Transverse vibrations of an Euler–Bernoulli uniform beam carrying two particles in-span
Introduction
Several publications are available on the transverse vibrations of a uniform Euler–Bernoulli beam carrying one particle in-span. Some of them were listed and commented by Naguleswaran [1]. A commonly occurring situation in civil and mechanical engineering consists of a uniform beam subjected to various ‘complications’ at several ‘locations’ along its length. The ‘complications’ may be various combinations of particles, heavy bodies, spring-mass systems, resilient supports, axial tension, step or continuous change in beam flexural rigidity, etc. A few selected references on vibrations of such systems are briefly reviewed here. Pan [2] treated the free and forced vibrations of a simply supported beam carrying several heavy bodies but did not present any results. Jacquot and Gibson [3] considered the effects of discrete masses and elastic supports on the vibrations of a beam. They obtained the natural frequencies by expanding the mode shape in terms of well known characteristic beam functions—a form of Galerkin's method. They stated that such ‘problems do not lend themselves to classical eigenvalue techniques’. Kounadis [4] extended the work in Ref. [2] and presented some results on the vibrations of a beam resiliently supported at one end and carrying heavy bodies of negligible axial dimension. Laura et al. [5] used Rayleigh–Ritz method to study the vibration of beams carrying several particles and subjected to an axial force. de Irasar et al. [6] extended the work in Ref. [5] to include in-span supports. Gürgöze [7] used the ‘normal mode summation’ technique to obtain the approximate fundamental frequency of a restrained beam carrying several particles. Gürgöze [8] extended the work in Ref. [7] to include heavy bodies. Bapat and Bapat [9] pointed out that for a beam with n locations, it was possible to express the frequency equation as a 4(n+1)th order determinant equated to zero. They pointed out the computational difficulties experienced and added that the determinant “in general… can be very complex even for n=2”. Hence they used the ‘transfer matrix method’ to solve the problem and among the results presented were the frequencies of a cantilever carrying three equispaced particles. Ercoli and Laura [10] investigated the problem using the Jacquot's method described in Ref. [3]. Liu and Yeh [11] used the Rayleigh–Ritz method to obtain the natural frequencies of various types of non-uniform beams carrying up to two particles. Lui et al. [12] used Laplace transforms to study the vibration of a beam with elastically restrained ends and carrying up to three particles. Kim and Dickinson [13] considered several types of problems, including the title problem using Rayleigh–Ritz method. They used the term ‘complications’ referred to earlier in the present paper. Liu and Huang [14] was concerned with the vibrations of a beam hinged by a rotational spring at one end and carrying a particle at the beam tip and a particle in-span. Kim and Dickinson [15] extended the work in Ref. [13]. Wu and Lin [16] used the ‘mode superposition method’ to study the vibrations of a cantilever carrying several particles. They stated that “it is impractical to obtain the closed form solution for natural frequencieswith more than two masses…”. Gürgöze [17] used the ‘Lagrange multipliers method’ to study the vibrations of a cantilever with a particle and a spring-mass system at the tip. Gürgöze [18] extended the work in Ref. [17] to include the effect of an in-span elastic support. Auciello and Maurizi [19] extended the work in Ref. [14] to include tapered beams. Refs. [14], [17], [19] are special cases of the title problem. Wu and Chou [20] extended the method developed in Ref. [16] to study the vibrations of a uniform cantilever carrying any number of elastically mounted particles. Gürgöze [21] used the ‘assumed mode method’ to study the vibrations of a cantilever carrying several spring-mass systems. Nalim and Grossi [22] presented a general algorithm based on Rayleigh–Ritz method for tapered beams on resilient end supports and carrying several particles. Low [23] obtained the Dunkerley fundamental frequency of a uniform beam carrying two particles. He presented the ‘error’ of the Dunkerley solution, which were up to 50% for certain system parameters. It is outside the scope of the present paper to comment on Dunkerley frequencies. Gürgöze and Inceoglu [24] investigated the possibility of preserving the fundamental frequency of a beam despite mass attachments.
The present paper is concerned with the vibrations of a uniform Euler–Bernoulli beam carrying two particles—the end supports being combinations of the classical types. In the text, the segment of the beam from the left end to the first particle is referred to as the first portion, in between the two particles as the second portion and from the second particle to the right end as the third portion. The beam mode shape will be the superposition of the mode shapes of the three portions. The mode shape of each portion has 4 constants of integration i.e. a total of 12 for the three portions. It is necessary to satisfy: the boundary conditions; continuity of deflection and continuity of slope at the two ‘locations’; and compatibility of bending moments and compatibility of forces acting on the two particles. The 12 linear equations yielded the frequency equation as a 12th order determinant equated to zero. Technically it should be possible to calculate the roots of the frequency equation without first expanding the determinant but the writer encountered computational difficulties—possibly the same sort mentioned in Refs. [9], [16]. The writer overcame these difficulties with some preliminary algebraic simplifications. Two of the four constants of integration in the mode shape of the first portion of the beam were eliminated when the boundary conditions at the left end was utilized. Similarly two constants of integration were eliminated from the mode shape of the third portion using the boundary conditions at the right end. Reduction of the mode shape of the second portion from one containing four constants of integration into one with the same two constants of integration of the first portion achieved further simplification. Here the continuity of deflection and continuity of slope at the first location; and compatibility of moments and compatibility of forces acting on first particle were used. It was possible to express the two constants of integration in the mode shape function of the third portion in terms of the two constants of integration of the first portion from continuity of deflection and continuity of slope at the second location. One was now left with two constants of integration. Consideration of compatibility of moments and compatibility of forces acting on the second particle resulted in the frequency equation as a 2nd order determinant equated to zero. The natural frequencies of the system were obtained from the roots of this frequency equation.
The first three natural frequency parameters of the title problem are tabulated for 16 combinations of cl, pn, sl or fr boundary conditions and several sets of mass and location parameters.
The method developed in the present work may be easily extended to cases when the beam was on resilient end supports and/or the particles are replaced with rigid bodies and/or springs. If the particles are elastically attached to the beam, additional co-ordinates will be required to analyse the system and this problem is considerably more difficult to solve.
Section snippets
Theory
Fig. 1a shows a uniform Euler–Bernoulli beam O1O3 of flexural rigidity EI, mass per unit length m, and length (R1+R2+R3)L carrying the first particle of mass δ1mL at axial coordinate R1L from O1 and the second particle of mass δ2mL at axial coordinate R3L from O3. δ1 and δ2 are the particle mass parameters and R1, R2 and R3 are the particle location parameters. Without loss of generality one may choose
To write the equations of transverse vibrations of the system, three coordinate
Numerical experiments
Numerical experiments to tackle the problem when the frequency equation was retained in the 12th order determinantal form were not successful. It took several seconds to evaluate the determinant (writer's subroutine/s) and hence the ‘root-search’ scheme either took too long or were often simply unsuccessful. The 8th order determinantal frequency equation derived following Eq. (10), did not alleviate the numerical problems. The 4th order determinantal frequency equation derived following Eq. (14)
Conclusions
The frequency equation of a uniform beam carrying two particles was expressed in closed form as a 2nd order determinant equated to zero as follows. Firstly the mode shape of the first portion of the beam was expressed in a form that contained two constants multiplied by an associated mode shape function. This was followed by the reduction of the mode shapes of the other two portions in terms of the same two constants but with different mode shape functions. The frequency equation will follow in
Acknowledgements
The help given by Malathy Naguleswaran in the preparation of this manuscript is gratefully acknowledged.
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