Group Title: analytical dynamics of the woodpecker problem
Title: The Analytical dynamics of the woodpecker problem
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Title: The Analytical dynamics of the woodpecker problem
Physical Description: vi, 69 leaves. : illus. ; 28 cm.
Language: English
Creator: Roy, Russell Albert, 1942-
Publication Date: 1974
Copyright Date: 1974
 Subjects
Subject: Oscillations   ( lcsh )
Engineering Sciences thesis Ph. D
Dissertations, Academic -- Engineering Sciences -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaf 68.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Russell A. Roy.
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Bibliographic ID: UF00097557
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: alephbibnum - 000580709
oclc - 14076153
notis - ADA8814

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THE ANALYTICAL DYNAMICS OF
THE WOODPECKER PROBLEM







By





RUSSELL A. ROY


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1974













ACKNOWLEDGMENTS


The author wishes to thank many people who have helped

him over the years. His parents provided help and support in

spite of many setbacks. Several of his friends helped him

considerably during his work in the Physics Department on

Project Sisyphus.

Special thanks are extended to the chairman of the

author's committee, Dr. Knox Millsaps, for his suggestion of

a topic and his encouragement in completing the work. The

other members of his committee also offered many helpful

suggestions.

Appreciation is extended to many people at the Orlando

Division of the Martin Marietta Corporation, especially Max

Farrow, Avery Owen, and Al Roy, who were generous with time

and equipment.

The author's wife, Eileen, has been unfailingly patient

and supportive for many years, so much so that it would have

been impossible to complete the work without her.

Finally, it is necessary to thank Terence (185-159 B.C.)

who said:

Ita vita est hominum, quasi cum ludas tesseris;
si illud quod maxime opus est jactu non cadit, illud,
quod cecidit forte, id arte ut corrigas.
















TABLE OF CONTENTS


ACKNOWLEDGMENTS .

LIST OF SYMBOLS .

ABSTRACT . . .

CHAPTER

I INTRODUCTION

II THEORY . .

III EXPERIMENTAL

IV COMMENTS .

APPENDIX A ..

APPENDIX B . .

APPENDIX C . .

LIST OF REFERENCES

BIOGRAPHICAL SKETCH


Page

ii

iv

vi


RESULTS

. . .

. . .

. . .

. . .

. . .


iii













LIST OF SYMBOLS


MB mass of the bird, grams

MSL mass of the sleeve, grams

M mass of the spring, grams

a dimensionless factor describing an effective
spring mass

k spring constant, dynes/cm.

F frictional force opposing the motion of the spring,
0 dynes

1 frequency, radians/sec.

4a arithmetic decrement of the spring per cycle, cm.

xI displacement of the bird mass with respect to the
sleeve, cm.

I represents an idealized impulse delivered to the
bird mass twice each cycle, dynes

x the distance of the bird mass from equilibrium,
o caused by the rotation of the sleeve on the shaft,
also where the impulses occur (xI = x ), cm.

T dimensionless variable related to time (T = wt)

x dimensionless variable related to displacement
(x = x1/x0)

f dimensionless variable related to friction
S (fo = F /kx )

I dimensionless variable related to the impulses
S (I = I/kx )
O O
b dimensionless variable related to those positions
o of the bird mass with respect to the sleeve where
it locks and unlocks and where the impulses take
Dlace.








LIST OF SYMBOLS Continued


R radius of circular arc in phase space

v dimensionless speed in phase space
2
h dimensionless energy associated with energy given
to the bird mass during an impulse

A the real amplitude of the motion of the bird mass, cm.

y velocity of bird mass in phase space

Ad the amount the sleeve slips down the shaft each
half cycle as a result of the unlocking of the
sleeve, cm.








Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



THE ANALYTICAL DYNAMICS OF
THE WOODPECKER PROBLEM


By

Russell A. Roy

December, 1974


Chairman: Knox A. Millsaps
Major Department: Aerospace Engineering



A toy, usually in the form of a small woodpecker, is

shown to be a problem in the area of self-excited oscilla-

tions. An analysis of the problem by J. P. Den Hartog is

discussed and shown to be without solution. An alternative

theory is proposed and its results compared with those of

experiment.













CHAPTER I

INTRODUCTION



In the celebrated course at M.I.T. on nonlinear mechanics

J. P. Den Hartog demonstrates a toy "woodpecker" to his class.

The toy can be idealized as shown in Figure I. The solution

is given as a problem on the course's final exam and Den Har-

tog's analysis is found in the Study Guide [1] for the course.

The problem and his analysis are included in all pertinent de-

tail in Appendix A. His analysis concludes with the equation

fA() = 2 + B 2 62 = 0 (1-1)
where

a = A2(cosh Ap-l) (1-2)

B = A(A(-sinh-A) (1-3)

y = (A2+2)(cosh Ap-l) A( sinh A( (1-4)

6 = -A sinh AM + A2 cosh A( (1-5)

with A and ) defined in Appendix A. The solution of (1-1)

would mean the solution to the analysis, but this has not been

done until this point. In fact, Den Hartog remarks [1] that,

"This I have not done, and if anyone ever does it, I shall be

grateful if he or she will send me the solution."

Letting

x = cosh A(











































(a) (b)


Figure I





3


and

y = sinh A(


and substituting these into (1-2), (1-3), (1-4), and (1-5),

and thence into (1-1) we have, after canceling and gathering

terms:


fA() = A4 2 A4 2 x 4A3 y + 4A3 xy

+ A2(-4x +8x-4- y 2) + A(40xy-40y) 4x2 + 8x 4.


The right-hand side can be factored and then both sides

divided by x-1, giving:

S( -A 2 22
Ax- A42(-x-1) + 4A3 y -A2[4(x-1) + x- ]
x-1 x-1

+ 4A(y 4(x-1).


This expression can be regrouped and divided by 4:

_fA A44 2 3
4x1) = -x(- -- 4+ A2+1) + y(A 3 +Ao)

A4~2 2 2
-- (1 + A2) + A + 1.
(x-1)A
2 2 2
Noting that x -y = 1 or y = (x+l)(x-l) and dividing by

A2+1 we get

fA() A4 2 x+1
fA) A- [x+1 + 2 ] + A(y (x-1).
4(x-1)(A +1) 4(A +1) A

Rearranging this results in

fA() A 2 -6)
= -- (x+1) + Acy (x-1) fA((). (1-6)
4(x-1)(A +1)








Clearly fA( ) and fA(j) have the same nontrivial roots

so we need only examine f*(j), the interesting factor of

fA(]). By letting z = Aq, fA*() can be written as

fA(q) = z sinh z ( 2(cosh z+l) (cosh z-1). (1-7)


Equation (1-7) can be explored by expanding sinhz and cosh z

in their familiar power series,

3 5 2n+1
z z z
sinh z = z + + + + + + z
3! 5! (2n+1)! +
2 4 2n
cosh z = 1 + + + + +
2! 4! 2n!

and then substituting them into (1-7), obtaining

f2 2 z2 z4 z4 z4
fA ) = (z 2 + 4-2! -4
6 6 6
z z z
+ T5-! T 60 +

2n+2 2n+2 2(n+l)
Z z z
(2n+l) 4(2n!) 2(n+l) ) +

The first two groupings are equal to zero and the third

grouping is negative for all values of z except for z = 0,
th
when it is zero. The n grouping can be written as

2n+2
z 1 1 1
2n! 2n+ 4 (2n+2) (2n+l)]

and will be negative for all n > 2 for all nonzero z. Thus

f*(O) is negative for all z with the exception of the physic-

ally trivial case of z = 0. One must conclude that the Den

Hartog analysis is not satisfactory and the basic physical

model must be reexamined.








Several models of the toy woodpecker were built and

examined. The first models built were like that shown in

Figure I-a. These were not stable and could not be con-

structed in such a way as to produce periodic motion. While

this may not be impossible, it seems quite difficult, with

several parameters being critical. After several hundred

variations were attempted unsuccessfully, this model was

abandoned, and one like that in Figure I-b was built. It was

made using a spring from a ball point pen, a piece of broken

capacitor, a scrap of wooden dowel, and a shaft of aluminum

rod, and delighted the author by working perfectly the first

time it was tried. By observing the motion for various con-

figurations several things became apparent. To make sure of

what was happening, the motion was videotaped and then the

tape was played one "frame" at a time, allowing intervals of

one-sixtieth of a second to be observed. The following

features of the motion were clear:

1. The "bird" mass oscillates up and down, above and

below the sleeve. When this mass is above or below the

sleeve by a certain amount, the sleeve is locked on the shaft

due to friction. In between these points the sleeve is un-

locked and drops a short distance.

2. "Hard" excitation is necessary to start the motion.

That is, a certain threshold energy input was necessary to

start the characteristic slip-stick motion.

3. The motion is periodic with no change in frequency

from start to finish.








4. The sleeve locks in both possible orientations.

5. The frequency of the system is not the same as the

natural frequency of the bird mass on the spring with the

sleeve locked, but somewhat less.

6. When the sleeve is locked and the natural motion

observed the decay in the amplitude is not geometric but

appears to be arithmetic.

7. The sleeve moves a short distance down (about 0.15

centimeter) twice each cycle.

8. Running the woodpecker in a vacuum rather than in a

normal atmosphere resulted in no change in frequency.

9. The steady state motion is independent of the initial

conditions.

10. None of the physical parameters appear to be critical.

11. When the system is started with a very large initial

displacement the sleeve is observed to lock and unlock, but

not to drop for several cycles.

The above observations indicate that this problem falls

in the area of self-excited oscillations. More specifically,

it falls in the area of mechanical clocks. The theory

describing mechanical clocks has been worked out, most

notably by A. A. Andronov and coworkers [2,3] although other

people [4,5] have discussed it as well. As Andronov points

out [2, p. 168], "A clock is an oscillating system which

maintains oscillations whose amplitude is independent of the

initial conditions. To start the clock a large initial im-

pulse is usually needed. If the initial impulse is too








small, then the clock comes to rest again." He also notes

[2, p. 168] that

Any clock mechanism can be broadly divided into
three parts: (1) an oscillating system, for exam-
ple, a pendulum, a balancewheel, etc., (2) a source
of energy such as a weight or spring, (3) a trigger
mechanism, connecting the oscillating structure
with the energy source. For fixed positions of the
oscillating system (referred to, for the sake of
brevity, as the pendulum) the trigger mechanism
acts and energy is given to the pendulum during a
short impulse. In a good clock the impulse is of
very short duration. The trigger mechanism usually
acts twice during a period close to the position
of equilibrium where the velocity is greatest. It
is important to note that the instant of time when
the trigger mechanism begins to act is entirely
determined by the position of the pendulum. In
addition, the manner of its action and the magni-
tude of the impulse also depend on the state of
the pendulum. Consequently, all forces which
arise in the mechanism depend only on the positions
and velocities of the separate parts of the system,
and not on the time. Thus the clock is an
autonomous system.

There can be no question that the toy woodpecker is in

fact a mechanical clock, and can be analyzed with the aid of

clock theory. In Chapter II, the theory of the woodpecker,

now revealed as a cuckoo, will be discussed, and in Chapter

III a comparison of the theory with some experimental results

will be given. Chapter IV includes a summary of this work


and its conclusions.













CHAPTER II

THEORY


Self-excited oscillations form an important class of

phenomena including some electronic oscillators, buzzers,

electric bells, wind and string musical instruments, brake

"squeal," some kinds of aerodynamic flutter, and so forth.

These phenomena are characterized by definite oscillations

which are properties of a specific system and not dependent

on the initial conditions. One of the most interesting

aspects of these systems is that they can generate a periodic

process from a nonperiodic source.

Clocks are a distinct class of mechanical self-excited

oscillations, and consist of three parts: (1) an oscillating

system, (2) an energy source, and (3) a trigger mechanism or

escapement connecting the two. Clock theory has been worked

out [2-5] and, although the woodpecker is different in

several ways from the clocks discussed in the literature, the

theory can be suitably extended to describe it.

The physical features of the motion mentioned in Chapter

I should now be more carefully examined in the context of

developing the most descriptive mathematical model. Hard

excitation is necessary to start the motion. Soft excitation

is defined as when an arbitrarily small initial displacement

is needed to start the motion and hard excitation as when an

8








initial displacement (or other appropriate energy input) must

be larger than a certain amount to start the motion. The

hard excitation observed implies that the friction present

is either of the Coulomb type (dry friction), or dominated

by Coulomb friction. A summary of an argument by Andronov

indicating that Coulomb friction (compared to linear or

velocity dependent friction) is the more appropriate type for

clocks is given in Appendix B.

The actual motion of the system is not complex and can

be analyzed in a series of several steps. Beginning with the

bird mass in the maximum upward position with respect to the

sleeve, and with the sleeve locked the features of the motion

are as follows:

1. The bird begins to descend and continues until it is

a short distance above the horizontal.

2. At this point the sleeve unlocks and the whole sys-

tem begins to descend.

3. The bird continues to descend with respect to the

sleeve and when the bird is a short distance below the hori-

zontal the sleeve locks. At this point from the frame of

reference of the bird it would appear that its velocity with

respect to the sleeve suddenly increases. The idealization

of this is an impulse applied to the bird at the instant the

sleeve locks.

4. The bird continues to its downward extreme and then

starts back up.








5. When the bird is just below the horizontal the sleeve

unlocks and as the bird continues up with respect to the

sleeve, the system again begins to descend.

6. The bird continues up with respect to the sleeve

and when it is just above the horizontal the sleeve locks

and again this can be idealized as an impulse applied to the

bird.

7. The bird continues to its upward maximum and the

cycle begins again.

The general form of the differential equation describ-

ing the motion can now be written as


MX1 + kxI = +F + I (2-1)


where each of the terms will be discussed below.

The mass, M, is equal to MB + aMs for that part of the

motion when the sleeve is locked. The mass of the bird is

MB and Ms is the mass of the spring. The quantity a is a
correction factor used to introduce an effective spring mass.

It would be measured by locking the sleeve and then measuring

the frequency of the bird for various values of MB. These

values would then be fitted by the curve


f + kM (2-2)
27 M S I

using least squares to get the best value for a.

During that part of the motion when the system is falling

M would be given by








(MB + aMS)MSL
M B + S + MSL (2-3)
B S SL

where MSL is the mass of the sleeve. For this situation the

natural frequency would be

2 k
k = = (2-4)


with M given by Equation (2-3).

The spring constant, k, has the usual definition and

would be measured by the normal static means. The Hookes

Law approximation is made and must be substantiated by exper-

imental results. Although the present theory is based on

phase plane analysis with linear equations pieced together

on the phase plane, it would be possible to extend it to in-

clude a nonlinear term such as kx if it were necessary. To

handle such a term an approach such as the "phase-plane delta"

method [6, p. 244] would have to be used.

The term Fo is a quantity describing the frictional

losses within the spring. The sign is chosen as opposite to

that of xl. It would be measured by locking the sleeve and

measuring the arithmetic decrement of the amplitude per cycle,

4a, and noting [2, p. 153] that


F = amw2 = ak. (2-5)
0

As far as the impulse term, I, is concerned, one of two

assumptions is usually made in clock theory: that either

(1) momentum or (2) energy is conserved during the impulse.

Since in the woodpecker a uniform amount of gravitational








potential energy enters the system during each cycle, the

second assumption seems to be the more reasonable one. It

is also the one made more frequently in clock theory [6,

p. 168].

There is a slight amount of rotational motion present

due to the sleeve locking in an "up" and a "down" position.

It is possible to consider a treatment with two variables,

one linear and one angular, to describe this motion. However,

because the angular motion is relatively slight, it is pos-

sible to do an analysis with one variable by approximating

the effect of the rotational motion. To do this, the motion

is divided into three regions:


MX1 + kx1 = +F + I for -x0 < x1 <_ X (2-6)


MXk + kx' = +F x' = x + x for x > x (2-7)
1 1 0 1 1 0 1 0

M3' + kx' = +F, x' = x x for x1 < x0 (2-8)


Where the impulses are pictured as occurring at xl = x0, and

at this value of xl, the position of the bird is changed by

x0, an amount which corrects xl for the slight rotational

motion of the sleeve.

The equations (2-6), (2-7), and (2-8) can be simplified

by a change of variable. First the equations are divided by

kx0, and then let

k 1/2

and

T = wt (2-9)





13


so that


M x1-
x x (2-10)
0

where now we have a dimensionless time, T. Also let


f (2-11)

and
x0
b0 0 1.00, (2-12)

and

10 k (2-13)

so that now (2-6), (2-7), and (2-8) take the form


S+ x = f0 + I0 for -1 < x < +1 (2-14)

R + x = f0 + b0 for x > 1 (2-15)

x + x = +f0 b0 for x < 1 (2-16)


This motion can be shown with a phase plane diagram. The

velocity, y, of the bird with respect to the sleeve is repre-

sented by the vertical axis with upwards (positive) y corre-

sponding to downward motion of the bird. The x axis repre-

sents the displacement of the bird with respect to the sleeve

which is also positive for downward motion of the bird.

There are two impulses per cycle which in this model are con-

sidered to be of equal magnitude, occur symmetrically in each

half-cycle and take no time. The phase diagram is shown in

Figure II. The convention used in clock theory is to follow







































































Figure II


,v I b',&








the phase path in a clockwise sense. At point S the bird is

at its maximum negative (upwards) displacement and its

velocity is zero. The bird then begins to speed up as it

falls. At x equal to minus one the sleeve unlocks and the

system falls, the bird going slower with respect to the

sleeve than before. At x equal to plus one the sleeve locks,

an impulse is delivered to the bird and it continues to its

positive (downward) extremum at f. This process repeats it-

self in the other direction ending at S. The phase paths in

the various parts ot the motion are portions of circles

centered at (0,f0) or (0,f0+b0). This can be seen by

letting y = k; then equation (2-14), for example, can be

written as


y + x = f0 (x i 1)

or

y 1+ x = f


Rearranging, we have


ydy = (f0-x)dx

1 R2 2
and letting (R -f0) be a constant of integration, we get

after integration

2 2 2 2
y = 2f0x x + R- f0

or

y2 + (x-f)2= R2 (2-17)


which is a circle in the phase plane centered at (0,-f0).








For periodic motion we require that the speed at point A

be the same as that at point B:


vA = B (2-18)

This condition will allow the frequency and amplitude of the

motion to be calculated. As shown in Figure III, we have

2 2 2
R2 = 2 + (1-b +f 2 (2-19)
1 A o (2-19)

R + 2fo = R (2-20)

and

R2 (1-f 2 = R (b +f 1)2 (2-21)
3 o 2 o o

Also, we see that


(v)2 + (f+1)2 = R (2-22)

and

(vA)2 + h = v2 (2-23)

where


A(y2) = h2 (2-24)

is the change in the square of the speed in dimensionless

units associated with each impulse. The equations above must

be solved to obtain a relation between vA and vB. Substituting

first (2-22) and then (2-21) into (2-23) we obtain


2 = h2 (l+f)2 R2 + (1-f) (b +f -1)2
B o 2 o o o


and then substituting (2-20) into this, we have




















































Figure III








2 2 2 _2 2
Sh- (1+f ) + (-f ) (- (b+f-l) + R 4fR + 4f1.
B o o o o 1 o

2
Using (2-19) for R1 one gets


2 = V + (+b+f )2 + h (+f)2 + (1-f)2
B A 0 0 0 0

(b +f -)2 + 4(b +f ) 4(b +f )R .


Setting

2 2 2
VB = VA = v


and canceling, we get


4f R = 4f2 + h2 4b f (2-25)
ol 0 0 0oo

Dividing by 4fo, squaring both sides, and again using (2-20)

results in

7 h2 b h 4
2 2 2 b0 h 2 ho
v + (1+b +f ) = f 2f b + + + + .
o o o oo 2 o 2f 1f
S 16f

Rearranging terms gives

S 4 2 b h2
v= + 2 o- + 2b 1 2f (2-26)
2 2 2T_ o o
16f o

It is still necessary to determine h to be able to cal-

culate v. By noting that the system drops a specific dis-

tance, Ad, once each half cycle,the energy input during the

impulse can be calculated by assuming it is due to this

change in gravitational potential energy:


g(Ad) = (A2 (2-27)
glad2








where

2 2 2
(v)2 (x1)2 x) 1

is the change in the square of the physical speed as a result

of the impulse. This must still be related to the dimension-
2
less quantity h2. By definition


2 d x 1 2 d x1 2
h = T (--) 2 UT x1


and using equation (2-9) we have


h2 _1 d X 2 1 d x1 2
O O

Then substituting in the definition of w, we get

2 M 2 *2
h = --2 [(x2? (x1)]
kx

or

2 M 2
h 2 -- [(Av) ]
kx

and using (2-26), we finally get

2 M
h = [2g(Ad)]. (2-28)
kx
o

The mass, M, in this equation is MB + aMs and does not in-

clude the sleeve mass since the energy the sleeve picks up

will be lost when it is stopped by the friction between it

and the shaft.








Now that h has been obtained it is possible to solve for

v and then to solve for R1, R2, and R3 using the equations

developed above. The frequency of the motion can be calcu-

lated by noting "that the representative points move along

the phase paths with an angular velocity equal to unity"

[2, p. 197]. Thus the time of transit in T units along one

of the semicircular arcs is equal to the angle subtended at

its center by the arc. As shown in Figure III, the relevant

angles are a, 8, 6, and y, and the period in T units would

be given by


T = 2(a + B + 6 + y + -). (2-29)


By inspection of Figure III the angles above can be easily

obtained in terms of known quantities:

v (2-30)
tan a = l-b +f
O O
b +f -1
sin B o (2-31)
2
1-f
sin 6 o (2-32)
3
l+f
sin y (2-33)


In physical units the period would be

M+ M MMSL 1
T 2(a++ ( B s 1/2 2(61)(k(ML /2
T = 2(++ -)( ) + 2(6+y)(k(M+MSL)")


and the frequency would be 1/T.








The energy balance is an important aspect of this motion.

For regular, periodic motion the energy input per cycle must

equal the losses per cycle. The energy input for the bird

and spring is on the order of 2(MB+ Ms)g(Ad) per cycle and

the energy losses per cycle are on the order of 4f A or 4akA
o
where A is the amplitude of the motion measured from the

equilibrium point. Equating these two energies and solving

for the amplitude, we have


A Mg(Ad)-3
2ak (2-34)

To the extent that the energy balance is correct, this rela-

tion should predict the amplitude. This balance is basic to

the theory and (2-33) can be derived another way by noting

that the normalized amplitude, Ao, where


A = A/x ,


can be derived from the phase diagram, Figure III, as


A R + b f
o 1 o o

Using (2-25) for R1 we have

A h2
A T
o 0

Substituting (2-5), (2-11), and (2-28) into this results in


A Mg(Ad)
x 2akx
O O


which is the same as (2-34).





22


Before turning to an experimental exploration of this

theory, some general points about it can be made. The theory

predicts that the frequency will be less than the natural

frequency of the bird on the spring with the sleeve locked.

We can understand why this is so physically because the peak

to peak amplitude of the woodpecker will be 4b0 longer than

the amplitude of the natural system, other things being equal,

and this will take longer.

We also note the necessity for at least one dynamical

measurement, Ad being the most appropriate. In this it is

like the Den Hartog theory where f, the coefficient of fric-

tion between the sleeve and the shaft, is idealized as having

the same value for motion as for rest. In fact, this would

be unlikely and some sort of dynamic measurement would be

necessary to establish some appropriate average value for f.

In the present theory it is possible to get around this

necessity for a dynamic measurement, at least theoretically,

although some approximations will be necessary. First

assume that when the sleeve unlocks it falls freely with an

acceleration of g. Since it starts from rest we have


Ad = gt2 (2-35)


where t will correspond to that part of the period when the

system is unlocked:


S= (6+ M MSL 1/2
t = +y) k(M+M (2-36)








Now

l+f
-1 o
6 = sin
R3

and

-I 1-fo
y = sin R
R3

and these can be expanded in a power series:

3 5
sin x =x x+ .
3! 5!

By taking only the first two terms of the series (this and

other approximations remain to be justified in Chapter III)

we have

1-fo l+fo (1-fo) 3 (+f) 3
S+y R + R 3 3- -
3 3 6R3 6R
3 3
and

2+6f2
2 o
6 + Y = R 3 -
3 6R3
3

It is further assumed that f is small and terms of the order
2
f and higher can be neglected. So we have
0

6 + y = 2 (2-37)
3 3R

By substituting (2-20) and (2-19) and then (2-26) into (2-21)

we obtain,after some algebra

2 8f 16f 16f 32f 1/2
R3 = -0o + + (2-38)
o h o h h h








2
Again, terms of the order f and above are neglected, result-

ing in


2 8f /
h o 1/2
R3 h 1- 2)
o h


(2-39)


Expanding this in a series and keeping only the first two

terms gives approximately

2 4f
R3 = 4T (1 2--)
o h

or


h2
R3 =-


Substituting this into (2-37) yields

8f f
So 16 o
h -4f (h -2-4f )
O O


The second term is neglected and substituting this for

6 + y in (2-36) gives


Ad g SL o 2
2 k(M+Ms) 11 2 'f


Letting

L M M S 1/2
2 k(M+MSL)

and

2 B(Ad)
h = B(Ad)


(2-40)


(2-41)


where








B (2gM
kx

and putting these into (2-41) gives

64Lf2
2 o
(BAd-4fo) = (2-42)


This relation can be solved numerically to give a Ad for a

particular MB. In Chapter III this equation will be explored

with specific data to determine its validity.

Although only the steady state motion has been explored

here it would not be difficult to extend the phase plane

analysis to cover various types of transients present when

the motion begins. Several references (see, for example

[6, p. 274,196], [2, p.178], and [3, p. 13]) discuss various

types of transient analysis.













CHAPTER III

EXPERIMENTAL RESULTS



The present work was originally intended to be entirely

theoretical in nature, consisting of completing the analysis

begun by Den Hartog and then exploring various physical situ-

ations with the completed theory. Because it was necessary to

redo the theory, it was felt to be prudent to include experi-

mental results with the new theory so that a complete and con-

sistent picture would be present for the first time.

Because the original thrust of this work was not experi-

mental, the measurements were made in most cases on the basis

of rather limited resources. Fortunately, most of the physi-

cal quantities required are not difficult to measure and the

results form a consistent pattern which unquestionably sub-

stantiates the theory.

The frequencies of the woodpecker model for various bird

masses were measured first. They were measured by counting

forty or fifty cycles and timing them with a stopwatch. Many

runs were made with the same bird mass with different numbers

of cycles (for example, 10, 20, 40 and 50 cycles) to see if

the woodpecker slowed down or speeded up as it moved down the

shaft. No such effect was observed. These measurements as

well as many others were also made through the use of a video








tape arrangement made available for this work from the Orlando

Division of the Martin-Marietta Corporation. The woodpecker

was taped in front of a numbered centimenter grid with a

digital clock reading in hundredths of a second also in the

picture, because the video tape could be played back and

"framed" to show a still picture of the motion every sixtieth

of a second. Appendix C contains some examples of data col-

lected from the video tape.

Table I shows the average frequency, fe, obtained for

various bird masses, MB. The masses were measured to an

accuracy of about 0.05 gram. The individual frequency measure-

ments were only precise to about 0.1 Hertz, but because large

numbers of measurements were made the error estimate was cut

in half.



Table I

M + 0.05 (grams) f 0.05 (Hertz)

0.00 4.32
10.15 2.95
11.50 2.83
12.00 2.77
13.50 2.70
16.20 2.59
18.10 2.49
20.70 2.42
21.20 2.32
24.55 2.21




The spring constant, k, was measured in the usual way,

measuring deflection for various loads. There was some scat-

ter and a slight indication of spring "hardening" at large









deflections. Table II gives the values of the loads in grams

and the respective deflections.


Load 0.05
(grams)

2.00
3.00
4.00
5.00
5.93
6.00
7.00
8.00
8.43
9.00
9.68
10.00
10.15
10.90
11.00
11.37
12.19
13.00
15.00
17.71
18.96
20.18
21.45


Table II

Deflection 0.05
(centimeters)

0.40
0.60
0.75
0.85
1.10
1.10
1.25
1.45
1.50
1.50
1.68
1.70
1.90
1.87
1.75
2.10
2.04
2.10
2.30
3.01
3.22
3.44
3.54


The average k for these values is 5.738 grams per centimeter

or 5,623 dynes per centimeter. A graph of the loading versus

deflection is shown in Figure IV.

The effective spring mass coefficient, a, could now be

determined using the value for k obtained above. This quan-

tity was determined by locking the sleeve in place and measur-

ing the frequency, f, for various MB. Then a was determined

from the equation







































c)
14-



I -
7 2-
II





J
7





4
5 -








I I
I 2


DEFLECTION IN CM.


Figure IV








f 1 ( k 1/2
2 77 IMB +aMS

or, solving for a:


k B
a = (3-1)
4 2 2M MS


The mass of the spring, MS, is 5.1 grams and the following

values for a were obtained:


MB + 0.05 f + 0.01 a
(grams) (Hertz)

9.70 3.19 0.84
11.30 3.03 0.83
12.75 2.86 0.91
17.75 2.54 0.85
18.99 2.46 0.90



The average value of a is 0.87 and this was the value used.

The quantity Ad was measured for each of a variety of

bird masses by looking at the total distance the system

dropped for a certain number of cycles and then averaging to

find the distance per half cycle. By inspection of the video

tape it was determined that the drops were uniform although

the drops when the bird mass was going down were slightly

different from those when it was going up. Table III summar-

izes the values of Ad for the respective bird masses.

The most difficult quantity to measure accurately was 4a,

the arithmetic decrement. This quantity was measured by lock-

ing the sleeve in place, starting the bird mass oscillating,

and then measuring the amplitude from the equilibrium point








Table III


MB 0.05
(grams)

0.00
10.15
11.50
12.00
13.50
16.20
18.10
20.70
21.20
24.55


Ad 0.005
(centimeters)

0.231
0.182
0.175
0.173
0.166
0.152
0.143
0.130
0.128
0.112


for each half cycle. During these measurements the bird mass

would begin processing which made the vertical amplitude

appear to decay more quickly on the video tape than it actu-

ally was. To avoid as much error as possible, only the first

five or six extrema were considered. For a bird mass of 12.75

grams, for example, the following values were observed:


Run I

Amplitude
(centimeters)

5.179
5.083
4.986
4.839
4.451
4.562
4.104
3.939

Run II

6.12
5.93
5.98
5.83
5.75
5.32
5.59


2a
(centimeters)


-0.096
-0.097
-0.147
-0.388
+0.111
-0.458
-0.165




-0.19
+0.05
-0.15
-0.08
-0.43
+0.29


Extrema


up
down
up
down
up
down
up
down



down
up
down
up
down
up
down








Of course, many more runs were made and analyzed but these

data are shown as examples of the data obtained, showing the

experimental scatter and the precessional effects. It was

not possible to make a defensible statement with regard to

the possible variation of a with MB. Since it was not

desired to fit a to theoretical data, a value of 0.044 centi-

meters was assigned to all values of MB.

It is possible that the damping could be linear (propor-

tional to velocity in some way) rather than constant. The

woodpecker system was run in a partial vacuum (<0.1 atm) to

see if there was any velocity dependent damping due to air

resistance. No effect was observed and the system ran exactly

the same (frequency and amplitude) as it did in a normal

environment. It is still possible that the losses in the

spring could be due to linear friction and this should be

explored.

In the case of linear friction the ratio of two succes-

sive maxima would be given [6, p. 201] by


n+1l -2Tv
- e (3-2)
x
n
or

v Ln n (3-3)
27r n+l

where v is called the damping ratio, and where the damping is

small. In the case of constant friction we have


xn xn+/2 = 2a = constant.


(3-4)








Substituting some experimental values into (3-3) we have

1 5.18
v Ln .
27T 4.99

or v = 0.0059 to the correct order of magnitude. With this

value of v, e-2 7is 0.964. If we expand e-2~v in a power

series, consider only the first two terms, and substitute

in the argument we have


e 27 = 1-0.037 = 0.963.


Hence, to a good approximation


n+l
l v (3-5)
n
an d

xn xn+ = vx (3-6)


for a full cycle and


x x+/2 n (3-7)


for a half cycle. In the woodpecker we note that at the end

of each half cycle the impulse restores the amplitude. In

fact, it is this exact relationship of energy balance which

gives the woodpecker its clocklike behavior. We note the

resemblance of (3-7) and (3-4) for small v and the wood-

pecker's physical situation and conclude that whatever linear

friction is present, if any, can be treated as constant to a

good approximation.

The final experimental quantity necessary to be measured

is x the displacement of the bird mass with respect to the





34


horizontal at which the sleeve locks and unlocks and at which

the impulses take place. This was measured statically by

rotating the sleeve up and down to its two locked positions

without letting the sleeve move vertically and measuring the

resulting displacement of the bird mass. These measurements

give an x0 of 0.75 0.01 centimeter.

The final results of the frequency calculations, a sum-

mary of data values used to get them, and the experimental

frequencies are shown in Table IV.


Tab le

k = 5,623 dynes/cm
a = 0.044 cm

MB (grams) d (cm)

0.00 0.231
10.15 0.182
11.50 0.175
12.00 0.173
13.50 0.166
16.20 0.152
18.10 0.143
20.70 0.130
21.20 0.128
24.55 0.112


IV

xo = 0.75 cm
aMSL = 4.43 grams

f f % error
---e-
4.50 4.32 -4.1
2.90 2.95 +1.6
2.85 2.83 -0.7
2.81 2.77 -1.4
2.70 2.70 0.0
2.53 2.59 +2.4
2.43 2.49 +2.4
2.30 2.42 +4.9
2.28 2.32 +1.7
2.15 2.21 +2.8


The agreement seems to be very good. By considering

the values of the amplitude predicted from (2-34),

A Mg(Ad)
2 ak

we can understand better one source of error in the frequency

calculations. The quantity a is the most inaccurate in (2-34)









and is most responsible for error. Table V shows the calcu-

lated amplitudes compared with experimentally observed ones.

The experimental amplitudes here are taken from the equilibrium

position and obtained from a least squares fit to five values.

The agreement is not bad, although it could be improved, most

probably by using more accurate values of a.



Table V


MB (grams)


10.15
11.50
12.00
13.50
16.20
18.10
20.70
21.20
24.55


Theoretical
A (cm)

5.26
5.52
5.63
5.89
6.21
6.38
6.47
6.50
6.43


Experimental
A (cm)

5.40
5.49
5.52
5.62
5.80
5.93
6.11
6.14
6.36


The last step in the calculations will be to examine the

validity of the approximate formula for Ad derived at the end

of Chapter II:

64cf2
(BAd-4f )2 = (2-42)
(Bdlo) ad


where


B= 2M
kx

and

g2 M MSL 1/2
2 k(MSL+M)


% error


2.6
0.5
2.0
4.9
7.1
7.6
5.9
5.8
1.1


~








Calculating Ad for an MB of 20.70 grams one gets a value

of 0.23 centimeters, which is higher than the observed value,

and with this value of 0.23 a frequency of 2.34 Hertz. This

is higher than the original calculated frequency but is in

better agreement with the observed frequency of 2.42 Hertz.

Each step of the derivation of (2-42) can be examined to

see how appropriate the approximations are. With the original

data for an MB of 20.70 grams, the actual value of R3 is

found to be 7.626. The approximate formula (2-40),

h2
R = 1
3
0

gives a value of 7.622 which is in excellent agreement. The

actual value of 6+y is found to be 0.263, whereas the approxi-

mate formula

8f
6 + y = O
h -2_4f

gives a value of 0.262 which is also in very good agreement.

The main source of error responsible for the two values of Ad

not agreeing any better is probably in

1 2
Ad = 2 gt2


in that the system is not in free fall but accelerating at

less than g due to the presence of some friction between the

sleeve and the shaft even when the sleeve is unlocked. A

smaller value of the acceleration would make the calculated

value of Ad smaller which is the correct direction. An








accurate measurement of the actual friction between the shaft

and the sleeve would call for rather sophisticated dynamic

measurements and is probably not worth doing in the sense that

Ad can be measured directly much more easily. In fact, it is

a characteristic of this type of problem that the friction

present is idiosyncratic and complex. It must usually be

idealized to a large extent for a class of problems rather

than dealt with for a specific case. In this sense the analy-

sis here seems to be one of the few cases where a real situa-

tion involving impact maintained oscillations is studied

completely.













CHAPTER IV

COMMENTS



This work begins with an analysis by J. P. Den Hartog

and shows that this approach to solving the problem of the

"toy woodpecker" is not satisfactory. Several possible reasons

for the failure of this analysis are discussed although a com-

plete "negative" analysis indicating the exact reasons for

the failure of the model and proposed solution has not been

done. This work proposes a different physical model for the

woodpecker than that chosen by Den Hartog and one which corre-

sponds more closely to the toys actually observed. This

physical model is then analyzed mathematically and the fre-

quency and amplitude of the motion predicted for various

physical situations.

A major step in the theoretical development is to recog-

nize the toy woodpecker as belonging to a class of phenomena

known as mechanical clocks. Clock theory has been worked out,

most notably by the Russian, A. A. Andronov, and his coworkers.

This theory was extended and applied to the toy woodpecker.

The method involves piecewise linear differential equations

matched together on the phase plane and is very suitable for

handling the impacts assumed in the model.








Actual working models of the toy were built and measure-

ments of several variables of the models were taken and com-

pared with the theoretical results. The agreement is very

good and it is clear that the theory successfully describes

the physical motion of the toy woodpecker. The calculations

and measurements are not difficult to do for the most part

and it is suggested that the toy woodpecker model, appropri-

ately constructed, would make a useful mechanical analogue

for dealing empirically with various types of self-excited,

impact maintained oscillations described by equations like

(2-1).

Although the analysis is necessarily more complex, the

theory can be extended to cover various other situations with-

out too much difficulty. If, for example, we consider a situ-

ation with a nonlinear spring we could have


MK + klx + k2x3 = F (4-1)


This could be analyzed in the same fashion as that in Chapter

II if some changes are made. The arithmetic decrement must

be recomputed for example. This can easily be done. The

potential energy for a spring with the nonlinear spring con-

stant is

2 4
kx kx
V + 2 (4-2)
2 4

The work, W, spent in overcoming friction will not depend on

the velocity (for Coulomb friction) but only on the path.





40


If xol is the amplitude, measured from equilibrium, of the

first maxima and xo2 the amplitude of the first minima, the

work done in overcoming friction will be


w = Ix0ll+X02 1)F.


(4-3)


The potential energies at x01 and x02 are


2
klx01
1 2


k- 2 ^
k2x01
4


and


2
klx02
2 2


k2x02
2


from (4-2). By conservation of energy we know that


V1 V2 = V


or, substituting in we have


2 4 2
klx1 k3x02 k02
2 4 2


4
k3xx02x02)F
4 (IX011+I x021)F


For k2 equal to zero we have the situation for the linear case,


IX01 -Ix02 = 2aL
^Ol1 ~ k~- ^L


(4-4)


with


2aL I1l 1x021


(4-5)


as noted in Chapter II.


For k2 not equal to zero we have


S 2 2 2 4
- (x01-x02 (x01-x02) = (1x011+1x011)F





41


Canceling like factors results in:

k k
21 (Ix01 IX02 I + -l4x 1 0 I2)( x01 Ix02|) = F.(4-6)


This can be factored and written as

2 X1 2+ 2 2F
(Iol 1 -Ix021)[1 + 2k (Ixol l +x02 l)]- k


Substituting in (4-5) and defining


a. F (4-7)
N. L.
1

as the nonlinear value for the arithmetic decrement, we get

2 2+ 12
N.L. = aL[l + 2k (lx012x02 2 (4-8)
1

as the formula relating a to the observed amplitudes.

It is possible to solve (4-1) on the phase plane

although a more involved process of geometrical construction

is required. First, the equation is rewritten in the form

S k2 3 F
2+ M- x + k- x + = 0


and letting

2 k1
P2 M1 (4-8)

we have

2 k2 3 F 2 2
2 +[p x px] + p x = 0

or


x + p2(x+6) = 0


(4-9)








where


6 2 [M x 3+ ] (4-10)
P

Equation (4-9) is now in the standard form for solution by

the phase-plane delta method [6, p. 244]. This method in-

volves approximating 6 as a constant over short intervals of

the motion and constructing the phase paths as short arcs

approximating the true path.

Linear friction can also be included in a very straight-

forward way if it is proportional to the simple velocity

[2, p. 170]. For linear damping of a type proportional to

xn where n is not equal to one and may be fractional, more

sophisticated methods may be useful although the phase-plane

delta method can still be used. For small nonlinearities a

variety of methods are available [3,4] and can be applied as

necessary. On the whole, it is remarkable that the toy wood-

pecker can be dealt with as effectively as it is without

requiring more advanced methods.













APPENDIX A


The problem is stated by Den Hartog as follows: The toy

woodpecker demonstrated at the end of the last lecture can be

idealized as shown in Figure A-I. The sleeve m of length Z

fits loosely around the fixed stem. The birdie (mass M)

tilts the sleeve slightly within the clearance, so that point

contact occurs at A and B. When M stands still, this is a

self-locking frictional system. When M moves upwards, the

torque of the sleeve is released and the sleeve slips down a

bit. Assume the sleeve mass m (including the arm to the left)

to have its center of gravity in the center of the vertical

stem. Let the friction coefficient be f, the same value for

motion as for rest. Let x be the downward displacement of

the sleeve and let y be the downward displacement of M rela-

tive to the sleeve m, measured from the position of static

equilibrium. The system is self-locking at rest (y = 0) and

x then is constant in time. For y negative (upwards) there

may be slip.

(a) Discuss the static equilibrium problem and write an

expression for the net downward force on m. This expression

is used only when positive; for negative values the friction

reverses sign. Here is the nonlinearity!

(b) Set up the two differential equations of motion of

the system. There will be two such pairs: one good while m
43



















K


(a)


Figure A-I








is slipping (variables xl,Y1); another good when m is at

rest (variables x2,y2).

(c) As you have seen in the model demonstration at the

end of Lecture 23, there exists a steady, periodic, motion

which is shown in Figure A-II in its general characteristics.

The period is w t = 6 divided into two parts. First the

sleeve slips from A (ant = 0) until B(wnt = o0), when is

seizes by friction. Between o < wnt < 6 the sleeve m is

locked on the stem and only the bird M moves (the y shown

above is positive, i.e., the bird is down, binding the sleeve

on the stem). Since in general 60 is not equal to 2r (but

somewhat greater probably) the natural frequency is not a .

Study Figure A-II until you clearly understand its physical

implications.

(d) Now, by exact, piecewise-linear method, write the

general solution of the four equations for x1,Y1 (good from

A to B above) and for x2,Y2 (good from B to C). Since all

individual equations are linear, this can be done without

great difficulty; the solution contains many (up to 6 or 7)

integration constants C1, C2, ....

(e) Set up the boundary (continuity) equations at the

points A, B, and C (wnt = 0, o o ). These will involve the

unknowns C1, C2, ..., o 0. Check and polish until you

have as many condition equations as there are unknowns.















































. = nt


Figure A-II








Solution

(a) When the sleeve is just ready to slip downwards, but

does not move yet, it must be in static equilibrium under

forces as shown in Figure A-III.

Moment equilibrium about the center requires:


(Mg + ky)a = Nk

or

N = (Mg + ky)a/k.


The vertically downward force on the sleeve then is


(M+m)g + ky 2fN = (M+m)g + ky 2 (Mg+ky)

( m 2fa 2fa
= Mg + m 2a + ky 1 (A-0)

M a 1
The toy's dimensions are approximately m = 10; = 4; f 4'

and it is clearly self-locking when not vibrating, i.e., for

y = 0. Then the downward force must be negative or at least

not positive;

m 2fa 1 2 9
1+ - 1 + 2 -1 < 0.


With these values, the downward driving force (a) becomes

-9_ Mg ky, which means that it drives downwards only for

negative, i.e., upward values of y. This is physically clear,

because for upward y, the normal force N is diminished and

the friction lock is broken.

(b) and (c) Now consider expression (A-0) to be the down-

ward driving force on the sleeve during slipping motion

































----


N






















tTVJ


Figure A-III


Mg + Ky


N


'I





49


(x > 0). Whenever x = 0, the expression (a) is meaningless,

because the friction is no longer fN, but smaller than that.

The differential equations of motion are:


mo = Mg(l + fa + kyl( 2a) (A-l)
for x > 0
(from A to B)
M(II + 1) + kyl = 0 (A-2)



for 0 = 0 (A-3)
for x = 0
(from B to C)
My2 + ky2 = 0 (A-4)

(d) To solve these differential equations (all linear),
2 k
use the notation W = 2 This wn is the natural frequency

of the bird on its spring when the sleeve is fixed and not

slipping; it is not the frequency of the periodic motion

including slip. Substitute x1 from equation (A-l) into

equation (A-2), divided by M:

2 M m 2fa k 2fa
S+ wn Y1 +- g( + M ) + m l -1(1) = 0

or

2 M 2fa M 2fa
l + wn Y[I + m- (1 )] = -g[1 + m- ( -- )].

Now let

M 2fa 2
1 + ( 2fa -A

M a 1
a constant of the system. (For = 10; = 4; f = we have

-A2 = -9 or A = +3). Then:







2 2
Y (A n) Y1 = Ag

with the solution:


Y1= CleAnt + C2e-A2nt g2 (A-5)
n

To find xl, we return to equation (A-2):


xl = -Yl Wn Y

R = -A2w2 [C eAwnt + C2e-Awnt] 2 [ g
X1 = -A1Wn n "

l = -(A2 + 1)wn2[CleALnt + C2e-Ant] + g

A2 1 [CeAwnt + Ce-Ant] gt2 + Ct + C. (A-6)
X 2 3 t 4

Now we pass to equations (A-3) and (A-4) for the x2,Y2 func-

tions for the no-slip domain. We leave equation (A-3)

without further work out, and integrate equation (A-4) to:

Y2 = C5 sin w t + C6 cos n t. (A-7)

The equations (A-5), (A-6), (A-3), and (A-7) are the answer

to part (d).

(e) Now the boundary conditions. Referring to the figure

on page 46, we see that the conditions for the y motion are:


(Yl) t= = (Y2) t=6 (A-8)
n n o

(Yl)w t=O = (Y2w t=6 (A-9)

(l) t = (Y2) =o (A-10)
n o n o








(y t = o (2) t= (A-11)
1 cm t=c = b2)w t=4
n o n o

The geometric conditions for the x motion are:


(Xl) t=o = 0 (A-12)
n

(xl)w t=0 0 (A-13)
n o

There is one more (and very important) condition for the x

motion of a mechanical nature. Just before A (before wnt = 0)

the sleeve was self-locking, but at point A it just ceases to

lock. The net driving force at A then just breaks away from

zero, so that


(:l)1 t=0 = 0. (A-14)
n

The acceleration just before B ( nt = 0 ) is not necessarily

zero. Summarizing, we have seven condition equations (A-8)

to (A-14) and apparently we have eight unknowns:


C1,C2 'C3,C4 C5 ,C6 ,o ,6 .


But we notice that the constant C4 only occurs in equation

(A-6) for the xl motion. The seven condition equations (A-8)

to (A-14) do not mention xl itself, but only x1 and xl, and

we see from equation (A-6) that the constant C4 drops out in

the differentiation. Thus, we have seven unknowns


C1,C2 ,C3,C5 ,C6,' o,6


in the seven equations (A-8) to (A-14). In what follows the

letters 4o and 6 occur many times; from now on we will








just write and 6 without subscripts, for simplicity. Sub-

stitution of equations (A-5), (A-6), (A-3), and (A-7) into

the condition equations (A-8) to (A-14) gives:


C1 + C2 - = C sin 6 + C6 cos 6
1 2 2 5 6
n

A(C1 C2) = C5 cos C6 sin O


CleA + C2eA - = C5 sin 4 + C6 cos 4
n


A CleA


- A C2e-A = C5 cos q C6 sin


-(A2 + 1) (C1 C2) + C3 = 0

0 3
-(A2 + 1) (Cle C2e-A 3 = 0
n

-(A2 + 1) 2(C1 + C2) + g = 0


Eliminate C3 between (A-12) and (A-13), leading to:


A +1 [C1(l-eA) C2 (-e-A)] + -- = 0.

n

Solve for C5 and C6 from (A-8) and (A-0):


(A-8)


(A-9)


(A-10)


(A-11)


(A-12)


(A-13)


(A-14)


(A-12*)


C5 = (Cl + C2 -)sin 6 + A(C1
n


C6 = (C1 + C2


- )cos 6 A(C1
n


- C2)cos 6


- C2) sin 6


and substitute these first into (A-10), and later into (A-1l)


CleAQ+C2e-AQ = (C1+C2- -g-) (sin sin 4 + cos 6 cos 4)
n n


+ A(C1-C2)(cos 0 sin 4 sin 6 sin P).








CleA1+C 2e-A 2 = (C+C- 2)cos(e-)-A(C -C2)sin(e-0).
n n (A-10*)

A CleA-_A C2 e- = (C1+C2 --) (sin 0 cos 4 cos a sinp)
W
n

+ A(C1-C2)(cos 0 cos p + sin 9 sin q)

A C2eA-A C2e-AO = (C1+C2 2-)sin(6-))+A(C1-C2)cos(e-c).
n (A-11*)

By this time the set of seven equations (A-8) to (A-14) have

been whittled down to four equations (A-10*), (A-11*),

(A-12*) and (A-14), involving the four variables Cl, C2',

and (6-0). We now solve for C1 and C2 from (A-14) and (A-12*):


C g 1-Ac-e-A
1 W 2 (A2 +1) 2-eAO -eA
1 2 2 A_ -A4


2 (A2+1) 2-e A-e
n







gC +A-2A+ -e
21 2 2-eAe -AO





P (A +1)


Then we substitute C1 and C2 and the two combinations

C1 +C2) and (C1-C2) first into equation (A-10*) and later

into (A-llA):
into (A-11*):








eA-Ace A- eA+Ae -Al 2(A21
P A -A + P A -A p(A+)
2-e -e 2-e A-e-Ac


2-eA_ -eeA


where we have used the abbreviation:

S= e p.

Dividing all terms by p and multiplying them with the denomi-

nator of most of the terms gives:

eA AqeA 1 + e-A1 + Aqe-AP = 1 = (A2+1)(2-eA-e-AP)


= -A2(2-eA-e-A )cos 4 2 -2A2+AeAAe-Ae-A)sin n

or

2 cosh Ap 2Ap sinh A 2 A2(1-cosh Ap) 2(1-cosh A4)

= -2A(1-cosh A))cos 4 + 2A(Ap-sinh Ac)sin 4


or, divided by 2:

[-2 -A2 + (A2+2)cosh AO Ap sinh Ap]

= [A2(cosh Ap-l)]cos 4 + [A(Ap-sinh Ap)]sin 4. (A-10**)

In the same manner, we find equation (A-ll**):

[-A sinh A1 + A2p cosh AO]

= -[A2(cosh A--l)]sin 4 + [A(Ap-sinh Ap)]cos 4. (A-ll**)

These are two equations in the unknowns p and 4, and they

have the form:








y = a cos + + B sin i

6= -a sin I + 6 cos '


where a, 6, Y, and 6 are complicated functions of '. We now

eliminate P from the above pair by squaring each one of them

and then adding:


2 + 62 = (a2 + 2)(cos2' + sin2i).


Our final result in the analytical part of the investigation

thus is:

2 + B2 y2 62 = 0,


where

a = A2(cosh A'-l)

6 = A(A'-sinh A')

y = (A2+2)(cosh A'-1) A' sinh A'

6 = -A sinh A' + A2 cosh A'.

Further analytical work does not seem to yield any simplifi-

cation, so that we now turn to numerical computation.

The quantity A, which must be positive, is a character-

istic parameter of the system, expressing the amount of dry

friction locking. For A = 0 [see equation (A-0)] the fric-

tion f is so small that the sleeve is just ready to slide

down with zero acceleration. For A > 0, there is a self-

locking at rest. The range of practical values of A appears

to be from 0 to 3. For A = 0, all four expressions a to 6 go

to zero, so that no direct conclusions can be drawn (we have








to employ a limit process). Hence, take


A = 0.1, 1, 2, 3.


Then put in P in steps of one degree from 0 to 360 degrees

and find the zeros of a2 + 2 Y2 It is possible

that several such zeros will be found. Since a squaring

operation has occurred at the end of our analysis, not all

of the P-solutions may be valid. For each q, the motion has

to be computed and inspected. This I have not done, and if

anyone ever does it, I shall be grateful if he or she will

send me the solution.


This concludes Den Hartog's remarks. There are probably

several reasons why the analysis is not more fruitful. With-

out considering any of the experimental results as to the

relative stability of the two models in Figure A-I, we can

see in the light of Chapter II that several things should have

been included in the above analysis:

1) Losses in the spring. These are assuredly present

and do influence the motion. When a larger than necessary

displacement is used to start the motion, most of the excess

energy seems to be dissipated in the spring as the sleeve is

not observed to start dropping for several cycles.

2) The impulsive nature of the motion when the sleeve

locks. In this light boundary condition (A-ll) is incorrect

and probably should read


(1)w t= + = (Y2 t=
n o n o





57


where v is related to the drop of the system as in Chapter

II.

3) Energy balance considerations. These would relate

directly to whether stability is theoretically possible since

if energy input per cycle is not equal to energy loss per

cycle the motion will either delay and cease, or grow uncon-

trollably.













APPENDIX B


Let us consider a situation where there are two constant

energy impulses delivered each period and let the losses be

due to linear (velocity dependent) friction. Let the loga-

rithmic decrement per half cycle be denoted by d, and let the

phase velocity be y. Just after an impulse let the phase

velocity be y1. A half cycle later it will be


y = -d2 (B-1)

Just after this an impulse occurs which changes the velocity:

2 2 = h2 (B-2
y2 Yo = h(B-2)

The quantity h2 is a constant representing a constant energy

input during the impulses. Just after the impulse the

velocity is thus


Y2 = [Y2 + h2]1/2

or

y2 = [12e-d + h2]1/2 (B-3)

using (B-l). We require that yl equal y2 for periodic

motion so that we have

h (B-4)
yl = 2 = Y -d 1/2 (B-4)
[1 -e ]








This defines a closed phase path corresponding to periodic

oscillations. The motion can be further investigated by con-

structing the graph of the sequence function (B-3) [4, p.

433], sometimes called "Lamerery's diagram" [2, p. 161]. This

is shown in Figure B-I. The vertical axis plots values of

Y2 and the horizontal axis, yl. For a stationary oscilla-
tion yl equals y2 and this is represented by a straight line

through the origin. If the oscillations are not stationary

the sequence of amplitudes can be determined by (B-3) and

Figure B-I. If the initial amplitude is y', then the subse-

quent amplitude will be y2, and the corrected amplitude will

be y2 on the yl axis. The next amplitude will be yl on the

Y2 axis and the sequence of amplitudes will continue until y
is reached, at which point a stable limit cycle exists.

Physically this corresponds to a situation where initially

the energy introduced is greater than that dissipated and the

oscillations grow in amplitude until a stable energy balance

is reached.

The important point to note is that y' can be made

arbitrarily small and the oscillations will still grow until

a stable limit cycle is reached. This implies soft excitation

is possible, a phenomenon which is not observed physically.

In fact, Andronov concludes his analysis [2, p. 172] with the

statement, "Again, a model with linear friction does not

explain the need for an initial finite impulse to start the

clock and must therefore be rejected. It is necessary to





























YZ =l t e-d -t IJ


Figure B-I








assume that the clock is a self-oscillating system with dry

friction."

The fact that Coulomb friction is appropriate can be

seen by considering a phase trajectory of the actual wood-

pecker. This is shown in Figure B-II. This trajectory

terminates on the x axis between the points (-b -f ,O) and

(b +f ,O) which characterizes the zone of static friction.

When a trajectory does not emerge from this zone the motion

ceases. Physically, this corresponds to the situation where

the amplitude of the bird mass is not sufficient to cause the

sleeve to unlock initially. The quantity b0 used in Chapter

II can now be seen as a parameter describing the friction

lock of the sleeve as far as its effect on the motion is con-

cerned.



















































-4.





-1%-(


III


A
-b~i** -hb


I II


I'.


Figure B-II


" L --













APPENDIX C


The following are examples of data collected from the

videotape made of the woodpecker motion. The times listed

were read from an electronic timer with a digital readout

and are accurate to 0.005 second. The hundredths position

was sometimes difficult to read. The elapsed time is more

definite with each reading occurring one-sixtieth of a second

after the last. The position readings refer to positions in

centimeters 0.15 read from a labeled grid behind the wood-

pecker. Vertical displacements are labeled y and horizontal

ones, x. The first series of readings are for a bird mass of

11.30 grams.


Sleeve y

0.2
0.2
0.2
0.2
0.2
0.2
0.3
0.4
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.6
0.6


Bird x

7.2
7.9
8.9
10.0
10.6
11.1
11.3
11.3
11.2
10.0
11.0
11.0
11.2
11.3
11.1
10.7
9.8


Bird y Time

8.8 3.80
8.2 3.82
7.4 3.84
5.9 3.86
4.8 3.87
3.2 3.89
3.0 3.91
1.0 3.92
0.1 3.94
-0.7 3.96
-0.7 3.97
-0.2 3.99
0.2 4.01
1.6 4.02
3.1 4.04
4.9 4.06
6.3 4.07
(continued)








Sleeve v Bird x Bird y Time

0.6 8.7 7.8 4.09
0.6 8.0 8.5 4.11
0.6 6.9 9.1 4.12
0.6 6.8 9.2 4.14
0.6 6.1 9.1 4.16
0.7 8.0 8.5 4.17
0.7 9.0 7.6 4.19
0.7 9.9 6.1 4.21
0.7 10.8 4.9 4.22
0.7 11.0 3.5 4.24
0.7 11.2 2.3 4.26
0.8 11.2 1.2 4.27
0.85 11.0 0.3 4.29
0.85 10.9 -0.1 4.31
0.85 10.8 -0.2 4.32
0.85 11.0 0.1 4.34
0.85 11.2 0.9 4.36
0.85 11.4 2.0 4.37
0.90 10.1 3.5 4.39
1.00 10.9 5.2 4.41
1.00 10.1 7.0 4.42
1.00 8.9 8.2 4.44
1.00 7.9 9.0 4.46
1.00 7.1 9.8 4.47
1.00 6.8 9.8 4.49
1.00 7.1 9.7 4.51
1.00 7.9 9.0 4.52
1.00 8.8 8.2 4.54
1.00 10.0 6.9 4.56
1.00 10.8 5.2 4.57
1.00 11.1 4.0 4.59
1.00 11.6 2.9 4.61
1.10 11.5 1.8 4.63



The following data were taken for a bird mass of 11.30


grams with the sleeve locked in place.



Bird x Bird y


12.1
12.0
12.0
11.9
12.0
12.0
12.0


7.3
6.5
6.0
5.9
6.2
6.8
7.7


Time


2.08
2.10
2.11
2.13
2.15
2.17
2.18
(continued)








Bird x Bird y Time

11.7 9.0 2.20
11.1 10.1 2.21
10.5 11.1 2.23
9.8 12.0 2.25
9.0 12.9 2.27
8.7 13.2 2.28
8.6 13.2 2.30
8.9 13.0 2.31
9.6 12.4 2.33
10.2 11.4 2.35
11.0 10.4 2.37
11.4 9.4 2.38
11.9 8.3 2.40
12.0 7.3 2.42
12.0 6.9 2.43
12.0 6.4 2.45
11.9 6.1 2.47
12.0 6.3 2.48
12.0 6.9 2.50



Figure C-I shows a graph of some of the data points for

the position of the sleeve versus time and Figure C-II shows

a graph of the speed of the sleeve versus time obtained by


calculating the slope of the line in Figure C-I.






66





































a-
IS







F












LI 'I 5 I I < I 1 I 0I 0 I V SW 20 10 o


cS<3mI3l33 A A-3a-s






67






































I t


















oII 01 b g L 7

C)Nc---/ (;N21?W'uaa-Lt tqDa7













LIST OF REFERENCES


1. Den Hartog, Jacob P., Nonlinear Vibrations Study Guide,
Massachusetts Institute of Technology, Cambridge, Mass.,
1972.

2. Andronov, A. A., Vitt, A. A., and Khaikin, S. E., Theory
of Oscillators, Addison-Wesley Publishing Company, Inc.,
Reading, Mass., 1966.

3. Butenin, N. V., Elements of the Theory of Nonlinear
Oscillations, Blaisdell Publishing Company, New York,
1965.

4. Minorsky, N., Introduction to Non-Linear Mechanics,
Edwards Brothers, Inc., Ann Arbor, Michigan, 1947.

5. Den Hartog, Jacob P., Mechanical Vibrations, McGraw-Hill
Book Company, Inc., New York, 1956.

6. Jacobsen, Lydik S., and Ayre, Robert S., Engineering
Vibrations, McGraw-Hill Book Company, Inc., New York,
1958.













BIOGRAPHICAL SKETCH


Russell A. Roy was born February 7, 1942, in Norristown,

Pennsylvania. He completed most of his secondary education

in Pennsylvania and was graduated from Winter Park High School

in Winter Park, Florida, in June, 1960. He received a B.A.

in physics from Swarthmore College in 1964 and an M.S. in

physics from the University of Florida in 1967. He has a

wife, Eileen, and two children, Kevin and Katy.








I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequa in scope and quality,
as a dissertation for the de re D- Philosophy.




K6 t T. Rll 11 ran
Pro essor of ering Science,
Mech nics A space
Engine g


I certify that I have read this study and that in my
opinion it conforms to acceptable stan yards of scholarly
presentation and is fully adequate, i scope and quality,
as a dissertation for the degree of cctor of Philosophy.




awrence E. Malvern
Professor of Engineering Science,
Mechanics, and Aerospace
Engineering


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.




Robert L. Sierakowski
Professor of Engineering Science,
Mechanics, and Aerospace
Engineering


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.




Richard L. Fearn
Assistant Professor of Engineer-
ing Science, Mechanics, and
Aerospace Engineering








I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.




Vasile M. Pop6v
Professor of Mathematics


This dissertation was submitted to the Graduate Faculty of
the College of Engineering and to the Graduate Council, and
was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.

December, 1974




e College of Engineering


Dean, Graduate School




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