1. 3 General forced response
So far, all of the driving forces have been sine or cosine excitations
In this chapter we examine the response to any form of excitation such as
Impulse
Sums of sines and cosines
Any integrable function
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Linear Superposition allows us to break up complicated forces into sums of simpler forces, compute the response and add to get the total solution
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3. 3.1 Impulse Response Function
F(t)
Impulse excitation
t -e
t +e t
e is a small positive number
Figure 3.1
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From sophomore dynamics The impulse imparted to an object is equal to the change in the objects momentum
F(t)
i.e. area under pulse
t -e
t +e t
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We use the properties of impulse to define the impulse function:
Dirac Delta function
Equal impulses
F(t) t
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6. The effect of an impulse on a spring-mass-damper is related to its change in momentum.
Just after impulse
Just before impulse
Thus the response to impulse with zero IC is equal to
the free response with IC: x0=0 and v0 =FDt/m
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7. Recall that the free response to just non zero initial conditions is:
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8. So for an underdamped system the impulse response is (x0 = 0)k
x(t) c 1
0.5
h(t)
-0.5 -1 10 20 30 40
Time
Response to an impulse at t = 0, and zero initial conditions
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The response to an impulse is thus defined in terms of the impulse response function, h(t).
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t=0 h1
For example: If two pulses occur at two different times then their impulse responses will superimpose -1 10 20 30 40 1
t=10 h2 -1 10 20 30 40 1
h1+h2 -1 10 20 30 40
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Time
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11. Consider the undamped impulse response
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12. Example 3.1.1Design a camera mount with a vibration constraint
Consider example of the security camera
again only this time with an impulsive load
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Using the stiffness and mass parameters of Example 2.1.3, does the system stay with in vibration limits if hit by a 1 kg bird traveling at 72 kmh?
The natural frequency of the camera system is
From equations (3.7) and (3.8) with  = 0, the impulsive response is:
The magnitude of the response due to the impulse is thus
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Next compute the momentum of the bird to complete the magnitude calculation:
Next use this value in the expression for the
maximum value:
This max value exceeds the camera tolerance
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15. Example 3.1.2: two impacts, zero initial conditions. (double hit)
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16. Example 3.1.2 two impacts and initial conditions
Note, no need to redo constants of integration
for impulse excitation (others, yes)
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17. Computation of the response to first impulse:
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18. Total Response for 0< t < 4
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19. Next compute the response to the second impulse:
Here the Heaviside step function is used
to “turn on” the response to the impulse
at t = 4 seconds.
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20. To get the total response add the partial solutions:
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21. 3.2 Response to an Arbitrary Input
The response to general force, F(t), can be viewed as a series
of impulses of magnitude F(ti)Dt
Response at time t due to the ith impulse zero IC
xi(t) = [F(ti)Dt ].h(t-ti) for t>ti ti xi t
F(t)
Impulses
F(ti)
(3.12) t
t1,t2 ,t3 ti
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22. Properties of convolution integrals: It is symmetric meaning:
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The convolution integral, or Duhamel integral, for underdamped systems is:
The response to any integrable force can be computed with
either of these forms
Which form to use depends on which is easiest to compute
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24. Example 3.2.1: Step function input
Figure 3.6 Step function To solve apply (3.13):
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26. Integrating (use a table, code or calculator) yields the solution:
Fig 3.7
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Example: undamped oscillator under IC and constant force
For an undamped system: F0 t1 t2
The homogeneous solution is
x(t) m
Good until the applied force acts at t1, then: k
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28. Next compute the solution between t1 and t2
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29. Now compute the solution for time greater than t2
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30. Total solution is superposition:
0.3
0.2
Displacement x(t)
0.1
-0.1 2 4 6 8 10
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Time (s)
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31. Example 3.2.3: Static versus dynamic load
This has max value of , twice the static load
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32. Numerical simulation and plotting
At the end of this chapter, numerical simulation is used to solve the problems of this section.
Numerical simulation is often easier then computing these integrals
It is wise to check the two approaches against each other by plotting the analytical solution and numerical solution on the same graph
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33. 3.3 Response to an Arbitrary Periodic Input
We have solutions to sine and cosine inputs.
What about periodic but non-harmonic inputs?
We know that periodic functions can be represented by a series of sines and cosines (Fourier)
Response is superposition of as many RHS terms as you think are necessary to represent the forcing function accurately 2 T
1.5 1
0.5
Displacement x(t)
-0.5 -1
-1.5 -2 2 4 6
Time (s)
Figure 3.11
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34. Recall the Fourier Series Definition:
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35. The terms of the Fourier series satisfy orthogonality conditions:
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36. Fourier Series Example
F(t)
Step 1: find the F.S. and determine how many terms you need F0 t1
t2=T
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37. Fourier Series Example
1.2 1
F(t)
2 coefficients
0.8
10 coefficients
100 coefficients
0.6
Force F(t)
0.4
0.2
-0.2
0.5 1
1.5 2
2.5 3
3.5 4
4.5 5
Time (s)
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38. Having obtained the FS of input
The next step is to find responses to each term of the FS
And then, just add them up!
Danger!!: Resonance occurs whenever a multiple of excitation frequency equals the natural frequency.
You may excite at 100rad/s and observe resonance while natural frequency is 500rad/s!!
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39. Solution as a series of sines and cosines to
The solution can be written as a summation
Solutions calculated from equations of motion (see section Example 3.3.2)
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3.4 Transform Methods
An alternative to solving the previous problems, similar to section 2.3
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Laplace Transform
Laplace transformation
Laplace transforms are very useful because they change differential equations into simple algebraic equations
Examples of Laplace transforms (see page 216) in book)
f(t)
F(s)
Step function, u(t)
e-at
sin(w t )
1/s
1/(s+a)
w / ( s2 + w 2)
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Laplace Transform
Example: Laplace transform of a step function u(t)
u(t) t
Example: Laplace transform of e-at
e-at t
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43. Laplace Transforms of Derivatives
Laplace transform of the derivative of a function
Integration by parts gives,
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44. Laplace Transform Procedures
Laplace transform of the integral of a function
Steps in using the Laplace transformation to solve DE’s
Find differential equations
Find Laplace transform of equations
Rearrange equations in terms of variable of interest
Convert back into time domain to find resulting response (inverse transform using tables)
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45. Laplace Transform Shift Property
Note these shift properties in t and s spaces...
thus
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Example 3.4.3: compute the forced response of a spring mass system to a step input using LT
Compare this to the solution given in (3.18)
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Fourier Transform
From Fourier series of non-periodic functions
Allow period to go to infinity
Similar to Laplace Transform
Useful for random inputs
Corresponding inverse transform
Fourier transform of the unit impulse response is the frequency response function 1 2 3 4
-0.1
-0.05
0.05
0.1
Time (s)
h(t) w n
=2 and M=1
Fourier Transform 1 2 3 4 5
-20
-10 10 20
Frequency (Hz)
Normalized H(w) (dB)
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