A 2dof system is shown m1 lb-sec2in c10 lb-sein k10000 lbin

Question

Problem - A 2DOF system is shown (M=1 lb-sec²/in, c=10 lb-se/in, k=10000 lb/in);

a) Draw the free body diagram and write the two equations describing its motion.

b) Write the equations of motion in matrix format.

c) Determine the natural frequencies for the two modes by hand calculations and Matlab(eig).

d) Determine the mode shapes for the two modes by hand calculations and Matlab(eig).

e) Show that the modes are orthogonal w.r.t. the mass and stiffness by hand calculations.

f) Find the response of the system to the initial conditions as (x1=1in; x1'=0.5 in/s; x2=2, and x2'=- 0.5). For this problem, perform an analytical solution.

g) Use ODE45 to perform a numerical solution for part f. Plot the response in Matlab and overlap the two graphs (analytical and numerical) for displacement and velocity of the two masses.

h) Identify an initial displacement that will excite (1) the first mode only (2) the second mode only. Simulate your Matlab model with your identified initial condition and plot the response.

i) From a mode super position approach, write the analytical steady state response due to sinusoidal excitation of Fsin(0.9ω₁t) applied on the 2M mass at DOF 1; write the solution for each modal DOF of the system and the combined response. Plot the results in Matlab.

You need to use transfer the modal to Modal Space (where the two degrees of freedom are independent) and solve each equation using analytical solution for 1DOF systems. Then you transfer the response from modal space to physical space.

j) From a mode superposition approach, write the analytical steady state response due to sinusoidal excitation of Fsin(0.5(ω₁ +ω₂)t) applied on the 4M mass at DOF 2. Write the solution for each modal DOF of the system and the combined response. You need to use analytical solution for 1DOF systems. Plot the response in Matlab.

Add figures and plots and make sure you refer to them in the text.