vibration of a rotating cantilever beam with an
TRANSCRIPT
Scholars' Mine Scholars' Mine
Masters Theses Student Theses and Dissertations
1970
Vibration of a rotating cantilever beam with an independently Vibration of a rotating cantilever beam with an independently
rotating disk on the free end rotating disk on the free end
Darrell Blaine Crimmins
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VIBRATION OF A ROTATING CANTILEVER BEAM
WITH N~ INDEPENDENTLY ROTATING DISK
ON THE FREE END
BY
Darrell Blaine Crimmins, 1946-
A
THESIS
submitted to the faculty of
THE UNIVERSITY OF MISSOURI-ROLLA
in partial fulfillment of the requirements for the
Degree of
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
Rolla, Missouri
1970
Approved by
~~~(advisor)
T2485 c.l 85 pages
ii
ABSTRACT
This thesis provides a vibration analysis of a rotating
cantilever beam with an independently rotating thin circu
lar disk on the free end. The exact differential equations
of the system as defined by classical Bernoulli-Euler beam
theory are written using the methods of the calculus of
variations. The exact equations are not solved, but two
different approximations are found by assuming a cubic
polynomial deflection curve and applying the equation of
Lagrange.
The solutions are restricted to small deflections of
the beam and a shaft stiffness which permits a deflection
in only a single plane. Nonlinear differential equations
result in the second approximation and are solved by a
digitai analog simulation. The nonlinear equations are
then linearized using only the dominant terms. Using the
linearized equations, the first two natural frequencies
and their respective amplitude ratios are solved for in a
general computer program that can be applied to many
different free vibration beam problems.
The results show that the fundamental mode frequency
decreases with increasing tip mass and increasing beam ro
tational speed which results in instability at high speeds.
The relative spin of the disk with respect to the beam has
no effect at zero beam rotation, but the effect of the
relative spin of the disk increases as the beam rotation
iii
increases. The results obtained follow the trend reported
in other works for limiting cases of this problem.
ACKNOWLEDGEMENTS
The author wishes to extend his sincere thanks to
his advisor, Dr. Clark Barker, for his guidance and
suggestion of the topic. The author also wishes to
thank Dr. Richard Rocke for his advice and valuable
assistance, and Alice Crangle for typing the thesis.
iv
v
TABLE OF CONTENTS
Page
ABSTRACT. . • . • • • . . . • . • . . . . . . . . . . . . . . . . . . • . • . . . . . . . . . . . . . . ii
ACKNOWLEDGEMENTS. . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i v
LIST OF FIGURES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
NOMENCLATURE ...•..•...•................................. vii
I. INTRODUCTION. . . . . . • . . . • . . . . . . . . . . . . . . . . . . . . . . . . l
II. THE REVIEW OF LITERATURE ....................... 2
III. DEVELOPMENT OF THE DIFFERENTIAL EQUATIONS ...... 5
IV. FIRST APPROXIMATE SOLUTION ..................... 17
V. SECOND APPROXIMATE SOLUTION .................... 26
A. Linearization of Second Approximation ...... 32
B. Stability Consideration .................... 36
VI. CHARACTERISTICS OF THE SOLUTION ................ 38
VII. CONCLUSIONS ...•................................ 44
VIII. APPENDICES ..................................... 47
A. General Computer Program for First and
Second Approximations. . . . . . . . . . . . . . . . . . . . . . 4 8
B. Digital Analog Simulation of Unlinearized
Second Approximation ....................... 54
c. Comparison of the Results for the Special
case of Q=O, Qs=O, and md=mb=M ............. 70
D. Comparison of the Results to the Exact
Equations for a Vibrating Beam ............. 73
IX. BIBLIOGRAPHY. . . . . • . • . . . . . • . . . . . . . . . . . . . . . . . . . . . 7 5
X. VITA •..... •. • ... • ....... • • • ..... • .............. 77
vi
LIST OF FIGURES
Figure Page
1. Rotating Cantilever Beam with an Independently
Rotating Disk Attached............................ 6
2. Relationship Between the X21 Y 21 z2 System and the
X 1 1 Y 1 1 Z 1 S y stern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3. Geometric Relationships for Functions of B .•...... 27
4. First Natural Frequency versus Angular Speed for
R = 3 ~n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 0
5. Second Natural Frequency versus Angular Speed for
R = 3 1n • . . . • . . . . . • . . . • . . . . . . . • . . . • . . • . . . . . • . . . . . . 41
6. First Natural Frequency versus Angular Speed for
R = 2 1.n • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 4 2
7. Second Natural Frequency versus Angular Speed for
8.
9.
R = 2 in ......................................... .
Analog Simulation Elements ....................... .
Analog Circuit for Equations (B-23) and (B-24)
10. Analog Circuit for Equations (B-23) and (B-24)
cont . ............................................ .
11. Analog Circuit for Equations (B-23) and (B-24)
cont . ............................. · . · . · · ...... · · ..
12. Analog Circuit for Equations (B-23) and (B-24)
con t ............................................. .
13. Analog Circuit for Equations (B-23) and (B-24)
43
60
61
62
63
64
cont .............................................. 65
14. Analog Circuit for Equations (B-23) and (B-24)
cont .............................................. 66
vii
NOMENCLATURE
n - Magnitude of the angular velocity of beam
ns - Magnitude of the angular velocity of the disk with
respect to the beam
i,j,k- Unit vectors along an X,Y,Z coordinate system
S - Angle of bending
L - Lagrangian function
Tb - Kinetic energy of the beam
Tt Translational kinetic energy of the disk
T Rotational kinetic energy of the disk r
V Potential or strain energy of the beam
Vm - Velocity magnitude of an element drn of the shaft
I.. - Mass moment of inertia of the disk about the ii ll
axis
w. - Angular velocity vector in the i direction l
p - Mass per unit length of the beam
L - Length of the beam
E - Modulus of elasticity of the beam
I Area moment of inertia
K - Stiffness parameter EI
t - Time
- Indicates derivatives with respect to time, or
vector dot products
R - Radius of the disk
td Thickness of the disk
b
h
y
w
a. 1
E. 1
B. 1
D. 1
- Width of the beam
- Height of the beam
- Weight density of steel lb/in3
- Natural frequency
- Function of time
Deflection of beam in the Y1 ,z 1 plane
- Amplitude of time function
- Constants
- Constants
- Constants
- Mass of the disk
- Mass of the beam
viii
I. INTRODUCTION
The design and analysis of almost all modern machinery
involves rotating shafts, beams, and gears. The whirling
of beams has also been increasingly used in the explora-
tion of space. The vibration analysis of a rotating
cantilever beam with an independently rotating disk on
the free end is performed in this thesis because it may
represent many of these structures.
The analysis presented in this thesis treats the
rotating cantilever beam as a conservative system and
restricts the deflection to a single plane. Continuous
properties and small amplitudes of vibration are also re
quired. The total kinetic and potential energy of the
system is written and Lagrange's equation applied to ob
tain the differential equations of motion. Approximate
solutions for the first two natural frequencies and princi
pal mode amplitude ratios are then found.
1
~ One of the most important features of this thesis is
the Fortran program which solves a linearized approximate
solution, for rotating cantilever beams with rotating tip
mass and limiting cases, for the first two natural fre
quencies and principal mode amplitude ratios. This computer
program may be used without any knowledge of the solution
by substituting into it the parameters and dimensions of a
given system.
2
II. THE PEVIEW OF LITERATURE
Beams are necessary in the construction of many things.
The early study of beams lead to the development of the well
known Bernoulli-Euler beam equation (1)*. Wagner (2) de-
scribes the large amplitude free vibration of beams deviat
ing from the classical theory of vibration based on restric
tive assumptions, such as small dynamic deflections. The
vibration of beams carrying masses has also been extensive
ly studied (3,4,5,6,7,8,9). Baker(3) solves the classical
Bernoulli-Euler beam equation for the case of uniform beams
with masses at the midpoint and supported at each end.
Chen (4) introduced a new formulation to the problem of
vibrating beams carrying masses in which the Dirac o -function is used in the differential equations of motion
to describe the effect of a concentrated mass. Eigen
functions are obtained by separation of variables and the
application of Laplace transforms to the ordinary differ
ential equations. Pan (5) treats the transverse vibration
of an Euler beam carrying a system of heavy bodies.
Srinivasan (10) undertook the study of the rotating
mass on a shaft as a vibration absorber in an effort to
widen the range over which many conventional absorbers are
restricted. In this study the governing differential
equations of motion are derived using Lagrangian techniques.
* Numbers in parentheses refer to the Bibliography.
Jones (11) treats a syncronous vibration absorber both an
alytically and experimentally.
In recent years, several papers have been published
3
on vibration of rotating beams and rotating beams with tip
masses (12,13,14,15,16,17,18). Jones and Buta (12) de
scribe the vibration of a whirling beam allowing for vibra
tion in the plane of rotation, as well as the perpendicular
to the plane of rotation. Using a method of undeformed
coordinates a prediction of certain instabilities, not
previously noticed, was reported. Lo and Renbarger (13)
and Boyce (14) have also contributed to the study of rotat
ing beams of constant cross section and properties. Jones
and Buta (15) treated the axial vibrations of a whirling
bar using undeformed or Lagrangian coordinates. They
found that whirling lowers the natural frequency, and when
the angular velocity of the bar approaches certain critical
values, static resonance, or instabilities occur, and the
axial displacements everywhere in the bar tend to become
unboundedly large. The study of rotating beams has been
stimulated in the last few years because of their increased
use as antenna elements of artificial communication satel-
lites and on re-entry spacecraft. Craig (16) studied the
vibration of a rotating beam with a tip mass. Using
Hamilton's principle in the determination of a variational
principle, it is stated to be applicable to large displace
ments of an unstressed beam or small displacements in a
beam initially stressed. The results of the study agree
closely to that done by Boyce and Randleman (17) in that
the effect of the tip mass was seen to have a greater
4
effect on the second natural frequency than on the funda
mental mode. Also the first natural frequency was seen to
decrease with the addition of tip mass while this was not
necessarily so for the second natural frequency, especially
at high rotational speeds. In an earlier study by Cohen,
Boyce, and Randleman (18) using a method involving Rayleigh's
quotient they concluded that both numerator and denominator
were increased by increasing tip mass but that the funda
mental mode frequency always decreased. The frequencies of
the higher modes seemed to be unaffected by the rotary
motion.
5
III. DEVELOPMENT OF THE DIFFERENTIAL EQUATIONS
The differential equations of motion for a rotating
cantilever beam with an independently rotating disk attach
ed on the free end may be derived by considering the po
tential and kinetic energy of the system and applying
Lagrange's equation (19). Since Lagrange's equation in
the form to be used is valid only for a conservative system
no damping internally or externally will be considered.
The form of the potential energy equation also restricts
the solution to small deflections of the beam (20).
Consider a cantilever beam fixed in a set of bearings
with an independently rotating rigid circular disk attach
ed on the free end, as shown in figure 1. It should be
noted in figure 1 that the angle S is different for differ
ent sections of the beam. Let the x1 ,Y1 ,z 1 system be fixed
to the beam and rotate with it at some constant rate ~
relative to the X,Y,Z inertial reference system. Restrict
the stiffness of the beam, such that a deflection is per
mitted only in the Y1 , z1 plane. Let the x2 ,Y2 ,z 2 system
be fixed to the beam at the disk end and rotate, such that
Y2 and z2 remain in the Y1 ,z1 plane. From figure 1 it
can be seen that
( 1)
and
n
z,z1
dm
.r:::=_--=- 1 I I: i IJ))Ji f&i;;;;;aoa..C
~yl
Figure 1. Rotating Cantilever Beam with Independently Rotating Disk Attached.
y2
I .,. I y I yl
0'\
Q beam/XYZ = Qj + S i 2 (2)
where Q is the magnitude of the angular velocity of the s
disk with respect to the beam. Therefore
, (3)
where S(L) denotes S to be evaluated at Y1 =L.
The x2 ,Y2 ,z 2 system is related to the x1 ,Y1 ,z1 system,
as shown in figure 2. The unit vector relationship be-
( 4)
Since
A A
jl - j (5)
then
j = j2 cos s - k2 sin s , ( 6)
and
Q disk/XYZ
{7)
7
Figure 2. Relationship Between the x2 ,Y 2 ,z 2
System and the x1 ,Y1 ,z 1 System.
y 2
8
9
To apply the equation of Lagrange, the Lagrangian
function L, which is the total kinetic energy of the system
minus the potential energy, must be found.
L = T - V (8)
The Lagrangian function for this problem is made up of
four energy terms, as follows:
1. The strain energy of the beam.
2. The kinetic energy of the beam.
3. The translational kinetic energy of the disk.
4. The rotational kinetic energy of the disk.
The strain energy of the beam is expressed in the form (20)
v = (9)
To find the kinetic energy of the beam, a velocity vector
for an element dm of the beam is required.
v rn
Therefore, the kinetic energy of the beam is
L
f 0
(10)
(ll)
10
Since <v > 2 is m v m dotted on itself,
<v > 2 v . v a zl 2 z2 n2 = = <a:t> + (12) m m m 1 ,
and L
1 = 2 p J ( 13) 0
The translational kinetic energy of the disk is
(14)
The velocity of the center of mass of the disk is
vern= (15)
azl where z 1 (L) and (~)L denote the function evaluated at
yl=L.
Since the velocity needed to calculate the translational
kinetic energy must be expressed in terms of a Newtonian
reference frame, k 2 must be written in terms of jl and k 1 .
From figure 2 it can be seen that
(16)
Since
11
( 17)
then
-vern ( 18)
and
azl 2 2 2 + <at:> [cos 13 + sin 13].
L
(19)
Therefore,
(20)
and the translational kinetic energy of the disk may be
written
( 21)
The rotational kinetic energy of the disk may be expressed
in general form as (19)
For a thin circular disk of mass md and radius R,
2 mdR
= -4-
and
(22)
( 2 3)
From equation 6 it can be seen that
A
wy2 = (ns+ncosS)j 2 ,
and
w = (-nsinS)k2 . z2
Therefore,
and
,..,2 . 2(3 = ~£ s1.n .
Substituting into equation 22 and simp1ifing yields
( 2 4)
(25)
( 2 6)
( 2 7)
(2 8)
( 29)
( 30)
(31)
The Lagrangian function may now be written using equation
13,17,21 and 31 as (32)
0
L 2
12
2 1 ·2 2 2 EI [(ns+ncosS) + 2 cs(L)+n sin S)l- ~ f a z1 2
( 2 ) dyl. 0 dyl
13
The exact differential equations of motion for the
system may be written by applying Hamilton's principle (20).
Hamilton's principal states that a conservative mechanical
system moves from configuration 1 to configuration 2 in
such a manner that the function
( 33)
is a minimum. The integral I may be minimized by methods
of the calculus of variations (20) . The Lagrangian func-
tion as written in equation 12 is seen to contain four
distinct parts, the integral of which each may be minimized
independently. Therefore,
where
I 1
1 = 2 p
Jt2 [ (ns H6cos(3) 2
tl
(34)
(35)
(36)
1 • 2 2 2 + 2 ((3(L) H"t sin (3)] dt,
( 37)
and
EI =2
14
( 38)
For a single integral of the form Jx2F{x,y,y',y", ... y<n))dx xl
the Euler equation of the variational problem may be
written (20)
(39)
where F in this case is the Lagrangian function, I is in-
tegrated with respect to x, andy' is dy/dx.· Applying the
Euler equation yields,
and
min
For a double integral of the form f f R
(40)
(41)
F(x,y,w,w ,w ,w , X XX XY
w )dxdy the Euler equation of the variational problem may yy
be written (20)
15
0 I (42)
where F is the Lagrangian function, I is integrated with
respect to x and then y, and the variable subscript on w
refers to partial derivatives with respect to that variable.
Applying the Euler equation yields
2 2 . = P (n -
m~n
and
min = EI
a zl 2) I
at ( 4 3)
( 4 4)
These minimum integrals may now be summed and set equal to
zero to give the exact differential equations of motion for
the system.
2 •• 2(5}5} +5} cosS)]-S(L)] - EI
s ( 4 5)
Since this partial differential equation would be quite
difficult to solve, an approximate solution to the problem
is presented. It may be noted that if md and n are set
equal to zero in equation 45 the classical Bernoulli
Euler beam equation will result.
16
IV. FIRST APPROXIMATE SOLUTION
Since the exact equation for the system would be
very difficult to solve, an approximate solution of the
Rayleigh, Rayleigh-Ritz form can be found by assuming
the shape of the deflection curve. Let this deflection
curve be of the form:
(46)
where the a's are functions of time. For a cantilever
17
beam the deflection and slope must be zero at the constrain-
ing bearings, so the boundary conditions are
zl. (O,t) = 0 ( 4 7)
and
a z1 ( 0, t) = 0
ayl ( 4 8)
Applying these boundary conditions shows that the deflection
curve reduces to:
(49)
The Lagrangian function, as written in equation 8 may now
be evaluated. The following partial derivatives are neces-
sary in this evaluation:
18
azl 2 3 at = a2yl + a3yl (50)
a zl 2 "2 4 . . 5 . 2 6 (-)= a2yl + 2a2a3yl + a3yl (51) at ,
8z1 2a2yl +
2 ayl
= 3a3yl (52)
azl 2 2 2 3 2 4 <ay-> = 4a2yl + 12a2a 3y 1 + 9a3yl (53) , 1
a 2 z 1 2a2 + 6a3yl 2 = (54) ,
ayl
and
a 2 z 2 ( 1)2 24a2 a 3y 1 + 2 2 = 4a2 + 36a3y 1 (55) 2
ayl
The strain energy of the beam, as written in equation 13,
becomes
EI JL 2 2 2 v = 2 (4a2+24a2a 3y 1 + 36a3y 1 )dy1 (56)
0
and
2 2 2 3 v = EI(2a2L+6a2 a 3L +6a 3L ) (57)
The kinetic energy of the beam, as written in equation 17,
becomes
and
(59)
The translational kinetic energy of the disk, as written
in equation 21, becomes
(60)
and
( 61)
To evaluate the rotational kinetic energy of the disk
assume S is a small angle so that
cos s ~ l
and
sin S - tan
Also assume that S
Therefore,
T r
is small compared to ~ and ~ s
(62)
( 6 3)
(64)
19
20
and
(65)
The Lagrangian function is now defined as
Lagrange's equation for a conservative system is (19)
i = 1,2,3, ... ( 6 7)
where q. is a generalized coordinate. For this particular 1
case qi = a 2 and a 3 • To obtain the equations of motion the
following derivatives are required,
5 4 . L 6 5 . ( pL L ) + (-p-- + L ) = 5 + md a2 6 md a3 , ( 6 8)
(69)
21
(70)
( 71)
and
(p n 2L 7 2 6 9 2 2 4 3 ~ 6 7 + mds-2 L + 4 mdR s-2 L - 12 EIL )a3 . (73)
Substituting these derivatives into equation 67 yields the
differential equations of motion. Rearranging and dividing
through by L 4 and L5 the differential equations are
(74)
and 2 6EI 3 2
(pL + md)a2 + (~ + mdL)a3 + l- - ( .l2.b_ + md (1+2 R ) ) 6 7 L3 6 L2
2 [12EI 2 9 R2 2
s-2 ] a2 + (~ + md (L+ 4 L ))Q ]a3 = 0. ( 7 5) L2 7
22
These differential equations take the form
0 (76)
and
( 7 7)
Substituting pL for the mass of the beam and K for the
stiffness EI, the constants in equations 76 and 77 are
mb ( 78) Bl = 5 + md
B2 ~L
(79) = -5- + mdL
[~ + 2
B3 4k md(l + ~)]s-22 (80) = L3 5 L2
~L 3 2 6k ~)]r.l2 (81) B4 = L2
- [-6- + md (L + 2 L
Dl mb
+ md (82) = 6
mbL D2 = -7- + mdL (83)
mb 3 2 6k + md (1+ ~)] s-22 (84)
D3 = L3
[- 2 6 L2
IlbL 9 2
12k +md(L + L)Jr22 (85) D4 =
L2 [- 4 7 L
Equations 76 and 77 may now be written in matrix form.
+ = 0
(86)
where the natural frequencies and mode shapes of the beam
can be found by assuming
( 8 7)
and
(88)
which yields for the non-trivial case the frequency deter-
minant
= 0 ( 8 9)
and amplitude ratio
= ( 9 0)
2 where the two values of w found from the frequency deter-
minant are to be substituted. The frequency determinant
yields the polynomial in w
23
24
( 91)
Equation 91 takes the form
(92)
where
(93)
( 9 4)
and
( 9 5)
Substituting into the quadradic formula
2 E2+ J E~ - 4E1E 3 wl = -2E 1
(96)
and
-E +J E2 - 4E1E 3 2 2 2 w2 = 2E 1
( 9 7)
The first two natural frequencies and their respective
amplitude ratios may be found for various beams, disks, and
values of ~ by substituting the desired parameter into the
general computer program found in Appendix A.
25
This computer program evaluates the B.,D., and E. ~ ~ ~
constants which are then used to solve equations 90,96,
and 97.
It should be noted that B is probably not small com
pared to n and n so that the results from this approximas
tion will not give exceptionally good results, especially
for the second natural frequency. A second approximation
to be presented on the next page should be used even for
limiting cases.
Since the relative spin of the disk does not appear in
the first approximation a second approximation must be made
in an effort to retain this effect.
V. SECOND APPROXIMATE SOLUTION
In the first approximate solution the assumptions
made in evaluating the rotational kinetic energy of the
disk caused ~ , the spin of the disk relative to the beam, s
to vanish. Since S and S were evaluated assuming small
angle theory a higher order approximation may be used to
retain the effect of the relative spin of the disk with
respect to the beam.
To find expressions for 13 1 cos 13 1 and sin S consider
the geometric relationship shown in figure 3. It can be
seen that
tan s dZl
= dyl ( 9 8)
(3 dyl
cos = dS I (99)
and
sin (3 dz1
= dS (100)
26
Taking the derivative of tan S with respect to time yields
S(L) 2 (101)
yl=L .
2 2 2 it follows that Since ds = dyl + dz 1 azl
sin (3= d1l (102)
J 1 + (~)2 y -L ayl 1- I
Figure 3. Geometric Relationships for Functions of S
27
cos s 1 =
I 1 + azl 2
Cay-> yl = L (103) 1
and
2 1 cos s = azl 2
1 + < ayl> L (104) yl =
Using the assumed deflection curve and the partial deriva-
tives listed on page 18, the functions of S necessary to
write the rotational kinetic energy of the disk are
and
• 2(.) s1n ...., =
cos s =
2 cos s =
,
1
1
(lOS)
(106)
(10 7)
(10 8)
28
Substituting into Equation 31 the rotational kinetic energy
is
T r
2 = mdR (2f.l 2 +
--8- s 4~/f.l
s +
+.
(109)
The Lagrangian function for the second approximate solution
is the same as for the first approximation except for the
29
T term. Therefore, from equations 66 and 109 the Lagrangian r
function is
f.l 2 +
4f.lf.l s
2 3 + 6a3L ) • (110)
To apply Lagrange's equation, the following derivatives
are required to write the equations of motion:
=
+
(2~~ s
(lll)
(112)
30
and
ClL = aa 3
+
(2~W s
31
(113)
2 6 2 7 + n 2 )+(ps-2 6L + mn 2L5 - 6EIL2 )a2 + (p~ L + n 2L 6 12EIL3 ) ~~ ~~ 7 m~~ - a3.
(114)
Simplifing and substituting pL for the mass of the beam and
k for the stiffness EI the differential equations are
32
+
(115)
and
(116)
A. Linearization of Second Approximation
Equations 115 and 116 are nonlinear simultaneous
differential equations and cannot be solved directly as was
33
the case for the first approximation. However, they can
be linearized by neglecting certain terms that do not appear
to be dominant. For small values of a 2 and a 3 it is assumed
that
( 117)
and
(118)
The validity of these assumptions may be checked by program-
ming equations 115 and 116 on the Pactolus digital analog
simulator. This program is explained in detail in Appendix
B.
With these assumptions equation 115 becomes
(119)
and equation 116 becomes
34
( 2DD + Q2) + 8kL - 2 mbn2L4 2 n2 4) s 21 ~" - 3 mduG L a3 = 0 . ( 12 0)
These differential equations take the form
Bla2 + B2a3 + B3a2 + B4a3 = 0 ( 121)
and
Dla2 + D2a3 + D3a2 + D4a3 = 0 (122)
The constants in equations 121 and 122 are
3 2 mbL 3 ( 12 3) Bl = mdR L + --+ mdL 5
4 3 mdR2L2
mbL 4 (124) B2 = 2 + -6- + mdL
2 (2QQ + s-22) + 4k mbs-22L3
- m Q2L3 B3 = mdR L -
s 5 d
(125)
(126)
35
3
Dl mdR2 L + mbL 2 3 = --+ 3 mdL 9 , ( 12 7)
D2 3 mdR2L2 2 4 2 ~4 = 2 + 21 mbL + 3 mdL ( 12 8)
(129)
and
(130)
Completing the solution from this point is identical to the
first approximation. The amplitude ratio is
=
and the natural frequencies are
and
where
E 2 + ~ E~ - 4E1 E 3
-2E1
-E 2 +~E~- 4E 1 E 3
2El
(131)
(132)
( 13 3)
36
(134)
(135)
and
(136)
The first two natural frequencies and their respective
principal mode amplitude ratios may be found for various
beams, disks, ~'s, and ~ 's by substituting the desired s
parameters into the general computer program found in
Appendix A. This program evaluates the B., D., and E. con-~ ~ ~
stants which are them used to solve equations 131, 132, and
133.
B. Stability Consideration
By observing the results of many different beam and
disk configurations, it was seen that the first natural
frequency of a rotating cantilever beam is lowered by in-
creasing the angular velocity of the beam. If this angular
velocity is large enough, then the natural frequency will
reach zero. This implies that for a particular configuration
a critical speed exists such that an inherent instability is
present. The critical speed may be found by setting the first
natural frequency equal to zero. Therefore,
0 = 'E 22 4E E v - 1 3
(137)
which is the condition for instability.
The critical speed for any value of n may be found s
be an iterative process in the general computer program
found in Appendix A. The critical speeds of a particular
37
configuration may be seen in Figures 4 and 6 of chapter VI.
38
VI. THE CHARACTERISTICS OF THE SOLUTION
The characteristics of the solution are best illustrated
by observing one of the examples that have been studied.
Assume a beam and disk to have the following dimensions
and properties:
1. Material: Steel
2. E 30 X 10 6 lb/in 2
3. y . 3 lb/in 3 =
4 . I 1/6 in 4 =
5. md = .0220 slug
6. mb = .0155 slug
7. R = 3 in.
8. L = 10 in.
Substituting these values into the general program for
the second approximation yields the numerical results. Figure
4 shows the first natural frequency decreases as Q increases
and that the relative spin of the disk is capable of widen-
ing the range of stability. Figure 5 shows the second
natural frequency increases slightly, with Q having more s
effect than Q.
To illustrate the effect of a decrease in tip mass let
R=2 in. and md=.0049 slug. Figure 6 shows the first natural
frequency behaving the same as the 3" disk with Qs again hav
ing a control on stability. The second natural frequency
however, as seen in Figure 7, does not always increase, as
was the case in the first example. The relative spin of the
39
disk again has more effect than the speed of the beam.
The triangles on Figure 4 show the results on the
Pactolus program for the case of R = 3 in. It can be
seen that as ~ increases the result is very close to that
given by the linearized second approximation.
Limiting cases of the general problem are found in
Appendix C and Appendix D. Appendix C compares the results
obtained from the general computer program for the case of
a cantilever beam with a tip mass. Appendix D compares the
general computer program results for the case of a simple
cantilever beam.
800
700
600
..... u 500 (J) til
.......... '0 rei 400 H --r-l :3
300
200
100
0 0 100 200 300 400 500 600 700 800 900
Q (rad/sec)
Figure 4. First Natural Frequency Versus Angular Speed for R=3 in.
~ 0
5840 l 5820
5800
5780
5760 ..... C)
5740 QJ Ul
57201 ........, rcl !tl 5700 H ....-
~N 5680
5660
5640
56201 5600
5580
0
1 rt 8 =2000
/ rt =1000 I s
/~ r rt =0 s -
100 200 300 400 500 600 700 800 rt (rad/sec)
Figure 5. Second Natural Frequency Versus Angular Speed for R=3 in.
~ ,_.
1300
1200
1100
1000
900 ..... u (1) 800 Ul
" 'U ttl 700 H -'
rl 600 ~
500
400
300
200
100
0 I
0
~ =2000 s
~ =0 s
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400
~(rad/sec)
Figure 6. First Natural Frequency Versus Angular Speed for R=2 in.
~ N
11080
11060
0
~ 11040 '-... '0 ctj 1-1
rt(rad/sec)
Q =1000 s
Q =0 s
Figure 7. Second Natural Frequency Versus Angular Speed for R=2 in.
"'" w
44
VII. CONCLUSIONS
In this thesis, vibration of a rotating cantilever
beam with an independently, rotating thin disk on the free
end has been studied. The results obtained seem to agree
with the results reported in previous work for limiting
cases of this problem. The first natural frequency was
found to always decrease with an increase in rotational
speed of the beam and with an addition of mass at the free
end. The second natural frequency does not always increase,
and the effect of the beam rotation is not nearly as great.
The relative spin of the disk with respect to the beam has
no effect on the natural frequencies when the beam is not
rotating, because ns appears only as a product of n. This
probably happens because the solutions are restricted to
small angles, and the spinning disk effectively moves in
only a vertical plane. The relative spin of the disk
always has a greater effect as the beam speed increases.
As the rotation of the beam lowers the natural frequency,
the relative spin of the disk always raises the natural
frequencies, especially at high beam rotation. If the beam
rotation is allowed to be large enough, the first natural
frequency will reach zero and a condition of instability
will exist. Since the relative spin of the disk always
raises the natural frequency, it was found to be able to
increase the critical beam speed.
45
The assumptions made to linearize the second approxi
mation are easily checked by the method shown in Appendix
B. However, it has been found that the effect of the
non-linearities is small so that the results are reasonable
2 "2 2 . . 3 "2 4 even when (~s + 2~~s) : aa2L +24a2 a 3L +18a 3L
The problem was formulated with the restriction that
the beam stiffness allows a deflection in only one plane
and that small deflections of the beam are assumed. Since
the deflections are not small near the unstable condition,
the solutions are not rigidly valid but do give a good indi
cation of what to expect.
The general computer program found in Appendix A is
written so that anyone with a similar problem would only
have to insert the appropriate data cards into the deck
and no detailed knowledge of the solution is required.
This program solves equations 96, 97, and 90 for the
first and second natural frequencies and their respective
principal mode amplitude ratios for the first approximate
solution. It also solves equations 131, 132, and 133 for
the first and second natural frequencies and their respec-
tive principal mode amplitude ratios for the linearized
second approximation. Equation 137 is also solved for
the critical beam speed for any value of ~s·
During the literature search solutions were found for
rotating beams with tip masses but none showed solutions
that would not have to be thoroughly studies or programmed
in an attempt to use them. It is hoped that this thesis
will provide a convenient and quick approximate solution
to those with this type of problem.
46
47
VIII. APPENDICES
APPENDIX A
General Computer Program for
First and Second Approximations
48
GENERAL COMPUTER PROGRAM FOR
FIRST AND SECOND APPROXIMATIONS
49
This general computer program solves equation 96, 97,
and 90 for the first and second natural frequencies and
their respective principal mode amplitude ratios for the
first approximate solution. It also solves equations 131,
132, and 133 for the first and second natural frequencies
and their respective principal mode amplitude ratios for
the linearized second approximation. Equation 137 is also
solved for the critical beam speed for any value of ~s·
NOMENCLATURE:
ST
z
R
PS
p
MB
MD
w
ws
WMIN
WMAX
WI
The stiffness parameter K
The length of the beam
The radius of the disk
The mass of the beam
The mass of the disk
The mass of the beam (Print
The mass of the disk (Print
The angular velocity of the
The angular velocity of the
spect to the beam Sl s
The minimum value of ~
The maximum value of ~
The incremental value of Sl
out)
out)
beam ~
disk with re-
WSMIN
WSMAX
WSI
(B (I) I I=ll 4)
C(l)
(D(I) I I=ll4)
(E (I) I I=l I 4)
F (1)
F(2)
F ( 3)
F ( 4)
Ratio 1
Ratio 2
N
N=l 1
N=2 I
N=3 I
The minimum vlaue of Q s
The maximum value of Q s
The incremental value of Qs
50
The coefficients of equation 76 or equation
121
The conditional equation for instability
The coefficients of equation 77 or equation
122
The coefficients of equation 92
The second natural frequency squared
The first natural frequency squared
The second natural frequency
The first natural frequency
The amplitude ratio for the second mode
The amplitude ratio for the first mode
An index designating the solution desired
and the iterative process
The program will solve the second approxi-
mate solution over any range of Q and Q s
as designated
The program will solve the second approxi-
mate solution for the case when Q =0 over s
any range of Q.
The program will solve the first approxi
mate solution for any value of Q. Qs does
not appear in this solution.
N=4, The program will calculate the critical
speed for any beam configuration and
relative spin of the disk.
1 600
DIMENSION B (10) ,C (10) ,D {10) ,E (10) ,F (10) WRITE ( 3 , 6 0 0 ) FORMAT(/) READ(l,l00,END=42)ST,Z,R,P,PS,N FORMAT(Fl2.0,4Fl0.5,110) READ(l,l04)WMIN,WMAX,WI,WXMIN,WSMAX,WSI FORMAT ( 6Fl0. 0)
100
104 K=O WRITE(3,500)ST,Z,R 1 P 1 PS 1 N
500 FORMAT(25X 1 1 K=' 1 Fll.0 1 5X, 1 L=' 1 F5.2 1 5X 1 'R= 1 1 F4.2 1 5X 1
I MD= I I F 6 . 4 I 5 X I I
CMB= 1 1 F6.4,5X 1 1 N= 1 1 Il,/) ~v=WMIN
IF(4-N)800,97 1 800 800 WRITE(3 1 400)W 400 FORMAT(52X 1 1 W= 1 ,F7.1 1 /)
IF(2-N)92,93,94 93 IF(l-K)91,91,94 94 WRITE(3 1 300)
300 FORMAT(l9X 1 NATURAL FREQUENCY AMPLITUDE RATIO NATURAL FREQUENCY
c AMPLITUDE RATIO I I 7X, I ws I )
K=K+l 91 WS=O.
IF(2-N)99,2,97 97 WS=WSMIN
51
2 B(l)=(P*R*R*Z)+((PS*Z**3)/S.)+P*Z**3 B(2)=(1.5*P*R*R*Z*Z)+((PS*Z**4)/6.)+P*Z**4 B(3)=(2.*W*WS+W*W)*(P*R*R*Z) B(3)=B(3)+(4.*ST)-((PS*W*W*Z**3)/5.)-P*W*W*Z**3 B(4)=(2.*W*WS+W*W)*(l.S*P*R*R*Z*Z) B(4)=B(4)+(6.*ST*Z)-((PS*W*W*Z**4)/6.)-P*W*W*Z**4 D(l)=(P*R*R*Z)+((PS*Z**3)/9.)+((2.*P*Z**3/3.) D(2)=((3.*P*R*R*Z*Z)/2.)+((2.*PS*Z**4)/21.)+((2.*P*Z**4)
/3.) D(3)=(2.*W*WS+W*W)*(P*R*R*Z) D(3)=D(3)+(4.*ST)-((PS*W*W*Z**3)/9.)-((2.*P*W*W*Z**3)
/3.) D(4)=(2.*W*WS+W*W)*((3.*P*R*R*Z*Z)/2.)
52
D(4}=D(4}+(8.*ST*Z}-((2.*PS*W*W*Z**4)I21.}-((2.*P*W*W*Z **4}13.}
95 E(1}=B(1}*D(2)-B(2)*D(l) E(2)=B(4)*D(1)+B(2)*D(3}-B(3)*D(2)-B(1}*D(4} E(3)=B(3)*D(4)-B(4)*D(3} IF(4-N)43,108,43
43 F(1)=(SQRT(E(2)*E(2)-4.*E(1)*E(3))-E(2))1(2.*E(1)) F ( 2) = ( SQRT. (E ( 2) *E ( 2} -4 • *E ( 1) *E ( 3) ) + E ( 2} ) I (-2 • *E ( 1) ) IF(O.+F(2))3,4,4
3 WRITE(3,700)W,WS 700 FORMAT(I,l5X, 1 FREQUENCY 1 IS IMAGINARY WHEN W= 1 ,F7.1, 1
AND WS= I ,F C7.1,1) F(2)=-F(2)
4 F(O.+F(1)}6,7,7 6 WRITE(3,101)W,WS
101 FORMAT(I,15X, 1 FREQUENCY 2 IS IMAGINARY WHEN W= 1 ,F7.1, 1
AND WS= I ,F C7.1,1)
F(1)=-F(1) 7 CONTINUE
F ( 3) =SQRT (F (1}) F(4)=SQRT(F(2)) RATI01=(B(1)*F(1)-B(3))1(B(4}-B(2}*F(1)) RATI02=(B(1)*F(2)-B(3))1(B(4)-B(2}*F(2}) WRITE(3,200)F(4) ,RATI02,F(3) ,RATI01,WS
200 FORMAT(20X,F10.2,F18.5,10X,F10.2,F18.5,10X,F7.1} IF(2-N)38,38,96
96 WS=WS+WSI IF(WSMAX-WS)38,38,2
38 W=W+WI IF(WMAX-W)40,40,800
99 B(1)=PSI5.+P B(2)=((PS*Z)I6.)+P*Z B(3)=((4.*ST}I(Z**3}} B (3) =B (3)- ( ((PSIS.} +P+ ( (P*R**2) I (Z**2}}} * (W**2}} B(4)=((6.*ST)I(Z**2}} B (4) =B (4)- ( ( ( (PS*Z} 16.} + (P*Z} + ( (3. *P*R**2) I (2. *Z})}
*(W**2)} D(1)=PSI6.+P D(2}=((PS*Z)I7.)+(P*Z) D(3}=((6.*ST}I(D**3}} D(3}=D(3)- ( ( (PSI6.)+P+ ( (3.*P*R**2)1(2.*Z**2}}) * (W**2)) D(4)=((12.*ST)I(Z**2}) D ( 4) =D ( 4} - ( ( ( (PS * z) 17. ) + (P* z} + ( ( 9 . *P * R* * 2} I ( 4 • * z) ) )
*(W**2)) GO TO 95
92 IF(1-K)91,91,82 82 WRITE(3,102)
102 FORMAT(19X 1 NATURAL FREQUENCY AMPLITUDE RATIO NATURAL FREQUENCY
C AMPLITUDE RATIO') K=K+1 GO TO 91
40 GO TO 1 108 C(1}=(SQRT(E(2)**2-(4.*E(1)*E(3)))+E(2))/(-2.*E)1)
IF ( 0 . +C ( 1) ) 4 4, 4 4 , 4 5 44 WRITE(3,1-7)W,WS
107 FORMAT(40X,'CRITICAL SPEED=',F8.2,3X,'WHEN WS=',F7.1) GO TO 1
45 W=W+1. GO TO 2
42 STOP END
/DATA
53
APPENDIX B
Digital Analog Simulation of
Unlinearized Second Approximation
54
DIGITAL ANALOG SIMULATION OF
UNLINEARIZED SECOND APPROXIMATION
Equations 110 and 111 are nonlinear differential
equations. A numerical solution to these equations may
be found by the digital analog simulation program called
55
Pactolus. Let the following terms from these equations be
written
cl 2 = mdR L, (B-1)
c2 3 2 2 = 2" mdR L , (B-2)
~L3 3 c3 = -5- + mdL , (B-3)
4 mbL 4
c4 = -6- + mdL , (B-4)
mbQ2L3 2 3
c5 = 4K - - mdQ L , 5 (B-5)
mbQ2L4 2 4 c6 = 6KL - - mdQ L , 6
(B-6)
3 mbL 2 3
c7 = -9- + 3 mdL , (B-7)
2 4 2 4 ca = 21 mbL + 3 mdL , (B-8)
Q2L3 ~ 2 2 3
c9 = 4K - - 3 mdQ L , 9 (B-9)
2 mbQ2L4 2 2 4 c10 = 8KL - 21 3 mdQ L , (B-10)
56
F(a2 ,a3 ) (1 + 4a~L 2 + 3 9a~L4 ) 2 , (B-11) = 12a2 a 3L +
and,
. . h 4a~L2 3 9a~L4 f(a2 ,a3 ,a2 ,a3 ) = 2QQS + + 12a2a 3L +
+ Q2 + (B-12)
With these substitutions equations 115 and 116 reduce
to
.. (C1+c 3F(a2 ,a3))a2 + (C2+c 4F(a2 ,a3 ))a3 + (C1 f(a 2 ,a3 ,
(B-13)
and,
. . (C1+c 7F(a2 ,a3))a2+(C2+c 8F(a2 ,a3 ))a3+(C 1f(a 2 ,a3 ,a2 ,a3 )+
(B-14)
Equations (B-13) and (B-14) take the form
(B-15)
and,
where
w4
ws
and,
w6
= c 1 f(a2 ,a3 ,a2a 3 )+C5F(a2 ,a3 )
c 1+c 3F(a2 ,a3 )
. . = c2f(a 2 ,a3 ,a2 ,a3 )+C6F(a2 ,a3 )
c 1+c 3F(a2 ,a3 )
c 1+c 7F(a2 ,a3 ) = c 2+c 8F(a2 ,a3 > I
c 1 f(a2 ,a3 ,a2 ,a3 )+C 9F(a2 ,a3 ) = c 2+c8F(a2 ,a3 )
c 2 f(a 2 ,a3 ,a2 ,a3 )+C10F(a2 ,a3 ) = c 2+c 8F(a2 ,a3 )
(B-16)
(B-17)
, (B-18)
, (B-19)
(B-2 0)
(B-21)
(B-22)
Eliminating a 3 from equation (B-15) and a 2 from equation
(B-16) they take the form
57
58
wlw5-w2 ) a2
wlw6-w3 a2 = ( + ( 1-w w ) a3 l-w1w4 1 4
(B-23)
and
w2w4-w5 w3w4-w6 )a3 a3 = <1-w w ) a2 + (
l 4 l-w1w4 (B-2 4)
The analog simulation may now be programmed using the elements
shown in Figure 8. The analog circuit is drawn in the same
manner as a typical analog circuit, except that the output
is positive with respect to the input for a given element
and that element 76 is reserved for time. The analog circuit
representing equations (B-23) and (B-24) is shown in Figures
9 through 14. The program as presented on punched cards is
shown following Figure 14.
It should be noted that the initial conditions on a 2
and a 3 need to be the amplitude ratio of the desired principal
mode. If arbitrary initial conditions are used, both modes
are excited, and they will appear superimposed upon each
other. The assumptions made in linearizing the equations
may be conveniently checked in this manner. If the amplitude
ratio for the point in question is substituted into the
program, a sine wave output will result if the assumptions
were valid. If the assumptions do not hold then the degree
of error may be estimated by the deviation from a pure
sine wave. 2 However, even at low speeds where (0. + 20.0. ) is s
small, good results have been obtained for several
examples indicating that the nonlinearities of the system
do not have a large effect.
59
60
NAME TYPE SYMBOL DESCRIPTION
Half Power H ei--EJ>-- e e = {e-: 0 0 ~
el pl
e 0 =P1 +} (e1+e 2P 2+e 3P 3)dt Integrator I eo
Constant K CiP- e e =P 0 0 1
el~ Weighted w e2 P W . e eo=Plel+P2e2+P3e3 Summer e3 0
Multiplier X
el=®>-e2 eo eo=ele2
el
b> eo=el+e2+e3 Summer + e2 e 0
e3
Divider I e2~ el / eo eo=el/e2
[;> Sign Inverter - e. eo eo =-e. ~
~
Gain G e. ®1 e e =P 1e. ~ 0 0 ~
Figure 8. Analog Simulation Elements
61
8
L2 9
L L4 K
1 8 10 10
1
11 md m L 3
d 2 12
11
15
22
17
16
G 23 K KL 23
K 18 5 18
5
21
Q
6 19 19
28
2.0 2QQ G 24
20 24
Figure 9. Analog Circuit for Equations (B-23) and (B-24)
8 9
10 15 11
1 22 23
12
17
5
11
18
16
21
28
17
19 24
c2 25
1/9
2/3
-5/9
c10
Figure 10. Analog Circuit for Equations (B-23) and (B-24) cont.
c 7F
62
8 9
10 15 25
37
38
39
40
41
42
43
44
19 24
15
41
25
42
43
44
63
106
19
~--------------------------------------------------------~ 24
Figure 11. Analog Circuit for Equations (B-23) and (B-24) cont.
64
8 9
10
108
55
57
62
73
64
75
57 79
58
80
59
106
----------------------------------------------------------· 19 ------------------------------------------------------------· 24
Figure 12. Analog Circuit for Equations (B-23) and (B-24) cont.
65
8
9
.-------------------~------------~----_.10 108
~----------------~--------------~------------------•19 (2rlrl + r.l 2 )
s
111
~------------------~------------------------------------~24
Figure 13. Analog Circuit for Equations (B-23) and (B-24) cont.
8
91
102
106
19
24
9
•
•
109 110
Figure 14. Analog Circuit for Equations (B-23) and (B-24) cont.
66
67
PACTOLUS DIGITAL ANALOG SIMULATOR PROGRAM
CONFG CONFIGURATION SPECIFICATION
BLOCK TYPE INPUT 1 INPUT 2 INPUT 3 1 K 0 0 0 2 K 0 0 0 3 K 0 0 0 4 K 0 0 0 5 K 0 0 0 6 K 0 0 0 7 K 0 0 0 8 X 1 1 0 9 X 8 1 0
10 X 9 1 0 11 X 2 9 0 12 X 11 1 0 13 X 3 3 0 14 X 13 2 0 15 X 14 1 0 16 X 4 9 0 17 X 16 1 0 18 X 5 1 0 19 X 6 6 0 20 X 7 6 0 21 X 19 23 0 22 G 15 0 0 23 G 16 0 0 24 G 20 0 0 25 X 22 1 0 26 X 19 11 0 27 X 19 17 0 28 X 19 12 0 29 + 11 23 0 30 w 12 17 0 31 w 5 21 26 32 w 18 27 28 33 w 16 11 0 34 w 12 17 0 35 w 21 5 26
36 w 28 27 18
37 X 29 106 0
38 X 30 106 0
39 X 31 106 0
40 X 32 106 0
41 X 33 106 0
42 X 34 106 0
43 X 35 106 0
44 X 36 106 0 108 0
45 X 15 25 108 0
46 X + 15 37 0
47
68
48 + 25 38 0 49 + 45 39 0 50 + 46 40 0 51 + 15 41 0 52 + 25 42 0 53 + 45 43 0 54 + 46 44 0 55 I 48 47 0 56 I 49 47 0 57 I 51 52 0 58 I 53 52 0 59 I 54 52 0 60 58 0 0 61 59 0 0 62 X 57 56 0 63 56 0 0 64 I 50 47 0 65 X 58 55 0 66 X 59 55 0 67 X 64 57 0 68 64 0 0 69 X 57 55 0 70 w 71 69 0 71 K 0 0 0 72 + 62 60 0 73 I 72 70 0 74 + 67 61 0 75 I 74 70 0 77 + 63 65 0 78 + 68 66 0 79 I 77 70 0 80 I 78 70 0 81 X 79 88 0 82 X 80 98 0 83 X 73 88 0 84 X 75 98 0 85 + 81 82 0 86 + 83 84 0 87 I 85 0 0 88 I 87 0 0 89 X 88 88 0 90 X 87 87 0 91 X 90 8 0 92 X 89 8 0 93 X 97 87 0 94 X 98 88 0 95 X 94 9 0 96 X 93 9 0 97 I 86 0 0 98 I 97 0 0
69
99 X 98 98 0 100 X 99 10 0 101 X 97 97 0 102 X 101 100 0 103 w 91 96 102 104 w 92 95 100 105 + 104 71 0 106 X 105 105 0 107 I 103 105 0 108 + 107 110 19 109 H 105 0 0 110 X 109 24 0
END 111 + 19 24 0 ICPAR
INITIAL CONDITIONS AND PARAMETERS BLOCK IC/PARl PAR2 PAR3
1 Length 2 Mass of Disk 3 Radius 4 Mass of Beam 5 Stiffness K 6 Speed of Beam 7 Relative Speed of Disk
22 1.5 0 0 23 .2 0 0 24 2.0 0 0 30 1.0 0.16667 0 31 4.0 -1.0 -1.0 32 6.0 -0.16667 -1.0 33 0.1111 0.66667 0 34 0.66667 0.09524 0 35 -0.55556 4.0 -0.66667 36 -0.66667 -0.09524 8.0 70 1.0 -1.0 0 71 1.0 0 0 88 Amplitude of a2 98 Amplitude of a~4.0 103 8.0 18.0
END 104 4.0 12.0 9.0 TIMES
INTEGRATION INTERVAL TOTAL TIME SAMPLE TIME INCREMENT
END OUTPT
OUTPUT 1 2 3 4 5 6 7 8 END 88 98 105 111 103 107 0 0
APPENDIX C
Comparison of the Results for the
Special Case of ~=0, ~s=O, and md=~=M
70
COMPARISON OF THE RESULTS FOR THE
SPECIAL CASE OF ~=0, ~s=O, and md=mb=M
71
The first natural frequency of a uniformly loaded can-
tilever beam with a concentrated mass M at the free end
equal to the mass of the uniform beam is treated in example
7.3-3 by Thomson (1). The solution is arrived by substi-
tuting into Dunkerley's formula the frequency equation for
the uniformly loaded beam by itself and the frequency
equation for the concentrated mass attached to a weightless
cantilever beam. 'rhe result is
2.41 ( EI3) J.VIL
(C-1)
This result is then compared to the frequency equation
obtained by Thomson (1) by Rayleigh's method which is
= 2.43 (EI3) ML
(C-2)
For this special case the mass of the disk must be equal
to the mass of the shaft (assuming the disk and the shaft
to be of the same material) so that
bhL
To obtain a numerical comparison let the material be
steel and the parameters have the following values:
(C-3)
td = 1 in y = .3 lb/in 3
R - 3 ~n E = 30 10 6 lb/in 2 X
b 1 in I = 1/6 in 4 = .
h = 2 in
(C-4)
With these parameters the value of L is calculated to be
rrR2 t L = ~ = 14.13 in. (C-5)
and
M = bhLy = 022 slu . g. g (C-6)
Therefore the frequency obtained by Thomson's (1) equation
is
and
6 = 2.41 ( 5 x 10 3 ) = 1.94 x l0 6 (rad/sec) 2
.022(14.13)
w1= 440. rad/sec .
(C-7)
(C-8)
This result compares favorably with the natural frequency
72
calculated from the general program. For the first approxi-
mate solution w1 =442.09 rad/sec and for the second approxi
mation w1=437.73 rad/sec.
APPENDIX D
Comparison of the Results to the
Exact Equations for a Vibrating Beam
73
COMPARISON OF THE RESULTS TO THE
EXACT EQUATIONS FOR A VIBRATING BEAM
74
The natural frequencies for the free vibration of a
cantilever beam have been found by many methods. The exact
natural frequencies are
and
w1 = 3. 515 J EI 3 ~L
= 22.034 J EI 3 mbL
(D-1)
(D-2)
Using the same beam and properties, as the example in
Appendix C, the exact natural frequencies are
= 3.515 J 5 X 10 6
(.022) (14.13) 3 = 1001. rad/sec (D-3)
and
J 5 X 10 6 w = 22.034
2 (.022) (14.13) 3 = 6350. rad/sec.
(D-4)
These results agree with those calculated from the general
program especially for the fundamental mode. The approxi-
mate first natural frequency was calculated to be 1002.7
rad/sec while the second natural frequency was not as close
at 9879.2 rad/sec.
75
IX. BIBLIOGRAPHY
1. Thomson, W.T. {1965) Vibration theory and applications Prentice-Hall, Inc., Englewood Cliffs, N.J. p. 273-276.
2. Wagner, H. {1965) Large amplitude free vibrations of a beam, Trans. ASME 32E {Journal of Applied Mechanics) 4, p. 887-892.
3. Baker, W.E. {1964) Vibration frequencies for uniform beams with central masses, Trans. ASME 31E {Journal of Applied Mechanics) 2, p. 335-337.
4. Chen, Y. {1963) On the vibration of beams or rods carrying a concentrated mass. Trans. ASME 85E {Journal of Applied Mechanics) 30, p. 310-311.
5. Pan, H.H. {1965) Transverse Vibration of an Euler beam Carrying a System of heavy bodies, Trans. ASME 32E {Journal of Applied Mechanics) 2, p. 434-437.
6. Hoppman, W.H. {1952) Forced Lateral Vibrations of beams Carrying a Concentrated Mass. Trans. ASME 74E {Journal of Applied Mechanics) 19, p. 301-307.
7. Durvasula, s. {1966) Vibration of a Uniform Cantilever Beam Carrying a Concentrated Mass and Moment of Inertia at the tip. Journal of the Aeronautical Society of India, 18, 1, p. 17-25.
8. Arnba-Rao, C.L. {1966) Method of Calculation of Frequencies of Partially fixed beams Carrying Masses. Journal of the Acoustical Society of America 40, 2, p. 367-371.
9. Srinath, L.S., and Das, Y.C. {1967) Vibration of beams Carrying Mass. Trans. ASME 34E {Journal of Applied Mechanics) 3, p. 784-785.
10. Srinivasan, A.V. {1968) Analytical and Experimental Studies on Gyroscopic Vibration Absorbers {Part 1). Prepared under contract No. NASW-1394 by Kaman Aircraft, Division of Kaman Corporation, Bloomfield, Connecticut.
76
11. Jones, R. (1967) The Gyroscopic Vibration Absorber. Trans. ASME 89B (Journal of Engineering for Industry) 4, p. 706-712.
12. Jones, J.P., and Bhuta, P.G. (1963) Vibrations of a Whirling Rayleigh beam. Journal of the Acoustical Society of America 35, 7, p. 994-1002.
13. Lo, H., and Renbarger, J. (1950) Bending Vibrations of a rotating beam. Proc. First u.s. Natl. Cong. Applied Mech. p. 75-79.
14. Boyce, W.E. (1954) Vibrations of rotating Beams of Constant section. Proc. Second U.S. Natl. Cong. Applied Mech., p. 165-173.
15. Bhuta, P.G., and Jones, J.P. (1963) Axial Vibrations of a Whirling Bar. Journal of the Acoustical Society of America 35, p. 217-221.
16. Craig, R.R., Jr., (1963) Rotating beam with Tip Mass Analyzed by a Variational Method. Journal of the Acoustical Society of America 35, 7, p. 990-993.
17. Boyce, W.E., and Randleman, G.H. (1960) Vibration of Rotating Beams with Tip Mass. Rensselaer Polytech. Inst., Math. Dept. Rep. 39.
18. Cohen, H., Boyce, W.E., and Randleman, G.H. (1958) Vibration of a Uniform Rotating Beam with Tip Mass. Rensselaer Polytech. Inst., Math. Dept. Rep. 13.
19. Greenwood, D.T. (1965) Principles of Dynamics. Prentice-Hall, Inc., Englewood Cliffs, N.J. p. 256-257, p. 296.
20. Langhaar, H.L. (1962) Energy Methods in Applied Mechanics. John Wiley and Sons, Inc., New York, p. 40-41, 234-239, 82, 96.
X. VITA
Darrell Blaine Crimmins was born on October l, 1946,
in Oklahoma City, Oklahoma. He received his primary and
secondary education in Cobden, Illinois. He received his
college education from the University of Missouri-Rolla,
Rolla, Missouri, where he received the Degree Bachelor of
Science in Mechanical Engineering in January 1969.
77
He has been enrolled as a graduate student in the
graduate school of the University of Missouri-Rolla, Rolla,
Missouri, since January 1969.
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