The problem of the vibration power flow through coupled substructures when one of the substructures is subjected to a distributed load is addressed. In all the work performed thus far, point force excitation was considered. However, in the case of the excitation of an aircraft fuselage, distributed loading on the whole surface of a panel can be as important as the excitation from directly applied forces at defined locations on the structures. Thus using a mobility power flow approach, expressions are developed for the transmission of vibrational power between two coupled plate substructures in an L configuration, with one of the surfaces of one of the plate substructures being subjected to a distributed load. The types of distributed loads that are considered are a force load with an arbitrary function in space and a distributed load similar to that from acoustic excitation

Cuschieri, J. M.

English

NASA Technical Reports Server

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APPLICATION OF THE 
MOBILITY POWER FLOW APPROACH 
TO STRUCTURAL RESPONSE 
FROM DISTRIBUTED LOADING 
J. M. Cuschieri 
Center for Acoustics and Vibration 
Department of Ocean Engineering 
Florida Atlantic University 
Boca Raton, Florida 33431 
December, 1988 
Progress Report 
Grant Number NAG-1-685 
Submitted to 
NASA Langley Research Center 
Hampton, VA 23665 
https://ntrs.nasa.gov/search.jsp?R=19890004446 2020-03-20T04:19:44+00:00Z
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FOREWARD 
This report describes the work performed between September 
1988 to December 1988 under Research Grant Number NAG-1-685, 
entitled, IWse of Energy Accountancy and Power Flow Techniques 
for Aircraft Noise Transmission". This is the fifth progress 
report under this research grant in which most of the work done 
thus far addresses the problem of the characterization of the 
vibrational power flow. During this latest phase of the work, a 
Ph.D. Thesis was completed with the title, Vibrational Power 
Flow in Thick Connected Plates1', the Abstract of which will be 
published in the Journal of the Acoustical Society of America in 
the Technical Notes and Research Briefs section. The work in 
this thesis was also partially funded by an ONR Fellowship which 
paid for the stipend of the graduate student. 
The author would like to acknowledge the graduate students 
who assisted in the work on this project, the Department of Ocean 
Engineering and most importantly the financial support by the 
Structural Acoustics Branch of the NASA Langley Research Center. 
Submitted by 
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J. M. Cuschieri 
Principal Investigator 
2 
ABSTRACT 
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This report investigates the problem of the vibration power 
flow through coupled plate substructures when one of the 
substructures is subjected to a distributed load. The approach 
used is based on the mobility power flow method. Two types of 
distributed loading are considered. The first part of the report 
deals with a distributed mechanical load which is independent of 
the response of the structure. Power flow expressions are 
presented for this type of loading. In the second part of the 
report, the analysis for distributed load which is influenced by 
the motion of the structure, as in the case of acoustic 
excitation is presented. Results for power flow when the plate 
substructures are subjected to acoustic excitation will be 
published in future reports. 
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1. INTRODUCTION 
The previous work using the mobility power flow (MPF) 
approach has, in general, been restricted to point excited 
structures [l-51. However in the case of excitation of an 
aircraft fuselage distributed loading on the surface of a panel 
can be as important as excitation from directly applied forces at 
defined locations on the structure. The term ttmobility power 
flow" is being used here to distinguish this technique from other 
techniques on power flow. In this approach, the vibrational 
power flow is determined using mobility expressions. There are 
other methods which are similar to either finite element analysis 
[6] or statistical energy analysis where the results are 
frequency or spatial averaged [7,8]. In this report the 
formalism of the power flow expressions for two coupled plate 
substructures with a distributed load on one of the substructures 
is presented. 
Two types of loadings are considered. They are a distributed 
force load of arbitrary shape and a distributed load created by 
an obliquely incidence acoustic waves on the surface of one of 
the plate substructures. The difference between these two types 
of distributed loads is that in the mechanical loading case the 
applied loading is independent of the response of the structure 
while in the case of the excitation from an incident acoustic 
wave, the loading is dependent on the response of the structure 
due to the presence of the scattered acoustic waves. In this 
latter type of loading coupled expressions must be developed for 
the excitation and response. This, to some extent; can be viewed 
4 
as an introduction into the development of expressions for the 
power flow (acoustical and vibrational), using a mobility power 
flow approach, when coupling exists between a structure and the 
surrounding medium as in the case of the interior of an aircraft 
fuselage. 
2. DISTRIBUTED FORCE 
The development of power flow and mobility expressions for 
the distributed force loading, with arbitrary dependence on the x 
and y coordinates, is very similar to the work reported 
previously for point loading [ 2 , 4 ] .  The differences are that the 
presentation of the mobility functions can be generalized to 
apply for any loading condition. Using the configuration and 
axes definition as shown in figure 1, the modal input mobility 
function for the junction, and the transfer mobility function 
between the junction and any point on the plate surface, both 
functions being defined as the response per unit applied moment 
for  every mode m along the junction, are given by [ 2 , 4 ] .  
I s in(k2y) s inh ( kly ) !-- s i n (  k2b) s i n h (  klb) i r , ( Y )  = j 5 2  (Y) = m T m * j F  
5 
where rm is the mode m component of the edge moment, kland k2 are 
defined by: 
2 2 2 k l = 2 k  + k  ; 
X Y  
k = k  
2 Y  
k - m?r/a ; 
X 
2 
Y X - k  ( 3 )  
um is the mode m component of the plate surface displacement, Om 
is the mode m component of the angular displacement at the edge 
of the plate and D" and ph are the plate flexural rigidity and 
surface density respectively. In the above equations it is 
assumed that the two plate substructures are identical. 
Note that in equation (2) the transfer mobility is a function 
of coordinate y. This is required since the load is distributed 
and defined everywhere along the y direction on the plate 
surf ace. 
The above modal mobility expressions are for a system with an 
applied edge moment, which represents one of the configurations 
for which mobility expressions are derived [2]. The second 
configuration to be considered is the one which represents the 
external loading to the global structure. Considering first the 
distributed load, this can be decomposed into its modal 
components as follows: 
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a 
F = - 2 lo Fo f(x) s in[  F] dx 
m 
a 
b 
F - - 2 lo f(y) s in[  T] dy 
n 
b 
( 5 )  
This load decomposition will be used to define the mobility 
functions for the second plate configuration. Therefore, the 
modal mobility for the input load, Mlmn defined as the ratio of 
mode (m,n) component of the transverse velocity on the surface of 
the plate, to an applied distributed surface load with the same 
distribution as the eigenfunction f o r  mode m,n. is given by: 
mn F 
mn 2 27rph 
where 
1 
2lr 
f* = -
mn 
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and Umn is the mode m,n component of the plate surface 
displacement. The expression for the transfer mobility, M21m 
defined as the ratio of the mode m component of a plate edge 
angular velocity, to the mode m component of an applied 
distributed load on the surface of the plate is given by; 
n ( - 1 1 ~  n F 
2 lm 
m F 
.L [ f m ; ) 2 - f  2 
n 2phb 
The evaluation of the above mobility functions is obtained 
from consideration of the equations of motion for a plate 
structure. 
The expression for the input and transmitted power are the 
same as in the case of point loading [2] except that all mobility 
functions are modal expressions; 
mn 
Real 1 IF,,I2 MI 1 
2 
- Power -  
input  
m,n 
a 
m - l M 2  m + M 3  m* 2 
m and n are respectively the mode number for the x and y 
directions, that is m is along the junction and n perpendicular 
to the junction, and a and b are the dimensions of the plates 
(figure 1). 
For a point load, the force is in this case described 
mathematically by; 
and 
4 mnx 
F - -  mn 0 
ab c a  
sin 
where xo and yo are the x and y coordinates of the point of 
application of the load. If the load is applied at the center of 
the plate (xo = a/2 and yo = b/2), then only odd values for m an 
n are allowed. Substituting in equations (11) and (12), the 
results obtained for the input and transferred power, when the 
load is in the center of the source plate, are shown in figure 
(2). Comparing these results to those obtained in reference 
[13], the two sets of results are identical. This verifies the 
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formulation of the power flow expressions in the form shown in 
equations (11) and (12). 
For a uniform distributed load described by: 
F(X,Y) = Fo 
f(y) = 1.0 
and 
0 
16 F 
.rr2 (2m + 1) (2n + 1) 
F -  rnn 
The solution in this case follows in the same way as for the 
point load case. The results for the power flow are shown in 
figure ( 3 ) .  
Comparing the results for the power flow obtained for the 
point load with the results obtained for the distributed load, 
the following similarities and differences can be observed. 
First, the modes excited by the two types of loading are 
identical. This is expected since both loading conditions are 
symmetrical about the center of the source plate. Second, the 
general level of the power flow for the distributed load case 
decreases with frequency, while that for the point load does not 
decrease. The reason for this can be mathematically described by 
the inverse dependency of the modal components of the distributed 
applied load on the mode number for the case of the distributed 
10 
load (equation (17)). Physically this implies that the higher 
frequency modal components of the load are suppressed. The 
results for the distributed load would be similar to those for 
excitation by normal incidence acoustic plane waves, if the 
scattered pressure component is neglected. 
3 .  EXCITATION BY INCIDENT ACOUSTIC WAVES 
In dealing with this form of distributed excitation, the 
scattering of the incident acoustic wave due to the response of 
the receiving structure must be included in the analysis [9]. 
The scattered acoustic field is a function of the response of the 
structure, and therefore, the loading on the structure consists 
of two components; one component is from the incident pressure 
being blocked by the surface and the second component is the 
influence of the scattered acoustic wave. The component 
associated with the blocked incident pressure is independent of 
the motion of the structure except for the phase differences 
between structure response and surface motion. The component 
associated with the scattered pressure is a direct function of 
the motion of the structure and therefore the loading becomes 
structure response dependent. Therefore to deal with this type 
of distributed loading some modifications to the basic 
formulation of the expressions for the vibrational power input 
and power flow are required. To demonstrate the differences in 
the formulation, the case of a point load which is a function of 
the response velocity at the point of application for arbitrary 
structure configuration is discussed first. 
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Consider the configuration shown in figure 4 ,  where the input 
load F is a function of the response velocity. The governing 
equations of motion for this configuration are as follows [l]; 
V = F Mi + FA M i 2  18. 
20. 
but 
F = A+BV 21. 
Where A and B are two constants and all the other terms have 
the same definition as in previous reports in this series [l] . 
Equations (19) and ( 2 0 )  can be used to solve for the junction 
force FA in terms of the input load F, and then substituted into 
equation (18) to solve for the velocity V. In the case where F 
is a function of V, the result f o r  V must be substituted back 
into the expression for F and then used to solve for FA and VA. 
Alternatively V and FA are expressed in terms of the constants A 
and B and the mobility functions. That is, 
-M12 
M2+M3 2 2 .  
12 
and 
23 
The input and transmitted power, Pi and Pt respectively, are 
then given by: 
1 
( A  + BV) FA [ 24. 
Applying this approach to the case of the acoustic 
excitation, the expressions for the input and transfer modal 
mobility functions for an input edge moment (equations 1 and 2) 
are still applicable. For the input mobility of an acoustic 
excited plate structure, consider an incident acoustic wave at an 
angle of 4 to the normal to the plate, where the trace of the 
acoustic wave on the plane of the plate makes an angle 0 to the 
x-axis (figure 5). The trace of the incident pressure on the 
surface of the panel is given by; 
-jk s i n  0 s i n  4 y .-jk s i n  4 cos 0 x P ( X , Y )  = Pi e 25 .  
where k is the acoustic wavenumber, and pi is the magnitude of 
the incident pressure. 
The component of the input load associated with the blocked 
incident pressure is equal to twice the incident pressure [9] at 
the surface of the plate. The scattered component can then be 
13 
formulated based on the in vacuo response of the plate and is 
controlled by the amplitude of this response. The response that 
must be considered in determining the scattered component is the 
response of the global structure that is a simultaneous solution 
must be obtained for the forcing and the response of the global 
structure. 
Let the surface velocity of the receiver plate be described 
by ; 
2 6 .  
The scattered pressure can be obtained by equating the plate 
surface velocity to the particle velocity of the acoustic medium 
at the interface. Using a spatial transform to represent the 
response of the whole plate, the scattered pressure component for 
mode q is given by; 
2 7 .  
where 
where k is the acoustic wavenumber. The positive or negative 
sign is selected to satisfy the far field conditions. The total 
modal pressure decomposed into the wavenumber /I,. acting on the 
14 
plate surface is given by the summation of the blocked component 
and the scattered component. 
29. 
where 
- jky s in4  sin0 - jpy  dr -jkxsin4 cos0 s in[  y ] c ~ x ]  e e 
3 0 .  
and 
31. 
The terms of the summation for which q z m represent 
intramodal coupling which is negligible for light fluid loading 
provided the modal density of the structure is low. Equations 
(30) and (31) define the terms A and B in matrix equation (22). 
To solve for the velocity of the source substructure; 
where 
3 2 .  
3 3  
2 phw' D* 
D* 
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which is obtained from the plate equations of motion, and M12(p)m 
is the spatial fourier transform of equation (2). From 
continuity of motion at the junction, 
and therefore 
34 .  
35.  
The integrand of this last expression contains a V(p)m term 
and therefore when substituting equation (35) in equation (32) to 
solve for V ( p ) ,  the solution can only be obtained numerically. 
From the solution for V ( p ) ,  the input and transmitted vibrational 
power calculated. To evaluate the input power it is not 
necessary to inverse Fourier transform the expression for V(P)m 
and P(p)m since an integral over the structure surface is 
required. This would be equivalent, by Parceval's Theorem, to an 
integral over the entire p space. For the transmitted power this 
is obtained from the relationship; 
Power transmitted 
and then summed f o r  all modes m. 
35 
16 
A complete solution to these equations has not been achieved 
and therefore no results will be presented in this progress 
report. Results for the L-shaped plate structure, including a 
parametric evaluation and comparison to the results for the point 
force loading, will be presented in a following progress report. 
4 .  CONCLUSION 
The mobility power flow formalism for two coupled structures 
with excitation from distributed loading, including acoustic 
excitation, on one of the components of the coupled structure has 
been discussed. For a distributed loading which is independent 
of the motion of the structure, the mobility power flow formalism 
is relatively straight forward and very much similar to that of 
point loading. For the case of acoustic excitation, where the 
magnitude of the excitation is controlled by the response of the 
global structure, the formalism is much more complex. Because of 
the coupling of the response to the excitation, the only method 
available for obtaining a solution is through a numerical 
approach. This is similar to conclusions reached by other 
authors who investigated the problem of acoustic excitation 
where, unless the mode shapes of vibration of the global 
structure are a priori known, the solution is always obtained 
numerically. This does not in any way limit the application of 
the mobility power flow method to multiple coupled structures, 
since the complexities arise mainly for the source structure. 
Results for the analysis presented in this report as applied 
to an L-shaped plate will be reported in the future together with 
17 
. 
the experimental evaluation of the results for both point loading 
and distributed loading. 
1. 
2. 
3. 
4 .  
5. 
6. 
7. 
8. 
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REFERENCES 
J.M. Cuschieri, ItPower Flow As A Complement to Statistical 
Energy Analysis and Finite Element Analysis", ASME 
Publication NCA-Vol 3, 58-61, (1987). 
J.M. Cuschieri, ttExtension of Vibrational Power Flow 
Techniques To Two-Dimensional Structurestt, NASA Contract 
Report 181710, September, 1988. 
J.M. Cuschieri, "Parametric and Experimental Analysis 
Using A Power Flow Approachvt, NASA Contract Report 181990, 
February, 1990. 
J.M. Cuschieri, ttPower Flow Analysis Of Two Coupled Plates 
With Arbitrary Characteristics", NASA Contract Report 
182033, June, 1990. 
R. J. Pinnington, R.G. White, IIPower Flow Through Machine 
Isolators to Resonant and Non-Resonant Beamstf, Journal of 
Sound and Vibration, 75, 179-197, 1981. 
D.J. Nefske, S.H. Sung, IIPower Flow Finite Element 
Analysis of Dynamic Systems: Basic Theory and Application 
to Beamsv1 ASME Statistical Energy Analysis Symposium 
proceedings, NCA - Vol. 3, 47-54 ,  1987. 
L.D. Pope, J . F .  Wilby, "Band Limited Power Flow Into 
Enclosuresll, Journal of the Acoustical Society of America, 
Vol. 62 ( 4 ) ,  906-911, 1977. 
L.D. Pope, J.F. Wilby, "Band Limited Power Flow Into 
Enclosures IT", Journal of the Acoustical Society of 
America, Vol. 67 (3), 823 - 826, 1980. 
F.J. Fahy, IISound and Structural - Radiation Transmission 
and Response", Academic Press, 1985. 
18 
/I Material: Aluminium 
Figure 1 L-shaped plate structure showing plate 
c h a ra cte rist ics 
V 
e 
t+ -70.0 
-80.0 
5 -90.0 
-100.0 -! -- 
0 200 400 600 800 1000 
FREQUENCY H t  
Flgve 2. Normalized power flow results for point load- --- : Power input; ---: power transfer; -.-: power ratio. 
0.0 
p3 -10.0 
3 -20.0 
-30.0 
-40.0 
8 -50.0 
E -60.0 
+a -70.0 A 
-80.0 
-90.0 
-100.0 
0 
a 
I \ I  
0 200 400 600 800 1000 
FREQUENCY Hz 
F i v e  3. Normalized power flow results for distributed loading 
: power Input; ---: power transfer: -.-: power ratio. --- 
* -  
I 
L 1 
=A+BV 
F A  F A  
figure 4. Source-receiver model where F is a function 
of t h e  response at the point of application. 
/ - - - - I -  / 
Figure 5. lncedent acoustic pressure at oblique angle #. 


Application of the mobility power flow approach to structural response from distributed loading

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