## 1 Introduction  
Microscaled materials and microscaled structures have attracted considerable interest as sensors, resonators, vibrators, and attenuators [1, 2, 3]. Studies on small materials and structures with vibrations are indispensable in the current micro/nano-technology [4], and they are among the challenging tasks which micro-mechanical engineers have to face. The lateral vibration dominates the beam-like microstructures with an oscillation, of which some can be modeled as the lateral excited vibrations of fully clamped microbeams according to the external excitation and boundary constraints. In fact, the microstructures are often subjected to time-dependent external excitation, and, also, the fully clamped boundary is one of the most common end restrictions, both of which are easily seen in micro-electro-mechanical systems (MEMS) and other devices at the microscale [5]. However, in recent years, much attention has been paid to the free vibration of microscaled materials and structures [6, 7, 8, 9, 10, 11, 12, 13, 14]. For example, [PERSON] et al. [6] investigated the free vibration of circular nanotubes subjected to both the torsional deformation and axial motion. The nonlinear history of strains is considered, and an infinite-order differential equation of motion is obtained via a variational principle. [PERSON] et al. [7] established a functionally graded Timoshenko beam model to reveal the size-dependent properties, including the strain gradient and nonlocal scale effects in its free vibration. [PERSON] and [PERSON] [9] examined the free vibration of rectangular nanoplates using a refined shear deformation theory. Both the nonlocal strain and nonlocal stress were considered, and the Rayleigh-Ritz method was employed to solve the governing equations. The nonlocal strain and stress approaches were compared, and the effects of various parameters were demonstrated. [PERSON] and [PERSON] [13] presented the elastic buckling and free vibration of functionally graded microbeams via the stress-strain integral-differential equations that were solved by a Laplace transform technique. As aforementioned, a large amount of studies regarding the free vibrational behaviors of beam-like microstructures can be found, but the excited vibration analyses of beam-like microstructures have not been investigated fully, and this is due to the lack of an excited vibration model containing intrinsic scale effects. Nevertheless, excited vibration is the main form of structural vibration in practical engineering [15]. With respect to a beam-like microstructure, it heavily depends on whether or not the intrinsic characteristic effects are taken into account, because the characteristic effects, such as the scale effect, cause changes in its lateral vibration. The process of discovering how to characterize the scale effect in microstructures is an important problem in micro-mechanics [16]. To this end, some modified non-classical continuum theories have been developed, of which the nonlocal theory is the most widely used size-dependent theory [17]. The nonlocal theory was first proposed by [PERSON] et al. [18; 19] to solve the stress distribution at a crack tip and wave propagation problems. It assumes that the stress at a point in the continuum is a function of strains at all points, and that is different from the classical continuum theories, because the latter considers the stress at a point as a function of the strain at only that point. The nonlocal theory belongs to a category with long-range interaction between atoms, but the classical continuum theories are just the opposite. During the past several decades, the nonlocal theory has been applied extensively in micro/nano-mechanics [20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30]. [PERSON] et al. [21] examined the vibration characteristics in wave propagation of a pre-stressed double-walled carbon nanotube that was modeled as a Euler-Bernoulli beam model. The nonlocal theory was employed, and the van der Waals force was taken into consideration.