APRESENTATION
ON
TRANSLATIONAL AND ROTIONAL SYSTEM
PRESENTED BY Vipin Kumar Maurya ROLL NO. 1604341510
CONTENTS1. Translational mechanical system2. 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
TRANSLATIONAL MECHANICAL SYSTEMSBACKGROUND AND BASICS VARIABLES
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)
Generally
Where f is the force applied and dx is the displacement.For constant forces
PowerPower 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.
Viscous frictionFriction, 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.
Stiffness elementsAny mechanical element which undergoes a change in shape
when subjected to a force, can be characterized by a stiffness element .
PulleysPulleys are often used in systems because they can change the
direction of motion in a translational system.The pulley is a nonlinear element.
Interconnection LawD’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 :
Law of Reaction forcesLaw of Reaction forces is Newton’s Third Law of motion often
applied to junctions of elements
Law for Displacements
Deriving the system model Example - Simple mass-spring-damper system.
INTRODUCTION OF ROTATIONAL SYSTEMA 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.
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
ELEMENT LAWS OF ROTATIONAL SYSTEMThere are three element laws of rotational
system.1. Moment of Inertia2. Viscous friction3. Rotational Stiffness
Moment of InertiaAs 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
Viscous frictionviscous friction would be occure when two
rotating bodies are separate by a film of oil (see below), or when rotational damping elements are employed
Rotational StiffnessRotational stiffness is usually associated with
a torsional spring (mainspring of a clock), or with a relatively thin, flexible shaft
GearsIdeal gears have1. No inertia 2. No friction 3. No stored energy 4. Perfect meshing of teeth
Interconnection Laws of Rotational systemD’Alembert’s LawLaw of Reaction TorquesLaw of Angular Displacements
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.
Law of Reaction TorquesFor 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
Law of Angular Displacements Algebraic sum of angular displacement
around any closed path is equal to zero
Obtaining the system model of Rotational systemProblem Given: Input , 𝑎(t) 𝜏 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
1. Draw Free-body diagram:
2. Apply D’Alembert’s Law
3. Define state variables
In state-variable form: