Mode Shape Equations Research Articles

Exact general solutions for the mode shapes of longitudinally vibrating non-uniform rods via Lie symmetries

A Lie symmetry method-based approach is proposed for systematically computing general solutions in closed-form for the mode shape equation of non-uniform and unconventional vibrating rods. The mode shape equation is modeled by the elementary rod theory, addressing polynomial, exponential, trigonometric, and hyperbolic cross-section variations. The method provides algorithmic order-reduction steps for solving the investigated mode shape equation, producing a first-order Riccati equation whose integration reveals the aimed solutions for the problem. Illustrative examples are presented, including original solutions in closed-form as well as solutions previously obtained in the literature by other approaches. Mode shapes from general solutions with appropriate rod boundary conditions are also considered for different examples. • Original general solutions establish new exact mode shapes of non-uniform rods. • The Lie symmetry method allows addressing new cross-section variations systematically. • Mode shapes derive from the general solutions with rod boundary conditions. • Lie symmetry applications have been highlighted for further problems of interest.

Modal analysis of deepwater drilling riser in freestanding disconnected mode

As a further development to the work published on survivability study of deepwater drilling riser in freestanding mode, this paper presents the modal analysis of drilling riser freestanding mode (DRFM) for the purpose of studying the natural frequencies and mode shapes in order to avoid resonance which threatens the survivability of the riser. The vibration equation of DRFM is solved using separating the variables method to obtain mode shape equation which is then solved numerically using Newton's method. The vortex shedding frequencies obtained for DRFM under extreme current condition show the region of DRFM necessary for damping. The key influencing parameters such as length, buoyancy can upthrust ratio (BCUR), bending stiffness, and wall thickness were analyzed for the DRFM. Results show that the length, BCUR, bending stiffness, and wall thickness have significance effects on the natural frequency and natural period of DRFM. The effects of these parameters are minimal for first four mode numbers but grew significantly as the mode number increases for natural frequency but with a reverse case for natural period. The mode shapes of DRFM are not affected by these parameters. The vibrational study of the DRFM has been validated with Abaqus simulation.

Formulae for the frequency equations of beam-column systemcarrying a fluid storage tank

In this work, a mathematical model of beam-column system carrying a double eccentric end mass system is investigated, and solved analytically based on the exact solution analysis. The model considers the case in which the double eccentric end mass is a rigid storage tank containing fluid. Both Timoshenko and Bernoulli-Euler beam bending theories are considered. Equation of motion, general solution and boundary conditions for the present system model are developed and presented in dimensional and non-dimensional format. Several important non-dimensional design parameters are introduced. Symbolic and/or explicit formulae of the frequency and mode shape equations are formulated. To the authors knowledge, the present reduced closed form symbolic and explicit frequency equations have not appeared in literature. For different applications, the results are validated using commercial finite element package, namely ANSYS. The beam-column system investigated in this paper is significant for many engineering applications, especially, in mechanical and structural systems.

Dynamic Response of a Damped Euler–Bernoulli Beam Having Elastically Restrained Boundary Supports

While doing vibration analysis in terms of the dynamic response of a beam, the end conditions, in most of the cases, are assumed to be fixed, simply supported, free or sliding imposing an ideal condition of end support displacement. But, in real engineering structures, the end supports are non-ideal (for example welded, riveted etc.) and allow certain degree of translational or rotational motion of the support ends. To simulate the effect of real engineering boundary states, a combination of linear and torsional springs has been considered at the two boundaries of the beam structure. The theoretical analysis of free undamped beam structure with elastically restrained end conditions has been undertaken using approximation of the modal displacement as a Fourier cosine series to generate natural frequencies and corresponding mode shape equations. The effect of variation in stiffness values of the boundary springs on the vibration characteristics (natural frequency, mode shapes and dynamic response) has been presented for vibration conditions for the free undamped beam structure. Thereafter, forced undamped and damped (taking combined effects of both viscous and structural damping) beam structures have been theoretically analysed to generate response equation. Finally, numerical assessment has been undertaken to present the effect of damping ratio, forcing frequency and force magnitude on the response of a nearly clamped beam and has been compared with the response of an undamped beam under similar forcing and boundary condition. The results indicate considerable effects of real boundary condition, on the vibration characteristics and dynamic response of beam structures thus cannot be ignored in design of real engineering structures. Further, the presence of damping does not change the vibration characteristics beam structures, but is useful in achieving steady-state condition of response for a harmonic force input. The technique presented provides a simplified and convenient tool for obtaining vibration response for real engineering boundary supports. Further, the beam model can be extended for analysis of plate structures considering non-ideal boundary conditions.

A theoretical model for the vibration of fuel rod with multi spans supported by springs

The prediction of fuel rod vibration is of importance in the designing of reactor core. A precise simulation of the vibration amplitude of fuel rods is necessary. It is disappointed that the mode shape equation of a spring-supported fuel rod with multi spans is not available in previous literature, due to the singularity of coefficient matrix. In this work, a theoretical model for the vibration of spring-supported rod with multi spans is established. The fuel rod is assumed to be a homogeneous beam. The flexural displacement of the rod is represented as the combination of Fourier series and auxiliary trigonometric functions in order to overcome the discontinuity and singularity of mode shape. The vibration frequency and mode shape are obtained. The calculation results are in satisfactory with theoretical results. The variation of spring constant and its effect could be simulated.

A comparative evaluation of some solution methods in free vibration analysis of elastically supported beams

This paper presents a comparison between two analytic methods for the determination of natural frequencies and mode shapes of Euler-Bernoulli beams. The subject under scrutiny is the problem of a beam supported by an arbitrary number of translational springs of varying stiffness, which is solved first by the method of Laplace transform and then by the Green’s function method. The two methods are compared on the level of algebraic equivalence of the resulting formulas and these are compared to the results obtained by FEM analysis for the case when the beam is supported with a single spring. It can be shown that both methods result in equivalent algebraic expressions, whose results are to be regarded as accurate for any given set of boundary conditions. This is also verified by means of FEM analysis, whose solutions converged to almost identical values. Hence, the two methods are found to be equally accurate for the calculation of natural frequencies and natural modes. This paper also brings a new formulation for the mode shape equation, obtained by the Laplace transform method, which to the best of the authors’ knowledge has not been reported in existing literature. Additional comments about the advantages and disadvantages of the two employed analytical methods are given from the standpoint of mathematical structure and practicality of implementation, which is also supplemented by a comment on the comparative advantages and disadvantages of FEM analysis.

Using local stiffness indicator to examine the effect of honeycombs on the flexural stiffness of reinforced concrete beams

Experimental modal analysis was performed on RC beams with honeycombs. The mode shapes were extracted, and the eigenvectors were used in determining the mode shape equations using nonlinear regression. The equation used in the regression was the generalized solution of transverse vibration of a Bernoulli–Euler prismatic beam. By utilizing the regressed phase variable, the global flexural stiffness was evaluated. It was observed that the global stiffness dropped with increasing volumes of the honeycombs in the beam. The results were compared with values computed using the secant modulus from the load–deflection plot obtained upon loading at each load stage and the trend was similar. The local stiffness indicator was used successfully to locate the region of the honeycombs.

Examining the trend in loss of flexural stiffness of simply supported RC beams with various crack severity using model updating

The flexural stiffness of simply supported cracked reinforced concrete beams was determined by model updating. The beams were 150mm wide, 250mm deep and 2200mm long. Different FE models were created which include a datum and models with a single crack at three different locations along the length of the beam. The mode shape equation was obtained by using non-linear regression. The equation used in the regression was the generalized solution of transverse vibration of a prismatic beam. Local flexural stiffness, EI, at each coordinate point was derived by substituting the regressed data by using the centered-finite-divided-difference formula. Experimental modal analysis was performed on a control beam and beams with load-induced cracks at predetermined loading. Results from FE analyses showed the trend in the loss of stiffness was similar to the results obtained on the experimental beams. The more severe the damage, the higher the loss of stiffness and the loss patterns are similar for damage at different locations along the beam. The updating technique is able to indicate the trend in the loss of stiffness as a result of cracks of varying severity in the RC beams showing good agreement with experimental results.

Crack damage detection of reinforced concrete beams using local stiffness indicator

This paper described the application of the generalized solution for transverse vibration to detect crack damages in reinforced concrete beams. A single crack was induced in a full-scale reinforced concrete beam by application of a point load. The load was increased in stages to obtain different crack heights to represent the extent and severity of the defect. Experimental modal analysis was performed on the beam prior to application of the load and after each load stage. The mode shape equation for the beams was obtained by using nonlinear regression. Global flexural stiffness was derived by utilizing the regressed variablel into the equation for transverse vibration of a Bernoulli-Euler prismatic beam. Local flexural stiffness at each coordinate point was derived by substituting the regressed data at that point and by using the centered-finite-divided-difference formula. The global stiffness decreased with increased severity of the crack in the beam. The results were compared with values computed using the secant modulus from the load-deflection plot obtained upon loading at each load stage and the trend was similar. The proposed algorithm could form the basis of a technique for structural health monitoring of load induced damaged reinforced concrete structures. Key words: crack damage, defect severity, local stiffness indicator, reinforced concrete beam, structural health monitoring.

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation are studied analytically. The axial speed, as the source of parametric vibrations, is assumed to involve a mean speed, along with small harmonic variations. The method of multiple scales is applied to the governing non-linear equation of motion and then the natural frequencies and mode shape equations of the system are derived using the equation of order one, and satisfying the compatibility conditions. Using the equation of order epsilon, the solvability conditions are obtained for three distinct cases of axial acceleration frequency. For all cases, the stability areas of system are constructed analytically. Finally, some numerical simulations are presented to highlight the effects of system parameters on vibration, natural frequencies, frequency–response curves, stability, and bifurcation points of the system.

Non-linear parametric vibration and stability analysis for two dynamic models of axially moving Timoshenko beams

Non-linear parametric vibration and stability of an axially moving Timoshenko beam are considered for two dynamic models; the first one, with considering only the transverse displacement and the second one, with considering both longitudinal and transverse displacements. The set of non-linear partial-differential equations of both models are derived using an energy approach. The method of multiple scales is applied directly to both models, and using the equation order one, the mode shape equations and natural frequencies are obtained. Then, for the equation order epsilon, the solvability conditions are considered for the resonance case and the stability boundaries are formulated analytically via Routh–Hurwitz criterion. Eventually, some numerical examples are provided to show the differences in the behavior of the above-mentioned non-linear models.

A continuous vibration analysis model for cables with sag and bending stiffness

The paper reports an investigation on a continuous model for dynamics of large-diameter sagged cables, taking bending stiffness and sag extensibility into account. Following the classical guidelines of cable dynamics, and by making some simplifications, a continuous model is found in this work; frequency and mode shape equations have been obtained in non-dimensional form. Comparison with Finite Element (FE) and Finite Difference (FD) models has been made to validate the proposed method. Moreover, it allows us to represent local effects at the end zones of cable correctly, if clamped boundary conditions are assumed. To obtain comparable results, the FE model would require a large subdivision, with consequently, a large number of elements. The proposed eigenfunctions allow us to solve the dynamic problem without a large computational effort compared to FE or FD approaches. Finally, in order to show the usefulness of the developed continuous model, an application on the main cables of the designed Messina Strait Bridge has been made.

Study on the Vibration and Stress of Copper Tubes in Refrigerators and Air Conditioners.

In this work the vibration and stress of copper tubes caused by operation of compressors in refrigerators and air conditioners are considered. Natural frequencies and stresses caused by forced deformation were calculated for both sides of fixed straight tubes of 6 mm to 32 mm diameter. Simple equations relating the natural frequency and stress to diameter, length and displacement were derived. Calculation equation for the stress of a tube by the torsional displacement of the compressor at starting stage was obtained with a compressor and a straight tube model. Furthermore, stress of the tube at resonance was calculated using the mode shape equation and a simple equation relating stress to natural frequency and maximum displacement or acceleration. Tube design can facilitated through use of these equations relating frequency and stress of the tubes. Vibration transmission to the case and the noise caused by the vibration of the case will be discussed in a forthcoming report.

Lateral Vibration Of A Centrifugally Tensioned Uniform Euler-Bernoulli Beam

This paper describes the lateral vibration of a uniform Euler-Bernoulli beam that is doubly symmetric in cross-section and attached radially to the outside of a rotating hub. It is assumed that a principal axis of the beam is parallel to the axis of rotation and thus the out-of-plane and in-plane vibrations are uncoupled. The equation of motion is derived on the basis that the attachment at the hub is radially restrained. The general solution of the mode shape equation is expressed as the superposition of four linearly independent functions. Clamped, pinned and free boundary conditions are considered. It is shown that the natural frequencies depend not only on the natural and/or geometric boundary conditions but also on which of the two boundaries is radially free. The first three dimensionless natural frequencies are tabulated for out-of-plane vibration for six combinations of the simple boundary conditions and for a range of offset parameters and dimensionless rotational speed. From the tables it is possible to deduce the dimensionless natural frequencies for in-plane vibrations. For parameters not listed, interpolated results are accurate to within 0·3%. It is hoped that the tabulated results will serve as data for development of frequencies for problems with more complicated flexural rigidity and/or mass distribution.

Vibration in the Two Principal Planes of a Non-Uniform Beam of Rectangular Cross-Section, One Side of Which Varies as the Square Root of the Axial Co-ordinate

The transverse vibration of a type of slender beam of continuously varying rectangular cross-section is considered in this paper. One side of the cross-section is constant, while the other side is proportional to the square root of the axial co-ordinate. It is assumed that the beam dimensions are such that application of the Euler-Bernoulli theory of bending is valid. Thus the vibration in the two principal planes will be uncoupled. The two mode shape equations are fourth order linear differential equations with variable coefficients and with regular singularity. Analytical solutions of the two mode shape equations are derived, based on the method of Frobenius. Combinations of ideal clamped, pinned, sliding and free boundary conditions are considered. The sharp end of a "complete" beam cannot sustain a bending moment or shearing force and hence must always remain free. The large end may be clamped, pinned, sliding or free. The frequency equations for a "complete" beam is the determinant of a 2 × 2 matrix equated to zero. For a truncated beam any combinations of the four boundary conditions is possible and the frequency equations are the determinant of a 4 × 4 matrix equated to zero. The first three dimensionless natural frequencies for vibration in the two principal planes are tabulated for 16 combinations of the four boundary conditions and truncation factors from 0·05 to 0·7. To preserve the accuracy of the frequencies to six figures, the results are presented in tabular form. It is hoped that they will serve as benchmarks.

A Direct Solution for the Transverse Vibration of Euler-Bernoulli Wedge and Cone Beams

A number of publications have appeared on the transverse vibration of "complete" and truncated wedge and cone beams. The mode shape equations are linear differential equations with regular singularity. The natural frequencies of "complete" beams are found in several publications. For truncated beams several numerical solutions have been proposed and the natural frequencies documented (in most cases) for cantilever wedge and cone beams clamped at the large end. Analytical solutions of the mode shape equations have been derived based on Bessel functions. Only in one publication are the natural frequencies tabulated, for nine combinations of clamped, pinned and free boundary conditions, but the accuracy of the frequencies is limited to two figures after the decimal point for the fundamental frequency, and the higher mode frequencies are rounded to integers. The results are not suitable for judging the quality of the various approximate methods. In this paper a direct solution of the mode shape equation is presented. The sliding boundary condition is also considered and the first three dimensionless natural frequencies computed for 16 combinations of the boundary conditions are tabulated for values of truncation factors from 0·05 to 0·7. The tabulated results are benchmarks, and frequencies for truncation factors not in the tables may be interpolated to within 0·1% accuracy.

Vibration of an Euler-Bernoulli beam of constant depth and with linearly varying breadth

The transverse vibration of a homogeneous Euler-Bernoulli beam of constant depth and linearly varying breadth is investigated in this paper. A direct analytical solution is presented for the mode shape equation. The constraints imposed by the boundary conditions yield the frequency equation. The four ideal boundary conditions considered are clamped, pinned, sliding and free. For a complete beam (sharp ended) it is a necessary condition that the sharp end remain free, and as a result the frequency equations for clamped, pinned, sliding or free boundary conditions at the large end are the determinant of a 2 × 2 matrix equated to zero. A truncated beam is considered as part of a hypothetical complete beam. The frequency equation for truncated beam is the determinant of a 4 × 4 matrix equated to zero. The roots of the frequency equations are determined by an iterative procedure. The first four natural frequencies of complete beam and the first three frequencies of truncated beams for 16 combinations of the ideal boundary conditions for various truncation factors are presented. The results are benchmarks and are therefore presented in tabular form to preserve the accuracy to six figures.

The effect of pretwisting on the buckling behaviour of slender columns

The equations governing the buckling of a pretwisted column are derived as a subset of the general stability equations for a spatial rod. Numerically exact solutions to these fourth-order differential equations are obtained by determining the analytic forms of the buckled mode shape equations, and calculating the buckling loads and mode shape coefficients using a simple iterative technique. Plots are provided to detail the effect of pretwist on the buckling strength for both statically determinate and indeterminate configurations. These plots indicate that, for statically indeterminate configurations, a pretwisted column may be more than twice as strong as the corresponding prismatic column for the first buckling mode. It is also shown that, generally, the pretwisted buckled mode shapes strongly resemble their prismatic counterparts, and may be treated as effectively planar.

Axisymmetric vibration of point-supported circular plates

The problem of axisymmetric vibration of point-supported circular plates has been investigated. Both elastic and rigid point supports have been considered. The modal expansion technique together with the receptance method have been employed to formulate the frequency and mode shape equations. Numerical results for natural frequencies for elastic and rigid point supports are presented and a number of mode shapes for various cases are shown.

Free vibration of circular plates with elastic or rigid interior support

In this work reported in this paper the problem of free vibration of a circular plate with an elastic or rigid interior support distributed uniformly around the plate at an arbitrary radius has been considered. The case of an elastic or rigid point support at the plate center has also been studied. The frequency and mode shape equations have been formulated by using the modal expansion technique and the receptance method. Numerical results for natural frequencies are presented, and wherever it has been possible the results obtained are compared with results given in other papers in the literature.