Browsing by Subject "Pendulum"
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Item Linear and autoparametric modal analysis of aeroelastic structural systems(Texas Tech University, 1984-05) Woodall, Tommy DaleThis investigation deals with the linear modal analysis and autoparametric interaction of aeroelastic systems such as an airplane fuselage and wing with fuel storage. The mathematical modeling is derived by applying Lagrange's equations taking into consideration the Christoffel symbol of the first kind to account for the nonlinear coupling of the system coordinates, velocities, and accelerations. The linear modal analysis will be obtained by considering the linear, conservative portion of the equations of motion. The normal mode frequencies and the associated mode shapes are obtained in terms of the system parameters. The main objective of the linear analysis is to explore the critical regions of autoparametric (or internal) resonance conditions, £kiwi=0 (where ki are integers and wi are the normal mode frequencies). The results show that for certain system parameters the condition of internal resonance is satisfied. The dynamic behavior of the structure in the neighborhood of internal resonance conditions is obtained by considering the nonlinear coupling of the normal modes. The asymptotic approximation technique due to Struble is employed. Three groups of internal and normal resonance conditions are obtained from the secular terms of the first-order perturbational equations. The transient and steady-state responses cure obtained numerically by using the IBM Continuous System Modeling Program (CSMP) with double precision Milne integration. The transient response shows a build up in the interacted modes to a level which exceeds the steady-state response. In addition, the excited mode is suppressed by virtue of the nonlinear feedback of other modes. Under certain conditions, the steady-state response is derived analytically. It is concluded that the nonlinear modal analysis reveals certain types of response characteristics which cannot be interpreted within the framework of the linear theory of small oscillations.Item Theoretical and experimental investigation of the dynamics and bifurcations of an impacting spherical pendulum with large deflection(Texas Tech University, 1993-12) Garza, SantosThe inverted spherical pendulum has been a common engineering paradigm for strong focusing mechanisms. The spherical pendulum has been used to damp irregular motions in helicopters and on space stations as well as for many other appHcations. An inverted impacting spherical pendulum with large deflection was investigated. The model was designed to approximate an ideal pendulum, with the pendulum bob contributing the vast majority of the mass moment of inertia of the system. Two types of bearing mechanisms and tracking devices were designed for the system, one of which had a low Coulomb damping coefficient and the other with a high Coulomb damping coefficient. In the development of the dynamics theory a thorough analysis of the air damping was performed. The analytical solution assumed that the velocity squared term and the Coulomb term dominated the system damping. This broke with conventional analysis which had linear velocity terms dominating the damping of the pendulum. The system description consisted of the equations of motion, steady state motions. Type I and II subharmonics, and inversion criteria, AU simulations were run at twenty times the natural frequency of the pendulum to maximize shaker output. Power outputs from 0 to 125 mm-htz were studied both numerically and experimentally. Additional numerical simulations in the region 0 to 600 mm-htz were performed to determine extended system dynamics. Numerical simulations were performed using Butcher's fifth-order Runge-Kutta approximations for speed and accuracy. Fractal dimensions and Lyapunov exponents were derived using a new computational scheme that reduced the time necessary to find these components by an order of magnitude. This paper will discuss the characteristic response of the system for various coefficients of restitution, mass moment of inertias. Coulomb damping coefficients and forcing amphtudes. It will be shown that though the pendulum can move in three dimensions, it will act as a plane pendulum in its response. Furthermore, it will be shown that the system responded in a Type I response at the lower values of forcing amplitude, had a mixed Type I/II response and then exhibited a subharmonic Type II response at higher values of forcing amplitude. Numerical simulations showed that at large values of forcing amplitude the system exhibited a loss of stabihty after a harmonic region and then stably inverted.