Offset Center of Mass

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

     y_w = y + d sinθ

This motion will generate a force based on the mass, m₂ and acceleration of the wheel, d²y_m/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

The homogeneous equation is

and has the solution

where

The particular solution can be found by substituting

     y_P(t) = D sin(ωt - φ)

into the differential equation

The constants, D and φ, can be found by separating the terms involving cos and sin:,

The total solution is the sum of the particular and homogeneous parts,

     y(t) = y_c(t) + y_P(t)

Substituting gives,

The initial conditions, y(t=0), and dy(t=0)/dt, are used to determine the constants B and C,

This gives

The motion of the wheel can now be plotted using the appropriate system parameters. The results for wheel speeds of 90, 120, and 150 rpm are graphed below.