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Large amplitude dynamic response of a diagonal line-loaded rectangular plate
Journal
of Sound and
LARGE
Vibration
AMPLITUDE
(1984)
96(l),
23-27
DYNAMIC
LINE-LOADED
RESPONSE
RECTANGULAR
OF A DIAGONAL PLATE
S. K. CHAUDHURI Department
of Mathematics,
(Received
Acharya
4 July
B. N. Seal College,
Coach-Behar,
1983, and in revised,form
25 October
In this paper the large amplitude dynamic response of a diagonal plate is investigated. The well-known approximate equations of
West Bengal,
India
1983)
line-loaded Berger
[I]
rectangular are used.
1. INTRODUCTION Loads applied to structural systems invariably change with time. If the time variations are small and occur over an extended interval the inertial effects may be neglected and the behaviour of the structure can be approximately determined from consideration of equilibrium and material properties. In modern aircraft and spacecraft vehicles, however, rapid time variations of loadings occur and are significant for the proper design of the structure. Inertial effects must be included and the resulting dynamic behaviour of the system must be determined as a function of time from the equations of motion. Many of the structural problems in the strength and stability of plates, however, cannot be analyzed adequately by using linear theory, since the plate deflection does not remain small in comparison with its thickness. It is therefore of importance to establish the influence of such large amplitudes of deflection upon the response of the system to pulse load excitations. For moderately large deflections of plates von Karman [2] proposed coupled non-linear partial differential equations. Large amplitude vibrations of various elastic plates subsequently have been investigated by Nowinski [3], Vendhan and Das [4] and Batter [S]. Berger proposed a simplified method to analyze large deflections of circular and rectangular plates. Nash and Modeer [6] extended Berger’s method to the dynamic case and analyzed quite elegantly the large amplitude free vibrations of circular and rectangular plates. It has been observed that Berger’s method can be sufficiently accurate both for clamped as well as simply supported edge condition. Problems concerning diagonally loaded plates exist in engineering in supersonic wing theory, where the wing is subjected to a shock wave caused by the corner effect, or in ship design where the structure can be loaded diagonally by severe waves due to storms. This type of non-linear problem has not been previously investigated, to the author’s knowledge. In this paper the large amplitude dynamic response of a diagonal line-loaded rectagular plate is analyzed. Berger’s approximate equations are used. The numerical results for the non-linear and linear responses of the plate when excited by an exponentially decaying pulse are shown graphically. The corresponding linear case results are due to Stanisic [7]. 23 0022-460X/84/
I70023
+05
$03.00/O
IQ 1984
Academic
Press
Inc.
(London)
ILimited
24
S. K.
C‘HAUDHURI
2. ANALYSIS A rectangular ing on position
plate, simply supported along the edges, subjected to a line-load dependand time, is illustrated in Figure 1. Berger’s approximate equations
b
Figure
governing
the vibrations
I. Geometry
of such plates
of the plate and loading.
can be written
as
v4WF(t)-p2F3(f)V2W+(12/h2~2)~(f)W=q(x,y, where ,B2 is determined
t)/D,
(1)
by
(1 II0 and [*=ph3/12D. For the fundamental
bcdxdy={;job;[(g)2 0 mode of vibration
dx dy
(2)
one takes (3)
Clearly
W satisfies
simply
supported’edge
dx,
Y, r) =
P(x, y) = QoH[Y - {(bla)x
conditions.
Also let
Q(r)Wx, YL -
&II- H[Y -
(4) {(bla)x
+
~11.
(5)
Here Q(t) is purely a function of time, QO is the amplitude of the load and E is a small displacement. P(x, y) is a function representing the line loading. H(. . .) is the Heaviside step function, so that
(6) H[y-{(b/a)x+&}]=
0, Putting
equation
(3) in equation
y2(b’a)x+E
y<(b/a)x+E
(2) gives
~2=(3~2/2){(1/.2)+(l/b2)}.
(8)
By applying Galerkin’s method to equation (I), using equations (3) and (4), and keeping in mind the relation (8) one obtains the following equation determining the time function F( 2):
‘F3(() =g; QoQ(r).
(9)
LARGE
AMPLITUDE
LOADED
PLATE
2.5
RESPONSE
To determine the response of the rectangular plate to an exponentially decaying pressure of the form PO e-“’ one puts Q(t) = e-“’ and P, = (ht2/3D)( e/b)Q” = (h.$*/3D)(~/u)Q,( l/h), where A = b/a. Then equation (9) can be put in the form F(t) +o*F(
2) + .z,w’F~( t) = PO e-“‘,
(10)
E, = 1.5, and (Y is a constant.
where w2=(~4/12)h2~2{(l/a2)+(l/b2)}2, equation (10) is given by [5]
PO
F(c)=(~~+~~)
of
+&iql(tl),
eC”‘-cos&+~sinwS,
[
The solution
1
5, = t/(1 -~&,P;/(cz+W2)W~} P&II2 41(5l) =
cm2
2
w2-3a
+&2)3
320~
~*-a*)(a~-3co~)-224a~w* +3( %(a2- 3~‘)’ + 16a2w2}w2
cos (41)
1
I 3a
3a(a2-3w2) 32w4 -
2w2
3cY 9a2+w2
24(~~-cx*)(uw~+l2~1~~(~~-3~~) 2{(ff2 -3w*)*
+ 16a2w2}w2
+24~r~~~-3a((~~-3w*)(w~-n*) ~w~{((Y~-~o~)~ -32W2(~!+W2)X
[ (wZ-3a’)
+y (CY*--a~‘) W
sin 3~5,
+2(d+w*)3[(
x[{2w(u(a2
-05,
(41) 1sin
cos 3W5
1 P;
3P; -2(a2+w*)3
+ 16cu’w2}
e-3uflW2
-
e
(cY2+w’)3(9a2+w*)
a2 - 3~‘)’ + 16a2w2]
- 30~) +4aw(w2
- (Y’)} sin 2~5,
+{8~‘w2-(w2-a2)(~*-3w2)}~~s2~~,] _3P$0 4(a2
eeZuEl . +“2)3
‘ln
06l’
When el is taken as zero (instead of 1.5 as here) the linear agreement with that given in reference [7]. 3. NUMERICAL
RESULTS
AND
frequency
w is an excellent
DISCUSSION
Figure 2 shows, in graphical form, calculated results for F(T) Vs. T (T = wt/ r) for both linear and non-linear responses of a square and a rectagular plate. For the calculations the following data was used: h2,f2/a4 = 1; a =O*Ol; (h,$*/3D)(e/a)Q,= 15. From Figure 2 the following observations can be made.
26
S.
K.
C‘HAlJDHLiRI
O-6
-0-81
/
0
I
I
03
0.6
I
0.9
I .2
I5
T Figue 2. Dynamic response showing F( 7) VS.7. A = b/a = 1: -, square plate (linear) ; - X--, square (non-linear); A = b/a =0.75: --O-, rectangular plate (linear); -0--, rectangular plate (non-linear).
plate
(1) The non-linear dynamic response is greater than the linear one under the diagonal loading. This is true for both square and rectangular plates. The reason for this is that the membrane stresses arising in the non-linear analysis are not effective in stiffening the plate for this special type of diagonal loading: i.e., a decaying pulse excitation. (2) It is interesting to note the difference between the dynamic responses of the square and rectangular plates under this type of diagonal loading. For a square plate both linear and non-linear responses are greater than those of the rectangular plate. This is due to the fact that the region of loading in the case of a square plate is larger than that of a rectangular plate. (3) It is also interesting to note that the non-linear response fluctuates between negative and positive values. The linear analysis will also behave in the same way for higher loadings than that used in these calculations.
TABLE
Dynamic
response
1
F, (P,,) for difirent
load amplitudes
PO
10
20
30
0.05414
0.15482
0.5023
PO 0.01
(linear)
0.1
4.8675 x lO-5
4.8675 x 1O-4
4.8706 x IO--’
4.8674 x lO-5
4.8662 x lO-4
4.7444 x 1O-3
F(P,)
(non-linear)
1.0
-0.07963
-1.14917
-6.12226
Table 1 shows the variations of the dynamic response F,(P,) of a square plate for different values of the load amplitude PO. Here F, (PO) has been obtained by replacing t by 7 (= cot/r) and considering P,, as variable in the expression for F(t). The following data was used in these calculations: w = 5*6982; LY= 0.01, T = 0.1. It can be seen that the response for the non-linear case is significantly different from that for the linear one at higher loadings.
LARGE
AMPLITUDE
LOADED
PLATE RESPONSE
21
ACKNOWLEDGMENT The author wishes to thank Dr B. Banerjee, Head of the Department of Mathematics, Jalpaiguri Government Engineering College, Jalpaiguri, for his guidance in the preparation of this paper.
REFERENCES 1955 Journal of Applied Mechanics 22, 465-470. A new approach to an analysis of large deflection of elastic plate. 2. T. VON KARMAN 1910 Encyklopadie der Mathematischen Wissenschaften, Volume IV, p. 349. Leipzig. 3. J. L. NOWINSKI 1972 MRC Technical Summary Report No. 34, Mathematics Research Centre, U.S. Army, University of Wisconsin. Note on an analysis of large deflections of rectangular plate. 4. C. P. VENDHAN and Y. C. DAS 1975 Journal ofSound and Vibration 39, 147-157. Application of Rayleigh-Ritz and Galerkin methods to non-linear vibration of plate. 5. H. F. BAUER 1968 Journal of Applied Mechanics 35, 47-52. Non-linear dynamic response of elastic plates due to pulse excitation. 6. W. A. NASH and J. MODEER 1960 in Proceedings of the Symposium of the Theory of Thin Elastic Shells, pp. 331-354. Amsterdam: North-Holland Publishing Co. Certain approximate analysis of the non-linear behaviour of plates and shallow shells. 7. M. M. STANISK 1977 American Institute of Aeronautics and Astronautics Journal 15, 1804-1806. Dynamic response of a diagonally loaded rectangular plate. I. H. M. BERGER
of Sound and
LARGE
Vibration
AMPLITUDE
(1984)
96(l),
23-27
DYNAMIC
LINE-LOADED
RESPONSE
RECTANGULAR
OF A DIAGONAL PLATE
S. K. CHAUDHURI Department
of Mathematics,
(Received
Acharya
4 July
B. N. Seal College,
Coach-Behar,
1983, and in revised,form
25 October
In this paper the large amplitude dynamic response of a diagonal plate is investigated. The well-known approximate equations of
West Bengal,
India
1983)
line-loaded Berger
[I]
rectangular are used.
1. INTRODUCTION Loads applied to structural systems invariably change with time. If the time variations are small and occur over an extended interval the inertial effects may be neglected and the behaviour of the structure can be approximately determined from consideration of equilibrium and material properties. In modern aircraft and spacecraft vehicles, however, rapid time variations of loadings occur and are significant for the proper design of the structure. Inertial effects must be included and the resulting dynamic behaviour of the system must be determined as a function of time from the equations of motion. Many of the structural problems in the strength and stability of plates, however, cannot be analyzed adequately by using linear theory, since the plate deflection does not remain small in comparison with its thickness. It is therefore of importance to establish the influence of such large amplitudes of deflection upon the response of the system to pulse load excitations. For moderately large deflections of plates von Karman [2] proposed coupled non-linear partial differential equations. Large amplitude vibrations of various elastic plates subsequently have been investigated by Nowinski [3], Vendhan and Das [4] and Batter [S]. Berger proposed a simplified method to analyze large deflections of circular and rectangular plates. Nash and Modeer [6] extended Berger’s method to the dynamic case and analyzed quite elegantly the large amplitude free vibrations of circular and rectangular plates. It has been observed that Berger’s method can be sufficiently accurate both for clamped as well as simply supported edge condition. Problems concerning diagonally loaded plates exist in engineering in supersonic wing theory, where the wing is subjected to a shock wave caused by the corner effect, or in ship design where the structure can be loaded diagonally by severe waves due to storms. This type of non-linear problem has not been previously investigated, to the author’s knowledge. In this paper the large amplitude dynamic response of a diagonal line-loaded rectagular plate is analyzed. Berger’s approximate equations are used. The numerical results for the non-linear and linear responses of the plate when excited by an exponentially decaying pulse are shown graphically. The corresponding linear case results are due to Stanisic [7]. 23 0022-460X/84/
I70023
+05
$03.00/O
IQ 1984
Academic
Press
Inc.
(London)
ILimited
24
S. K.
C‘HAUDHURI
2. ANALYSIS A rectangular ing on position
plate, simply supported along the edges, subjected to a line-load dependand time, is illustrated in Figure 1. Berger’s approximate equations
b
Figure
governing
the vibrations
I. Geometry
of such plates
of the plate and loading.
can be written
as
v4WF(t)-p2F3(f)V2W+(12/h2~2)~(f)W=q(x,y, where ,B2 is determined
t)/D,
(1)
by
(1 II0 and [*=ph3/12D. For the fundamental
bcdxdy={;job;[(g)2 0 mode of vibration
dx dy
(2)
one takes (3)
Clearly
W satisfies
simply
supported’edge
dx,
Y, r) =
P(x, y) = QoH[Y - {(bla)x
conditions.
Also let
Q(r)Wx, YL -
&II- H[Y -
(4) {(bla)x
+
~11.
(5)
Here Q(t) is purely a function of time, QO is the amplitude of the load and E is a small displacement. P(x, y) is a function representing the line loading. H(. . .) is the Heaviside step function, so that
(6) H[y-{(b/a)x+&}]=
0, Putting
equation
(3) in equation
y2(b’a)x+E
y<(b/a)x+E
(2) gives
~2=(3~2/2){(1/.2)+(l/b2)}.
(8)
By applying Galerkin’s method to equation (I), using equations (3) and (4), and keeping in mind the relation (8) one obtains the following equation determining the time function F( 2):
‘F3(() =g; QoQ(r).
(9)
LARGE
AMPLITUDE
LOADED
PLATE
2.5
RESPONSE
To determine the response of the rectangular plate to an exponentially decaying pressure of the form PO e-“’ one puts Q(t) = e-“’ and P, = (ht2/3D)( e/b)Q” = (h.$*/3D)(~/u)Q,( l/h), where A = b/a. Then equation (9) can be put in the form F(t) +o*F(
2) + .z,w’F~( t) = PO e-“‘,
(10)
E, = 1.5, and (Y is a constant.
where w2=(~4/12)h2~2{(l/a2)+(l/b2)}2, equation (10) is given by [5]
PO
F(c)=(~~+~~)
of
+&iql(tl),
eC”‘-cos&+~sinwS,
[
The solution
1
5, = t/(1 -~&,P;/(cz+W2)W~} P&II2 41(5l) =
cm2
2
w2-3a
+&2)3
320~
~*-a*)(a~-3co~)-224a~w* +3( %(a2- 3~‘)’ + 16a2w2}w2
cos (41)
1
I 3a
3a(a2-3w2) 32w4 -
2w2
3cY 9a2+w2
24(~~-cx*)(uw~+l2~1~~(~~-3~~) 2{(ff2 -3w*)*
+ 16a2w2}w2
+24~r~~~-3a((~~-3w*)(w~-n*) ~w~{((Y~-~o~)~ -32W2(~!+W2)X
[ (wZ-3a’)
+y (CY*--a~‘) W
sin 3~5,
+2(d+w*)3[(
x[{2w(u(a2
-05,
(41) 1sin
cos 3W5
1 P;
3P; -2(a2+w*)3
+ 16cu’w2}
e-3uflW2
-
e
(cY2+w’)3(9a2+w*)
a2 - 3~‘)’ + 16a2w2]
- 30~) +4aw(w2
- (Y’)} sin 2~5,
+{8~‘w2-(w2-a2)(~*-3w2)}~~s2~~,] _3P$0 4(a2
eeZuEl . +“2)3
‘ln
06l’
When el is taken as zero (instead of 1.5 as here) the linear agreement with that given in reference [7]. 3. NUMERICAL
RESULTS
AND
frequency
w is an excellent
DISCUSSION
Figure 2 shows, in graphical form, calculated results for F(T) Vs. T (T = wt/ r) for both linear and non-linear responses of a square and a rectagular plate. For the calculations the following data was used: h2,f2/a4 = 1; a =O*Ol; (h,$*/3D)(e/a)Q,= 15. From Figure 2 the following observations can be made.
26
S.
K.
C‘HAlJDHLiRI
O-6
-0-81
/
0
I
I
03
0.6
I
0.9
I .2
I5
T Figue 2. Dynamic response showing F( 7) VS.7. A = b/a = 1: -, square plate (linear) ; - X--, square (non-linear); A = b/a =0.75: --O-, rectangular plate (linear); -0--, rectangular plate (non-linear).
plate
(1) The non-linear dynamic response is greater than the linear one under the diagonal loading. This is true for both square and rectangular plates. The reason for this is that the membrane stresses arising in the non-linear analysis are not effective in stiffening the plate for this special type of diagonal loading: i.e., a decaying pulse excitation. (2) It is interesting to note the difference between the dynamic responses of the square and rectangular plates under this type of diagonal loading. For a square plate both linear and non-linear responses are greater than those of the rectangular plate. This is due to the fact that the region of loading in the case of a square plate is larger than that of a rectangular plate. (3) It is also interesting to note that the non-linear response fluctuates between negative and positive values. The linear analysis will also behave in the same way for higher loadings than that used in these calculations.
TABLE
Dynamic
response
1
F, (P,,) for difirent
load amplitudes
PO
10
20
30
0.05414
0.15482
0.5023
PO 0.01
(linear)
0.1
4.8675 x lO-5
4.8675 x 1O-4
4.8706 x IO--’
4.8674 x lO-5
4.8662 x lO-4
4.7444 x 1O-3
F(P,)
(non-linear)
1.0
-0.07963
-1.14917
-6.12226
Table 1 shows the variations of the dynamic response F,(P,) of a square plate for different values of the load amplitude PO. Here F, (PO) has been obtained by replacing t by 7 (= cot/r) and considering P,, as variable in the expression for F(t). The following data was used in these calculations: w = 5*6982; LY= 0.01, T = 0.1. It can be seen that the response for the non-linear case is significantly different from that for the linear one at higher loadings.
LARGE
AMPLITUDE
LOADED
PLATE RESPONSE
21
ACKNOWLEDGMENT The author wishes to thank Dr B. Banerjee, Head of the Department of Mathematics, Jalpaiguri Government Engineering College, Jalpaiguri, for his guidance in the preparation of this paper.
REFERENCES 1955 Journal of Applied Mechanics 22, 465-470. A new approach to an analysis of large deflection of elastic plate. 2. T. VON KARMAN 1910 Encyklopadie der Mathematischen Wissenschaften, Volume IV, p. 349. Leipzig. 3. J. L. NOWINSKI 1972 MRC Technical Summary Report No. 34, Mathematics Research Centre, U.S. Army, University of Wisconsin. Note on an analysis of large deflections of rectangular plate. 4. C. P. VENDHAN and Y. C. DAS 1975 Journal ofSound and Vibration 39, 147-157. Application of Rayleigh-Ritz and Galerkin methods to non-linear vibration of plate. 5. H. F. BAUER 1968 Journal of Applied Mechanics 35, 47-52. Non-linear dynamic response of elastic plates due to pulse excitation. 6. W. A. NASH and J. MODEER 1960 in Proceedings of the Symposium of the Theory of Thin Elastic Shells, pp. 331-354. Amsterdam: North-Holland Publishing Co. Certain approximate analysis of the non-linear behaviour of plates and shallow shells. 7. M. M. STANISK 1977 American Institute of Aeronautics and Astronautics Journal 15, 1804-1806. Dynamic response of a diagonally loaded rectangular plate. I. H. M. BERGER