Chapter 2 - Free Vibration of 1 DOF Systems with Friction (Coulomb Damping)

Question 1

What can Free Vibration of 1 DOF Systems with Friction (Coulomb Damping) result in?

Answer 1

Can result in the system not stopping at the neutral point.

Question 2

Eqn for total energy?

Answer 2

E = T + U or more formally:
sumT + sumU = E
T=KE and U=PE

Question 3

Eqn relating T and U if a block is moved from one state to another?

Answer 3

sumT1 + sumU(1->2) = sumT2
1/2.m.v^2 + (P.x - mu.N.x) = 1/2.m.v^2
==> P.x = mu.N.x (WHICH IS NOT STRICTLY TRUE)

Question 4

Can you draw the FBD’s for these two states? (1&2)?

Answer 4

YES OR NO

Question 5

Why is P.x = mu.N.x not strictly true? How do we get heating?

Answer 5

The driving force moves through x, but the friction force moves through a slightly smaller distance x(bar) due to microscopic deformations (asperities) of the structure. DRAW DIAGRAM. Actual friction force is therefore mu.N.x(bar). Thus, a small amount of energy mu.N(x-x(bar)) is available to be released as heat after all.

Question 6

Equilibrium eqn for vibrating systems with friction?

Answer 6

F - kx = mx(double dot)

mx(double dot) + kx = F

Question 7

Complimentary function of this eqn?

Answer 7

x=Asin(w_nt)+Bcos(w_nt)

Question 8

P.I for this eqn?

Answer 8

x= F/k

Question 9

using initial conditions x(0)=X and x(dot)(0)=0 gives us the overall behaviour of our system (at least during the first half-cycle of the motion) as? (EOM)

Answer 9

x = (X-F/k)cos(w_nt) + F/k

Question 10

Can you draw the graph of the response of the system?

Answer 10

YES OR NO

Question 11

When does the motion stop?

Answer 11

as soon as the mass comes to rest in the dead zone (a max or min occurs in the dead zone)

Question 12

How wide is the dead zone?

Answer 12

Friction force = F, spring force = kx as long as F=kx system will rest and not start again ==> dead zone is 2F/k wide.