dynamic vibrations and stresses in composite elastic plates

Received 30 December 1968 12.7

Dynamic Vibrations and Stresses in Composite Elastic Plates

M. H. COBBLE

Department of Mechanical Engineering, New Mexico State University, Las Cruces, New Mexico 88001

The equation for longitudinal displacement in each of k sections of a composite elastic solid is solved. The elastic composite consists of k discrete plates (separately isotropic), each of different material, joined solidly at their k--1 interfaces. The composite plates are initially unstressed, and are acted upon, at their two mutual external boundaries, by separate arbitrary time-dependent stresses. The internal boundary conditions maintain across each interface the equality of displacement and stress. The solution is obtained using a new orthogonality relationship. The solution for the displacement is given for arbitrary k.

LIST OF SYMBOLS

I a[ determinant ai compression wave velocity, feet/second A o, Bgj, Co constants Vn, ZUn •4 in, Bin constants Xin a•r elements of a determinant x lb[ determinant ½nj constant z Cnj* derived constant fig D determinant V2 Ei Young's modulus, pounds/foot s Fj(t) function of time i, j, k integers /•n Lgj(x) function of x nj constant m integer Nn sum of integrals n integer p integer t time, seconds Ui derived displacement in the ith section

Un(t) function of time Vg(x), Wg(x) functions of x

constants

eigenfunction, dimensionless distance, feet displacement in the ith section, feet term in matrix equation ratio of properties, dimensionless Laplacean operator integers eigenvalue, 1/sec density in the ith section, pounds

second2/foot 4 component of the normal stress in the

x, y and z directions, pounds/foot s derived eigenfunction terms, dimension-

less

Lam• constants in the ith section, pounds/foot 2

INTRODUCTION

Longitudinal vibration in an elastic (isotropic) solid of a single material is discussed in the book, Dynamics of Vibration by Volterra and Zachmanoglou, 1 and they give a small bibliography on the subject. Churchill?

1 E. Volterra and E. C. Zachmanoglou, Dynamics of Vibration (Charles E. Merrill Publishing Company, Columbus, Ohio, 1965), pp. 293-303, 424.

• R. V. Churchill, Operational Mathematics (McGraw-Hill Book Co., New York, 1958), 2nd ed., pp. 199-257.

in his book, Operational Mathematics, treats the problem of longitudinal vibration of a solid (isotropic) of one material, and he also solves two component problems in conduction heat transfer, using complex variable methods and residue theory as well as Laplace trans- forms. Cinelli, 3 in a recent paper, treated the dynamic vibration and stress problem in cylinders and spheres

a G. Cinelli, "Dynamic Vibrations and Stresses in Elastic Cylinders and Spheres," J. Appl. Mech. 32, Ser. E, No. 4, 825-830 (1966).

The Journal of the Acoustical Society of America 1175

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07:23:48

O'lx(X , ,t) : F•(t) bY• (X• ,t) = F'i -•.-•

X I X 2 X 3 X 4 X•_ I X•

M. H. COBBLE

O-•x(X•+ I , t) = F2(t)

• J E•+I E k

Xi+ I Xi+2 X k

= Ek •"•k (Xk,I, t)

Xk+l

FIG. 1. Elastic composite of k sections with different arbitrary time-dependent stresses on the external boundaries.

(one isotropic component), in which the boundary stresses were two different arbitrary functions of time, when initially the cylinders and spheres were unstressed. The solutions are based on a new type of finite Hankel transform. Composite vibration problems in any coordinate system do not appear to have been treated very extensively.

In this paper, a method is developed to find the displacement in each of k elastic (separately isotropic) plates joined solidly at their k--1 interfaces and subject to different arbitrary stresses on the two external surfaces. The solution is ultimately developed by means of a new orthogonality relationship and is based on the work of Vodicka, 4.s who first developed the method to handle heat-conduction problems. Tittle 6 has pointed out that this method could be used to solve problems based on the wave equation and Schr6dinger's equation. The method reduces to a standard Sturm-Liouville procedure when k--1, pro- viding classical solutions for these problems.

I. PROBLEM

The equation for the displacement in the ith section of k solidly joined plates (separately isotropic) is

½,t)= (o,/ot9 Xi•X•Xi+I, i = 1, 2, 3, ..., k, t>_O, (1)

where ai is the compression-wave velocity in the ith section in feet/second and ¾• is the lon,gitudinal displacement in the ith section in feet. The plates are subject to the initial conditions

i=1, 2, 3, ..., k. (2)

The plates are subject to the external boundary conditions (see Fig. 1)

Ex(OY•/Ox)(x•,t)=Fz(t)=o'•(xx,t) (3)

•V. Vodicka, "Warmeleitung in geschichteten Kugel und ZylinderkSrpern," Schweiz. Arch. 10, 297-304 (1950).

• V. Vodicka, "Eindimensionale Warmeleitung in geschichteten K6rpern," Math. Nachr. 14, 47-55 (1955).

• C. W. Tittle, "Boundary Value Problems in Composite Media," J. Appl. Phys. 36, No. 4, 1486-1488 (1965).

1176 Volume 46 Number 5 (Part 2) 1969

and

œk(OYk/Ox)(xk+•,t)=F,.(t)=•rkx(Xk+•,t), (4)

where Ei is Young's modulus in the ith section in pounds/foot 2 and •ix (x,t) is the component of the normal stress in the ith section in pounds/foot 2. The plates are subject to the internal boundary conditions

Y•(x•+x,t)= Yg+•(xg+•,t), i= 1, 2, 3,..., k (5)

and

(o ½,+ = (o r,+ ½,+ i=1, 2, 3, ..., k. (6)

To obtain homogeneous external boundary conditions, following the Mindlin-Goodman 7 procedure, let

j----1

x•_<x_<x•+•, i=1,2,3,...,k, t_>0, (7)

and further, let

Lo(x)=Cox2/2+A ijx+Bgj, xi_<x_<xi+•, i=l, 2,3,...,k,j=l, 2. (8)

Substitution of Eqs. 7 and 8 into Eq. ! gives

./=1

0 2 Ui 2

OF •=•

xi_<x_<xi+•, i=l, 2, 3, ..., k,

Initial conditions Ui(x,t) are represented as 2

c,½,o) = - E = j=l

x•_<x<x•+•, i=l, 2, 3, ..., k

t>_0. (9)

* R. D. Mindlin and L. E. Goodman, "Beam Vibrations with Time-Dependent Boundary Conditions," J. Appl. Mech. 16, 377-380 (1950).

(lO)


07:23:48

VIBRATIONS AND• STRESSES IN COMPOSITE PLATES

and

OU½ , ½,o) = -22 = Ot

xi_<x_<x•+•, i= 1, 2, 3, ß ß., k;

external boundary conditions, Ui(x,t), as

= 0= internal boundary conditions, Ug(x,t), as

U•(x•+x,t)= Ug +• (x•+•,t), i= l, 2, 3, ..., k--1

and

E,( O U•/ Ox) (x,+ x,t) = E,+ x (0 V,+ •/ Ox) (x,+ x,t) , i=l, 2, 3, ..., k-l;

external boundary conditions, Li•(x), as

= ,

(11)

(12)

(13)

(14)

(15)

external boundary conditions, L•2(x), as

Zx:'(x•) = 0, Z•:'(x•+•) =

and internal boundary conditions, Lq(x), as

Lq(xi+ •) = L<+ • (x•+ x) , i=1, 2, 3, "',k--l,j=l, 2

and

EiLi/ (xi+ •) = El+ •Li+ •,/ (xi+ •) , i=1, 2, 3, ..., k-l, j=l, 2.

(16)

(17)

(18)

II. SOLUTION, Lil(X)

When we use Eq. 8 in the boundary conditions (external and internal), we get a system of 2k linear nonhomogeneous equations for determining the 2k constants A•x and Cid. Using the matrix form, the 2k simultaneous equations can be written as

x• 1 0 0 0 0 ...

x2•/2 x• -x•/2 -x•. 0 0 ... fi•x• fi• --x• -1 0 0 ...

0 0 x•/2 x• -x•/2 -x• ... 0 0 fi2x• fi2 -x• -1 ...

0 0 0 0 0 0 ... 0 0 0 0 0 0 ... 0 0 0 0 0 0 ...

o o o o o o o o

o o o o

o o o o o o o o

x•'/2 x• -x•V2 -x• fik-•Xk fik_• -x• -1

0 0 x•+• 1

C 11

11

21

21

ß

-1 1

kl

1/E• B2•--B•

0

B•--B2• 0

oo•

B•i--B•_•,• o

o

(19)

Using matrix notation, these equations may be written as

[a]{z}={p}, (20)

where the elements a•r, •, •'= 1, 2, 3, ..., 2k are evident from Eq. 19, and where

and

z2r-•=Cr•, z2r=Ar•, r=l, 2, 3, ..., k (21)

p•=l/E•, P2•=O, p2r_• =0, •=2,3,4, ...,k-1 p2r=Br+•.•--Br•, •=1, 2, 3, ..., k--1. (22)

Since Bg• represents an arbitrary constant in the expression for L•(x), we may set all the Bi•=0, i- 1, 2, 3, ..., k, with no loss in generality. Further, in Eq. 20, if the determinant of the system [a I •0, then the system of equations has a unique solution given by

D• D•. Dr z•= , z2=--,'", zr=--,'", z•.•= , (23)

[a] ]a[ [a[

where Dr is the determinant formed by replacing the elements (a•,.,a•.,.,aor,.. ',a•r) of the rth column, by P•,P•,Pa," ',P•, respectively. In this way, all the C• and A u are found.

III. SOLUTION, Liu(x)

Development here is the same as shown above, with the second subscript 2 replacing 1. The argument is the same except that

p•=O, p:•= 1/E•. (24)

The result is that all the A g: and Ci2 are foundß

IV. SOLUTION, U•(x,t)

The solution to Eq. 9 is found by assuming the series

c,½,t) = Z (t)x. ½), n=O

XiSX.__•Xi+I, i= 1, 2, 3, "', k, t>_ O, (25)

where u• (t) is a function to be determined by using the initial conditions, and the functions Xg•(x), x•_<x _<x½+l, i= 1, 2, 3, ..., k are the eigenfunctions of the following eigenvalue problem'

d2Xi•(x)

dx •

i=1, 2, 3, ..-, k, (26)



07:23:48

M. H. COBBLE

with the external and internal boundary conditions

X,,' (xx)= 0= X•,' (x•+x), (27)

Xg•(x,+•) =X,+•.,•(x,+•), i= 1, 2, 3, ..-, k--l, (28)

and

EiX in' (xi+ 1) = F4+ iX i+ 1, n' (xi+ 1), i=1,2,3, ..-,k--l, (29)

where t• is the eigenvalue of the problem. The solution to Eq. 26 is

X,o(x)=A,o+B,ox, n=0 (30) and

X,•(x) = A ,,• cos(3t,•x/ai)-f-B,,• sin (t•x/a,), n=l, 2, 3, .-., (31)

but for simplicity in further developments, we may write

Xg,•(x)=Ai•,i,•(x)+Bi•k•(x), n=0, 1, 2,..-. (32)

If we substitute Eq. 32 in the boundary conditions shown by Eqs. 27-29, we get a system of 2k linear homogeneous equations for determining the 2k inte- gration constants A• and B•.

In the theory of linear homogeneous equations, this system of equations has a nontrivial solution when its determinant is equal to zero. Setting the determinant equal to zero gives the transcendental equation

...... D= . ....... 0 (33)

ß -.-

for determining the eigenvalues t•. Equation 33 has an infinite number of roots

t•= 0 t• t•. ( ..... (34)

and, for each of these roots, there are corresponding values of A i• and B• in the eigenfunctions Xi,•(x) that can be determined within a multiple of an arbitrary constant; because of the form of Eq. 25 and a following normalization process, the solution is unaffected by the value of this constant, which, with complete generality, can be equated to unity so that Xi,•(x) is then completely specified.

The functions Xim(X) and X•(x), -1, 2, 3, .-., k, m, n, integer, are orthogonal in the domain Xl, xk+•. The orthogonality condition of these

functions is

/Xi+l { const, k PiJx i XimXindx: i=1 0, (35)

m•n,

where pi is the density in the ith section in pounds second•/foot 4. This identity, Eq. 35, can be shown by first integrating by parts and then applying internal and external boundary conditions.

Since Lgi(x) satisfies Dirichlet's conditions, we may expand the following in an infinite series of the eigenfunctions

Wi(x)-- Z wnXin(x),

• •0

Lij(•)-- Z lnjXin(•),

(36)

ai2Cii--Cij * = Z 6•j*Xi• (x). n--•O

Multiplying the left and right sides of Eqs. 36 by ogXg•(x), integrating with respect to x from xg to xg+•, summing each of the identities for all the values of i, and using the orthogonality relation of Eq. 35, we obtain

Wn---- Pi Wi(x)Xin(x)dx, N• i=• Jxi

n=0, 1, 2, ..., (37)

with corresponding expressions for Vn and l•. Addi- tionally, we find that

1 k

605*------ Y] piai2Cij(Xi+l--Xi), n=0, (38) N0 i=1

c•*=0, n=l, 2, 3, ..-, (39)

fXi+l Nn:•.iJxi W,n2(X)dx. (40) i=1

Using Eqs. 25 and 36 in Eq. 9 and applying the initial conditions Eqs. 10 and 11 gives the solution of Ug(x,t), as

+ • w• cosiest+--sint•t---- n=l •n

1 Z l• f F/'(t--r) sin3t,•rdr X,,•(x) /.tn 5=1 Jo

x•_<x_<x•+•, i=l, 2, 3, ..., k, t>O, (41)

1178 Volume 46 Number 5 (Part 2) 1969


07:23:48

where

VIBRATIONS AND STRESSES IN COMPOSITE PLATES

w• = - • l•F•(O), v• = -- • l•F/ (0), n =0, 1, 2, ß ... (42)

In Eq. 41, the following holds'

X,0(x) = 1, n=0, (43)

and, from Eq. 32, for n= 1, 2, 3, ...

At,,=l, Ai,,= [b=i_a[/[b[, i=2, 3, 4, ..., k, (44)

and

B•- [b•_•l/Ibl, i- •, 2, 3, ..., k. (45)

We obtain lb[ and [ac], r- •, 2, 3, ..., 2&- •, from the elements in the determinant D, the transcendental Eq. 33. If we designate b•r, •, •= 1, 2, 3, ..., 2k as an element of D, then [b[ is given by

lb[ = bla ß ß ß bl,ak

ß

ß o •0 (46)

and is of order 2k-- 1. The determinant [br[, •= 1, 2, 3, ..., 2k--1 is formed by replacing the •th column in [hi, by the column (--bn, --bat, --bat, "', formed from the corresponding elements in D.

V. SOLUTION, Y•(x,t)

All the components of Ug(x,t) and Lgs(x) are now known, and so displacement in each section as given by Eq. 7 is now completely specified. The solution for Yg(x,t) gives the longitudinal displacement in any of the k elastic plates (separately isotropic) that are solidly joined at their k--1 interfaces. The plates have no initial displacement (initially unstressed), and no initial velocity, but are subject to different arbitrary time-dependent stresses on either external boundary of the composite.

vI. STRESS

The stress in the ith section is given by

O-,x ½,t)= •:,(a •,/a:•) ½,t)= (x,+ 2•,) (a •,/a:•) (•,t) xg_<x<_x•+•, i=l, 2, 3, ..., k, t>_O, (47)

and by

•,.• (•,t) = •,z (•,t)= x,(a •,/a•) ½,t) OCi•__X__•OCi+I, i = 1, 2, 3, ..., k, t>_O, (48)

where Xi, ui are Lam• constants in the ith section in pounds/foot s, and Yi(x,t) is given by Eq. 7.



07:23:48

dynamic vibrations and stresses in composite elastic plates

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