Use of equivalent mass method for free vibration analyses of a beam carrying multiple two‐dof spring–mass systems with inertia effect of the helical springs considered

Abstract

SUMMARYThis paper investigates the free vibration characteristics of a beam carrying multiple two-degree-offreedom (two-dof) spring-mass systems (i.e. the loaded beam). Unlike the existing literature to neglect the inertia effect of the helical springs of each spring-mass system, this paper takes the last inertia effect into consideration. To this end, a technique to replace each two-dof spring-mass system by a set of rigidly attached equivalent masses is presented, so that the free vibration characteristics of a loaded beam can be predicted from those of the same beam carrying multiple rigidly attached equivalent masses. In which, the equation of motion of the loaded beam is derived analytically by means of the expansion theorem (or the mode superposition method) incorporated with the natural frequencies and the mode shapes of the bare beam (i.e. the beam carrying nothing). In addition, the mass and stiffness matrices including the inertia effect of the helical springs of a two-dof spring-mass system, required by the conventional finite element method (FEM), are also derived. All the numerical results obtained from the presented equivalent mass method (EMM) are compared with those obtained from FEM and satisfactory agreement is achieved. Because the equivalent masses of each two-dof spring-mass system are dependent on the magnitudes of its lumped mass, spring constant and spring mass, the presented EMM provides an effective technique for evaluating the overall inertia effect of the two-dof spring-mass systems attached to the beam. Furthermore, if the total number of two-dof spring-mass systems attached to the beam is large, then the order of the overall property matrices for the equation of motion of the loaded beam in EMM is much less than that in FEM and the computer storage memory required by the former is also much less than that required by the latter.

Free vibration analyses of simply supported beams carrying multiple point masses and spring-mass systems with mass of each helical spring considered

“…The literature concerning free vibration analyses of uniform or non-uniform beams carrying single or multiple spring-mass systems is plenty [1][2][3][4][5][6][7][8][9][10][11][12][13]. However, among the existing reports [1][2][3][4][5][6][7][8][9][10][11][12][13], most of them neglect the mass of helical spring of each spring-mass system except Refs.…”

“…However, among the existing reports [1][2][3][4][5][6][7][8][9][10][11][12][13], most of them neglect the mass of helical spring of each spring-mass system except Refs. [10][11][12][13]. In Refs.…”

“…In Refs. [12,13], the property matrices of a spring-mass system with inertial effect of its helical spring considered are determined first and then the conventional finite element method (FEM) is used to study the influence of masses of the helical springs on the dynamic characteristics of the entire vibrating system. In this paper, in addition to the fundamental frequency, the other lowest four natural frequencies and the lowest five mode shapes of the simply supported uniform beams carrying arbitrary number of point masses and spring-mass systems are determined with mass of each helical spring considered by using two analytical methods (Methods 1 and 2).…”

Free vibration analysis of elastically supported Timoshenko columns with attached masses by transfer matrix and finite element methods

“…Wu and Chiang (2004) studied about natural frequencies and mode shapes of the double-tapered wedge beams carrying multiple point masses. Wu (2006) investigated free vibration characteristics of a beam carrying multiple two-dof spring-mass systems. Wu and Chen (2001) studied free vibration analysis of a Timoshenko beam carrying multiple spring-mass systems by using the numerical assembly technique.…”

Consideration of the masses of helical springs in forced vibrations of damped combined systems

“…As a first step to fill this gap, Gürgöze (2005) derived the frequency equation of a classical combined system consisting of a cantilevered beam to the tip of which is attached a helical springmass system, the novelty being that the helical spring is modeled as a longitudinally vibrating elastic rod (James et al, 1994). Wu (2005) has taken into account the inertia effects of the helical springs, for free vibrations analysis of a Bernoulli-Euler beam carrying multiple two-degree-of freedom systems by using equivalent mass method. Gürgöze et al (2006) dealt with the determination of the frequency equation of a Bernoulli-Euler beam simply supported at both ends, to which is attached in-span a longitudinally vibrating elastic rod with a tip mass, representing a helical springmass system with mass of the helical spring considered.…”