translational and rotational system

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: