Module 19: Solve a Relative Acceleration Problem

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Engenharia de Sistema em Movimento: A dinâmica de partículas e corpos em movimento 2D

190 classificações

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Georgia Institute of Technology

Engenharia de Sistema em Movimento: A dinâmica de partículas e corpos em movimento 2D

190 classificações

This course is an introduction to the study of bodies in motion as applied to engineering systems and structures. We will study the dynamics of particle motion and bodies in rigid planar (2D) motion. This will consist of both the kinematics and kinetics of motion. Kinematics deals with the geometrical aspects of motion describing position, velocity, and acceleration, all as a function of time. Kinetics is the study of forces acting on these bodies and how it affects their motion. --------------------------- Recommended Background: To be successful in the course you will need to have mastered basic engineering mechanics concepts and to have successfully completed my courses en titled an “Introduction to Engineering Mechanics” and “Applications in Engineering Mechanics.” We will apply many of the engineering fundamentals learned in those classes and you will need those skills before attempting this course. --------------------------- Suggested Readings: While no specific textbook is required, this course is designed to be compatible with any standard engineering dynamics textbook. You will find a book like this useful as a reference and for completing additional practice problems to enhance your learning of the material. --------------------------- The copyright of all content and materials in this course are owned by either the Georgia Tech Research Corporation or Dr. Wayne Whiteman. By participating in the course or using the content or materials, whether in whole or in part, you agree that you may download and use any content and/or material in this course for your own personal, non-commercial use only in a manner consistent with a student of any academic course. Any other use of the content and materials, including use by other academic universities or entities, is prohibited without express written permission of the Georgia Tech Research Corporation. Interested parties may contact Dr. Wayne Whiteman directly for information regarding the procedure to obtain a non-exclusive license.

Na lição

Planar (2D) Rigid Body Kinematics I

In this section students will learn about planar (2D) rigid body kinematics, relative velocity equation, rotation about a fixed axis, instantaneous center of zero velocity, and relative acceleration equations.

Module 14: Define Rigid Body Kinematics; Identify three types of Planar Rigid Body Motion; Derive Relative Velocity Equation11:12

Module 15: Solve a Relative Velocity problem9:20

Module 16: Define and Locate the Instantaneous Center of Zero Velocity (IC)10:21

Module 17: Solve an Instantaneous Center of Zero Velocity (IC) Problem9:30

Module 18: Define Angular Acceleration; Derive the Relative Acceleration Equation7:54

Module 19: Solve a Relative Acceleration Problem9:39

Conheça os instrutores

Dr. Wayne Whiteman, PE
Senior Academic Professional
Woodruff School of Mechanical Engineering

[BLANK_AUDIO]

Hi, this is Module 19 of Two Dimensional Dynamics.

Previously we looked at the velocities for two dimensional motion of bodies.

We looked at the velocity between, the linear velocity between two points

on a body as related to the, the angular velocity of the body.

Now we're going to extend that for accelerations of

a body in, in, in planar 2D motion.

And so, here was the, the theory that we did before.

And, so, we're going to go ahead and apply this

acceleration equation to this problem which we did for velocities.

And so we have omega one is ten, clockwise.

Omega two is 2.42 radians per second, clockwise.

We found that from the velocity problem.

And we have the angular acceleration is

five radians per second counter-clockwise, on body B1.

And we want to find again the linear translation of, of piston B.

So I'm going to go ahead and apply the,

the the acceleration equation we came up with.

And so I've got the acceleration of the unknown point A is equal

to the acceleration of the known point, O, which is zero, because it's a pin,

plus, we're working on body O B, or O A. We're going from

knowns to unknowns here. And so I'm going to have theta double

dot B 1. And that's in the k

direction. Crossed with the position vector from the

known to the unknown, or from O to A. And then I've got minus theta

dot B1 squared r from O to A.

And so let's go ahead and put those values in, so I've got the acceleration of A is

equal to, well, we do know that the acceleration

at, of O is zero, because it's a pin.

And then I have plus, theta double dot K, for body B1 is given.

It's alpha one, it's five radians per second.

Counter-clockwise is going to be in the positive

K direction, so this is going to be

5K crossed with r from O to A, is, I

go minus 3 in the X direction plus 4 in the Y direction.

So minus 3i plus 4j, minus, now,

theta dot B1 squared.

The magnitude of the angular velocity of bar B 1 is 10, so this is going to be

10 squared times r from O to A, that's again minus

3i, plus 4j.

If you do that multiplication, I'll let you do that on your own, you'll

get the acceleration of A now, is equal to 280

i minus 415 j inches

per second squared. So let's continue on from there.

Here's our acceleration of A, which we just found.

So I've gone from this known to this unknown, which is now known.

And now I want to go down to the acceleration of B which

is unknown, and, and what we're trying to solve for in the problem.

So, I've got, the acceleration of B now,

is going to be equal to the acceleration

of A plus, theta double dot B2 k,

crossed with r, now from our known

to our unknown, r A to B minus theta

dot B2 squared times r from A to B.

Okay, do I know anything special about the acceleration of B to reduce the unknowns?

Again B is moving in rectilinear translation, so it can only have

one component of acceleration, and that's going to be in the idirection, so I

can reduce A as a vector to A sub B in only the i direction,

equals A sub A, which I found, which is 280i

minus 415j, plus, now I don't know what theta

double dot B2 is, so that's going to be an unknown.

So I've got theta double dot B 2 in the

k direction, crossed with, now r from A to B.

We're going to have to use a little bit of geometry here.

We know this distance is 4 inches, because this is 13.

And this is 13, this is a 4 on 3 triangle with 5 inch

hypotenuse, so that's 4 inches. So if you do your, your Pythagorean's

theorem you'll find that this side is 12.4 inches.

And so in going from A to B, I go 12.4 in the i direction and

minus 4 in the j direction, so this is 12.4i

minus 4j. And that takes care of that term.

I still have my, my radial or normal acceleration term.

So that's going to be minus the,

the magnitude of B2's velocity is 2.42. So I'm going to get 2.42 squared

and that's going to be times r again, so that's

12.4i minus 4j. And

again I'm going to let you do that tho-, that mathematics, multiply that all out.

This is the result you should come up with.

AB in the i direction equals 207 plus 4

theta double dot B2 in the i

direction, equals minus 392 plus

12.4 theta double dot B2 in the j direction.

And so you should be able to do that, that, that mathematics.

'Kay, so here's that result. And my question

is, okay, we have one equation, a vector equation, and we have two unknowns.

How can I solve for those two unknowns?

And you should be used to this now.

We're going to go ahead and equate components.

Let's start by equating the j components.

[SOUND] And for the j components, I get

theta double dot B2 equals 392

over 12.4 equals 31.6 radians

per second squared. And then I'll do the i components.

Why don't you go ahead and do the i components on your

own and come on back and see if you got the answer correct.

And for the i components, I've got AB is equal to

207 plus 4 times theta double dot B2.

But I just found theta double dot B2 was equal to 31.6, and

so this equals 333, or is a vector A of B is equal to 333 all in the

i direction, inches per second squared. And we've solved our problem.

[SOUND]

As I've said all along, practice is going to

make you better and better and these engineering problems.

So I have an

angular velocity and angular acceleration problem for you to complete now.

You have to find, at the instance shown, the angular acceleration of B2.

I always like to put these problems in terms of,

of real world systems and you recall back well, no you

may not recall back, but you hopefully you've been to amusement

park rides and you may have seen this amusement park ride.

Which is of, like a

pirate ship going back and forth.

And you can see that that's somewhat related.

It's not exact, but here's somewhat of the similar configuration here.

If we think of this as being the

passenger compartment for the, the, pirate ship ride.

This would be a fixed vertical poll.

In the, in the case that I have here, it's not fixed.

But, you can see that it's a similar type configuration.

And, you know why is that important, why, why would we even

care about accelerations on amusement park rides?

And so what, what you should say is that you know, the human body can only

sustain so many G's of acceleration before you black out or have problems.

And so you know when, if you were a designer or a, an engineer working

on these amusement park rides, you'd want to

make sure that you stay within realistic standards.

And so these,

these have real world applications, these problems and, I think

it's important to remember that, and we'll see you next time.

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