ME 563 Mechanical Vibrations Lecture #18 - Purdue …deadams/ME563/lecture1810.pdf · ME 563...
Transcript of ME 563 Mechanical Vibrations Lecture #18 - Purdue …deadams/ME563/lecture1810.pdf · ME 563...
![Page 1: ME 563 Mechanical Vibrations Lecture #18 - Purdue …deadams/ME563/lecture1810.pdf · ME 563 Mechanical Vibrations Lecture #18 Multiple Degree of Freedom Frequency Response Functions](https://reader031.fdocuments.in/reader031/viewer/2022022010/5afcf5f57f8b9a864d8cd569/html5/thumbnails/1.jpg)
ME 563 Mechanical Vibrations
Lecture #18 Multiple Degree of Freedom
Frequency Response Functions
![Page 2: ME 563 Mechanical Vibrations Lecture #18 - Purdue …deadams/ME563/lecture1810.pdf · ME 563 Mechanical Vibrations Lecture #18 Multiple Degree of Freedom Frequency Response Functions](https://reader031.fdocuments.in/reader031/viewer/2022022010/5afcf5f57f8b9a864d8cd569/html5/thumbnails/2.jpg)
Frequency Response 1
When we considered a single degree of freedom system with one input force and one output response, the relationship between the steady state response and force was written as:
This system is called a single input single output (SISO) system.
This relationship can also be developed for multiple degree of freedom systems with more than one input and output. In this type of multiple input multiple output (MIMO) system, there is more than one frequency response function.
![Page 3: ME 563 Mechanical Vibrations Lecture #18 - Purdue …deadams/ME563/lecture1810.pdf · ME 563 Mechanical Vibrations Lecture #18 Multiple Degree of Freedom Frequency Response Functions](https://reader031.fdocuments.in/reader031/viewer/2022022010/5afcf5f57f8b9a864d8cd569/html5/thumbnails/3.jpg)
Frequency Response 2
![Page 4: ME 563 Mechanical Vibrations Lecture #18 - Purdue …deadams/ME563/lecture1810.pdf · ME 563 Mechanical Vibrations Lecture #18 Multiple Degree of Freedom Frequency Response Functions](https://reader031.fdocuments.in/reader031/viewer/2022022010/5afcf5f57f8b9a864d8cd569/html5/thumbnails/4.jpg)
MDOF System 3
Consider the system shown below.
The equations of motion are given by:
€
M 00 M
˙ ̇ x 1˙ ̇ x 2
+2C −C−C C
˙ x 1˙ x 2
+ 2K −K−K K
x1
x2
=f1
f2
![Page 5: ME 563 Mechanical Vibrations Lecture #18 - Purdue …deadams/ME563/lecture1810.pdf · ME 563 Mechanical Vibrations Lecture #18 Multiple Degree of Freedom Frequency Response Functions](https://reader031.fdocuments.in/reader031/viewer/2022022010/5afcf5f57f8b9a864d8cd569/html5/thumbnails/5.jpg)
Transfer Functions 4
After applying the Laplace transform, to these equations of motion, the following transfer function relationships are found:
Note that the denominator Δ(s) is the same for every transfer function; i.e., natural frequencies are properties of the system.
![Page 6: ME 563 Mechanical Vibrations Lecture #18 - Purdue …deadams/ME563/lecture1810.pdf · ME 563 Mechanical Vibrations Lecture #18 Multiple Degree of Freedom Frequency Response Functions](https://reader031.fdocuments.in/reader031/viewer/2022022010/5afcf5f57f8b9a864d8cd569/html5/thumbnails/6.jpg)
Transfer Functions 5
The transfer functions are written using subscripts, where the first subscript denotes the response degree of freedom and the second subscript denotes the input degree of freedom:
To obtain the frequency response functions, we substitute s=jω as we did previously for single degree of freedom case:
![Page 7: ME 563 Mechanical Vibrations Lecture #18 - Purdue …deadams/ME563/lecture1810.pdf · ME 563 Mechanical Vibrations Lecture #18 Multiple Degree of Freedom Frequency Response Functions](https://reader031.fdocuments.in/reader031/viewer/2022022010/5afcf5f57f8b9a864d8cd569/html5/thumbnails/7.jpg)
Frequency Response Functions 6
These frequency response functions can be utilized to calculate the sinusoidal responses of a multiple degree of freedom system as we did in the single degree of freedom case. €
X1( jω)X2( jω)
=H11( jω) H12( jω)H21( jω) H22( jω)
F1( jω)F2( jω)
where H11( jω) = K −Mω 2 + jωC( ) /Δ( jω)
H12( jω) = H21( jω) = K + jωC( ) /Δ( jω)
H22( jω) = 2K −Mω 2 + jω2C( ) /Δ( jω)
![Page 8: ME 563 Mechanical Vibrations Lecture #18 - Purdue …deadams/ME563/lecture1810.pdf · ME 563 Mechanical Vibrations Lecture #18 Multiple Degree of Freedom Frequency Response Functions](https://reader031.fdocuments.in/reader031/viewer/2022022010/5afcf5f57f8b9a864d8cd569/html5/thumbnails/8.jpg)
Using the FRFs 7
![Page 9: ME 563 Mechanical Vibrations Lecture #18 - Purdue …deadams/ME563/lecture1810.pdf · ME 563 Mechanical Vibrations Lecture #18 Multiple Degree of Freedom Frequency Response Functions](https://reader031.fdocuments.in/reader031/viewer/2022022010/5afcf5f57f8b9a864d8cd569/html5/thumbnails/9.jpg)
Plots of FRFs 8
![Page 10: ME 563 Mechanical Vibrations Lecture #18 - Purdue …deadams/ME563/lecture1810.pdf · ME 563 Mechanical Vibrations Lecture #18 Multiple Degree of Freedom Frequency Response Functions](https://reader031.fdocuments.in/reader031/viewer/2022022010/5afcf5f57f8b9a864d8cd569/html5/thumbnails/10.jpg)
Notes about Plots of FRFs 9
Low frequency response corresponds to static response.
Resonant frequencies are the same for all FRFs.
Anti-resonant frequencies (zeros) only occur in driving point FRFs (for which input and output are the same).
Phase recovers after a resonance in driving point FRF but is lost through both resonant frequencies in cross point FRFs.
Cross point FRFs are equal so long as the system is conservative and absolute coordinates are utilized.
![Page 11: ME 563 Mechanical Vibrations Lecture #18 - Purdue …deadams/ME563/lecture1810.pdf · ME 563 Mechanical Vibrations Lecture #18 Multiple Degree of Freedom Frequency Response Functions](https://reader031.fdocuments.in/reader031/viewer/2022022010/5afcf5f57f8b9a864d8cd569/html5/thumbnails/11.jpg)
Sensitivity of FRFs 10
![Page 12: ME 563 Mechanical Vibrations Lecture #18 - Purdue …deadams/ME563/lecture1810.pdf · ME 563 Mechanical Vibrations Lecture #18 Multiple Degree of Freedom Frequency Response Functions](https://reader031.fdocuments.in/reader031/viewer/2022022010/5afcf5f57f8b9a864d8cd569/html5/thumbnails/12.jpg)
Sensitivity of FRFs 11
![Page 13: ME 563 Mechanical Vibrations Lecture #18 - Purdue …deadams/ME563/lecture1810.pdf · ME 563 Mechanical Vibrations Lecture #18 Multiple Degree of Freedom Frequency Response Functions](https://reader031.fdocuments.in/reader031/viewer/2022022010/5afcf5f57f8b9a864d8cd569/html5/thumbnails/13.jpg)
Sensitivity of FRFs 12