Probabilistic Approach to Mechanisms

307 Pages·1984·4.377 MB·1-301\307

by  B.Z. Sandler (Eds.)

STUDIES IN APPLIED MECHANICS
1. Mechanics and Strength of Materials (Skalmierski)
2. Nonlinear Differential Equations (Fuöik and Kufner)
3. Mathematical Theory of Elastic and Elastico-Plastic Bodies
An Introduction (Neöasand Hlavaöek)
4. Variational, Incremental and Energy Methods
in Solid Mechanics and Shell Theory (Mason)
5. Mechanics of Structured Media, Parts A and Β (Selvadurai, Editor)
6. Mechanics of Material Behavior (Dvorak and Shield, Editors)
7. Mechanics of Granular Materials: New Models and Constitutive
Relations (Jenkins and Satake, Editors)
8. Probabilistic Approach to Mechanisms (Sandler)
STUDIES IN APPLIED MECHANICS 8
Probabilistic Approach
to Mechanisms
Β. Ζ. Sandler
Mechanical Engineering Department, Ben-Gurion
University of the Negev, Beer-Sheva, Israel
ELSEVIER
Amsterdam — Oxford — New York — Tokyo 1984
ELSEVIER SCIENCE PUBLISHERS B.V.
Molenwerf 1
P.O. Box 211, 1000 AE Amsterdam, The Netherlands
Distributors for the United States and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY INC.
52, Vanderbilt Avenue
New York, NY 10017
ISBN 0-444-42306-0 (Vol. 8)
ISBN 0-444-41758-3 (Series)
© Elsevier Science Publishers B.V., 1984
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or
transmitted in any form or by any means, electronic, mechanical, photocopying, recording or other-
wise, without the prior written permission of the publisher, Elsevier Science Publishers B.V., P.O. Box
330, 1000 AH Amsterdam, The Netherlands
Printed in The Netherlands
ν
ACKNOWLEDGEMENT
I express my deep gratitude and appreciation to my colleagues at the
Mechanical Engineering Department of the Ben Gurion University of the Negev for
their spiritual support and efficient cooperation in accumulating the material
for this book and assistance in writing it.
I express especially sincere gratitude and thanks to those who transformed
my poor Russian English into something readable, who edited, typed and helped
through their useful advice in manuscript writing. My thanks to Mrs. Inez
Muerinik, Mrs. Marion Milner and Mrs. Tzila Barneis. And, of course, many,
many thanks to my wonderful family and my wife whose warmth and humor helps me
in all aspects of my life.
VII
P R E F A CE
Probabilistics is a powerful tool for the processing of experimental and
computational data. As such, it has been in use for more than a century for
estimating the reliability of experimental and computational results in many
scientific and technological fields.
During the past 30-40 years, classical probabilistics have been reinforced by
the development of random function theory, particularly by Norbert Wiener and
Andrey Kolmogorov. Random function theory which is the very heart of cybernet-
ics was originally created to meet the needs of automatic control theory and
optimization. Later, especially in the 60s, the beginning of the era of
missiles and space investigation, random function theory was applied to problems
connected with mechanical vibrations of bodies and shells.
At that stage, probabilistics was almost never used in the investigation of
the kinematics and dynamics of mechanisms: on the contrary, situations in which
random factors were inherent were largely dealt with by deterministic means.
For instance, accuracy problems in linkages, cams, or gears were often reduced
to the common classical forms by making determined assumptions about the shape,
frequency, or nature of the errors. Classical deterministic analyses and syn-
thesis methods are attractive because of their ability, at least in principle,
to provide complete numerical solutions. On the other hand, a statistical
approach may in some cases shorten the computation process because this
technique inherently requires less complete information.
Although a great deal of work has been devoted to perfecting deterministic
computation techniques, and brilliant results have been achieved, there is a
certain sphere of problems for which the probabilistic approach can be fruit-
fully applied. These problems include, firstly, the estimation of the influence
of random factors on the action of a particular mechanism. Secondly, the
spectral theory of random processes can be effectively applied for estimating
the dynamic behavior of mechanical systems. Thirdly, the use of probabilities
can sometimes obviate the need for a master pattern. Lastly, the higher-order
kinematic pairs to which cam and gear mechanisms belong can be effectively
investigated by means of a probabilistic approach where conventional determin-
istic methods are practically useless (this book shows, for example, how random
function theory can be applied to the problems caused by random errors in cam
profiles and gear pitch). In summary, application of random function and
VIII
probabilistic theory deal with the concept of kinematic and dynamic accuracy of
mechanisms.
This book also deals with some other technical applications of probabilistic
theory, including those relating to pneumatic and hydraulic mechanisms and
rolling bearings. Kinematic and dynamic treatments are presented for both
linear and nonlinear cases. The text discusses both the analysis and synthesis
aspects of the mechanisms described. (Synthesis problems imply the optimiza-
tion of the system in line with fixed criteria.) The approach presented
illustrates that the spectral concept of random function theory is a powerful
tool for the optimal synthesis procedure even when the excitation is determined.
Optimization problems led the author to consider the possibility of the
creation of adaptive mechanisms, i.e. to the possibility of automatically
adapting the transform function of a mechanical system to the variations of a
randomly changing excitation. Some examples of automatic vibration control are
discussed and illustrated.
The possibility of obtaining reliable results for mass phenomena has
facilitated the development of a measurement technique which differs in essence
from classical measuring methods in that it does not require a master part.
Results are obtained by processing the statistical characteristics gathered by
comparison of the measured elements. Some examples of the application of this
probabilistic technique are given in the book.
This book consists of seven chapters, the material being divided as follows.
In Chapter I the main concepts of probability theory (sections 1-24) and random
function theory (sections 25-37) are discussed. This chapter may be omitted by
those readers who are familiar with the subject. Chapter II introduces the
general concepts of kinematics and dynamics of mechanisms and shows the major
general dependences and error transformations. Chapter III and IV are devoted
to the problems of kinematic and dynamic accuracy of cam and gear mechanisms,
respectively. In Chapter V we consider nonlinear kinematic and dynamic
problems, with emphasis on the influence of the backlash on the action of
mechanisms. Chapter VI is devoted to some special applications of statistics
to pneumatic, hydraulic, and belt-drive mechanisms. Finally, Chapter VII
deals with the automatic vibration control aspect of the creation of adaptive
mechanisms.
B.Z. Sandler
1
CHAPTER I
CONCEPTS OF PROBABILITY
1. INTRODUCTION
This chapter is devoted to describing the main concepts and definitions in
probability theory that the reader of this book will encounter. It does not
aspire to replace first-hand basic acquaintance with probability theory but is
aimed at helping the reader to become familiar with the designations and for-
mulas used consistently by the author. The intention is to refresh the memory
rather than to teach. Logical sequence is not always strictly observed: in
order to keep the listing brief, concepts are sometimes discussed before their
definition is given.
The chapter covers two main topics:
(1) Random variables (sections 2 to 24)
(2) Random functions (sections 25 to 37).
2. RANDOM VARIABLES
A variable - a quantity of unpredicted value - is generally defined as random.
In practice, a random variable changes within a specific range, and a definite
probability is associated with each value (for discrete variables) or with a
definite set of values (for continuous variables).
For example:
(1) Deviation of the real size of a machine part from the nominal value denoted
in the designer's working drawings.
(2) Deviation of the acceleration of a cam follower at some specific moment
from the designed value.
We will now give a number of definitions, each of which will be illustrated
with an example:
(1) Random events (RE)
e.g., whether or not a part will break in the course of a specific period of
time.
(2) Discrete random variables (DRV)
e.g., the number of rebounds of a relay contact during the process of closing.
(3) Continuous random variables (CRY)
e.g., the thickness of a tooth of a gear wheel.
2
(4) Random functions (RF)
e.g., the profile of a cam; the trace of a ball bearing.
(5) A function of a random variable (FRY)
e.g., the movement of a cam follower as a function of the rotation angle of the
cam.
3. EVENTS
Random variables and probability are closely connected with the concept of an
"event".
Mechanical examples of events are:
(1) acceleration exceeding some specified allowed value;
(2) breakage or nonbreakage of a machine part;
(3) a specific ball in a ball bearing coming into contact with the upper ring
of a thrust bearing;
(4) occurrence of a crack in a tooth of a wheel.
An event is a fact which as a result of an experiment may or may not occur.
For instance, the appearance of a part with a size bigger than predicted is an
event. We can express an "event" in terms of numbers.
We will now list the following group of definitions:
Impossible events - those which can never occur:
The probability of the occurrence of an impossible event P=0.
Trustworthy or sure events - those which always occur:
The probability of the occurrence of a trustworthy event P=l.
A complete whole group of events - a group of events, one of which always takes
place as a result of an experiment:
The probability of the occurrence of one of these events is P=l.
Mutually exclusive (incompatible) events - events which cannot occur simul-
taneously:
Thus, if Pj, P , P^,..., P^ are separate probabilities of such events, the
2
probability Ρ that one of these events will occur is
Ρ = P. + P + ... + Ρ
12 0 η
It should be emphasized that the sum of all probabilities of all possible out-
comes always equals 1.
Events not mutually exclusive - those which can occur separately and simul-
taneously:
In this case, the probability of obtaining event A or Β is given as
Ρ(A + Β) = Ρ(A) + P(B) - Ρ(AB)
where AB - the appearance of both events.
3
Independent events - events the occurrence of which does not depend on the
others :
If P^, 3>···Ρ> p a er t en probabilities of independent events, the prob-
n
ability Ρ that all of these events will occur in a trial is
Ρ = P, · Ρ · Ρ · ... · Ρ
12 3 η
Dependent events - events the occurrence of each of which depends on the
situation of one vis-a-vis the others:
Here, the concept of conditional probability has to be introduced. If P^ is the
probability of the occurrence of the first event and P^, P^,..., P^ are the
conditional probabilities of the occurrences of the subsequent events 2, 3,...,η
the probability Ρ that all subsequent events will occur in the specified order
is
Ρ = P.1 - Pi - Pi - ... - P'
12 3 η
4. PROBABILITY
In practice, we define probability as the ratio of successful outcomes to all
possible outcomes in an experiment or measurement:
Ν
where N is the number of successes and Nj is the number of outcomes of all
s
kinds or number of trials. As was stated above, the probability of an imposs-
ible event equals 0; the probability of a trustworthy event equals 1.
Hence, it is true that:
0 < Ρ(A) < 1,
where P(A) is the probability of a certain outcome A.
5. DISTRIBUTION
The distribution of a random variable describes its behavior: it connects
the values of the variable with their respective probabilities.
6. DISTRIBUTION OF A DISCRETE VARIABLE
The random variable is better described by a cumulative distribution func-
tion (CDF). This function F(x) describes the probability that X is equal or
less than a stated value x:
F(x) = P(X $ x)
4
If the random variable X changes over a range from x, to x then
?
F(x) =0 for X < χ
1
F(x) = 1 for X > x
2
F(x)
x
0
X α ß X
2
Fig. 1. A typical cumulative distribution function.
Figure 1 shows a typical CDF. Practical problems often demand a solution
to the question: "What is the probability that the random value falls within an
interval between α and 3?"
α < X < 3
i.e., we seek the probability Ρ(α < X < 3)·
With regard to the following events, we can state that:
for event A X < 3
for event Β X < α
for event C α < X < 3
Taking into account that A = Β + C, we can write
P(X < 3) = P(X < α) + P(a < X < 3)
or (1.6.1)
Ρ(α $ Χ < 3) = F(3) - F(α)
The probability that the random value falls within the given interval equals the
increment of the distribution function on this interval.
If α ^ 3 we find the probability P(X = 3). From equation (1.6.1) it follows
that only in the case that F has a jump at point 3 do we obtain a result which
equals the jump of the distribution function at this point.
Obviously, if the distribution function is continuous at point 3 then
P(X = 3) =0. The probability of the occurrence of any separate fixed value of
a continuous random variable equals 0.