Table Of ContentMathematical and Analytical Techniques with
Applications to Engineering
Livija Cveticanin
Dynamics of
Bodies with
Time-Variable
Mass
Mathematical and Analytical Techniques
with Applications to Engineering
The importance of mathematics in the study of problems arising from the real
world, and the increasing success with which it has been used to model situations
ranging from the purely deterministic to the stochastic, in all areas of today’s
Physical Sciences and Engineering, is well established. The progress in applicable
mathematics has been brought about by the extension and development of many
important analytical approaches and techniques, in areas both old and new,
frequently aided by the use of computers without which the solution of realistic
problemsinmodernPhysicalSciencesandEngineeringwouldotherwisehavebeen
impossible.Thepurposeoftheseriesistomakeavailableauthoritative,uptodate,
and self-contained accounts of some of the most important and useful of these
analytical approaches and techniques. Each volume in the series will provide a
detailed introduction toa specific subjectarea of current importance, and then will
go beyond this by reviewing recent contributions, thereby serving as a valuable
reference source.
More information about this series at http://www.springer.com/series/7311
Livija Cveticanin
Dynamics of Bodies
with Time-Variable Mass
123
Livija Cveticanin
Faculty of Technical Sciences
University of NoviSad
NoviSad
Serbia
ISSN 1559-7458 ISSN 1559-7466 (electronic)
Mathematical andAnalytical Techniques with Applications to Engineering
ISBN978-3-319-22055-0 ISBN978-3-319-22056-7 (eBook)
DOI 10.1007/978-3-319-22056-7
LibraryofCongressControlNumber:2015946063
SpringerChamHeidelbergNewYorkDordrechtLondon
©SpringerInternationalPublishingSwitzerland2016
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
foranyerrorsoromissionsthatmayhavebeenmade.
Printedonacid-freepaper
SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia
(www.springer.com)
My family has provided the inspiration for
this work and I gratefully dedicate it to them.
To:
my mother Manci and father Jenci,
husband Miroslav,
son Dragan, and
granddaughters Nikoleta, Nora and Ema.
Preface
This book is the result of my long-time investigation and interest in the field of
dynamics of mass variable systems. In this book, the terms ‘mass variable system’
and ‘mass variation’ include change of mass, but also variation of moment of
inertia.
Asignificantnumberofrealsystemsexistwheremassandmomentofinertiaare
varying in time. Let us mention some of them: celestial bodies (earth, moon), but
also machines like centrifuges, pumps, fans, turbines, sieves, transportation
mechanisms and rotors used in textile, cable, paper industry, etc. Mass variation
affects some special phenomena which are not known in dynamics of bodies with
constant mass. The book treats these occurrences and has an intention to explain
them. The problem of mass variation is divided into two groups: discontinual and
continual.
For discontinual mass variation, a particle or a body is separated or is added to
aninitialsystemduringinfinitesimalshorttime.Thistypeofmassvariationoccurs
in nature, but also in machines and mechanisms in everyday use. Due to this
occurrence,velocityandangularvelocityhaveajump-likevariationwhichmaybe
the reason for serious malfunctions and even harmful damages in machines. In the
text, methods for obtaining dynamic characteristics of the final system after mass
variation are treated. Procedures are based on the well-known principle of
momentum and principle of angular momentum. The problem is also solved using
statements of analytic mechanics. Theoretical results are applied for a rotor which
part is separated, for in-plane separation of the beam, for particle separation of a
four-particle system and a shovel disjoint of a wind turbine. Knowledge of
dynamics of discontinual mass variation provides us valuable information about
motion of systems and gives thepossibility of preconception of motion properties.
In systems with continual mass variation, mass-time function is a continual and
monotone one. Due to mass variation, reactive force occurs which depends on
relativevelocityofmassadditionorseparation.Inthisbookthereactivetorque,asa
result of time variation of the moment of inertia and relative angular velocity of
body variation, is introduced. Influence of reactive force and of reactive torque on
motion of the system is analysed. Results of investigation are applied for a mass
vii
viii Preface
variable rotor: a drum on which the band is winding up. This model is useful in
machines in textile, paper, cable industry, etc. Research on energy distribution in
this type of nonconservative systems is very important. Lagrange’s equations of
motion, which include generalized force due to reactive force and reactive torque,
areformedandconsidered.Asoneoftheresults,energyvariationofmassvariable
system is obtained. Noether’s theorem and Killing’s equations are adopted for
determining of first integrals and conditions for existence of conservation law in
such nonconservative system.
In the book, special attention is directed toward oscillatory motion of mass
variable systems. In investigation, the nonlinear properties of the system are
included. Methods for determination of oscillations of nonlinear systems with
variable mass are developed. Analytic solving procedures, based on the Ateb and
trigonometric functions, are introduced. Interaction between reactive force and
damping force is determined. Vibrations of truly nonlinear oscillator, Duffing
oscillator, oscillator with fifth order of nonlinearity, and of Van der Pol oscillator
arepresented.Theoreticalresultsareappliedonsolvingtheproblemofvibrationof
a Jeffcott rotor and a two-body system with nonlinear connection.
The book is an extension of Dynamics of bodies with constant mass to
Dynamics of bodies with mass variation, with specifications of phenomena which
aretheresultofmasschangeintime.Thebookissuitabletobeusedbyresearchers
in dynamics of mass variable systems in all fields of science, for example:
mechanics, mechanical devices, electromechanical systems, electric circuits,
physics, chemistry, biology, medicine, etc. Besides, it is applicable for engineers
andtechniciansforsolvingpracticalproblemsinmatterandtoapprovedesignsand
eliminatethepossiblecollapseofmachines.Thebookisassignedalsoasatextbook
for students in dynamics of mass variable systems.
This book is the extended version of my MTA dissertation awarded from the
Hungarian Academy of Sciences.
Novi Sad Livija Cveticanin
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Linear and Angular Momentums . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Linear Momentum for Body Separation. . . . . . . . . . . . . . . . . . 12
2.2 Angular Momentum for Body Separation . . . . . . . . . . . . . . . . 13
2.3 Linear Momentum for Body Addition. . . . . . . . . . . . . . . . . . . 14
2.4 Angular Momentum for Body Addition. . . . . . . . . . . . . . . . . . 15
3 Discontinual Mass Variation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Discontinual Mass Separation. . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.1 Velocity After Mass Separation . . . . . . . . . . . . . . . . . 21
3.1.2 Angular Velocity After Mass Separation . . . . . . . . . . . 22
3.2 Discontinual Mass Addition. . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.1 Velocity After Mass Addition. . . . . . . . . . . . . . . . . . . 23
3.2.2 Angular Velocity After Mass Addition . . . . . . . . . . . . 24
3.2.3 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 In-Plane Discontinual Mass Variation . . . . . . . . . . . . . . . . . . . 25
3.4 In-Plane Body Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4.1 Some Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.5 In-Plane Addition of the Body. . . . . . . . . . . . . . . . . . . . . . . . 33
3.6 Analytical Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.6.1 Lagrange-D’Alembert Principle . . . . . . . . . . . . . . . . . 36
3.6.2 Virtual Work of the Impulse of the Impact
Force and Torque. . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.6.3 Modified Lagrange’s Equations . . . . . . . . . . . . . . . . . 38
3.6.4 Kinetic Energy Variation for the In-Line
Particle Mass Variation. . . . . . . . . . . . . . . . . . . . . . . 43
ix
x Contents
3.7 Separation of a Part of the Rotor . . . . . . . . . . . . . . . . . . . . . . 45
3.7.1 Motion of the Initial Rotor. . . . . . . . . . . . . . . . . . . . . 46
3.7.2 Rotor Separation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.7.3 Motion of the Rotor After Mass Separation . . . . . . . . . 49
3.8 In-Plane Separation of a Beam. . . . . . . . . . . . . . . . . . . . . . . . 50
3.9 Separation from a Four-Particle-System. . . . . . . . . . . . . . . . . . 55
3.9.1 In-Plane Particle Separation from a System . . . . . . . . . 57
3.9.2 Dynamics of the System After Particle Separation. . . . . 59
3.9.3 Friction Force is Dominant . . . . . . . . . . . . . . . . . . . . 60
3.9.4 Damping Force Acts. . . . . . . . . . . . . . . . . . . . . . . . . 67
3.9.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.10 Mass Variation of a Wind Turbine . . . . . . . . . . . . . . . . . . . . . 76
3.10.1 Dynamics of a Wind Turbine Due to Bending
of the Shovel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.10.2 Dynamics of a Wind Turbine During Separating
of the Shovel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4 Continual Mass Variation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.1 Dynamics of the Mass Variable Body. . . . . . . . . . . . . . . . . . . 86
4.1.1 Discussion of Differential Equations of Motion . . . . . . 90
4.2 Band is Winding up on a Drum. . . . . . . . . . . . . . . . . . . . . . . 92
4.2.1 Geometric and Physical Properties
of the Drum with Band. . . . . . . . . . . . . . . . . . . . . . . 93
4.2.2 Forces Acting on the System . . . . . . . . . . . . . . . . . . . 94
4.2.3 Shaft is Rigid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.2.4 Shaft is Elastic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3 Analytical Mechanics for Mass Variation. . . . . . . . . . . . . . . . . 99
4.4 Lagrange-D’Alembert Principle . . . . . . . . . . . . . . . . . . . . . . . 100
4.5 Lagrange’s Second Order Differential Equations. . . . . . . . . . . . 100
4.6 Lagrange’s Equation for Body with Variable Mass. . . . . . . . . . 103
4.7 Lagrange’s Equation and Lagrangian . . . . . . . . . . . . . . . . . . . 106
4.8 Conservation Laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.9 Conservation Law of Energy Type. . . . . . . . . . . . . . . . . . . . . 108
4.9.1 Ciolkowsky Problem. . . . . . . . . . . . . . . . . . . . . . . . . 109
4.10 Conservation Law of Arbitrary Type. . . . . . . . . . . . . . . . . . . . 114
4.10.1 Killing’s Equations. . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.10.2 Conservation Law for a Vibrating Machine
with Variable Mass. . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.11 Conservation Law for Truly Nonconservative System. . . . . . . . 122
4.11.1 Conservation Law for an Oscillator
with Variable Mass. . . . . . . . . . . . . . . . . . . . . . . . . . 123
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
