Philip D. Cha

Imposing nodes at arbitrary locations for general elastic structures during harmonic excitations

Abstract Spring–mass systems are frequently used as vibration absorbers to minimize excess vibration in structural systems. In this paper, sprung masses are used to impose the points of zero vibration for general elastic structures during forced harmonic excitations. For convenience, such points are referred to as nodes. When the oscillator attachment locations and the node locations coincide (or are collocated), it is always possible to select the spring–mass parameters such that multiple nodes are induced at any desired locations along the structure for any excitation frequency. When the oscillators and the node locations are not collocated, however, it is only possible to induce nodes at certain locations along the elastic structure for a given driving frequency. Moreover, when the desired node locations are closely spaced, it is possible to specify a region of nearly zero amplitudes for a particular driving frequency, effectively quenching vibration in that region. A procedure to guide the proper selection of the spring–mass parameters in order to induce multiple nodes is outlined in detail, and numerical experiments are performed to verify the utility of the proposed scheme of imposing nodes at multiple locations during harmonic excitations.

Quenching vibration along a harmonically excited linear structure using lumped masses

Undamped oscillators consisting of spring-mass systems are commonly deployed as vibration absorbers to quench excess vibration in structural systems. While masses can be easily made to specification, it is often costly and more difficult to manufacture springs with the desired stiffness values precisely. In this paper, lumped masses only are used to suppress vibration by imposing points of zero displacement or nodes anywhere along an arbitrarily supported elastic structure during harmonic excitations. An efficient approach is developed that can be used to tune the required lumped-mass parameters and their attachment locations so that nodes are induced at the desired locations. Instead of solving for lumped masses directly, one solves for the restoring forces exerted by the lumped masses, which can be easily obtained using Gauss elimination. These restoring forces are then used to tune the required mass parameters. Design plots are proposed that show all the feasible attachment locations, and numerical experiments are performed to validate the proposed scheme of quenching vibration using lumped masses only.

Eigenfrequencies of an Arbitrarily Supported Beam Carrying Multiple In-Span Elastic Rod-Mass Systems

Many vibrating mechanical and structural systems consist of a host structure to which a number of continuous attachments are mounted. In this paper, an efficient method is proposed for determining the eigenfrequencies of an arbitrarily supported Euler–Bernoulli beam with multiple in-span helical spring-mass systems, where the mass of the helical spring is considered. For modeling purposes, each helical spring can be modeled as an axially vibrating elastic rod. The traditional approach of using the eigenfunctions of the beam and rod in the assumed modes method often leads to an intolerably slow convergence rate. To expedite convergence, a spatially linear-varying function that corresponds to the static deformed shape of a rod is included in the series expansion for the rod. The proposed approach is systematic to apply, easy to code, computationally efficient, and can be easily modified to accommodate various beam boundary conditions. Numerical experiments show that with the addition of a spatially linear-varying function, the proposed scheme converges very quickly with the exact solution.

Mitigating Vibration Along an Arbitrarily Supported Elastic Structure Using Multiple Two-Degree-of-Freedom Oscillators

Simple spring-mass systems are often deployed as vibration absorbers to quench excess vibration in structural systems. In this paper, multiple two-degree-of-freedom oscillators that translate and rotate are used to mitigate vibration by imposing points of zero displacement, or nodes, along any arbitrarily supported elastic structure during harmonic excitations. Nodes can often be enforced along an elastic structure by attaching suitably chosen two-degree-of-freedom oscillators. In application, however, the actual selection of the oscillator parameters also depends on the tolerable translational and rotational vibration amplitudes of the attached oscillators, because if these vibration amplitudes are large, then theoretically feasible solutions would not be practical to implement. In this paper, an efficient approach is developed that can be used to tune the oscillator parameters that are required to induce nodes, while satisfying the tolerable vibration amplitudes of the oscillators. Instead of solving for the oscillator parameters directly, the restoring forces exerted by the springs are computed instead. The proposed approach is simple to apply, efficient to solve, and more importantly, allows one to easily impose the tolerable translational and rotational vibration amplitudes of the two-degree-of-freedom oscillators. A design guide for choosing the required oscillator parameters is outlined, and numerical experiments are performed to validate the proposed scheme of imposing nodes along a structure at multiple locations during harmonic excitations.

Eigenvalues and Eigenvalue Sensitivities of a Beam Supported by Viscoelastic Solids

The eigenvalues and the first and second-order eigenvalue sensitivities of a uniform Euler–Bernoulli beam supported by the standard linear solid model for viscoelastic solids are studied in detail. A method is proposed that yields the approximate eigenvalues and allows the formulation of a frequency equation that can be used to obtain approximate eigenvalue sensitivities. The eigenvalue sensitivities are further exploited to solve for the perturbed eigenvalues due to system modifications, using both a first- and second-order Taylor series expansion. The proposed method is easy to formulate, systematic to apply, and simple to code. Numerical experiments consisting of various beams supported by a single or multiple viscoelastic solids validated the proposed scheme and showed that the approximate eigenvalues and their sensitivities closely track the exact results.

Applying Sherman-Morrison-Woodbury Formulas to Analyze the Free and Forced Responses of a Linear Structure Carrying Lumped Elements

A simple approach is proposed that can be used to analyze the free and forced responses of a combined system, consisting of an arbitrarily supported continuous structure carrying any number of lumped attachments. The assumed modes method is utilized to formulate the equations of motion, which conveniently leads to a form that allows one to exploit the Sherman-Morrison or the Sherman-Morrison-Woodbury formulas to compute the natural frequencies and frequency response of the combined system. Rather than solving a generalized eigenvalue problem to obtain the natural frequencies of the system, a frequency equation is formulated whose solution can be easily solved either numerically or graphically. In order to determine the response of the structure to a harmonic input, a method is formulated that leads to a reduced matrix whose inverse yields the same result as the traditional method, which requires the inversion of a larger matrix. The proposed scheme is easy to code, computationally efficient, and can be easily modified to accommodate arbitrarily supported continuous linear structures that carry any number of miscellaneous lumped attachments.

Imposing points of zero displacement and zero slopes on a plate subjected to steady-state harmonic excitation

In this paper, a simple and effective method to enforce fixed nodes, or points of zero displacement and zero slope, on an arbitrarily supported rectangular plate subjected to steady-state harmonic excitations is developed. This is achieved by attaching properly tuned translational and rotational oscillators at specified locations. The governing equations of the combined system are first derived using the assumed-modes method. By enforcing the conditions of zero displacements and zero slopes simultaneously, a set of constraint equations are formulated, from which the oscillator parameters can be determined. When the attachment locations coincide with the desired fixed node locations, it is always possible to select the oscillator parameters such that one or multiple fixed nodes are induced at any locations on the plate for any excitation frequency. When the attachment and the desired node locations are not collocated, it is only possible to induce nodes at certain locations on the plate. When the fixed node locations are judiciously chosen, a selected region of the plate can be made to remain nearly stationary. Thus, the proposed method provides a simple and yet effective means to passively control excessive vibrations.

Using a Characteristic Force Approach to Determine the Eigensolutions of an Arbitrarily Supported Linear Structure Carrying Lumped Attachments

In this paper a characteristic force approach is developed that can be used to determine the eigenvalues and mode shapes of any arbitrarily supported linear structure carrying various lumped attachments, including a lumped mass, rotary inertia, grounded translational or torsional spring, grounded translational or torsional viscous damper, and an undamped or damped oscillator with or without a rigid body degree of freedom. Using the proposed approach, each lumped element is first replaced by the load, either a force or moment, that it exerts on the linear structure, thus transforming the free vibration problem into a forced vibration one. By expressing the deflection of the linear structure in terms of these forces or moments and enforcing the compatibility conditions at each attachment points, the roots of the characteristic determinant of these loads can be graphically or numerically solved to find the eigenvalues of the combined system. Once the eigenvalues have been found, the corresponding mode shapes can be readily obtained. The proposed method is easy to code, systematic to apply, and can be easily modified to accommodate any arbitrarily supported one- or two-dimensional linear structure carrying various lumped attachments.

Improved modal convergence using the assumed modes method for rods carrying various lumped elements

In this paper, the assumed modes method is used to determine the modes of vibration of an arbitrarily supported uniform and nonuniform rods carrying various lumped elements, including a lumped mass, a grounded spring, a grounded viscous damper, and an undamped and damped oscillator. In applying the assumed modes method, the set of trial functions used in the expansion can be arbitrary as long as they satisfy the geometric boundary conditions of the system. In practice, the trial functions are often selected to correspond to the eigenfunctions of the bare uniform rod. Numerical experiments show that while this set of trial functions converges to the exact results, the rate of convergence can be exceedingly slow. In order to expedite modal convergence, the eigenfunctions are augmented with piecewise linear functions that capture the slope discontinuities of the mode shapes at the attachment locations due to the presence of the lumped elements. The results obtained using the two sets of trial functions are compared with those obtained exactly. It is shown that including the piecewise linear functions significantly improves the accuracy of the modes of vibration of the system while drastically reducing the computational time and effort.

Perturbation Methods for the Eigencharacteristics of Symmetric and Asymmetric Systems

Dynamic analysis for a vibratory system typically begins with an evaluation of its eigencharacteristics. However, when design changes are introduced, the eigensolutions of the system change and thus must be recomputed. In this paper, three different methods based on the eigenvalue perturbation theory are introduced to analyze the effects of modifications without performing a potentially time-consuming and costly reanalysis. They will be referred to as the straightforward perturbation method, the incremental perturbation method, and the triple product method. In the straightforward perturbation method, the eigenvalue perturbation theory is used to formulate a first-order and a second-order approximation of the eigensolutions of symmetric and asymmetric systems. In the incremental perturbation method, the straightforward approach is extended to analyze systems with large perturbations using an iterative scheme. Finally, in the triple product method, the accuracy of the approximate eigenvalues is significantly improved by exploiting the orthogonality conditions of the perturbed eigenvectors. All three methods require only the eigensolutions of the nominal or unperturbed system, and in application, they involve simple matrix multiplications. Numerical experiments show that the proposed methods achieve accurate results for systems with and without damping and for systems with symmetric and asymmetric system matrices.