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

     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



The homogeneous equation is


and has the solution




The particular solution can be found by substituting

     yP(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) = yc(t) + yP(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.


Results for Different RPMs
Practice Homework and Test problems now available in the 'Eng Dynamics' mobile app
Includes over 400 free problems with complete detailed solutions.
Available at the Google Play Store and Apple App Store.