Tilting a mass suspended on 2 springs

Question

I was trying to model the following problem: There is a solid brick shaped body, with center of mass $(x,y,z)$. We put this body onto 2 springs. For making the problem easier, we cut out the slice from the brick, which contains its center of mass, and parallel to one of its side. So we are in 2D now. The springs are located under the 2 corners of the same side. The other ends of the springs are on the ground. Now, I want to find out the distortion of the springs under the body's mass. Furthermore, I would like to know the distortion of the springs if I tilt the ground. If we described the problem with solid beams instead of springs, (which has no distortion) I could calculate the force on each beams. But with this spring configuration I find the problem quite difficult, since the forces on each springs are function of the tilt angle, but the tilt angle of the mass is function of the springs' distortion as well. And the springs also have to change their angle where they are attached to the ground. I am asking for some papers which describe this problem.

Answer 1

It depends on the number of the degrees of freedom of your system. But if for simplicity, we imagine there are fixed railings going perpendicular to the ground and points $D$ and $C$ can only move along the lines $h$ and $g$, but not laterally. Then the system has two degrees of freedom, which can be chosen as the lengths of the springs $l_g$ and $l_h$.

Now if we fix the values $l_g$ and $l_h$, we can calculate the potential energy of the springs $$U_s = \frac{k_g}2(l_g-l_{g0})^2+\frac{k_h}2(l_h-l_{h0})^2.$$ Given the ground is tilted by some angle $\varphi$, we can calculate the height of the centre of mass $H_G$ as a some function of $l_h$, $l_g$ and $\varphi$: $H_G=f(l_h, l_g, \varphi)$. This is a purely geometrical problem. Thus, we know the full potential energy $U(l_h,l_g)=U_s+MgH_G$. In the equilibrium the potenital energy is minimal, so it should be true that: $$ \frac{\partial U}{\partial l_g} = 0,\qquad \frac{\partial U}{\partial l_h} = 0. $$ Solving these two equations for $l_g$ and $l_h$ will give you the lengths of the springs as functions of $\varphi$. And knowing the lengths of the springs, you can find the forces if needed.