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 - THEORY For conservative systems (i.e. no friction or energy lose), the equation of motion can be found from the conservation of energy, which states that the sum of kinetic and potential energies is a constant,      T + V = constant Taking the time derivative gives      d(T + V)/dt = 0 From this, the equation of motion for vibrations can be determined. Development of Energy Equation To understand how energy methods can be used, a simple mechanical system with linear and rotational kinetic, spring potential, and gravity potential energy will be analyzed. For the system shown, summing the kinetic and potential energies gives       Here, C is an arbitrary constant (total energy). Also, the displacement variable, yo, needs to be related to the rotation variable, θ since only one variable can be used with the energy method.      yo = r θ The moment of inertia for the pulley is      I1 = 1/2 m1 r2 Substitute these equations in the total energy equation gives,       Take the first time derivative and rearrange,       Since the velocity is not zero at all times, the equation of motion is At static equilibrium, the acceleration is zero, so the static equilibrium position is      yo = m2 g/k Rewriting the equation of motion in the new coordinate      y = yo - m2 g/k gives the familiar equation       where the natural frequency ωn is       The general solution is      y = A sinωnt + B cosωnt The constants A and B are determined from the initial conditions:

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