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 Marble Path   Marble Path   Angle θ related to φ Most problems in dynamics can be solved by using energy methods. This method is particularly helpful with complex systems. Energy will be used for the problem to demonstrate how it can be used. First, the total energy, both potential and kinetic, needs to be determined for the system. The gravitational potential energy V of the marble is      V = mgh The kinetic energy of the marble, including both linear and rotational, is       One problem is that there is three variables, h, v, and dφ/dt that describe the marble motion and position. Energy methods can only have one unknown variable, so the three must be related. The marble's vertical location h is directly related to angle θ by      h = (R - r) (1 - cosθ) The marble linear velocity is related to the angular velocity dθ/dt through the arc as      Finally, the angle φ can be expressed in terms of the angle θ through the relationship       Adding potential and kinetic energy (total energy), and using h, v, and dφ/dt relationships give Solution Results: Initial Angle = -10o Solution Results: Initial Angle = -35o Solution Results: Initial Angle = -60o I = 2/5 mr2 Substituting I, and simplifying gives The total energy, E, is constant regardless of the marble location. Thus, the first derivative with respective to time gives,       Rearranging gives,       If the motion is restricted to small oscillations, sinθ can be approximated by θ and the equation of motion can be simplified to       and the natural frequency ωn is       The general solution to this equation is      θ(t) = A sinωnt + B sinωnt where       Results are given for initial angles of -10o, -35o and -60o degrees. Note that for comparison, the results have been non-dimensionalized with respect to the initial conditions.

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