Vibrational Characteristics of Advanced Composite Laminates With a Hole.

Two simple finite element methods for Reissner–Mindlin plates with clamped boundary condition

In this paper, the large amplitude vibrations of clamped-clamped thin viscoelastic rectangular plates due to a concentrated transversal harmonic load are investigated both theoretically and experimentally. Clamped boundary condition on all edges and von Kármán nonlinear strain-displacement relationships are considered while rotary inertia, geometric imperfections, and shear deformation are neglected. In the theoretical study, the viscoelastic behaviour of the material is modelled using the Kelvin-Voigt model. In-plane loads applied during the assembly of the plate are taken into account and clamped boundary conditions are modelled using artificial rotational springs. The nonlinear ordinary differential equations for the considered Kelvin-Voigt model are obtained using the generalized energy approach. These equations contain quadratic and cubic nonlinear viscoelasticity terms in addition to quadratic and cubic stiffness terms. Non-dimensionalization of variables is carried out and each second order equation is converted into two first order equations. The resulting system of equations is solved using AUTO (software based on the arclength continuation method that allows bifurcation analysis), to get the frequency-response curves at various force levels. Moreover numerical time integration of equations was also performed using the fourth-order Runge-Kutta method to understand the time response of the structure. In the experimental study, two rubber plates with different material and thicknesses were considered; a silicone plate with 0.0015 m thickness and a neoprene plate with 0.003 m thickness. The plates were fixed on a heavy rectangular metal frame thereby ensuring the clamped boundary condition on all edges. Linear experimental modal analysis was carried out as a first step to estimate the mode shapes and natural frequencies. In the second step, the nonlinear vibration response of the plate around its first resonance was measured at various harmonic force levels. At each force level, the amplitude of the harmonic excitation was kept constant by LMS Data Acquisition System and Test.Lab Stepped Sine software module while slowly varying the frequency of excitation to get the frequency-response curves. Laser Doppler Vibrometry was used to measure the response from the plate as it eliminates the possible mass loading effect introduced by any contact type sensors. A maximum amplitude of more than three times the thickness of the plate was achieved. The nonlinear response curves showed a typical hardening type nonlinearity along with sudden jumps as expected for plates. Experimental frequency-response curves were compared with theoretical results and a good agreement was found. The influence of nonlinear viscoelastic damping terms was clearly noticed on the response curves of the plate. The retardation time, measured in seconds decreases with increasing excitation force and larger amplitude vibrations.

In this paper, an analysis of free vibration in functionally graded nanoplate is presented. Third-order shear deformation plate theory is used to reach more accuracy in results. Small-scale effects are investigated using Eringen`s nonlocal theory. The governing equations of motion are obtained by Hamilton`s principle. It is assumed that the properties of nanoplates vary through their thicknesses according to a volume fraction power law distribution. The finite element method (FEM) is presented to model the functionally graded nanoplate and solve mathematical equations accurately. The finite element formulation for HSDT nanoplate is also presented. Natural frequencies of FG nanoplate with various boundary conditions are compared with available results in the literature. At the end some numerical results are presented to evaluate the influence of different parameters, such as power law index, nonlocal parameter, aspect ratio and aspect of length to thickness of nanoplate. In addition, all combinations of simply supported and clamped boundary conditions are considered.

This paper investigates the nonlinear coupled radial-axial vibration of single-walled carbon nanotubes (SWCNTs) based on numerical methods. Two coupled partial differential equations that govern the nonlinear coupled radial-axial vibration for such nanotube are derived using nonlocal doublet mechanics (DM) theory. To obtain the nonlinear natural frequencies in coupled radial-axial vibration mode, these equations are solved using Homotopy perturbation method (HPM). It is found that the coupled radial-axial vibrational frequencies are complicated due to coupling between two vibration modes. The influences of some commonly used boundary conditions, changes in vibration modes and variations of the nanotubes geometrical parameters on the nonlinear coupled radial-axial vibration characteristics of SWCNTs are discussed. It was shown that boundary conditions and maximum vibration velocity play significant roles in the nonlinear coupled radial-axial vibration response of SWCNTs. It was shown that unlike the linear one, the nonlinear natural frequencies are dependent to maximum vibration velocity. Increasing the maximum vibration velocity increases the natural frequency of vibration compared to the prediction of the linear model. However, with increase in tube length, the effect of the maximum vibration velocity on the natural frequencies decreases. It was also shown that the amount and variation of nonlinear natural frequencies are more apparent in higher vibration modes and two clamped boundary conditions. To show the accuracy and capability of this method, the results obtained herein are compared with the fourth order Runge-Kuta numerical results and also with the other available results and good agreement is observed. It is notable that the results generated herein are new and can be served as a benchmark for future works.

Waves caused by steady ship motion are analyzed by the finite element method on the basis of the concept of the superposition of analytical and finite element solutions. The radiation condition is substituted by assuming appropriate radiating waves. Wave motions in a certain domain Ωa around the ship is assumed as a linear combination of the radiating waves and certain finite element solutions.The amplitude of the radiating waves can be determined by the condition of continuity on the surface of the domain Ωa, which can be expressed as the form of a single collocation equation.The finite element formulation can be developed by the same way as linear elastic problems, and its analysis can be performed easily by the use of general-purpose-programs for structural analysis. Numerical results obtained by the present method is compared with those by the previous authors, and they show good agreement.

In this paper, we discuss the dynamics of beams with multiple delaminations. The objective of the study is to use the dynamic response of the delaminated beams in health monitoring of composite structures. As a first step, we present experimental results on the vibrations of a cantilever beam with multiple delaminations. These experiments are conducted by exciting the beam with piezoelectric actuators. The response is detected by using a high-speed video camera to study the existence of delamination opening modes in a beam with multiple delaminations. Then, an analysis of natural frequencies and mode shapes, for a delaminated beam, is presented by using analytical techniques. The measured dynamic responses of the beam with multiple delaminations are also presented. The experimentally detected nonlinear response is discussed. Introduction Delaminations or interlayer cracks and impact damages are typical defects observed in composite structures. The presence of these defects changes the dynamic characteristics of the structure . Some of the specific changes in the observed dynamic characteristics include changes in natural frequencies, changes in the mode shapes and changes in damping ratios. These changes can be used to identify the existence, location and magnitude of a delamination or an impact damage, before they can grow to their critical sizes. Studies, to date, are concerned with the dynamics of beams and plates with a single delamination. In this paper, we specifically discuss the structural dynamics of a beam with multiple delaminations. This information can be used to develop health-monitoring techniques for composite beams with multiple delaminations. In an earlier paper, Luo and Hanagud studied the effects of defects, like saw-cuts, impact damage and delamination, on the structural dynamic characteristics of composite beams. They have shown that perturbations in natural frequencies and modes of a beam, due to defects, can be used for health monitoring of structures. In particular, it has been shown that an integral equation that considers perturbations in curvature modes and natural frequencies can be very effectively used to detect local defects. To distinguish delaminations from other types of monitored (identified) defects Luo and Hanagud have discussed nonlinear effects like superharmonic response in the dynamic response of a harmonically excited delaminated beam. They considered only a single delamination. In this paper we study beams with multiple delaminations. More specifically, we discuss dynamic response and the changes in natural frequencies, mode shapes due to multiple delaminations in a composite beam. One of the earliest models for vibration analysis of composite beams with delaminations was proposed by Ramkumar et al. This model simply used four Timoshenko beams connected at the delamination edges to model a composite beam with one throughwidth delamination. The predicted frequencies, based on this model, were consistently lower than the results of experimental measurements. Wang et al. improved the analytical solution by including coupling between flexural and axial vibrations of the delaminated sublaminates. Using an isotropic beam with splits and the classical beam model, they found that the calculated natural frequencies were closer to experimental results. With similar considerations, Nagesh Gummadi and Hanagud formulated a finite element solution for arbitrary composite beams. In the finite element models, they considered a classical composite beam model as well as the beam model with high order shear deformations. Mujumdar and Suryanarayan have proposed a model which imposed a constraint between the delaminated sublaminates to force them to have the same flexural displacement. This model was unable to * Ph.D. Student, School of Aerospace Engineering † Professor, School of Aerospace Engineering, AIAA Member Copyright © 1999 The American Institute of Aeronautics and Astronautics Inc. All rights reserved

Effect of geometric nonlinearity on the free flexural vibrations of moderately thick rectangular plates

The present study deals with both experimental and numerical investigation on the free vibration behavior of laminated composite plates subjected to varying temperature and moisture. A simple laminated plate model is developed for the vibration of composite plates subjected to hygrothermal loading. A computer program based on finite element method in the MATLAB environment is developed to perform all necessary computations. The woven fiber glass/epoxy specimens were hygrothermally conditioned in a humidity cabinet where the conditions were maintained at temperatures of 300–425 K and relative humidity ranging from 0% to 1.0% for moisture concentrations. Quantitative results are presented to show the effects of geometry, material and lamination parameters of woven fiber laminate on the free vibration of composite plates for different temperature and moisture concentrations. The numerical and experimental results show that there is reduction in natural frequency in Hz with the increase in temperature and moisture concentration for laminates with simply supported and clamped boundary conditions with different aspect ratio and side-to-thickness ratios.

Free vibration and stability analysis of piezolaminated plates is presented using the finite element method based on higher order shear deformation theory. The distribution of electric potential is assumed as linear through the thickness for each piezoelectric layer. The variation of the temperature field is neglected for the orthotropic layers of the laminate and for the piezoelectric layer. An eight node quadratic isoparametric quad element is employed in the finite element formulation. Numerous parametric studies are conducted for free vibration and stability analysis of a piezolaminated plate with different electric conditions of the piezolayer. The emphasis is on examining the influence of boundary conditions, ply orientation and plate aspect ratio on the free vibration and stability analysis of simply supported piezoelectric laminated plates. The results are verified with other numerical solutions available in the literature and are found to be in good agreement.

<p>The Dynamic Finite Element (DFE) theory is employed to calculate the natural frequencies and mode shapes of three-layered sandwich beams. Three formulations are developed and tested using a number of different numerical cases. The first two theories pertain to the analysis of straight beams while the third one is developed for the analysis of curved beams. The results from all three derivations agree well with published data that uses Classical Finite Element (FEM) theory and the Dynamic Stiffness Method (DSM) in the analysis of the free vibration behaviour of sandwich beams. For all test cases, the DFE results agree well with the published results as well as the FEM developed as an additional benchmark for the DFE. The results of the research are very encouraging and demonstrate that DFE is a valuable structural analysis tool that can be used in conjunction with FEM and DSM. </p>

This paper presents the formulation of a finite element model and vibration frequency analysis of a fluid filled pipe having variable cross sections. The finite element method with consideration of Coriolis force developed in [1] was adopted for frequency analysis of a pipe in this study. The stiffness matrix, the c-matrix (Coriolis force) and mass (for dynamic analysis) matrix that contain all parameters of the fluids properties and flow conditions have been developed. The numerical model was employed to simulate the dynamic performance of the piping system with the specific configurations for an application. A critical relationship between the natural frequencies and pipe geometry has been established. The results of frequencies analysis of the piping system gave us an insight whether a resonance frequency might occur.

A simple and novel finite element (FE) formulation is proposed to study the thermal post-buckling of composite and FGM columns with axially immovable ends and operating in severe thermal environment. A linear eigenvalue analysis gives the critical buckling temperature but practically the buckled columns can withstand additional thermal load beyond critical temperature, which can be obtained using von-Karman geometric nonlinearity, applicable for moderately large deflections. In the present study, the solution of the non-linear post-buckling problem is obtained by treating it as a linear eigenvalue problem using the concept of effective stiffness. Here, the total degrees of freedom (dof) of the discretized column are reduced and the post-buckling load is obtained without the need for iterative analysis. Comparison of the numerical results obtained from this FE formulation is in very good agreement with those obtained from the earlier FE formulations.

Based on the Bernoulli–Euler theory, the vibration characteristics of a clamped-elastic pinned beam under the elastic constraint and the compressive axial loads are derived and verified by finite element method. The results of examples show that the natural frequency of the beam decreases with loads increase and the frequency increases with the increase of the elastic constraint stiffness. When the constraint stiffness increases from 104 to 108 N/m, the first-order natural frequency becomes 4.24 times, the second-order natural frequency becomes 2.19 times, and the third-order natural frequency becomes 1.57 times; when the constraint is weak, the loads change is mainly reflected in the first mode shape. When the constraint stiffness is 104 N/m, the first-order natural frequency decreases by 20%, the second-order modal natural frequency changes by 3%, and the third-order natural frequency changes by less than 1%. The range of the elastic constraints with significant changes in the natural frequencies of the higher-order modes is larger. When the first-order frequency is taken to 100EI/l3, the change tends to be flat, the second-order frequency is about five times that of the first-order, and the range of the third-order frequency is 15 times or more.

In modern aerospace technology, strapdown inertial navigation systems (SINS) are widely used, using fiber-optic (FOG) or ring laser (CLG) gyroscopes. During the operation of such systems, the sensitivity axes are rotated relative to the basic coordinate system. The resulting angles between the axes of the base coordinate system and the axes of sensitivity of the navigation system (non-orthogonality) are one of the factors leading to an increase in the measurement errors of the device, which affects the measurement accuracy. During operation, the system is affected by vibrations of various nature, the impact of which can contribute to the appearance of non-orthogonality. The purpose of this work is to determine the maximum permissible vibration amplitudes affecting the SINS body according to the permissible values ​​of the deviation of the FOG sensitivity axes for two variants of the SINS layout. An approach to determining the permissible amplitudes of an external harmonic impact on the unit of a strapdown inertial navigation system based on fiber-optic or ring laser gyroscopes is considered. A design scheme, mathematical and finite element models for calculating natural frequencies and forced oscillations of a strapdown inertial navigation system unit have been developed. In various frequency ranges, numerical calculations have determined the boundary values ​​of the amplitudes of the external harmonic impact on the base of specific configurations of the SINS assembly. It has been established that dangerous states take place in the region of the 1st natural frequency of the system, as well as near higher frequencies. Comparison of the results for design options 1 and 2 allows us to conclude that in order to weaken the effect of vibrations on the accuracy of the SINS unit, it is advisable that the lowest natural vibration frequencies for the SINS assembly be as high as possible (more than 1000 Hz).
 Key words: vibration; fiber optic gyroscope; strapdown inertial navigation system; finite element method; natural frequencies and modes of vibration.

This paper deals with the study of the flapwise bending vibration and dynamic stability of rotating functionally graded material plates in thermal environments. A finite element formulation is derived for modal and dynamic stability analyses of rotating functionally graded material plates using first-order shear deformation theory. Temperature-dependent material properties of the plates are considered in the analysis and a simple power law is assumed for composition of constituent materials to vary along the thickness direction. The same power law is also proposed in thermal environments for temperature variation across the thickness of the plate. Some numerical results obtained from the present method are compared with numerical results available in the literature and are found to be in good agreement. Parametric investigation is carried out thoroughly to study the effect of the temperature rise, hub radius, and rotational speed on vibration and the dynamic stability of rotating plate in thermal environment. Bolotin’s method is used to generate the boundaries of stability and instability regions. These instability regions are plotted in the parameter space with the nondimensional dynamic load and excitation frequency. It is observed that the natural frequencies reduce with an increase in temperature rise. Increase in rotational speed and hub radius results in increase of natural frequencies of vibration. The rise in temperature leads to reduction in the dynamic stability of plate. Increase in rotational speed and hub radius enhances the dynamic stability of the rotating plate.