 Ch 10. Vibrations Multimedia Engineering Dynamics Free Vibs. Undamped Free Vibs. Damped Forced Vibration Energy Method
 Chapter - Particle - 1. General Motion 2. Force & Accel. 3. Energy 4. Momentum - Rigid Body - 5. General Motion 6. Force & Accel. 7. Energy 8. Momentum 9. 3-D Motion 10. Vibrations Appendix Basic Math Units Basic Equations Sections Search eBooks Dynamics Statics Mechanics Fluids Thermodynamics Math Author(s): Kurt Gramoll ©Kurt Gramoll DYNAMICS - CASE STUDY SOLUTION 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 where 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

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