机械振动加州大学UniversityofCaliforniaChapter2FreeVibration.docVIP

-Chapter 2 Free Vibration No external excitation Objectives: Equation of motion Natural frequency Damping Principle of conservation of energy 2.1 VIBRATION MODEL Mass: [kg], [lb(s2/in] Massless spring: linear spring with, k [N/m], [lb/in] Viscous damping: , c [N/m/s], [lb/in/s] 2.2 EQS OF MOTION: NATURAL FREQUENCY Simple Undamped Spring-Mass System: Assumed to move only along the vertical direction ( is the deformation of the spring in the static equilibrium position. (2.2.1) (2.2.2) With the static equilibrium position as reference for x, the resultant force on m is simply the spring force due to the displacement x. (2.2.3) Eq.(2.2.2) can be written as . (2.2.6) The natural period of the oscillation (2.2.7) and the natural frequency is (2.2.8) In terms of (, (2.2.9) [Example 2.2.1] A -kg mass is suspended by a spring having a stiffness of 0.1533 N/mm. Determine its natural frequency in cycles per second. Determine its static deflection. [Example 2.2.2] Determine the natural frequency of the mass M on the end of cantilever beam of negligible mass shown in Fig. 2.2.2. [Example 2.2.3] An automobile wheel and tire are suspended by a steel rod 0.50 cm in diameter and 2 m long, as shown in Fig. 2.2.3. When the wheel is given an angular displacement and released, it makes 10 oscillations in 30.2 s. Determine the polar moment of inertia of the wheel and tire. [Example 2.2.4] Figure 2.2.4 shows a uniform bar pivoted about point O with springs of equal stiffness k at each end. The bar is horizontal in the equilibrium position with spring forces P1 and P2. Determine the equation of motion and its natural frequency. 2.3 ENERGY METHOD The differential equation motion by the principle of conservation of energy The kinetic energy T: Stored in the mass by virtue of its velocity The potential energy U: Stored in the form of strain energy in elastic deformation or