Large Amplitude Vibrations of Completely Free Rectangular Plates

2012 ◽  
Author(s):  
F. Alijani ◽  
M. Amabili
Keyword(s):  
Shear Deformation Theory ◽  
Energy Functional ◽  
Harmonic Excitation ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Geometric Imperfections ◽  
Natural Modes ◽  
Modes Of Vibration ◽  
Collocation Scheme ◽  
Large Amplitude Vibrations

Nonlinear forced vibrations of completely free rectangular plates are studied using multi-modal Lagrangian approach. Nonlinear higher-order shear deformation theory is used and the nonlinear response to transverse harmonic excitation in the frequency neighborhood of the fundamental mode is investigated. Geometric imperfections are taken into account. The analysis is carried out in two steps. First, the plate displacements and rotations are expanded in terms of Chebyshev polynomials and a linear analysis is conducted to obtain the natural frequencies and mode shapes. Then, the energy functional is discretized by using the natural modes of vibration and a system of nonlinear ordinary differential equations with cubic and quadratic nonlinear terms is obtained. A pseudo arc-length continuation and collocation scheme is employed to carry out a bifurcation analysis. The effect of number of modes retained in the approximation, thickness ratio and geometric imperfections on the trend of nonlinearity is discussed.

Large Amplitude Vibrations of Laminated Rectangular Plates With Free Edges

2013 ◽  
Author(s):  
F. Alijani ◽  
M. Amabili
Keyword(s):  
Shear Deformation Theory ◽  
Laminated Composite ◽  
Energy Functional ◽  
Harmonic Excitation ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Natural Modes ◽  
Modes Of Vibration ◽  
Collocation Scheme ◽  
Large Amplitude Vibrations

Nonlinear forced vibrations of completely free laminated composite rectangular plates are studied using multi-modal Lagrangian approach. Nonlinear higher-order shear deformation theory is used and the nonlinear response to transverse harmonic excitation in the frequency neighborhood of the fundamental mode is investigated. The numerical analysis is conducted in two steps. First, the plate displacements and rotations are expanded in terms of Chebyshev polynomials and a linear analysis is performed to obtain the natural frequencies and mode shapes. Then, the energy functional is discretized by using the natural modes of vibration and a system of nonlinear ordinary differential equations with cubic and quadratic nonlinear terms is obtained. A pseudo arc-length continuation and collocation scheme is employed to carry out a bifurcation analysis. A consistent reduced-order model necessary to capture the nonlinear dynamics of the plate is developed and the effect of number of modes retained in the numerical model is discussed.

A new semi-analytical solution method for free vibration analysis of composite rectangular plates with general edge constraints coupled with single piezoelectric layer

2018 ◽  
Vol 29 (20) ◽  
pp. 3873-3889 ◽  
Author(s):  
Mehdi Baghaee ◽  
Amin Farrokhabadi ◽  
Ramazan-Ali Jafari-Talookolaei
Keyword(s):  
Boundary Conditions ◽  
Piezoelectric Layer ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Composite Plates ◽  
Energy Functional ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Generalized Displacements

In this article, a new approach is presented to study the free vibrations of rectangular composite plates coupled with single piezoelectric layer. The laminated plate with general stacking sequences is subjected to the elastic edge restraints. Based on the first-order shear deformation theory and Hamilton’s principle, the equations of the motion along with boundary conditions of the problem are deduced. To solve the problem, generalized displacements as well as general electric potentials are expanded using the Legendre polynomial series as the base functions. Then, the kinetic and potential energies of the problem are obtained. Afterwards, by means of Lagrange multipliers all the boundary conditions have been added to the energies to form the functional. This energy functional is extremised to get the natural frequencies and mode shapes of the problem through generalized eigenvalue problem. Credibility of the proposed method is verified by comparing the obtained results with those achieved by other theories and finite element method.

scholarly journals Dynamic testing of a modern concrete bridge

1979 ◽  
Vol 6 (3) ◽  
pp. 447-455 ◽  
Author(s):  
J. H. Rainer ◽  
G. Pernica
Keyword(s):  
Natural Frequencies ◽  
Dynamic Testing ◽  
Dynamic Properties ◽  
Mode Shapes ◽  
Total Width ◽  
Concrete Bridge ◽  
Reinforced Concrete Bridge ◽  
Natural Modes ◽  
Modes Of Vibration ◽  
Good Agreement

A posttensioned reinforced concrete bridge, slated for demolition, was tested to obtain its dynamic properties. The 10 year old bridge consisted of a continuous flat slab deck of variable thickness having a total width of 103 ft (31.39 m) and spans of 28 ft 6 in. (8.69 m), 71 ft 0 in. (21.64 m), and 42 ft 6 in. (12.95 m). The entire bridge was skewed 10°50′ and the deck was slightly curved in plan.The mode shapes, natural frequencies, and damping ratios for the lowest five natural modes of vibration were determined using sinusoidal forcing functions from an electrohydraulic shaker. These modes, located at 5.7, 6.4, 8.7, 12.0, and 17.4 Hz, were found to be highly dependent on the lateral properties of the bridge deck. Damping ratios were determined from the widths of resonance peaks. The modal properties from the steady state excitation were compared with those obtained from measurements of traffic-induced vibrations and good agreement was found between the two methods.

scholarly journals Boundary Characteristic Orthogonal Polynomials in the Study of Transverse Vibrations of Nonhomogeneous Rectangular Plates with Bilinear Thickness Variation

2012 ◽  
Vol 19 (3) ◽  
pp. 349-364 ◽  
Author(s):  
R. Lal ◽  
Yajuvindra Kumar
Keyword(s):  
Orthogonal Polynomials ◽  
Ritz Method ◽  
Cartesian Product ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Rectangular Plates ◽  
Transverse Vibrations ◽  
Modes Of Vibration ◽  
Boundary Characteristic

The free transverse vibrations of thin nonhomogeneous rectangular plates of variable thickness have been studied using boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method. Gram-Schmidt process has been used to generate these orthogonal polynomials in two variables. The thickness variation is bidirectional and is the cartesian product of linear variations along two concurrent edges of the plate. The nonhomogeneity of the plate is assumed to arise due to linear variations in Young's modulus and density of the plate material with the in-plane coordinates. Numerical results have been computed for four different combinations of clamped, simply supported and free edges. Effect of the nonhomogeneity and thickness variation with varying values of aspect ratio on the natural frequencies of vibration is illustrated for the first three modes of vibration. Three dimensional mode shapes for all the four boundary conditions have been presented. A comparison of results with those available in the literature has been made.

Thermal Effects on Geometrically Nonlinear Vibrations of Rectangular Plates With Fixed Ends

2009 ◽  
Author(s):  
M. Amabili ◽  
S. Carra
Keyword(s):  
Stainless Steel ◽  
Coefficient Of Thermal Expansion ◽  
Harmonic Excitation ◽  
304 Stainless Steel ◽  
Effect Of Temperature ◽  
Thermal Loads ◽  
Rectangular Plates ◽  
Aisi 304 ◽  
Temperature Changes ◽  
Large Amplitude Vibrations

Large-amplitude vibrations of rectangular plates subjected to harmonic excitation and thermal loads are investigated. The von Ka´rma´n nonlinear strain-displacement relationships are used to describe the geometric nonlinearity and the effect of thermal loads are included. Calculations are performed for a AISI 304 stainless-steel thin plate which has been tested experimentally. The plate is fixed to a AISI 410 stainless-steel frame which has a different coefficient of thermal expansion. Both calculations and experiments show a very large effect of temperature changes of the order of one degree Celsius on the nonlinear results.

Modal Characteristics of Swept Cantilevered Plates

Volume 2 ◽  
2004 ◽  
Author(s):  
Naveed A. Din ◽  
S. Olutunde Oyadiji
Keyword(s):  
Natural Frequencies ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Modal Data ◽  
Nodal Lines ◽  
Modes Of Vibration ◽  
Cantilevered Plates ◽  
Definition Of ◽  
Data Frequency

The aim of this paper is to produce modal data which can be used to synthesise assumed shapes for use in Rayleigh-Ritz approximations of the free vibrations of cantilevered swept plates. The modal data is generated via the use of the FEA technique to predict the natural frequencies and mode shapes of aluminium alloy plates of aspect ratio 2.0 and of swept angles varying from 0° to 20° in steps of 2°. The first fifty natural frequencies and mode shapes of swept cantilevered plates were calculated using ABAQUS FE programme which includes the ABAQUS/CAE pre-and post-processor. To classify each mode shape, the number of nodal lines i along the x-axis and the number of nodal lines j along the y-axis were defined. This definition worked fine for uniform rectangular plates of zero swept angles and also for the first few modes of the swept plates. But as the number of modes of the swept plates increased, this definition became difficult to apply. Similar mode shapes of the various swept angles were put into families and groups headed by the i and j definition of the uniform rectangular plate design. From this modal data, frequency charts, which showed the variation of the dimensionless natural frequencies of the swept plates with swept angle, were constructed. These charts can be used to deduce the types of modes of vibration, whether bending or torsion, of a vibrating swept plate and to synthesis accurately the assumed shapes for use in the prediction of the vibration characteristics of swept plates using the Rayleigh-Ritz approach.

Mode shapes and frequencies of thin rectangular plates with arbitrarily varying non-homogeneity along two concurrent edges

2016 ◽  
Vol 23 (17) ◽  
pp. 2841-2865 ◽  
Author(s):  
Roshan Lal ◽  
Renu Saini
Keyword(s):  
Eigenvalue Problems ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Transverse Vibrations ◽  
Modes Of Vibration ◽  
Generalized Differential

Analysis and numerical results are presented for free transverse vibrations of isotropic rectangular plates having arbitrarily varying non-homogeneity with the in-plane coordinates along the two concurrent edges on the basis of Kirchhoff plate theory. For the non-homogeneity, a general type of variation for Young’s modulus and density of the plate material has been assumed. Generalized differential quadrature method has been used to obtain the eigenvalue problem for such model of plates for four different combinations of boundary conditions at the edges namely, (i) fully clamped, (ii) two opposite edges are clamped and other two are simply supported, (iii) two opposite edges are clamped and other two are free, and (iv) two opposite edges are simply supported and other two are free. By solving these eigenvalue problems using software MATLAB, the lowest three eigenvalues have been reported as the first three natural frequencies for the first three modes of vibration. The effect of various plate parameters on the vibration characteristics has been analysed. Three dimensional mode shapes have been plotted. A comparison of results with those available in literature has been presented.

Robot Manipulator Trajectory Synthesis for Minimal Vibrational Excitation

1997 ◽  
Author(s):  
W. Kim ◽  
J. Rastegar
Keyword(s):  
Robot Manipulator ◽  
Vibrational Excitation ◽  
High Harmonic ◽  
Harmonic Excitation ◽  
Tracking Accuracy ◽  
Natural Modes ◽  
Starting Point ◽  
Modes Of Vibration ◽  
Operating Space ◽  
Harmonic Components

Abstract As a robot manipulator is forced to track a given trajectory, the required actuating torques (forces) may excite the natural modes of vibration of the system. Due to their nonlinear dynamics, internally and externally induced high harmonic excitation torques are generally generated even though such harmonics have been eliminated from the synthesized trajectories and filtered from the drive inputs. It is therefore desirable to synthesize trajectories such that the actuating torques required to realize them do not contain higher harmonic components with significant amplitudes. In this paper, a systematic method is presented for synthesizing such trajectories. With such trajectories, a robot manipulator can operate at higher speeds and achieve higher tracking accuracy with suppressed residual vibration. It is shown that in general and for a given starting point, such trajectories can only be synthesized to a portion of the operating space of the manipulator. The method is developed based on the Trajectory Pattern Method (TPM). The application of the method to optimal trajectory synthesis for a plane 2R manipulator is presented.

Effects of Geometric Imperfections on Large-Amplitude Vibrations of Rectangular Plates With Hysteresis Damping

1984 ◽  
Vol 51 (1) ◽  
pp. 216-220 ◽  
Author(s):  
David Hui
Keyword(s):  
Large Amplitude ◽  
Plate Thickness ◽  
Rectangular Plates ◽  
Structural Damping ◽  
Geometric Imperfections ◽  
Geometric Imperfection ◽  
Forcing Function ◽  
Initial Geometric Imperfections ◽  
Spatial Shape ◽  
Large Amplitude Vibrations

The present paper deals with the effects of initial geometric imperfections on large-amplitude vibrations of simply supported rectangular plates. The vibration mode, the geometric imperfection, and the forcing function are taken to be of the same spatial shape. It is found that geometric imperfections of the order of a fraction of the plate thickness may significantly raise the free linear vibration frequencies. Furthermore, contrary to the commonly accepted theory that large-amplitude vibrations of plates are of the hardening type, the presence of small geometric imperfections may cause the plate to exhibit a soft-spring behavior. The effects of hysteresis (structural) damping on the vibration amplitude are also examined.

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