2. CONTENTS
1. Translational mechanical system
2. Interconnection law
3. Introduction of rotational system
4. Variable of rotational system
5. Element law of rotational system
6. Interconnection law of rotational system
7. Obtaining the system model of Rotational system
8. References
4. x; v; a; f are all functions of time, although time dependence
normally dropped (i.e. we write x instead of x(t) etc.)
As normal
Work is a scalar quantity but can be either
Positive (work is begin done, energy is being dissipated)
Negative (energy is being supplied)
5. Generally
Where f is the force applied and dx is the displacement.
For constant forces
6. Power
Power is, roughly, the work done per unit time (hence a scalar
too)
Element Laws
( w(t0) is work done up to t0 )
The first step in obtaining the model of a system is to write down a
mathematical relationship that governs the Well-known formulae
which have been covered elsewhere.
7. Viscous friction
Friction, in a variety of forms, is commonly encountered in
mechanical systems. Depending on the nature of the friction
involved, the mathematical model of a friction element may
take a variety of forms. In this course we mainly consider
viscous friction and in this case a friction element is an
element where there are an algebra relationship between the
relative velocities of two bodies and the force exerted.
8. Stiffness elements
Any mechanical element which undergoes a change in shape
when subjected to a force, can be characterized by a stiffness
element .
Pulleys
Pulleys are often used in systems because they can change the
direction of motion in a translational system.
The pulley is a nonlinear element.
9. Interconnection Law
D’Alembert’s Law
D’Alembert’s Law is essentially a re-statement of Newton’s
2nd Law in a more convenient form. For a constant mass we
have :
10. Law of Reaction forces
Law of Reaction forces is Newton’s Third Law of motion often
applied to junctions of elements
Law for Displacements
12. INTRODUCTION OF ROTATIONAL SYSTEM
A transformation of a coordinate system in which the new
axes have a specified angular displacement from their
original position while the origin remains fixed. This type
of transformation is known as rotation transformation and
this motion is known as rotational motion.
13. VARIABLES OF ROTATIONAL SYSTEM
Symbol Variable Units
θ Angular displacement radian
ω Angular velocity rads-1
α Angular acceleration rads-2
T Torque Newton-metre
14. ELEMENT LAWS OF ROTATIONAL SYSTEM
There are three element laws of rotational system.
1. Moment of Inertia
2. Viscous friction
3. Rotational Stiffness
15. Moment of Inertia
As per Newton’s Second Law for rotational bodies
Jω is the angular momentum of body
is the net torque applied about the fixed axis of
rotation system.
J is moment of inertia
16. Viscous friction
viscous friction would be occure when two rotating
bodies are separate by a film of oil (see below), or
when rotational damping elements are employed
17. Rotational Stiffness
Rotational stiffness is usually associated with a
torsional spring (mainspring of a clock), or with a
relatively thin, flexible shaft
18. Gears
Ideal gears have
1. No inertia
2. No friction
3. No stored energy
4. Perfect meshing of teeth
19. Interconnection Laws of Rotational system
D’Alembert’s Law
Law of Reaction Torques
Law of Angular Displacements
20. D’Alembert’s Law
D’Alembert’s Law for rotational systems is essentially
a re-statement of Newton’s 2nd Law but this time for
rotating bodies. For a constant moment of Inertia we
have
Where sum of external torques’ acting
on
body.
21. Law of Reaction Torques
For two bodies rotating about the same axis, any
torque exerted by one element on another is a
accompanied by a reaction torque of equal
magnitude and opposite direction
22. Law of Angular Displacements
Algebraic sum of angular displacement around any
closed path is equal to zero
23. Obtaining the system model of Rotational system
Problem Given:
Input , 𝜏𝑎(t)
Outputs
Angular velocity of the disk (ω)
Counter clock-wise torque exerted by disc on
flexible shaft.
Derive the state variable model of the system
