Analytical Methods in Vibrations

Optimizing the natural frequencies of axially functionally graded beams and arches

Nonlinear dynamic analysis near resonance of a beam-ring structure for modeling circular truss antenna under time-dependent thermal excitation

The study of free in-plane vibration of curved beams, using different beam theories, is more challenging than that of straight beams, since the structural deformations in curved beams depend not on...

Free in-plane vibration of curved beam structures: A tutorial and the state of the art:

Classical gyroscopic continua include axially moving materials and spinning structures. The gyroscopic effect of the axially moving material is pronounced via the gyroscopic coupling among the basis functions in the same motional direction. On the other hand, the gyroscopic coupling of the spinning structure acts in the two different directions of motion. In this paper, we study the dynamics of a beam with both axial moving motion and spinning motion as a prototype of bi-gyroscopic continua. The influence of bi-gyroscopic effects on the natural frequencies, modes, and stability is investigated by an analytical method applied to the discretized equations of the axially moving and spinning beam. Distinct bifurcation series of the eigenvalues and corresponding physical interpretations are discussed by numerical display of the modal motions. The complex modes describing both whirling motions and traveling waves are investigated in detail for such bi-gyroscopic system. New interesting phenomena have been analyzed numerically and important conclusions have been drawn for such bi-gyroscopic system.

Dynamics of a beam with both axial moving and spinning motion: An example of bi-gyroscopic continua

Free in-plane vibration analysis of a curved beam (arch) with arbitrary various concentrated elements

The critical speeds of flexible rotors are important information for many engineers; thus, there is great amount of literature concerning this subject. The existing literature reveals that the dynamic problems of rotor-bearing systems are solved by the step-by-step integration process [1], transfer matrix method (TMM) [2], analytical method [3–5], assumed mode method [6], hybrid method [7], frequency-dependent TMM [8] or the finite element method (FEM) [9,10]. Besides, Yamamoto and Ishida [11] have introduced the applications of the analytical methods, the TMM and FEM, to the linear and nonlinear dynamics of multidisk rotor-bearing systems.

From the foregoing literature reviews, one sees that all existing techniques for the analysis of whirling motions are the approximate approaches except for the analytical method presented by Eshleman and Eubanks [4] and that introduced by Yamamoto and Ishida [11]. In theory, the solution of Eshleman and Eubanks [4] is an exact one, however, it is only for the whirling speeds of a rotating shaft carrying “one” disk and the corresponding “whirling mode shapes” are not considered. Thus, the purpose of this paper is to extend and modify the aforementioned technique, so that the lowest five (or higher) forward and backward whirling speeds and the associated mode shapes for a shaft carrying any number of disks with various boundary conditions can be easily obtained. To this end, the transverse displacement of each shaft cross-section is represented by a complex number and then the equation of motion, the continuity equations for the deformations, the equilibrium equations for the forces (and moments), and the associated boundary conditions are derived in terms of the complex numbers in which the effects of each rigid disk mounted on the spin shaft are replaced by a lumped mass and a frequency-dependent equivalent mass moment of inertia. Finally, the method for obtaining the natural frequencies and mode shapes of a stationary beam carrying multiple concentrated elements [12–14] is used to determine the forward and backward whirling speeds and mode shapes of a spinning shaft mounted by arbitrary rigid disks. In addition to a comparison with the existing literature, most of the results obtained from the presented method are also compared with those obtained from the FEM by using the technique shown in the Appendix with the property matrices of each shaft element and each rigid disk given by Nelson and McVaugh [9] and Przemieniecki [15].

A structural system may be considered as the “continuous” system or “discrete” system and the solution obtained from the former is called the “closed-form” or “exact” solution, while that from the latter is called the “approximate” solution [16]. Thus, the results obtained from the proposed method are the “exact” solutions and may be the “benchmark” for evaluating the accuracy of the other “approximate” solutions such as those obtained from the FEM.

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Analytical solution for whirling speeds and mode shapes of a distributed-mass shaft with arbitrary rigid disks

Although the exact solutions for the free vibration problems regarding most of the non-uniform beams are not yet obtainable, this is not true for the special case when the equation of motion of a non-uniform beam can be transformed into that of an equivalent uniform beam. The nonlinearly tapered beam studied in this paper is a single-tapered beam with constant depth h0 and varying width b(x) along its length in the form ( ) b x  4 0[1 ( )] b x L   , where b0 is the minimum width,  is the taper constant, x is the axial coordinate and L is the total beam length. For the case of no concentrated elements (CEs) attaching to it, the exact solution for its lowest several natural frequencies and the associated mode shapes has been appeared in the existing literature, however, the exact solution for the free vibrations of the last tapered beam carrying various CEs in various boundary conditions (BCs) is not found yet due to complexity of the problem. This is the reason why this paper aims at studying the title problem by using the continuous-mass transfer matrix method (CTMM). It is different from the general uniform (or multi-step) beam carrying various CEs in that the nonlinearly tapered beam itself as well as the attached translational and rotational CEs must all be transformed into the equivalent ones in the derivations. In addition to the solution accuracy, one of the salient merits of the proposed method is that the order of the characteristicequation matrix keeps constant (4  4) and does not increase with the total number of the CEs or the beam segments such as in the conventional finite element method (FEM), so that it needs Fig. 1. The vortex wind generator developed by Vortex Bladeless (2015). less than 0.2% of the CPU time required by the FEM to achieve the exact solutions. The CEs on the nonlinearly tapered beam include lumped masses (with eccentricities and rotary inertias), translational springs and rotational springs. The formulation of this paper is available for various classical or non-classical BCs. In addition to comparing with the existing available data, most of the numerical results obtained from the proposed method are also compared with those of the FEM and good agreement is achieved.

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Jong Shyong Wu

Papers

Free in-plane vibration analysis of a curved beam (arch) with arbitrary various concentrated elements

Analytical solution for whirling speeds and mode shapes of a distributed-mass shaft with arbitrary rigid disks

Free vibration analysis of a nonlinearly tapered beam carrying arbitrary concentrated elements by using the continuous-mass transfer matrix method