# 3-DOF Mass-Spring System: Analytical Solution ## **Step 1: Formulate the Equation of Motion** Given: - Masses: \( m_1 = m_2 = m_3 = 4\,\text{kg} \) - Spring constants: \( k_1 = k_2 = k_3 = 4\,\text{N/m} \) - Initial conditions: \( x_1() = .1\,\text{m},\ \dot{x}_1() = \) \( x_2() = \dot{x}_2() = \) \( x_3() = \dot{x}_3() = \) - \( \leq t \leq 20\,\text{s} \), increment = .1s ### **System Matrices** The equations of motion for the system: \[ M\ddot{x} + Kx = \] Where: \[ M = \begin{bmatrix} 4 & & \\ & 4 & \\ & & 4 \end{bmatrix} \] \[ K = \begin{bmatrix} 8 & -4 & \\ -4 & 8 & -4 \\ & -4 & 4 \end{bmatrix} \] (The stiffness matrix is constructed considering each mass's connection to the springs.) ### **Normalized Equation** \[ \ddot{x} + M^{-1}Kx = \] \[ M^{-1}K = \frac{1}{4}K \] \[ \Rightarrow \ddot{x} + A x = ,\ \text{where}\ A = \frac{1}{4}K \] \[ A = \begin{bmatrix} 2 & -1 & \\ -1 & 2 & -1 \\ & -1 & 1 \end{bmatrix} \] --- ## **Step 2: Find Solution Using Modal Analysis** The general solution for a system with initial displacement on \(x_1\) only: ### **a. Find Eigenvalues and Eigenvectors** Solve \( \det(A - \lambda I) = \): Eigenvalues (natural frequencies squared), \( \lambda_i = \omega_i^2 \), are: \[ \lambda_1 \approx .438,\quad \lambda_2 = 2,\quad \lambda_3 \approx 2.561 \] So, \[ \omega_1 \approx .662\,\text{rad/s},\quad \omega_2 \approx 1.414\,\text{rad/s},\quad \omega_3 \approx 1.6\,\text{rad/s} \] ### **b. Write General Solution** The response is: \[ x(t) = \sum_{i=1}^{3} \phi_i \left[ A_i \cos(\omega_i t) + B_i \sin(\omega_i t) \right] \] Where \( \phi_i \) are mode shapes (eigenvectors), and \( A_i, B_i \) are constants found using initial conditions. Given initial velocity is zero, only cosines will remain, and constants are found by projecting initial displacement onto mode shapes. ### **c. Example Calculation** Let's denote the normalized eigenvectors as: \[ \Phi = \begin{bmatrix} \phi_{11} & \phi_{12} & \phi_{13} \\ \phi_{21} & \phi_{22} & \phi_{23} \\ \phi_{31} & \phi_{32} & \phi_{33} \end{bmatrix} \] Let initial displacement be \( x() = [.1, , ]^T \). Project onto modes: \[ A_i = \phi_i^T x() \] The full time response is then a linear combination of these modes. --- ## **Step 3: Plot the Results** Below is a sample Python code to plot \( x_1(t), x_2(t), x_3(t) \): ```python import numpy as np import matplotlib.pyplot as plt from scipy.linalg import eigh # System matrices M = np.eye(3) * 4 K = np.array([[8, -4, ], [-4, 8, -4], [, -4, 4]]) # Eigenvalues and eigenvectors w2, phi = eigh(K, M) w = np.sqrt(w2) # Initial conditions x = np.array([.1, , ]) v = np.zeros(3) # Modal coordinates A = phi.T @ x B = phi.T @ v / w # Time vector t = np.arange(, 20.1, .1) X = np.zeros((3, len(t))) # Time response for i in range(3): X += np.outer(phi[:, i], A[i] * np.cos(w[i] * t) + B[i] * np.sin(w[i] * t)) plt.plot(t, X[], label='$x_1(t)$') plt.plot(t, X[1], label='$x_2(t)$') plt.plot(t, X[2], label='$x_3(t)$') plt.xlabel('Time (s)') plt.ylabel('Displacement (m)') plt.title('Response of 3-DOF System') plt.legend() plt.grid(True) plt.show() ``` --- ## **Final Answer** - The motion of each mass as a function of time is a **combination of three harmonic motions** (normal modes) with frequencies approximately \( .662 \), \( 1.414 \), and \( 1.6 \) rad/s. - The **time histories** can be plotted as shown above, with \( x_1(t) \) starting at .1 m and the others at zero. --- ### **Summary Table** | Step | Action | Result | |------|--------|--------| | 1 | Write equations, find matrices | \( M, K \) and normalized system | | 2 | Solve eigenproblem, apply initial conditions, plot | Time history for \( x_1, x_2, x_3 \) | --- **Example Plot Output:**  *Illustrative plot of responses \(x_1(t)\), \(x_2(t)\), \(x_3(t)\).*