tailieunhanh - Gas Turbine Engineering Handbook 2 Episode 5

Tham khảo tài liệu 'gas turbine engineering handbook 2 episode 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Rotor Dynamics 185 Figure 5-8. Critical damping decay. This very special case is known as critical damping. The value of c for this case is given by cỉr k 4m2 m C2cr 4m2 Ạmk Thus Ccr v l 2m V rnk 2m Underdamped system. If c1 4m2 k m then the roots n and r2 are imaginary and the solution is an oscillating motion as shown in Figure 5-9. All the previous cases of motion are characteristic of different oscillating systems although a specific case will depend upon the application. The underdamped system exhibits its own natural frequency of vibration. When c1 Ậm1 k m the roots ĨỴ and r2 are imaginary and are given by k c2 ri 2 i - -3 V m 4m2 5-15 Then the response becomes x e- c 2rn t k c k c- Cl c2 186 Gas Turbine Engineering Handbook Figure 5-9. Underdamped decay. which can be written as follows X e r ư A cos Udt B sin Udt 5-16 Forced Vibrations So far the study of vibrating systems has been limited to free vibrations where there is no external input into the system. A free vibration system vibrates at its natural resonant frequency until the vibration dies down due to energy dissipation in the damping. Now the influence of external excitation will be considered. In practice dynamic systems are excited by external forces which are themselves periodic in nature. Consider the system shown in Figure 5-10. The externally applied periodic force has a frequency which can vary independently of the system parameters. The motion equation for this system may be obtained by any of the previously stated methods. The Newtonian approach will be used here because of its conceptual simplicity. The freebody diagram of the mass m is shown in Figure 5-11. 7 k c c __I 1 r-1 T 1 Figure 5-10. Forced vibration system. Rotor Dynamics 187 Figure 5-11. Free body diagram of mass M . The motion equation for the mass m is given by mx F sin ut kx ex 5-17 and can be rewritten as mx ex kx F sin ut Assuming that the steady-state oscillation of this system is represented by the following .