Response Of Linear SDOF Systems To Harmonic Excitation

1. Response Of Linear SDOF Systems To Harmonic Excitation Many dynamic systems are subjected to harmonic (sinusoidal) excitation. Rotating machinery is an example. Objectives • Learn how o find response to harmonic excitation. • Understand concept of resonance

2. Undamped systems • Response = harmonic wave with frequency ωn +harmonic wave with frequency of excitation • The first component is due to initial conditions plus the force. • The second component is due to the force

3. General equation for response to force

4. Harmonic Response Of Undamped System natural frequency=1 rad/sec, excitation frequency=2 rad/sec, x(0)=0.01 m, xd(0)=0.01

5. Harmonic Response Of Undamped System natural frequency= 1 rad/sec, excitation frequency=0.95 rad/seczero initial displacement and velocity

6. Harmonic Response Of Undamped System natural frequency= 1 rad/sec, excitation frequency=0.9999 rad/seczero initial displacement and velocity

7. Harmonic Response Of Undamped System natural frequency= 1 rad/sec, excitation frequency=20 rad/seczero initial displacement and velocity

8. Observations • Response=harmonic wave with frequency ωn +harmonic wave with frequency of excitation • When excitation frequency is almost equal to natural frequency, vibration amplitude is very large. Rapid oscillation with slowly varying amplitude. • When excitation frequency is equal to the systems natural frequency, vibration amplitude= . This condition is called resonance. • When excitation frequency>>natural frequency, vibration amplitude is very small. The reason is that the system is too slow to follow the excitation.

9. Damped systems Free vibration response (also called transient response) Particular response (steady-state response) • Use equations that we learnt in the chapter for free vibration response for transient response. Transient dies out with time. • Response converges to particular response with time.

10. Steady-state response • We often ignore transient response and focus on steady-state response. • Steady-state is response is solution of equation of motion: • From experience we know that:

11. Find coefficients A and B by substituting assumed equation for response in equation of motion and solving for these coefficients.

12. Sine-in, sine-out property: Steady-state response to sine wave is also sine wave with same frequency as the excitation. Amplitude of response equal to quasi-static response times dynamic amplification factor. This is true for any value of damping ratio

14. Natural frequency 5 rad/sec, excitation frequency, 1 rad/sec

15. Natural frequency 5 rad/sec, excitation frequency, 5 rad/sec

16. Natural frequency 5 rad/sec, excitation frequency, 20 rad/sec

17. Effect of frequency and damping on amplitude Natural freq.=5 rad/sec. Amplitude of excitation=1. =0.1 =0.5 =

18. Effect of frequency and damping on phase angle.Natural freq.=5 rad/sec. Amplitude of excitation=1.

19. Observations • Response amplitude depends only on damping ratio, , and ratio of frequency of excitation over natural frequency, /n. • /n small, response amplitude = F/k (quasi-static response) • /n much greater than one, response amplitude is close to zero • /n =1, and  small (less than 0.1), response amplitude very large. If =0, response amplitude= • If <0.707 response amplitude is maximum for • If >0.707, response is maximum for =0. • To reduce response amplitude a) change /n so that it is as far away from 1 as possible, or b) increase damping.

20. Observations (continued) • Phase angle is always negative. • For small /n, phase angle is almost zero. • For /n =1, phase angle is -900. • For large /n phase angle tends to -1800.