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Technical

Memorandum

109097

Free Vibration of Thermally Loaded Panels Including Initial Imperfections and Post-Buckling Effects

K. D. Murphy Duke

University,

and

L. N. Virgin

Durham,

North

Carolina

S. A. Rizzi NASA

Langley

Research

Center,

Hampton,

Virginia

(NASA-TM-109097) FREE VIBRATION THERMALLY LOADED PANELS INCLUDING INITIAL IMPERFECTIONS AND POST-BUCKLING EFFECTS (NASA. Langley Research Center) 13 p

OF

N94-29461

Unclas

G3/71 April

1994

National Aeronautics and Space Administration Langley Research Center Hampton, Virginia 23681-0001

0003805

"I,

°

_

FREE

VIBRATION

OF THERMALLY

IMPERFECTIONS

Kevin

LOADED

AND

D. Murphy

Department Duke

and

Lawrence

of Mechanical

University,

NASA

INITIAL

N. Virgin

NC 27708-0300

A. Rizzi

Langley

Acoustics

INCLUDING EFFECTS

Engineering

Durham,

Stephen Structural

PANELS

POST-BUCKLING

Research

Branch,

Center

Hampton,

VA 23681-0001

ABSTRACT A combined

theoretical

amplitude

free

uniform

heating.

ary elasticity,

reveal

experimental

approach

characteristics

Included initial

experiment

and

vibration

in this

study

imperfections

excellent

of fully

and

is developed

clamped

are the effects

to consider

panels

under

of higher

post-buckling.

the

modes,

Comparisons

the

small

influence

in-plane

between

of

bound-

theory

and

and

shell

agreement.

INTRODUCTION In high

performance

structures

subject

and

loading,

while

the

ordering

insight [1], i.e.

the

understanding Early on simply

state

stability

eigenvalues frequencies

the

temperatures

frequencies

steady

relative

negative

applications,

to elevated natural

the

to the

computing

aerospace

dynamic

is a primary

determine

the

response

in terms

of these

oscillations

imply

an unstable

of these

structures

buckled

beams

Along

of the

high

by

ratio

damping response

Clearly,

and

a realistic

under

give

Walker

model

is a first

structures

panels

they

Croll

temperatures

of continuous

and high aspect

with

transient

In addition,

as described

response. at

of plate

concern.

character of resonances.

the entire response of the system. studies in the vibration characteristics

supported

behavior

for

step

to

concentrated

uniaxial

sion [2], [3], [4] and [5]. More recently, Gray and Mei [6] used a finite element consider this problem for flat plates with either simply supported or clamped

compresanalysis to boundary

conditions. The

current

paper

uses Galerkin's

method

to develop

an analytic

the dynamics on the other

of fully clamped (out of plane) panels. The in-plane hand, may be set to allow in-plane displacements

accomplished

by restricting

in-plane

displacements.

initial function

edge geometric

are

presented

edge

stresses

Also incorporated

into

imperfections.

of temperature,

experimental

the in-plane

study and

Finally,

spanning

the

was also conducted compared

to the

the pre

behavior

and

of the

the natural

findings.

describing

boundary conditions, at the edges. This is

to be proportional this model

post-buckled

to measure

theoretical

model

are

to the

the

natural regime,

effects

average of small

frequencies

as a

is considered.

An

frequencies.

These

results

THEORETICAL Equation The

of Motion

von Karman

used

in this

and

nonlinear

study.

Solution

partial

It may

APPROACH Technique

differential

be written

equation

0_07 \0_07 + 0_07/j + 7 where the

w is the

Airy

addition,

lateral

displacement

of the

stress

function,

C is the

_ and

r/ are the

nondimensional

governing

in nondimensional

system

form

the panel's

behavior

is

[7]

+ cb-v_+ _P = 0

panel,

w0 is the

damping

and

AP

initial is the

(1)

imperfection, external

F

is

loading.

In

coordinates x

y

a_ and a_ and b_ are the length and width of the panel, respectively. The associated compatibility equation relating the in-plane stress resultants to the lateral displacements takes the form

(2) where

v is Poisson's

ratio.

In an attempt procedure For this

is taken

to obtain which

procedure,

displacement

For a complete a solution

expands

a one

mode

field are assumed

on the initial

to take w0ff,7)

discussion

equations

see Dowell

for the

displacement

field,

method

developed

by Ventres

imperfection the

of these

and

a two step

a nine-mode

and

solution

Dowell

expansion

[7]. [8].

for the

form

= a_(_)_(,) 3

3

w(_,,, ,) = _ _ a,j(_)¢,(_)%(,)

(3)

i----1 j----I

where _(_) and q)j(7) are spatial beam mode shapes which satisfy zero slope boundary conditions for a fully clamped panel. Throughout study

the

modes

were assumed

the zero deflection, the course of this

to be

@i({) = cos([/-

117r_) - cos([/+

117r{)

%(r/) = cos([j - 1]Trr/)- cos([j + 117r7) The solution

first step

is found

by

is to obtain substituting

a solution the

to the compatibility

assumed

displacement

(4)

equation. field

The

(Equation

particular (3))

into

Equation (2) and writing

Fp as an expansion

in terms

of the

spatial

modes,

q)i(¢)

and

_j(r/). As a result, the particular solution does not contribute to the in-plane load at the boundaries (since the mode shapes are zero there). The homogeneous solution, Fh, must

account

"average"

for the

contribution

force-displacement

at the

boundary

002

edges.

-

0¢ 2 =where

the

A's are the in-plane

are in-plane allows

edge

for finite

procedure panel

Equation motion,

in-plane

second

step

(3),

the

and

solution This

Airy

stress

(1).

The resulting

and

integrated

and

leaves

Acdr/

1¢,

And¢

boundaries.

the following

the

is to carry the

is then

domain.

[M] is a coupled

mass matrix,

Galerkin's

partial

on wo,

equation

of

removes

functions, all spatial

C is a scalar

stiffness

term,

= fi

damping

(6) term

fi is an excitation

(assuming

containing differential

the nine modal coefficients, aij. This is a set of nine coupled equations governing the modal coefficients.

vector

and

proportional 2" is a vector

nonlinear

ordinary

Frequencies

First,

consider

damping

the

natural

and excitation

by expanding xeq = 0, and the form

frequencies

terms

the nonlinear discarding

at the

in Equation

stiffness

higher

[d] is the

Jacobian

ambient

order

terms

representing

temperature.

(6), the resulting

in a Taylor

[M]_

given

method

by the modal

integration

f

where

of the

for w and

differential

damping),

Natural

is a nonlinear

K's

set of equations

[M]_ + C[M]_: + f(_) where

the

therefore,

description

expressions

multiplied

This

an

in [9].

out

the

and

formulation,

A detailed

substituting

expression over

thickness

This

F, is provided

F, into

by enforcing

form

h_ is the panel

procedure

involves

the

(5)

springs).

function,

function,

(I),(r/),

dependence

at the

takes

I(_

in-plane

Airy stress

in the

of motion.

Equation

_r(¢)

the

is accomplished

This

displacements,

(distributed

equation and

_zz

displacements

for determining

The the

edge

stiffnesses

This

condition.

the

series about

[10]. The

After

set of equations the static

resulting

system

omitting

the

are linearized

equilibrium

vector,

of equations

= -[J]_

take (7)

linearized

stiffness.

The

elements

of [d] are

by J,j -

From

Equations

nine

eigenvalues,

(7) and

(8), the

associated

with

standard the

(s)

Of' I' Oxj _q dynamic

equilibrium

merically.

3

eigenvalue configuration,

problem

results

are then

[11]. The

computed

nu-

250

200 ©

15o g

100

rr

50

Z

0 50

100

150

200

250

150

200

250

!

0.15

g "_

O. 1

e_

0.05

..

......................

-

.....

.-*_-°'°"

r5 0

•

i

0

•

f

50

i

100 _T

(deg.

F)

Figure 1: Typical flat plate (--) and initially imperfect loads and comparison with Finite Element Results (*).

The

temperature

by setting

the

equations

is then

figuration

at the

is then

entire

linearized about this temperature

nonlinear

solved

This

indicates

and

to zero.

unstable

This

temperature

as an initial state

new

equilibrium

and

using

guess

a loss of stability

of the flat

eigenvector

(the

eigenvector

plate

The panel under consideration = 1.25) and was made of AISI

la.

encountered begin critical

to rise

For

the

perfectly

at a change again.

temperature

The

flat

panel,

is

flat panel (i.e. one eigenvalue

equilibrium

for

no initial becomes

configuration

[1].

must be found using the Newton-Raphson the initial guess vector in the direction associated

the

of AT_

loss of stability displayed

stiffness

with

the

negative

eigenvalue),

equilibrium configuration the post-buckled natural

had dimensions of 15in x 12in x 0.125in 321 stainless steel. The material properties

in temperature

is clearly

the

con-

are computed

steel include E = 28x106psi, u = 0.33, c_ = 9.6x10 -6 (in/in)/°F and A typical frequency result displaying the behavior of the fundamental in Figure

algebraic

equilibrium

Next,

a new set of eigenvalues

in this scheme occurs for an initially above procedure is carried out until

is found

coupled the

vector.

to thermal

vector

set of nonlinear

This allows the Newton-Raphson routine to find the non-zero about which the system will be linearized in order to obtain frequencies. ratio

(- -) responses

algorithm

As a result, the new stable equilibrium position scheme. This is accomplished by perturbing of the

the

a Newton-Raphson

the new equilibrium and so on.

The one deviation imperfection). Here, the negative.

stiffness

with

previous

incremented

plate

of the

by Figure

fiequencies _155.25 flat

plate

lb which

decrease °F

after

shows

p = 0.291b/in 3. mode is shown

until

which

equilibrium

(aspect of this

the

instability frequencies

position

the center

is

at this

deflection

as

a function This the

of temperature.

case shows system

never

a panel

has a finite

a distinct

become

Also shown

are the

with an initial displacement

bifurcation,

center

"stiffening" deflection

even below

as in the

flat

plate

effects

of initial

imperfections.

of 0.038in

(see AT

= 0). Because

ATcr (as displayed

in Figure

configuration,

the

and

temperatures

for the initially

flat

panel.

EXPERIMENTAL

See Figure

there

eigenvalues

negative [12]. This is properly displayed by Figure la. In addition, a finite element package was used to independently

at discrete

lb),

verify

la and

is

never

these

results

lb.

APPROACH

Facilities The

experimental

Apparatus testing

portion

(TAFA)

of this work

[13] at NASA

small panels The acoustic

was conducted

Langley.

TAFA

in the

Thermal

is a progressive

Acoustic

wave

tube

subject to thermal and/or acoustic excitations. excitation in TAFA is provided by a set of air modulators

an exponential horn to a 6ft x 6ft x lft test section. Within the test section, mounted on a side wall and, thus, are subject to a grazing acoustic load. capable

of providing

both

Overall

sound

pressure

Directly

across

sinusoidal

levels

and

between

broadband

125dB

excitation

and

165dB

test

specimen

Fatigue facility

for

coupled

by

specimens are The system is

in the range

of 40-500Hz.

can be generated

in the

TAFA

facility. quartz

window

loads

behind

to the

maximum

which

panel.

heat

In addition, produce

the chamber

the

are a set of ten

Using

flux

from

all 10 lamps,

of 45 BTU/(ft

a low velocity

2 sec).

mean

quartz

this

boundary

for this

experiment

conditions

of an

Thin

by the fact that both result in an unavoidable during

heating.

and

very

steps

were

nearly

can

provided

were made expansion

to over

a

2000

°F.

convection

section,

zero

frame

thermal

of generating

natural

the

The

x lin thick

to provide

be heated

to minimize

panel.

taken

used

is capable

distribution [13]. side wall of the test

clamped

frame and panel and undesirable

Several

panels

flow is employed

ideally

units

configuration

a more spatially uniform temperature The mounting frame, attached to the

specifically

lamp

is an 18in x 28in

was

deflection,

design

and

designed zero

was

slope

complicated

of the same material. This would of the frame along with the panel

to minimize

this

effect.

Insulating

blanket

material (Min-K) was placed on the inside wall of the test section surrounding to minimize thermal conduction through the wall to the support frame. Zircar,

the panel a ceramic

insulation

minimize

material,

conduction

was

placed

the

panel

between

the inside

of the frame

radiation

from

water heat

cooled up,

the

one series of the

The

facility

test

was

panel

using

seemed

and

and

frame

a water

of the

thermocouples

10 °F.

with

several

were used

although while

thermocouples

the

on

thermal

temperature

heating.

ambient

to determine

was mounted

of the largest the

and,

appreciable

to help

channel

experiments,

to 315 °F above

gages

the

in the vicinity

up about

instrumented strain

Finally,

to prevent

was heated

panel

cooling

course

only went

was

test

frame.

the

monitored taken

system

distribution

the

the

continuous

During

the panel

cooled

and

to provide

panel.

safeguards

of tests,

water

temperature

the

between

of the frame

did

For instance,

in

the

temperature

to ascertain

the dynamic

the

response.

Temperature data from the panel and mounting frame wererecordedcontinuouslyon a computer. By adjustingthe lampbankenergydistribution, a nearly uniform temperature field wasobtained on the panel. Dynamic strain measurementswererecordedon a multi-channelspectrum analyzer. Frequencyresponsefunctionsweregeneratedbetweenthe strains and the acoustic load, as measuredby a pair of microphones,to determinethe resonantfrequencies.In addition to the strain gages,a scanninglaservibrometer "+-asusedto determinethe outof-plane RMS velocity distribution over the panel [14]. The laser vibrometer gives a measure

of the instantaneous

between

the reference

by directing the laser mirrors. A PC based over

a series

and

stored

distribution

the

Beginning to excite

frequencies.

The

RMS

structure

shape

frequencies peak

was

i th resonant vibrometer the mode

ment

are out

one

is computed

on this

it's mode appear

information

shape.

in the

considered

frequency, scan

This

panel

related

of phase

in the

point

a rectified

made

by 90°).

response.

Using

to show

Indeed,

a narrow

the rms

velocity

was carried

shape,

the

out again.

response

By identifying

as a function

section

wall each at

i th mode.

From

this and

in Figure

2a and

was

increased

can

that

excitation

velocity

each frequency

of temperature,

fact

difficulty

in the

field.

(since

are shown

test

this

acoustic

primarily

was evident scans

the

band

to associate

by the

To resolve

to oscillate

frequency

Two typical

it is impossible

is complicated

response.

was forced

to that

alone,

problem

frequency

individually.

the

was then

shape

procedure

ing mode degree

at each

of its resonances,

(2,1) and the (3,2) modes, respectively, are clearly visible. Once all the peaks had been identified, the temperature

above

shift

panel using a set of positioning to move the beam sequentially

velocity

at

the doppler

may be determined

was obtained.

will also be present

which

field,

Based

with

resonances

laser

the

information

Procedure

frequency

frequency the

mode

the

by measuring

Spatial

many points over the system [14] was used

on the panel.

the

at a point

beam.

at the ambient temperature, a low level, broadband acoustic input was used the panel. The frequency response functions were generated and used to iden-

non-panel has

response

reflected

By exciting

resembling

tify resonant a given

points" PC.

Experimental

and

spot across acquisition

of "grid on

velocity

beam

A

velocity displace2b where and

the

and its correspond-

be inferred

with

a high

of confidence.

RESULTS The material

following

properties

temperature

This While

is not

as described

3a presents that due

this effect

not change

were

obtained

using

previously.

a panel

At the

time

with

the

of the

same

dimensions

experiment,

the

and

ambient

was 67.5 °F,

Figure It is evident

results

the

none

the experimental of the

to the

less than

would

tend

character

frequency

experimental perfectly

to decrease

of the results.

results

frequencies clamped

out

the frequencies Therefore,

as a function

come

of plane at a given

the fact

of temperature.

particularly

that

close

boundary

to zero.

conditions.

temperature, the experimental

it would

a)

b)

Figure

2: The

rms velocity

field as measured

(2,a) mode and b) the (3,2) mode.

using

the

Vibration

Pattern

Imager.

a) the

a) 1000 (2,3)

(1,3) 800 : (3,2)

_D

600 ,(3,1) (2,2)

Z

( 1,2)

400

(2,1)

(1,1)

200

l

00

50

t

i

100

150 AT

(deg.

t

200

250

F)

b)

(2,3) 1000 (1,3)

800 (3,2)

600

(3,1)

(2,2) (1,2)

400

(2,1)

(1,1)

200

0 0

50

100

150 AT

Pigure the right results:

3: Natural

frequencies

give an indication initially

flat

(--)

of the panel imperfect

mode•

(- -).

250

F)

as a function

of the dominant and

(deg.

200

of temperature.

a) experimental

results,

Designations b) theoretical

on

frequenciesdo not drop to zero implies the presenceof an initial imperfection, which would be expectedin the presentsystem. Figure 3b showsthe theoreticalfrequency-temperatureresults for the flat plate and the initially imperfect panel. Theseresultsweredeterminedusing a finite in-plane edgestiffness,whosevalue wasbasedon matchingthe theoreticalcritical buckling temperature with experimental observations, Experimentally, the finite edge stiffnessis, most likely, due to the inability of the ceramicinsulation material to prevent in-plane displacements.Basedon this figure, the flat panelbucklesin the (1,1) modeat a critical temperature of ATcr ,_155.25 °F. The results for the imperfect panel are based on the assumption with

that

the

an amplitude

center

of the

imperfection

an

plate

(15%

A comparison tively,

the

for the

limited

of modes.

number

theoretical

Quantitative modes. The evident.

Both

quency near

3a and

Furthermore,

a temperature

Based fact

natural

Figure

a more

results

between

becomes

expected

(3)) at the

improve

the

higher

modes

shown

to the

a significant

gap

between

and

(2,2)

of the

(3,1)

is predicted

appear

that

There

be-

in Figure three

(2,1)

3.

or four modes

and

loci occurs

a

is

(1,2)

fre-

experimentally

albeit

at a lower

tem-

contains the essential dynamics as a function of temperature.

there

However,

are no appreciable there

theoretical

is about

first

the

with

agreement

for all the

by theory

theoretical model of the frequencies

and

increasingly

would be limited

Quantita-

in a study

modes

experiment

temperature.

scale.

agreement

and

experimental

frequency

Equation

of 0.019in

is revealing.

theory

it might

with the

(see

for the should

of 142.5 °F and

frequencies

with

these

crossing

figures,

4 compares

reasonable

frequency

results

°F). Clearly, the for the behavior

on these

mode deflection

results

is entirely

additional

3b show

the

increase

perature (AT =115 which are responsible in the

Including

This

This

experimental

similarities

Figure

loci.

(1,1)

theoretical

agreement.

frequencies.

from

qualitative

and

are in good

comparisons

of the

to an initial

thickness).

higher and

shape

corresponds

experimental

(1,1) frequencies

the

This

of the

of the

less impressive tween

is in the

= 0.038.

are,

and

results

to highlight

for the

a 20% variation

variations

(1,1)

in the

this

mode

on

fundamental

temperature.

DISCUSSION This behavior

paper

presented

of the natural Comparisons

ment

between

more

noticably.

of the higher

modes,

between lower

This

theoretical The

flection

the

a combined

frequencies

is partly

implying

differences

Considerable

initial

imperfection.

and

that

to accurately

effort

was spent

Because

K_ and

higher to the

there

the model the

subject

experiment

The

attributed

However,

between

of our inability

setup.

theory

eigenvalues.

analysis.

theoretical

of a panel

captured

model K,

were

study

relatively

and the based

quantitative

on the

other

numerical

on static

agreediffer

used

in the

course

agreement,

even

dynamics results

conditions

in-plane

the

hand,

few modes

qualitative

the relevant

into

loads.

show reasonable

the boundary

determining

experimental

eigenvalues,

is excellent

experimental

and

to thermal

of the are

in the

problem.

partly

a re-

of the experimental

boundary information

stiffness over

and

the

a range

250

200

50

0

I

0

50

100

150 AT

Figure 4: fundamental

Comparison mode.

of temperatures, any

given

els dynamic factors

Furthermore,

on the edges, behavior,

as prestress

tributions

and

does

several

have

develops AT,

the

in-plane

magnitude) fast)

and

method

increments consider

over,

of motion

may

boundary

for computing (see Figure

other

aspects

la and

the the

the panels element

into the

stiffnesses, Also,

of the dynamic

(Equation

effects

terms

technique

frequencies

Finally, problem,

the results of this study provide important space, (T,w), for the forced problem.

10

analytic

as the

imperfection

particularly

most notably

information

This Because

is an efficient

this formulation about

may

at

the

easily

a portion

are such dis-

formulation this

study

expressions,

temperature (both

rise,

shape

and

(computationally

for small

the forced

pan-

temperature

thickness.

(1)) using

of such

exactly

differences

analysis.

K_, If n, the initial

Galerkin

natural lb).

to these

a finite

the

influence

nonuniform

through

for the panel

be gained

so on.

say,

for

force-displacement

may significantly

clamping),

(o)

to match

uses an "average"

which

during

frequencies

are not expected

Also contributing

(induced

in temperature

advantages

experimental

analysis

effects,

are not considered.

variations

insight

the

boundary

in the panel

the equations

additional

and

results

because

local

250

(deg. F)

theoretical

the frequency-temperature

AT.

condition

of the

200

temperature

be extended case.

In addition,

of the parameter

to

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NASA

Technical

Memorandum

Ca-

REPORT

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12.REPORT DArE

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I

i

J April 1994

I Technical Memorandum

4, TITLE AND SUBTITLE

S. FUNDING NUMBERS

Free Vibration of Thermally Loaded Panels Including Initial Imperfections and Post-Buckling Effects 6.

WU 505-63-50-10

AUTHOR(S)

K. D. Murphy L. N, Virgin S. A. Rizzi 7.

PERFORMING

ORGANIZATION

NAME(B)

AND

8.

ADDRESS(ES)

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ORGANIZATION

NUMBER

NASA Langley Research Center Hampton, VA 23681-0001

9.

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National Aeronautics and Space Administration Washington, DC 20546

/ MONITORING

REPORT

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NASA TM-109097

I1. SUPPLEMENTARY NOTES Murphy and Virgin: Duke University, Durham, NC Rizzi: NASA Langley Research Center, Hampton, VA Note: Paper submitted for presentation at the Fifth International Conference on Recent Advances in Structural Dynamics, Southampton, England, 7118-21/94. 12¢ DISTRIBUTION / AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

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13.

ABSTRACT

(Maximum

-- 71

200

words)

A combined theoretical and experimental approach is developed to consider the small amplitude free vibration characteristics of fully clamped panels under the influence of uniform heating. Included in this study are the effects of higher modes, in-plane boundary elasticity, initial imperfections and post-buckling. Comparisons between theory and experiment reveal excellent agreement.

14.

SUBJECT

15.

TERMS

Thermal Buckling; Initial Imperfection; Free Vibration; Natural Frequencies

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