The diagram below represents a three degree of freedom system, with the following equation of motion:

Where is the mass matrix and  is the stiffness matrix.

Assume that  kg,  kg,  N/m,  N/m and  N/m.

What line(s) of MATLAB code can be used to calculate the eigenvalues and mode shapes of the system from the mass and stiffness matrix?

Hint: The output must be a diagonal matrix, , containing the eigenvalues and an orthogonal matrix, , holding the mode shapes in its columns. There may be more than one correct answer.

The plots (A to F) represent the response of the mass in Task 1, , for different damping values when  kg, N/m and =1.5 m.

Using the Simulink model that you created, select the plots that describes the response of the mass for each of the following damping values:

Using the Simulink model that you created for Task 1, select the damping coefficient and initial height of the mass that produces the velocity profile shown below.

Assume that  kg and  N/m.

The single degree of freedom system in the figure below is subjected to forced vibrations to better understand its dynamic behaviour. 

Match one of the forced vibration responses (A - J) to each of the following forcing functions and damping values.

Assume that 

Use the Simulink model that you created for Task 1 to select the system parameters and initial conditions that yields the free vibration response shown in the figure below.

Use the Simulink model that you created for Task 2 to select the system parameters and initial conditions that yields the free vibration response shown in the figure below.

Suppose that you want to understand the behaviour of the single degree of freedom system shown in the figure below.  A mass

The system is subjected to some initial conditions and you want to determine the maximum positive displacement (

Select the correct values for 

As in the previous question, assume that a mass

Select all the possible combinations of viscous damping and friction that will result in a response with the following characteristics:

• The maximum positive displacement of the mass is less than 0.04 m.
• The maximum negative displacement of the mass is less than -0.08 m (i.e. the maximum negative displacement of the mass is between -0.08 m and 0 m).
• The mass stops oscillating before 1 s.

Assume that 

As in the previous question, we are interested in the forced vibrations of a single degree of freedom.

Select the vibration response (A - J) that best matches each of the following descriptions:

Use the Simulink model that you created for Task 2 with the following system parameters and initial conditions:

What is the minimum change that must be made to the viscous damping coefficient,