 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 Fluids Math Mechanics Statics Thermodynamics Author(s): Kurt Gramoll ©Kurt Gramoll DYNAMICS - CASE STUDY SOLUTION Problem Parameters A free-body diagram can be used to identify all the forces on a the platform. The weight, mg, will act on the spring but the spring force will counter-act the weight, so the weight actually cancels. The spring deflection will cause a force, ky, that will accelerate the platform. The final equation of motion becomes, The general solution of this differential equation is       y(t) = A sinωnt + B cosωnt where ωn represents the natural frequency and is defined as For the two cases, k1 = 5,000 and k2 = 10,000, the natural frequencies are found:      case 1:   ωn1 = 2.24 rad/s      case 2:   ωn2 = 3.16 rad/s The initial conditions for this particular situation are      y(0) = -20 ft      dy(0)/dt = 0 These two conditions can be used to determined the constants A and B. The final equation becomes      y(t) = -20 cosωnt Case 1 Results Case 1 Results The velocity equation is determined by taking the time derivative of the displacement equation,      dy/dt = -20 ωn sinωnt Likewise, the acceleration equation is found taking the second time derivative of the displacement,      d2y/dt2 = -20 ωn2 cosωnt Therefore, the maximum acceleration is directly related to the square of the natural frequency, or This can be converted to g forces by dividing by the acceleration of gravity,      gmax = 20/32.2 ωn2 = 0.621ωn2 g's The motion of the platform and its acceleration are plotted at the left for the two spring constants of 5,000 and 10,000 lb/ft. The 10,000 lb/ft case is unacceptable since the acceleration exceeds the safety limit of 4 g's.

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