## 2 Function Principle Dynamic Load Identification Method  
According to the law of conservation of energy, the work done by the external force to the system equals the variation of the energy of the system, which includes kinetic energy, potential energy, and dissipation energy. In the multiple-degree-of-freedom vibration system, from \(t_{1}\) to \(t_{2}\), the work done by the external force to the system equals the sum of the variation of the kinetic energy, the potential energy, and the dissipated energy of the system, which is written as [23]  
\[W=\Delta T+\Delta V+D \tag{1}\]  
where \(W,\Delta T,\Delta V\), and \(D\) refer to, respectively, the work done by the external force, the variation of the kinetic energy, the variation of the potential energy, and the dissipated energy generated by the damp in this period. The work done by the external force to the system is [23]  
\[W=\int_{t_{1}}^{t_{2}}f(t)dx(t) \tag{2}\]  
where \(f(t)\) and \(x(t)\) represent the external force and the displacement. The expressions of the kinetic energy and the potential energy are [23]  
\[T=\frac{1}{2}m\dot{u}^{2} \tag{3}\]  
\[\mathrm{V=}\frac{1}{2}k\Delta u^{2} \tag{4}\]  
The expression of the dissipated energy is [23]  
\[D=\int_{t_{1}}^{t_{2}}ci^{2}dt \tag{5}\]  
where \(\dot{u}\) represents the velocity of the certain mass and \(\Delta u\) represents a certain spring deformation. Suppose in a tiny period of time, the magnitude of external force \(f\) remain constant, donated as \(\overline{f}\); then, the work done by the external forces in this segment is  
\[W=\overline{f}\Delta u \tag{6}\]  
where \(\Delta u\) represents the displacement of external load application point. Time-Domain Identification Method of Dynamic Load Based on Function Principle for Single-Degree-of-Freedom System  
Figure 1 presents a single-degree-of-freedom system, in which the mass is represented by \(m\) and the spring stiffness is noted \(K\). The external force acting on the system is unknown, and the displacement and velocity of the system can be measured. Assuming a zero initial state and the system is in equilibrium position, the entire duration of the observation is \(T\), and divided into \(n\) tiny periods with the interval time \(\Delta t\), then \(n=T/\Delta t\). In one period of the time that starts at \(t_{1}\) and ends at \(t_{2}\), the variation of the energy of the system is  
\[\Delta E=\frac{1}{2}m\dot{u}(t_{2})^{2}+\frac{1}{2}ku(t_{2})^{2}-[\frac{1}{2} m\dot{u}(t_{1})^{2}+\frac{1}{2}ku(t_{1})^{2}]+\int_{t_{1}}^{t_{2}}ci^{2}dt \tag{7}\]  
where \(u\) and \(\dot{u}\) represent the displacement and velocity of the mass block, respectively. During this short period, because the external load changes continuously and the amount of change is small, the external load remains constant. Then, the external force in this short period is  
\[W=\overline{f}\Delta u \tag{8}\]  
\[\Delta u=u(t_{2})-u(t_{1}) \tag{9}\]  
According to the law of conservation of energy, we can obtain the equation  
\[W=\Delta E \tag{10}\]  
\[W=\int_{t_{1}}^{t_{2}}f(t)dx(t) \tag{2}\]0 Single-degree-of-freedom system. The equation has only one unknown number \(\overline{f}\), so the external load on the micro segment can be obtained  
\[W=\int_{t_{1}}^{t_{2}}f(t)dx(t) \tag{2}\]1  
Similarly, apply the above method on each micro-period of time to obtain the external load value of each micro segment. Fit all the obtained external loads to a curve to obtain the external load value in the entire time domain. The flow chart of load identification for the single-degree-of-freedom system is shown as \[W=\int_{t_{1}}^{t_{2}}f(t)dx(t) \tag{2}\]2  
Time-Domain Identification Method of Multi-Point Dynamic Loads Based on Function Principle for Multiple-Degree-of-Freedom System  
For multi-point loads acting on a multiple-degree-of-freedom system, we take the two external loads of the three-degrees-of-freedom system shown in Figure 3 as an example and apply external loads on the mass \(m_{2}\) and \(m_{3}\).