Samir A. Emam

Free vibration analysis of functionally graded size-dependent nanobeams

Abstract This paper presents free vibration analysis of functionally graded (FG) size-dependent nanobeams using finite element method. The size-dependent FG nanobeam is investigated on the basis of the nonlocal continuum model. The nonlocal elastic behavior is described by the differential constitutive model of Eringen, which enables the present model to become effective in the analysis and design of nanosensors and nanoactuators. The material properties of FG nanobeams are assumed to vary through the thickness according to a power law. The nanobeam is modeled according to Euler–Bernoulli beam theory and its equations of motion are derived using Hamilton’s principle. The finite element method is used to discretize the model and obtain a numerical approximation of the equation of motion. The model is validated by comparing the obtained results with benchmark results. Numerical results are presented to show the significance of the material distribution profile, nonlocal effect, and boundary conditions on the dynamic characteristics of nanobeams.

Mathematical modeling of gearbox including defects with experimental verification

A nine degree-of-freedom model of one stage gear system is presented in this research work. The gearbox structure is coupled with the vibration of the gear shaft. The model considers gear size, errors, and faults. The model includes varying meshing stiffness and a realistic representation of the gear transmission error (TE) and gear faults. Gear TE is modeled as a displacement excitation. The model equations are solved using Matlab and using parameters representing a real experimental gearbox rig. Experimental and simulated data are compared for different operating speeds, torque loads, and gear cracks. The simulation results are in good agreement with the experimental ones. The authors believe that the model presented here can be used in studying gear faults and would be very useful in developing gear fault monitoring techniques.

A Review on Bistable Composite Laminates for Morphing and Energy Harvesting

Bistable composite laminates have received a considerable attention due to their fabulous behavior and potential for morphing and energy harvesting. A bistable or multistable laminate is a type of composite structure that exhibits multiple stable static configurations. The characterization of unsymmetric fiber-reinforced laminated composite plates as a bistable structure is well established and quantitatively determined after about 30 years of research. As predicting cured shapes of unsymmetric composite laminates became well identified, attention was directed to the design of these structures for morphing applications. Bistable composite laminates have attracted researchers as a morphing structure because a bistable structure settles at one of its equilibrium positions without demanding continuous power to remain there. If the structure is triggered to leave an equilibrium position, it will snap or jump to the other equilibrium position. The snapthrough response is highly geometrically nonlinear. With the increased demand for broadband vibration energy harvesters, bistable composite laminates, which are able to gain large-amplitude vibrations in snapthrough motion, have recently attracted attention. This paper aims to summarize, review, and assess references and findings concerned with the response of bistable composite laminates for morphing and energy harvesting to date. It also highlights the remaining challenges and possible future research work as research in bistable composites transitions from phenomena to application.

Dynamics of a Flexible-hub Geometrically Nonlinear Beam with a Tip Mass

The dynamics of a flexible-hub geometrically nonlinear beam carrying a tip mass is presented. The hub-beam system is assumed to move in plane and the hub is restrained by a translational and a rotational spring. Hamilton’s principle is used to derive the equations of motion and their boundary conditions. A flexural model that takes into account the geometrical coupling between the axial and lateral deformations and ignores the axial deformation and its time derivatives is obtained. An exact solution for the natural frequencies and mode shapes of the free vibration problem is obtained. Using these mode shapes, a reduced-order model of the system is obtained using the Galerkin’s method. The dynamic response of the system using the present low-order model shows excellent agreement with the recent finite-element solutions available in the literature.

An Exact Solution for the Natural Frequencies of Flexible Beams Undergoing Overall Motions

We investigate the free vibrations of a flexible beam undergoing an overall two-dimensional motion. The beam is modeled using the Euler-Bernoulli beam theory. An exact solution for the natural frequencies and corresponding mode shapes of the beam is obtained. The model can be extended to beams undergoing three-dimensional motions.

A Generalized Model of Electrically Actuated, Microbeam-Based MEMS Devices

We present a model of electrically actuated microbeam-based MEMS devices incorporating the nonlinearities associated with moderately large displacements and electric forces. The model can handle any capacitor configuration disposing of the complete electrode-overlapping (parallel-plate theory) restriction. The boundary-value problem describing the static deflection of the microbeam under the electrostatic loading 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. Results generated by our model for the parallel-plate case are in agreement with published results. Our results show that the underlying assumptions of the closed-form formula of the parallelplate case underestimates the electrostatic force and leads to an overestimation of the pull-in voltage. The model provides an analytical tool to predict the static and dynamic response of any electrically actuated MEMS device based on clamped-clamped microbeams.Copyright

Postbuckling and Free Vibrations of Composite Beams

An exact solution for the postbuckling configurations of composite beams is presented. The equations governing the axial and transverse vibrations of a composite laminated beam accounting for the midplane stretching are presented. The inplane inertia and damping are neglected, and hence the two equations are reduced to a single equation governing the transverse vibrations. This equation is a nonlinear fourth-order partial-integral differential equation. We find that the governing equation for the postbuckling of a symmetric or antisymmetric composite beam has the same form as that of a metallic beam. A closed-form solution for the postbuckling configurations due to a given axial load beyond the critical buckling load is obtained. We followed Nayfeh, Anderson, and Kreider and exactly solved the linear vibration problem around the first buckled configuration to obtain the fundamental natural frequencies and their corresponding mode shapes using different fiber orientations. Characteristic curves showing variations of the maximum static deflection and the fundamental natural frequency of postbuckling vibrations with the applied axial load for a variety of fiber orientations are presented. We find out that the line-up orientation of the laminate strongly affects the static buckled configuration and the fundamental natural frequency. The ratio of the axial stiffness to the bending stiffness is a crucial parameter in the analysis. This parameter can be used to help design and optimize the composite beams behavior in the postbuckling domain.Copyright

On Identification of Leaky Pipeline Parameters via Monte Carlo Simulation

In this study, a numerical simulation using inverse pipeline analysis is conducted for the localization and quantification of leak using minimum number of measurement points of a leaky pipeline. The friction coefficient and leak cross section area were also identified by window marching technique using only the pressure head at one end of the pipeline and the flow rate measurements at the other end of the pipeline. The Monte Carlo simulation approach is applied to identify the pipeline parameters by minimizing an objective function that represents the mismatch between the measured and numerically modeled pipeline variables. The simulation results are able to identify and localize the leak location as well as the friction coefficient and leak across section area. The effect of additive measurement noise on the leak localization accuracy is also investigated.Copyright

A Static and Nonlinear Dynamic Analysis of Resonant Microbeams

An investigation into the response of microbeams to DC and AC electric actuation is presented. The beam is modeled according to the Euler-Bernoulli beam theory and small strains and moderate rotation approximations are assumed. The governing equation is a nonlinear integral-partial-differential equation in space and time. The model accounts for mid-plane stretching, applied axial load, DC electrostatic forces, and AC harmonic forces. A reduced-order model based on the Galerkin discretization technique is introduced to simulate the behavior of microswitches and resonant sensors. The static behavior of the microbeam under electrostatic forces is studied and compared to the results available in the literature. The dynamic behavior of resonant microbeams under AC harmonic forces is investigated. An analytical solution for the vibration modes and natural frequencies of the microbeam around its statically deflected position is obtained. A shooting method is used to numerically integrate the nonlinear discretized equations and obtain periodic orbits of the response. The stability of these periodic orbits is investigated using Floquet theory. The sensitivity of the device to small-amplitude excitations is also investigated.Copyright

AN INVESTIGATION INTO THE GLOBAL DYNAMICS OF A BUCKLED BEAM IN THE CASE OF A ONE-TO-ONE INTERNAL RESONANCE

We investigate the nonlinear vibrations of a clamped-clamped buckled beam in the case of a one-to-one internal resonance between the first and second modes when one of them is externally excited by a primary resonance. To examine whether these two modes are nonlinearly coupled, we use the method of multiple scales to directly attack the partial-differential equation and associated boundary conditions and obtain the equations governing the modulation of their amplitudes and phases. We find that the two modes are nonlinearly coupled. To investigate the large-amplitude dynamics, we use a multi-mode Galerkin discretization to reduce the partial-differential equation, in space and time, governing the nonlinear vibrations of the buckled beam into a set of nonlinearly coupled ordinary-differential equations in time only. We use a shooting method to compute periodic orbits of the discretized equations and Floquet theory to investigate the stability of these solution and their bifurcations. We report theoretically and experimentally an energy transfer from the first mode, which is externally excited by a primary resonance, to the second mode.