- Email: [email protected]
Prediction of limiting response of panel-cavity coupled transient oscillations
J. Sound Vib. (1968) 8 (l), 97-102
PREDICTION
OF LIMITING
COUPLED
RESPONSE
TRANSIENT
OF PANEL- CAVITY
OSCILLATIONS
G. D. WHITEHOUSE Louisiana State University, Baton Rouge, Louisiana, U.S.A.
AND R. L. LOWJXRY Oklahoma State University, Stillwater,
Oklahoma,
U.S.A.
(Received 5 October 1967)
Results from instrumentated test houses during controlled supersonic flights indicate that cavity resonances are common during a sonic boom. These pressure oscillations for lightly damped systems can persist for many cycles. The analysis and subsequent prediction of the structural response of a cavity-panel coupled system is quite difficult if one considers the continuous structure and accounts for the odd geometry. The derivation of a general expression to account for the various configurations is virtually impossible and would be limited for practical use. This article suggests a prediction of the upper limit for a panelcavity coupled system based on the parameters of the two separate systems. This method of prediction shows for a lightly damned panel coupled with a lightly damned cavity of about the same natural frequency, the upper bound of the amplification ratio is on the order of 16to 1.
1. INTRODUCTION Measurements taken from pressure microphones and displacement transducers used in the Oklahoma City, Oklahoma, sonic boom tests in 1964 gave indications that cavity resonances can occur during sonic boom flights. In the records made in Test House Number 5, corresponding to 4,5,6,7 and 8 of July 28 it was observed that the motion of one window in the living room was in phase with the motion of the ceiling. That is, the ceiling moved upward, as the window moved outward, at a frequency of 5 Hz. Since the natural frequencies of the window and the ceiling were found to be 25 and 10 Hz respectively, it was concluded that the room cavity was resonating as a large Helmholtz resonator. In this particular situation the observed strain of the window was not excessive due to the large difference in the natural frequencies of the cavity and of the window. Many other incidents are known and could be cited, in which large panel displacements of windows and flexible ceilings have caused permanent damage due to overpressures of aircraft flying at supersonic speeds. The problem of transient panel-cavity coupled oscillations will be increasingly important when scheduled supersonic commercial flights are made in this country. The analysis and subsequently the prediction of structural damage due to sonic boom overpressures is difficult. Innumerable physical configurations consisting of odd geometric shapes exist in typical residential dwellings. If an attempt were made to arrive at a general analysis to exactly predict criteria for failure, it would be extremely time-consuming and very difficult. A more practical and feasible approach might be to formulate some criteria, based on the 7 97
98
G. D. WHITEHOUSE AND R. L. LOWRY
physical parameters of the system, for the prediction of the upper limit or limiting response of a panel-cavity coupled system. Figure 1 shows the actual pressure recordings of two low-frequency microphones located inside and outside a typical residential dwelling used as a test house during a sonic boom test flight. The outside microphone recorded a typical sonic boom overpressure signature while
Figure 1. Sonic boom record for Oklahoma City tests.
the inside microphone indicated persisting pressure oscillations existing within the rooms. This figure illustrating the residual cavity oscillations was not an isolated incident but occurred frequently throughout the six-month test period. 2. ANALYSIS
A window or panel mounted in the wall of a room having an external passage or opening can be approximated by a simple two-degree-of-freedom system. For that matter, a cavity having two panels, but no exterior passageway can be represented by essentially the same model. The response of either system to any external pressure excitation can easily be calculated for any one specific case. However, it is much more difficult to derive meaningful transient response spectra for the undamped two-degree-of-freedom system than it is for the onedegree-of-freedom system. Further, if all of the various damping mechanisms are included, such as the acoustic radiation, fluid friction, mechanical hysteresis, and air leakage, it is impossible to arrive at the maximum response conditions without resorting to the sheer brute force technique of making a large number of parametric calculations. In terms of structural response to the sonic boom, what may be of prime importance is a knowledge of the greatest upper bound of the displacement or strain in the panel. This result can be approximated by assuming that the two-degree-of-freedom system is divided into two separate systems each having one degree of freedom. This will be a valid assumption if the masses of the two subsystems are of contrasting magnitude. Then the natural frequencies of the uncoupled systems would not be appreciably different from the fundamental frequency of the combined system. If the damping ratio of each subsystem is varied over the range observed in many measurements of representative systems, say, 0.01 to 0.1, the response of the smaller of the subsystems can easily be calculated by considering the damped transient motion of the first system as the input. If one considers the panel as an idealized simple damped oscillator, capable of being described solely in the fundamental mode of vibration, with the excitation forcef(f) as shown in Figure 2, the well-known
differential
equation
of motion
results :
PANEL-CAVITY
TRANSIENT
99
OSCILLATIONS
The solution of the differential equation by the use of Laplace transforms with zero initial conditions is given in the complex domain as
f:(s>lm +) Then the displacement-time equation (2) or
= (s2 + 250s + 02) .
response of the system in the time domain is the inverse of
40
=
3-l
ts2
+
25ws
+
w2)
(3)
-
A logical assumption for f(t), considering the pressure trace of the inside microphone (Figure l), would be a damped sinusoidal function. Of course, this neglects the coupling of the systems and oversimplifies the system in some respect. However, it will be a very accurate idealization in that the limiting response will serve as an upper limit. The coupled system cannot achieve a higher transient response than the response of the simple system with a
--____
__ __. Time kec
1 i :-‘
Figure 2. Damped sinusoidal input. damped sinusoidal forcing function. The magnitude of the damped sine wave, F,, cannot exceed the maximax value of a system having one degree of freedom. For a true N wave input the maximax response is approximately 2.1 times the static response for the same pressure. The forcing function as illustrated in Figure 2 can be represented by f(t) = F,, evar sin wd t,
(4)
where F, = forcing amplitude coefficient, wd = damped angular forcing frequency, a = decay 5 rate = li& or cJ;rod,and & = damping ratio, The transfo:med expression for (4) is
m=
Few
(s + a)2 +
co:
(5)
where wd represents the value for 2~17~. Substitution of expression (5) into expression (2) yields the final form for the displacementtime response in the complex domain :
Fowd 3s)
=
(s
+
42
+
l/m w;
‘($2
+
52ws + w2) *
(6)
100
G. D. WHITEHOUSE AND R. L. LOWERY
The inverse of expression (6) can be obtained in a straightforward the property of convolution and Borel’s theorem:
manner making use of
x(t) = 9-1 X(S)= 9-r h,(s) .hz(S)= h,(t)* h,(t)
(7)
where,
FOWd
Us)=(s + cc>* + of! and
h*(s)= The displacement-time
l
s*+2&Js+o*
response can now be written as
x(t) = j h,(t - T)h2(T)dT.
(10)
0
The displacement-time
response is the evaluation of the expression . e-cwo-T)sin (wdT) . sin uJ~
(t -
T)
d7.
(11)
The integration of expression (11) poses no problem, although the algebra becomes somewhat tedious. After integrating and non-dimensionalizing, the expression for the dynamic response factor is given by [y sin At - A cos At] + eYt 2[72 + AZ]
A + eYt[y sin Bt - Bcos Bt] + 2[y2 f B2] 2[9 + A*] + B + 2[y* + B2] I
-cosol/l
-<*t
x
x ey,[AsinAt+ycosAt]_eyt[BsinBt+ycosBt] [
-2[y21:A2]+2[y2:
where A=w,-wm,
2[y*+ B*]
2[y* + AZ]
B= wd + wl/l
-
B*]
-
(12)
<*, and y = -[CZ- &J].
3. RESULTS
Figures 3, 4, 5 and 6 are plots of the response factor (D.R.F.) (non-dimensional) of the system as a function of the frequency ratio for various values of the damping factor, both of the system and the decaying type input. From the plots of the dynamic response ratio it can be seen that for lightly damped systems, large amplitudes can be achieved. Lightly damped cavities such as large enclosures with hard interior surfaces would be the type that persistent pressure oscillations can exist. If the cavity fundamental frequency is about the same as the natural frequency of a coupled structural member, such as a window, then large amplitudes can be expected. In Figure 3, with an input damping factor of about 0.01, and a damping factor of a window in the order of O-1a dynamic
PANEL-CAVITY
TRANSIENT
OSCILLATIONS
1011
response of nearly 4-Ois observed. The damping factor of 0.1for a window is not unreasonable as some experimental measurements from free vibration tests (see reference 1) have shown that large windows can have damping ratios as small as 5 = O-08. Figures 4 through 6 further illustrate that if either the input forcing function persists for a few cycles or the structural system, such as a window or flexible ceiling, is lightly damped, then large amplitudes might occur.
W/oJd
wh,j
Figure 3. Frequency effect on D.R.F., 5, = 0.01. Figure 4. Frequency effect on D.R.F., & = 0.05.
0
0.5
I.0
I.5
2-O
2.5
Figure 5. Frequency effect on D.R.F., & = 0.2. Figure 6. Frequency effect on D.R.F., & = 0.1.
4. CONCLUSIONS It should be noted that the D.R.F. used in this article is based on the simple relationship of x/(I;,/k). The overall D.R.F. could be higher if it was based on the input area times the input pressure of the boom.
.3.0
102
G. D. WHITEHOUSEAND R. L. LOWERY
Future standards for structural design will have to account for the possibility of lightly damped coupled systems. The onset of the supersonic aircraft will cause considerable attention to this area, and new methods for fastening of windows, ceilings and other flexible structural members must be contemplated. REFERENCES 1. ANDREWSASSOCIATES, INC. and HUDGINS,THOMPSON,BALL and ASSOCIATES, INC. 1965 Final Report on Studies of Structural Response to Sonic Booms. Federal Aviation Agency. 2. J. D. SIMPSON1966 Unpublished Thesis, Oklahoma State University. The transient response of a Helmholtz resonator with application to sonic boom studies. 3. W. T. THOMPSON 1962 Laplace Transformation. 2nd. ed. New York: Prentice-Hall. 4. G. D. WHITEHOUSE 1967 Unpublished Thesis, Oklahoma State University. Coupled and uncoupled panel response to sonic boom type inputs.
APPENDIX : NOMENCLATURE? m, c, k fW,fW Fo wd
%: w T
mass, damping coefficient and spring constant in simple system forcing function-time dependent, forcing function transformed forcing amplitude coefficient damped angular forcing frequency damping ratio of simple system damping ratio of damped sinusoid forcing input undamped natural frequency of simple system dummy variable used in time integration
PREDICTION
OF LIMITING
COUPLED
RESPONSE
TRANSIENT
OF PANEL- CAVITY
OSCILLATIONS
G. D. WHITEHOUSE Louisiana State University, Baton Rouge, Louisiana, U.S.A.
AND R. L. LOWJXRY Oklahoma State University, Stillwater,
Oklahoma,
U.S.A.
(Received 5 October 1967)
Results from instrumentated test houses during controlled supersonic flights indicate that cavity resonances are common during a sonic boom. These pressure oscillations for lightly damped systems can persist for many cycles. The analysis and subsequent prediction of the structural response of a cavity-panel coupled system is quite difficult if one considers the continuous structure and accounts for the odd geometry. The derivation of a general expression to account for the various configurations is virtually impossible and would be limited for practical use. This article suggests a prediction of the upper limit for a panelcavity coupled system based on the parameters of the two separate systems. This method of prediction shows for a lightly damned panel coupled with a lightly damned cavity of about the same natural frequency, the upper bound of the amplification ratio is on the order of 16to 1.
1. INTRODUCTION Measurements taken from pressure microphones and displacement transducers used in the Oklahoma City, Oklahoma, sonic boom tests in 1964 gave indications that cavity resonances can occur during sonic boom flights. In the records made in Test House Number 5, corresponding to 4,5,6,7 and 8 of July 28 it was observed that the motion of one window in the living room was in phase with the motion of the ceiling. That is, the ceiling moved upward, as the window moved outward, at a frequency of 5 Hz. Since the natural frequencies of the window and the ceiling were found to be 25 and 10 Hz respectively, it was concluded that the room cavity was resonating as a large Helmholtz resonator. In this particular situation the observed strain of the window was not excessive due to the large difference in the natural frequencies of the cavity and of the window. Many other incidents are known and could be cited, in which large panel displacements of windows and flexible ceilings have caused permanent damage due to overpressures of aircraft flying at supersonic speeds. The problem of transient panel-cavity coupled oscillations will be increasingly important when scheduled supersonic commercial flights are made in this country. The analysis and subsequently the prediction of structural damage due to sonic boom overpressures is difficult. Innumerable physical configurations consisting of odd geometric shapes exist in typical residential dwellings. If an attempt were made to arrive at a general analysis to exactly predict criteria for failure, it would be extremely time-consuming and very difficult. A more practical and feasible approach might be to formulate some criteria, based on the 7 97
98
G. D. WHITEHOUSE AND R. L. LOWRY
physical parameters of the system, for the prediction of the upper limit or limiting response of a panel-cavity coupled system. Figure 1 shows the actual pressure recordings of two low-frequency microphones located inside and outside a typical residential dwelling used as a test house during a sonic boom test flight. The outside microphone recorded a typical sonic boom overpressure signature while
Figure 1. Sonic boom record for Oklahoma City tests.
the inside microphone indicated persisting pressure oscillations existing within the rooms. This figure illustrating the residual cavity oscillations was not an isolated incident but occurred frequently throughout the six-month test period. 2. ANALYSIS
A window or panel mounted in the wall of a room having an external passage or opening can be approximated by a simple two-degree-of-freedom system. For that matter, a cavity having two panels, but no exterior passageway can be represented by essentially the same model. The response of either system to any external pressure excitation can easily be calculated for any one specific case. However, it is much more difficult to derive meaningful transient response spectra for the undamped two-degree-of-freedom system than it is for the onedegree-of-freedom system. Further, if all of the various damping mechanisms are included, such as the acoustic radiation, fluid friction, mechanical hysteresis, and air leakage, it is impossible to arrive at the maximum response conditions without resorting to the sheer brute force technique of making a large number of parametric calculations. In terms of structural response to the sonic boom, what may be of prime importance is a knowledge of the greatest upper bound of the displacement or strain in the panel. This result can be approximated by assuming that the two-degree-of-freedom system is divided into two separate systems each having one degree of freedom. This will be a valid assumption if the masses of the two subsystems are of contrasting magnitude. Then the natural frequencies of the uncoupled systems would not be appreciably different from the fundamental frequency of the combined system. If the damping ratio of each subsystem is varied over the range observed in many measurements of representative systems, say, 0.01 to 0.1, the response of the smaller of the subsystems can easily be calculated by considering the damped transient motion of the first system as the input. If one considers the panel as an idealized simple damped oscillator, capable of being described solely in the fundamental mode of vibration, with the excitation forcef(f) as shown in Figure 2, the well-known
differential
equation
of motion
results :
PANEL-CAVITY
TRANSIENT
99
OSCILLATIONS
The solution of the differential equation by the use of Laplace transforms with zero initial conditions is given in the complex domain as
f:(s>lm +) Then the displacement-time equation (2) or
= (s2 + 250s + 02) .
response of the system in the time domain is the inverse of
40
=
3-l
ts2
+
25ws
+
w2)
(3)
-
A logical assumption for f(t), considering the pressure trace of the inside microphone (Figure l), would be a damped sinusoidal function. Of course, this neglects the coupling of the systems and oversimplifies the system in some respect. However, it will be a very accurate idealization in that the limiting response will serve as an upper limit. The coupled system cannot achieve a higher transient response than the response of the simple system with a
--____
__ __. Time kec
1 i :-‘
Figure 2. Damped sinusoidal input. damped sinusoidal forcing function. The magnitude of the damped sine wave, F,, cannot exceed the maximax value of a system having one degree of freedom. For a true N wave input the maximax response is approximately 2.1 times the static response for the same pressure. The forcing function as illustrated in Figure 2 can be represented by f(t) = F,, evar sin wd t,
(4)
where F, = forcing amplitude coefficient, wd = damped angular forcing frequency, a = decay 5 rate = li& or cJ;rod,and & = damping ratio, The transfo:med expression for (4) is
m=
Few
(s + a)2 +
co:
(5)
where wd represents the value for 2~17~. Substitution of expression (5) into expression (2) yields the final form for the displacementtime response in the complex domain :
Fowd 3s)
=
(s
+
42
+
l/m w;
‘($2
+
52ws + w2) *
(6)
100
G. D. WHITEHOUSE AND R. L. LOWERY
The inverse of expression (6) can be obtained in a straightforward the property of convolution and Borel’s theorem:
manner making use of
x(t) = 9-1 X(S)= 9-r h,(s) .hz(S)= h,(t)* h,(t)
(7)
where,
FOWd
Us)=(s + cc>* + of! and
h*(s)= The displacement-time
l
s*+2&Js+o*
response can now be written as
x(t) = j h,(t - T)h2(T)dT.
(10)
0
The displacement-time
response is the evaluation of the expression . e-cwo-T)sin (wdT) . sin uJ~
(t -
T)
d7.
(11)
The integration of expression (11) poses no problem, although the algebra becomes somewhat tedious. After integrating and non-dimensionalizing, the expression for the dynamic response factor is given by [y sin At - A cos At] + eYt 2[72 + AZ]
A + eYt[y sin Bt - Bcos Bt] + 2[y2 f B2] 2[9 + A*] + B + 2[y* + B2] I
-cosol/l
-<*t
x
x ey,[AsinAt+ycosAt]_eyt[BsinBt+ycosBt] [
-2[y21:A2]+2[y2:
where A=w,-wm,
2[y*+ B*]
2[y* + AZ]
B= wd + wl/l
-
B*]
-
(12)
<*, and y = -[CZ- &J].
3. RESULTS
Figures 3, 4, 5 and 6 are plots of the response factor (D.R.F.) (non-dimensional) of the system as a function of the frequency ratio for various values of the damping factor, both of the system and the decaying type input. From the plots of the dynamic response ratio it can be seen that for lightly damped systems, large amplitudes can be achieved. Lightly damped cavities such as large enclosures with hard interior surfaces would be the type that persistent pressure oscillations can exist. If the cavity fundamental frequency is about the same as the natural frequency of a coupled structural member, such as a window, then large amplitudes can be expected. In Figure 3, with an input damping factor of about 0.01, and a damping factor of a window in the order of O-1a dynamic
PANEL-CAVITY
TRANSIENT
OSCILLATIONS
1011
response of nearly 4-Ois observed. The damping factor of 0.1for a window is not unreasonable as some experimental measurements from free vibration tests (see reference 1) have shown that large windows can have damping ratios as small as 5 = O-08. Figures 4 through 6 further illustrate that if either the input forcing function persists for a few cycles or the structural system, such as a window or flexible ceiling, is lightly damped, then large amplitudes might occur.
W/oJd
wh,j
Figure 3. Frequency effect on D.R.F., 5, = 0.01. Figure 4. Frequency effect on D.R.F., & = 0.05.
0
0.5
I.0
I.5
2-O
2.5
Figure 5. Frequency effect on D.R.F., & = 0.2. Figure 6. Frequency effect on D.R.F., & = 0.1.
4. CONCLUSIONS It should be noted that the D.R.F. used in this article is based on the simple relationship of x/(I;,/k). The overall D.R.F. could be higher if it was based on the input area times the input pressure of the boom.
.3.0
102
G. D. WHITEHOUSEAND R. L. LOWERY
Future standards for structural design will have to account for the possibility of lightly damped coupled systems. The onset of the supersonic aircraft will cause considerable attention to this area, and new methods for fastening of windows, ceilings and other flexible structural members must be contemplated. REFERENCES 1. ANDREWSASSOCIATES, INC. and HUDGINS,THOMPSON,BALL and ASSOCIATES, INC. 1965 Final Report on Studies of Structural Response to Sonic Booms. Federal Aviation Agency. 2. J. D. SIMPSON1966 Unpublished Thesis, Oklahoma State University. The transient response of a Helmholtz resonator with application to sonic boom studies. 3. W. T. THOMPSON 1962 Laplace Transformation. 2nd. ed. New York: Prentice-Hall. 4. G. D. WHITEHOUSE 1967 Unpublished Thesis, Oklahoma State University. Coupled and uncoupled panel response to sonic boom type inputs.
APPENDIX : NOMENCLATURE? m, c, k fW,fW Fo wd
%: w T
mass, damping coefficient and spring constant in simple system forcing function-time dependent, forcing function transformed forcing amplitude coefficient damped angular forcing frequency damping ratio of simple system damping ratio of damped sinusoid forcing input undamped natural frequency of simple system dummy variable used in time integration