Question 5.22: A Two-Degree-of-Freedom Mass–Spring System Consider the two-...
A Two-Degree-of-Freedom Mass–Spring System
Consider the two-degree-of-freedom mass–spring system shown in Figure 5.107. The mass block m_1 and the spring k_1 represent a rotating machine, which is subjected to a harmonic disturbance force f = 40\text{sin}(7πt) \text{N} due to a rotating unbalanced mass. The mass block m_2 and the spring k_2 represent a vibration absorber (see Section 9.3 for more details), which is designed to reduce the displacement of the machine. The mathematical model of the system is given by a set of ordinary differential equations
m_1\ddot{x}_1 + (k_1 + k_2)x_1 – k_2x_2= f
m_2\ddot{x}_2 – k_2x_1 + k_2x_2 = 0
where m_1 = 6 \text{kg}, k_1 = 6000 \text{N/m}, m_2 = 1.65 \text{kg}, and k_2 = 800 \text{N/m}. Assume zero initial conditions.
a. Build a Simulink model of the system based on the differential equations of motion and find the displacement outputs x_1(t) and x_2(t).
b. Convert the ordinary differential equations to the state-space form and repeat Part (a).
c. Build a Simscape model of the physical system and find the displacement outputs x_1(t) and x_2(t).
a. Solving for the highest derivatives of the output x_1 and x_2, respectively, gives
\ddot{x}_1 = \frac{1}{m_1}[f – (k_1 + k_2)x_1 +k_2x_2]
\ddot{x}_2 = \frac{1}{m_2} (k_2x_1 + k_2x_2)
The corresponding Simulink block diagram is shown in Figure 5.108, where four Integrator blocks are included to obtain the signals \dot{x}_1, x_1, \dot{x}_2, and x_2. A Sine Wave block is used to model the harmonic disturbance force f = 40\text{sin}(7πt) \text{N}. Double-click the block and type 40 for the Amplitude and 7*pi for the Frequency to define the disturbance input.
b. Define the state, the input, and the output vectors as
\mathbf{x} = \begin{Bmatrix}x_1 \\ x_2 \\ x_3 \\ x_4 \end{Bmatrix} = \begin{Bmatrix}x_1 \\ x_2 \\ \dot{x}_1 \\ \dot{x}_2 \end{Bmatrix} , u = f , \mathbf{y} = \begin{Bmatrix}x_1 \\ x_2 \ \end{Bmatrix}
The state-space representation is
\begin{Bmatrix}\dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \\ \dot{x}_4 \end{Bmatrix} = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -\frac{k_1 +k_2}{m_1} & \frac{k_2}{m_1} & 0 & 0 \\ \frac{k_2}{m_2} & -\frac{k_2}{m_2} & 0 & 0 \end{bmatrix} \begin{Bmatrix}x_1 \\ x_2 \\ x_3 \\ x_4 \end{Bmatrix} + \begin{bmatrix} 0 \\ 0 \\ \frac{1}{m_1} \\ 0 \end{bmatrix}u, \mathbf{y} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}\begin{Bmatrix}x_1 \\ x_2 \\ x_3 \\ x_4 \end{Bmatrix}
The Simulink block diagram built based on the state-space form is shown in Figure 5.109, in which a State-Space block is used to represent the mass–spring system. Same as Part (a), a Sine Wave block is used to model the input. Double-click the State-Space block and define the matrices A, B, C, and D, which is a 2 × 1 zero vector. The parameter of Initial conditions is a 4 × 1 zero vector. The bar-shaped block in Figure 5.109 is called Demux, which can be found in the library of Signal Routing and is used to split the vector signal y into two signals x_1 and x_2.
c. The Simscape block diagram corresponding to the physical system is shown in Figure 5.110, which can be created by following the steps similar to those in Example 5.21.
