The motion is generated by a rotating object that is off balance. This causes the wheel to move up and down. This motion can be be modeled by first examining the vertical motion of the wheel's center of mass. In terms of the rotation angle, θ, the displacement is
yw = y + d sinθ
This motion will generate a force based on the mass, m2 and acceleration of the wheel, d2ym/dt2.
This motion is resisted by the spring and damping cylinder. After summing all forces, including the overall acceleration mass m1, gives
This can be rearranged to give