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Caltech Authors (147.820 recursos)
Repository of works by Caltech published authors.
Group = Dynamics Laboratory
Repository of works by Caltech published authors.
Group = Dynamics Laboratory
Vakakis, Alexander; Caughey, Thomas
In this work we examine the free and forced oscillation of a class of two-Degree-of Freedom Undampted Nonlinear Systems. We examine the existence and stability of Similar Normal Modes of the Unsymmetric system and we analyse the subharmonic periodic orbits of the weakly coupled symmetric oscillator. Furthermore, exact solutions for the steady state forced motions of the strongly nonlinear system are derived and specific applications for systems with cubic nonlinearities are given. Finally we give a methodology for examining the Nonsimilar Normal Modes of such systems and we apply the analysis to the case of a weakly coupled system with...
O'Kelly, Michael Edmond James
A general review of normal mode theory as applied to the vibration of linear damped lumped parameter bilateral systems is presented. It is shown that systems possessing classical damping may always be solved by the method developed by Rayleigh. However, for more general type non-classical damping the method proposed by F. A. Foss must be used, The main differences between classical and non-classical normal modes are noted. A non-classically damped system which does not possess a mode type solution is solved by La place Transform techniques.
The effect of damping on the natural frequencies of a linear system is discussed. It...
Rocke, Richard Dale
The use of transmission matrices and lumped parameter models for describing continuous systems is the subject of this study. Nonuniform continuous systems which play important roles in practical vibration problems, e.g., torsional oscillations in bars, transverse bending vibrations of beams, etc., are of primary importance.
A new approach for deriving closed form transmission matrices is applied to- several classes of non-uniform continuous segments of one dimensional and beam systems. A power series expansion method is presented for determining approximate transmission matrices of any order for segments of non-uniform systems whose solutions can not be found in closed form. This direct series...
Cronin, Donald Leslie
The response of linear, viscous damped systems to excitations having time-varying frequency is the subject of exact and approximate analyses, which are supplemented by an analog computer study of single degree of freedom system response to excitations having frequencies depending linearly and exponentially on time.
The technique of small perturbations and the methods of stationary phase and saddle-point integration, as well as a novel bounding procedure, are utilized to derive approximate expressions characterizing the system response envelope -- particularly near resonances -- for the general time-varying excitation frequency.
Descriptive measurements of system resonant behavior recorded during the course of the analog study--maximum...
Atkinson, John David
[Contains mathematical notation that does not convert: see report for the correct formula.]
The Fokker-Planck (FP) equation is used to develop a general method for finding the spectral density for a class of randomly excited first order systems. This class consists of systems satisfying stochastic differential equations of form m . . . where f and the . . . are piecewise linear functions (not necessarily continuous), and the . . . are stationary Gaussian white noise. For such systems, it is shown how the Laplace -transformed FP equation can be solved for the transformed transition probability density. By manipulation of...
Tso, Wai Keung
A study of the coupled torsional and bending vibrations of thin-walled beams of asymmetric open section is made. The formal solution to Gere's theory for the case of a monosymmetric section under general loading conditions and boundary conditions is presented.
A higher order theory including the effect of shear strain induced by bending and warping of the beam is derived. Spectrum curves of the higher order theory are compared with those from the elementary theory for various boundary conditions for a special family of monosymmetric sections. A study is made to assess the effect of the shape of the cross section...
Dickerson, John Randall
Sufficient stability criteria for classes of parametrically excited differential equations are developed and applied to example problems of a dynamical nature.
Stability requirements are presented in terms of 1) the modulus of the amplitude of the parametric terms, 2) the modulus of the integral of the parametric terms and 3) the modulus of the derivative of the parametric terms.
The methods employed to show stability are Liapunov's Direct Method and the Gronwall Lemma. The type of stability is generally referred to as asymptotic stability in the sense of Liapunov.
The results indicate that if the equation of the system with...
Crede, Charles E
This report is in the nature of a progress report; it sets forth in some detail the status of the research on the several tasks as of the date of the report. In general, the research was incomplete on that date and no conclusions are presented. The report is written primarily to keep parsonnel of the sponsoring agency apprised of work that has been accomplished. It is not intended for outside distribution or publication.
This is the first annual report under a continuing research project whose objective is to gain a better understanding of the failure of equipment when subjected...
Masri, Sami Faiz
A study is made of the general behavior of a single particle impact damper, with the main emphasis on symmetric 2 impacts/cycle motion. The exact solution for this case is derived analytically and its asymptotically stable regions are determined. The stability analysis defines the zones where the modulus of all the eigenvalues of a certain matrix relating conditions after each of two consecutive impacts is less than unity.
Results of the analysis are supplemented and verified by experimental studies with a mechanical model and an analog computer. Additional numerical investigations are made with a digital computer.
It is found that, under practically...
Malhotra, R. K.
Free and forced oscillations in oscillators governed by the equation [MATHEMATICAL NOTATION GOES HERE. VIEW IT IN THE DOCUMENT] are studied with appropriate constraints on [MATHEMATICAL NOTATION GOES HERE. VIEW IT IN THE DOCUMENT]. Theorems are proved on the existence and uniqueness of stable periodic solutions for free oscillations using the Poincaré-Bendixson theory in the phase-plane. There follow several examples to illustrate the theorems and limit cycles are obtained for these examples by the Liénard construction. A result on the existence of periodic solutions in the forced case is obtained by use of Brouwer's fixed point theorem. The part on...
Lutes, Loren Daniel
Many practical cases of uniform beams carrying discrete loads cannot be accurately approximated either as single degree of freedom systems, or as simple uniform beams. A mathematical analysis of such systems is presented, where Bernoulli-Euler theory is used for the beam, and the effects of both the mass and the rotational inertia of the discrete load are included.
Numerical values of various normal mode factors are presented for seven particular systems -- namely: a fixed-free beam with the discrete load at the tip, and fixed-fixed and hinged-hinged beams with the discrete load at the mid-point, at the one-third point, and at...
Hu, Paul Yu-fei
Analytical and experimental investigations are made of the response of linear systems subject to magnitude-limited Gaussian broadband random excitation. A mathematical analysis for determining the statistical properties of this excitation is developed. Experimental studies on the probabilistic response of linear systems with magnitude-limited input are also presented.
Secondly the peak characteristics of the response of linear systems subject to Gaussian broadband random excitation are investigated. It is shown that the number of peaks per unit time of the response of a single degree of freedom system increases as the frequency bandwidth of the excitation increases. Analytical and experimental techniques are developed...
Caughey, Thomas Kirk; O'Kelly, Michael Edmond James
The usual treatment of linearly damped lumped parameter systems assumes that the system equations may be transformed to a symmetrical set of equations. This assumption is justified in passive systems. However, in many problems of interest to aeronautical and electrical engineers the system equations cannot be transformed to a symmetric set of equations. One case in point is the analysis of an aircraft wing under flutter conditions. That non-symmetric systems are physically realizable will be understood when one remembers that it is possible to build any non-symmetric system using an active analog computer. It is the purpose of this report...
Zaremba, Slawomir M
The applicability of a dynamical systems approach to the analysis of gearbox vibration signatures is investigated. The signal acquired from a standard one-step helical gearbox is analyzed and the existence of the low-dimensional nonlinear and chaotic behavior is examined. For this purpose, the criteria of broadband spectrum, sensitivity to initial conditions, positive Lyapunov exponents, and short-term dynamical predictability are applied. The largest Lyapunov exponent is also used to quantify the predictability of the measured time series and a surrogate data test is performed to confirm that the analyzed signal is unlikely to correspond to a linear stochastic time-invariant model. To...
Smith, Paul Wesley
An approach is developed that simplifies calculation of the dynamic characteristics of a self-acting, gas-lubricated slider bearing system. This technique avoids a lengthy simultaneous solution of the equations of motion of the slider and the time-dependent Reynolds' equation, while providing additional design information that is otherwise unobtainable.
The equilibrium pressure distribution in the gas film is obtained using the Bunov-Galerkin formulation of the finite element method. By considering small perturbations of the slider bearing system about equilibrium, two coupled, second-order partial differential equations are derived, which define the in-phase and out-of-phase perturbation pressures in the gas film. These perturbation pressures are...
Jones, Nicholas Patrick
When a body is exposed to a flowing fluid, oscillations can occur due to one or more of several different mechanisms. The resulting large amplitudes of motion and fatigue are potential sources of structural failure. Furthermore, the drag force on a structure can be increased due to the effectively larger cross-sectional area presented to the flow from the oscillation. A complete understanding of the nature of such vibration is essential in the design of many civil and mechanical engineering systems.
Previous solutions to the vortex-induced vibration problem were primarily based on modal analysis, using a one- or two-mode approximation. Use of...
Reinhall, Per Gustaf
A difference equation with a cubic nonlinearity is examined. Using a phase plane analysis, both quasi-periodic and chaotically behaving solutions am found. The chaotic behavior is investigated in relation to heteroclinic and homoclinic oscillations of stable and unstable solution manifolds emanating from unstable periodic points. Certain criteria are developed which govern the existence of the stochastic behavior. An approximate solution technique is developed giving expressions for the quasi-periodic solutions close to a stable periodic point and the accuracy of these expressions are investigated. The stability of the solutions is examined and approximate local stability criteria are obtained. Stochastic excitation of...
Blevins, Robert Dilworth
Models are developed for both multi-degree-of-freedom aerodynamic galloping and vortex induced oscillation of bluff structures. These models are useful in the analysis of elastic structures exposed to a steady fluid flow.
An asymptotic method, based on the approximation of Bogoliubov and Mitropolsky, is developed for the analysis of the autonomous, internally resonant, nonlinear differential equations produced by the models. It is shown that the solutions of these systems can be divided into two classes by the nature of the secular terms arising in the perturbation equations.
A model for multi-degree-of-freedom galloping is developed by modeling the aerodynamic forces on the structure as...
Moeller, Thomas Lee
This thesis analyzes the dynamics of a spinning elastic disk. The disk rotates at a constant angular velocity and is acted upon by a load consisting of a mass distributed over a finite area of the disk, a spring, and a dashpot. Using a finite mode approximation, the equation of motion of the transverse deflection of the disk is written as a system of ordinary differential equations with constant coefficients.
Analysis of the eigenvalues of the finite mode approximation yields four distinct types of instability. An instability occurs due to the stiffness of the load, terminal instabilities occur due to both...
Cronin, Donald Leslie
The response of linear, viscous damped systems to excitations having time-varying frequency is the subject of exact and approximate analyses, which are supplemented by an analog computer study of single degree of freedom system response to excitations having frequencies depending linearly and exponentially on time.
The technique of small perturbations and the methods of stationary phase and saddle-point integration, as well as a novel bounding procedure, are utilized to derive approximate expressions characterizing the system response envelope -- particularly near resonances -- for the general time-varying excitation frequency.
Descriptive measurements of system resonant behavior recorded during the course of the analog study...