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Instability of a periodically moving plate
Journal
of Sound and Vibration (1980) 68(2),
INSTABILITY
18 l-l
86
OF A PERIODICALLY
MOVING PLATE
J. ZAJ4CZKOWSKl Lodz Technical University, Lode, Zwirki 36, Poland
AND G. YAMADA Faculty of Engineering, Hokkaido University, Sapporo, Japan (Received 9 May 1979)
The parametric instability of the in-plane motion of a rectangular plate of periodically varying length is investigated. The boundaries of the instability regions are found and plotted. 1.
INTRODUCTION
The parametric instability of plates has been investigated by a number of authors. EvanIwanowski [l] has given a wide discussion of the problem as well as a survey of publications on the subject. The problem of the instability of the motion of beams of periodically varying length has been discussed in reference [2]. The purpose of this paper is to investigate the instability of the in-plane motion of a rectangular plate of periodically varying length.
2.
EQUATIONS OF MOTION
The plate considered, shown in Figure 1, is moving periodically in the x-direction, the support being motionless. The motion results in a varying length of the plate. The two opposite edges parallel to the x-axis are simply supported; the left edge is free and the
Figure 1. Moving plate of periodically
varying
length.
181 0022-460X/80/020181
+06 $02.00/O
C 1980 Academic
Press Inc. (London)
Limited
182
J. ZAJl$ZKOWSKI
AND G. YAMADA
right edge is clamped. The driving force is applied to the right edge of the plate which is sliding in the motionless support. The coefficient of friction at the supports where relative motion occurs is assumed to be equal to zero. According to the linear theory [3] the equation of motion of the plate is (a list of symbols is given in the Appendix)
(1) The transverse loading, normal force, shearing forces and bending and twisting moments are given by
a%/at2,
qW = -mp
N, =
Q, = aiwax + aM,,iaY, M, = -
Dp(a2w/aX2+V
ey = aMylay - aMxylax,
a2Wfay2),
MXy = -M,,,
a%,jat2,
-mpx
My
= D,(i
=
-V)
- Dp(a2W/ay2 a2w/ax
+ V d2W/dX2),
ay,
(2)
where w denotes the transverse deflection of the plate, mp the mass per unit area, L, the length of the plate in the x-direction, and D, = Eh3/(12- 12~~) is the plate stiffness, h being the plate thickness and v Poisson’s ratio. The boundary conditions are assumed to be as follows: w(L,,y,t)
= 0,
Q,P,y,t) = 0,
(wax)
(L,, Y,t) = 0,
MAO,Y, t) = M,,(O, Y, t) = 0,
w(x,O, t) = w(x, L,, t) = 0,
M,(x,O,t)
= M,(x,L,,t)
= 0.
(3)
The width of the plate L, in the y-direction is constant. The length of the plate in the xdirection is taken to be a periodic function of time of the form L, = L,q(t)
= L,(l-acoswt),
(4)
where L, denotes the mean length of the plate, a is the dimensionless amplitude of the inplane motion, and o is the radian frequency of the excitation. To proceed further in the analysis, the above equations are subjected to the following transformations: 5 = XlL,,
? =
YlL2,
9 =
a/2,
w(x, Y,4 = w,
rl99).
(5)
The transverse acceleration of the element of the plate then takes the form
W) ur> 49=C&G% 4, ..,-G&, 1
The transverse deflection of the plate V(& q, 9) is to be expanded into the following series:
41
i
= ZT.
(7)
i G(Q)
The functions Z, are taken to be of the form
(8)
PARAMETRIC
INSTABILITY
OF PLATES
183
where the Xi are the eigenfunctions of the beam when one side is clamped and the Y;:are the eigenfunctions of the beam when it is simply supported, for i = 1,2 and j = 1,2. In order to impose the boundary conditions it is convenient to apply Galerkin’s method to the equation of the motion in the form (1) = 0.
(9)
Integration by parts of equation (9) with the use of equations (2)-(7) and imposing boundary conditions (3) on the resultant expression give the set of ordinary differential equations Dd’T 2 dq d92+--Ecp d9 +
dT d9
,z4j)j’Le (s4A,+s2G,~2+A,~4)-;~B P
+
0
(10)
where s = Lo/L2 is the length-to-width ratio and the constant matrices are given by A, =
A, = 1
A, =
la2zT
a2z
-----ddr sso o a5 all a5 all
drl,
A, =
B=
G = v(A,+A,)
+ 2(1-v)A,.
(11)
Equation (10) can be rewritten in the form of equation (2) of reference [4] : 1”P’“‘T = 0
(12)
k2 = 4D,/o%n, L,,
(13)
1 n=o.z
where
the operator PC”)is defined by
a(") mecos
d2 2m9 __ d9’
(14)
184
J. ZAJ.$CZKOWSKI
AND ‘3. YAMADA
and the matrix coeffkients of the trigonometric series are given by #’ = (1 + 5~’+ 15a4/8) D ,
a\:) = -(5a+
a\“,’= -(5a3/2+5aS/16)D,
a$ = (5~’+ 5a4/2) D, ag) = 5a4 D/8,
a$) = -usD/16, S’,” = -(8u2 +4u4)E
SC) = (4u+6u3 +as/2)E,
Sk”,’= -2u4E,
pi”,’= (6u3 + 3u5/4) E,
- 2u2C,
#I\;,’= u&4,
y’P,’= -(4a + 18u3 + 5u5/2) B - (3a3 + u5/2) C,
@ = (8a2 + 6a4) B + (2u2 + 3a4/2) C, ~(20) c = 8(u2+a4)B
15a3/2+5as/8)D,
y$“,’= -(6a3+5u5/4)B
+ (3u3+as/4)C,
y\“,’= 2a4 B - 3a4 C/2,
y\‘,’= -a”(B-C)/4,
yb” = A + sZ(1+3a2/2)G
+ s4(1+5a2+15a4/8)A,,
r’:,’ = -uA - s2(3a+3a3/4)G
- s4(5a+ 15a3/2+5u5/8)A,,
y’;?,’= %‘a2 G/2 + 5s4(u2 +u4/2) A,,
y\‘,’= -?a3
G/i - 5s4(a3/2+us/16) A,,
yi”’ = 5s4u4 A,/8,
y:t,’ = -s4a5A,/16.
(15)
Coeffkients not listed above are equal to zero.
3. INSTABILITY
ANALYSIS
The instability regions, having periodic solutions on the boundaries, were found from equation (24) of reference [4], with c x 80~ + 10 for i = 1,2 and j = 1,2 in expression (8). There was no coupling between the symmetric and antisymmetric modes and the instability domain for s = 1 and s = 2 in the areas investigated was determined by the symmetric mode. The combined resonances of the symmetric modes were estimated by using equation (25) of reference [4], and the combined resonances associated with the antisymmetric modes for s = 0.5 and s = 1.0 were found by utilizing the results obtained for the symmetric modes for s = 1.0 and s = 2.0, respectively. The natural frequencies of a motionless plate of length Z., are
wij = 1;Jlipipg
(16)
The parameters ;iij were calculated from the equation (s4A,+s2G+A-X41)X
= 0,
and are shown in Table 1. TABLE
1
Eigenvulues of motionless plate
0.5 1.0 2.0
2.39 3.57 6.49
5.00 5.78 804
3.57 6.49 12.66
5.78 8J-M 13.58
(17)
PARAMETRIC
INSTABILITY
185
OF PLATES
Figure 2. Stability chart for a plate undergoing in-plane periodic motion. L,/L, = 0.5, I;, , = 2.39; (b) L,/L, = 1, X,, = 3.57; (c) L,/L, = 2.1, I = 6.49.
L,(t) = L,(l
-
a cos
wt). (a)
Poisson’s ratio was taken to be v = 0.3. Figures 2(a), (b) and(c) show the instability regions (shaded areas) in the parameter plane of Jmand a for s = 0.5, s = 1 and s = 2, respectively. The broken lines indicate the boundaries of the combined resonances which were found approximately. The stability chart for s = 2 is much different from that for s = 0.5. This is caused by the fact that for plates that are long in the x-direction the influence of the simply supported edges parallel to the x-axis is dominant over the influence of the relatively faroff clamped and free edges, while the opposite is true for plates long in the y-direction. In particular, for an infinitely wide plate (S = 0) equation (10) governing the motion of the plate tends to the equation governing the motion of a beam clamped at one end, as studied in reference [2]. The influence of the second symmetric mode in the x-direction in the areas investigated is significant for long plates and becomes small for wide plates.
ACKNOWLEDGMENT
This work was sponsored by Hokkaido University. The authors would like to acknowledge the guidance of Professor T. Irie of Hokkaido University. Acknowledgment is also due to Professor J. Zakrzewski of Lodz Technical University for his encouragement and support in undertaking of the study.
.I. ZAJ.$-ZKOWSKI
186
AND G. YAMADA
REFERENCES 1. R. M. EVAN-IWANOWSKI1976 Resonance Oscillations in Mechanical Systems. AmsterdamOxford-New York : Elsevier Scientific Publishing Company. 2. J. ZAJ~CZKOWSKIand J. LIPI~SKI 1979 Journal of Sound and Vibration 63, 9-18. Instability of the motion of a beam of periodically varying length. 3. S. P. TIMOSHENKO and S. WOINOWSKY-KRIEGER 1959 Theory of Plates and Shells. New York: McGraw-Hill Book Company, Inc. 4. J. ZAJ~CZKOWSKIand J. LIPI~KI 1979 Journal of Sound and Vibration 63, l-7. Vibrations of parametrically excited systems.
APPENDIX:
MAIN SYMBOLS
dimensionless amplitude of the in-plane motion of the plate plate thickness length of the plate in x-direction width of the plate in y-direction mean length of the plate integer mass per unit area of the plate transverse loading length-to-width ratio time transverse deflection of the plate in-plane co-ordinates constant R x R matrices A,, A,, A,B,C,D,E,G constant R x R matrices D, = Eh3/(12- 121”) plate stiffness ED Young’s modulus of elasticity unit matrix bending moments twisting moments normal force operator shearing forces number of elements of vector Z column R-vector transverse deflection of the plate eigenfunctions of one side clamped beam eigenfunctions of simply supported beam row R-vector transpose of the vector Z elements of the vector Z dimensionless time eigenvalues of motionless plate dimensionless in-plane co-ordinates periodic function of time w radian frequency of excitation @j natural frequencies of motionless plate of length Lo
A,,‘i:
of Sound and Vibration (1980) 68(2),
INSTABILITY
18 l-l
86
OF A PERIODICALLY
MOVING PLATE
J. ZAJ4CZKOWSKl Lodz Technical University, Lode, Zwirki 36, Poland
AND G. YAMADA Faculty of Engineering, Hokkaido University, Sapporo, Japan (Received 9 May 1979)
The parametric instability of the in-plane motion of a rectangular plate of periodically varying length is investigated. The boundaries of the instability regions are found and plotted. 1.
INTRODUCTION
The parametric instability of plates has been investigated by a number of authors. EvanIwanowski [l] has given a wide discussion of the problem as well as a survey of publications on the subject. The problem of the instability of the motion of beams of periodically varying length has been discussed in reference [2]. The purpose of this paper is to investigate the instability of the in-plane motion of a rectangular plate of periodically varying length.
2.
EQUATIONS OF MOTION
The plate considered, shown in Figure 1, is moving periodically in the x-direction, the support being motionless. The motion results in a varying length of the plate. The two opposite edges parallel to the x-axis are simply supported; the left edge is free and the
Figure 1. Moving plate of periodically
varying
length.
181 0022-460X/80/020181
+06 $02.00/O
C 1980 Academic
Press Inc. (London)
Limited
182
J. ZAJl$ZKOWSKI
AND G. YAMADA
right edge is clamped. The driving force is applied to the right edge of the plate which is sliding in the motionless support. The coefficient of friction at the supports where relative motion occurs is assumed to be equal to zero. According to the linear theory [3] the equation of motion of the plate is (a list of symbols is given in the Appendix)
(1) The transverse loading, normal force, shearing forces and bending and twisting moments are given by
a%/at2,
qW = -mp
N, =
Q, = aiwax + aM,,iaY, M, = -
Dp(a2w/aX2+V
ey = aMylay - aMxylax,
a2Wfay2),
MXy = -M,,,
a%,jat2,
-mpx
My
= D,(i
=
-V)
- Dp(a2W/ay2 a2w/ax
+ V d2W/dX2),
ay,
(2)
where w denotes the transverse deflection of the plate, mp the mass per unit area, L, the length of the plate in the x-direction, and D, = Eh3/(12- 12~~) is the plate stiffness, h being the plate thickness and v Poisson’s ratio. The boundary conditions are assumed to be as follows: w(L,,y,t)
= 0,
Q,P,y,t) = 0,
(wax)
(L,, Y,t) = 0,
MAO,Y, t) = M,,(O, Y, t) = 0,
w(x,O, t) = w(x, L,, t) = 0,
M,(x,O,t)
= M,(x,L,,t)
= 0.
(3)
The width of the plate L, in the y-direction is constant. The length of the plate in the xdirection is taken to be a periodic function of time of the form L, = L,q(t)
= L,(l-acoswt),
(4)
where L, denotes the mean length of the plate, a is the dimensionless amplitude of the inplane motion, and o is the radian frequency of the excitation. To proceed further in the analysis, the above equations are subjected to the following transformations: 5 = XlL,,
? =
YlL2,
9 =
a/2,
w(x, Y,4 = w,
rl99).
(5)
The transverse acceleration of the element of the plate then takes the form
W) ur> 49=C&G% 4, ..,-G&, 1
The transverse deflection of the plate V(& q, 9) is to be expanded into the following series:
41
i
= ZT.
(7)
i G(Q)
The functions Z, are taken to be of the form
(8)
PARAMETRIC
INSTABILITY
OF PLATES
183
where the Xi are the eigenfunctions of the beam when one side is clamped and the Y;:are the eigenfunctions of the beam when it is simply supported, for i = 1,2 and j = 1,2. In order to impose the boundary conditions it is convenient to apply Galerkin’s method to the equation of the motion in the form (1) = 0.
(9)
Integration by parts of equation (9) with the use of equations (2)-(7) and imposing boundary conditions (3) on the resultant expression give the set of ordinary differential equations Dd’T 2 dq d92+--Ecp d9 +
dT d9
,z4j)j’Le (s4A,+s2G,~2+A,~4)-;~B P
+
0
(10)
where s = Lo/L2 is the length-to-width ratio and the constant matrices are given by A, =
A, = 1
A, =
la2zT
a2z
-----ddr sso o a5 all a5 all
drl,
A, =
B=
G = v(A,+A,)
+ 2(1-v)A,.
(11)
Equation (10) can be rewritten in the form of equation (2) of reference [4] : 1”P’“‘T = 0
(12)
k2 = 4D,/o%n, L,,
(13)
1 n=o.z
where
the operator PC”)is defined by
a(") mecos
d2 2m9 __ d9’
(14)
184
J. ZAJ.$CZKOWSKI
AND ‘3. YAMADA
and the matrix coeffkients of the trigonometric series are given by #’ = (1 + 5~’+ 15a4/8) D ,
a\:) = -(5a+
a\“,’= -(5a3/2+5aS/16)D,
a$ = (5~’+ 5a4/2) D, ag) = 5a4 D/8,
a$) = -usD/16, S’,” = -(8u2 +4u4)E
SC) = (4u+6u3 +as/2)E,
Sk”,’= -2u4E,
pi”,’= (6u3 + 3u5/4) E,
- 2u2C,
#I\;,’= u&4,
y’P,’= -(4a + 18u3 + 5u5/2) B - (3a3 + u5/2) C,
@ = (8a2 + 6a4) B + (2u2 + 3a4/2) C, ~(20) c = 8(u2+a4)B
15a3/2+5as/8)D,
y$“,’= -(6a3+5u5/4)B
+ (3u3+as/4)C,
y\“,’= 2a4 B - 3a4 C/2,
y\‘,’= -a”(B-C)/4,
yb” = A + sZ(1+3a2/2)G
+ s4(1+5a2+15a4/8)A,,
r’:,’ = -uA - s2(3a+3a3/4)G
- s4(5a+ 15a3/2+5u5/8)A,,
y’;?,’= %‘a2 G/2 + 5s4(u2 +u4/2) A,,
y\‘,’= -?a3
G/i - 5s4(a3/2+us/16) A,,
yi”’ = 5s4u4 A,/8,
y:t,’ = -s4a5A,/16.
(15)
Coeffkients not listed above are equal to zero.
3. INSTABILITY
ANALYSIS
The instability regions, having periodic solutions on the boundaries, were found from equation (24) of reference [4], with c x 80~ + 10 for i = 1,2 and j = 1,2 in expression (8). There was no coupling between the symmetric and antisymmetric modes and the instability domain for s = 1 and s = 2 in the areas investigated was determined by the symmetric mode. The combined resonances of the symmetric modes were estimated by using equation (25) of reference [4], and the combined resonances associated with the antisymmetric modes for s = 0.5 and s = 1.0 were found by utilizing the results obtained for the symmetric modes for s = 1.0 and s = 2.0, respectively. The natural frequencies of a motionless plate of length Z., are
wij = 1;Jlipipg
(16)
The parameters ;iij were calculated from the equation (s4A,+s2G+A-X41)X
= 0,
and are shown in Table 1. TABLE
1
Eigenvulues of motionless plate
0.5 1.0 2.0
2.39 3.57 6.49
5.00 5.78 804
3.57 6.49 12.66
5.78 8J-M 13.58
(17)
PARAMETRIC
INSTABILITY
185
OF PLATES
Figure 2. Stability chart for a plate undergoing in-plane periodic motion. L,/L, = 0.5, I;, , = 2.39; (b) L,/L, = 1, X,, = 3.57; (c) L,/L, = 2.1, I = 6.49.
L,(t) = L,(l
-
a cos
wt). (a)
Poisson’s ratio was taken to be v = 0.3. Figures 2(a), (b) and(c) show the instability regions (shaded areas) in the parameter plane of Jmand a for s = 0.5, s = 1 and s = 2, respectively. The broken lines indicate the boundaries of the combined resonances which were found approximately. The stability chart for s = 2 is much different from that for s = 0.5. This is caused by the fact that for plates that are long in the x-direction the influence of the simply supported edges parallel to the x-axis is dominant over the influence of the relatively faroff clamped and free edges, while the opposite is true for plates long in the y-direction. In particular, for an infinitely wide plate (S = 0) equation (10) governing the motion of the plate tends to the equation governing the motion of a beam clamped at one end, as studied in reference [2]. The influence of the second symmetric mode in the x-direction in the areas investigated is significant for long plates and becomes small for wide plates.
ACKNOWLEDGMENT
This work was sponsored by Hokkaido University. The authors would like to acknowledge the guidance of Professor T. Irie of Hokkaido University. Acknowledgment is also due to Professor J. Zakrzewski of Lodz Technical University for his encouragement and support in undertaking of the study.
.I. ZAJ.$-ZKOWSKI
186
AND G. YAMADA
REFERENCES 1. R. M. EVAN-IWANOWSKI1976 Resonance Oscillations in Mechanical Systems. AmsterdamOxford-New York : Elsevier Scientific Publishing Company. 2. J. ZAJ~CZKOWSKIand J. LIPI~SKI 1979 Journal of Sound and Vibration 63, 9-18. Instability of the motion of a beam of periodically varying length. 3. S. P. TIMOSHENKO and S. WOINOWSKY-KRIEGER 1959 Theory of Plates and Shells. New York: McGraw-Hill Book Company, Inc. 4. J. ZAJ~CZKOWSKIand J. LIPI~KI 1979 Journal of Sound and Vibration 63, l-7. Vibrations of parametrically excited systems.
APPENDIX:
MAIN SYMBOLS
dimensionless amplitude of the in-plane motion of the plate plate thickness length of the plate in x-direction width of the plate in y-direction mean length of the plate integer mass per unit area of the plate transverse loading length-to-width ratio time transverse deflection of the plate in-plane co-ordinates constant R x R matrices A,, A,, A,B,C,D,E,G constant R x R matrices D, = Eh3/(12- 121”) plate stiffness ED Young’s modulus of elasticity unit matrix bending moments twisting moments normal force operator shearing forces number of elements of vector Z column R-vector transverse deflection of the plate eigenfunctions of one side clamped beam eigenfunctions of simply supported beam row R-vector transpose of the vector Z elements of the vector Z dimensionless time eigenvalues of motionless plate dimensionless in-plane co-ordinates periodic function of time w radian frequency of excitation @j natural frequencies of motionless plate of length Lo
A,,‘i: