Ali H. Nayfeh

Linear and nonlinear structural mechanics

Preface. 1. Introduction. 1.1 Structural Elements. 1.2 Nonlinearities. 1.3 Composite Materials. 1.4 Damping. 1.5 Dynamic Characteristics of Linear Discrete Systems. 1.6 Dynamic Characteristics of Nonlinear Discrete Systems 1.7 Analyses of Linear Continuous Systems. 1.8 Analyses of Nonlinear Continuous Systems. 2. Elasticity. 2.1 Principles of Dynamics. 2.2 Strain--Displacement Relations. 2.3 Transformation of Strains and Stresses. 2.4 Stress--Strain Relations. 2.5 Governing Equations. 3. Strings and Cables. 3.1 Modeling of Taut Strings. 3.2 Reduction of String Model to Two Equations. 3.3 Nonlinear Response of Strings. 3.4 Modeling of Cables. 3.5 Reduction of Cable Model to Two Equations. 3.6 Natural Frequencies and Modes of Cables. 3.7 Discretization of the Cable Equations. 3.8 Single--Mode Response with Direct Approach. 3.9 Single--Mode Response with Discretization Approach. 3.10 Extensional Bars. 4. Beams. 4.1 Introduction. 4.2 Linear Euler--Bernoulli Beam Theory. 4.3 Linear Shear--Deformable Beam Theories. 4.4 Mathematics for Nonlinear Modeling. 4.5 Nonlinear 2--D Euler--Bernoulli Beam Theory. 4.6 Nonlinear 3--D Euler--Bernoulli Beam Theory. 4.7 Nonlinear 3--D Curved Beam Theory Accounting for Warpings. 5. Dynamics of Beams. 5.1 Parametrically Excited Cantilever Beams. 5.2 Transversely Excited Cantilever Beams. 5.3 Clamped--Clamped Buckled Beams. 5.4 Microbeams. 6. Surface Analysis. 6.1 Initial Curvatures. 6.2 Inplane Strains and Deformed Curvatures. 6.3 Orthogonal Virtual Rotations. 6.4 Variation of Curvatures. 6.5 Local Displacements and Jaumann Strains. 7. Plates. 7.1 Introduction. 7.2 Linear Classical Plate Theory. 7.3 Linear Shear--Deformable Plate Theories. 7.4 Nonlinear Classical Plate Theory. 7.5 Nonlinear Modeling of Rectangular Surfaces. 7.6 General Nonlinear Classical Plate Theory. 7.7 Nonlinear Shear--Deformable Plate Theory. 7.8 Nonlinear Layerwise Shear--Deformable Plate Theory. 8. Dynamics of Plates. 8.1 Linear Vibrations of Rectangular Plates. 8.2 Linear Vibrations of Membranes. 8.3 Linear Vibrations of Circular and Annular Plates. 8.4 Nonlinear Vibrations of Circular and Annular Plates. 8.5 Nonlinear Vibrations of Rotating Disks. 8.6 Nonlinear Vibrations of Near--Square Plates. 8.7 Micropumps. 8.8 Thermally Loaded Plates. 9. Shells. 9.1 Introduction. 9.2 Linear Classical Shell Theory. 9.3 Linear Shear--Deformable Shell Theories. 9.4 Nonlinear Classical Theory for Double--Curved Shells. 9.5 Nonlinear Shear--Deformable Theories for Circular Cylindrical Shells. 9.6 Nonlinear Layerwise Shear--Deformable Shell Theory. 9.7 Nonlinear Dynamics of Infinitely Long Circular Cylindrical Shells. 9.8 Nonlinear Dynamics of Axisymmetric Motion of Closed Spherical Shells. Bibliography. Subject Index.

A reduced-order model for electrically actuated microbeam-based MEMS

We present an analytical approach and a reduced-order model (macromodel) to investigate the behavior of electrically actuated microbeam-based MEMS. The macromodel provides an effective and accurate design tool for this class of MEMS devices. The macromodel is obtained by discretizing the distributed-parameter system using a Galerkin procedure into a finite-degree-of-freedom system consisting of ordinary-differential equations in time. The macromodel accounts for moderately large deflections, dynamic loads, and the coupling between the mechanical and electrical forces. It accounts for linear and nonlinear elastic restoring forces and the nonlinear electric forces generated by the capacitors. A new technique is developed to represent the electric force in the equations of motion. The new approach allows the use of few linear-undamped mode shapes of a microbeam in its straight position as basis functions in a Galerkin procedure. The macromodel is validated by comparing its results with experimental results and finite-element solutions available in the literature. Our approach shows attractive features compared to finite-element softwares used in the literature. It is robust over the whole device operation range up to the instability limit of the device (i.e., pull-in). Moreover, it has low computational cost and allows for an easier understanding of the influence of the various design parameters. As a result, it can be of significant benefit to the development of MEMS design software.

Characterization of the mechanical behavior of an electrically actuated microbeam

We present a nonlinear model of electrically actuated microbeams accounting for the electrostatic forcing of the air gap capacitor, the restoring force of the microbeam and the axial load applied to the microbeam. The boundary-value problem describing the static deflection of the microbeam under the electrostatic force due to a dc polarization voltage is solved numerically. The eigenvalue problem describing the vibration of the microbeam around its statically deflected position is solved numerically for the natural frequencies and mode shapes. Comparison of results generated by our model to the experimental results shows excellent agreement, thus verifying the model. Our results show that failure to account for mid-plane stretching in the microbeam restoring force leads to an underestimation of the stability limits. It also shows that the ratio of the width of the air gap to the microbeam thickness can be tuned to extend the domain of the linear relationship between the dc polarization voltage and the fundamental natural frequency. This fact and the ability of the nonlinear model to accurately predict the natural frequencies for any dc polarization voltage allow designers to use a wider range of dc polarization voltages in resonators.

Dynamics and Control of Cranes: A Review:

We review crane models available in the literature, classify them, and discuss their applications and limitations. A generalized formulation of the most widely used crane model is analyzed using the method of multiple scales. We also review crane control strategies in the literature, classify them, and discuss their applications and limitations. In conclusion, we recommend appropriate models and control criteria for various crane applications and suggest directions for further work.

Theoretical and experimental investigation of the fast- and slow-scale instabilities of a DC-DC converter

We use an exact formulation based on nonlinear maps to investigate both the fast-scale and slow-scale instabilities of a voltage-mode buck converter operating in the continuous conduction mode and its interaction with a filter. Comparing the results of the exact model with those of the averaged model shows the shortcomings of the latter in predicting fast-scale instabilities. We show the impact of parasitics on the onset of chaos using a high-frequency model. The experimentally validated theoretical results of this paper provide an improved understanding of the dynamics of the converter beyond the linear regime and this may lead to less conservative control design and newer applications.

A NEW APPROACH TO THE MODELING AND SIMULATION OF FLEXIBLE MICROSTRUCTURES UNDER THE EFFECT OF SQUEEZE-FILM DAMPING

We present a new approach to the modeling and simulation of flexible microstructures under the effect of squeeze-film damping. Our approach utilizes the compressible Reynolds equation coupled with the equation governing the plate deflection. The model accounts for the electrostatic forcing of the capacitor airgap, the restoring force of the microplate and the applied in-plane loads. It also accounts for the slip condition of the flow at very low pressures. Perturbation methods are used to derive an analytical expression for the pressure distribution in terms of the structural mode shapes. This expression is substituted into the plate equation, which is solved in turn using a finite-element method for the structural mode shapes, the pressure distributions, the natural frequencies and the quality factors. We apply the new approach to a variety of rectangular and circular plates and present the final expressions for the pressure distributions and quality factors. Our theoretically calculated quality factors are in excellent agreement with available experimental data and hence our methodology can be used to simulate accurately the dynamics of flexible microstructures and predict their quality factors under a wide range of gas pressures. Because the pressure distribution is related analytically to the deflection, the dimension of the coupled structural-fluidic problem and hence the number of global variables needed to describe the dynamics of the system is reduced considerably. Consequently, the new approach can be significant to the development of computationally efficient CAD tools for microelectromechanical systems.

Modal Interactions in Dynamical and Structural Systems

The authors review theoretical and experimental studies of the influence of modal interactions on the nonlinear response of harmonically excited structural and dynamical systems. In particular, they discuss the response of pendulums, ships, rings, shells, arches, beam structures, surface waves, and the similarities in the qualitative behavior of these systems. The systems are characterized by quadratic nonlinearities which may lead to two-to-one and combination autoparametric resonances. These resonances give rise to a coupling between the modes involved in the resonance leading to nonlinear periodic, quasi-periodic, and chaotic motions.

Nonlinear Interactions: Analytical, Computational, and Experimental Methods

Two-To-One Internal Resonance One-To-One Internal Resonance Three-To-One Internal Resonance Combination Resonances Systems with Widely Spaced Modes Multiple Internal Resonances Nonlinear Normal Modes Bibliography Subject Index.

Nonparallel stability of boundary‐layer flows

The spatial stability of two‐dimensional incompressible boundary‐layer flows is analyzed using the method of multiple scales. The analysis takes into account the streamwise variations of the mean flow, the disturbance amplitude, and the wavenumber. The theory is applied to the Blasius and the Falkner–Skan flows. For the Blasius flow, the nonparallel analytical results are in good agreement with the experimental data. The results show that the nonparallel effects increase as the pressure gradient decreases.

Dynamics of MEMS resonators under superharmonic and subharmonic excitations

We present an analysis and simulations for the dynamics of electrically actuated microbeams under secondary resonance excitations. The presented model and methodology enable simulation of the transient and steady-state dynamics of microbeams undergoing small or large motions. The microbeams are excited by a dc electrostatic force and an ac harmonic force with a frequency tuned near twice their fundamental natural frequencies (subharmonic excitation of order one-half) or half their fundamental natural frequencies (superharmonic excitation of order two). In the case of superharmonic excitation, we present results showing the effect of varying the dc bias, the damping and the ac excitation amplitude on the frequency–response curves. In the case of subharmonic excitation, we show that, once the subharmonic resonance is activated, all frequency–response curves reach pull-in, regardless of the magnitude of the ac forcing. We conclude that the quality factor has a limited influence on the frequency response in this case. This result and the fact that the frequency–response curves have very steep passband-to-stopband transitions make the combination of a dc voltage and a subharmonic excitation of order one-half a promising candidate for designing improved high-sensitive RF MEMS filters. For both excitation methods, we show that the dynamic pull-in instability can occur at an electric load much lower than a purely dc voltage and of the same order of magnitude as that in the case of primary-resonance excitation.