Equation of equilibrium of a SDOF

[Andrew_unique]

Hi guys! I would really need some help with a problem I have over here:
how do I you get the equations of equilibrium for the scheme in picture of the single degree of freedom system? I am having some trouble understanding why I have m1*Hc*(H*theta_0) mostly, but also about why I have moment of inertia I_i*rotation or mass*displacement; shouldn't it be m*a and I*angular acceleration? or does the multiplication with angular frequency make this? I appreciate all answers related to this topic.

Yours truly,
Andrew

 

[Illmatic1]

The equation of equilibrium for the single degree of freedom system in the picture can be written as follows:m1*Hc*(H*theta_0)= m2*g*L + I_i*ω^2*θ - m2*a*Lwhere m1 is the mass of the object, Hc is the spring constant, H is the difference between the initial and final displacement of the spring, theta_0 is the initial angle of the system, m2 is the mass of the other object, g is the gravitational acceleration, L is the length of the string, I_i is the moment of inertia, ω is the angular frequency, θ is the angle of the system, and a is the acceleration of the other object. The equation relates the forces acting on the system to the motion of the system. The left-hand side of the equation represents the spring force, which is equal to the product of the mass of the object, the spring constant, and the difference between the initial and final displacement of the spring. The right-hand side of the equation contains the gravitational force, the rotational force due to the moment of inertia, and the linear force due to the acceleration of the other object. By solving this equation, you can determine the motion of the system, given the forces acting on it.