## 3 Simulation Example
### Multiple-Degree-of-Freedom System  
Case 1: In this example, we take the multiple-degree-of-freedom system shown in Figure 3 as a research object and identify the dynamic load on it. We set the stiffness of the system to \(k_{1}=4\) N/m, \(k_{2}=5\) N/m and \(k_{3}=6\) N/m, the mass to \(m_{1}=3\) kg, \(m_{2}=2\) kg, \(m_{3}=1\) kg, and the damping to \(c=0.01\). Apply the external load \(f=8\cos(t)\sin(3t)\) on the mass \(m_{3}\). The external force represents a kind of relatively complex force, whose amplitude changes harmoniously. The similar performance is also able to be obtained for other functions. The time interval from \(t_{1}\) to \(t_{2}\) is \(\Delta t=0.02\) s. The initial state is 0, and observation lasts for 10 s. The identification results are as follows.  
Figure 7 presents that the load identified by the algorithm is highly consistent with the actual load applied to the system, with a deviation of less than 5%, which proves the feasibility of the method for the load identification of multiple-degree-of-freedom systems.  
The previous example verifies the feasibility of the method for identifying a single load of a multiple-degree-of-freedom system. Next, we will use another example to discuss the feasibility of identifying a multiple-degree-of-freedom system under multiple loads. Let us also take the three-degrees-of-freedom system shown in Figure 3 as an example.  
Case 2: Assume that the stiffness of the system is \(k_{1}=4\) N/m, \(k_{2}=5\) N/m, \(k_{3}=6\) N/m; the mass is \(m_{1}=3\) kg, \(m_{2}=2\) kg, \(m_{3}=4\) kg; and the damping is \(c=0.01\). The forces  
Figure 6: Load identification error comparison with Newmark method.  
Figure 7: Multiple-degree-of-freedom system load-identification results.  
applied on mass \(m_{2}\) and \(m_{3}\) are \(f_{1}=10\sin(8\pi t)\) and \(f_{2}=15\sin(2\pi t)\), respectively. The time interval from \(t_{1}\) to \(t_{2}\) is \(\Delta t=0.02\) s. The initial state is 0, and observation lasts for 10 s. The identification results are as follows.  
From the identification results of two loading points of three degrees of freedom identified in Figures 8 and 9, each excitation is identified with a derivation less than 5%, which proves the feasibility of the method for multi-loads identification of multiple-degree-of-freedom system.  
Case 3: Assume that the system parameters are the same as in Case 2. the loads applied on the mass \(m_{2}\) and \(m_{3}\) are \(f_{1}=50\sin(2\pi t)\) and \(f_{2}=100\sin(\pi t)\), respectively. The time interval from \(t_{1}\) to \(t_{2}\) is \(\Delta t=0.02\) s. The initial state is 0, and observation lasts for 20 s. To discuss the effect of the noise level on the performance of the proposed method, 5%, 10%, and 15% random noises are added to the response signal, respectively. The load-identification results are shown in Figures 10 and 11. Table 1 shows the errors of load identification with different noise levels. In the case of response with 15% noise, the maximum value of errors is 4.12%. The accuracy of load identification is still in the acceptable range. This also proves that the proposed method has good noise immunity.  
Figure 8: \(f_{1}\) load-identification results.  
Figure 9: \(f_{2}\) load-identification results.