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 m1 and the spring k1 represent a rotating machine, which is subjected to a harmonic disturbance force f = 40sin(7πt) N due to a rotating unbalanced mass. The mass

Question

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 [latex]m_1[/latex] and the spring [latex]k_1[/latex] represent a rotating machine, which is subjected to a harmonic disturbance force [latex]f = 40 ext{sin}(7πt) ext{N}[/latex] due to a rotating unbalanced mass. The mass block [latex]m_2[/latex] and the spring [latex]k_2[/latex] 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

[latex]m_1\ddot{x}_1 + (k_1 + k_2)x_1 - k_2x_2= f[/latex]

[latex]m_2\ddot{x}_2 - k_2x_1 + k_2x_2 = 0[/latex]

where [latex]m_1 = 6 ext{kg}, k_1 = 6000 ext{N/m}, m_2 = 1.65 ext{kg}[/latex], and [latex]k_2 = 800 ext{N/m}[/latex]. Assume zero initial conditions.
a. Build a Simulink model of the system based on the differential equations of motion and find the displacement outputs [latex]x_1(t)[/latex] and [latex]x_2(t)[/latex].
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 [latex]x_1(t)[/latex] and [latex]x_2(t)[/latex].

Accepted Answer

a. Solving for the highest derivatives of the output [latex]x_1[/latex] and [latex]x_2[/latex], respectively, gives

[latex]\ddot{x}_1 = \frac{1}{m_1}[f - (k_1 + k_2)x_1 +k_2x_2][/latex]

[latex]\ddot{x}_2 = \frac{1}{m_2} (k_2x_1 + k_2x_2) [/latex]

The corresponding Simulink block diagram is shown in Figure 5.108, where four Integrator blocks are included to obtain the signals [latex]\dot{x}_1, x_1, \dot{x}_2[/latex], and [latex]x_2[/latex]. A Sine Wave block is used to model the harmonic disturbance force [latex]f = 40 ext{sin}(7πt) ext{N}[/latex]. Double-click the block and type [latex]40[/latex] for the Amplitude and [latex]7*pi[/latex] for the Frequency to define the disturbance input.

b. Define the state, the input, and the output vectors as

[latex]\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}[/latex]

The state-space representation is

[latex]\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}[/latex]

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 [latex]2 × 1[/latex] zero vector. The parameter of Initial conditions is a [latex]4 × 1[/latex] 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 [latex]x_1[/latex] and [latex]x_2[/latex].

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.

Question 5.22: A Two-Degree-of-Freedom Mass–Spring System Consider the two-...

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