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Problem #2 - Modeling, Mass-Spring System = 10 N/m, m₁ = Two masses are connected by springs in series as shown in Figure 1 below where k₁ = 20 N/m, k₂ 10 kg, and m₂ 5 kg. Solve the system of ODEs in matrix form using the determinant to obtain the characteristic equation (it will make sense to substitute > for w² when you get to your characteristic equation). Determine the eigenvalues (frequencies) and eigenvectors (modes) of oscillation. Assume that no damping is present and that the masses slide without friction. On your diagram, sketch the modes of oscillation (think of how the masses could oscillate with respect to each other). What determines which mode or combination of modes that the masses oscillate at? i.) ii.) iii.) iv.) v.) vi.) vii.) Sketch the system and include a free body diagram. Use a conservation equation to develop your differential equation model (you should end up with two ODEs to describe the motion of both masses). Write your system of ODEs in matrix form. Set up the eigenvalue problem and then determine the eigenvalues. Use the eigenvalues to determine the eigenvectors. Write the general solution for the position of each mass as a function of time. Sketch the modes of vibration using your eigenvectors. www m₁ ww M2

Problem #2 - Modeling, Mass-Spring System = 10 N/m, m₁ = Two masses are connected by springs in series as shown in Figure 1 below where k₁ = 20 N/m, k₂ 10 kg, and m₂ 5 kg. Solve the system of ODEs in matrix form using the determinant to obtain the characteristic equation (it will make sense to substitute > for w² when you get to your characteristic equation). Determine the eigenvalues (frequencies) and eigenvectors (modes) of oscillation. Assume that no damping is present and that the masses slide without friction. On your diagram, sketch the modes of oscillation (think of how the masses could oscillate with respect to each other). What determines which mode or combination of modes that the masses oscillate at? i.) ii.) iii.) iv.) v.) vi.) vii.) Sketch the system and include a free body diagram. Use a conservation equation to develop your differential equation model (you should end up with two ODEs to describe the motion of both masses). Write your system of ODEs in matrix form. Set up the eigenvalue problem and then determine the eigenvalues. Use the eigenvalues to determine the eigenvectors. Write the general solution for the position of each mass as a function of time. Sketch the modes of vibration using your eigenvectors. www m₁ ww M2

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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### Problem #2 – Modeling, Mass-Spring System Two masses are connected by springs in series as shown in Figure 1 below where \( k_1 = 20 \, N/m \), \( k_2 = 10 \, N/m \), \( m_1 = 10 \, kg \), and \( m_2 = 5 \, kg \). Solve the system of ODEs in **matrix form** using the determinant to obtain the characteristic equation (it will make sense to substitute \( \lambda \) for \( \omega^2 \) when you get to your characteristic equation). Determine the eigenvalues (frequencies) and eigenvectors (modes) of oscillation. Assume that no damping is present and that the masses slide without friction. On your diagram, sketch the modes of oscillation (think of how the masses could oscillate with respect to each other). What determines which mode or combination of modes that the masses oscillate at? i.) Sketch the system and include a free body diagram. ii.) Use a conservation equation to develop your differential equation model (you should end up with two ODEs to describe the motion of both masses). iii.) Write your system of ODEs in matrix form. iv.) Set up the eigenvalue problem and then determine the eigenvalues. v.) Use the eigenvalues to determine the eigenvectors. vi.) Write the general solution for the position of each mass as a function of time. vii.) Sketch the modes of vibration using your eigenvectors. --- #### Figure Description **Figure 1:** Schematic showing two masses connected by springs in series. Assume the springs have no damping and that the masses slide without friction. - **Diagram Description:** - There are two blocks representing masses \( m_1 \) and \( m_2 \). - Mass \( m_1 \) is connected to the left fixed support through a spring with spring constant \( k_1 \). - Mass \( m_1 \) is also connected to mass \( m_2 \) using a spring with spring constant \( k_2 \). - The block representing mass \( m_1 \) is labeled with \( m_1 \) (10 kg) and the block representing mass \( m_2 \) is labeled with \( m_2 \) (5 kg). This text provides a detailed scheme to address the mass-spring system involving understanding
Transcribed Image Text:### Problem #2 – Modeling, Mass-Spring System Two masses are connected by springs in series as shown in Figure 1 below where \( k_1 = 20 \, N/m \), \( k_2 = 10 \, N/m \), \( m_1 = 10 \, kg \), and \( m_2 = 5 \, kg \). Solve the system of ODEs in **matrix form** using the determinant to obtain the characteristic equation (it will make sense to substitute \( \lambda \) for \( \omega^2 \) when you get to your characteristic equation). Determine the eigenvalues (frequencies) and eigenvectors (modes) of oscillation. Assume that no damping is present and that the masses slide without friction. On your diagram, sketch the modes of oscillation (think of how the masses could oscillate with respect to each other). What determines which mode or combination of modes that the masses oscillate at? i.) Sketch the system and include a free body diagram. ii.) Use a conservation equation to develop your differential equation model (you should end up with two ODEs to describe the motion of both masses). iii.) Write your system of ODEs in matrix form. iv.) Set up the eigenvalue problem and then determine the eigenvalues. v.) Use the eigenvalues to determine the eigenvectors. vi.) Write the general solution for the position of each mass as a function of time. vii.) Sketch the modes of vibration using your eigenvectors. --- #### Figure Description **Figure 1:** Schematic showing two masses connected by springs in series. Assume the springs have no damping and that the masses slide without friction. - **Diagram Description:** - There are two blocks representing masses \( m_1 \) and \( m_2 \). - Mass \( m_1 \) is connected to the left fixed support through a spring with spring constant \( k_1 \). - Mass \( m_1 \) is also connected to mass \( m_2 \) using a spring with spring constant \( k_2 \). - The block representing mass \( m_1 \) is labeled with \( m_1 \) (10 kg) and the block representing mass \( m_2 \) is labeled with \( m_2 \) (5 kg). This text provides a detailed scheme to address the mass-spring system involving understanding
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