## 1 Introduction  
The double-beam system has wide applications in reality, such as floating-slab railway tracks [1], track-bridge systems [2; 3], sandwich beams composed of an interconnected layer [4], vibration absorbers [5; 6] and carbon nanotubes [7]. Studying modal properties (frequencies and mode shapes) of such a system contributes to understanding the mechanism of the system dynamics and lays the foundation for working out its dynamic response under external excitations by using Modal Superposition (MS) method. Seelig and Hoppmann II [8] presented the analytical frequencies and associated mode shapes of the elastically connected double-beam system consisting of two identical beams for the first time. Rao [9] studied the free vibrations of elastically connected parallel bars considering the effects of rotary inertia and shear deformation. Tadatake et al. [5] presented the natural frequencies and mode shapes of double-beam systems with two general beams by using a generalized method of finite integral transformation and the Laplace transformation. The same problem was also studied by Oniszczuk [10] by using the classic Bernoulli-Fourier method. Mirzabeigy et al. [11] presented explicit expressions for natural frequencies of double-beam systems with arbitrary boundary conditions. Han et al. [12] applied the dynamic stiffness method to establish the exact dynamic stiffness matrix and frequency equation for an axially-loaded double-beam system with arbitrary boundary conditions. The transcendental frequency equation was solved by an improved Wittrick-Williams algorithm. Hao et al. [13] investigated the free vibrations of double-beam systems with arbitrary boundary conditions by a modified Fourier-Ritz approach. Compared with the vibrations of a single beam under moving loads [14; 15; 16; 17], the vibrations of double-beam systems excited by moving loads are more complicated and have drawn much attention recently. The vibrations of a simply supported double-beam system with two identical beams interconnected by an elastic layer under the action of a fixed harmonic force were solved in the closed-form solutions by Vu et al. [18]. The connecting damping between two beams was ignored. The vibrations of the same system subjected to a moving constant force [19], a moving harmonic force [20] were studied analytically. The system under the action of a moving oscillator [21] was also studied numerically. Zhang and Ma [22] investigated the influence of compressive axi loads on the vibration of the system. Kessel [23] studied the resonances of a simply supported double-beam system with two general beams excited by a cyclic moving load. The connecting damping between two beams was ignored, which was then considered in Reference [24]. The analytical solutions for the forced vibrations of the same system were presented by Oniszczuk [25]. Chonan [26] investigated the vibration of a double-beam system interconnected by elastic springs subjected to an impulse load considering the mass of the springs. Pavlovic et al. [27] investigated the stability of a double-beam system subjected to random compressive axi forces. The forced vibration and buckling of a Rayleigh and Timoshenko double-beam system subjected to arbitrary continuous load with the effect of compressive axi loads were studied by Stojanovic and Kozic [28]. Kozic et al. [29] analyzed the free vibration and buckling of a double-beam system under axial loading joined by a Kerr-type layer. It should be noticed that all these studies regarding forced vibrations of double-beam systems are restricted to simply supported double-beam systems. Hamada et al. [5] pointed out the similarity between an elastically connected multi-beam system with same boundary conditions for all the beams (symmetric boundary conditions) and a multi-degree-of-freedom mass-spring system. The free vibrations of an elastically connected double-beam system, taking account of the axial forces on the beams were studied in References [30; 31].