#I f - -.. - - . . . . - -. . . - - NASA TECHNICAL NOTE NASA TN D-2831 -. (NASA CR OR TMX OR AD NUMBER) a. GPO PRICE S W PRICE(S1 $ 3h OB71 I Hard copy (HC) I Microfiche (MF) csf) I FREE A N D FORCED VIBRATIONS OF CANTILEVER BEAMS WITH VISCOUS DAMPING by Floyd J. Stanek George C. Marshall Space Flight Center Hzmtsville, Ala, NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. 0 JUNE 1965 https://ntrs.nasa.gov/search.jsp?R=19650017041 2018-06-07T13:08:19+00:00Z
Transcript
# I f
- -.. - - . . . . - -. . . - -
N A S A TECHNICAL NOTE N A S A TN D-2831 -.
(NASA CR OR TMX OR AD NUMBER)
a .
GPO PRICE S
W PRICE(S1 $ 3h OB71
I Hard copy (HC) I
Microfiche (MF) csf) I
FREE AND FORCED VIBRATIONS OF CANTILEVER BEAMS WITH VISCOUS DAMPING
by Floyd J. Stanek
George C. Marshall Space Flight Center Hzmtsville, Ala,
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. 0 JUNE 1965
Deflection of Cantilever Beam - Forced Vibration p = 5 . 0 a , = O . 2 ................................... 25
Deflection of Cantilever Beam - Forced Vibration P = i O . O a , = O . O ................................... 26
Deflection of Cantilever Beam - Forced Vibration P = i O . O a = O . 2 ................................... 27
Bending Moment in Cantilever Beam - Forced Vibration p = 5 . 0 a , = O . O ................................... . 2 8
Bending Moment in Cantilever Beam - Forced Vibration P = 5 . 0 a , = O . 2 ................................... 29
Bending Moment in Cantilever Beam - Forced Vibration ................................... p = i o . o a , = o . o I30
Bending Moment in Cantilever Beam - Forced Vibration p = i o * o a = o . 2 ................................... 31
Y +
It- + DEFINITION OF SYMBOLS
Free Vibration
77 1st Mode 2nd Mode
(a ) (b)
~ Forced
Vibration
7777777 r = Yo sin ut
FIGURE 1. CONFIGURATION OF A VIBRATING CANTILEVER BEAM
Symbol Definition
X axial coordinate of beam (in. )*'
L beam length (in. )
z = x / L dimensionless axial coordinate of beam
Y lateral deflection of beam (in. )
initial displacement of f r e e end of beam in free vibration (in. ) next page for definition in forced vibration).
(See top of
'iC These a re the most common set of units used in the U. S. A. ; however, any consistent set may be used as the formulas are presented in dimensionless form.
vi
4
Symbol
YO
W
I
E
g
K
Ld
P
C
t
8
h
a!
DEFINITION OF SYMBOLS (Concluded)
Definition
amplitude of lateral displacement of "fixed" end of beam in forced vibration (in. ) (See previous page for definition in f ree vibration) . weight of beam (lb)
moment of inertia of beam cross-section (in. *)
modulus of elasticity of the material of the beam (lb/in. 2,
acceleration of gravity. (in. /sec2) (taken equal to 386.4 in. /sec2 in sample examples ) 7-
=&p beam parameter in f ree vibration (sec-I)
circular frequency of excitomotor ( r a d s e c )
- - beam parameter in forced vibration (dimensionless)
coefficient of viscous damping (lb-sec/in. )
time (sec)
dimensionless time
Kt, in free vibration
u t , in forced vibration
characterist ic root; first five values tabulated in Section I1 (dimensionless)
damping factor (dimensionless)
cff in f ree vibration 2WK '
in forced vibration w w
vii
t b I '
FREE AND FORCED VIBRATIONS OF CANTILEVER BEAMS WITH VISCOUS DAMPING
SUMMARY
The fundamental theory for analyzing a n undamped vibrating cantilever beam is iresented in elementary texts on mechanical vibrations. The necessary formulas for a complete study of the state of motion and the behavior of a vibrating cantilever beam with viscous damping, however, have not been developed and presented fo r design.
This r e p o r t b o v i d e s the equations of motion for the free and the forced vibrations of a cantilever beam with viscous damping. The equations of motion include formulas for the bending moment, lateral shearing force, deflection, velocity, and the accelera- tion at any desired point of the beam for any chosen time. The coefficient of viscous dzmping is assumed to be c m s k ~ t thrcughcut the length r?f the h e m . !
_I.--.
'I&s assumed during f ree vibration that the free end of the beam is initially dis- placed some arb i t ra ry distance Yo and then released. The mode of vibration and natural frequency is defined by the value of a characteristic root. The first five values of this root are tabulated(in this report.
It is assumed during forced vibration that the normally fixed end of the cantilever beam igsubjected to a lateral displacement of the form y = Yo sin ut and that the zero slope is maintained. The equations of motion for forced vibration are for the steady-state condition only.,
The equations of motion are presented in dimensionless form in a convenient and usable manner. This report also contains the derivations, design curves to evaluate some of the parameters , and a set of dimensionless results for each of several sample examples. The intent of this report is to bridge the gap between theory and design.
SECTION I. INTRODUCTION
Necessary formulas fo r the state of motion for the free and the forced vibration of a cantilever beam are presented in this report. These formulas include necessary equations for evaluating bending moment, lateral shearing force , deflection, velocity, and acceleration at any desired point along the beam for any desired instant.
. > .
During free vibration it is assumed that the free end of the cantilever beam is initially displaced some arb i t ra ry distance Yo and then released. Natural frequency of the vibrating cantilever beam is dependent upon the mode of vibration, characterized by the value of a characterist ic root. The first five values of this root are included in this report.
During forced vibration it is assumed that the normally fixed end of the beam is subjected to a lateral displacement of the form y = Yo sin u t , where Yo is an a rb i t ra ry constant and o is the circular frequency of the excitomotor, and the zero slope is main- tained. Formulas for forced vibration are for the steady-state conditions only; that is, equations which define the state of motion after sufficient time has elapsed f o r the transient te rms to decay out of range of significant value.
Equations of motion a r e expressed in t e rms of dimensionless coordinates and parameters. These coordinates are axial z and t ime 8. These parameters are defined in the tabulation of the formulas for each type of vibration. One of these parameters is the damping factor a which is dependent upon the coefficient of viscous damping. This coefficient is assumed constant throughout the length of the beam. of uniform cross-section may be analyzed by the procedure presented.
Practically any beam
This procedure and the necessary formulas for each type of vibration are pre- sented in a convenient and usable manner in Section 11. Design curves fo r the evaluation of the beam parameters and the damping factor are presented in Section 111. Also, in Section I11 is a set of dimensionless resul ts for the deflection and bending moment for each of several sample cases. The derivation of the formulas is presented in the Ap- pendix. It is not necessary to understand this derivation to understand its application.
SECTION 11. APPLICATION
A. FREE VIBRATION
i. Procedure. The necessary formulas for the state of motion, bending moment, and lateral shearing force in a cantilever beam during free vibration are tabu- lated in this section.
The evaluation of these formulas is straightforward when done in a step-wise manner. A brief account of the physical aspects of some of these steps is given in the remainder of this section.
The free vibration of a cantilever beam is characterized by the mode of vibration. This mode of vibration is defined by the value of the characterist ic root A. The values of the first five roots are given in Formula 3. The first two modes of vibration are shown in Figure 1 parts (a) and (b) in the definition of symbols.
2
c
Values of the characterist ic roots are solutions of a characterist ic equation representing four boundary conditions; namely the lateral deflection and slope are zero at the fixed end, and the bending moment and lateral shearing force are zero at the f r e e end of the beam.
Only two parameters are required for the initial evaluation of the formulas; the damping factor CY (Formula 2) and the characteristic root A (Formula 3). The value of the beam parameter K (Formula I) is used to convert the dimensionless resul ts to physical quantities. This parameter ig a l so included in the definition of the damping factor a. The values of the parameters K and (Y may be obtained from the design curves given in Section III.
The values of A and CY determine the value of the vibration parameter y (Formula 4). The significance of this parameter is that the product yK is the natural c i rcular frequency of the cantilever beam for the mode of vibration characterized by the value of A. The value of y reduces to A2 when the beam is undamped (a = 0 ) .
The integration constant B (Formula 5) is one of two elements of a characterist ic vector corresponding to the particular value of the characterist ic root A. The other ele- ment (integration constant) does not appear in the formulas as it was taken equal to one.
The Z-function (Formula 6 ) is a function of the dimensionless axial coordinate z only. The value of the Z-function at z = 1 is the value of the constant Z , (Formula 7 ) . This constant appears in the T-function (Formula 8) ; a function of dimensionless t ime 8 only. The integration constants in the T-function were evaluated in t e rms of Z, by the initial conditions that the displacement is equal to Yo and the velocity (dy/dt) is zero at the free end of the beam when t = 0.
The product of the Z- and T-functions is the dimensionless deflection y/Yo (Formula 9) . The remaining resul ts , a lso dimensionless, are obtained by evaluating Formulas 10 through 13. A se t of dimensionless resul ts for the deflection and for the bending moment (Formulas 9 and 12, respectively) are given in Section 111.
The set of formulas f o r the f r ee vibration of a cantilever beam are now tabulated in the next subsection. The procedure and the formulas for forced vibrations are given after this tabulation.
2. Formulas.
Beam parameter
t
Damping factor
G i L 2WK C Y =
Characteristic root, A*
Mode A
I I. 87510 2 4.69409 3 7.85476 4 I O . 99554 5 14.13717
Vibration parameter
y = jT7-
Integration constant in Z-function
cosh A + cos A sinh A + sin A
B = -
The Z-function
Z = coshh Z - cos A z + B(sinh A z - sin A z)
The Z l constant
Z 1 = value of Z -function at z = I
= cosh A - cos A + B(sinh A - sin A )
The T-function
I ,ae T = - e ( y cos y8 + a sin ye ) YZI
( 3 )
(4 )
(7)
~ ~~~~~
yk A is the root of I + cosh A cos A = 0.
4
Deflection, y
-- - Z T Y O
Velocity, V
Acceleration, A
Bending moment,
M L2 -- - A2 [ cosh Az + cos Az + B( sinhhz + sin hz) ] T
Shearing force, Q
BL3 = A3 [ sinh Az - sin Az + B (cosh Az + cos Az)] T E IYo
B. FORCED VIBRATION
i. Procedure. The necessary formulas for the state of motion, bending moment, and lateral shearing force during the steady-state condition of the forced vibra- tion of a cantilever beam a r e given. in this section. The normally fixed end of the canti- lever beam is subjected to a lateral displacement y = Yo sin ut, but the zero slope is maintained. A steady-state condition is established- when sufficient time has elapsed for the transient te rms to decay out of range of significant value.
In the case of forced vibration, the beam parameter p (Formula 14) is dimen- sionless and equivalent to @ where K is the beam parameter for free vibration (Formula i in the previous subsection). The damping factor a! for forced vibration (Formula 15) is not the same as fo r f r ee vibration (Formula 2) however, it is sti l l dimensionless.
The parameters p and a! are the only two parameters required for evaluation of a complete set of dimensionless results. The value of the parameter p may be obtained from the design curves given in Section 111. The results are obtained in a step-wise manner. The physical aspects of some of these steps and the nomenclature used in this presentation are explained in the following discussion.
5
The parameters 0 and (Y define the four vibration parameters- $I, p, a , and b (Formula 16) . The vibration parameters a and b define the symbols si and t i ( i = I , 2 , 3 , 4) which, in turn, define the symbols Si j and Ti j (Formula 17) . The sym- bols S.. and T.. are used to evaluate the elements of a four by four matrix (Formula 18).
1.l 1J
The solution of this matrix equation evaluates the integration constants A , B, C , and D and in turn, evaluates the four integration constants E , F, G, and H (Formula 19) . Various linear combinations of these eight integration constants will be required later. The presentation of these combinations is explained below (in the following paragraphs).
The general form of the expression for the dimensionless deflection y/Yo is as follows
C y/y0 = zS sine + z case, S C where 0 is the dimensionless time ut, Z and Z are functions of the dimensionless
axial coordinate z only. The significance of the superscripts s and c becomes apparent when each is correlated with the s ine and the cosine t e r m , respectively.
S The Z function is defined as follows:
S Z = A cosh az cos bz + B sinh az s in bz +
C sinh a z cos bz + D cosh az s in bz +
E cosh bz cos a z + F sinh bz s in az +
G sinh bz cos az + H sinh bz s in a z ,
where the symbols A through H are the eight integration constants evaluated by Formulas 17 and 18.
S The expression fo r Z is transformed to the following matrix equation
+ [cosh bz sinh bzl NS S Z = [cosh az sinh az]MS
S S where the eight integration constants are grouped into the matrices M and N as follows
6
, C S The form of the Z function is the same as the form of the Z function, except fo r a different set of eight integration constants. This set is designated by the super- scr ipt c. The expression for the derivative dnZ/dzn, of any order n , is also the same form as the Z-function, except with a different set of eight constants. The second and the third derivative of Z with respect to z is required for the evaluation of Formulas 25 and 26. An integer subscript is added to provide the scripted symbols Z j M j and Ni, j
i' i' where j is s o r c and the integer i designates the order of the derivative with respect to z. This scripted notation on the matrices M and N also applies to each of the elements within the matrix.
The non-scripted symbols A through H defined by Formulas 18 and 19 are equiva- lent to the corresponding scripted symbols with j = s and i = 0. The value of each of the other scripted symbols (that is, the elements of the other matr ices M and N) is obtained by taking certain linear combinations of the original eight integration constants A through H. This linear combination is presented as the linear combination of two, 2 by 2 ma- trices (Formula 20) . This is illustrated below for the elements A t and GF (see matrix equation for M t and N! in Formula 20) .
S A, = A cos2+ + B sin2+ = A' cos20 + BE s in2+, 0
and, s im ilarly
C S S
0 0 G3 = F sin3# - E cos3+ = F sin3+ - E cos30
Af ter the evaluation of a l l the scripted elements (constants) , the desired result is obtained by evaluating the appropriate of Formulas 22 through 26; noting that the pay- ticular Z { function is established by substituting the corresponding matrices M i and N i into Formula 21.
The initial results are dimensionless as defined by the expression on the left of the equal sign in each of Formulas 22 through 26. The dimensionless t ime is converted to physical t ime with the relationship 8 = ut. A set of dimensionless values for the deflec- tion and bending moment fo r several examples are presented in Section 111.
The formulas fo r forced vibration of a cantilever beam a r e tabulated in the remainder of this section. The equations of motion presented in this section are derived in the Appendix,
7
I.
2. Formulas.
Beam parameter, p
Damping factor, a I a!= 9 2
WW
Vibration parameters
i $I =-pan-' (Y
p = 6 ( I + a2)li
( 16) a = p cos @
b = p sin @
Definition of symbols for the elements of the matrix to evaluate the first four integration constants.
st = cosh a ti = cash b
s2 = sinh a
~3 = C O S b
s4 = s in b
t 2 = sinh b
t3 = cos a
t4 = sin a
s13 = s1s3
s14 = s1s4
s23 = s2s3
Sa = ~ 2 ~ 4
Tl3 = t l t3
Ti4 = t l t 4
T23 = t2t3
T24 = t2t4
8
~~
The matrix equation
Integration constants E , F, G, and H
s c a n d N i . C ( i = 0, 2, and3) Matrices MS , Ni , M i , i
M2 S = c'os 2@ z] + sin 24 E
9
M3 S = cos 3@ I] + s i n 3 @ E 4 S
N3 = -sin 3@
( 2 0 Concluded)
j The Z . - function ( i = 0, 2, 3; j = s , c)*c
Z . j 1 = [cosh az sinh az ] M i [r: 1
+ [cosh bz sinh bz] NI [r: 11 (21 )
Deflection, y
C
0 0 LL = zS sin e + z cos e 0
Y
Velocity, V
S C
0 0 = Z cos 8 - Z s in 8 V
a yo
Acceleration, A
'le The non-scripted symbols in Formulas 18, 19,and 20 are equivalent to the scripted symbols when j = s m d i = 0.
10
Bending moment,
- M L2 E1 Y
= p2 sin e + zC cos e ) 2 2
0
Shearing force, Q
C
3 3 QL' = p3 (zs sin 0 + z cos e)
E1 Y 0
SECTION ID. SAMPLE RESULTS
A set of dimensionless values for the deflection and bending moment of a vibrating cantilever barn are presented h- this sectinn; Design curves for the evaluation of the beam parameters K and p and for the damping factor a for free vibration are also given. The use of these curves is illustrated in each Figure.
The set of dimensionless resul ts fo r the cases of free vibration immediately follow Figure 4. These cases are for the f i r s t two modes of vibration (first two values of A) for each of the damping factor 01 equal to 0 and to 0 . 2 . These resul ts were obtained by evaluating Formulas 9 and 12 of Section I1 and are presented in Tables iA through 8A.
Similar data fo r the cases of forced vibration are presented after Table 8A. These cases are for the beam'parameter p equal to 5 and to 10 for each value of the damping factor (Y for forced vibration equal to 0 and 0.2. These resul ts were obtained by evalu- ating Formulas 22 and 25 of Section I1 and are presented in Tables i B through 8B.
The format of each table is the same; with the dimensionless time coordinate 8 ( in angular degrees) in the first column and with the dimensionless axial coordinate z as columnar headings of the remaining columns. In the case of forced vibration, it is recalled that the formulas are for the steady state condition only. That is, the point of zero time ( 8 = 0) is merely a beginning reference point of a repetitious cycle. This explains why the results a r e not zero everywhere along the beam at 8 = 0 when the beam is subjected to viscous damping ( a ) # 0) . The derivation of the formulas in Section I1 is given in the Appendix which follows Table 8B.
I1
.
FIGURE 2. THE BEAM PARAMETER K
1000
900
800
700 a $3 0 0 Q) m k
a m aJ
Q) 600
4
5 9 ' 500
3 2 0
u bD 400 E: x 0 x w
*I+
3 OC
20(
101
30 sec-l
Exciting frequency = 250 cps
K = 12 sec-1 i Read: = 11.5
I 5 10 15 20 25, 30 35 0
p - Dimensionless
FIGURE 3. THE BEAM PARAMETER P
13
0.5
0.4
0. 3
0 .2
0.1
0 0 -
0. 05
0.10
0. 15
0.20
0.25
/5/
Data for illustrative example:
C = 0.15 lb-sec/in.
W = 20 lbs
K = i o sec-l
Read: Q = 0.145 . I I
FIGURE 4. THE DAMPING FACTOR (Y FOR FREE VIBRATION
14
TABLES
15
0 b
0-4
0' b
co b
r- b
9 b
Ln b
4 . n b
(v . .-I b
0
0
-
m m m a m m b b
N m
coco a m b b
0 0 N O r-r- . . 0 0 o'r- m m b b
. 4 m 9 Q 4 4 b b
m r - m m m c u b b
m N N N N N b b
9.4 m m .44 b b
m . 4 99 0 0 b b
99
0 0 A 4
b b
0 0 0 0 0 0
3 0 b b
9 r 9 0 c o r b .
9(T Q O F Q b b
cow N d a m b .
.4r- m a AI+ . . (T9 (TN m m b .
4-0 @if N N b b
m N m a 4 4 b b
C0.0 I + m A 0 b .
m m m a 0 0 b b
Q . 4
0 0 .4d
b b
0 0 0 0 0 0
0 0 b b
oco o m L A N b b
. 4 m m N Q N
* e
Nr- 9 C O mI+ b b
LnN m m N 4 b b
o m N.4 m d b b
mr- 9 0 3 40 b b
4 - m d m 40 b b
coo a m 0 0 b b
4 9 m d 0 0 b b
c o 4 0 0 0 0 b b
0 0 0 0 0 0
0 0 b .
Oco o m O N
0 1 b b
o m O N O N
0 1 b b
or- 0 0 3 0.4
0 1 b b
O N o m 0.4
0 1 b b
o m 0.4 0.4
0 1 b b
or- o m 0 0
0 1 b b
o m o m 0 0
0 1 b b
o m o m 0 0
0 1 b b
0 9 04 0 0
0 1 b b
o i f 0 0 0 0
0 1 b b
0 0 0 0 0 0
0 0 b b
mr- m o Q r - I I b b
.4m m o 4-9
I I b b
N N 9.4 m m
I I b b
mr- m 4 N *
I I . . 09
r u m
I I
m r u b b
m o 9 i f 4 N
I I b b
Q N Pta
I I
.4d b b
a 3 9 a m 0 0
I I b b
dLn m i f 0 0
I I b b
a).+ 0 - 0 0
I I b .
0 0 0 0 0 0
0 0 b b
o m 0 9 00'
I I b .
Cvm a m c o c o I I b b
m o N O r r I I b b
0 0 m r m l n I I b b
d m 9 i f 4-4
I I b b
m r m r v m m b b
I I
m N N N N N
I I . . 9.4 m m
I I
4.4 b b
m 4 99 0 0
I I b b
99
0 0
I I
4 4
b b
0 0 0 0 0 0
0 0 b b
oco o m O N
0 b b
o m O N O N
0 b b
or- oco O d
0 b b
O N o m 0 4
0 b b
o m 04 0.4
0 . .
or- oco 0 0
0 b b
o m o m 0 0
0 b b
o m o m 0 0
0 b b
09 0 4 0 0
0 b b
o * 0 0 0 0
0 b .
0 0 0 0 0 0
0 0 b b
mr- m o iff- b b
.4m m 0 4.0 . . N N 9.4 m m b b
m r m . 4 N Q b b
O Q m N r u m b b
m o 9 Q .4N b b
\ t N 49 4 4 b b
a 9 90' 0 0 b b
. 4 m m a 0 0 b .
c o d 0 4 0 0 b b
0 0 0 0 00
00 b .
16
- 0 b "
o' b
Q3 b
IC b
Q b
m b
it b
m b
N b
d b
0
0 b
- z -
0'9 0 ' Q Do' b b
NIT Q m c o c o b b
n o N O r-r- b b
0 0 o'r- m m b b
dln Q i t *it b b
o'co m N m m b b
o'N P J N N N b .
Q d m m d " b b
m d 9 9 0 0 b b
Q Q
0 0 " 4
0 .
0 0 0 0 0 0
0 0 b b
a * Q d cor- b b
c o 9 i t 4 r - 9 b b
o'co N d Q m b b
m N d N m * b b
00' O N i t m b b
i t w o ' f N N b b
w i t o 'Q 4.4 b b
cor- d0' 4 0 b b
m m m * 0 0 b b
i t 4
0 0 4 d
b b
0 0 0 0 0 0
0 0 b b
r - d do' O N b b
9" .fm * N b b
In" r - d m N b b
m N or- m " b b
c o i t m m N " b b
m c o I C 0 d o b b
c o 9 d 9 d o b b
00' i -m 0 0 b b
m c o m " 0 0 b b
c o i t 0 0 0 0 b b
0 0 0 0 0 0
0 0 b b
N m m c o 0 4 b b
I
Q c o i t m 0 -
I b b
r - m m m 0 4 b b
I
o m m o O d b b
I
itit N c o 0 0 b .
I
r - N t - 4 9 0 0
I . b
4 N d * 0 0
I b b
r - m O N 0 0
I b b
m d O d
9 9 I
o m 0 0 0 0
0 1 b b
0 0 0 0 0 0
0 0 b b
o ' m o ' m m m I I b b
i t N 4-0 m l n I I b b
O N OIN N i t
I I 0 .
Q i t
N m I I b b
.fa, 0 3 9 d N
I I b b
m r - m 0 '
I 1
l - 4 4 b b
dit 0 ' m 0 4
I I b .
Q c T mr- 0 0
I I b b
mr- N m 0 0
I I b b
Go' 00 0 0
I I b b
0 0 0 0 0 0
00 b b
"r- N O F a
I I b b
N 9 No* 99
I I b b
m m N c o m m I 1 b b
9 9 Nr- e * I 1 b b
N N m r - n m
I I b b
*it e r - N N
I 1 b b
l n m aco
I I
d d b b
coo
0 "
I I b b
a d i t m 0 0
I I b b
N m
0 0
I I
" d
b b
0 0 0 0 0 0
0 0 b b
Q F m o coco I I b b
t-49 No\ r -9 I I b b
99 oca 9 m I I b b
itr- Ql(c it*
I I b b
m n J cor- m m I I b b
m * c o r - N N
I I b b
N u l 0 ' c o
I I
d d b b
4.0 d " d d b b
I I
m d m u 3 0 0
I I b b
i t m
0 0
I I
d l - 4
b b
0 0 0 0 0 0
0 0 b b
mr- No' r - m I I b b
Q L n Nt-4 .Om I I b b
Q m N m m i t I I b b
c o m N m i t f m
I I b b
4 - m m r - m N
I I b b
Q N i t 0 N N
I I b b
9 r - Q m
I I
"d b b
mt-4 crco 0 0
I I b .
Q C O * m 0 0
I I b b
N O
0 0
I I
t-4-
b .
0 0 0 0 0 0
0 0 b b
N m m a Q N
I I b b
m o r - d r n N I I b b
C Q Q dr- m d I I b b
m m m * Nt-4
I I b b
o ' N cT.-(
I 1
d d b b
. O N i t a 4 0
I I b b
0 ' m c T m 0 0
I t b b
0 ' m m m 0 0
I I b b
r - m N d 0 0
I I b b
Fit 0 0 0 0
I I b b
0 0 0 0 0 0
0 0 b b
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i f c c c \ lN I C 0
I C ' . .
I
O N A D i f a
I I
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or- i f c o
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m a * . d r '
R
8 d m a w u p: 0 crr
2 c w F4 p: w > w
z
R 0 R
u w I4 kl
5
I
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8
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r- .
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0
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b .
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0 b .
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b b
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. b
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b b
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. b
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or - 0 9 or( 0
b b
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0 . b
~
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b b
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0 1 . b
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b .
oco o m o m 0
b .
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0 b b
O N 0 4 o m 0
e .
0 9 0 0 o m 0
b .
0 0 o m 0 0
0 b .
or- 0 9 0 4
0 1 b b
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0 1 b b
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- 1 b b
I
o m 4 - m a4- I I b .
4 4 - 4
4-N 0 0
I I . b
r O ~m d r l
b b
O Q
o m m m . . rl
m N a 0 4.0
. b d
O N D d O m
b b
d Q D O m m . . coo D m 0 0
b b
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I I b b
24
0 . d
cr .
ca b
r b
9 .
m . 4
b
m b
N . d
b
0
0 b
$ 0 r m 4-0 '
b .
d d
m N P - 0 r o
b b
4
c o d 0 0 d d
b b
m c o a m 4 9
0 .
I I
Q C O t-r w 4 I 4 0 .
I
d c o r c o o m
I I
c o r O N
I I
c r m 0 0 cog\
I I
b b d d
m r b b
d d
b b
0 0 l -0 4-3
I I b b
Q1c Q m 4 0
I b b
oco o m O N
0 b b
m w m w o m N d
b b
com r Q c r r
0 .
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r. cr o m w 9 I I 0 .
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I I
0 .
4.4
m . 4 9 4 - m o
. b
d 4 I I
a m or- d l
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r C N O N N m
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m m m a m c r
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~
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b . 4.4
m N r o P O
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I
W d 0 0 d d
b b
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m c o a m 4 a
0 .
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b b
r-l
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0 .
d d
c o r
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b b d d
c r m 0 0 coo,
b .
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b b
Q F 4 - L n d o
b b
I
oco o m O N
0 1 b b
mcr c r r nJ4- N N I I
b .
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0 .
4 4 I I
coco coa 0 0
I I b b
P O 00 COG- . . d r - O N 4 m
b . 4.4
d m 4 N at- . . 4 4
( 3 4 - N c o 4 4
b b d d
m m O D m o
b .
m m oco mr-4
0 .
c o d m Q N i f
I I b .
c r r cro 4 r
I I 0 .
m N m w o m N 4 I I
w m P-a mt- I I
0 .
b .
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0 .
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0 .
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0 .
A d
m d 9 3 m o
0 .
d d
d Q a m or-
b *
d
9 0 r m Q N
0 .
P - N O N N m
I I b .
O N r c o r r
I I b .
o m 0 9 ocr . - . I 1
b b
I
25
- 3 . -4
0' .
m .
IC
9 .
m .
4- .
m .
N .
-l .
0
0 . - 7 -
300 3 0 3 m
3 1 0 .
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0 1 0 .
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04. 0 0 0 0
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0 0 .
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0 * .
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0 0 .
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0 .
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0 .
m o m l n 0 0
I I 0 .
m N Q N m m
I I * .
N N 4 - m m I I e *
c o d o.+
I I ? ?
+ l c 0 m m m
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~~
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m o N O r-r- I I 0 .
99
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0 .
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0 .
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~
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a m 04- m c o
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m o m m 0 0
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m n J 4- r - mr- I 1 0 .
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c a m 9* m m I I * .
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m d o d o r . . N d
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e a d d
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d l 0 .
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e .
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r- .
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m .
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0
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0 .
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0 .
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0 .
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m a m Q i n N
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m N m o "co
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in&
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0 .
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I 1 0 .
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0 .
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0 .
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0 .
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0 .
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0 .
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0 .
I
m.3- o m Nal
0 .
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0 0 0 .
d m A 9
0 .
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4 "
r - 4 Q o L n m m I I 0 .
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m r - Q Q m N
I . + . .
I
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I
or - o m P O
I " 0 .
I
m d - m4- 4 - m I I 0 .
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0 .
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0 . 4
m d o w 4 - 0 3
0 .
d 4
0 0 o c 00
0 0 0 .
CD" m 0 "
I I . .
cPco l n m I C 9
I I 0 .
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I t
0 .
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a 3 0 o w m m
d d 0 .
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0 .
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1 1
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d l I
0 .
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0 .
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0 . A"
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0 0 c o 00
G O . . o m L n c o " 0
I 1 0 .
m a r-r- il'N
I I 0 .
coco c o i f I C *
1 1 0 .
N o L n d D i n
I I . . r-6 a 3 3 ad-
I I . .
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I I 0 .
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0 .
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0 .
m r 4 m c o d - . . m a 4 - m 0 4 -
0 . r 4
- 0
-c
* .
co .
r- .
Q .
m . .f . m .
N .
-4 . 0
0 . - x -
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0 0 0 .
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0 1 0 .
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0 1 0 .
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0 1 0 .
o m 0-4 0 0
0 0 .
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0 0 .
0.f oo' 0-c
0 0 .
o m O N 0 0
0 0 .
0 9 0 4 - 0-4
0 1 0 .
o m O N O h
0 1 0 .
oco 0 9 0-4
0 0 .
0 0 0 0 0 0
0 0 0 .
O N 4 - w m N
I I 0 .
4 - m m-4 I C 9
I I 0 .
c o r O D 9 4 -
I I 0 .
m r - 4-m 0 0
0 .
m m 4-N a m
0 .
A N m m Q U I
a .
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0 .
o'CP coo'
I I
4 - m 0 .
o m N e
I I
. f m 0 .
N c o Q m 0 4 -
0 .
0 0 0 0 0 0
0 0 0 .
o m 0 0 0-4
0 0 .
o m O N O N
0 0 .
0 4 o w 0-4
0 0 .
o m O A 0 0
0 1 0 .
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0 1 0 .
0 4 - 00' 0-4
0 1 b .
o m O N 0 0
0 1 0 .
0 9 0 . f 0-4
0 m .
o m O N 0 4
0 0 .
oco 0 9 0-4
0 1 0 .
30
. . - 0 . 4
OI .
W
r-
9
m
it .
m .
N
4 .
0
0 . - z -
0 0 0 0 0 0
o c * .
Nr-
4 0 .or . . 0 0 ccm m c r
0 .
C P A F N - 0 . . r-m m m 4.4
0 .
I I
d - N ea2
I 1
m c r 0 .
Q O Q c r A 0 * . I
O F A d - " . . NOI r - d - mcJ
e .
0 0 mr- 0 0 * .
I
r - 0 a m a m I I m .
0 0 0 0 G O
0 0 . .
m m A 0 O A
I I a .
IC.-( 6 9 O N
I I 0 .
COW
.-(N
I 1
m a , . . O'd- m.-c A A * . I I
Fr- 0 9 0 4 e m I
I C 0 a m A m * .
r-0' 99 "
e .
0 9 4 m A 0 * .
I
r - 4 coo A N
I I * .
c o d m a m 4 I I * e
~
0 0 o c G O
0 0 . e
CGO 0 3
I I
m m . . O I C O c0.t at- I I . e
r-4. a 4 m 9 I I 0 .
39 o m 0 0
b .
I
4 0 c o r n . . N m 9 c o 99
e .
0 0 r - - 4 4 . . 4-43 N O d - m I I * .
m 4 99 i t d -
I I . . 09
a m m o \
0 .
0 0 c o 0 0
0 0 . e
a m 3 m m m
I I * .
~n lncl fir.
I I . .
CGN OId- l n m I I . .
i t r -
0 - r o
e .
m i t r m 9 9 * .
r - a ma, a m
e .
N c o 3 N 0 0 e -
l
i t m m a m m I I * .
~m N 9 i t m
I I * .
mc\l .-la2 lcr * *
0 0 o c c c c c . .
m cr: 4 0 c - * .
t - - 4 99 C N
m .
c c c c mcc .-IN * .
OId- m d d d . . r - r 0 9 O A
I . . r o mLn d m I 1 . .
r m a.0 N N
I I . .
0.0 4 m 4 0
a . I
r - 4 a2D A N * .
c o d m a mcc
e .
0 0 c c c c o c . .
L n l n 0: L n A N . . r - m mCO d - 0 . . O I d A N 4 - m . . m m c o d - c c
0 .
A N m r m 3 I I 0 .
cp- (re i t m I I . .
N O \ m . 4 N N
I I * .
d r n c o d A m . . a 3 r - m m d -
a .
9 - 3 m O N
1 1 . .
0 c c c c c c c . .
coo 0 3 m m
e .
DCO c@* at- . . rd- cccr m Q * .
i t a o m 0 0 . .
I
A G C C L n Ins
I I . .
~m a m Q-I?
I 1 a .
0 0 r4 - 4 . .
I 1
d - 9 N O d-Ln
a .
mcc 99 4-4 - * .
09 m a d - L n
I I b .
31
A PPE NDJX, DERIVATIONS
A. INTRODUCTION
The derivations of the formulas presented in Section I1 are given in this sec- tion. These derivations stem from the solution of a differential equation whose develop- ment for undamped vibration is presented in texts on elementary strength of materials and mechanical vibrations. The presentation in this report treats primarily the develop- ment of the formulas from the solution of this differential equation.
B. INITIAL DIFFERENTIAL EQUATION
A brief account of the development of the initial differential equation f o r a vibrating cantilever beam is given in this section. This development begins with the equation of equilibrium and the moment-curvature relationship written below. xc
I. Equilibrium Equations
aM - = Q - 2. Moment-Curvature Relationship
In the above relationships, Q is the lateral shearing force and x i s the bending moment in the beam. The symbols m and c are the mass and the viscous damping coef- ficient per unit length of the beam, respectively.
tions i and 2. This yields The initial differential equation is developed by eliminating and Q from Equa-
The formulas for each type of vibration are developed from a solution of this dif- ferential equation (Eq. 3). The first step in this development is the transformation of Equation 3 to dimensionless form. This transformation is slightly different for each type of vibration. This transformation and the subsequent derivations for each type of vibra- tion is given in separate subsections. The formulas for free vibration are developed in the next subsection.
These relationships for the undamped vibrations of a cantilever beam are developed and presented by W. T. Thomson, "Mechanical Vibrations, f f Prentice-Hall, Inc. Chapter 6. In this report, the positive direction of the coordinate y is changed.
1948,
32
C. FREE VIBRATION
The derivation of the formulas for free vibration is given in this section. This derivation begins by introducing the following coordinate transformations and symbol definitions.
2 - z k cff_ WL3 , ‘‘=2wK K - (4)
0 = Kt.
The above definition of K and a! is the same as Formulas 1 and 2 of Section 11.
The definitions and coordinate transformations in Equation 4 transform Equation 3 to the following form.
The solution of Equation 5 is obtained by the method of separation of variables, that i s , the deflection y is taken to be of the form
y/Yo = ZT, (6 )
where Z is a function of the dimensionless axial coordinate z only, T is a function of the dimensionless time 8 only and’Y is an arbitrary constant.
0
Substituting Equation 6 into Equation 5 and separating the variables yields the following two ordinary differential ,equations.
d4Z dz4 - - A d z = 0,
d2T + 2 - dT + A4T = 0, -- de dtl
where A is a constant, which will be evaluated la ter from the specific conditions imposed upon the vibrating beam.
Equation 7a is written below in a form which satisfies the boundary conditions that the deflection and slope a r e zero at the fixed end of the cantilever beam; in other words, z = dZ/dz = 0 at z = 0.
Z = A(cosh A z - cos A z ) + B(sinh A z - sin A z ) . (8)
33
The remaining two boundary conditions are that the bending moment and lateral ’ shearing force a r e zero at the free end of the beam. This corresponds to setting the second and the third derivatives of Z with respect to z equal to zero at z = 1. These conditions yield the following matrix equatiqn.
cosh h + cos h
s inhh - s i n h cosh + c o s h
s inhh+ s in A] I] = 0. ( 9 )
A non-trival solution of Equation 9 is obtained only when h is the characterist ic root of the coefficient matrix. Equating the deteriment of this coefficient matrix to zero, yields the following expression for the evaluation of the characteris tic root.
1 + cosh A COS A = 0. ( 10)
The first five roots of Equation 10 are tabulated in Section I1 as Formula 3. Accuracy greater than eight significant digits is required to obtain the higher roots be- cause of the characterist ic behavior of Equation 10 at these higher values. vibration is characterized by the value of the characterist ic root, A. a r e shown in Figure la and l b in the definition of symbols.
The mode of The first two modes
For each characterist ic root there corresponds a characterist ic vector whose elements are the integration constants A and B. is arbitrari ly taken equal to one. of Equation 9 ; this yields:
The value of the integration constant A The integration constant B is evaluated from the first
cosh h + cos h sinh A + sin h
B = - *
Equation 11 is Formula 5 of Section II. The evaluation of A and B completes the definition of the Z-function in Equation 8 and is Formula 6 of Section 11. Z , (Formula 7 of Section 11) is the value of the Z-function at z = 1 and will be used later. This discussion continues with the solution of Equation 7b.
The constant
The general solution of Equation 7b is
T = e - ( E c o s y 8 + F s i n y O ) , (12) ae
where E and F are integration constants, and y = 4 h4 - a2. During this solution it is assumed that h4 is greater than cr2 which must be t rue in order to have oscillatory motion.
It is assumed that the initial conditions for free vibration are that the deflection is equal to an arbitrary displacement Yo and that the velocity is equal to zero at the free end of the beam when t = 0; mathematically,
34
and
when 6 = 0 and z = I. y =Yo)
*= 0, de
Imposing these conditions upon Equaxion 6 yields
E = l /Z i ) and F = C ~ / Y Zi . (14)
Substituting Equations 14 into Equation 12 completes the definition of the T-function; that is,
- ae
Y Z I e T=- (Y COSY e + Q! sin re) )
which is Formula 8 of Section 11.
Introducing the above definitions of the Z-and the T-functions into Equation 6 com- pletes the derivation of Formula 9 of Section 11. The derivation of the remaining formulas for the free vibration of a cantilever beam (Formulas 10 through 13) are developed in a straightforward manner by taking the appropriate derivative of Formula 9. The defini- tions and the corresponding coordinate transformations for this are tabulated below.
,Velocity
Acceleration
Bending Moment
T & = g ! & -~ '0 - d2Z ax2 L~ az2 L2 dz2
- - M = E I
Shearing Force
T -&Yi=EI a3s - _ _ _ - ' 0 d3Z ax3 L~ az3 L3 dz3
- Q = E1
Formulas 10 through 13 a re verified by substituting and performing the operations indicated in Equations 16. These operations are elementary and Straightforward.
35
This concludes the discussion on the derivation of the formulas for the free vibia- tion of a cantilever beam presented in Section 11. The formulas for the forced vibration of a cantilever beam is given in the next subsection.
D. FORCED VIBRATION
The formulas for the forced vibration of a cantilever beam a r e derived in this section. This derivation begins with the initial differential Equation 3.
In the case of forced vibration, it is assumed that the normally fixed end of the cantilever beam is subjected to an exciting displacement of the form y = Yo sin u t , where Yo is a constant and w is the circular frequency of the excitomotor. sented in Section I1 for this type of vibration pertain only for the steady state condition. This condition is characterized mathematically by a particular ra ther than by the general solution of Equation 3.
The formulas pre-
The initial step for the derivation of this particular solution is the transformation of Equation 3 to dimensionless form, by introducing the following symbols and coordinate transformations.
c = - C L ' , e = u t , W x = z L , m = -
gL
a = Q . WL3 u2 -- - W 2 - '' EIg K2 ' WW
The above definitions for p and a are Formulas 14 and 15 of Section 11, respec- tively.
Imposing these definitions and transformations upon Equation 3 yields
= 0. (18)
A particular solution of Equation 18 is assumed to be of the form
(19) S C Y/Y = z sin o + z cos e ,
0
S C where Z and Z are functions of the axial coordinate z only and Yo is a constant.
Equation 19 is a solution of Equation 18 provided that the Z-functions satisfy the following two fourth-order ordinary differential equations simultaneously.
36
d4 Zs C - p b ( Z S + a ! Z ) = O ,
- - d4 Zc p (ZC - a!z S ) = 0. dz4
The following eighth-order differential equation is developed by combining Equa- tions 20 to eliminate zC.
d8Z -2p- d4Zs + P ( i + a * ) Z S = o . dz8 dz
The following symbols are introduced for writing the general solution of Equation 21.
i p = p ( I+ 0 2 ) (p = 4 tan-1 a!
a = p cos q5 b = p sin q5.
The definitions in Equations 22 are Formulas 16 and are determined by the roots The general solu- of the auxiliary equation of the eignth-order differential Equation 21.
tion of Equation 21, in matrix form is
cos bz
s in bz
S Z = [ cosh az sinh az] MS [ ] + [ cosh bz sinh bz] NS
where the eight integration constants are presented as the elements of the following two matrices M' and N'.
where the single superscript on the right is applicable to each of the elements in the matrix.
The form of the Zc function is identical to the form in Equation 23, except for a different set of eight integration constants. This set is designated with the superscript c. In order to satisfy Equations 20 simultaneously, one set of the eight integration con- stants must be dependent upon the other set. To show this and then to evaluate the re- maining eight integration constants it will become necessary to develop the first four derivatives of the Z-functions.
37
The form of the derivative of the Z-function, of any order , is the same form as Equation 23, except with a different set of eight constants. and the order of the derivative are designated with an integer subscript on the matr ices
M and N. This creates the scripted symbols Z . , M
i = 0, I, 2, 3, o r 4. scripted symbols when j = s and i = 0. relationship is for both as is the case in the following discussion.
These other sets of constants
j j j 1 i’ 1
The non-scripted symbols developed above are equivalent to the The superscripts s and c are omitted when the
and N., where j is s o r c and
The elements of the sequential matr ices (corresponding to the sequential deriva- t ives) a r e obtained from a recurrence formula. Two transformation matr ices , desig-
are introduced f o r writing this recurrence formula. natea with the symbols J and K,
J = [ 1, a n d K = -O -‘I. I O
The recurrence formulas for the sequential matrices M and N are written below.
M. = a J M i-1 + bMi-lK
N. = bJN
1
( i = I, 2, 3, 4) . + aNi lK 1 i-1 -
The elements of the matrices for i 2 2 are expressed in t e rms of the matrices Mo and No by a sequential substitution process. This process is illustrated for the first of Equation 26 with i = 1 and 2.
M i = a J M + b M K ,
Mz = aJMl + bMiK . 0 0
Before substituting the first into the second of Equation 27, the following proper- tv of matEices J and K is recognized.
where I is the identity matrix of order two.
Accordingly,
M2 = (a2 - b2) Mo + 2abJM K, 0
which, upon substituting for a and b from Equations 22 and using known trigonometric identities, transforms to the following form
38
M2 = p2 (cos 2@ M + sin 2@ JM K) . 0 0
The remaining matrices a re developed in a s imi l a r manner. The final form of matrices Mi and Ni ( i = I, 2 , 3 , 4) are tabulated below.
Mi = p (COS @ JMo + sin @ MoK)
Mz = P ~ ( C O S 24 M + sin 2$ JM K) 0 0
M~ = $(COS 3+ JM + sin 3+ M ~ K ) 0
M4 = p4(c0s 4c$ M + sin 4@ JM K) 0 0
N, = p ( s in @ J N + cos @ N K)
N2 = p2(-cos 2@ N + sin 2@ J N K)
0 0
0 0
N3 = - p 3 ( s . - ;I, 3@ J N + C 9 E 2j+ N E() 0 0
N4 = p4(c0s 4@ N - sin 46 JNoK) . 0
Any derivative of the Z-function is obtained by substituting the appropriate matrix The order of the derivative is designated by the integer M and N of Equations 30 into 23.
subscript .
The above relationships are used to evaluate the two sets of eight integration con- stants; that i s , the s-set and the c-set. The eight integration constants in each set are designated by the capital letters A through H and are presented as the elements of the matrices M and N; each with its corresponding superscript. The location of these con- stants within the matrices is as shown in Equations 24.
The integration constants in the c-set are expressed in t e rms of the s-set by substituting Equation 23, with the appropriate definition in Equation 30, into the first of Equations 20. This yields
C S N = - J N K, MC 0 = JM:K, and 0 0
or , in expanded form
39
The set of matrix equations in Formulas 20 in Section I1 is verified by substituting Equation 31 into the appropriate definition in Equations 30 and expanding the matrices and upon noting that the p2 and p3 t e rms are omitted in Formulas 20 as they are included in Formulas 25 and 26.
The eight integration constants in the set are evaluated by applying a set of four These conditions are tabulated be- boundary conditions to each of the two Z-functions.
low.
a t z = O
zs = 1, zc = 0
- - - 0 dZC dz
- - - 0 , dZS dz
d2ZC - = 0 1 d2ZS - - dz2 - '3 dz2
a t z = i - - d3ZS - = o J d3ZC
dz dz3 - '3
The f i r s t two of Equations 32a ensure that the displacement at z = 0 is of the form y = Yo sine, where Yo is the amplitude of the exciting displacement, (see Equation 19). The second two of Equations 32a ensure that the slope dy/dx is zero at z = 0. Equations 32b ensure that the bending moment and lateral shearing force are zero at z = 1.
The value of the Z-function at z = 0 (Eq. 23) reduces to
40
Z = A + E .
Equations 3% transform to the following set of equations.
s s 0 0
c c 0 0
s s
A + E = i
A + E = O
A , + E 1 = O
A, + E , = 0 . c c
(33)
Equation 34 are expressed in te rms of the s-set of integration constants by using Equation 31 and the appropriate definitions in Equations 24, 25, and 30. This result is written below.
Y
s s 0 0
A + E =I,
I S
0 0 B S - F = O ,
4
J -sin@ (Cz + H:) + cos+ (D" + G s ) = 0, 0 0
o r I alternatively for Equations 35b.
s s 0 0
s s 0 0
C + H = O
D + G = O .
rriting the valu The following symbols are introduced for of the Z-function at z = i , and are used in developing the expression for the remaining four boundary condi- tions in Equations 32b.
s2 = sinh a t z = s i n h b
~3 COS b t3 = cos a
s4 = sin b t 4 = s i n a
si3 = Ti3 = h t 3
si4 = sis4
S, = ~ 2 ~ 3
Sa = ~ 2 ~ 4
T i 4 = h t 4
T23 = t2t3
T u = t2 t4
The definitions in Equations 36 are the same as those of Formulas 17 in Section 11.
41
b
The value of the Z-function at z = I is written below in te rms of-the above sym- bols; first in matrix form and then in algebraic form.
Equations 37 are valid for any set of the scripted symbols introduced previously. In other words, the expression for any Z-function and any of its derivative is obtained by merely applying a consistent s e t of scr ipts to each of the symbols Z and A through H in Equations 37. To illustrate, the second derivative of Zs and Zc are written below.
where, upon substituting Equation 31 into the appropriate de then simplifying with Equation 28,
M i = p2 (cos 2 @ M t + sin 2 @ J MSK)
M: = p2 (-sin 2@MS + cos 2 @ J MSK)
0
0 0
nition of Equation 30 and
(39 )
S N2 = p2 (-cos 2@N: + sin 2 @ J NSK) 0
C S
0 N2 = p2 (sin 2@ N + cos 2@ J N:K) . The algebraic representation of the first two conditions in Equations 32b are ob-
tained by substituting the appropriate set in Equations 39 into Equations 38. These initial algebraic equations a r e simplified by a large degree by taking certain linear combinations. These linear combinations are illustrated below for Equations 38.
42
. t
The corresponding relationships for the last two of Equations 32b are developed in a s imi la r manner and are written below.
S - [ti t2] N~ K S C cos 34 Z3 - sin 34 Z3 = [si s21 J Mo
S C s in 3+ Z, + cos 3+ Z, = [si s21 M: K
Equations 40 and 41 are transformed to the matrix equation written below by sub- stituting Equations 24 for matrices It$ and NE and introducing the symbols Sij and T. . as
1J defined in Equations 36.
s23 r . -. AS
BS
CS
DS
0
0
0
0
T i 3 T23 T14
TZ4 T14 -T23
T23 T13 T24
. - ES
FS
GS
HS
0
0
0
0 . .
(42)
The elements of the second vector are related to the elements of the first vector in Equation 42 by the following expressions ( s e e Equatioiis 35a and 35c).
E = 1-A, F = B , G = - D , a d H = - C . (43)
The scripted notation is no longer required. Substituting Equations 43 for the elements of the second vector, transforms Equation 42 to the following four by four matrix.
+ s24 - s23 -I- T14 si4 + T23
-s24 + T24 T13 - T23
s23 - Ti4 s14 - T23 s13 T i 3 s24 -
-si4 T23 s23 - Ti4 -s24 T24 s i3 + 1 Equations 43 and 44 are Formulas 19 and 18 i i
- -
ection 11,
k] re spec t ive ly .
(44)
This verifies each of the Formulas 14 through 22 of Section II. The remaining formulas (23
43
through 26) a r e verified by taking appropriate derivatives of the expression for the beam deflection (Eq. 1 9 ) . forming the operations indicated in the following definitions.
This is done in a straightforward manner by substituting and per-
Velocity
S C * = w * = w ~ ( Z c o s e - z s i n e ) , at ae 0 0 0
v =
Acceleration
Bending Moment
Yo s C - M = E I & - E 1 a&-- - ( Z 2 s in 8 + Z, cos e ) , ax2 -3 a22 ~2
Shearing Force
o s C E1 Y a% EI a3v
ax L~ ax3 - ~3 Q = E I 3 = - (Z3 s in 8 + Z 3 cos e ) .
The first two of Equations 45 are Formulas 23 and 24, respectively. Formulas 25 and 26 are verified with the last two of Equations 45 by recognizing that the pn t e r m s appearing in Equations 30 are omitted in Formulas 20 but a r e included in Formulas 25 and 26.
George C. Marshall Space Flight Center National Aeronautics and Space Administration
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