# ME 431 Homework

## Homework 01

Now batting... Homework 01.

Lyric of the day
There is moonlight and moss in the trees,

On the second problem of HW1, am I ok if I use the small angle assumption? This is in regard to the deflection of k2 which connects to the wall and the outer radius of the disk. If I assume a small angle, I can say that the axis of the spring stretch is in the i direction, whereas if I don’t assume a small angle, the axis of stretching rotates.
Actually, you don’t need to use a small angle assumption. The cable to which the spring is attached is wrapped around the disk, so that the deflection of the spring is always in the i direction, regardless of the angle of rotation.
Thanks

For problem #2, is R2=2*R1?
Hi... not necessarily. You should just treat them as separate values.
No, r1 and r2 are just arbitrary values for the inner and outer radii...

Dr. Quinn,

Did you post the homework solutions for hw 01? I did not see them.

Dr. Quinn,

For problem 6, above the left spring constant symbol, k1, what is X denoting? Is it a symbol of the direction of force from spring/damper onto the mass? Is it the change in length from equilibrium of the spring/damper?

Same question for problem 5, where a fixed position is noted as X.

For problem 2, in the problem statement are you asking us to find the deflection of the spring from the complimentary solution set, or in other words when the differential equation describing the the deflection of the springs is equal to zero? How can we apply the next statement in the problem description that in the absence of gravity the springs are unstretched in their equilibrium equation?
In both problems 5 and 6, the coordinate x describes the displacement of the object. In problem 5 it is measured from the configuration of the system when the force F vanishes, and in problem 6 it is measuring from the equilibrium position.

In problem 2, I am asking for the stretch in the springs when the system is in equilibrium. I am also describing the initial stretch in each spring in the absence of the gravitational forces, which is zero.
Professor Quinn,

In your solutions you related the the displacement of the blocks as x1=-r*theta and x2=r*theta can I relate them opposite of that ? So, I would think x1=-r*theta and x2=r*theta

Thanks
A similiar questions for the angular momentum equation you derived the sum of moments to be (T1 r − T2 r) kˆ = IG¨θ k and I would derive it to be (T2 r - T1 r). I assumed this because T2>T1 and they are the same distance from where we summed the moments, and it agrees with your positive sign convection of theta . So, in other words, the two external forces applied on the disk are -T1 and -T2 in the j direction. If T2>T1 then the disk will tend to rotate in the ccw direction around the k hat direction. So, when I apply the angular momentum equation it yields T2 r - T1 r = IG ‘’θ k.
Dr Quinn,

Sorry I’ve go another question as well, How did you use the 3 equations from newtons second law to get the denominator :

theta dot = (W1-W2)r/(Ig + (m1+m2)r^2)

On the problem 1a solution, I am able to follow everything up to “Combining with the coordinate constraints, we may solve for theta double dot.” Then there is some magic that happens to get the resulting solution. No matter how I try to combine the equations, I am unable to get the same result. What do I do with x1 double dot, and x2 double dot?
I have the same question Dr. Quinn.
It’s a matter of solving the set of equations. I would take the second and third equations listed and solve for T1 and T2, then substitute back into the first equation shown. Finally, you can eliminate x1_ddot and x2_ddot by using the coordinate relations, which should provide the answer.
Problem 1a discussion:

I did exactly as you described, but I do not see how the x1_ddot and x2_ddot cancel out. My best guess was to take the second derivative of x1 and x2, which is 0. When I do that, I am still lost on where (m1 + m2)r^2 in the denominator comes from.

Professor can you elaborate further?

You have five equations. Three from momentum balance and constraint equations. Then, there are five unknowns (T1, T2, x1, x2, theta). Eliminate everything except theta and solve.

The second derivative of x1 is related to theta as
x1_ddot = r theta_ddot

Dr. Quinn,

Could you please explain how the forces acting on the block in Problem 6 are all negative? Thanks.

If x is positive then the forces are in the -i direction.