Vibration

This thesis deals with the analytical study of vibration of carbon nanotubes and graphene plates. First, a brief overview of the traditional Bresse-Timoshenko models for thick beams and Uflyand-Mindlin models for thick plates will be conducted. It has been shown in the literature that the conventionally utilized mechanical models overcorrect the shear effect and that of rotary inertia. To improve the situation, two alternative versions of theories of beams and plates are proposed. The first one is derived through the use of equilibrium equations and leads to a truncated governing differential equation in displacement. It is shown, by considering a power series expansion of the displacement, that this is asymptotically consistent at the second order. The second theory is based on slope inertia and results in the truncated equation with an additional sixth order derivative term. Then, these theories will be extended in order to take into account some scale effects such as interatomic interactions that cannot be neglected for nanomaterials. Thus, different approaches will be considered: phenomenological, asymptotic and continualized. The basic principle of continualized models is to build continuous equations starting from discrete equations and by using Taylor series expansions or Padé approximants. For each of the different models derived in this study, the natural frequencies will be determined, analytically when the closed-form solution is available, numerically when the solution is given through a characteristic equation. The objective of this work is to compare the models and to establish the eventual superiority of a model on others.

This dissertation is concerned with the relevant research in developing finite dimensional indirect adaptive schemes to control vibrations in flexible smart structures based on the finite element approximation of the infinite dimensional system. The advantage of this type of modeling is that the dominant modes of vibrations wherein the total energy is concentrated are accommodated thereby avoiding the so-called "spillover" phenomenon. Further, the mass, stiffness and damping coefficients associated with each element appear explicitly in the model facilitating the derivation of the ARMA parametric representation which is suitable for on-line estimation of the structural parameters. The state-space representation of the finite dimensional model is used to design an indirect linear quadratic self tuning regulator algorithm using the parameter estimation, indicated above. Further, a method to choose the control and state weighting matrices (required to design the controller) to yield a stable closed-loop system, is presented. Simulation results demonstrating the performance of the adaptive control system are presented. Another algorithm based on the model reference technique is also developed by considering the discrete time approximation of the finite dimensional model. This control algorithm in conjunction with the parameter estimation constitute an indirect model reference adaptive control system. Simulation results are presented to demonstrate the effect of the reference model parameters, which may impose certain constraints on the force requirements causing actuator saturation and thereby affecting the stability of the closed-loop system. In order to overcome the problem of using bulky and expensive sensors to measure transverse displacement and velocity, a new spatial recursive technique to estimate these variables alternatively by using a distributed set of (measured) strain data, is developed. Relevant algorithm enables the use of smart materials to sense the strain developed at various locations along the length of the structure leading to the development of flexible smart structures. Experimental results on the personal computer based control of vibrations in an aluminum beam using patches of polyvinyldene fluoride (PVDF), and lead zirconate titanate (PZT) as sensors and control actuators respectively, are furnished to demonstrate the feasibility of real-time implementation of the above mentioned control algorithms.

A mobility power flow approach is used to study the response of an infinitely-long cylindrical shell with an internal plate discontinuity. The shell is excited by either a ring radial force or by a plane acoustic wave. The junction between the shell and the internal plate is assumed to be radially pinned such that in-plane waves of the plate can be neglected. The junction forces are expressed in terms of the mobility functions of the plate and the shell. From knowledge of the junction forces and velocities, the power input, the power flow from the shell to the plate, the shell response and the radiated far-field scattered pressure are determined for the circumferential mode n = 0. The results show how the energy propagates from one structure to the other, and present a very clear picture of the characteristics of the scattering pattern from the junction forces.

A Statistical Energy Analysis (SEA) approach is used to investigate the vibrational behavior of a journal bearing. In developing the SEA model, consideration is given to the determination of coupling loss factors between non-conservatively coupled substructures. In the case of the journal bearing, the oil film between the rotating shaft and the bearing liner represents non-conservative coupling. The coupling loss factors are estimated using experimentally measured point mobility functions. The internal loss factors are directly measured with the bearing structure disassembled. Additionally, estimates for the coupling and internal loss factors are obtained in-situ using an energy ratio approach. Using the determined coupling and internal loss factors in an SEA model, estimates for the average mean square velocities on the surface of the bearing subcomponents are obtained for both static and dynamics conditions. The SEA estimates match well with directly measured results for the spatial average surface velocities at medium to high frequencies.

Structural intensity is propagated through a thick structure by both in-plane and out-of-plane (transverse) waves. These waves propagate at different phase speeds and therefore it is important to distinguish the components of the structural intensity associated with each wave type. To show the presence of these different wave components, experimental results are performed on a thick beam. Using a frequency-wavenumber analysis, the different waves and contributions to the structural intensity are identified. The significance of the contributions to the structural intensity are a function of both frequency and thickness of the structure. Using simulated measurements on a thick L-shaped plate, the relative importance between the in-plane and out-of-plane contributions to structural intensity as a function of frequency and thickness is demonstrated. It is shown that in-plane wave contributions increase in importance as frequency or thickness increases.

Proper application of matrix perturbation theory to the structural modification of vibrating systems yields accurate results without resolving the eigenvalue problem. For this reason it seems that the method may require less computer time and cost in performing iterative type design work. This thesis investigates the application of first-order perturbation techniques to a torsional shaft system subject to viscoelastic damping treatment modifications. The modifications are of general form in that the coating length and thickness are not restricted. Large modifications are built up by a series of small first-order perturbations. In addition to developing a better understanding of the usefulness of this application of perturbation theory, the goal of this paper is also to understand the nonproportional damping effects of partial viscoelastic coatings.

The objective of this thesis is the investigation of the transmission of vibrational power between beam-like and plate-like structures with joints using Statistical Energy Analysis (SEA) and an experimental approach. In the case of the L-shaped plate the influence of different structure parameters is investigated to determine possible ways to reduce both the transmitted power and the noise radiated by the receiver structure. The total transmitted power is measured and compared to theoretical predictions. Also power maps are generated which show the transmission of vibrational power from the source beam to the receiver beam. In the power flow measurements three different techniques are implemented with similar results which all match the analytical results.

Of the many methods of introducing damping in
vibrating structures, the dissipation of energy due to
interfacial slip can significantly increase the damping loss
factor. However, because of the lack of understanding and
other phenomena such as fretting corrosion and loss of
structural rigidity, friction damping is rarely used. A
study was thus undertaken to investigate this complex
phenomenon, with emphasis on trying to gain a better
understanding of friction damping with certain parameters
such as clamping pressure, frequency, magnitude of
excitation and surface finish. Although the non-linearities
associated with friction makes this mechanism difficult to
model mathematically, finite element (FE) analysis shows
some promise. Although the results obtained using an FE
model were not exactly comparable to the experimental
results, these analytical results did show the same general
trends as observed in the experiments.

A theoretical analysis for predicting the system loss factors and
natural frequencies of rectangular plates with complete and partial
constrained-layer damping treatments has been presented. This analysis
is based upon an energy approach to the free vibration of plates.
Results predicted were compared with those from experiments. Satisfactory
agreement has been reached. Both the theoretical and the
experimental results presented in this thesis indicate clearly that
partial constrained-layer damping treatments can provide effective, or
even superior, amounts of damping, and that their use can lead to
significant savings in material costs and weight.

The dynamic behavior of straight cantilever pipes conveying fluid is studied, establishing the conditions of stability for systems, which are only limited to move in a 2D-plane. Internal friction of pipe and the effect of the surrounding fluid are neglected. A universal stability curve showing boundary between the stable and unstable behaviors is constructed by finding solution to equation of motion by exact and high-dimensional approximate methods. Based on the Boobnov-Galerkin method, the critical velocities for the fluid are obtained by using both the eigenfunctions of a cantilever beam (beam functions), as well as the utilization of Duncan's functions. Stability of cantilever pipes with uniform and non-uniform elastic foundations of two types are considered and discussed. Special emphasis is placed on the investigation of the paradoxical behavior previously reported in the literature.