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INTRODUCTION TO SYSTEM THEORY
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D'Alembert's principle
Force-Voltage analogy
Force-current analogy
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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.
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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.
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• 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.
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Electric Analog of the mechanical system
It can be done by::
1. Force-voltage analogy
2. Force-Current analogy
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Force-voltage analogy
The force, f in mechanical system is analogous
to voltage, v in the electrical system in this
type of analogy.
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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.
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Example of f-v analogy
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mechanical system
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Electrical system analogous to mechanical system(f-v analogy)
Equation for mechanical system
Equation for electrical system (f-v analogy)
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Force-Current analogy
The force, f in mechanical system is analogous to current, i in the electrical system in this type
of analogy.
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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
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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.
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Example of f-i analogy
Mechanical system
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Electrical system analogous to mechanical system(f-i analogy)
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Equation for mechanical system
Equation for electrical system (f-i analogy)
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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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