nearly-periodic continuum to harmonic excitation. The focus of the analysis is the prototypical system of a two-span beam with a strong torsional spring at the intermediate pin support. The results of an exact eigenvalue analysis, not previously disclosed, are used to perform a modal analysis of the steady-state response to a harmonic concentrated force applied to the middle of one span. The modal equations are used to identify situations in which the force response is localized to one span, as well as the degree to which the location and magnitude of the peak displacement display parameter sensitivity. The effect of hysteretic loss on forced localization is discussed.

Forced Response of Continuous System With Hysteresis Loop Characteristics: System With Quadrilateral Hysteresis Loop Characteristics

2005 ◽  
Author(s):  
Shigeru Aoki ◽  
Takeshi Watanabe
Keyword(s):  
Approximate Solution ◽  
Steady State ◽  
Energy Loss ◽  
Hysteresis Loop ◽  
Nonlinear Response ◽  
Continuous System ◽  
Forced Response ◽  
Steady State Response ◽  
State Response ◽  
Discontinuous Line

This paper deals with steady-state response of a continuous system with collision characteristics. Considering the energy loss in a collision, an analytical method of approximate solution for the continuous system with symmetrical hysteresis loop characteristics is presented. The resonance curves of nonlinear response obtained from approximate solution are shown as discontinuous line, and are discussed the phenomenon.

scholarly journals Vibration Equation of Fractional Order Describing Viscoelasticity and Viscous Inertia

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 850-856 ◽  
Author(s):  
Jun-Sheng Duan ◽  
Yun-Yun Xu
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Harmonic Excitation ◽  
Steady State Response ◽  
Vibration System ◽  
Derivative Operator ◽  
Vibration Equation ◽  
State Response ◽  
Frequency Relation

Abstract The steady state response of a fractional order vibration system subject to harmonic excitation was studied by using the fractional derivative operator ${}_{-\infty} D_t^\beta,$where the order β is a real number satisfying 0 ≤ β ≤ 2. We derived that the fractional derivative contributes to the viscoelasticity if 0 < β < 1, while it contributes to the viscous inertia if 1 < β < 2. Thus the fractional derivative can represent the “spring-pot” element and also the “inerterpot” element proposed in the present article. The viscosity contribution coefficient, elasticity contribution coefficient, inertia contribution coefficient, amplitude-frequency relation, phase-frequency relation, and influence of the order are discussed in detail. The results show that fractional derivatives are applicable for characterizing the viscoelasticity and viscous inertia of materials.

Reduction of Harmonic Response of Cylindrical Shells

1975 ◽  
Vol 97 (4) ◽  
pp. 1371-1377 ◽  
Author(s):  
G. B. Warburton
Keyword(s):  
Cantilever Beams ◽  
Steady State Response ◽  
Optimum Conditions ◽  
Harmonic Force ◽  
Shell Surface ◽  
Single Degree Of Freedom ◽  
Harmonic Response ◽  
State Response ◽  
Single Degree ◽  
Mode Method

The normal mode method is used to investigate the reduction in the steady-state response of a simply supported cylindrical shell when conventional absorbers are attached to the shell. Two types of excitation are considered: (a) a single radial harmonic force, and (b) a harmonic pressure distributed over the shell surface. The effect upon response of varying the absorber parameters is studied. Optimum conditions for specific cases are obtained and compared with those required to minimize response when absorbers are added to cantilever beams and to the classical single degree of freedom system.

Analytical evaluation of track response in the vertical direction due to a moving load

2016 ◽  
Vol 23 (18) ◽  
pp. 2989-3006 ◽  
Author(s):  
Wlodzimierz Czyczula ◽  
Piotr Koziol ◽  
Dariusz Kudla ◽  
Sergiusz Lisowski
Keyword(s):  
Steady State ◽  
Moving Load ◽  
Vertical Direction ◽  
Vertical Displacement ◽  
Steady State Response ◽  
Concentrated Force ◽  
State Response ◽  
Weakly Coupled ◽  
Wide Range ◽  
Rail Deflection

In the literature, typical analytical track response models are composed of beams (which represent the rail) on viscoelastic or elastic foundations. The load is usually considered as a single concentrated force (constant or varying in time) moving with constant speed. Concentrated or distributed loads or multilayer track models have rarely been considered. One can find some interesting results concerning analysis of distributed loads and multilayer track structures that include both analytical and numerical approaches. However, there is a noticeable lack of sufficient comparison between track responses under concentrated or distributed load and between one and multilayer track models. One of the unique features of the present paper is a comparison of data obtained for a series of concentrated and distributed loads, which takes into account a wide range of track parameters and train speeds. One of the fundamental questions associated with the multilayer track model is the level of coupling between the rail and the vibrations of the sleepers. In this paper, it is proved that sleepers are weakly coupled with the rail if the track is without significant imperfections, and the steady-state response is analyzed for this case. In other words, sleeper vibrations do not influence the rail vibrations significantly. Therefore the track is analyzed by means of a two-stage model. The first step of this model determines rail vibration under a moving load, and then the sleeper vibration is calculated from previously obtained kinematic excitation. The model is verified by comparison of the obtained results with experimental data. Techniques based on Fourier series are applied to the solution of the steady-state track response. Another important problem associated with track response under moving loads arises from the analysis of the effect of longitudinal forces in rails on vertical displacement. It is shown that, in the case of the steady-state response, longitudinal forces do not influence rail displacements significantly and this observation remains correct for a wide range of track parameters and train speeds. The paper also analyzes the legitimacy of the statement that additional rail deflection between sleepers, compared to the continuous rail support, can be considered as a track imperfection.

Stability and Stationary Response of a Skew Jeffcott Rotor With Geometric Uncertainty

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Nicolas Driot ◽  
Alain Berlioz ◽  
Claude-Henri Lamarque
Keyword(s):  
Equations Of Motion ◽  
Dynamical Behavior ◽  
Forced Response ◽  
Stochastic Methods ◽  
Steady State Response ◽  
State Response ◽  
Internal Excitation ◽  
Geometric Uncertainty ◽  
Jeffcott Rotor ◽  
Support Motion

The aim of this work is to apply stochastic methods to investigate uncertain parameters of rotating machines with constant speed of rotation subjected to a support motion. As the geometry of the skew disk is not well defined, randomness is introduced and affects the amplitude of the internal excitation in the time-variant equations of motion. This causes uncertainty in dynamical behavior, leading us to investigate its robustness. Stability under uncertainty is first studied by introducing a transformation of coordinates (feasible in this case) to make the problem simpler. Then, at a point far from the unstable area, the random forced steady state response is computed from the original equations of motion. An analytical method provides the probability of instability, whereas Taguchi’s method is used to provide statistical moments of the forced response.

scholarly journals Structural Damage Detection Using Slopes of Longitudinal Vibration Shapes

2015 ◽  
Author(s):  
W. Xu ◽  
W. D. Zhu ◽  
S. A. Smith
Keyword(s):  
Steady State ◽  
Damage Detection ◽  
Structural Damage ◽  
Longitudinal Vibration ◽  
Harmonic Excitation ◽  
Mode Shapes ◽  
Noise Robustness ◽  
Steady State Response ◽  
Structural Damage Detection ◽  
State Response

While structural damage detection based on flexural vibration shapes, such as mode shapes and steady-state response shapes under harmonic excitation, has been well developed, little attention is paid to that based on longitudinal vibration shapes that also contain damage information. This study originally formulates a slope vibration shape for damage detection in bars using longitudinal vibration shapes. To enhance noise robustness of the method, a slope vibration shape is transformed to a multiscale slope vibration shape in a multiscale domain using wavelet transform, which has explicit physical implication, high damage sensitivity, and noise robustness. These advantages are demonstrated in numerical cases of damaged bars, and results show that multiscale slope vibration shapes can be used for identifying and locating damage in a noisy environment. A three-dimensional (3D) scanning laser vibrometer is used to measure the longitudinal steady-state response shape of an aluminum bar with damage due to reduced cross-sectional dimensions under harmonic excitation, and results show that the method can successfully identify and locate the damage. Slopes of longitudinal vibration shapes are shown to be suitable for damage detection in bars and have potential for applications in noisy environments.

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