On the transverse vibration of Timoshenko double-beam systems coupled with various discontinuities

Highlights

• Vibrations of Timoshenko double-beam systems with discontinuities are studied.

• Closed-form expressions of natural frequencies, mode shapes and FRFs are found.

• Connections and discontinuities can be systematically described in matrix form.

• Both translational and rotational effects of the connection springs are considered.

• The presented results are compared and validated with other investigations.

Abstract

An efficient analytic framework is developed in this paper to study transverse vibrations of double-beam systems used in numerous engineering applications, in which two parallel Timoshenko beams are connected by discrete springs and coupled with various discontinuities. By dividing the entire structure into a series of distinct components, and then systematically organizing compatibility and boundary conditions with matrix formulations, closed-form expressions for the exact natural frequencies, mode shapes and frequency response functions (FRFs) can be determined. In particular, the proposed model enables one to simultaneously consider (1) both the free and forced vibrations, (2) translational and rotational effects of the connection springs, (3) any boundary conditions, and (4) various combinations of discontinuities. Furthermore, regardless of the number of discontinuities and connections, the sizes of all associated matrices are never larger than 8×8. This results in a significant computational advantage, especially for the derivation of the FRFs. Some results are compared with previous publications and also with the finite element method to demonstrate the accuracy and versatility of the proposed method. Finally, parametric studies are performed for a practical example to illustrate the influences of parametric variabilities on the dynamic behaviors.

Introduction

The double-beam systems connected by discrete or continuous springs have been widely used in numerous aerospace, mechanical and civil engineering applications. Investigations on continuously joined double-beam systems have obtained plentiful achievement. Several recent papers are selectively quoted here [1], [2], [3], [4], [5] to provide further references on this subject. On the other hand, relatively few papers have been published on the dynamics of the double-beam systems interconnected by discrete springs [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], though such systems generally provide fundamental components of many commonly used engineering constructions, including flexible base structures in vessels, dual-rotor systems in gas turbine engines, suspension bridges, etc. Moreover, effective evaluation of the dynamic behavior has long been regarded as one of the most important tasks for design processes of double-beam systems. Hence, despite the potential mathematical complexities arising in the analytical solution of this coupled dynamic problem [13], the subject is worthy of investigation.

Several interesting theoretical works have been presented in recent years which demonstrate an emerging attention to double-beam systems interconnected by discrete springs. Dublin and Friedrich [6] presented some analytical expressions for forced vibrations of a double-beam system connected by two spring-dashpot subunits. Chonan [7] investigated the dynamical behaviors of a double-beam system connected with a set of independent springs subjected to an impulsive load employing Laplace transformations with respect to both time and space variables. Yamaguchi [8] studied the forced response of a beam coupled with a spring dashpot subunit and a small beam as a dynamic absorber, and discussed the effect of parameters. Kukla et al. [9] analyzed the bending vibration of axially loaded double-beam systems coupled by several translational springs using Green׳s function method. The authors tabulated results for possible combinations of classical boundary conditions. Shankar and Keane [10], [11] studied the vibration energies of two beams coupled by discrete springs based on finite element analysis and Green׳s function method. With both the classical approach [12] and Green׳s function method [13], Gürgöze et al. tackled the problem of a combined system composing of two clamped-free beams carrying tip masses and coupled with several double spring-mass systems. In their formulations, as the number of the spring-mass systems increases, the problem may become difficult to resolve, which limit the practical applicability of those solutions. Concerned with the same mechanical system in Refs. [12], [13], Gürgöze and Erol [14] formulated the generalized eigenvalue problem using the assumed modes method, where the eigenfrequencies can be simultaneously obtained. Gao and Cheng [15] studied vibration transmissions in a cantilever double-beam system connected by several actuators theoretically and experimentally. Modal expansion approach was used to analyze the effects of actuator parameters on resonance frequencies and vibrational power flow between coupled beams.

From the above literature review, one may clearly notice that (1) although efficient solutions were available on vibrations of double-beam systems with classical boundary conditions, general solutions concerning arbitrary boundary conditions (either classical cases or non-classical cases) have not yet been carried out, (2) only translational springs were considered in the connection spring subunits, rotational effects were seldom involved, (3) Euler beam theory (EBT) was preferable in the foregoing investigations to capture the physical insights, though fewer results were presented according to Timoshenko beam theory (TBT) to give a better approximation to the true behavior of beams, (4) the mentioned studies were majorly concluded upon establishing natural frequencies and modes, thus the dynamic response was seldom considered, and (5) in addition, it is also worthwhile mentioning that in the aforementioned works all the structures were modeled by two uniform beams connected by discrete spring subunits without any other attachment [6], [7], [8], [9], [10], [11], [12], [13], [14].

In realistic applications, however, double-beam systems are usually confronted with situations of mounting various equipments, such as engines, oscillators, isolators and so forth. Practically, these assemblies can be approximated by lumped discontinuities attached to the beams [16], [17], introducing additional analytical complexities to the solution of the problem. Unfortunately, to the authors׳ knowledge, no available report could be cited in the open literature concerning the free and forced vibration analysis of double-beam systems, made of two parallel beams coupled by discrete translational/rotational spring subunits and simultaneously coupled with multiple discontinuities. Accordingly, the present study concentrating on the above-mentioned issues aims to provide a systematic approach to the analysis of these types of structures and to offer a clear and concise way for designing such structures at the design stage.

Actually, free and forced vibrations of single beams with arbitrary discontinuities have been extensively investigated using several different analytical approaches [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]. However, many procedures do not efficiently scale to the double-beam systems since (1) the discontinuities contained in each beam prevent the direct application of common beam theories and (2) mechanical coupling between beams results in further analytical difficulties. As a result, the acquirement of exact solutions for double-beam systems with discontinuities is somewhat more complicated than that for single beams. Recently, Zhang et al. [28] presented a general solution for determining exact natural frequencies and mode shapes for a single non-uniformTimoshenko beam coupled with flexible attachments and discontinuities. Specifically, a general analytic framework [28] was established by analyzing the combined structure as a series of distinct sub-beams with continuity and compatibility enforced at discontinuity locations. Closed-from natural frequencies and mode shapes were obtained and numerically validated. Nevertheless, the dynamic response was not considered. Meanwhile, the authors note that the method of Ref. [28] is potentially extensible to the present study by treating the interrelated sub-beams in both beams as a whole in order to tackle the coupling problem.

Therefore, motivated by the efficient approach presented by Zhang et al. [28] and aimed at overcoming the limitations highlighted above, this paper contributes a general mathematical framework for analyzing transverse vibrations of double-beam systems, made of two Timoshenko beams connected by the discrete spring subunits and simultaneously coupled with multiple discontinuities, about which little is known in the literature. Consequently, the proposed technique enables one (i) to obtain closed form expressions for the accurate natural frequencies, mode shapes and frequency response function (FRF), (ii) to consider both translational and rotational effects in connection subunits and (iii) to accommodate double-beam systems with various combinations of discontinuities for any boundary conditions. It is also worth noting that, by organizing compatibility and boundary conditions systematically with the proposed approach, the size of resulting matrix determinant is never larger than 8×8, which leads to a significant computational advantage, especially for the calculation of the FRFs.

The remainder of the paper is structured as follows. The problem statement is given in Section 2. In 3 Derivation of eigenvalues and mode shapes, 4 Determination of frequency response functions, detailed information from the general to the specific is accessible on the derivations of closed-form solutions for natural frequencies, mode shapes and FRFs. Section 5 presents the model of FEM used for comparison. Comparative studies and parametric studies are made in Section 6 to demonstrate the practicability and accuracy of the developed procedure. Finally, concluding remarks are summarized in Section 7.