Free vibration analysis of a double-beam system joined by a mass-spring device

Abstract

In this paper, the free vibration analysis of a double-beam system is investigated. This structure is formed by two beams with elastic restraints at one end and free at the other end. These beams are connected by a mass-spring device. First, the related eigenvalue problem is established in the frequency domain, employing the well-known Fourier transform. Then, by utilizing eight boundary and compatibility conditions and solving two differential equations, the eigenvalues of the system are found and tabulated for different amounts of the parameters. Furthermore, some mode shapes of the mechanical system under study are plotted for various values of the springs' stiffness. In order to verify the results, some special cases are analyzed, and their outcomes are compared with the available ones.

An exact dynamic stiffness method for multibody systems consisting of beams and rigid-bodies

“…Therefore, methods predominantly based on numerical techniques are not always suitable for modal analysis in the high frequency range [11] . To circumvent the above problem, many analytical models have been proposed for beam structures connected to various types of attachments such as lumped concentrated mass [12][13][14][15][16] and/or rotatory inertia [17] , spring and damper [18] , single spring-mass [14,[19][20][21][22][23][24][25][26] , double spring-mass [27][28][29][30][31][32] , spring-mass chain [33,34] which all have wide ranging applications in engineering. The attachment as lumped mass used in these publications is by and large assumed to be a concentrated point mass without any consideration to the size or dimension of the mass or its mass moment of inertia.…”

“…Thus, there are some difficulties to apply these existing methods to complex multibody systems. For example, unlike the numerical method where the stiffness and mass matrices can be formulated separately [9] , almost all existing analytical methods [45][46][47][48][49][50]2,6,13,52,14,15,[22][23][24]26,27,[29][30][31]55,[32][33][34][35][37][38][39][40]42,44,61,62] apply the usual determinant method for non-trivial solution of the eigenvalue problem for which the determinant of the coefficient matrix vanishes. The determinant method needs the evaluation of the determinant numerically for one frequency at a time.…”

Dynamics of a beam with multiple spring/mass attachments using binary asymptotic admissible functions

“…The method was extended to n spring-mass-spring devices at different locations of the beam for all classical boundary conditions. Rezaiee and Mojtaba 13 studied the dynamics of two horizontal beams connected by a spring-mass system. Applying the Fourier transform to the coupled differential equations leads to an algebraic frequency equation.…”

Free response of a continuous vibrational system with attachments and/or discontinuities using segmented operational Tau method

“…The above equations are general descriptions of the free vibrational problem of many constrained systems such as beam/bars with mass–spring attachments, intermediate/resilient supports and/or subjected to opening cracks. During recent years, vast efforts have been made to solve special cases of the problem (1)–(3) through analytical techniques (Aydin, 2008; Bakhtiari-Nejad et al., 2014; Chaudhari and Maiti, 2000; Hassanpour et al., 2007; Kirk and Wiedemann, 2002); Li, 2000, 2002; Maiz et al., 2007; Mazanoglu and Kandemir-Mazanoglu, 2016; Naguleswaran, 2002, 2003a, 2003b, 2004; Rezaiee-Pajand and Hozhabrossadati, 2014; Shifrin and Ruotolo, 1999; Sourki and Hoseini, 2016; Wang and Qiao, 2007). Although analytical methods are very useful in solving equations (1–3), they are reported to have many limitations (Zhang, 2011).…”