Lumped Attachments Research Articles

A modified polynomial-based approach to obtaining the eigenvalues of a uniform Euler–Bernoulli beam carrying any number of attachments

Free vibration characteristics in uniform beams with several lumped attachments are an important problem in engineering applications that have to deal with mounting different equipment (e.g. motors, oscillators or engines) on a structural beam. In order to solve the lack of a generalized automatic procedure, this investigation presents a simple solving approach based on analytical means applied to a secular frequency equation for obtaining the natural frequencies of an arbitrarily supported single-span, or multi-span Euler–Bernoulli beam carrying any combination of miscellaneous attachments. The approach is obtained by solving a characteristic polynomial equation using a classical method for computing the roots of a polynomial. Interestingly, if the number of elements is greater than one, a pole-zero cancellation is needed, but it does not require manual interventions such as initial values and iteration. The mathematical approach is validated with bibliographic references and evaluated for accuracy and computational effectiveness. A good agreement is observed with relative error values practically negligible mostly ranging between 10−3 and 10−9 in the first five natural frequencies, which confirms the validity of the presented approach in this paper. The MatLab code that has been developed with the solving approach is freely available as a supplementary material to this paper. Additionally, a MatLab graphical user interface has also been developed in this work which allows to obtain the eigenvalues of a simply supported Euler–Bernoulli beam carrying an undetermined number of lumped elements. The graphical user interface is also available for download, along with help facilities to be run in a Windows operating system and detailed instructions to reproduce the case studies presented here. The proposed scheme (and also the MatLab graphical user interface) is very easy to code, and can be slightly modified to accommodate beams with arbitrary supports.

A sensitivity‐based approach to solving the inverse eigenvalue problem for linear structures carrying lumped attachments

SummaryThis paper presents an efficient algorithm for designing dynamical systems to exhibit a desired spectrum of eigenvalues. Focusing on combined systems of linear structures carrying various lumped element attachments, we apply the assumed‐modes method and the implicit function theorem to derive analytical expressions for eigenvalue sensitivities, which are used to efficiently determine the minimal set of structural modifications needed to achieve a set of desired eigenvalues. The proposed algorithm employs an adaptive step size, performs significantly better than existing approaches, and can be easily applied to a broad range of structures. Convergence properties and limitations on achievable eigenvalues are also discussed, and a number of case studies demonstrating the performance of the algorithm in a wide variety of different applications are also included.

Exact Frequency Equation of a Linear Structure Carrying Lumped Elements Using the Assumed Modes Method

Combined systems consisting of linear structures carrying lumped attachments have received considerable attention over the years. In this paper, the assumed modes method is first used to formulate the governing equations of the combined system, and the corresponding generalized eigenvalue problem is then manipulated into a frequency equation. As the number of modes used in the assumed modes method increases, the approximate eigenvalues converge to the exact solutions. Interestingly, under certain conditions, as the number of component modes goes to infinity, the infinite sum term in the frequency equation can be reduced to a finite sum using digamma function. The conditions that must be met in order to reduce an infinite sum to a finite sum are specified, and the closed-form expressions for the infinite sum are derived for certain linear structures. Knowing these expressions allows one to easily formulate the exact frequency equations of various combined systems, including a uniform fixed–fixed or fixed-free rod carrying lumped translational elements, a simply supported beam carrying any combination of lumped translational and torsional attachments, or a cantilever beam carrying lumped translational and/or torsional elements at the beam's tip. The scheme developed in this paper is easy to implement and simple to code. More importantly, numerical experiments show that the eigenvalues obtained using the proposed method match those found by solving a boundary value problem.

Sensitivity Analysis of an Axially Vibrating Elastic Rod with Several Lumped Attachments

In a recent study, the eigenvalue sensitivities of a combined system consisting of a linear structure with several lumped attachments were analyzed. An efficient approach was developed for the eigenvalue sensitivities with respect to the parameters of the attachments and their locations by using the implicit function theorem. In the study mentioned above, the theory was applied to a Bernoulli–Euler beam with several attachments. Motivated by this study, the methodology established there will be applied, in the present paper, to an axially vibrating rod with several lumped attachments. Hence, the utility of the methodology is justified through a quite different type of vibrational system. The method is shown as an efficient tool that can be taught in related graduate-level courses for the eigenvalue estimation of complex linear systems with several lumped attachments.

A Structured Approach to Solve the Inverse Eigenvalue Problem for a Beam with Added Mass

The problem of determining the eigenvalues of a vibrational system having multiple lumped attachments has been investigated extensively. However, most of the research conducted in this field focuses on determining the natural frequencies of the combined system assuming that the characteristics of the combined vibrational system are known (forward problem). A problem of great interest from the point of view of engineering design is the ability to impose certain frequencies on the vibrational system or to avoid certain frequencies by modifying the characteristics of the vibrational system (inverse problem). In this paper, a method to impose two natural frequencies on a dynamical system consisting of an Euler-Bernoulli beam and carrying a single mass attachment is evaluated.

Frequency Analysis of an Arbitrarily Supported, Exponentially Tapered Beam

In most introductory vibration textbooks, the vibration of non-uniform beams is not covered. In this paper, the free vibration of an arbitrarily supported Euler–Bernoulli beam of constant thickness and exponentially varying width is investigated. The ‘assumed modes‘ method is utilized to obtain the approximate natural frequencies of the tapered beam, where the basis functions used in the expansion correspond to the eigenfunctions of the uniform beam, including its rigid-body modes. Expressing the eigenfunctions in terms of exponentials, the generalized masses and stiffnesses for the non-uniform beam can all be obtained in closed form, which substantially simplifies the analysis. The proposed scheme is simple to code, efficient to apply and can be easily modified to accommodate a tapered beam with various boundary conditions. Additionally, with slight modifications, the proposed method can also be used to analyze the free vibration of a combined system consisting of an exponentially varying beam carrying lumped attachments. Numerical experiments validate the present approach, and show that the approximate natural frequencies converge very quickly to the exact solution.

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.

Frequency Response of a Combined Structure Using a Modified Finite Element Method

A modified finite element method is proposed to analyze the frequency response of an arbitrarily supported linear structure carrying various lumped attachments. This is accomplished by updating the finite element mass and stiffness matrices of the linear structure without any lumped attachments using its exact natural frequencies and mode shapes, so that the eigensolutions of the updated linear structure coincide with the exact eigendata. Once the updated matrices for the linear structure are found, the finite element assembling technique is then exploited to include the lumped attachments by adding their parameters to the appropriate elements in the modified mass and stiffness matrices. After assembling the global mass and stiffness matrices, the frequency response of the combined system can be easily obtained. Numerical experiments show that, using only a few finite elements, the proposed scheme returns natural frequencies and frequency responses that are nearly identical to those obtained by using a finite element model with a very fine mesh. The new method is easy to apply and efficient to use, and it can be extended to determine the frequency response of any combined linear structure over any specified range of excitation frequencies.

Improved eigenvalues for combined dynamical systems using a modified finite element discretization scheme

New approaches are presented to discretize an arbitrarily supported linear structure carrying various lumped attachments. Specifically, the exact eigendata, i.e., the exact natural frequencies and mode shapes, of the linear structure without the lumped attachments are first used to modify its finite element mass and stiffness matrix so that the eigensolutions of the discretized system coincide with the exact modes of vibration. This is achieved by identifying a set of minimum changes in the finite element system matrices and enforcing certain constraint conditions. Once the updated matrices for the linear structure are found, the finite element assembling technique is then used to include the lumped attachments by adding their parameters to the appropriate elements in the modified mass and stiffness matrices. Numerical experiments show that for the same number of elements, the proposed scheme returns higher natural frequencies that are substantially more accurate than those given by the finite element model. Alternatively, the proposed discretization scheme allows one to efficiently and accurately determine the higher natural frequencies of a combined system without increasing the number of elements in the finite element model.

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.

A general approach to formulating the frequency equation for a beam carrying miscellaneous attachments

Beams carrying various lumped attachments have been studied extensively over the years. In this paper, a simple approach is proposed that can be used to readily determine the eigenvalues of an arbitrarily supported single-span or multi-span beam carrying any combination of lumped mass, rotary inertia, grounded translational or torsional spring, grounded translational or torsional viscous damper, an undamped or damped oscillator with or without a rigid body degree of freedom. Rather than solving a generalized eigenvalue problem to obtain the eigenvalues of the system, a frequency equation is formulated instead whose solution can be easily solved either numerically or graphically. The proposed scheme is easy to code, and can be easily modified to accommodate beams with arbitrary supports and carrying any number of miscellaneous lumped attachments.