Free Vibration Modes Research Articles

NUMERICAL ANALYSIS OF ROTATIONALLY SYMMETRIC SHELLS BY THE MODAL SUPERPOSITION METHOD

The system of differential equations governing the analysis of rotationally symmetric shells under time-dependent or static surface loadings is formulated with the transverse, meridional, and circumferential displacements as the dependent variables. The thickness of the shell may vary, and four homogeneous boundary conditions may be prescribed at each boundary edge of the shell. The governing differential equations for each Fourier harmonic are obtained by use of Fourier series in the circumferential direction of the shell. Influence coefficients at each of the node points along the shell meridian are obtained for each Fourier component by employing ordinary finite difference representations for the meridional co-ordinate derivatives. With these influence coefficients, a set of homogeneous flexibility equations governing the free vibration characteristics of the shell is obtained and solved for the frequencies and mode shapes for each Fourier component. The solutions under time-dependent or static surface loadings and due to initial displacements and velocities are then obtained by expanding the solutions in terms of the modes of free vibration of the shell for each Fourier component. The solution for the total shell response is obtained by Fourier series summation. Solutions for typical shells have been found to be in excellent agreement with solutions by the method of temporal and spatial finite differences. Solutions for a parabolic shell are presented as an example. The solution is also presented for a published cylindrical shell example and is seen to be in excellent agreement with the published finite element method results.

Three-dimensional vibrations of thick spherical shell segments with variable thickness

A three-dimensional method of analysis is presented for determining the free vibration frequencies and mode shapes of spherical shell segments with variable thickness. Displacement components u φ , u z , and u θ in the meridional, normal, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the φ and z directions. Potential (strain) and kinetic energies of the spherical shell segment are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Novel numerical results are presented for thick spherical shell segments with constant or linearly varying thickness and completely free boundaries. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the spherical shell segments. The method is applicable to thin spherical shell segments, as well as thick and very thick ones.

Dynamic analysis of viscoelastic sandwich bems

Dynamic analysis of sandwich beams is performed. The external layers are modeled as beams on the basis of the Timoshenko model, and the internal layer possesses the characteristics of Winkler's viscoelastic one-directional base. The free vibrations are described by a homogeneous system of conjugate partial differential equations. After separation of variables in the system of differential equations, the boundary problem is solved and four complex sequences—a frequency sequence and a sequence of free-vibration modes—are obtained. The orthogonality of the complex modes of the free vibration is demonstrated. The free-vibration problem is solved under arbitrary initial conditions. The results of computations conducted for different sandwich beams with different parameters are presented.

Free vibration of the axially loaded system of two beams connected by a two-directional viscoelastic interlayer

In this paper, the analytical method has been used for solving a problem of free vibration with clamping of an axially loaded sandwich beam. The sandwich beam consists of two external layers connected by a viscoelastic two-directional Winkler interlayer. The upper external layer is described by the Bernoulli–Euler model, which is loaded by a constant axial force. The lower external layer is modeled as the Timoshenko model. The phenomenon of free vibration has been described using a homogenous system of conjugate partial differential equations. After separation of variables in the system of differential equations, the boundary problem has been solved and three complex equations for the definition of frequency and modes of free vibration have been obtained. The free-vibration problem for arbitrarily assumed initial conditions and various axial forces has been considered.

A Highly Accurate and Efficient Analytical Approach to Bridge Deck Free Vibration Analysis

The superposition method is employed to obtain an accurate analytical type solution for the free vibration frequencies and mode shapes of multi‐span bridge decks. Free edge conditions are imposed on the long edges running in the direction of the deck. Inter‐span support is of the simple (knife‐edge) type. The analysis is valid regardless of the number of spans or their individual lengths. Exact agreement is found when computed results are compared with known eigenvalues for bridge decks with all spans of equal length. Mode shapes and eigenvalues are presented for typical bridge decks of three and four span lengths. In each case torsional and non‐torsional modes are studied.

Three-Dimensional Vibration Analysis of Thick Shells of Revolution

A 3D method of analysis is presented for determining the free vibration frequencies and mode shapes of hollow bodies of revolution (i.e., thick shells), not limited to straight-line generators or constant thickness. The middle surface of the shell may have arbitrary curvatures, and the wall thickness may vary arbitrarily. Displacement components u;gf, uz, and u;gu in the meridional, normal, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the ϕ and z directions. Potential (strain) and kinetic energies of the entire body are formulated, and upper-bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials are increased, frequencies converge to the exact values. Novel numerical results are presented for two types of thick conical shells and thick spherical shell segments having linear thickness variations and completely free boundaries. Convergence to four-digit exactitude is demonstrated for the first five frequencies of both types of shells. The method is applicable to thin shells, as well as thick and very thick ones.

Three-dimensional vibrations of hollow cones and cylinders with linear thickness variations

A three-dimensional method of analysis is presented for determining the free vibration frequencies and mode shapes of hollow cones and cylinders with variable thickness. Displacement components us, uz, and uθ in the meridional, normal, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the s and z directions. Potential (strain) and kinetic energies of the cones and cylinders are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Novel numerical results are presented for thick, linearly tapered, hollow cones and cylinders with completely free boundaries. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the cones and cylinders. The method is applicable to thin cones and cylinders, as well as thick and very thick ones.

Experimental study of dynamic buckling of plates under fluid–solid slamming

This paper describes the experimental investigation of the elastic–plastic dynamic buckling properties of rectangular plates under in-plane fluid–solid slamming. Based on the observation of elastic–plastic dynamic response characteristics of the plates, a dynamic buckling criterion and a dynamic yielding criterion are defined. The corresponding critical impulse are determined from the experimental results for each tested plate. The effect of different boundary conditions on the elastic–plastic dynamic buckling properties of plates is also examined. The results indicate that dynamic buckling always takes place elastically for the types of rectangular plates tested under fluid–solid slamming. The dynamic buckling modes of the plates are governed by the plate fundamental transverse free vibration mode. It is also found that boundary conditions strongly affect the dynamic buckling properties of plates subjected to fluid–solid slamming loads. Strengthening plate boundary constraint is a very effective way to enhance the plates’ ability to resist dynamic buckling.

Estimating damage in a rod from changes in node positions

This paper is an analytical/experimental investigation of the effect of damage on the nodes of free vibration modes of a thin rod in longitudinal vibration. The damage, a notch, is simulated by a simple spring. It is shown that nodes move toward the notch. The direction and amount by which they move may be used to estimate the position and severity of the damage. Analytical results agree well with experimental tests.

A thickness-shear high voltage piezoelectric transformer

A high voltage piezoelectric transformer working with thickness-shear modes is proposed. For the analysis of the proposed transformer, one-dimensional equations for coupled thickness-shear and flexural motions of a piezoelectric beam with rectangular cross section are employed and are simplified by thickness-shear long wave approximation. Free vibration frequency equation and modes are obtained which exhibit the mechanism of the transformer. The analysis is for piezoelectric materials of arbitrary anisotropy. A ceramic thickness-shear transformer is discussed as an example.

Three-Dimensional Vibrations of Thick, Circular Rings with Isosceles Trapezoidal and Triangular Cross-Sections

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, circular rings with isosceles trapezoidal and triangular cross-sections. Displacement components us,uz, and uθ in the meridional, normal, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the ϕ and z directions. Potential (strain) and kinetic energies of the circular ring are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Novel numerical results are presented for the circular rings with isosceles trapezoidal and equilateral triangular cross-sections having completely free boundaries. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the rings. The method is applicable to thin rings, as well as thick and very thick ones. [S0739-3717(00)00702-9]

Free vibration analysis of elliptic and circular plates having rectangular orthotropy

The natural frequencies and modes of free vibration of specially orthotropic elliptic and circular plates are analysed using the Rayleigh-Ritz method. The assumed functions used are two-dimensional boundary characteristic orthogonal polynomials which are generated using the Gram-Schmidt orthogonalization procedure. The first five natural frequencies are reported here for various values of aspect ratio of the ellipse. Results are given for various boundary conditions at the edges i.e., the boundary may be any of clamped, simply-supported or fret. Numerical results are presented here for several orthotropic material properties. For rectilinear orthotropic circular plates, a few results are available in the existing literature, which are compared with the present results and are found to be in good agreement.

Free Vibration of the System of Two Timoshenko Beams Coupled by a Viscoelastic Interlayer

In this paper the uniform analytical method [3] has been used for solving a problem of free vibrations of continuous sandwich beam with damping. External layers are modelled as Timoshenko beams, while the internal layer possesses the characteristics of a viscoelastic, one-directional Winkler foundation. The phenomenon of free vibration has been described using a homogenous system of coupled partial differential equations. After separation of variables in the system of differential equations, the boundary problem has been solved and four complex sequences have been obtained: the sequences of frequencies, and the sequences of free vibration modes. Then, the property of orthogonality of complex free vibration modes has been demonstrated. The free vibration problem has been solved for arbitrarily assumed initial conditions.

FURTHER RESULTS FOR MODAL CHARACTERISTICS OF ROTATING TAPERED TIMOSHENKO BEAMS

The in-plane and out-of-plane modes of free vibration of a tapered Timoshenko beam mounted on the periphery of a rotating rigid hub are investigated. The finite element method is used to discretize the beam. This formulation permits unequal breadth and depth taper ratios as well as unequal element lengths. The effects of shear deformation, rotary inertia, hub radius, setting angle, and spinning rotation are considered. The generalized eigenvalue problem is defined using explicit expressions for the mass and stiffness matrices and numerical solutions are generated for a wide range of parameter variations. Explicit expressions of Southwell coefficients are presented for the first time for the case of rotating uniform and tapered Timoshenko beams. Comparisons are made wherever possible with exact solutions and other numerical results available in the literature. Extended results are obtained to serve as a benchmark solution for other numerical techniques and specialized applications.

Optimal joint positions and stiffness distribution for minimum mass frames with damping constraints

Optimal design of frames including cross-sectional dimensions (size parameters) and rigid joint positions between beams (configuration parameters) is treated in the paper. The optimal design corresponds to a minimal mass structure with constraints set on damping capacity of free vibration modes. The sensitivity analysis of distinct as well as multiple frequencies is performed analytically using a complex variable sensitivity method. The linking process of size and configuration variables is applied to generate different classes of optimal designs. The numerical algorithm is based on quadratic approximation of the objective function and linear approximation of active constraints. The examples are provided for 2, 12, and 124 beam frames.

FREE VIBRATION ANALYSIS OF RECTANGULAR PLATES WITH VARIABLE THICKNESS

An approximate method for analyzing the free vibration of thin and moderately thick rectangular plates with arbitrary variable thickness is proposed. The approximate method is based on the Green function of a rectangular plate. The Green function of a rectangular plate with arbitrary variable thickness is obtained as a discrete form solution for deflection of the plate with a concentrated load. The discrete form solution is obtained at each discrete point equally distributed on the plate. It is shown that the numerical solution for the Green function has good convergency and accuracy. By applying the Green function, the free vibration problem of the plate is translated into the eigenvalue problem of the matrix. The convergency and accuracy of the numerical solutions for the natural frequency parameter calculated by the proposed method are investigated, and the frequency parameters and their modes of free vibration are shown for some rectangular plates.

Localising A Notch In A Rod From Changes In Node Positions

This paper is an analytical/experimental investigation of the effect of damage on the nodes of free vibration modes of a thin rod in longitudinal vibration. The damage, a notch, is simulated by a simple spring. It is shown that nodes move towards the notch. The direction and amount by which they move may be used to estimate the position and severity of the damage Analytical results agree well with experimental tests.

FREE VIBRATION OF PARTIALLY CLAMPED RECTANGULAR PLATES WITH AND WITHOUT RIGID POINT SUPPORTS

An analytical procedure based on the method of superposition is described for obtaining the free vibration frequencies and mode shapes of partially clamped cantilevered rectangular plates with and without rigid point supports. The supports are of the type provided by bolts with stand-offs. Good agreement is obtained when computed results are compared with those obtained experimentally. While the solution to this problem is of general interest it is of particular interest to people working in the design of electronic circuit boards.

Electrical Equivalent Circuit of Loaded Thick Langevin Flexural Transducer

In this paper, an electrical equivalent circuit of a loaded thick Langevin flexural transducer is proposed. As there are two vibration components at the mechanical output terminal of a thick flexural transducer, that is, lateral and rotational components. A load, corresponding to each vibration component, applied to the transducer must also be considered separately. When it is assumed that the load is fluid and that the mechanical output port is the end surface of the flexural transducer, the rotary motion of the end surface only contributes to sound radiation into fluid. An analytical solution of Timoshenko's beam theory given by Miklowitz and the piezoelectric equation are applied to calculate the free vibration mode and the electrical equivalent components of transducer. Using this derived equivalent circuit, the behavior of the loaded flexural transducer is experimentally studied to prove the validity of the proposed circuit.

Electrical equivalent circuit of a thick Langevin-type flexural transducer

Since there are two vibration components at a mechanical output terminal of a thick flexural transducer, that is, lateral and rotational components, corresponding electrical ports have to be provided in the electrical equivalent circuit. A load applied to the transducer also has to be considered separately corresponding to each vibration component. To treat this situation with the electrical equivalent circuit, an analytical solution, namely, Timoshenko’s beam theory given by Miklowitz and the piezoelectric equation, is applied to calculate the free vibration mode and the electrical equivalent components of the transducer. Using this derived equivalent circuit, the behavior of the loaded flexural transducer is experimentally studied to prove the validity of the proposed circuit. This is the first trial to derive such an equivalent circuit that can treat two vibration components simultaneously.