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In-plane vibration of plates under external disturbances applied at singular points
Thin-Walled Structures 18 (1994) 83-95
In-Plane Vibration of Plates Under External Disturbances Applied at Singular Points
Bulent A. Ovunc Department of Civil Engineering, University of Southwestern Louisiana, Lafayette, LA, USA
ABSTRACT The external disturbances which initiate vibration in plates are due to a suddenly applied displacement at singular points of the plates, within an infinitesimal time duration. The second- or fourth-order differential equation for the dynamic displacement function are obtained in the real domain. Then they are expressed in the complex domain in terms of a dynamic displacement function. For the vibration of a given plate under a given external disturbance and boundary and initial conditions, the dynamic displacement function can be obtained from the differential equation of motion, provided that for the same plate under the same conditions, the static displacement function is known. The effects of additional masses, vibration in an elastic medium, and elastic supports have also been introduced in the formulation. Moreover, in the limit, when the natural circular frequency tends to zero, the dynamic displacement function tends to the static displacement function. Illustrative examples are provided for practical applications of the dynamic displacement function.
NOTATION Cf D E g
Friction coefficient between the plate and elastic medium Time independent part o f the dynamic displacement function Young's modulus Gravitational acceleration 83 Thin-Walled Structures 0263-8231/94/$07.00 ~ 1994 Elsevier Science Ltd, England. Printed in Great Britain.
84
h m
P
X,Y
// v
B. A. Ovunc
Plate thickness Additional mass on the plate Own mass of the plate Components of the displacement D, along x and y axes, respectively Time-independent part of the displacement components, u and v Constant of Lam6 Poisson's ratio Components of the displacement function D
INTRODUCTION Recently, the dynamic analysis of plates subjected to in-plane forces has become an interested field because of the many practical applications in engineering. The vibration of plates under arbitrary in-plane forces has been analyzed by using the perturbation method. ~ The vibration of plates was determined by boundary element methods. 2 A comprehensive review on the application of the boundary element method to the dynamic analysis of plates can be found in a recent publication. 3 A new boundary element approach for the deflection analysis of thin elastic plates on an elastic foundation has been presented with the conventional fundamental solution to the biharmonic differential operators. 4 The free vibration of circular plates subjected to a pair of static concentrated forces acting at opposite ends of a diameter has been obtained by using the Ritz method. 5 The dynamic analysis of thin shallow shells has employed the static fundamental solution of thin elastic flat plates. 6 A boundary element method has been developed for the dynamic analysis of plates subjected to in-plane forces by utilizing the fundamental solution of the static problem of the plate. 7 A new algorithm for free vibration in elastodynamics has been presented and has been implemented numerically to the free vibration of membranes and thin plates. 8 Herein, the components of the displacement along two independent directions are combined into a dynamic displacement function in the complex domain. The stresses and strains are expressed in terms of the dynamic displacement function. The differential equation for the dynamic displacement function is obtained from the differential equations of motion of an infinitesimal element. The effects of the additional masses and the interaction of the elastic medium are introduced in the formulation. For a given plate subjected to a disturbance, the integration of the
In-plane vibration of plates
85
differential equation is obtained in series form by considering the static solution of the same plate under the same boundary conditions.
DYNAMIC DISPLACEMENT FUNCTION FOR IN-PLANE VIBRATION In the derivation, it is assumed that the material is homogeneous and isotropic, the displacements and strains are infinitesimal, and stresses are within their proportional limits. The two second-order differential equations of motion have been previously expressed in the real domain in terms of the two components u(x, y), v(x, y) of the displacement function D(x, y),10 02U "k- ~ (1 "02U Ox 2 - VI~y 2 +
02V ~ ---5+ Oy
2
02V (l + V) OxOy
" 02V ~ (l-V)~x2+
02U (l+V)Ox---Oy
l-v2ph+md2u E
g
Cfu=0
dt 2
l - v2 ph + m d2v E
g
dt 2
Cry=0
If the components u, v of the dynamic displacement function D are written in separable variable form, u(x, y) = X(x, y ) f ( t )
(1)
v(x, y) = Y(x, y ) f ( t )
(2)
the time independent part of the equations of motion yield 02X+l(1 02X 1 - oZY 2 Ox 2 _ - v)-fff-iy2 + ~ ( l + v) o - ~ y + f l o X = O .
02y
1 02Y ~(1 " 02X 2 ~y2 + ~ ( 1 - v)-~x2 + +v)o-b-~y+floY=O
(3) (4)
where
f12__
1 - v 2 ph + rn °92 _ Cf
E
g
If the component X(x, y) or Y(x, y) of the dynamic displacement function D(x, y), can be eliminated from the above two equations of motion (eqns (3) and (4)), one has (Z 2 -- 1) A A X-F 2Zf12o A X q - f14o X = 0
(5)
(X 2 - 1 ) A A y + 2Zf12oA y + f l ' o Y = 0
(6)
B.A. Ovunc
86
where
z
--
3-v l+v
_ _ j 4~ fl°2-1+v
,
2
By changing the independent variables x and y from the real domain to z and 2 in to the complex domain, the time independent part of the dynamic displacement function D(z, 2), can be written as,
D(z, 2),
D(z, 2) = X + i Y and the differential equation of motion in the complex domain can be written by means of the differential equations of X and Y (eqns (5) and
(6)), as, 1)
(Z 2
_
04D
02D
Oz20 22 + 2Z f12 Oz 0-----~+ fl4D = 0
(7)
The second-order differential equations (eqns (3) and (4)) in terms of the components X and Y, can be expressed in the complex domain, as
o-7
+ z
+ ~2o ~ = 0
(8)
0(09~ - - ~ + ZOD) ~ -z + f l 2 o D = 0
(9)
~-~
The dynamic displacement function form.
D(z, 2), can be assumed to be in series
STRESSES IN T E R M S OF T H E D Y N A M I C D I S P L A C E M E N T FUNCTION In order to satisfy the given stress boundary conditions, the stresses are expressed in terms of the displacement function D(z, 2), as
oo+oo)
~x-2(l-v2 ) (l+v)\0 Z ~ 2(1
(oo ~z +
v2)
E ( 1 - v ) i [ OO r x l . - 2(1 v2) L- ~ or,
+(l-v)
,,0)
In-plane vibration of plates
87
(1) in cartesian coordinates O'x + O'y -
2
(OD+OD)
z - 1 \3-7z
(11)
3-2
OD ay - ax + 2i*xy = - 2 0--7
(12)
(2) in polar coordinates, { = pe i'~ ae + ap --
2 X-1
(0~
+
0~)
a,~ - ap + 2i%o = - 2 e 2i90D 04
(13)
The differential equation of motion of an infinitestimal element can be also expressed in terms of the stresses: 02 Oz 02 (ax + ay) +/12 (ox + ay) = 0
(14)
or as a fourth-order differential equation, one has
02 //2 0 //4 OqZ20~2 (ffx"~(ry)-]~'~-~ (O'x q- O'y) q- (O'x -47 O'y) = 0
(15)
BOUNDARY CONDITIONS The b o u n d a r y conditions are to be used to determine the integration constants stemming from the integration of the differential equations. The b o u n d a r y conditions may be related to the geometry or to the stresses along the b o u n d a r y of the plates. The b o u n d a r y conditions m a y be: - - Geometrical b o u n d a r y conditions on the displacements, (D)I s = u(s) + iv(s)q,~, = dl + id2
(16)
- - Stress b o u n d a r y conditions as (3 = s + it) 1 at -- iTtsls -- )~ - 1
(
-)
OD OD -~ +~
OD -- e2i° 0~Ls- - - f i + if2
(17)
where, s and t are the independent variables along the tangential and normal directions to the boundary, dl, d2, are known displacements and f l , f2 are k n o w n stresses along the boundaries.
88
B. A. Ovunc
DETERMINATION OF THE D Y N A M I C DISPLACEMENT FUNCTION
An approximate approach to obtain the dynamic displacement function is by assuming that the dynamic displacement function can be written in series form. The first term of the series is the static displacement function and O.,.t and ¢.~t being its component, one has,
2 #D (z, 2).~ = Z(o,., (z) - z ~,,.,(z) - ~st(Z)
(18)
and the following terms of the series can be obtained by means of the differential equations (8) and (9). Thus, differential equations (8) and (9) can be written as
o (0v2 ova)
O~" ~-z + Z 02 J = fl 2°D*t 0--2 \ Of-z + Z--~-z J
= fl°Ds'
By integrating the above two equations while eliminating D2, one obtains the second term of the series as
D 2 (z, 5) -
2 +((Z
(()~ -- 1)2 _}_ ()f jr_ 1 )2 ) ~ f ¢~(z) dz
--
1) 2 --
~2 (Z + 1) 2) ~., ~b(2) -- ((Z -- 1) 4- (Z + 1)) zf (p (z) d2
-((Z - l) - (Z + 1))ff~k(z)dzdz]
(19)
where ~2 _-/~2° / ( z 2 _ 1) The same procedure is repeated for each successive term of the dynamic displacement function D(z, ~). Thus, one has
D(z, 2) = D,.,(z, 2) + D2 (z, 2) + . . . 4- O/(z, 2) + . . . The dynamic displacement function thus determined is approximate. A n o t h e r approximate approach is obtained by introducing the expression of the s u m m a t i o n of the normal stresses (eqn (11)), into the differential equation (14), yielding 02 1 + v fl:o (~,~ + ~y) = 0 OzO-~z (~x + ~.v) + ~ - .
In-plane vibration of plates
89
Again, the solution can be assumed to be in series form, the first term of the series being the static solution, the second term of the series ~b2, can be determined from the following expression,
02 ( , l+v 2 , , OzO2 ~b2(z, 2) + ~b~(z, 5) ) + ~ fl o (th,t (z) + ths, (z)) = 0
(20)
where, ()' represents first derivative of the function (). The successive terms of the series are calculated through the same process. The approximate solutions provide information for the direct solution. Direct determination of the dynamic displacement function is performed by considering the information obtained from the above approximate solution, then by assuming the dynamic displacement function as
D(z, -g) = d O 1 (g, ~) + ~D2 (z, 2) + C~D 3 (2, 2) + ~ D 4 (z, 5)
(21)
where D, (z, 2) = q~(ez) - ~2 J"th(ez) dz +
J"J"~b(az) dz dz
(~Zv)3J"J"J"th(~z) dz dz dz + . . .
D2 (z, -2) = dp"(~z) - ctz dp'(otz) + ~ D3 (z, ~) = ~b(ctz) - ~z J"~9(~tz)dz + ~
dp(o~z)- ~ [
$(~z) dz +...
J"J"~k(ctz)dz dz
(3-~z.~)3J"J" J"~(az)dzdzdz +... D 4 (z, z) = I I ~/((XZ)dz dz - ~ I J"J"O(ez) dz dz dz
+~I
I S I O(~z)dzdzdzdz - . . .
The above defined dynamic displacement function fourth-order differential equation (7), provided that
D(z,-g) satisfies the
(g2 _ 1)0~4 _ 2 Z / j 2 0~2 q_ /j4 = 0
Solving the four roots of the parameter a of the above equation, one has 0~1 = ]~/V/'X" - l, 0~2 = --]~/v/X - l, a 3 = ]~/~/X"-~- l and O~4 = --]~/~/X -3r- 1
90
B.A. Ovunc
The type of displacement function depends upon the sign of the parameter fix
-
1 - -
E
v2 p h + m
- - ~ o - - C f
g
Thus there are three types of displacement functions, for f12 > 0, f12 < 0, and/~2 = 0. The third case corresponds to a transition stage. Moreover, this dynamic displacement function obtained from the fourth-order differential eqn 7, also satisfies the second-order differential equations (8) and (9). The constants ~4, ~ , cg and ~ which appear in the displacement function (eqn (21)), are determined from the boundary conditions of the plates. The natural circular frequencies are evaluated from the condition that the magnitude of the concentrated force applied at the disturbance point should be equal to zero.
APPLICATIONS The applications are for half and infinite planes and circular plates. The boundary conditions may be simply or elastically supported, fixed, or free all along the boundary, or an initial disturbance may be applied at a point either on the boundary or within the plate. The plate may be subjected to additional mass and may vibrate within a partly or totally elastic medium. Half plane subjected to a disturbance on its boundary
Let us consider a half plane under a suddenly applied constant displacement Do within an infinitesimal time period. The static displacement function due to an initial displacement applied to point O (see Fig. 1) is given by
Fig. 1. Half plane subjected to an initial disturbance.
In-plane vibration of plates
91
2#D~, (z, 2) = i ~ [ g ~ ( z ) + z ~b'(z) + q;(z) ] where 4~(z) = ln(z)
and
~O(z)= ln(z)
Substituting the two functions, 4>(z) and ~k(z) in the dynamic displacement function (eqn (21)), one can write its components as (~lZn ~1n+2Zn+l i f ( n + 1 ) + . . . ) Ol,(z, 2) -- M1, (czl)ln ( z ) - k , ~ O(n) (n + 1)!
Dzn(Z, 2)= mz,(~z)ln(z)- k. n! O(n)
n+2~n+l
n--n
D3n(Z,-~)= mln(~l) ln(z)-
~! I
Dan (z, ~) = M2, (~z) In (z) -
dp(n) ~l
n.
z ~(n+l)+...)
( n + 1)! CX~+2Zn+l Z
n--n ~
dp(n+l)+...)
(n+l)]
q5(n)
( n + 1)!
• (n+ 1)+...)
where the newly defined n'th order, Bessel like function M i , 11 sponding to the i'th root of the parameter ~ can be written as,
min - °~inZn n!
~p(n+j)
~in+2 _n+lz 2 "~-O~i-n+42n+2-~2 (n + 1) !
~7+6 n +323 ~-...
(n + 2)!
1 1 1 n + 1 +n~--~+n---~+
n, c o r r e -
(n + 3) ! 1
... + -n+j
The integration constants d , ~ , g, ~ are evaluated from the boundary condition (eqn (13)). Half plane subjected to a disturbance within its body (see Fig. 2) If the external singular disturbance is applied at a point within the half plane, the components ~b(z) and ~b(z) of the static displacement function are ~b(z) = - ( d
+ i ~ ) I n (z) - (g + i ~ ) I n (z - 2ip) + 2ip - z - 2ip
L9(z) = (cg _ i 9 ) In (z) + ( d - i ~ ) In (z - 2ip) + 2ip - z - 2ip The static displacement function being known, the dynamic displacement function, the natural circular frequencies co, the modal shapes, and forced vibration can be obtained by following the above-described steps.
92
B. A. Ovunc
#I
x
fDo
Fig. 2. Half plane subjected to an initial disturbance applied within its body.
Free-free vibration The dynamic displacement function can be assumed as D(z, -2) = ag~ Mn (z, -2) + °Mn+2 Mn+2 (z, 2) + iCgn+ 2 Mn+2 (z, 2)
--i~n Mn (z, 2~) where Mn (z, Y) and Mn+ 2 (z, 2) are the newly defined Bessel like functions o f the order n and n + 2 and ~ n , ~ + 2 , cg,+2, ~ , are the integration constants of the functions Mn (z, ~) and Mn+2 (z, 2). The dynamic displacement function written as D(z, 2-) = ag~ M, (z, 2) + ~n+2 Mn+2 (-% z) + iCOn+2 Un+2 (_7, ~)
--i~n M. (z, 5) satisfies the boundary conditions.
Circular plates Let us consider the vibration o f a solid circular plate. The circular plate is subjected to a set of two opposite displacements applied at the intersection o f a diameter with the circular boundary (see Fig. 3).
x
Fig. 3. Circular plate subjected to diametrical initial displacement.
In-plane vibration of plates
93
The c o m p o n e n t s ~b(z) and qJ (z) of the static displacement function have been given as,
dp(z)=-A[ln(R-z)-ln(-R-z)+ R] [l n ( R - z ) - I n ( - R - z ) R 0(z)=A R-z
~ - - -R
R+z
The static displacement function being known, the dynamic displacement function, natural circular frequencies, ~, modal shapes, and the forced vibration can be obtained by following the above-described steps. Approximate solutions for the vibration can be assumed by neglecting the c o m p o n e n t ~ (z, 2) in the displacement functions. The dynamic displacement function is determined from the differential equation of normal stresses (eqn (11)),
02 ( , , l+v 2 Oz02 4'2(z'2)+ck2(z'2))+--U-#°(4"s'(Z)+
4¢s,
(z))=o
Its integral can be obtained t h r o u g h BesseMike functions. The dynamic displacement function can be written as D (z, 2) = {~ (z, 2)} T {C}
(22)
where {e#(z, 2)} can be called a vector of the shape functions, whose nature depend u p o n the signs of rio, and {C} is the vector of integration constants, d , 9L g , @. F o r plates with singularity points, the shape function can be selected as {~ (z - a, 2 - ~}T = (j//~ (z -- a, 2 -- b) ~i~ln (Z - -
where j l i ((z - ~, z - b) and
a, 2 - b)
pi ((z -
jg2 (z - a, 2 - b) ~ ] (z - a, 2 - b))
a, ~ - b) (i = 1 or 2) are
d'[ j ((Z -- a, z -- b) = mln (ill (z - a), fll (2 - b)) - ( - 1) j M2 (f12 (z - a), f12 (2 - b))
~
((z - a, 2 - b) = p1 (fl, (z - a), fl, (2 - b)) _ ( _ 1)i p2 (f12 (z - a), f12 (2 - b))
For fl~=fl2o/(Z+l)>0
and
f l ~ = f l 2o / ( Z - 1 ) > 0
B. A. Ovunc
94
the functions M~(fli(z g a), fii (z - b)) and P n' ( fli (Z - a ) , fii (2- - - b)) a r e similar to the first- and second kind of Bessel function. Their expressions are as follows: -
M ' (=, 2-) : f17 (z - a) n Z
( - 1)k/~k (z - a) k (z_-#-L-~)k k ! ( n + k + 1)] PI, (z, 2-) = In (z - a) M pli (7 \ ~ , 2-) k=0
- f i t (z - a)" Z
( - 1)k//~k (z - a) k (z---Z-b) k ~p(n + k) 1)!
where a and b show the location of a disturbance point within or along the boundary of the plates. Similar to the Bessel functions, M~ (z, 2-) and P i (z, 2-) are related to their derivatives. 1° The vector of the integration constant { C } , which appears in the dynamic displacement function (eqn (2)), is determined by writing the displacement along the boundary, by solving for {C}, then substituting {C} in the expression for displacement. For the vibration of circular plates without any singularity point within or along the boundary, the shape function in the expression of the displacement can be written from eqn (22) (a = b = 0), as
=
(z, 2-)
(z, 2-)
2-)
2 (z, 2-))
In nondimensional polar coordinates, z = R p e i~, r = Rp, the recently defined Bessel functions reduce to ~.#,,J (p, 7~) = e in'~ (J~ (2~,, p) - ( - 1 ) / 4 (7,,, P)) :~, (p, O) = 2ei,,0 (In {e ~ i0~/,,i, P) Jn (2,i, 1O) q- ~.n (l~ni, P) - ( - 1) ~ [In (e i~ 2.~, p) J. (2.i, p) + ,L, (':%~, P)])
where, J~ is the Bessel function of order n and ~,i = Rfilni, 7,i = Rfl2,,i. The parameters n and i are related to the nodal diameters and nodal circles, respectively. CONCLUSIONS The differential equation of motion is derived for the in-plane vibrations of plates, in terms of the dynamic displacement function. The differential equations are expressed in the complex domain, so that they can be integrated directly. The dynamic displacement function is determined uniquely, in series form, through the integration of the differential equation of motion, provided that, for a given system, the static displacement function under the same b o u n d a r y conditions is known.
In-plane vibration of plates
95
REFERENCES 1. Bassily, S. F. & Dickinson, S. M., Vibration of plates subjected to arbitrary in-plane loads. J. Appl. Mech., 40 (1973) 1023-8. 2. Katsikadelis, J. T., A boundary element solution to the vibration problem of plates. J. Sound & Vib., 141(2) (1990) 313-22. 3. Manolis, G. & Beskos, D. E., Boundary Element Methods in Elasto-D)namics. C. Allen and Unwin Publ., London, UK, 1988. 4. Costa, J. A. & Brebbia, C. A., Bending of plates on elastic foundations using the boundary element method. In Proc., 2nd Int. Conf. on Variational Methods in Engineering, ed. C. A. Brebbia. Springer Verlag, London, 1985, pp. 5-23. 5. Ayoub, E. F. & Leissa, A. W., Free vibration and tension buckling of circular plates with diametral point forces. Joint Applied Mechanics/Bioengineering Conf., Ohio State University, June 16-19, 1991, Paper No. 91-APM-24. 6. Providakis, C. P. & Beskos, D. E., Free vibrations of shallow shells by boundary and internal elements. Comp. Meth. Appl. Mech. Eng., 92(1) 55-74. 7. Katsikadelis, J. T. & Sapountzakis, E. J., A BEM solution to dynamic analysis of plates subjected to in plane forces. A S M E Engineering Systems Design and Analysis, PD, 47(5) (1992) 41-8. 8. Wang, X. X., Qian, J. & Huang, M. G., A new algorithm and application to free vibration analysis using boundary/interior elements approach. J. Compt. & Struct., 32(1) (1989) 55, 61. 9. Ovunc, B. A., Semi infinite plate subjected to concentrated loads acting in its internal point, Vol. 28. IABSE, Zurich, Switzerland, 1968, pp. 121-5. 10. Ovunc, B. A., In-plane vibration of plates by continuous mass matrix method. J. Compt. & Struct., 21(4) (1985) 887-91. 11. Ovunc, B. A., Nonlinear free vibration of plates. J. Compt. & Struct., 21(4) (1985) 887-91.
In-Plane Vibration of Plates Under External Disturbances Applied at Singular Points
Bulent A. Ovunc Department of Civil Engineering, University of Southwestern Louisiana, Lafayette, LA, USA
ABSTRACT The external disturbances which initiate vibration in plates are due to a suddenly applied displacement at singular points of the plates, within an infinitesimal time duration. The second- or fourth-order differential equation for the dynamic displacement function are obtained in the real domain. Then they are expressed in the complex domain in terms of a dynamic displacement function. For the vibration of a given plate under a given external disturbance and boundary and initial conditions, the dynamic displacement function can be obtained from the differential equation of motion, provided that for the same plate under the same conditions, the static displacement function is known. The effects of additional masses, vibration in an elastic medium, and elastic supports have also been introduced in the formulation. Moreover, in the limit, when the natural circular frequency tends to zero, the dynamic displacement function tends to the static displacement function. Illustrative examples are provided for practical applications of the dynamic displacement function.
NOTATION Cf D E g
Friction coefficient between the plate and elastic medium Time independent part o f the dynamic displacement function Young's modulus Gravitational acceleration 83 Thin-Walled Structures 0263-8231/94/$07.00 ~ 1994 Elsevier Science Ltd, England. Printed in Great Britain.
84
h m
P
X,Y
// v
B. A. Ovunc
Plate thickness Additional mass on the plate Own mass of the plate Components of the displacement D, along x and y axes, respectively Time-independent part of the displacement components, u and v Constant of Lam6 Poisson's ratio Components of the displacement function D
INTRODUCTION Recently, the dynamic analysis of plates subjected to in-plane forces has become an interested field because of the many practical applications in engineering. The vibration of plates under arbitrary in-plane forces has been analyzed by using the perturbation method. ~ The vibration of plates was determined by boundary element methods. 2 A comprehensive review on the application of the boundary element method to the dynamic analysis of plates can be found in a recent publication. 3 A new boundary element approach for the deflection analysis of thin elastic plates on an elastic foundation has been presented with the conventional fundamental solution to the biharmonic differential operators. 4 The free vibration of circular plates subjected to a pair of static concentrated forces acting at opposite ends of a diameter has been obtained by using the Ritz method. 5 The dynamic analysis of thin shallow shells has employed the static fundamental solution of thin elastic flat plates. 6 A boundary element method has been developed for the dynamic analysis of plates subjected to in-plane forces by utilizing the fundamental solution of the static problem of the plate. 7 A new algorithm for free vibration in elastodynamics has been presented and has been implemented numerically to the free vibration of membranes and thin plates. 8 Herein, the components of the displacement along two independent directions are combined into a dynamic displacement function in the complex domain. The stresses and strains are expressed in terms of the dynamic displacement function. The differential equation for the dynamic displacement function is obtained from the differential equations of motion of an infinitesimal element. The effects of the additional masses and the interaction of the elastic medium are introduced in the formulation. For a given plate subjected to a disturbance, the integration of the
In-plane vibration of plates
85
differential equation is obtained in series form by considering the static solution of the same plate under the same boundary conditions.
DYNAMIC DISPLACEMENT FUNCTION FOR IN-PLANE VIBRATION In the derivation, it is assumed that the material is homogeneous and isotropic, the displacements and strains are infinitesimal, and stresses are within their proportional limits. The two second-order differential equations of motion have been previously expressed in the real domain in terms of the two components u(x, y), v(x, y) of the displacement function D(x, y),10 02U "k- ~ (1 "02U Ox 2 - VI~y 2 +
02V ~ ---5+ Oy
2
02V (l + V) OxOy
" 02V ~ (l-V)~x2+
02U (l+V)Ox---Oy
l-v2ph+md2u E
g
Cfu=0
dt 2
l - v2 ph + m d2v E
g
dt 2
Cry=0
If the components u, v of the dynamic displacement function D are written in separable variable form, u(x, y) = X(x, y ) f ( t )
(1)
v(x, y) = Y(x, y ) f ( t )
(2)
the time independent part of the equations of motion yield 02X+l(1 02X 1 - oZY 2 Ox 2 _ - v)-fff-iy2 + ~ ( l + v) o - ~ y + f l o X = O .
02y
1 02Y ~(1 " 02X 2 ~y2 + ~ ( 1 - v)-~x2 + +v)o-b-~y+floY=O
(3) (4)
where
f12__
1 - v 2 ph + rn °92 _ Cf
E
g
If the component X(x, y) or Y(x, y) of the dynamic displacement function D(x, y), can be eliminated from the above two equations of motion (eqns (3) and (4)), one has (Z 2 -- 1) A A X-F 2Zf12o A X q - f14o X = 0
(5)
(X 2 - 1 ) A A y + 2Zf12oA y + f l ' o Y = 0
(6)
B.A. Ovunc
86
where
z
--
3-v l+v
_ _ j 4~ fl°2-1+v
,
2
By changing the independent variables x and y from the real domain to z and 2 in to the complex domain, the time independent part of the dynamic displacement function D(z, 2), can be written as,
D(z, 2),
D(z, 2) = X + i Y and the differential equation of motion in the complex domain can be written by means of the differential equations of X and Y (eqns (5) and
(6)), as, 1)
(Z 2
_
04D
02D
Oz20 22 + 2Z f12 Oz 0-----~+ fl4D = 0
(7)
The second-order differential equations (eqns (3) and (4)) in terms of the components X and Y, can be expressed in the complex domain, as
o-7
+ z
+ ~2o ~ = 0
(8)
0(09~ - - ~ + ZOD) ~ -z + f l 2 o D = 0
(9)
~-~
The dynamic displacement function form.
D(z, 2), can be assumed to be in series
STRESSES IN T E R M S OF T H E D Y N A M I C D I S P L A C E M E N T FUNCTION In order to satisfy the given stress boundary conditions, the stresses are expressed in terms of the displacement function D(z, 2), as
oo+oo)
~x-2(l-v2 ) (l+v)\0 Z ~ 2(1
(oo ~z +
v2)
E ( 1 - v ) i [ OO r x l . - 2(1 v2) L- ~ or,
+(l-v)
,,0)
In-plane vibration of plates
87
(1) in cartesian coordinates O'x + O'y -
2
(OD+OD)
z - 1 \3-7z
(11)
3-2
OD ay - ax + 2i*xy = - 2 0--7
(12)
(2) in polar coordinates, { = pe i'~ ae + ap --
2 X-1
(0~
+
0~)
a,~ - ap + 2i%o = - 2 e 2i90D 04
(13)
The differential equation of motion of an infinitestimal element can be also expressed in terms of the stresses: 02 Oz 02 (ax + ay) +/12 (ox + ay) = 0
(14)
or as a fourth-order differential equation, one has
02 //2 0 //4 OqZ20~2 (ffx"~(ry)-]~'~-~ (O'x q- O'y) q- (O'x -47 O'y) = 0
(15)
BOUNDARY CONDITIONS The b o u n d a r y conditions are to be used to determine the integration constants stemming from the integration of the differential equations. The b o u n d a r y conditions may be related to the geometry or to the stresses along the b o u n d a r y of the plates. The b o u n d a r y conditions m a y be: - - Geometrical b o u n d a r y conditions on the displacements, (D)I s = u(s) + iv(s)q,~, = dl + id2
(16)
- - Stress b o u n d a r y conditions as (3 = s + it) 1 at -- iTtsls -- )~ - 1
(
-)
OD OD -~ +~
OD -- e2i° 0~Ls- - - f i + if2
(17)
where, s and t are the independent variables along the tangential and normal directions to the boundary, dl, d2, are known displacements and f l , f2 are k n o w n stresses along the boundaries.
88
B. A. Ovunc
DETERMINATION OF THE D Y N A M I C DISPLACEMENT FUNCTION
An approximate approach to obtain the dynamic displacement function is by assuming that the dynamic displacement function can be written in series form. The first term of the series is the static displacement function and O.,.t and ¢.~t being its component, one has,
2 #D (z, 2).~ = Z(o,., (z) - z ~,,.,(z) - ~st(Z)
(18)
and the following terms of the series can be obtained by means of the differential equations (8) and (9). Thus, differential equations (8) and (9) can be written as
o (0v2 ova)
O~" ~-z + Z 02 J = fl 2°D*t 0--2 \ Of-z + Z--~-z J
= fl°Ds'
By integrating the above two equations while eliminating D2, one obtains the second term of the series as
D 2 (z, 5) -
2 +((Z
(()~ -- 1)2 _}_ ()f jr_ 1 )2 ) ~ f ¢~(z) dz
--
1) 2 --
~2 (Z + 1) 2) ~., ~b(2) -- ((Z -- 1) 4- (Z + 1)) zf (p (z) d2
-((Z - l) - (Z + 1))ff~k(z)dzdz]
(19)
where ~2 _-/~2° / ( z 2 _ 1) The same procedure is repeated for each successive term of the dynamic displacement function D(z, ~). Thus, one has
D(z, 2) = D,.,(z, 2) + D2 (z, 2) + . . . 4- O/(z, 2) + . . . The dynamic displacement function thus determined is approximate. A n o t h e r approximate approach is obtained by introducing the expression of the s u m m a t i o n of the normal stresses (eqn (11)), into the differential equation (14), yielding 02 1 + v fl:o (~,~ + ~y) = 0 OzO-~z (~x + ~.v) + ~ - .
In-plane vibration of plates
89
Again, the solution can be assumed to be in series form, the first term of the series being the static solution, the second term of the series ~b2, can be determined from the following expression,
02 ( , l+v 2 , , OzO2 ~b2(z, 2) + ~b~(z, 5) ) + ~ fl o (th,t (z) + ths, (z)) = 0
(20)
where, ()' represents first derivative of the function (). The successive terms of the series are calculated through the same process. The approximate solutions provide information for the direct solution. Direct determination of the dynamic displacement function is performed by considering the information obtained from the above approximate solution, then by assuming the dynamic displacement function as
D(z, -g) = d O 1 (g, ~) + ~D2 (z, 2) + C~D 3 (2, 2) + ~ D 4 (z, 5)
(21)
where D, (z, 2) = q~(ez) - ~2 J"th(ez) dz +
J"J"~b(az) dz dz
(~Zv)3J"J"J"th(~z) dz dz dz + . . .
D2 (z, -2) = dp"(~z) - ctz dp'(otz) + ~ D3 (z, ~) = ~b(ctz) - ~z J"~9(~tz)dz + ~
dp(o~z)- ~ [
$(~z) dz +...
J"J"~k(ctz)dz dz
(3-~z.~)3J"J" J"~(az)dzdzdz +... D 4 (z, z) = I I ~/((XZ)dz dz - ~ I J"J"O(ez) dz dz dz
+~I
I S I O(~z)dzdzdzdz - . . .
The above defined dynamic displacement function fourth-order differential equation (7), provided that
D(z,-g) satisfies the
(g2 _ 1)0~4 _ 2 Z / j 2 0~2 q_ /j4 = 0
Solving the four roots of the parameter a of the above equation, one has 0~1 = ]~/V/'X" - l, 0~2 = --]~/v/X - l, a 3 = ]~/~/X"-~- l and O~4 = --]~/~/X -3r- 1
90
B.A. Ovunc
The type of displacement function depends upon the sign of the parameter fix
-
1 - -
E
v2 p h + m
- - ~ o - - C f
g
Thus there are three types of displacement functions, for f12 > 0, f12 < 0, and/~2 = 0. The third case corresponds to a transition stage. Moreover, this dynamic displacement function obtained from the fourth-order differential eqn 7, also satisfies the second-order differential equations (8) and (9). The constants ~4, ~ , cg and ~ which appear in the displacement function (eqn (21)), are determined from the boundary conditions of the plates. The natural circular frequencies are evaluated from the condition that the magnitude of the concentrated force applied at the disturbance point should be equal to zero.
APPLICATIONS The applications are for half and infinite planes and circular plates. The boundary conditions may be simply or elastically supported, fixed, or free all along the boundary, or an initial disturbance may be applied at a point either on the boundary or within the plate. The plate may be subjected to additional mass and may vibrate within a partly or totally elastic medium. Half plane subjected to a disturbance on its boundary
Let us consider a half plane under a suddenly applied constant displacement Do within an infinitesimal time period. The static displacement function due to an initial displacement applied to point O (see Fig. 1) is given by
Fig. 1. Half plane subjected to an initial disturbance.
In-plane vibration of plates
91
2#D~, (z, 2) = i ~ [ g ~ ( z ) + z ~b'(z) + q;(z) ] where 4~(z) = ln(z)
and
~O(z)= ln(z)
Substituting the two functions, 4>(z) and ~k(z) in the dynamic displacement function (eqn (21)), one can write its components as (~lZn ~1n+2Zn+l i f ( n + 1 ) + . . . ) Ol,(z, 2) -- M1, (czl)ln ( z ) - k , ~ O(n) (n + 1)!
Dzn(Z, 2)= mz,(~z)ln(z)- k. n! O(n)
n+2~n+l
n--n
D3n(Z,-~)= mln(~l) ln(z)-
~! I
Dan (z, ~) = M2, (~z) In (z) -
dp(n) ~l
n.
z ~(n+l)+...)
( n + 1)! CX~+2Zn+l Z
n--n ~
dp(n+l)+...)
(n+l)]
q5(n)
( n + 1)!
• (n+ 1)+...)
where the newly defined n'th order, Bessel like function M i , 11 sponding to the i'th root of the parameter ~ can be written as,
min - °~inZn n!
~p(n+j)
~in+2 _n+lz 2 "~-O~i-n+42n+2-~2 (n + 1) !
~7+6 n +323 ~-...
(n + 2)!
1 1 1 n + 1 +n~--~+n---~+
n, c o r r e -
(n + 3) ! 1
... + -n+j
The integration constants d , ~ , g, ~ are evaluated from the boundary condition (eqn (13)). Half plane subjected to a disturbance within its body (see Fig. 2) If the external singular disturbance is applied at a point within the half plane, the components ~b(z) and ~b(z) of the static displacement function are ~b(z) = - ( d
+ i ~ ) I n (z) - (g + i ~ ) I n (z - 2ip) + 2ip - z - 2ip
L9(z) = (cg _ i 9 ) In (z) + ( d - i ~ ) In (z - 2ip) + 2ip - z - 2ip The static displacement function being known, the dynamic displacement function, the natural circular frequencies co, the modal shapes, and forced vibration can be obtained by following the above-described steps.
92
B. A. Ovunc
#I
x
fDo
Fig. 2. Half plane subjected to an initial disturbance applied within its body.
Free-free vibration The dynamic displacement function can be assumed as D(z, -2) = ag~ Mn (z, -2) + °Mn+2 Mn+2 (z, 2) + iCgn+ 2 Mn+2 (z, 2)
--i~n Mn (z, 2~) where Mn (z, Y) and Mn+ 2 (z, 2) are the newly defined Bessel like functions o f the order n and n + 2 and ~ n , ~ + 2 , cg,+2, ~ , are the integration constants of the functions Mn (z, ~) and Mn+2 (z, 2). The dynamic displacement function written as D(z, 2-) = ag~ M, (z, 2) + ~n+2 Mn+2 (-% z) + iCOn+2 Un+2 (_7, ~)
--i~n M. (z, 5) satisfies the boundary conditions.
Circular plates Let us consider the vibration o f a solid circular plate. The circular plate is subjected to a set of two opposite displacements applied at the intersection o f a diameter with the circular boundary (see Fig. 3).
x
Fig. 3. Circular plate subjected to diametrical initial displacement.
In-plane vibration of plates
93
The c o m p o n e n t s ~b(z) and qJ (z) of the static displacement function have been given as,
dp(z)=-A[ln(R-z)-ln(-R-z)+ R] [l n ( R - z ) - I n ( - R - z ) R 0(z)=A R-z
~ - - -R
R+z
The static displacement function being known, the dynamic displacement function, natural circular frequencies, ~, modal shapes, and the forced vibration can be obtained by following the above-described steps. Approximate solutions for the vibration can be assumed by neglecting the c o m p o n e n t ~ (z, 2) in the displacement functions. The dynamic displacement function is determined from the differential equation of normal stresses (eqn (11)),
02 ( , , l+v 2 Oz02 4'2(z'2)+ck2(z'2))+--U-#°(4"s'(Z)+
4¢s,
(z))=o
Its integral can be obtained t h r o u g h BesseMike functions. The dynamic displacement function can be written as D (z, 2) = {~ (z, 2)} T {C}
(22)
where {e#(z, 2)} can be called a vector of the shape functions, whose nature depend u p o n the signs of rio, and {C} is the vector of integration constants, d , 9L g , @. F o r plates with singularity points, the shape function can be selected as {~ (z - a, 2 - ~}T = (j//~ (z -- a, 2 -- b) ~i~ln (Z - -
where j l i ((z - ~, z - b) and
a, 2 - b)
pi ((z -
jg2 (z - a, 2 - b) ~ ] (z - a, 2 - b))
a, ~ - b) (i = 1 or 2) are
d'[ j ((Z -- a, z -- b) = mln (ill (z - a), fll (2 - b)) - ( - 1) j M2 (f12 (z - a), f12 (2 - b))
~
((z - a, 2 - b) = p1 (fl, (z - a), fl, (2 - b)) _ ( _ 1)i p2 (f12 (z - a), f12 (2 - b))
For fl~=fl2o/(Z+l)>0
and
f l ~ = f l 2o / ( Z - 1 ) > 0
B. A. Ovunc
94
the functions M~(fli(z g a), fii (z - b)) and P n' ( fli (Z - a ) , fii (2- - - b)) a r e similar to the first- and second kind of Bessel function. Their expressions are as follows: -
M ' (=, 2-) : f17 (z - a) n Z
( - 1)k/~k (z - a) k (z_-#-L-~)k k ! ( n + k + 1)] PI, (z, 2-) = In (z - a) M pli (7 \ ~ , 2-) k=0
- f i t (z - a)" Z
( - 1)k//~k (z - a) k (z---Z-b) k ~p(n + k) 1)!
where a and b show the location of a disturbance point within or along the boundary of the plates. Similar to the Bessel functions, M~ (z, 2-) and P i (z, 2-) are related to their derivatives. 1° The vector of the integration constant { C } , which appears in the dynamic displacement function (eqn (2)), is determined by writing the displacement along the boundary, by solving for {C}, then substituting {C} in the expression for displacement. For the vibration of circular plates without any singularity point within or along the boundary, the shape function in the expression of the displacement can be written from eqn (22) (a = b = 0), as
=
(z, 2-)
(z, 2-)
2-)
2 (z, 2-))
In nondimensional polar coordinates, z = R p e i~, r = Rp, the recently defined Bessel functions reduce to ~.#,,J (p, 7~) = e in'~ (J~ (2~,, p) - ( - 1 ) / 4 (7,,, P)) :~, (p, O) = 2ei,,0 (In {e ~ i0~/,,i, P) Jn (2,i, 1O) q- ~.n (l~ni, P) - ( - 1) ~ [In (e i~ 2.~, p) J. (2.i, p) + ,L, (':%~, P)])
where, J~ is the Bessel function of order n and ~,i = Rfilni, 7,i = Rfl2,,i. The parameters n and i are related to the nodal diameters and nodal circles, respectively. CONCLUSIONS The differential equation of motion is derived for the in-plane vibrations of plates, in terms of the dynamic displacement function. The differential equations are expressed in the complex domain, so that they can be integrated directly. The dynamic displacement function is determined uniquely, in series form, through the integration of the differential equation of motion, provided that, for a given system, the static displacement function under the same b o u n d a r y conditions is known.
In-plane vibration of plates
95
REFERENCES 1. Bassily, S. F. & Dickinson, S. M., Vibration of plates subjected to arbitrary in-plane loads. J. Appl. Mech., 40 (1973) 1023-8. 2. Katsikadelis, J. T., A boundary element solution to the vibration problem of plates. J. Sound & Vib., 141(2) (1990) 313-22. 3. Manolis, G. & Beskos, D. E., Boundary Element Methods in Elasto-D)namics. C. Allen and Unwin Publ., London, UK, 1988. 4. Costa, J. A. & Brebbia, C. A., Bending of plates on elastic foundations using the boundary element method. In Proc., 2nd Int. Conf. on Variational Methods in Engineering, ed. C. A. Brebbia. Springer Verlag, London, 1985, pp. 5-23. 5. Ayoub, E. F. & Leissa, A. W., Free vibration and tension buckling of circular plates with diametral point forces. Joint Applied Mechanics/Bioengineering Conf., Ohio State University, June 16-19, 1991, Paper No. 91-APM-24. 6. Providakis, C. P. & Beskos, D. E., Free vibrations of shallow shells by boundary and internal elements. Comp. Meth. Appl. Mech. Eng., 92(1) 55-74. 7. Katsikadelis, J. T. & Sapountzakis, E. J., A BEM solution to dynamic analysis of plates subjected to in plane forces. A S M E Engineering Systems Design and Analysis, PD, 47(5) (1992) 41-8. 8. Wang, X. X., Qian, J. & Huang, M. G., A new algorithm and application to free vibration analysis using boundary/interior elements approach. J. Compt. & Struct., 32(1) (1989) 55, 61. 9. Ovunc, B. A., Semi infinite plate subjected to concentrated loads acting in its internal point, Vol. 28. IABSE, Zurich, Switzerland, 1968, pp. 121-5. 10. Ovunc, B. A., In-plane vibration of plates by continuous mass matrix method. J. Compt. & Struct., 21(4) (1985) 887-91. 11. Ovunc, B. A., Nonlinear free vibration of plates. J. Compt. & Struct., 21(4) (1985) 887-91.