# Solution to the Three-Degree-of-Freedom System Let's break down the process into **two main steps** with examples and provide the final answer (including a sample MATLAB code and a graph). --- ## **Step 1: Formulate the Equations and Initial Conditions** ### **Mass and Stiffness Matrices** Given: - \( m_1 = m_2 = m_3 = 4 \) kg - \( k_1 = k_2 = k_3 = 4 \) N/m The **mass matrix** \( M \) and **stiffness matrix** \( K \): \[ M = \begin{bmatrix} 4 & & \\ & 4 & \\ & & 4 \\ \end{bmatrix} \] \[ K = \begin{bmatrix} 8 & -4 & \\ -4 & 8 & -4 \\ & -4 & 4 \\ \end{bmatrix} \] ### **Initial Conditions** \[ x_1() = .1,\quad x_2() = ,\quad x_3() = \] \[ \dot{x}_1() = \dot{x}_2() = \dot{x}_3() = \] --- ## **Step 2: Solve & Plot Displacement and Velocity** ### **A. System of 2nd Order Equations** The equation: \[ M\ddot{x} + Kx = \] Convert to a system of 1st order ODEs: Let \( y = [x; \dot{x}] \): \[ \dot{y} = \begin{bmatrix} \dot{x} \\ -M^{-1}Kx \end{bmatrix} \] ### **B. MATLAB Example Code** ```matlab % Parameters m1 = 4; m2 = 4; m3 = 4; k1 = 4; k2 = 4; k3 = 4; M = diag([m1, m2, m3]); K = [k1+k2, -k2, ; -k2, k2+k3, -k3; , -k3, k3]; % ODE function f = @(t, y) [y(4:6); -M\K*y(1:3)]; % Initial conditions: [x1 x2 x3 dx1 dx2 dx3] y = [.1; ; ; ; ; ]; % Time span tspan = :.1:20; % Solve [t, y] = ode45(f, tspan, y); % Extract displacement and velocity for x2 x2 = y(:,2); v2 = y(:,5); % Plot figure; plot(t, x2, 'b', t, v2, 'r--'); xlabel('Time (s)'); ylabel('Responses 2'); title('Displacement x_2(t) and Velocity v_2(t)'); legend('x_2(t)', 'v_2(t)'); grid on; ``` --- ## **Final Answer: Example Output** ### **Graphical Output**  *Alt text: Graph showing displacement x2(t) and velocity v2(t) over 20 seconds.* --- ## **Summary** #### **Step 1:** - Write the mass and stiffness matrices. - Set up the initial conditions. #### **Step 2:** - Solve the ODEs using `ode45`. - Plot displacement and velocity as shown above. **This process answers part (a) and (b) from your question.** If you need the MATLAB code for parts (c)-(f), let me know!