Forced longitudinal vibration response of nonlocal strain gradient (NLSG) rods is scrutinized and compared with nonlocal (NL), strain gradient (SG), and classical (CL) rods. In this respect, size-dependent kinematics and extended Hamilton’s principle are utilized to derive governing equations of motion. For the first time, forced longitudinal vibratory behavior of non-classical and classical rods is presented including three different types of non-harmonic forcing functions: unit impulse, unit step, and unit ramp. Forcing terms are functions of both temporal and spatial domains. Nonlocal strain gradient and nonlocal rod models take into account the Laplacian of the external forcing term as well. This implies that in models including nonlocal effects (NL, NLSG-h, s, n), forcing term is reduced by the order of its Laplacian with respect to spatial variables. Effect of non-classical parameters over time-domain response of the system is analyzed in details. Results are compared with benchmark for the sake of evaluation. This implies the necessity of adopting proper values of non-classical theory and corresponding parameters through analysis of forced oscillations of small-scaled size-dependent rods. Such external excitations are modeled using spatial Dirac delta function as a concentrated force applying at the critical point of mode shape function. Findings of this research reveal the underlying impact of non-classical theorems along with corresponding non-classical parameters over the prediction of the response of size-dependent rod elements for wide range of applications such as in musical instruments, navigating equipment, gyroscopes, control of robotics, sensors and actuators, resonators, energy harvesters, and vibration control systems.

详细研究了非局部应变梯度（NLSG）杆的强迫纵向振动响应，并将其与非局部应变梯度（NL），应变梯度（SG）和经典（CL）杆进行了比较。在这方面，利用尺寸相关的运动学和扩展的汉密尔顿原理来导出运动的控制方程。首次介绍了非经典杆和经典杆的强制纵向振动行为，包括三种不同类型的非谐波强迫函数：单位脉冲，单位阶跃和单位斜率。强迫项是时域和空间域的函数。非局部应变梯度和非局部杆模型也考虑了外部强迫项的拉普拉斯算子。这意味着在包含非局部效应（NL，NLSG-h，s，n）的模型中，强迫项相对于空间变量按其拉普拉斯算子的顺序减少。详细分析了非经典参数对系统时域响应的影响。为了进行评估，将结果与基准进行比较。这意味着有必要通过分析小尺寸尺寸杆的强迫振动来采用非经典理论的适当值和相应的参数。使用空间狄拉克δ函数作为施加在模态形状函数的临界点上的集中力来对这种外部激励进行建模。这项研究的发现揭示了非古典定理以及相应的非古典参数对预测尺寸依赖的杆件在诸如乐器等各种应用中的响应的潜在影响。