Free Vibration Mode Shapes Research Articles

Buckling and Free Vibration Characteristics of a Uniformly Heated Isotropic Cylindrical Panel

In this paper buckling and free vibration characteristics of an isotropic cylindrical panel subjected to uniform temperature rise has been investigated using finite element method. The procedure involves the determination of critical buckling temperature, which is followed by modal analysis considering pre-stress due to the thermal field in the cylindrical panel. Detailed studies are carried out to analyze the influence of curvature ratio, thickness ratio and aspect ratio on the critical buckling temperature and free vibration behavior of an isotropic cylindrical panel. It has been found that as the curvature ratio and the thickness ratio increases the thermal buckling strength of the cylindrical panel decreases. It has also been found that free vibration frequencies reduce with an increase in temperature and the reduction is more significant for the lowest frequency mode. It is observed that free vibration mode shapes at ambient temperature changes with an increase in temperature.

Development of a transfer matrix method to obtain exact solutions for the dynamic characteristics of a twisted uniform beam

In this study, a novel transfer matrix method is developed to determine the exact natural frequencies and mode shapes of free vibrations in a twisted uniform beam. It is possible by applying the distributed mass and transcendental function while using a local coordinate system in the derivation of the transfer matrix method. In addition, the computed results in the proposed method for a twisted uniform beam are independent of the number of subdivisions, but the accuracy of the conventional transfer matrix method for a twisted uniform beam depends on the number of subdivided elements. To validate the accuracy of the proposed method, the computed results are compared with those reported in the existing literature, and relevant conclusions are drawn.

Analysis of thickness-shear and thickness-twist modes of AT-cut quartz acoustic wave resonator and filter

The thickness-shear (TS) and thickness-twist (TT) vibrations of partially electroded AT-cut quartz plates for acoustic wave resonator and filter applications are theoretically studied. The plates have structural variations in one of the two in-plane directions of the plates only. The scalar differential equations derived by Tiersten and Smythe for electroded and unelectroded AT-cut quartz plates are used, resulting in free vibration resonant frequencies and mode shapes for both fundamental and overtone families of modes. The trapped modes with vibrations, mainly confined in the electroded areas, are found to exist in both the resonator and the filter structures. The numerical results for the trapped modes are presented for different aspect ratios of electrodes and material properties, providing a reference to the design and optimization of quartz acoustic wave resonators and filters.

Application of 3-D Laser Scanning Vibrometer in Determination of Free Vibration Frequencies of Composite Plates with Damage

The paper deals with experimental estimation of the frequencies of free vibrations for composite plates with damage. Two different glass-epoxy composite plates with damage were tested and their mechanical behavior was compared with their non-damaged counterparts. Dynamical properties of the tested structures were examined with the Laser Scanning Vibrometer. Finite Element Analysis (FEA) was performed simultaneously for free vibration frequencies and mode shapes. The numerical results agreed well with the experiment.

Vibration analysis of a tapered laminated thick composite plate with ply drop-offs

In this study, vibration characteristics of a tapered laminated thick composite plate have been investigated using finite element method by including the shear deformation and rotary inertia effects. The governing differential equations of motion of a tapered laminated thick composite plate are presented in the finite element formulation based on first-order shear deformation theory for three types of taper configurations. The effectiveness of the developed finite element formulation in identifying the various dynamic properties of a tapered laminated thick composite plate is demonstrated by comparing natural frequencies evaluated using the present FEM with those obtained from the experimental measurements and presented in the available literature. Various parametric studies are also performed to investigate the effect of taper configurations, aspect ratio, taper angle, angle ply orientation and boundary conditions on free and forced vibration responses of the structures. The comparison of the transverse free vibration mode shapes of the uniform and tapered composite plates under various boundary conditions is also presented. The forced vibration response of a composite plate is investigated to study the dynamic response of tapered composite plate under the harmonic force excitation in various tapered configurations. It is concluded that the dynamic properties of laminated thick composite plates could be tailored by dropping off the plies to yield various tapered composite plate.

Free vibration analysis for shells of revolution based on p-version mixed finite element formulation

In this study, an effective p-version two-node mixed finite element is newly presented for predicting the free vibration frequencies and mode shapes of isotropic shells of revolution. The present element considering shear strains is based on Reissner–Mindlin shear deformation shell theory and Hellinger–Reissner variational principle. To improve the accuracy and resolve the numerical difficulties due to the spurious constraints, field-consistent stress parameters are employed corresponding to displacement shape functions with high-order hierarchical shape functions. The elimination of stress parameters and the reduction of the nodeless degrees by the Guyan reduction yield the standard stiffness and mass matrix. Results of the proposed element are compared with analytical, experimental and numerical solutions found in the literature. We can confirm a very satisfactory numerical behavior of the present p-version mixed element.

A special case of parametrically excited systems: Free vibration of delaminated composite beams

The problem of free vibration of delaminated composite beams is revisited in this paper by showing the existence of parametric excitation, bifurcation and stability. Why and when the delaminated part buckles during the vibration? To answer this question, first a finite element model is developed for the uncracked beam portion using the theory of layered beams and the so-called system of exact kinematic conditions. Second, a transition element is developed to establish the kinematic connection and continuity among the delaminated and uncracked portions. Thirdly, the delaminated beam portions are modeled by simple layered beam elements. The coupling between the flexural and longitudinal vibration is taken into consideration in the delaminated part and therefore it is shown that this part is loaded by periodic normal forces during the free vibration of the system. The time dependent stiffness because of the normal forces does not influence the frequencies globally at all, however, locally delamination buckling can take place, as well. The bifurcation and stability of the delaminated parts are analyzed using the harmonic balance method and the critical forces in delaminated layered beams under dynamic stability are determined. The free vibration mode shapes in the state of instability are predicted based on the critical dynamic force and the condition of constant arc length of the delaminated parts. Finally the phase plane portraits of the motion of the midpoint in the delaminated region are presented giving a global view on the behavior of composite beams under harmonic vibration.

VIBRATION OF BONES: A CASE STUDY ON HUMAN FEMUR

The frequency analysis of bones is a new tool to assess bone quality or integrity to characterize osteoporosis. The modal analysis can also be used to determine failure characteristics of remodeled bone in the fractured model. This study describes the numerical characterization of the modal analysis of the standardized femur model. The objective of the numerical procedure is to identify the natural frequencies and mode shapes of an unconstrained femur. The vibration modes of the human femur are studied by digital modal analysis and finite element simulation using ANSYS version 10 programs, with respect to femur dimensions and mechanical properties. The changing of the values of free vibration natural frequencies and mode shapes of the femur due to changing of the femur densities are studied. The results are compared to those obtained experimentally. The comparison of the results shows a good agreement, which indicates that the used model can be utilized in vibration analysis of bones.

Vibration analysis of a partially treated laminated composite magnetorheological fluid sandwich plate

In this study, the vibration responses of a partially treated laminated composite magnetorheological (MR) fluid sandwich plate have been investigated. The governing differential equations of motion for a partially treated laminated sandwich plate embedding MR fluid and rubber as the core layer and the laminated composite plate as the face layers are presented in finite element formulation. The validity of the developed finite element formulation is demonstrated by comparing the results in terms of natural frequencies derived from the present finite element formulation with the experimental measurements. Various configurations of a partially treated laminated composite MR fluid sandwich plate are considered to study the effect of location and size of the MR fluid segment under various boundary conditions. The effect of magnetic field on the variation of natural frequencies and loss factor of the partially treated laminated composite MR fluid sandwich plate are also analyzed for various configurations at different boundary conditions. The free vibration mode shapes of various configurations of a partially treated laminated composite MR fluid sandwich plate are also presented. The forced vibration responses of the various configuration of a partially treated laminated composite MR fluid sandwich plate are also analyzed under harmonic force excitations. This analysis suggests that the natural frequency, loss factor and transverse displacements of the partially treated laminated composite MR fluid sandwich plate are strongly influenced by the location and size of the MR fluid segment apart from the intensities of the applied magnetic field. The application of the partial treatment alters the deflection pattern of the sandwich plate, particularly the location of peak deflection, which shows that it can be applied to critical components of a large structure to realize a more efficient and compact vibration control mechanism with variable damping.

Dynamic characterization of a laminated composite magnetorheological fluid sandwich plate

This study investigates the dynamic properties of a laminated composite magnetorheological (MR) fluid sandwich plate. The governing differential equations of motion of a sandwich plate embedding a MR fluid layer as the core layer and laminated composite plates as the face layers are presented in a finite element formulation. The validity of the developed finite element formulation is demonstrated by comparing the results in terms of the natural frequencies derived from the present finite element formulation with those in the available literature. Various parametric studies are also performed to investigate the effect of a magnetic field on the variation of the natural frequencies and loss factors of the MR fluid composite sandwich plate under various boundary conditions. Furthermore, the effect of the thickness of the MR fluid layer and the ply orientation of the composite face layers on the variation of the natural frequencies and loss factors are studied. The free vibration mode shapes under various boundary conditions of a MR fluid laminated composite sandwich plate are also presented. The forced vibration response of a MR fluid composite plate is investigated to study the dynamic response of the sandwich plate under harmonic force excitations in various magnetic fields. The study suggests that the natural frequency increases with increasing magnetic field, irrespective of the boundary conditions. The reduction in peak deflection at each mode under a harmonic excitation force with variation of the applied magnetic field shows the effectiveness of the MR fluid layer in reducing the vibration amplitude of the composite sandwich plate.

Free vibration analysis of functionally graded beams using an exact plane elasticity approach

Exact natural frequencies of functionally graded beams are determined using plane elasticity theory. The analysis yields infinitely many frequencies. For verification purposes, a comparison with the existing beam theory results is performed and a close agreement is observed for slender members. The elasticity solutions are general in the sense that they are valid for slender members as well as short and thick structural elements. Both flexural and axial free vibration mode shapes are presented for top and bottom surfaces and the effect of the beam thickness is discussed. The exact results presented herein can be used as benchmarks for future research of free vibration behavior of short and thick functionally graded material beams.

Free vibrations of a trapezoidal plate with an internal line hinge.

This paper deals with a general variational formulation for the determination of natural frequencies and mode shapes of free vibrations of laminated thin plates of trapezoidal shape with an internal line hinge restrained against rotation. The analysis was carried out by using the kinematics corresponding to the classical laminated plate theory (CLPT). The eigenvalue problem is obtained by employing a combination of the Ritz method and the Lagrange multipliers method. The domain of the plate is transformed into a rectangular domain in the computational space by using nonorthogonal triangular coordinates and the transverse displacements are approximated with a set of simple polynomials automatically generated and expressed in the triangular coordinates. The developed algorithm allows obtaining approximate analytical solutions for mentioned plate with different geometries, aspect ratio, position of the line hinge, and boundary conditions including translational and rotational elastically restrained edges. It allows studying the influence of the mentioned line on the vibration frequencies and respective mode shapes. The algorithm can easily be programmed and it is numerically stable. Additionally, as a particular case, the results of triangular plates can be easily generated.

2D Elastodynamic Solution for the Impact Response of Laminated Composites

This paper presents a general, exact, two-dimensional (2D) elastodynamic analysis of the response of laminated composite panels subjected to transverse impact loading under conditions of planar deformation. The natural frequencies and mode shapes of free vibration are first extracted. Inspired by a transformation technique for solving a special class of partial differential equations, the forced vibration problem of an impacted laminated panel is solved using an eigenfunction expansion technique. Several examples are studied by varying the laminate lay-up and length-to-thickness ratio. The distributions of transverse stresses in the through-the-thickness direction are further compared with two one-dimensional theories, classical lamination theory (CLT) and first-order shear deformation theory (FSDT), showing the inadequacy of these theories and the necessity to establish a benchmark solution for 2D elastodynamics. The 2D elastodynamic theory that is formulated is also applicable for studying other multilayered structures subjected to arbitrary loading profiles.

Bending, buckling and free vibration characteristics of FG-CNT-reinforced polymer composite beam under non-uniform thermal load

Bending, buckling and free vibration behaviors of functionally graded (FG) carbon nanotube (CNT)-reinforced polymer composite beam under different non-uniform thermal loads have been analyzed using finite element method. Extended rule of mixture is used to obtain effective material property of the composite. Four different types of FG beam exposed to four different assumed one-dimensional temperature distributions along the length of the beam are analyzed. Parameters studies are carried out to investigate influences of the volume fraction of the carbon nanotube, functional grading and the nature of temperature variation on bending, buckling and free vibration characteristics. It is found that bending deflection reduces with increase in volume fraction of the CNT except for unsymmetrical functional graded beam. The static bending deflection and deformed shape of the beams are significantly influenced by the nature of temperature field. The critical buckling temperature of the beam with symmetric CNT distribution (where CNTs concentration is far from the neutral axis) is greater than other beams under different temperature fields and its value is less when the beams are exposed to uniform temperature rise above ambient temperature compared to other non-uniform temperature variations. However, the critical buckling temperature is not increasing significantly with increase in volume fraction of the CNT. The fundamental buckling mode shape is not sensitive to the nature of temperature variation but bending amplitude of the buckling mode shape is significantly influenced by functional grading of CNT and volume fraction of the CNT. The natural frequency of the beams reduces significantly with increase in temperature and the free vibration mode shapes are not influenced by temperature rise, nature of temperature variation and volume fraction of the CNT.

A new Orthonormal Polynomial Series Expansion Method in vibration analysis of thin beams with non-uniform thickness

In this article, OPSEM (Orthonormal Polynomial Series Expansion Method) is developed as a new computational approach for the evaluation of thin beams of variable thickness transverse vibration. Capability of the OPSEM in assessing the free vibration frequencies and mode shapes of an Euler–Bernoulli beam with varying thickness is discussed. Multispan continuous beams with various classical boundary conditions are included. Contribution of BOPs (Basic Orthonormal Polynomials) in capturing the beam vibrations is also illustrated in numerical examples to give a quantitative measure of convergence rate. Furthermore, OPSEM is adopted for the forced vibration of a thin beam caused by a moving mass. Dynamics of beams supported by flexible elastic base like free to free beam on elastic foundation are also regarded. Verifications are made via eigenfunction expansion method and GMLSM (Generalized Moving Least Square Method). The very close observed agreement between the results of the two recently mentioned methods and that of OPSEM can be regarded as a guarantee of validity for the newly introduced technique. In comparison with eigenfunction expansion method, the simplicity and handiness of OPSEM in coping with different boundary conditions of the beam can be considered as its benefit for engineering practitioners.

A general Ritz formulation for the free vibration analysis of thick trapezoidal and triangular laminated plates resting on elastic supports

A general variational formulation for the determination of natural frequencies and mode shapes of free vibration of symmetric laminated plates of trapezoidal and triangular shapes is presented in this work. The kinematics corresponding to the first-order shear deformation plate theory (FSDT) is used to take into account the effects of shear deformation and rotational inertia in the analysis.The developed approach is based on the Ritz method and the plate geometry is approximated by non-orthogonal right triangular co-ordinates. The transverse deflection and two rotations of the laminate are independently approximated by sets of simple polynomials.The algorithm allows obtaining approximate analytical solutions for laminated plates with different shapes, aspect ratio, number of layers, stacking sequence, angle of fiber orientation and boundary conditions including translational and rotational elastically restrained edges. The algorithm is simple to program and numerically stable.

Energy trapping of thickness-shear and thickness-twist modes in a partially electroded AT-cut quartz resonator

The thickness-shear and thickness-twist vibrations of a finite and partially electroded AT-cut quartz resonator are investigated. The equations of anisotropic elasticity are used with the omission of the small elastic constant c56. An analytical solution is obtained using Fourier series from which the free vibration resonant frequencies, mode shapes, and energy trapping are calculated and examined.

CO2 effects of Antarctic stratification, North Atlantic Intermediate Water formation, and subantarctic nutrient drawdown during the last ice age: Diagnosis and synthesis in a geochemical box model

In this study, a novel transfer matrix method is developed to determine the exact natural frequencies and mode shapes of free vibrations in a twisted uniform beam. It is possible by applying the distributed mass and transcendental function while using a local coordinate system in the derivation of the transfer matrix method. In addition, the computed results in the proposed method for a twisted uniform beam are independent of the number of subdivisions, but the accuracy of the conventional transfer matrix method for a twisted uniform beam depends on the number of subdivided elements. To validate the accuracy of the proposed method, the computed results are compared with those reported in the existing literature, and relevant conclusions are drawn.

Vibration analyses of FGM plates with in-plane material inhomogeneity by Ritz method

In this study, functionally graded plates which the properties of material varying through the in-plane direction is considered. The analysis is based on a five-degree-of-freedom shear deformable plate theory with different boundary conditions. The vibration solutions are obtained using the Ritz method and assumed displacement functions are in the form of the Chebyshev polynomials. The material properties are assumed to vary as a power form of the in-plane direction. The convergence and comparison studies demonstrate the accuracy and correctness of the present method. Effects of the different material composition, the Poisson ratio and the plate geometry (side–side, side–thickness) on the free vibration frequencies and mode shapes are investigated.

Influence of Adhesive Dimensions on the Transverse Free Vibration of Single-Lap Adhesive Cantilevered Beams

This paper studies the influence of adhesive dimensions on the transverse free vibration of the single-lap adhesive cantilevered beams. The researches are performed by employing software ansys .Efficient analytic results of natural frequencies and mode shapes of transverse free vibration of the beams are provided, corresponding to different adhesive dimensions of bonded thicknesses and bondlines length. Bondlines length has more significant influence on the transverse natural frequencies and the lap joint’s mode shapes of the beams than bonded thickness. The transverse natural frequencies decrease with a decrease in the bondlines length of adhesive, but do not appear to variation observably with a decrease in the bonded thickness. Bondlines length shorting, the lap joint has a sharper mode shape. Simultaneously, the lap joint of even mode shapes influences the dynamic response of the beams significantly. These results indicate a local crack in adhesive layers because of the existence of stress concentration.