As shown in the ?gure, a bar is supported by two springs and a damper. The bar's left end is free to move vertically and the entire bar can rotate around this point. The bar is uniform with mass M and length L giving a mass moment about the center of mass ML2/12. The ?rst spring (k1) and a load (F) act at the center of mass. The other spring (k2) and damper act at the end of the bar. Neglect gravity.
1.a Show that the equations governing the motion of this system can be written as
Assuming L = 2 m, k1 = 2000 N/m, k2 = 500 N/m, and M = 10 kg, determine the following:
1.b. The natural frequencies and mode shapes of the undamped system.
1.c. The damping coef?cient at which two eigenvalues ?rst become real numbers.
1.d. The free response (symbolically and graphically) when the damping ratio is 10% of what you found in 1.b., the left end of the bar is initially displaced 1 cm, and the right end is initially held at it's equilibrium position.
Assuming L = 2 m, k1 = 2000 N/m, k2 = 500 N/m, and M = 10 kg, determine the following:
1.b. The natural frequencies and mode shapes of the undamped system.
1.c. The damping coef?cient at which two eigenvalues ?rst become real numbers.
1.d. The free response (symbolically and graphically) when the damping ratio is 10% of what you found in 1.b., the left end of the bar is initially displaced 1 cm, and the right end is initially held at it's equilibrium position.
a. Show that the equations of motion can be written as
b. Using the following parameters:
k = 1000 N/m, kt = 200 N-m, m = 10 kg, J = 5/6 kg-m2, d = 1/5 m
determine the natural frequencies and sketch the mode shapes when a = 300 N.
c. Construct the homogeneous solution in terms of four constant that could be determined from initial conditions.
d. If a increases to 350 N, ?nd the eigenvalues and discuss how the system will behave if given an initial perturbation.
