1. CHAPTER 4: HARMONIC RESPONSE WITH A SINGLE DEGREE OF FREEDOM
Mechanical Vibration by Palm
CHAPTER 4: HARMONIC RESPONSE WITH A SINGLE DEGREE OF FREEDOM
Instructor: Dr Simin Nasseri, SPSU
© Copyright, 2010

3. Transient solution (Complementary) which  Dies out with time
Stead state solution (Particular)  Continues with time

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5. Dies out with time
Dies out with time
Continues with time

6. Transient solution (Complementary) which  Dies out with time
Stead state solution (Particular)  Continues with time
If we ignore the transient response which dies out with time, we will only have the steady state response.
x(t)=

7. The Magnitude Ratio is the magnitude of the steady state response divided by the magnitude of F0 of the forcing function:

8. Example 4.1-1: The equation of motion of a certain mass-spring-damper system is
1- Lets find the free response which is transient:

9. 2- Now lets obtain the steady-state response:

10. Using Matlab the response of the mass under the applied force is plotted here, which is the steady-state solution.
The mass oscillates with the frequency ω=4 rad/s and not with

11. Example 4.1-2: Obtain the forced response of the following model for ω=1.
1- Lets find the transient response:

12. 2- Now lets obtain the steady-state response:

13. 1- Transient response:

14. 2- Steady-state response:

15. Total response:

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Check page 212 for detailed calculations.

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19. Because

20. DANIEL J. INMAN, Engineering Vibration, Prentice Hall, Englewood Cliffs, New Jersey,1994

21. Meaning of a Peak frequency:
For every graph, the maximum occurs at a frequency called peak frequency (marked by red stars). This is important since is shows the maximum displacement of the system with any damping value c.

22. At high forcing frequencies ω>>ωn , The response amplitude approaches zero! Because of the system’s inertia. Inertia prevents the system to follow a rapidly varying forcing function.
In the above equation, when ω gets relatively large with respect to ωn, the denominator becomes large and the whole fraction will become small.

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When ς=0, the amplitude of response becomes infinite when r=1 (when ω= ωn). This is called RESONANCE.
With a good damping (greater values of ς) , even if r=1, resonance will not happen.
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24. The phase shift φ is exactly -90o at the resonant frequency.
When a system is resonating, the velocity is in phase with the forcing function. This causes the response x(t) to increase indefinitely if there is no damping.

25. At high frequencies, the phase angle is almost -180o.
The response lags behind the forcing function for r<1. For example: F=F0sin(ωt),  x(t)=Xsin(ωt-45)
The forcing function lags behind the response for r>1. For example: F=F0sin(ωt),  x(t)=Xsin(ωt-135)

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27. Therefore the following equations:
For a system with a small amount of damping, we can assume that we are solving the following equation (c = ς = 0):
Therefore the following equations:
will be changed into:
(1)

28. We can show that the free response form is (see page 217)
(2)
So the total forced response is (Eq 1 and Eq 2):
With initial displacement ( x(0) ) and velocity (v(0)) the equation takes the form of:

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30. When the forcing frequency ω is substantially different from the natural frequency ωn , the forced response looks like Fig
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31. If the forcing frequency ω is close to the natural frequency ωn, then , the amplitude of the oscillation increases and decreases periodically. This behavior is called beating.
Beat period
Vibration period
Fig 4_2_2

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