Looking for help with a statics problem

[MobiusMind]

So, I've been trying to figure this problem out for a good part of the day. If anyone could help me out or give me some guidance I would really appreciate it. I basically have a long cylinder sitting on some compression springs. The bottom of the cylinder is hollowed out and the springs are equally spaced along the bottom edge of the cylinder, say one spring every 10 degrees, and the springs are identical with the same spring constant. The center of gravity of the cylinder is located high on the cylinder and a little bit off of the centerline. Not enough that the cylinder will topple over, but just enough that the cylinder will sit at an angle. So what I am looking for is how to figure out what that angle will be and what the deflection in the springs will be, assuming everything is in equilibrium and there are no oscillations. It seems like it should be a simple statics problem, but I must be missing something. thanks

 

[nvn]

Calculate the moment of inertia of the springs around the cylinder base, then apply the cylinder force and moment due to gravity to the spring pattern. For 36 springs around the cylinder base, I currently got maximum spring deflection, delta = -0.02778*m*g*(4*b/d + 1)/k, and cylinder rotation angle, alpha = asin[0.2222*m*g*b/(k*d^2)], where m = cylinder and contents total mass (located at center of gravity, CG), g = 9.8067 m/s^2, d = cylinder diameter, b = horizontal distance between cylinder axial centerline and mass CG, and k = spring constant for each spring.

 

[MobiusMind]

Thanks nvn, I will try to duplicate your equations. Just curious, how would that scale if I were to use 200 springs instead of 36?

 

[tiny-tim]

Welcome to PF!

I basically have a long cylinder sitting on some compression springs. The bottom of the cylinder is hollowed out and the springs are equally spaced along the bottom edge of the cylinder, say one spring every 10 degrees, and the springs are identical with the same spring constant. The center of gravity of the cylinder is located high on the cylinder and a little bit off of the centerline. Not enough that the cylinder will topple over, but just enough that the cylinder will sit at an angle. So what I am looking for is how to figure out what that angle will be and what the deflection in the springs will be, assuming everything is in equilibrium and there are no oscillations. It seems like it should be a simple statics problem, but I must be missing something. thanks

Hi MobiusMind ! Welcome to PF!

Yes, it's a simple statics problem (so moment of inertia shouldn't come into it).

Hint: in problems ike this, draw a diagram, and give letters to all the unknowns.

In this case, the two unknowns are the angle of tilt (call it θ), and the compressed length of the lowest spring (or the highest, or the middle ones, whichever you chose, call it x).

Then you know the lengths of all the other springs, in terms of θ and x, so you can calculate their compression forces.

Then put those forces in the diagram, together with the weight, and use the usual statics rules to find θ.

 

[nvn]

MobiusMind: For 200 springs, I currently got maximum spring deflection, delta = -m*g*(4*b/d + 1)/(n*k), and cylinder rotation angle, alpha = asin[8*m*g*b/(n*k*d^2)], where n = total number of equally-spaced springs, and all other parameters are as defined in my first post.

 

[nvn]

MobiusMind: To be more general, maximum spring deflection is delta = -(4*P*b + 4*F*z + P*d)/(n*k*d), and cylinder rotation angle is alpha = asin[8*(P*b + F*z)/(n*k*d^2)], where P = resultant of all vertical applied loads (including cylinder and contents weight), F = resultant of all horizontal applied loads (if any), b = horizontal distance between cylinder axial centerline and load P, z = vertical distance between cylinder base and load F, d = cylinder diameter, n = total number of equally-spaced springs, and k = spring constant for each spring.

The above formulas assume moment F*z is in the same direction as moment P*b. If, instead, moment F*z can only be applied in the opposite direction of moment P*b, then input a negative value for F, in both of the above formulas.

Parameter b is obtained by computing the moment (about a horizontal axis at the cylinder base passing through the cylinder axial centerline) of all vertical applied loads (including the cylinder and contents weight), then dividing this moment by P. Likewise, z is obtained by computing the moment (about the same axis used for computing b) of all horizontal applied loads, then dividing this moment by F.