## 1 Introduction  
Single-beam structures, one-dimensional continuous systems with different excitations and various boundary conditions, have been investigated extensively for many decades. The solutions and theories of dynamic problems of a single-beam structure are perfect [1; 2; 3; 4; 5; 6; 7]. Based on previous studies on single beams, a double-beam system, consisting of two one-dimensional continuous beams connected by a layer, is suggested. In fact, double-beam systems have attracted much attention from researchers and engineers in the past decade. Based on the assumption that both beams of system are identical, [PERSON] and [PERSON] [8] studied the dynamic response, free vibration, and static buckling of two parallel beams with different boundary conditions and a viscoelastic material layer in between. Similarly, [PERSON] and [PERSON] et al. [9] introduced a method to analyze a double-beam system subject to harmonic excitation with boundary conditions that must be the same on the same side. [PERSON] and [PERSON] et al. [10] considered the dynamic response of a double-identical-string system traversed by a constant or harmonically oscillating moving load. Using a simple change of variables to decouple two governing equations describing the vibration of two beams, [PERSON] and [PERSON] [11] developed analytical solutions for the dynamic deflections of both beams under moving harmonic loads. However, many researchers reject this assumption due to the limitations of identical beam systems. Based on previous studies for double-string systems [12], [PERSON] [13]provided analytical solutions for the free and forced vibrations of an elastically connected complex double-beam system with a simply supported boundary condition. [PERSON] and [PERSON] [14] reported a spectral finite-element method for a general double-beam system with unequal flexural rigidities, unequal masses, and arbitrary boundary conditions to investigate the free vibration characteristics. [PERSON] and [PERSON] et al. [15] investigated the free and forced transverse vibrations of an elastically connected simply supported double-beam system under a compressive axial load. Furthermore, new advances have been made in the research of double-beam systems in recent years. [PERSON] and [PERSON] [16] proposed a novel state-space form to study the transverse vibrations of a double-beam system with inhomogeneous beams, arbitrary boundary, and rate-dependent constitutive law for the inner layer. [PERSON] and [PERSON] [17] developed the general analytical solutions of forced vibrations of beams subjected to compressive axial loading and arbitrarily distributed continuous loads. Using the direct Lyapunov method and simplifying the system, [PERSON] and [PERSON] et al. [18] investigated the stability and instability of a double-beam system under compressive axial loading. [PERSON] and [PERSON] et al. [19; 20] applied a modal-expansion method to determine the forced vibration responses of a double-beam system when interconnected by a viscoelastic layer and joined by a uniformly distributed connecting elastic layer. [PERSON] and [PERSON] [21] studied the dynamic response of a double-beam system under a moving oscillator and solved the problem using a single-step scheme. Double-beam systems play an important role in many fields of structural engineering, such as sandwich or composite beams, and nanostructures. Nanobeams as an important element in highly accurate small-scale devices, and are used as nanosensors, nanoresonators, nanoactuators, nanoswitches, etc. These structures have the advantage of size, scale, and significant mechanical behavior, which makes them applicable in different systems. [PERSON] [22] mentioned that the nanobeam is an important element and is being extensively used for reliable and computationally efficient analysis of nanostructures, namely nanosensors, nanoresonators, and nanoswitches. Double nanobeam systems with great application in nano-optomechanical systems and sensors are one of the main nanostructures being investigated [23].