Vol.2, No 10, 2000 pp. 1453-1454
DYNAMICS OF MACHINES WITH VARIABLE MASS
by L. Cvetičanin
Publisher: Gordon and Breach Science Publishers,
This book is designed to be a complete and integrated text on the dynamic
properties of machines, mechanisms and rotors with variable mass. New results
are presented from investigations based on the general dynamics of systems
with variable parameters. The book considers both weak and strong nonlinear
vibrations of these systems and chaotic phenomena are also discussed. The
conservation laws and adiabatic invariants for systems with variable mass
are formulated and the stability and instability conditions of motion are
The book contains seven chapters for which brief description are given
Chapter 1 describes various machines and rotors with time variable
mass and parameters such as centrifuges, sieves for sorting particles,
transportation mechanisms, lifting mechanisms, cranes, automatic weight
measuring instruments, and rotors used in the textile, cable and paper
Chapter 2 introduces modelling of machines with variable mass. The
classical laws of mechanics for constant mass are transformed for systems
with variable mass. The theory of rigid body is specially treated. The
Lagrange procedure for forming differential equations of motion for systems
with variable mass is developed.
Chapter 3 and 4 consider the oscillations of machines with variable
mass. Nonlinear oscillations of systems with time variable parameters are
denoted analytically and compared with numerical results. Systems considered
are not only those with small nonlinearity but also those with strong nonlinearity.
Differential equations with a complex deflection function are examined,
as they describe the motion of the majority of rotors with variable mass.
The methods adopted for solving such equations are the Bogolubov-Mitropolski
method, the elliptic-Krylov-Bogolubov method, the method of slow variable
amplitude and phase, and the method of multiple scales. The influence of
reactive force on the motion of the machine is investigated. It can be
seen that reactive force has a fundamental influence on dynamic behavior
of rotors and mechanisms with variable mass. Therefore, it cannot be neglected.
Chapter 5 treats the problem of chaos in nonlinear systems with variable
mass. The analytical Melnikov criteria are extended for such systems. Two
examples of real machines are considered: chaos in the crane mechanism
and chaos in the separator.
Chapter 6 is about the formulation of conservation laws and adiabatic
invariants for systems with variable mass. The first integrals are obtained
using the theory of Emmy Noether. Examples of a rotor with variable mass
and of a sieve for separating particles are also discussed. The adiabatic
invariants for systems with one- and two-degrees-of-freedom for different
types of no linearity are also considered. The adiabatic invariants are
applied for finding approximate analytical solutions for nonlinear differential
equations of motion.
Chapter 7 deals with problems of stability in systems with variable
mass. The direct Lyapunov theorems of stability, asymptotic stability and
instability are applied. Some examples of Lyapunov function formation for
systems with variable mass are shown. In addition some new theories of
stability based on the Lyapunov theory are presented and in spite of some
limitations, they are simpler than previous procedures and thus much more
convenient for engineers. The influence of reactive force on the stability
properties of the system is also analyzed.
This book aims to expand previous knowledge and is intended for engineers
and investigators dealing in the fields of mathematics, mechanics and physics
with the problems of real machines with variable mass.
Prof. Dr. Djordje Djukić, corresponding member
of Serbian Academy of Sciences and Arts