This problem (as well as for any problem in kinetics and vibrations) follows the O'Reilly four-step method (which we all know very well by now!). [The "Solve" step here is to set up the differential equation of motion (EOM) and to solve the differential EOM.] Let's review the four steps for this problem:
- FBD (shown above)
- Newton-Euler: sum M_C = 2*[(k/2)*x]*r = I_C*theta_dot_dot
- Kinematics: a_O = a_C + alpha x r_O/C - omega^2*r_O/C . Doing the math gives: x_dot_dot = - r*theta_dot_dot
- Solve: Combining N/E with Kinematics gives: (I_C/r^2)*x_dot_dot + k*x = 0. This is the differential EOM. You need to solve this EOM as Professor Nauman outlined in today's class. Use initial conditions of x(0) = x_0 and x_dot(0) = 0 in your solution.
sum F_x = -2*(k/2)*x - f = m*x_dot_dot
which gives:
f = -2*(k/2)*x - m*x_dot_dot
Since you know x(t), you now have f(t). For impending slip, you need to set the maximum value of f(t) equal to mu_s*m*g. From this you can find x_0.
Does this help?
10 comments:
Why did you sum the moments about point C? Why not use point O?
-John
So you can't take the equation for friction:
f = -k*x - m*x_dot_dot
solve the differential equation for x_dot_dot as:
x_dot_dot = (-k*x*r^2)/I_c
and substitute this equation into the friction equation giving:
f = -k*x - m*((-k*x*r^2)/I_c)
Then set f = mu_s*m*g and solve the resulting equation algebraically for x?
Does this not eliminate the need to solve the differential equation as we did in class?
You are correct; you could use either O or C for the moment equation.
I chose point C to eliminate some of the effort in deriving the EOM. Say you choose to use the moment equation about O:
sum M_O = -f*r = I_O*theta_dot_dot
This picks up the unknown friction force f. If we now use:
sum F_x = -f - 2*(k/2)*x = m*x_dot_dot
we get another equation involving f. We can combine the above equations to get:
2*(k/2)*x*r = I_O*theta_dot_dot-m*r*x_dot_dot
Using the kinematics along with the parallel axis theorem, this reduces to the same equation that we got directly by summing moments about C. Same equation, less work.
Does that help?
Evan,
I almost agree with you. What is "x" in your answer? x should be a function of time with x_0 being the amplitude. You find x(t) by solving the differential EOM.
Okay, so in your solution, you are assuming that at time t = 0, the spring is at its maximum stretch, and x = x_0?
This would be opposed to assuming that at time t = 0, the spring is in its equilibrium position as we did in class.
There are many sets of initial conditions that will give you a response for x(t) that has an amplitude of x_0. The set of initial conditions that I chose is just one of many.
You could start the system out with it at the static equilibrium position (which for this problem is x = 0). You can find an initial value for x_dot that would give you a response amplitude of x_0. That would take a little bit of calculations to find that initial condition.
Again, I just took an easy route.
Does this make sense?
Thanks! That makes more sense now.
-John
Oops, nevermind, in class we assumed that the spring is unstretched at x = 0, not t = 0.
Also, after reworking the problem the way you suggested, I arrived at the same answer I had gotten without actually solving the differential equation earlier. It may just be a fluke that my answer from before came out the same...
Yes, it is making sense now, thank you.
No, I don't think that it was a fluke. Any "disagreement" that I had with your approach is not with your method or answer, but whether you understood what your answer represents.
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