IST assignment

INTRODUCTION TO SYSTEM THEORY

Electricals and Electronics

D'Alembert's principle

Force-Voltage analogy

Force-current analogy



D'Alembert's principle, also known as the Lagrange–

d'Alembert principle, is a statement of the

fundamental classical laws of motion. It is named after

its discoverer, the French

physicist and mathematician Jean le Rond d’Alembert.

Introduction

• Statement

• The D’Alembert’s principle states that the sum of the

differences between the forces acting on a system and

the time derivatives of the momenta of the system

itself along a virtual displacement consistent with the

constraints of the system, is zero.



Where,F= is the applied force,∂r= is the virtual displacement of the system, consistent with the constraint m = is the mass of the particles in the system,a = is the acceleration of the particles in the system,m.a = together as products represent the time derivatives of the system momenta,i = s an integer used to indicate (via subscript) a variable corresponding to a particular particle

Another way of stating D’Alembert’s principle

• For any body, the algebraic sum of externally applied

forces and the forces resisting motion in any given

direction is zero.

• In rotational mechanical system, D’Alembert’s principle

can be stated as: For any body the algebraic sum of

externally applied torques and the torque resisting

rotation about any axis is zero.


• It is the dynamic analogue to the principle of virtual

work for applied forces in a static system and in fact is

more general than Hamilton’s principle, avoiding

restriction to holonomic systems. A holonomic

constraint depends only on the coordinates and time.

It does not depend on the velocities.


Electric Analog of the mechanical system

It can be done by::

1. Force-voltage analogy

2. Force-Current analogy


Force-voltage analogy

The force, f in mechanical system is analogous

to voltage, v in the electrical system in this

type of analogy.



Table of conversion for F-V analogyMechanical system Electrical system(f-v analogy)

Force, f Voltage, V

Velocity, u Current, I

Displacement, x Charge, q

Mass, M Inductance, L

Damping coefficient, D Resistance, R

Compliance, K capacitance, C

Rule for drawing f-v analogous circuits.

Each junction in the mechanical system corresponds to a

closed loop which consists of electrical excitations

sources and passive elements analogous to the

mechanical driving sources and passive elements

connected to the junction. All points on a rigid mass are

considered as the same junction.


Example of f-v analogy


mechanical system


Electrical system analogous to mechanical system(f-v analogy)

Equation for mechanical system

Equation for electrical system (f-v analogy)



Force-Current analogy

The force, f in mechanical system is analogous to current, i in the electrical system in this type

of analogy.


Table of conversion for f-i analogyMechanical system Electrical system(f-i analogy)

Force, f Current, i

Velocity, u Voltage, v

Displacement, x Flux linkage

Mass, M Capacitance, C

Damping coefficient, D Conductance, G

Compliance, K Inductance, L


Rule for drawing f-i analogous circuits

Each junction in the mechanical system corresponds to a node (junction) which joins

electrical excitation sources to passive elements analogous to the mechanical driving sources and to passive elements connected to the junction. All points on a rigid mass is always connected to the

ground.


Example of f-i analogy

Mechanical system


Electrical system analogous to mechanical system(f-i analogy)


Equation for mechanical system

Equation for electrical system (f-i analogy)


Usefulness of analogy• The behaviour of a system can be completely

predicted by knowing the behaviour of its analogous system.

• For example, the behaviour of the mechanical system can be easily predicted by what we know about the

simple analogous electrical circuit.


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