Joint Receptance for Rigid & Elastically Coupled Subsystems
Revision A
By Tom Irvine
Email: tom@vibrationdata.com
January 10, 2014
_____________________________________________________________________________________
Introduction
Consider a single-degree-of-freedom system S constructed from two rigidly-connected
subsystems A and B.
The combined system S is
x
ka
kb
ms
Fs
Figure 1.
Subsystem A is
Subsystem B is
xa
xb
ka
Fa
ma
Figure 2.
kb
Fb
mb
Figure 3.
The combined system and the subsystems are assumed to have modal damping which implies a dashpot.
1
Variables
mi
ki
i
i
mass
stiffness
modal damping
natural frequency
Fi
Hi
xi

applied force
receptance function (displacement/force)
Displacement
excitation frequency
The subscript indicates the system or subsytem
Receptance Coupling
The receptance function for subsystem A from Reference 1 is
1
H a  
ma


1


  2   2  j 2  
a
a 
 a
(1)
The receptance function for subsystem B is


1 
1

H b  
m b   2   2  j 2   
b
b 
 b
(2)
The joint receptance H j  is
H j  
H j  
1
ma mb
Ha Hb
Ha  Hb
(3)



1
1



2
2

2
2
     j 2        j 2   
a
a  b
b
b 
 a

1 
1
1


m a   2   2  j 2    m b
a
a 
 a


1


  2   2  j 2   
b
b 
 b
(4)
2
H j  



1
1




  2   2  j 2      2   2  j 2   
a
a  b
b
b 
 a




1
1

  ma 
mb 
  2   2  j 2   
  2   2  j 2  
a
a 
b
b 
 a
 b
H j  
H j  
1
2
2
m b  b   2  j 2 b  b   m a a   2  j 2 a a 




1
m a a  m b b  m a  m b 2  j 2m a  a a  m b  b b 
2
2
(5)
(6)
(7)
The system natural frequency s is
s 
ka  kb
ma  m b
f s  s /( 2)
(8)
(9)
The coupled system damping from Reference 2 is
s 
 a m a a   b m b  b
ma a 2  m bb 2 ma  m b 
(10)
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An example of rigid-coupling is given in Appendix A.
Elastic coupling is given in Appendix B.
References
1. T. Irvine, An Introduction to Frequency Response Functions, Vibrationdata, 2000.
2. T. Irvine, Notes on Damping in FRF Substructuring, Revision A, Vibrationdata, 2014.
APPENDIX A
Rigid Coupling Example
Consider the system in Figure 1 and the subsystems in Figures 2 and 3.
variables.
ma
4 lbm
mb
2 lbm
ka
1000 lbf/in
kb
8000 lbf/in
a
0.05
b
0.05
Assign the following
The system natural frequency per equation (9) is
fn= 121.1 Hz
(A-1)
The system damping per equation (10) is
 s  0.04
(A-2)
The receptance function is shown in Figure A-1.
4
Figure A-1.
The magnitude unit is (in/lbf).
5
Figure A-2.
An FRF curve-fit was performed on the real and imaginary components of the system receptance
functions. The curve-fit yields the expected results.
fn = 121.1 Hz
damping ratio = 0.041
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APPENDIX B
Elastic Coupling
Consider a single-degree-of-freedom system S constructed from two subsystems A and B
connected by a spring.
The combined system S is
x s, b
x s, a
ka
kb
kc
mb
ms
Fs, a
Fs, b
Figure B-1.
Subsystem A is
Subsystem B is
xa
xb
ka
Fa
ma
kb
Fb
mb
Figure B-2.
Figure B-3.
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The receptance function for subsystem A is
H a  

1 
1


m a   2   2  j 2  
a
a 
 a
(B-1)
The receptance function for subsystem A plus the coupling spring is
H ac  
1
ma

 1
1


  2   2  j 2    k c
a
a 
 a
(B-2)
The receptance function for system B is


1 
1

H b  
2

2
m b     j 2   
b
b 
 b
(B-3)
The joint receptance H s, b  at the interface between the coupling spring and subsystem b is
H s, b  
H acH b
H ac  H b
(B-4)
Elastic Coupling Example
Consider the system in Figure B-1 and the subsystems in Figures B-2 and B-3. Assign the following
variables.
ma
4 lbm
mb
2 lbm
ka
1000 lbf/in
kb
8000 lbf/in
a
0.05
b
0.05
kc
2000 lbf/in
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Figure B-4.
The curve in Figure B-4 was calculated using equation (B-4).
The magnitude unit is (in/lbf).
The system modal parameters are
Parameter
Mode 1
Mode 2
Natural Frequency (Hz)
79
224
Damping Ratio
0.034
0.044
The modal parameters were extacted from the FRF using the curve-fit shown in Figure B-5.
The results were confirmed in a separate two-degree-of-freedom modal analysis.
9
Figure B-5.
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