The classical method, required for its solution, the application of boundary conditions to the solution of the beam equation. Except for the case cf the beam with one concentrated load at the center, it was not considered a practical solution. The transcendental equation obtained in the solution of the unsymmetrical case, considered in part B, was found too cumbersome to handle. It was not attempted in parts C and D. The Rayleigh Method proved to be a simple, accurate and reasonably rapid method for all cases considered. The Dunkerley Equation gave very satisfactory results for parts A, B, and C. It was rapid to use, accurate and in most cases the data could be found in prepared tabulations. Results were inaccurate for the two span beam, indicating the necessity for caution in its application to multi-span beams. The Ritz Method, which is a refinement of the Rayleigh Method, proved to be exceedingly accurate when applied to the beam with the single concentrated load. However, it was found, that as the number of terms in the assumed deflection equation increased, the work became more time consuming. It was used only in parts A and B. The Influence Coefficient Method and the application of D'Alembert's Principle, which methods are quite similar, proved to be simple, accurate, and rapid. However, as the number of degrees of freedom increased, the degree of the algebraic equation increased, which complicated the solution. The Iteration Method is probably the method to be used if the number of degrees of freedom exceeds three. As the number of modes increases the number of iterations would increase, but the individual operations in themselves would remain simple. This method proved simple and accurate to use. For the cases considered, it was more time consuming to use than either the Influence Coefficient Method or the application of D'Alembert's Principle. However, for higher degree situations, it should prove to be a more practical method. / Master of Science

Lai, Kim Fung = 幼樑之振動與力學特性 / 黎劍鋒. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 99-102). / Abstracts in English and Chinese. / Lai, Kim Fung = You liang zhi zhen dong yu li xue te xing / Li, Jianfeng. / Chapter I --- Vibrations of Timoshenko Beams --- p.1 / Chapter 1 --- Introduction --- p.2 / Chapter 1.1 --- Overview --- p.2 / Chapter 1.2 --- Simple theory of static beam bending --- p.6 / Chapter 1.3 --- Foundation of problem --- p.7 / Chapter 1.4 --- Literature review --- p.12 / Chapter 1.4.1 --- Euler-Bernoulli Beam Theory (EBBT) --- p.12 / Chapter 1.4.2 --- Timoshenko Beam Theory (TBT) --- p.16 / Chapter 1.5 --- Preview of our results --- p.20 / Chapter 2 --- 3-D problem --- p.22 / Chapter 2.1 --- Elastic theory --- p.23 / Chapter 2.2 --- Boundary conditions --- p.24 / Chapter 2.3 --- Plane waves in uniform thin beams --- p.25 / Chapter 2.4 --- Solving order-by-order analytically --- p.26 / Chapter 2.5 --- Minimization approach --- p.36 / Chapter 3 --- 2-D problem --- p.50 / Chapter 3.1 --- Boundary conditions and effective moduli --- p.51 / Chapter 3.2 --- Expansion for thin beams --- p.54 / Chapter 3.3 --- Plane waves in uniform thin beam --- p.56 / Chapter 3.4 --- Boundary conditions --- p.57 / Chapter 3.5 --- Truncation --- p.58 / Chapter 3.6 --- Numerical solution --- p.58 / Chapter 3.7 --- Analytic results for soft mode --- p.60 / Chapter 3.8 --- EBBT and TBT for 2-D problem --- p.62 / Chapter 3.9 --- Analytic results for hard mode at q = 0 --- p.64 / Chapter 3.10 --- Higher-order corrections for hard mode --- p.66 / Chapter 4 --- Summary --- p.71 / Chapter II --- Vibrations of Single-Walled Carbon nanotubes --- p.73 / Chapter 5 --- Introduction --- p.74 / Chapter 5.1 --- General properties --- p.74 / Chapter 5.2 --- Graphene sheet --- p.76 / Chapter 5.3 --- Rolling up a graphene sheet --- p.78 / Chapter 5.4 --- Foundation of problem --- p.79 / Chapter 5.5 --- Literature review --- p.79 / Chapter 5.6 --- Preview of our results --- p.80 / Chapter 6 --- Structure and strain energy under zero stress --- p.81 / Chapter 6.1 --- Description of the structure --- p.81 / Chapter 6.2 --- Description of the strain energy --- p.83 / Chapter 6.3 --- Minimization of energy --- p.86 / Chapter 7 --- SWCNT under strain --- p.89 / Chapter 7.1 --- Subject to an axial strain --- p.89 / Chapter 7.2 --- Subject to a radial strain --- p.94 / Chapter 7.3 --- Subject to a torsional strain --- p.95 / Chapter 8 --- Summary --- p.98 / Bibliography --- p.99 / Chapter A --- "Expressing elastic moduli G, λ and M in terms of Y andv" --- p.103 / Chapter B --- Simplification of the functional E to a neat expression --- p.105 / Chapter C --- Expressing effective elastic moduli G' and M' in terms of Y' and v' --- p.106 / Chapter D --- Illustration of the lowest non-trivial truncation --- p.107 / Chapter E --- The proof of Self-adjointness of H(q) --- p.109 / Chapter F --- Proof of the identity KeVec= KeVel --- p.112

In this work, several studies into the dynamic response of structures are made. In all the studies there is an interaction between the theoretical and experimental work that lead to important results. In the first study, previous theoretical results for the single-mode response of a parametrically excited cantilever beam are validated. Of special interest is that the often ignored nonlinear curvature is stronger than the nonlinear inertia for the first mode. Also, the addition of quadratic damping to the model improves the agreement between the theoretical and experimental results. In the second study, multi-mode responses of a slender cantilever beam are observed and characterized. Here, frequency spectra, pseudo-phase planes, Poincare sections, and dimension values are used to distinguish among periodic, quasi-periodic, and chaotic motions. Also, physical interpretations of the modal interactions are made. In the third study, a theoretical investigation into a previously unreported modal interaction between high-frequency and low-frequency modes that is observed in some experiments is conducted. This modal interaction involves the complete response of the first mode and modulations associated with the third and fourth modes of the beam. A model that captures this type of modal interaction is developed. In the fourth study, the natural frequencies and mode shapes of several composite plates are experimentally determined and compared with a linear finite-element analysis. The objective of the work is to provide accurate experimental natural frequencies of several composite plates that can be used to validate future theoretical developments. / Ph. D.

The response characteristics of beams and plates made from composite laminates are strongly affected by the shear deformations of their layers. However, incorporation of the shear deformation further complicates the equations of motion and their analysis. As a result the vibration analysis of such structures have been limited to simple free vibration studies such as determination of their frequencies. The forced vibration problems of these structures have been solved by exact methods for only some very simple cases. In this study, a generalized modal approach is presented to solve more general vibration problems of composite beams and plates. The coupled systems of partial differential equations, representing the equations of motion, are uncoupled into modal equations by utilizing the eigenfunctions of the system and its adjoint. A method is presented to obtain these eigenfunctions for beams with arbitrary boundary conditions and for plates with Levy-type boundary conditions. The forced vibration solutions obtained by this method are then used to calculate the random response characteristics of beams and plates subjected to spatially and temporally correlated random loads. In the analysis of beams, both symmetric cross-ply and angle-ply configurations have been considered. In the symmetric cross-ply configuration with no torsional loads, of course, the warping effects are absent. The angle-ply case, however, includes torsion-warping effects and coupled bending-torsion motions. A simple displacement field is introduced to reflect warping in the third-order shear deformation theory. In the analysis of plates also two configurations of the laminates have been considered: symmetric cross-ply and antisymmetric angle-ply. At this time, these are the only two configurations which can be solved by the closed-form modal analysis approach for the Levy-type boundary conditions. In both cases of the beams and plates, the numerical results with and without shear deformations are obtained and compared. The results for no shear deformation are obtained with the classical lamination theory. The results have also been obtained for the first-order shear deformation theory with a somewhat simpler displacement field which has been commonly used in the past by several investigators. The numerical results are obtained for the global response quantities such as frequencies, displacements and crossing rates as well as for the local response quantities such as normal and shear stresses across a cross section. The numerical results obtained with various deformation theories for the frequencies as well as response quantities are compared to evaluate the effect of the shear deformations. For thicker and rigid beams and plates, one observes large differences in the frequencies and responses obtained with (the first- and third-order shear deformation theories) and without consideration of shear deformations (classical theory). For the frequencies and global responses both the first- and third-order theories give about the same results. But for the local response quantities, the results obtained with the two shear deformation theories can also be quite different in some cases. In any case, the results clearly point out the importance of including the shear deformations in thick and rigidly constrained composite beams and plates. Although, in this study only uniform cross section or uniform thickness beams and plates have been considered, it is felt that the eigenfunctions developed herein can also be conveniently utilized with advantage as comparison functions in approximate Rayleigh-Ritz type of approaches to analyze non uniform structures. / Ph. D.

Several researchers engaged in family housing have recently become concerned about the vibrational behavior of residential floors. This concern resulted in a need for methods of sensing floor vibrations. Some investigators have sensed floor vibrations with electric resistance strain gauges bonded to the underside of the floor joists. These experiments using strain gages as vibration sensing devices resulted in a need to be able to determine the vibration amplitude (or deflection) from strain vibration data. The objectives of this project were to theoretically and experimentally determine the relationship between midspan flexural strain and midspan deflection of freely vibrating wood beams with various end conditions. Theoretical strain-deflection relationships of freely vibrating wood beams with pinned-end and fixed-end conditions were derived from vibration theory. Free vibration tests on three wood beams with pinned-ends and fixed-ends gave results which were in agreement with theory. The theoretical relationship between the end rigidity and natural frequency of beams with semi-rigid end connections was derived. Vibration tests performed on wood beams with semi-rigid end connections produced frequency-rigidity results which agreed with theory within five percent. The semi-rigid end connections were achieved by using a torsion bar on each end designed so that the beam would have a static behavior midway between pinned-end conditions and fixed-end conditions. However, it was found both theoretically and experimentally that these torsion bars resulted in a dynamic behavior (strain-deflection ratio and frequency) much closer to pinned-end conditions than to fixed-end conditions. It was established that the strain-deflection relationship of freely vibrating wood beams can be predicted from vibration theory if the rigidity of the end connections is known. / Master of Science

The nonlinear stochastic vibrations of a beam with a varying cross-section are investigated. The nonlinearity is caused by midplane stretching and cubic in nature, and the forcing function is wide band white noise. The analysis is carried out by expanding the deflection curve in terms of the undamped linear modes. Substituting this expansion into the partial differential equation yields a set of ordinary differential equations in terms of the modal response functions, which are coupled through the nonlinear terms. The normal modes are found by the finite element method. The differential equations are then converted to a set of Ito's equations, from which a set of first-order differential equations for the response joint moments is found using the Fokker-Planck equation. These equations form an infinite hierarchy which is closed by the quasi-moment method. The solution is investigated near an internal resonance condition and the effects of higher order cumulants in the closure scheme and of additional modes to the expansion arc considered. It is shown that the second order solution is inadequate in the presence of internal resonances, but the fourth order solution proves to be adequate. The one mode approximation underestimates the nonlinear stiffening, and a multiple mode approach is necessary. It is also shown that the effect of an internal resonance of the stochastic vibration is to transfer of energy from the higher modes involved to the lower modes involved. / M.S.

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1982. / Microfiche copy available in Archives and Barker. / Includes bibliographical references. / by Peter David McQuade. / M.S.

Much progress has been made in recent years in active control of sound radiation from vibrating structures. Reduction of the far-field acoustic radiation can be obtained by directly modifying the response of the structure by applying structural inputs rather than by adding acoustic sources. Discontinuities, which are present in many structures are often important in terms of sound radiation due to wave scattering behavior at their location. In this thesis, an edge or boundary type discontinuity (clamped edge) and a point discontinuity (blocking mass) are analytically studied in terms of sound radiation. When subsonic vibrational waves impinge on these discontinuities, large scattered sound levels are radiated. Active control is then achieved by applying either control forces, which approximate shakers, or pairs of control moments, which approximate piezoelectric actuators, near the discontinuity. Active control of sound radiation from a simply-supported beam is also examined. For a single frequency, the flexural response of the beam subject to an incident wave or an input force (disturbance) and to control forces or control moments is expressed in terms of waves of both propagating and near-field types. The far-field radiated pressure is then evaluated in terms of the structural response, using Rayleigh's formula or a stationary phase approach, depending upon the application. The control force and control moment magnitudes are determined by optimizing a quadratic cost function, which is directly related to the control performance. On determining the optimal control complex amplitudes, these can be resubstituted in the constitutive equations for the system under study and the minimized radiated fields can be evaluated. High attenuation in radiated sound power and radiated acoustic pressure is found to be possible when one or two active control actuators are located near the discontinuity, as is shown to be mostly associated with local changes in beam response near the discontinuity.. The effect of the control actuators on the farfield radiated pressure, the wavenumber spectrum, the flexural displacement and the near-field time averaged intensity and pressure distributions are studied in order to further understand the control mechanisms. The influence of the near-field structural waves is investigated as well. Some experimental results are presented for comparison. / Ph. D.

The effect of the shape of piezoceramic actuators on the vibration response of a simply supported beam is investigated. An equation is derived to convert between the shape of the piezoceramic actuator and the resulting moment distribution caused on the structure. A beam simulation program is then created to model the vibrations caused by various shaped moment distributions exciting a simply supported beam. The length of the moment distribution is iterated from the length of the beam to zero length, within the program, to show the trends in modal amplitudes. The amplitude of each mode is then plotted for each length of the moment distribution. An equation is then derived to explain the resulting minimums and maximums of the modal amplitudes. The equation is shown to be a useful tool in designing shapes to meet specific control criteria. An example is given showing how the shape of the actuator can be designed to give superior performance for specific control criteria than a traditional rectangular shape. Two possible actuator shapes are shown for the situation. One shape is optimized for the given control criteria by causing the maximum response for the critical mode. The results from the beam simulation for both shapes are shown. The shape of the actuator may now be used as a variable in the cost function for control optimization. / Master of Science

Position and vibration control of a flexible beam is studied analytically and in the laboratory. Two different motor types are compared as actuators throughout the thesis: a standard voltage controlled motor and a torque controlled motor. The experimental beam is controlled with a dc-servo motor at its base and is instrumented with strain gages and a potentiometer. The control law is a form of linear, direct-output feedback. State estimators augment the control law to provide rate information that is not available from the instrumentation. Accurate modeling of the system’s inherent damping characteristics is achieved by analyzing experimental data. Gains were iterated yielding minimum-gain norm and minimum-sensitivity norm solutions to meet imposed eigenvalue placement constraints. Results for the two solutions and the two systems are compared and contrasted. Experimental verification of analytical results is hampered by unmodeled system non-linearities. Several attempts at bypassing these obstacles are shown. Finally, conclusions and recommendations are made. / Master of Science / incomplete_metadata