The four steps:
- FBD's - shown above for crank and block individually
- Newton/Euler - For crank: sum M_O = -T*a - (c*y_dot)*b = I_O*alpha = 0. For block: sum F_x = T - mg - kx = m*x_dot_dot
- Kinematics: Need to relate y_dot to x_dot.
- Solve: Combine the two Newton/Euler equations by eliminating T and enforce Kinematics from Step 3. This gives single differential equation of motion (EOM).
From EOM found in Step 4, identify damping ratio. Recall discussion in today's lecture on this.
Let us know if you have any questions.
7 comments:
When I solved for the EOM, I followed the steps given and had two acceleration terms (x_dot_dot) and (g). In the EOM how do we consolidate them into one term? I guess this is just a notation question. Right now I have it as
(-m(x_dot_dot)-(m(g)))ra
No need to consolidate the two terms. Leave the "g" as an inhomogeneous term in the EOM.
Once you have the motion equation, to find the dampening ratio do you set:
c*x-dot*(stuff) = 2*eta*omega-n*x-dot?
I did this because it seemed we used this equation in class to solve for eta.
Thanks.
That's that same thing I did. Only I didn't include the x_dot's. I just left them out, but they would cancel out anyway in your eqn. Then just solve for the damping ratio zeta.
So should our EoM have a forcing function (g) on the right side of the equation?
I noticed that in article 8/2 the book shows how to eliminate this term by using equilibrium position as reference instead of the position where the spring itself is unstretched.
I don't believe there is a forcing function for this problem. Forcing functions will be made apparent in the book's figure (see tomorrow's homework problem, 8/48).
The professor's FBD shows all of the forces on the system. So if you are having trouble distinguishing forces and free bodies, refer to the image on the main page.
Good luck.
That's correct, this system only has internal forces, and a forcing function is defined by an external force.
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