Translational and Rotational system
-
Upload
vipin-maurya -
Category
Engineering
-
view
113 -
download
7
Transcript of Translational and Rotational system
![Page 1: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/1.jpg)
APRESENTATION
ON
TRANSLATIONAL AND ROTIONAL SYSTEM
PRESENTED BY Vipin Kumar Maurya ROLL NO. 1604341510
![Page 2: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/2.jpg)
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
![Page 3: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/3.jpg)
TRANSLATIONAL MECHANICAL SYSTEMSBACKGROUND AND BASICS VARIABLES
![Page 4: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/4.jpg)
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)
![Page 5: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/5.jpg)
Generally
Where f is the force applied and dx is the displacement.For constant forces
![Page 6: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/6.jpg)
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.
![Page 7: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/7.jpg)
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.
![Page 8: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/8.jpg)
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.
![Page 9: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/9.jpg)
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 :
![Page 10: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/10.jpg)
Law of Reaction forcesLaw of Reaction forces is Newton’s Third Law of motion often
applied to junctions of elements
Law for Displacements
![Page 11: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/11.jpg)
Deriving the system model Example - Simple mass-spring-damper system.
![Page 12: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/12.jpg)
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.
![Page 13: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/13.jpg)
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
![Page 14: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/14.jpg)
ELEMENT LAWS OF ROTATIONAL SYSTEMThere are three element laws of rotational
system.1. Moment of Inertia2. Viscous friction3. Rotational Stiffness
![Page 15: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/15.jpg)
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
![Page 16: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/16.jpg)
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
![Page 17: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/17.jpg)
Rotational StiffnessRotational stiffness is usually associated with
a torsional spring (mainspring of a clock), or with a relatively thin, flexible shaft
![Page 18: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/18.jpg)
GearsIdeal gears have1. No inertia 2. No friction 3. No stored energy 4. Perfect meshing of teeth
![Page 19: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/19.jpg)
Interconnection Laws of Rotational systemD’Alembert’s LawLaw of Reaction TorquesLaw of Angular Displacements
![Page 20: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/20.jpg)
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.
![Page 21: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/21.jpg)
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
![Page 22: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/22.jpg)
Law of Angular Displacements Algebraic sum of angular displacement
around any closed path is equal to zero
![Page 23: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/23.jpg)
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
![Page 24: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/24.jpg)
1. Draw Free-body diagram:
![Page 25: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/25.jpg)
2. Apply D’Alembert’s Law
3. Define state variables
In state-variable form:
![Page 26: Translational and Rotational system](https://reader034.fdocuments.net/reader034/viewer/2022042502/5878edd71a28abfa038b712f/html5/thumbnails/26.jpg)