Oscillatory damped distributed parameter systems

MECHANICS RESEARCH COMMUNICATIONS Vol.9(2),I01-i07,1982. Printed in the USA 0093-6413/82/020101-07503.00/0 C o p y r i g h t(c) 1982 Pergamon Press Ltd

OSCILLATORY DAMPED DISTRIBUTED PARAMETER SYSTEMS

D. J. Inman Department of Mechanical and Aerospace Engineering State University of New York at Buffalo Buffalo, New York 14260 U.S.A. (Received 17 August 1981; accepted for print 19 February 1982)

R e c e n t l y N i c h o l s o n [1], M u l l e r [2] and I n m a n and A n d r y [3] pres e n t e d s o m e r e s u l t s on the free v i b r a t i o n s of v i s c o u s l y d a m p e d lumped parameter systems. Their work presented sufficient conditions for each of the m o d e s to oscillate. The result p r e s e n t e d here e x t e n d s this c o n c e p t to d i s t r i b u t e d p a r a m e t e r s y s t e m s and p r e s e n t s a s u f f i c i e n t c o n d i t i o n for the t i m e r e s p o n s e to be oscillatory. This work does not depend on finite approximations to the describing partial differential equation. The result is applicable to a limited class of vibrating systems (real, selfadjoint positive definite linear systems). Examples are included.

P r o b l e m ~

a

n

d

~

The class of vibration problems discussed here are those that can be represented

by a system

of linear partial differential

equa-

tions of the f o r m

utt(x,t)

+ Ll[Ut(x,t)]

+ L2[u(x,t)]

= 0 in n (i)

B[u(x,t)]

= 0

on

a)

indicates

~n

where (')t

partial

differentiation

of

(.) w i t h

respect to the time t;

b)

is a bounded, boundary

open r e g i o n

~; i01

in R n, n=l,2

or 3 w i t h

102

D.J. INMAN c)

L 1 and L 2 are real l i n e a r tors

of

order

adjoint fined

the

satisfying

tives

B is a real

definite

set

of

up to the

linear

~

and

in

L2(~)

opera-

are

on the d o m a i n

functions

order

self-

D(L) with

and having partial

spatial

up to m a x ( n l , n 2 ) - i

differential

respectively,

Bu = 0 on

in L2(~)

order

n2

positive

to be

norm

d)

and

n I and

spatial

deunit

deriva-

max(2nl,n2);

differential

and r e f l e c t s

operator

boundary

of

condi-

tions. Here,

L2(~)

square inner

denotes

integrable product

]lull

the

space

of all

in the L e b e s q u e

defined

real

sense

functions

over

~

which

are

norm

and

, with

by

= ( f u 2 (x)dx) I/2

%

: f u(x)v(x)dx

F

respectively.

n Note

than the solution

that

of

the f u n c t i o n s

(i) demands.

The

in D(L)

solution

m a y be s m o o t h e r of

(i) is assumed

to be of the f o r m

u(x,t) where

the

=

[ an(t)@n(X) n=l

set

in L2(~).

{@n(X)~= 1

Each

of

the

is

Physical

include

vibration

branes

free

in various

a

real

functions

differentiable. the

(2)

systems

orthonormal a n(t)

set

is a s s u m e d

adequately

of s t r i n g s ,

of

functions

to be

described

beams,

plates

twice by

(i)

and m e m -

configurations.

Result

Following lumped

mass

the

result

systems

of M ~ l l e r the

[2] and I n m a n

following

theorem

and A n d r y

holds

for

[3] for

distributed

OSCILLATORY DAMPED DISTRIBUTED PARAMETER SYSTEMS mass

systems

operator

(i) under

the

restrictions

(4L2-LI 2) is positive

functions

definite

an(t ) in (2) will be decaying

listed

on D(L)

103

above:

If the

then each

of the

oscillations

of the form

a n (t ) = Ane-~nt cos(~nt + C n) where

A n and C n are constants

and I n and ~n are determined

(3)

determined

from

initial conditions

by L 1 and L 2.

To see that this is true consider

solutions

to (i) of the form

u n (x,t) =a n (t)@ n (x) where

each

functions

(4)

an(t ) has at least {~n(X)}n=l

are

will

then c o n v e r g e

into

(i), pre-multiplying

the following ~n(t) where

with

+ <~n,LI~n>

for

complete

each

<~n,LI@n>

of

determined

and

The

series

in

Substitution

integrating

an(t ) + <~n,L2~n>

an(t)

differentiation order

coefficients

(2)

of (4)

over ~ yields

an(t ).

(5)

respect

to time.

differential

equation

an initial

Since

are real numbers

= 0

with

ordinary

representing

functions

and <~n,L2@n>

of (i).

and the set of

for each an(t):

is a second

the

in D(L).

by @n(X)

the dot i n d i c a t e s

constant

derivatives

to the s o l u t i o n

expression

This expression

two

L 1 and

value problem L2

are

real,

and the form of an(t) is

by the sign of the discriminant

<~n,Ll~n> 2 - 4<~n,L2~n>

(6)

for each n. The proof now follows directly from the Cauchy inequality to the f u n c t i o n s

@n(X) and Ll~n(X),

i.e.

<~n,LI~n> 2 < <~n,LI2~n> where the self-adjointness side. that

By h y p o t h e s i s

applied

(7) of L 1 has been used on the right hand

(4L2-LI 2) is p o s i t i v e

definite

on D(L) so

104

D.J. INMAN

<@n,Ll2@n>

< 4<@n,L2@n>

(8)

Combining these two inequalities <~n,LI~n> 2 - 4<~n,L2~n> for all 0n(X) inant

in D(L).

(6), h e n c e

imaginary

part

(8)) yields

< 0

(9)

The left side of (9) is just the d i s c i m -

each

of

((7) and

of the an(t)

its

associated

of

(5) w i l l

have a n o n - z e r o

characteristic

root.

The

real

part of the root is given by -<~n,bl~n> 2 s i n c e L 1 is a s s u m e d

to be p o s i t i v e

d e c a y and each an(t) has the f o r m operator

4L2-LI 2

functions

an(t)

Two

examples

is

positive

given

closed form solution

definite

a known

For the f i r s t e x a m p l e

exact solution. solution

consider

in a s u r r o u n d i n g

that is proportional

constant

7> 0.

D(L)

The first example

Hence, each

if the of

the

is simple

with a

of the r e s u l t

can be

The second example

and i l l u s t r a t e s

result when one cannot easily compute

motion

(3).

on

so that the v a l i d i t y

not h a v e a c l o s e d f o r m

membrane

in

all the r o o t s w i l l

is underdamped.

are presented.

checked against

definite

the use of the

the exact solution.

the t r a n s v e r s e

medium

does

furnishing

vibrations resistance

of a

to the

to the velocity with proportionality

The e q u a t i o n of m o t i o n

is

utt + 2yu t - V2u = 0 on

(i0) u(x,t) where and

= 0 on ~

V 2 is the two dimensional

u = U(Xl,X2,t)

direction

is

perpendicular

the

biharmonic

deflection

of

operator, the

x = (Xl,X2)

membrane

to the xl-x 2 plane, ~ .

in

the

The b o u n d a r y

~

OSCILLATORY DAMPED DISTRIBUTED PARAMETER SYSTEMS represents

one or m o r e c u r v e s

eigenvalue

of -v 2.

variables

in the plane.

The s o l u t i o n

(see r e f e r e n c e

Let

105

An 2 be the n th

can be f o u n d by s e p a r a t i o n

[4] for i n s t a n c e )

and y i e l d s

of

an(t)

of

the form a n(t)

= e-Tt(AnCOS(~nt

+ Cn )

(ii)

as long as 7 < In is satisfied. In t e r m s of the t h e o r y p r e s e n t e d

h e r e the o p e r a t o r s

are

domain

L 1 = 27

functions

on

L 2 = - V 2.

u(x)

derivatives shows

and

such

that

The u(x)

up to the second

D(L)

= 0 on ~

order

is

and

in L 2(~).

simple

so

that

structure

its d e f i n i t e n e s s The eigenvalues

the

result

can

be

of the o p e r a t o r is d e t e r m i n e d

the

u(x)

set has

Simple

that L 1 and L 2 are both positive definite

D(L),

of i n t e r e s t of

all

partial

integration

and self-adjoint

applied.

Because

4 L 2 - L I 2 on D(L)

of

the

in t h i s case,

by the s i g n of its e i g e n v a l u e s .

are c o m p u t e d to be

4 (In2-~ ) where

In 2 are

In > 7

again

of -V 2 on D(L).

for all n, 4 L 2 - L I 2 is p o s i t i v e

presented

definite

h e r e s t a t e s that the f u n c t i o n s

f o r m g i v e n in (3). solution

the e i g e n v a l u e s

given

by

The last example

inertia,

an(t ) s h o u l d be of the

T h i s is in p e r f e c t a g r e e m e n t

w i t h the e x a c t

(ii). illustrates

vibrations

of a b e a m

denoted I(x),

the

use of the theorem

with

non-uniform

modulus.

a positive

Consider

area

moment

the of

The equation of motion is

utt(x,t ) - 2CUtxx(X,t ) + [EI(x)Uxx(X,t)]xx c is

stated here

and with a velocity dependent force resis-

ting the bending moment.

where

if

and the r e s u l t

w h e n the e x a c t s o l u t i o n is not e a s i l y c a l c u l a t e d . bending

Thus,

damping

A hinged-hinged

constant

configuration

and

= 0 on E is

yields

(0,i)

the

the

elastic

following

106

D.J. INMAN

boundary

conditions

u(o,t)

= u(l,t)

I(O)Uxx(O,t)

= 0

= I (1) Uxx (l, t) = 0.

Let the area m o m e n t 0 < m < for

all

x

be bounded,

i.e.

I(x) ! M < -

in

particular

of inertia

[0,1],

where

then the last

Uxx(0,t) The o p e r a t o r s

= Uxx(l,t)

m

and

M

are

set of b o u n d a r y

known

constants.

conditions

In

becomes

= 0.

L 1 and L 2 are

~2 L 1 = -2c ~

(')

~2 ~2 (" L 2 = E ~x-~/[I(x) ~x-~/ )]. The

domain

D(L)

the b o u n d a r y u(o)

Simple and To

u(1)

=

u"(o)

=

derivatives

integration

positive consider

in D(L)

set of all

functions

in L 2 [0,i]

and

by parts

definite the

u"(1)

=

in L 2 [0,I] shows

0

up t h r o u g h that

the

definiteness

of

an a r b i t r a r y

1

(v")2dx - c 2 f (v")2dx}.

0 0 > m this b e c o m e s 1 < v , ( 4 L 2 - L l 2 ) V > > 4(Em-c 2) f (v")2dx. 0 Thus, if E m > c 2, the o p e r a t o r 4 L 2 - L I 2 is p o s i t i v e

Much

of

the

the

v(x)

consider

(v") 2 > 0 and

of

order.

L 1 and L 2 are s e l f - a d j o i n t

4L2-LI 2 choose

1

each

fourth

on D(L).

= 4{El Since

satisfying

conditions

=

and having

is the

I(x)

functions

work

on

an(t ) should

be d e c a y i n g

distributed

parameter

definite

and

oscillations.

systems

involves

OSCILLATORY DAMPED DISTRIBUTED PARAMETER SYSTEMS approximate

solutions

result

does

here

not d e p e n d

useful in determining finite dimensional

lo

2. 3.

in the f o r m on any

of t r u n c a t i o n s such

107 of

(2).

approximation

The

but is

the proper form of an(t) to be used in any

approximation of the solution u(x,t).

D.W. Nicholson, Mech. Res. Comm., 5, 147 (1978) P.C. Muller, Mech. Res. Comm., 6, 81 (1979) D.J. I n m a n a n d A.N. A n d r y , Jr., J. Appl. Mech.

47,

927

(1980) .

I. Stakgold, B o u n d a r y V a l u e P r o b l e m s of M a t h e m a t i c a l Physics, Vol. 2, p. 281, MacMillian Co. (1967)