Dynamic Equations and System-model of a Simple System

Dynamic Equations and System-model of a Simple System

• It is quite necessary to analyze electro-mechanical conversion system for predicting the performance and/or for monitoring the system. A simple system is being taken up here to deal with dynamic equations and a simple model with its components is being related to the system. The details will vary from system to system, and accordingly the equations will vary.
• Figure shows different components of such a system meant for electrical to mechanical conversion. On one side, an electrical source feeds the device at the ‘electrical port’. On the other side, a force f_e is developed at the ‘mechanical port’. Mechanical load is connected to this port.

(a) At Electrical Port : A voltage source is shown to feed the device. r is its effective internal resistance. At the electrical port, the inputs are λ (= flux linkage with the coil) and i. From λ , the voltage induced in the coil can always be evaluated.

(b) Role of the Conversion device : With these inputs, the device converts the energy into

mechanical form, and is available as a force f_e (in case of linear motions), and, displacement x

measured from a suitable reference.

(c) At the Mechanical Port : The possible items are: spring, damper, mass and an applied

mechanical force. Their natural and simple dependence on displacement x and its derivatives are

indicated below:

(i) Spring: Force required to overcome spring elongation is proportional to the

displacement x.

(ii) Damper: Force required to overcome damping action in the system is proportional to

derivative of x.

(iii) Mass: Force required to overcome acceleration of mass is proportional to second

derivative of x.

(iv) Applied force, f_o : This has to be overcome by f_e. In terms of an equation, these terms

are related as follows:

            f_e = k_s(x – x_o) + Bx + Mx + f_o

where

    k_s = spring constant

    x_o = value of x for unstretched spring

    B = damping constant

    M = Mass to be accelerated

    f_o = External mechanical force applied to the system.