# Free Vibration Analysis of Triclinic Nanobeams Based on the Differential Quadrature Method
## 1 Introduction  
In the past decades, different analytical and numerical approaches have been applied in literature to study the structural response of even more complicated structures [1; 2; 3; 4; 5; 6]. Based on the literature, it is well known that many analytical solutions based on the Navier approximations cannot satisfy the governing equations of the problem, such that numerical approaches are usually required. In this context, the differential quadrature method (DQM) has been increasingly applied in several works and demanding applications as a powerful and efficient numerical method [7; 8; 9; 10; 11; 12], due to its beneficial properties. This method, indeed, is user-friendly for different engineering problems and it features a high accuracy even with few grid points and a reduced computational effort, (see [7; 8; 9; 10; 11]). In most cases, the DQM has been used to study the dynamic, static or stability response of structures such as beams, plates, and shells, whereas any application of the DQM in literature has focused to nanostructures made of anisotropic materials, as typically occurs in the actual nature of materials due to their mechanical properties, with different elastic components in each direction. If an isotropic behavior is related to a certain uniformity along all the orientations, on the other hand, anisotropy refers to situations where properties vary systematically. For example, a triclinic material features different properties in different directions, with 21 elastic constants and three components of the propagation vector [13]. Due to the complexity of anisotropic material models, a large amount of simplification in most works in the literature is generally based on isotropic material assumptions. To date, only a few studies have focused on the mechanical response of anisotropic structures. More specifically, in [14; 15; 16] the authorsanalyzed the vibration response of square plates made of orthotropic, monoclinic and hexagonal materials for different boundary conditions, whereby the recent works [17; 18; 19; 20] have focused on the wave propagation and buckling response of anisotropic materials at the nanoscale. The development of nanotechnology has led to significant contributions in the scientific community, due to its advantages for many practical applications, i.e., water purification, medical applications, electronic and mechanical systems, among many. Therefore, the research and development of these novel materials has received special attention in the last decades, especially at a nanoscale level, where classical theories are inapplicable and can fail. Hence, different methods, i.e., experimental tests, molecular dynamics (MD) simulations and non-classical mathematical formulations, have been proposed as alternative ways to predict the behavior of nanomaterials [21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35]. In work by [PERSON] and [PERSON] [36], a nonlocal strain gradient theory was proposed as an alternative non-classical method to capture both the hardening and softening stiffness mechanisms of nanostructured systems. Moreover, [PERSON] and [PERSON] [37] proposed the application of a nonlocal strain gradient model to overcome the reported paradox in nonlocal cantilever beams subjected to a point load, whereby in [38] the authors accounted for the effect of three small-scale parameters within the model, while checking for its accuracy with respect to some MD-based results for carbon nanotubes. Further numerical predictions about the size-dependent behavior of functionally graded materials and structures can be found in [39; 40; 41; 42; 43; 44; 45; 46], in the presence or not of porosities, in accordance to the nonlocal strain gradient model of elasticity. Based on limitations from the literature, in the current work the free vibrations of triclinic nanobeams with varying thickness along the length are studied for the first time, while applying the Timoshenko beam theory in conjunction with the nonlocal strain gradient model.