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 nonlinear dynamical behavior of a laterally compressed, flat, composite panel subjected to a high supersonic flow is analyzed. The structural model considers a higher-order shear deformation theory which also includes the effect of the transverse normal stress and satisfies the traction-free condition on both faces of the panel. The possibility of small initial imperfections and in-plane edge restraints are also considered. Aerodynamic loads based on the third-order piston theory are used and the panel flutter equations are derived via Galerkin’s method. Periodic solutions and their bifurcations are obtained by using a predictor-corrector type of numerical integration method, i.e., the Shooting Method, in conjunction with the Arclength Continuation Method for the static solution. For the perfect panel, the amplitudes and frequency of flutter obtained by the Shooting Method are shown to compare well with results from the Method of Multiple Scales when linear aerodynamics is considered and compressive loads are absent. It is seen that the presence of aerodynamic nonlinearities could result in the hard flutter phenomenon, i.e., a violent transition from the undisturbed equilibrium state to that of finite motions which may occur for pre-critical speeds also. Results show that linear aerodynamics correctly predicts the immediate post-flutter behavior of thin panels only. When compressive edge loads or edge restraints are applied, in certain cases multiple periodic solutions are found to coexist with the stable static solution, or multiple buckled states are possible. Thus it is seen that the panel may remain buckled beyond the flutter boundary, or it may flutter within the region where buck-led states exist. Furthermore, the presence of edge restraints normal to the flow tends to stabilize the panel by decreasing the flutter amplitudes and the possibility of hard flutter. Nonperiodic motions (i.e., quasiperiodic and chaotic) of the buckled panel are found to exist, and their associated Lyapunov exponents are calculated. The effects of transverse shear flexibility, aerodynamic nonlinearities, initial imperfections, and in-plane edge restraints on the stability boundaries are also studied. It is observed that the classical plate theory over-predicts the instability loads, and only the shear deformation theory correctly models the panel which is flexible in transverse shear. When aerodynamic nonlinearities are considered, multiple flutter speeds may exist. / Ph. D.

No description available.

<p> The problem of a thin, orthotropic skew plate of linearly varying thickness for vibration and buckling analyses is formulated under the assumptions of small-deflection theory of plates. Using the dimensionless oblique coordinates, the deflection surface of the plate is expressed as a polynomial series, each term of which satisfying the required polar symmetry conditions, and the natural frequencies are computed using Galerkin method. As is required in Galerkin method, the assumed deflection function satisfies all the boundary conditions on all the edges of the plate. For the skew plate, clamped on all the four edges, numerical results for the first few natural frequencies are presented for various combinations of aspect ratio, skew angle and taper parameter. Convergence study has been made for typical configuration of the plate and the limited available data is inserted therein along with the computed results, for comparison.</p> / Thesis / Master of Engineering (MEngr)

No description available.

This work discusses the deformations of various unsymmetric composite panels due to thermal and mechanical loads. Chapter 2 focuses on the warpage of large unsymmetric curved composite panels due manufacturing anomalies. These panels are subjected to a temperature change of -280°F to simulate the cooling from the autoclave cure temperature. Sixteen layer quasi-isotropic, axial-stiff, and circumferentially-stiff laminates are considered. These panels are intended to be symmetric laminates, but are slightly unsymmetric due to the manufacturing anomalies. Rayleigh-Ritz and finite-element models are developed to predict the deformations. Initially, to serve as a basis for comparison, warpage effects due to orthotropic thermal expansion properties in perfect panels are investigated and are found to produce deformations not captured in two-dimensional theories. This is followed by the investigation of the effects of ply misalignments. Ply misalignments of 5° are incorporated into the laminate, one layer at a time, to produce unsymmetric laminates. It is found that ply misalignments produce warpages much larger than those induced by orthotropic thermal expansion properties. Next, unsymmetric laminates resulting from ply thickness variations are investigated. Layers 10% thicker than nominal are incorporated into the laminate, one layer at a time, while the remaining layers are of uniform thickness. Due to the change in fiber volume fraction of the thicker layers, corresponding material properties are modified to reflect this change. The results show that ply thickness variations cause warpages of about 25-50% of those induced by ply misalignments. Finally, warpage of panels due to nonuniform cooling due to inplane thermal gradients during cure is investigated. A thermal gradient of 0.1°F/in. is used to construct six inplane distributions. It is found that the warpages induced by thermal gradients are very small. The warpages are negligible with respect to those induced by ply thickness variations or ply misalignments. Deformations induced by thermal gradients depend primarily on the magnitude of the thermal gradient, but not on the pattern of distribution. Overall, ply misalignments cause the most warpage, followed by ply thickness variations. Important variables for these imperfections are, the through-thickness location of the imperfections, the orientation of the layer containing the imperfections, and the lamination sequence. All cases show that geometric nonlinearities are important to accurately predict the deformations induced by these imperfections. Chapter 3 discusses the deformations of composite plates that are intentionally fabricated to be unsymmetric. Such plates, if flat, might be considered in applications where bending-stretching coupling effects can be used to advantage. It is assumed the laminates are cured at an elevated temperature and then cooled 280°F. Significant deformations result because of the high level of asymmetry in the laminate construction. Accordingly, geometric nonlinearities are included in the models. Four cross-ply laminates and three angle-ply laminates are considered. Four-term and 14-term Rayleigh-Ritz models are developed, together with finite-element models to model the deformations. Actual specimens were constructed and the deformations measured to compare with predictions. The results show that agreement between predictions and the experimental results are good. The 14-term Rayleigh-Ritz model is found to be the most useful due to its ability to find multiple solutions, its physical basis, and computational efficiency. Chapter 4 discusses the deformations of initially flat aluminum, symmetric, and unsymmetric composite plates due to axial endshortening under various boundary conditions, the aluminum and symmetric plates serving as a baseline. Seven plates are considered, each with three boundary condition combinations, namely, clamped ends and sides (CL-CL), clamped ends with simply-supported sides (CL-SS), and simply-supported ends and sides (SS-SS). Generally, the boundary conditions play a key role in the deformation characteristics of the plates. The aluminum and symmetric cross-ply plates have no out-of-plane deformations until classic buckling, or primary instability, then each exhibits two stable solutions. Each also exhibits secondary instability that results in two stable solutions. The symmetric laminates show less of a dependence on the boundary conditions compared to the unsymmetric laminates. Unsymmetric laminates show a mixture of characteristics. Some cases exhibit primary instability, other cases do not. Some cases exhibit secondary instability, while some case do not. The unsymmetric cross-ply laminates have only one stable solution after secondary buckling, while most other laminates and boundary condition combinations have two stable solutions. It is interesting to note that for the unbalanced unsymmetric [302/90/0]2T laminate, the boundary conditions controlled the sign of the out-of-plane deflection from the onset of axial endshortening. Generally speaking, the CL-CL cases carry the most load, followed by the CL-SS, and then the SS-SS cases. Like all the problems discussed in Chapter 2 and 3, geometric nonlinearities are found to be important for this case as well. / Ph. D.

The Layerwise Shell Theory is used to model discretely stiffened laminated composite cylindrical shells for stress, vibration, pre-buckling and post-buckling analysis. The layerwise theory reduces a three-dimensional problem to a two-dimensional problem by expanding the three-dimensional displacement field as a function of a surface-wise two-dimensional displacement field and a one-dimensional interpolation through the shell thickness. Any required degree of accuracy can be obtained by an appropriate, independent selection of the one-dimensional interpolation functions through the thickness and the two-dimensional interpolation of the variables on the surface. Using a layerwise format, discrete axial and circumferential stiffeners are modeled as two-dimensional beam elements. Similar displacement fields are prescribed for both the stiffener and shell elements. The contribution of the stiffeners to the membrane stretching, bending and twisting stiffnesses of the laminated shell is accounted for by forcing compatibility of strains and equilibrium of forces between the stiffeners and the shell skin. The layerwise theory is also used to model initial imperfections of the unstressed configuration. A finite element scheme of the layerwise model is developed and applied here to investigate the effect of imperfections on the response of laminated cylindrical shells. Using a finite element model of the layerwise theory for shells and shell stiffener elements, the accuracy and reliability of the elements is investigated through a wide variety of examples. The examples include laminated stiffened and unstiffened plates and shells and stand-alone beams under different types of external destabilizing loads. Finally, the study identifies the particular types of problems where the layerwise elements possess a clear advantage and superiority over the conventional equivalent single-layer models. / Ph. D.

A knowledge of the natural frequencies of a vibrating plate is of great importance if an effective design is to be made which will prevent critical conditions of heavy vibration from occurring. Those frequencies which are associated with the symmetric modes are especially important. Many approximate methods have been devised to determine these natural frequencies. In this dissertation a method of frequency determination is suggested through an integral equation approach. The plate vibration problem is formulated as a problem in the solution of a homogeneous, linear Fredholm integral equation of the second kind in which the kernel is either symmetric or can be made so by a convenient transformation. The integral equation, as formulated, satisfies the boundary conditions in that it includes Green's function of the plate which is a solution of the isolated force problem. Three approximate methods for solving the integral equation are described mathematically and then applied to three elementary examples. The three methods used ares 1) method of successive approximation, 2) method of collocation and 3) the trace or the kernel. It is shown that using the trace of the kernel always gives a lower bound to the frequency and is particularly useful for the determination of the fundamental frequency. After solving the three elementary problems the integral equation approach is made to the uniform circular cantilever plate where the frequency is approximated both by collocation and by the use of the trace of the kernel. The first and second approximate mode shapes are then derived and shown graphically. The results are seen to compare favorably with results obtained from the Rayleigh-Ritz method. Finally, the fundamental frequency is determined for the circular, stepped cantilever plate and the clamped elliptical plate. For the stepped plate fundamental frequency curves are drawn for various positions and magnitudes of the step. The fundamental frequency curve of the clamped elliptical plate is drawn as a function of the eccentricity of the ellipse. A frequency obtained from experiment is reported along with a calculated value determined from the Rayleigh-Ritz method. It is seen that the integral equation approach is about 19% below the experimental value whereas the Rayleigh-Ritz method gives a fundamental frequency about 27% above the experimental value. / Ph. D.

Ph. D.

Ph. D.