STATE SPACE MODEL OF THE MECHANICAL SUBSYSTEM OF THE ELEVATOR MECHATRONIC SYSTEMS

Abstract

One of the typical examples of mechatronic systems are modern elevators. At the same time, elevators
are complex mechatronic systems that must ensure a high level of precision in operation, as well as comfort and
safety for users. An integral part of the elevator mechatronic system (EMS) is the mechanical subsystem, which, like
any mechanical structure, is characterized by the existence of resonant frequencies. In order for the EMS to achieve
high performance, the control signals must have a rich spectrum that also contains resonant frequencies. In order to
determine the resonant frequencies, we can use computer simulations. For computer simulations, a mathematical
model of the mechanical subsystem of the EMS is necessary. This paper presents the derived dynamic model of the
mechanical subsystem of the EMS which is in the form of state space for its easier dynamic analysis by computer
simulations. The mechanical scheme which is considered in this paper consists of an elevator car and a
counterweight as two lumped masses and two overhead sheaves and one drive sheave as three inertia masses.
Masses are coupled by the steel wire rope. The dynamic model of wire ropes can be expressed in several ways. The
rope model as an inelastic rigid body may be satisfactory in some applications but it is not suitable for derivation of
a dynamic model. Hook's ideal elastic body and Standard model are often used in modeling, but in the case
described in this paper they are not adequate. Hook's ideal elastic body has no modeled damping, and the Standard
Model is too complex to parameterize. For derivation of the dynamic model of the mechanical subsystem of the
EMS, a steel wire rope model is presented with a spring of great stiffness and with damping. Actually, Calvin’s
model is used and proved to be completely satisfactory. A 5 DOF model, i.e. five differential equations, was
obtained, which, based on Cauchy's principle and by introducing variable states of the system, were adapted to be
presented in the matrix form. Derived model it has been tested and verified through computational simulations and
compared with experimental results obtained on small-scale real model of the EMS. The existence of resonant
vibrations in the EMS is proved and presented. The step change of the drive torque is used for time response of the
speed. Results clearly shows that the speed response has a damped oscillation. Frequencies of oscillations are equal
to resonant frequency of the EMS mechanical subsystem. For frequency analysis Bode plots are used, providing
much more information than time response plots. Results shows that the EMS has one resonant frequency that
depends only on the structure of the EMS and does not depend on the load in the EMS’s cabin. In order to be able to
determine the resonant frequency or the frequency range in which the resonant frequency is located, it is necessary
to use the presented dynamic model of the EMS mechanical subsystem. It is a simple and very quick tool and a
mandatory first step to synthesize the speed and position control of the EMS.