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Unformatted text preview: Chapter 2 Single DegreeofFreedom Vibration: Discrete Models Problems for Section 2.2 &Math Modeling: Deterministic 1. The beam in Figure 2.26 vibrates as a result of loading not shown. State the necessary assumptions to reduce this problem to a one degreeoffreedom oscillator. Then derive the equation of motion. Figure 2.26: Vibrating beam supported by springs. Solution: In order for the system to be approximated by a one DOF oscillator, one spatial coordi nate is required to de&ne its displacement. We need to assume that (i) the beam does not deform but moves as a rigid body, and (ii) the de¡ection at each end of the beam is the same, implying a symmetric loading (or only a moment is applied and the rotation at one end is equal and opposite to that at the other end). Under these limitations, a SDOF system can be used to model the behavior of this beam. Let F ( t ) be the external force acting upward on the beam. Then, using Newton¢s second law, + " X F vertical = F ( t ) & 2 kx = m d 2 x dt 2 giving the equation of motion m d 2 x dt 2 + 2 kx = F ( t ) : The system has the natural frequency ! n = p 2 k=m: 1 8 CHAPTER 2 SDOF VIBRATION: AN INTRODUCTION 6. For each idealized model in Figures 2.30 to 2.33, draw a freebody diagram and derive the equation of motion using (a) Newton&s second law of motion and (b) the energy method. The block in Figure 2.31 slides on a frictionless surface. State whether the oscillations are linear or nonlinear. Determine the natural frequency of each model. Figure 2.30: Oscillating mass. Figure 2.31: Sliding oscillation of mass. Figure 2.32: A simple pendulum. Figure 2.33: Torsional vibration. Solution: Fig.2.30 For an oscillator vibrating in the gravitational direction x , about static equi librium, we have + " X F x = m & x & kx & kx = m & x: Therefore, the equation of motion is m & x + ( k + k ) x = 0 with natural frequency ! n = p 2 k=m: 8 CHAPTER 2 SDOF VIBRATION: AN INTRODUCTION 6. For each idealized model in Figures 2.30 to 2.33, draw a freebody diagram and derive the equation of motion using (a) Newton&s second law of motion and (b) the energy method. The block in Figure 2.31 slides on a frictionless surface. State whether the oscillations are linear or nonlinear. Determine the natural frequency of each model. Figure 2.30: Oscillating mass. Figure 2.31: Sliding oscillation of mass. Figure 2.32: A simple pendulum. Figure 2.33: Torsional vibration. Solution: Fig.2.30 For an oscillator vibrating in the gravitational direction x , about static equi librium, we have + " X F x = m & x & kx & kx = m & x: Therefore, the equation of motion is m & x + ( k + k ) x = 0 with natural frequency ! n = p 2 k=m: Using an energy approach to the same problem, we ¡rst ¡nd the kinetic and potential energies, and 9 then apply the principle of energy conservation T + V = constant, T = 1 2 m _ x 2 V = 1 2 ( k + k ) x 2 1 2 m _ x 2 + 1 2 (2 k ) x 2 = const: Di/erentiating the last expression with respect to time, we have m _ x...
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 Fall '11
 Benaroya
 Energy, Kinetic Energy, Moment Of Inertia

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