San Fernando Valley State College
VIBRATION OF A POINT SUPPORTED RECTANGULAR PLATE
\\
A thesis submitted in partial satisfaction o£ the
requirements £or the degree o£ Master o£ Science in
Engineering
by
Jerry L. Crone
J
July, 1970
The thesis of Jerry
L.
Crone is apprpved:
...
.J
�
Committee Cha irman
San Fernando Valley State College
June, 1970
ii
·"
Acknowledgements
It is a pleasure £or the author to acknowledge the
support and encouragement given by his wife during the
course o£ this study.
Without her assistance, this
thesis would not have been possible.
The author extends thanks and gratitude to the
faculty and sta££ o£ the Engineering School o£ San
Fernando Valley State College.
Special thanks go to his_ thesis committee chairman,
Professor Leonard Spunt, £or his assistance and
guidance during the preparation o£ this thesis and
during the entire course o£ this Master's program.
iii
Index
Page
i
ii
iii
v
vi
vii
Description
Title page
Approval page
Acknowledgements
List of figures
Nomenclature
Abstract
1
Introduction
4
Theoretica-l developement
12
Experimental procedure
18
Results and conclusions
25
Bibliography
26
Appendix A - Experimental and theoretical
data
38
Appendix B - Theoretical equations
41
Appendix C
-
J
Alternate solution technique
iv
List of Figures
Figure
Number
Page
1
6
2
11
Graphical relationship of k to d for
the a/b ratios tested
3
13
Forcing function amplitude input
curve
4
15
Plate and holding fixture
5
17
Test equipment functional block
diagram
6
21
Graphical representation of
theoretical and experimental data
for � square steel plate
7
31
Data plot for a 2 by 5 rectangular
steel plate
8
32
Data plot for a 3 by 5 rectangular
steel plate
9
33
Data plot for a 3.5 by 5 rectangular
steel plate
10
34
Data plot for a 5 by 5 square
aluminum plate
11
35
Data plot for a 2 by 5 rectangular
aluminum plate
12
36
Data plot for a 3 by 5 rectangular
aluminum plate
13
37
Data plot for a 3.5 by 5 rectangular
aluminum plate
Description
Plan view of plate geometry
v
)
Nomenclature
a - length
b - length
-
h - plate thickness
w
deflection
E - Young's modulus
u - Poisson's ratio
9 - acrtan b/a
d - support location from plate corner
m - a/ (a
2dcos9)
n - b/ (b
2dsin9)
x
- coordinate location----
y - coordinate location
--
(> - density
D
V
K
plate stiffness
3
2
Eh /1 2{1 - u )
strain energy
)
kinetic energy
k - constant determined by f (m, n)
t - time
I
total system energy
1T- 3.1415
f - frequency {Hertz)
- ----------- ··----------------- ·
-
---·--
- ·---·------ --------
--- - --- -
-
-
vi
--- -----
-
--
-
-
·
-- ----- ---
ABSTRACT
VIBRATION OF A POINT SUPPORTED RECTANGULAR PLATE
by
Jerry L. Crone
Master o£ Science in Engineering
June, 1970
This thesis is devoted to the determination o£ the
£irs� fundamental frequency o£ a rectangular plate
supported at four discre-fe points.
The support location
is limited to an "on diagonal" configuration.
A general expression in determinate form is
developed for calculating frequency using the Ritz
method as applied to an energy technique.
i
Values £ot
various plate parameters are determined and compa<e
with experimentally determined frequencies.
i
The frequencies found analytically were in good
agreement with experimental values for a/b ratios less
than 2.5.
Graphical results are presented for several
test configurations with theoretical frequency curves
as a comparative base.
All results are tabulated in the
appendix.
vii
Introduction
The primary problem in elastokinetics is the
determination o£ the fundamental vibration�l-mod�- o£
an elastic medium when subjected to time and space
dependant displacements.
For most common structural
configurations, with well defined boundary conditions,
standardized solutions exist and have been tatiulated
in parametric form as an aid to the working engineer.
However, engineering evaluation o£ systems and sub­
systems is at best a unique art form requ } ring many
assumptions and approximations in the formulation o£
solutions £or structural responses to various int�rnal
and external stimuli.
As a result, discrete problems
have either been ignored or have been catagorized into
very general groupings and presented in parametric
forms.
Although these solutions are extremely important,
the working engineer is often required to produce a
design with more �tringent constraints than the existing
solutions allow.
Classically, the engineer would
approximate a solution, fabricate the system, and test
to determine the response to the required stimulus.
I£
a failure occurred, a modification would be proposed and
another test e££ected.
This process would be repeated
1
----- - -�-- -
2
until a working model was developed.
The nature of such
an approach is often successful but also often results
in excessive cost, weight, and redundancy.
This results
from the inability to recognize the boundary conditions,
lack of adequate time and ineffective working solutions.
The course of study for this thesis is a result of
difficulties encountered with practical problems for
which standard solutions have not been tabulated and
only very general equations have been systematically
stated in standard texts and reference materials.
Subject matter of this study is restricted to
rectangular plates, supported at four discrete points
'
and subjected to sinusoi_dal forcing functions normal
to the plane of the plate.
The objective was to determine
the first natural frequency of the plate as- the support
A
configuration was varied along the plate diagonals.
constant "g" forcing function was utilized for the
I
I
I
experimental verification so that alternate material�
and plate configurations could be compared on an
equivalent basis.
I
The theoretical developement of the
particular solutions is compared with the experimental
results and the applicability of the solutions is
evaluated�
The method utilized to develope the theoretical
solutions is a well known approach. The application
3
and extension o£ their usage is discussed and compared
with an alternate solution technique.
Results o£
theoretical calculations and experimental values are
presented in graphical and tabular form.
The general
solution is presented in the appendix £or reference and
is represented by three linear equations in a frequency
parameter and three undetermined coefficients to be
solved £or in determinate form.
Presentation in this
manner allows £or a solution to general plate problems
conforming to the constraints established in the
theoretical developement.
Calculated values are only
£or the plates experimentally evaluated during the
course o£ this study.
J
4
Theoretical Developement
Plate structures with clamped and simply supported
boundaries have been extensively investigated and
theoretical solutions have been well documented with
experimental results. · Combinations of these boundary
conditions and certain free edge configurations have
also been documented and established into traditional
solutions.
For this study, point supports are utilized
and a general equation is developed for determining the
natural frequencies by using an assumed deflection
expre,ssion inserted into an energy approach as presented
by Kirk (ref. 1).
This technique is the Rayliegh-Ritz
energy method and the Ritz method is used to determine
the frequencies.
Assume a thin elastic plate (h({a or b) of general
I
dimensions and that the effects of shear and rotatory
inertia are negligible.
;aI
The coordinate axes are par llel
to the edges of the plate and the origin of the
.
coordinate axes is coincident with the centroid of the
plate.
The coordinates of the supports are to be located
on the plate diagonals and in such a manner that for any
particular configuration the supports are equidistant
from the coordinate axes origin.
any a/b ratio.
The plate may assume
The values used for this study do not
excede· 2. 5 since the test equipment represented a physical
i
l
limitation.
It can be also shown that' values in excess of
this ratio rapidly converge to the standard beam solution.
The coordinates of the supports are to be defined as:
X =
a/2 - dcos
8
and
y
=
b/2 - dsin
-.-(1) ' ( 2)
e�-
where d_is the_dimension from the corner of the plate to
the support as measured on the diagonal of the plate, and
9
is the angle determined by
e
=
arctan b/a.
illustrates the derivation of these terms.
Figure 1
�·
Using classical plate theory, Timoshenko (ref. 2)
/
defines the strain energy (V) for a rectangular plate as:
c2
2
2
2
a/2
b/2
& w � w +
� 2 + � 2+
)
V = D/2
(
(
)
2u(
)
2
Sx2" S y2
6 y2
a/2 -b/2 S x
�
�
·
2(1 -- u)
� 2w 2l
) dx dy
<sx oy
J
(3)
The limits of Timoshenko's equation have been modified
to reflect the coordinate system used for this developement.
The kinetic energy (K) can be written as:
b/2
a/2
('h
Sw 2
( 4)
K =
(Jt) dx dy
2
-a/2
b/2
Expressions (3) and (4) define the total energy states
j_ /
of the plate system during dynamic displacements.
Applying Hamilton's principle to these equations, the
total energy expression for the system can be written in
the following form:
j2Tfj'W"
6
( K
0
-
v
) dt
=
0
The Ritz method utilizes an assumed deflection
(5)
6
y
'
'
'
'
'
d.
0...
Plan view of a typical plate
configuration
FIGURE 1
J
'
'
j.
l
!
i
i
-- ·-·-·--·-··--- --------·-··--··-------··. ····--·-···--··--·--·
---·
_!
····--· -····· --·-·-·····--·
7
equation which must satisfy the physical constraints o£ the
system.
This type o£ expression is generally written as:
w(x, y, t)
(6)
By the application o£ Hamilton's principle to equations
(3) and (4), a total energy expression (I) £or the plate
b-
system can be written as: I
j_ /_
a/2
D
b/2
/
2
Q hw-
::
{b/2
a/2
.
2
(W) dxdy 2
2
S w
u)(
ox6y
S;:;: ::�
P
2
2
S w
s w 2
s w 2
(
) +(-) + 2u(
(' 2
2
2
2
-a/2 -b/2 ox
Sx &y
oy
)_
_
dxdy
=
0
) jJ
(7)
For polar coordinate representation o£ the above
equation, see reference 5.
I£ the mode shape o£ the system is assumed to be o£ the
form, w(x, y, t)
W(x, y)coswt, (re£. 5) , the energy
a/2 b/2
2
2
W dxdy
expression (7) can be rewritten as: I ::: ehur
- / -b/2
? ·2
a/2 b/ 2
2
2
S w 2+
s w 2
w � w +
) +(
)
- n
)
) 2(1
2u(f
2
2
2
2
Sy
Sx 0y
x
-a/2 -b/2
.
2
� w
(
) dxdy = o (8)
oxSy
j_
=
_
/
}
_
·
/_ /_
::
(
LS
With equation (8),
J
J
the natural frequency o£ the system
can be determined i£ an appropriate expression £or w(x, y, t)
is assumed.
Since the maximum kinetic energy occurs when
the plate 1s at the plane position, an expression W(x,y)
need only be assumed.
The expression must naturally satisfy
the geometric constraints on the system o£ no deflection at
the support points.
hypothesized.
Several such expressions could be
The expression to be used is a modification
o£ an equation proposed by Tso (re£. 3).
This deflection
8
.. - .. �
-.
. '"" · � · ��-· ·-� -·-· .
·-
• ...... ··-·-· -.-----·-----·-·----
.•• ······�··· .�·-·--- �.
expression is written as: W (x,y)
B (2 -
4m
2 2
2 2
4n y
)
2
b
�
a
where m and
n
+
C (1
=
··;-�--
A (cos-a-
2·2
.
4m x
) (1
2
a
_
..
+
-·7'"
- -��;- · :· · · -- -- ···1
�
--
cos�)
2 2
4n y
)
2
b
.
- '" -
--,
+
are defined as:
m
=
a/ (a
2dcos 9)
(10)
n
=
b/ (b
2dsin 9)
(11)
.
be determined from the minimiz ing conditions of :
=
o , Sr/&B
o,
oi/SC
=
o
(12) (13) (14)
It is easily verified that the assumed deflection
expression satisfies the geometric constraints by
substitution�of (10), (11), (1), and (2) into equation (9)
and simplifing the results.
I
I
l
!
!
Alternate expressions were
proposed by Tso and they also will satisfy the system
constraints.
�
(9)
and A, B, and C are undetermined coefficients which can
Sr/&A
.
-
j
!
However, the expression utilized herein
yields the most compatible results with experimentation
since the Ritz method improves in accuracy with the addition
of terms to the assumed deflection equation.
The use of
alternate expressions would not serve any purpose for this
study and therefore is considered redundant and expression
(9 ) will be the only equation used.
Defining equation (3) as G and equation (4) as P,
equation (8) can be written in a simplified form as:
( G
where a
n
-
2
eh�
P )
Dg
=
o
implies the undetermined coefficients in the
assumed deflection expression.
(15)
i·-·-
' --
9
I£ the indicated integration and differentiation
is performed, three equations linear in A, B, and C
result .
These equations are contained in the appendix
of this text.
Since the number o£ terms become quite large,
substitution of individual configuration constants
into these expressions at this time results in
a
simplified algebraic expression in terms o£ constants
and a frequency parameter.
Using the Ritz technique, the determinate o£ these
expressions can be equated to zero (ref. 4).
The
undetermined coefficients need not be solved £or since
frequency is contained within the determinate.
A cubic
equation results when the determinate is expanded.
The
lowest, real :root o£ this expression can be used to solve
£or the frequency.
The other real roots are frequency
parameters £or higher modes o£ vibration and not required
£or the purposes o£ this study.
Although the author did not utilize a computer
technique for the solution o£ the equations, the
;·
I
i
1
forms presented in the appendix could readily be tailored
£or such applications.
The equations presented in the appendix do not
have the m and n terms as previously defined. By
substituting the values £or m and n into their respective
'
10
definitions, it becomes apparent that they are equal for
any particular value of d.
The dependancy of
m
and n on
d is shown for the a/b ratios used during the
experimental verification of the developed equations.
Since
m
and n are equal, they are defined as k and plotted
with d as the abcissa and the resulting curves are labeled
for the applicable a/b ratio.
Figure 2 represents this
relationship for the four ratios tested.
I
J
11
1.0
-�-.--u-=-•-••---�L -�-�
to
2.. o
'=
3.0
=-·�=��!.
4.0
Relationship o:f k to d :for the a/b ratios tested
FIGURE 2
12
Experimental Procedure
To develope a representative base £or experimental
verification o£ the theoretical equations, twenty
different plate configurations were utilized.
The
composition o£ the test specimens consisted o£; four
a/b ratios, three material types, two thicknesses, and
twenty-nine different mounting configurations.
A fixture was adapted to an Unholtz-Dickie vibration
excitation head.
All mounting configuration provisions
were contained on the single fixture.
Two accelerometers
were mounted on the pla��s to provide signals to a
multi-channel recorder.
A reference accelerometer was
mounted on the shaker head to be used as a-reference
on the recorder output.
Coupled with the recorder was
a single trace oscilloscope.
/
The manual fine adjust on
the shaker control panel was used to develope peak
accelerations by visually monitoring the plate
accelerometer outputs on the oscilloscope.
I
/
,
By switching
from one accelerometer to the other, the frequency and
peak acceleration could be ascertained.
The frequency
and "gn output were manually recorded £rom the control
panel meters and automatically recorded on the multichannel recorder.
This procedure provided a cross
13
oouer.. E
Rl"t\9l.Liu0£
INC. \&G:-5
,OJ
.001
J
•0001
�����--._�����----�
t
tlER.T.t
Forcing £unction input curve
FIGURE 3
I
r
14
I
·�, <-v.•-·-�•
•··�---··-•-,.-·•·•-�- ·--�··><·"'
• ,�,��·-•••'
reference as to the accuracy o£ the two instruments;
Correlation o£ data proved the equipment calibration
to be accurate within three hertz and
±
. 16g in
acceleration.
The test procedure consisted o£ first mounting the
accelerometers on the plate surface.
One was mounted
at the plate centroid and the other approximately . 25
inches in £rom the edge o£ the plate and on the
X
coordinate axis as previously defined in this text.
When this was accomplished, the vibration control
console was activated and a manual sweep was initiated.
The sweep was started at a low frequency ( normally
around 60 hertz ) increasing-the magnitude until a
resonant peak was observed on the .oscilloscope.
When
the peak acceleration was determined using the fine
control, the accelerometer outputs were switched £or
a frequency comparison.
Once the lowest frequency
was established, the automatic recorder was activated
and the readouts on the shaker control panel were
recorded.
This process was repeated three times for each
mounting configuration and the data was averaged.
The
complete procedure was repeated for each plate and
mounting configuration.
The "g" input to the plate
was held constant for all test conditions.
The
amplitude naturally varies with frequency for a constant
1.5
AC C. E. I-E! �0 MEP"li '­
L.GO:.I\TJ (HJ 5
F"l X TUR.£
Typical plate and holding fixture orientation
with accelerometer locations indicated
J
FIGURE 4
--------·-····-·-····-----
--·------·
--
----------··------···--------------- , -
.
·····-······-·-··-------·---- - --·--
. - ·-
I
16
g input.
The curve in figure 3 represents the double
amplitude sinusoidal input as a function of frequency.
Figure 4 represents a typical fixture and plate
arrangement with the accelerometer locations indicated
and a blowup of the hardware stackup of one of the
support points.
Figure 5 is a functional block diagram representative
of the test facility and the equipment used.
Total testing time was approximately twenty-six hours
including the initial setup time.
A data table is
included in the appendix and associated theoretical
values for each test condition is included adjacent
to the experimental freq�ency for easy comparison .
_,
)
17
CONTROL
SJNC.t...E
COt-!S.OLE
TRI'\t.£
-
o<.ape
V\ ���'it ON
HE. AD
--
--
-
AUTO
5 wr-rc H '"'<i
JUr.!CT tON
P.E CO A.D E: P,.
Test equipment functional block diagram
FIGURE 5
J
\.
' ---- --�- .
18
Results and Conclusions
Substitution o£ the plate parameters into the terms
listed in Appendix C derived £rom the partial
differentiation o£ the energy expression greatly reduced
the algebraic manipulation and simpl�£ied the
determination o£ the frequency pararo.eter Jl .
It can be
seen that a cubic equation results when the Ritz method
is applied and the resulting determinate expanded.
Since the objective o£ this thesis was to determine
the first fundamental frequency of the plate systems,
solut·ions for the undetermined coefficients was not
necessary because the frequency could be determined
using the roots of the cubic equation.
How�ver, a brief
discussion as to the approach used to calculate A, B, and
t
C will be included for reference.
i
If three linear equations represent solutions o£
;
given system and they all are equal to zero, Cramer s
rule (ref. 10) states that the resultant determinate is
equal to zero and the trivial solution of A
is implied.
=
B
=
C
=
0
This would indicate that no solution of
physical importance exists.
However, i£ one or more o£
the equations is a linear combination of the others, a
solution does exist.
By definition, if the determinate
is equal to zero, at least one of the equations must
be
19
deleting any row and expanding the remaining terms by
minors.
If the terms in Appendix B are denoted by:
r l
1
1
the values of A, B, and C can be written in determinate
x
y 2
1 1
1 1
and
B =
form as:
A =
22 x2
Y2 22
x y
1 1
=
c
x2 Y 2
When the determinates are evaluated, substitution
of the coefficients into the energy equation and carrying
out the indicated integration, the energy state of the
system can be determined fot_?ny time during a dynamic
displacement.
The resultant deflection equation can be
substituted into dynamic stress equations such as
presented in references 11 and 12.
To outline such a
developement would be quite extensive and beyond the
scope of this study.
The inclusion of these references
is made only to illustrate the versatility and importance
of this technique.
Frequencies calculated using the equations listed in
the appendix are conservative as 1s any approach using
an approximation technique.
This means the predicted
frequencies are always greater in magnitude than those
found by experimental methods.
This is basically due to
the assumption of an inexact deflection expression.
It can
20
readily be seen that assuming deflection expressions of
multiple terms which satisfy the system constraints can
result in equations which become quite cumbersome and in
which the algebraic manipulation is subject to error if
only in accounting.
An error may not significantly
effect the results since this is an approximation method
but this is a false indication because the boundary
conditions will not be satisfied when the appropriate
substitutions are made.
Fundamental frequencies determined by the theoretical
expressions developed in the course of this study were
of significant value over a specific range of support
locations.
Figure 6 represents a typical plot of the
theoretical solution as a solid line and the respective
test results as discrete points.
This par�icular plot
is for a square plate and can be compared with the results
presented in reference 1.
Comparison indicates that
i
i
the equations developed do in £act degenerate to a form
similar to that stated by Kirk.
That is, when a =
I
I
1'
the frequencies determined by using Kirkts equation
and frequency parameter are near the values calculated
in this report.
The basic difference is due to the
slightly different values used for Poisson's ra�io.
Additional curves for other a/b ratios and a tabular
presentation o£ all calculated and experimentally
21
-100
300
. !
!
2.00
)
Experimental
Theoretical
1.0
2.0
Experimental and theoretical plot of values
for a 5 by 5 inch square steel plate, .030 inches
thick using Poisson's ratio as .26
FIGURE 6
i
I
22
determined frequencies are contained in Appendix A.
Value for Poisson's ratio as applied to the equations
were taken from reference 6.
These vaiues h ave a
definite effect on the determination o:f the :ftequeriey
parameter and can effect the result by as much as two
percent.
The values from reference 6 used herein were,
u = .26 for steel,
.33 for brass,
and
.36
fot aiumirium;
Inspection of the graphical representations indicat�
a significant departure of the experimental results from
the theoretical values for d in excess o:f 1.4.
This
result is typical within fifteen percent ior ali the a/H
ratios tested.
The accuracy decreased as the a/b ratio
increased as might be expected.
However, the vaiues were
still applicable for support configurations near the
corners of the plate.
As
generai statement, t-il�
a
theoretical values can predict with reasonable accuracY"
the response of the plate if the support configurati 9ri
. I
dimension d does not exceed twenty percent of the longer
plate side.
.
/.
Beyond th•J.S poJ.nt•, th e accuracy can d evJ.ate
as much as sixty percent.
•
-
This result- is graphical-ly·
obvious by inspection of the curves in Appendix: k.·
Of all the tests performed, only one point exceededthe theoretically determined frequency.
A reason for:
this deviation cannot be postulated and additional- test::r
might establish the data as- false.
Inherent with t h�
'
--
23
test procedures were several potential error sources
which would have contributed to the seperation o:f the
experimental frequencies :from the theoret ical.
The first
and most predominate source would be t hat the supports
Due
were not true points but represented a :finite area.
to physical limitations, true point supports were not
possible.
The approximate area at e�ch support location
was .025 inches squared.
This area is small but would
represent a tendency for the :frequency to deviate :from
the theoret ical.
Intuitively, it would be assumed that
the resonance point might be increased by such a
configuration and the values would approach closer to
the calculated :frequencies.
This is only conjecture
but could be approximated by using the finite difference
approach outlined in Appendix C .
Other miQor error
sources were present such as inability to visually
establish the peak acceleration on the oscilloscope
)
I
and dimensional errors in t he fabrication of the test
specimens.
However, it is £e1t by t he author that
these sources were minimal and in general the results
were acceptable.
Having established the applicability o:f the energy
approach to plate configurations other than square for
a reasonable range o:f point supports is only as valid
as the assumed deflection expression.
The equation used
was quite extensive and it is felt t o be adequat e :for
\
-
24
the value of this approach and completed the study as
intended.
However, other point support configurations
are still to be evaluated and such a project would
represent a challenge to anyone involved in a program
of similar complexi"ty.
J
I
25
Bibliography
1.
Kirk,, C. L. , 11 A note on the lowest natural frequency
of a square plate supported at the corners 11
Aeronautical Society Journal, Vol . 66, pp. 240241, 1962
2.
Timoshenko and Woinowsky - Krieger, Theory o£ Plates
and Shells, 2nd Ed., pp. 342-346, McGraw-Hill
Book Co. , 1959
3.
Tso, W. K. , 1 1 On the fundamental frequency o£ a £our
point supported square plate 11, AIAA Journal,
Vol . 4, No . 4, pp. 733-735, Apr. 1966
4.
Timoshenko, Vibration Problems in Engineering, 3rd
Ed. , pp. 380-385, Van Nostrand Co ., 1955
5.
Flugge, Handbook o£ Engineering Mechanics, 1st Ed .,
PP• 61-21 - 61-24 and 65-31 - 65-32,
McGraw�Hill Book Co., 1962
6.
Roark, Formulas £or Stress and Strain, 3rd Ed.,
pp. 370-371, McGraw-Hill Book Co., 1 954
7.
Timoshenko and Gere, Theory o£ Elastic Stability,
2nd Ed ., pp . 92-94, McGraw-Hill Book Co. , 1961
8.
Nowacki, W., Dynamics o£ Elastic Systems, 1st Ed.,
pp. 198-255, J . Wiley and Sons Inc. , 1963 i
9.
Cox and Boxer, 11 Vibration o£ rectangular plates
point-supported at the corners 11, The Aeronautical Quarterly, Vol. 61, pp .45-5, ,
'
Feb. 1960
--
j
i
10.
Taylor, Calculus with Analytic Geometry, 4th Ed.,
pp. 524-538, Prentice Hall Book Co., 1 960
11.
Reismann, H. , 11 Forced motion o£ elastic plates 11,
Journal o£ Applied Mechanics, pp. 510-515,
Sept. 1968
12.
Ungar, Eric E. , 11 Maximum Stresses in beams and plates
vibrating at resonance 11, Journal o£
Engineering £or Industry, pp. 149-155, Feb. 1962
Appendix A
Tabulated Experimental and Theoretical Results
Included in the following tables and graphs are the
displacement amplitudes recorded during resonance during
the execution of the tests.
Although not
a
part of this
thesis, the information is included for interest and
reference.
If desired, the displacements could be
verified theoretically by solving the equations in
Appendix B for the undetermined coefficients A, B, and C
substituting the resulting values into the appropriate
deflection expression with the coordinates of the
accelerometers as defined herein .
Note that the
frequency parameter 1l. is in radians per second and has
been divided by 27.r prior to tabulation so that frequency
in hertz could be compared directly with the experim�ntal
values.
'
!
The collli�s of the tables have the followin�
L
connotation and dimensions : d is the support locat� n in
inches,
fNl<;is
the theoretical frequency in hertz,f is the
experimental frequency in hertz,
$0
�·
max. is the deflection
at resonance at the plate centroid, and
� max.
I.a,
is the
deflection approximately . 25 inches in from the plate
edge on the x axis as previously.
27
]
I
I
I
TABLE 1
Theoretical and Experimental Data
l
Support
Location d
in inches
Plate
Type
i
bo
Disp.
inches
b Q./2.
147
140
.031
.006
Alum.
.·70
211
201
.040
.004
1.05
275
265
.028
.007
1.40
400
378
.020
.007
1.75
433
405
.012
.003
2 .10
352
268
. 009
. 007
.030
.35
173
201*
.025
. 002
Alum.
-. 70
300
294
.033
.001
1 .05
412
401
.020
. 005
u = .36
1.40
508
496
.014
.oo6
a = 5"
1.75
584
557
.005
.006
2 .10
437
342
.002
.007
.030
.35
204
201
.023
.003
Alum.
.70
307
303
.031
.002
a/b = 1.66
1 .05
476
448
.026
.005
u = .36
1.40
614
585
.017
.008
a = 5"
1. 75
648
610
.008
. 009
2 .10
563
438
.006
.008
.030
.35
.. 248
243
.021
.002
Alum.
.70
370
342
.028
.004
a/b = 2. 50
1.05
611
468
. 022
.009
u = .36
1 .40
924
730
. 017
.013
a =
1. 75
815
605
.011
.012
2.10
-
-
a/b
=
l a/b
!
Disp.
inches
.35
.
=
1.00
�36
:a = 5"
'
'
Experimental
Frequency
in hertz
£
ne
.030
u
!
Theoretical
Frequency
in hertz
£
nc
=
5"
1.42
-
-
-
28
r·--
------"
------- _______..__________________
I
-·
-------------- ---------:.:- ------- ------------------------------�-- --..-:----- --_
--� ---- - -------------
----- -
TABLE 1 continued
I
i Plate
Type
d
:
.051
.35
269
262
.025
.007
Brass
.70
374
366
.031
.005
a/b = 1.42
1.05
531
508
.026
.009
u = .33
"
Ia = 5
1.40
633
559
.019
.012
1.75
884
778
.013
.012
2.10
751
562
.009
.015
.051
.35
280
273
.021
.005
Brass
.70
402
395
.027
.005
a/b = 1.66
1.05
587
554
.026
.010
u = .33
a = 5"
1.40
741
547
.021
.013
1. 75
1076
963
.014
.017
2.10
812
502
.007
.019
.35
328
307
.022
.004
!
i
f
f
nc
C a/2
ne
--
.051
Brass
--
.70
472
442
.025
.004
a/b = 2.50
1.05
705
632
.019
.006
u = .33
a = 5"
1.40
943
663
.011
.013
1.75
1213
839
.004
.019
2.10
-
-
-
-
.030
.35
163
160
-- .040
.007
Steel
.70
211
208
.042
.005
a/b = 1.00
1.05
285
278
.037
.011
u = .26
a = 5"
1.40
411
397
.019
.017
1.75
459
439
.011
.021
2.10
373
296
.008
.024
.030
.35
205
201
.039
.005
Steel
.70
308
302
.039
.006
a/b = 1.42
1.05
446
437
.032
.009
u = .-26
a =� 5"
1.40
541
498
.024
.011
1.75
601
583
.013
.013
� .10
512
342
.008
.018
-
'
-
29
-�-F,_.,__, -�-· -· o �· ••--• •
•
•.
TABLE 1 continued
Plate
Type
i
£
d
nc
£
ne
.060
.35
335
324
.036
.006
Alum.
.70
461
435
.041
.003
1.05
605
572
.031
.008
.36
1.40
776
743
.027
.012
5"
1.75
802
746
.011
.014
2.10
621
483
.007
.015
.060
.35
463
452
.027
.005
Alum.
.70
581
568
.033
.003
1.05
714
694
.024
.010
.36
1.40
975
847
.019
.016
5"
1_.75
1038
883
.009
.017
2.10
843
612
.005
.017
.060
.35
599
587
.026
.005
Alum.
.70
653
613
.034
.004
1.05
847
795
.026
.007
.
a/b
u
=
a
=
�/b
tu
Ia
=
=
1.00
=
=
1.42
ia/b
=
tu
a
.36
1.40
988
936
.018
.015
5"
1.75
1126
951
.011
.018
2.10
808
694
.004
.020
.060
.35
741
682
.023
.005
Alum.
.70
863
774
.029
.008
1.05
972
889
.019
.014
1188
1054
.012
.022
1294
1181
=
=
a/b
:u
=
a: =
=
1.66
2.50
.36
1.40
5"
1.75
2.10
.
-
-
·-
-
-
-
-
.051
.35
201
194
.023
.008
Brass
.70
283
267
.026
.007
1.05
378
360
.022
.011
.33
1.40
518
475
.019
.015
5"
1.75
621
515
.013
.017
2.10
495
374
.012
.018
a/b
u
a
=
=
=
1.00
30
TABLE 1 continued
Plate
Type
d
I
I
nc
8
ne
a/2
.. 030
.35
220
213
.041
.004
Steel
.70
321
314
.040
.006
1.05
505
·465
.038
.006
1.40
628
558
.027
.013
1.75
651
630
.016
.016
2.10
594
487
.011
.017
.030
.35
275
251
.037
.004
Steel
.70
397
358
.039
.005
a/b = 2.50
1.05
654
501
.031
.009
u = .26
1.40
947
811
.020
.015
a = 5"
1.75
863
674
.009
.021
2.10
-
-
a/b = 1.66
iu = .26
·� = 5"
-
-
31
fm,
Ht\Ril.
eoo
700
600
5'00
L)OO
l
j
i
300
•
-------
Experimenta
Theoretical
2.0
1.0
Experimental and theoretical plot o£ values
£or a 2 by 5 inch rectangular steel plate, . 030 in.
thick using Poisson1s ratio as .26
. FIGURE 7
fm.
HE.�TZ.
700
600
soo
400
'300
I
200
IOO
•
J.O
�
Experiment
Theoretical
I
2..0
�d.
tN<.l\E�
Experimental and theoretical plot o£ values
£or a 3 by 5 inch rectangular steel plate, .030 inches
thick using Poisson1s ratio as . 26
FIGURE 8
33
fl'l'\
HEP-.1'2.
700-
600
500
�00
•
'300
I
l
200
Experiment
Theoretical
d.
100
INtN£S
2.0
LO
Experimental and theoretical plot of values
for a 3.5 by 5 inch rectangular steel plate, .030 inches
thick using Poisson's ratio as .26
FIGURE 9
--- --- ---------------------·- -------r·-----
-------------------- ----- - - -------------------•----
-----·------------------------ ----------.-.
-- ----------- -------···--- ··-------�
34
400
300
•
2.00
l
' Experiment l
Theoretical
/00 �------�---b--� �
/.0
2.0
INCHES
Experimental and theoretical plot o£ values
£or a 5 by 5 square aluminum plate, �030 inches thick
using Poisson's ratio as .36
FIGURE 10
35
"700
600
500-
400
300
I
2oo
•
IOO
. J
E"A:per1mental
Theoretical
....-�----..
--'---------......!...
�-�
,
cl
2. 0
lliC!iSS
f. 0
Experimental and theoretical plot of values
£or a 3.5 by 5 inch rectangular aluminum plate, .030
inches thick using PoissonTs ratio as .36
FIGURE 11
36
1oo
Soo
2oo
o
. J
ExperJ.mental
Theoretical
Experimental and theoretical plot o£ values
£or a 3 by 5 inch rectangular aluminum plate, .030 inches
thick using Poissonfs ratio as .36
FIGURE 12
37
700
600
5oo
.qoo
300
•
j
Experimental
Theoretical
200
I .o
Experimental and theoretical plot of values
for a 2 by 5 inch rectangular aluminum plate, .030 inches
thick using Poisson1s ratio as .36
FIGURE 1 3
Appendix B
Theoretical Equations
Below are listed the equations developed by using
the minimizing conditions as defined in the main body o£
this thesis.
Although these expressions would lend
themselves to computerization, such a course was not
followed £or this study.
The application o£ these
equations are quite general subject to the constraints
discussed in the conclusion section o£ this thesis.
[ k1i
SI/!A
5
a b
A (--;:[4
+
3
8abJ1.(S
8
[ k�
16ab
16ab
11-
5
�)(1
kJr
k41T4
2
cos
2
kir sin k"tt2
+
1
sin
kn-
kit
-+
2
(sin
3
32u b
(- +
b
3 a
k-rr-
k�£_!__
2
+
3 3
2
16ua b
sin
6 6
k 1r
2k
+
1
k.;
kn.
2
. 2
22 s1n -"2"
k--rt3
k-rr
�) sin 2
-+
+
sin
klT)
�
2
k1t")
�k J - 11""cos
2
-
�
kit
2
c
+
�:?
+ b)
(a
3
2
3 uk b
-·
-{-·
a
11'3
+
sin
kit
2
--
kll>
16kab
sin ------3n:�2
+
3
a)
k-nsin
-2
b
+
128u (b
a
·n- 4
"----�
SI/6B
�-rr+
2
Sln
·
�
16ab.
O. .
-- (s1n
-·
k"t.-
If'
+
6
256uk
�.33.1- - 4 abD..(1
3
2��(.!.?
+ b�)
3 a
3
-···
sin
+
2
3
-- (a
3
k'll"
�
+ b)
128u (b
rr4 a
--·
256u b
(
5 a
k1t'
· -�
8k
-�
2
3n-
3
�
·--
3
+
+
3
2
·-·
�J-
k-rr 2
41<:
3
LID.
----1
��4+
ab
2 231<4+
- k + 9
0
64kab
1<17"'
"'3::;:;sin 2·-·
+
sin kn2
cos
cos
4
2
256uk k + 29}5_)
1
16abD.(- - ab
2
3
1 8
3
-a- ) cos !'5:!!:
2
b
+ ba )
3
3
a
sin kn2b)
--�
k11"'
2
S I/SC
A
a
b
-· )
3
32u b
3 a
kll-
+ �-�- ( -·- +
�E: t!
3
32u b
+ 1<;3(a
3
1<
_321.:
. (!:?_
1'1'3
a
+
+
3
�) sin k-rrb
2
8ab�·(
k
D.. .
s1n t�
1T2 fik
�
2
-·
�
J
j
3
a
k1l"'
b) sin 2 +
5
+rr- 2
k
4k
3
40
B
r� 28k4 (b:3
L
+
a )
b3
+
:1. �81<_:6(_1:?--3
�3
...
-·
+
a )
�3
+
256uk
ab
4
--�
+
.•
Appendix C
Alternate Approach
A more expedient method of solving for eigenvalues
has been developed and presented in an article by Cox
and Boxer (ref. 9).
The technique used to develope
tabular values of frequencies for rectangular plates
supported at the corners was based on a finite difference
approach and digital computer methods.
Since this thesis
was originally intended to be based on this approach, a
brief description will be made of their presentation and
the reasons for not_proceding with this method delineated.
Cox and Boxer made the usual assumptions as to the
neglection of shear �ffects and rotatory inertia effects
Poisson's ratio of .3 was used
on the plate response.
ln their analysis of the rectangular plate response.
This
is an average value for most common materials but does not
have to be;assumed but could in fact be inserted as data
in the computer program.
was:
t�
v
2
"
v
2
W
e
.
h
The general motion equation used
2
"4:1'
- --n
W =
O
where the parameters are as defined ln the body of this
.
-
thesis.-. This- -equation is then written in difference form
for a series of nodals which disect the subject plate into
a matrix of equally sized rectangles with the nodes at the
vertices.
The equation can then be written in matrix
..•
-
.
.
,-�-------------------:--------�---------·-----�-�-··----------,·--------·-···--····
II
i
.
.
.
4
.
-
( n A - /l . I ) W
form as :
l.
where I is a unit matrix, n
=
=
0
..
.
.•
a/h (h being the length of
each small rectangle), and W is a column matrix of
deflections and {t...
=
l.
.
'W"
=
n
2 4
� hto1 a
D
therefore,
D ) i/2
2 (� . 1/2 ( -)
1
� ha
4
This form is valid for various a/b ratios since the
column matrix satisfies these variations.
This method
was utilized to determine the eigen-frequencies and
dynamin displacements for several plate configurations.
For these tabulations see reference 9.
The accuracy of this method is generally adequate for
most engineering applications and can be made to converge
to classical solution values by using very small values
of h.
The only limitation to this method is .the capacity
of the digital computer in use.
The authors of reference 9 stated that the first
fundamental frequency calculated by their technique
approaches the values listed in reference
4 within a
fraction of a percent._ For higher resonant frequencies,
the accuracy becomes more degraded with the third or
fourth mode being a practical limit for their technique.
To extrapolate this technique to support conditions
other than at the corners required only a slight
modification to their technique.
Because this did not
represent a significant endeavor and theoretical values
;
had been experimentally verified, this approach was
discarded for the purposes of this thesis and the energy
approach substituted since this particular method had
not been extensively pursued.
J