Dynamics of a Mechanical Drive

Abstract

The equations derived in Chap. 1

$$ J\frac{{d\omega }} {{dt}} = m_M (\omega ,\varepsilon ,y_M ,t) - m_L (\omega ,\varepsilon ,y_L ,t), $$

((2.1))

$$ \frac{{d\varepsilon }} {{dt}} = \omega , $$

((2.2))

describe the dynamic behaviour of a mechanical drive with constant inertia in steady state condition and during transients. Stiff coupling between the different parts of the drive is assumed so that all partial masses may be lumped into one common inertia. The equations are written as state equations for the continuous state variables ω, ε involving energy storage, e. g. [38,88]). Only mechanical transients are considered; a more detailed description would have to take into account the electrical transients defined by additional state variables and differential equations. The same is true for the load torque m_L which depends on dynamic effects in the load, such as a machine tool or an elevator. Also the control inputs y_M, y_L to the actuators on the motor and load side have to be included. Fig. 2.1 shows a block diagram, representing the interactions of the mechanical system in graphical form. The output variables of the two integrators are the continuous state variables, characterising the energy state of the system at any instant. Linear transfer elements, such as integrators with fixed time constants, are depicted by blocks with single frames containing a figure of the step response. A block with double frame denominates a nonlinear function; if it represents an instantaneous, i. e. static, nonlinearity, its characteristic function is indicated. The nonlinear blocks in Fig. 2.1 may contain additional dynamic states described by differential equations.