Module 5: Determine the Angular Acceleration for a Moving Reference Frame Relative to another Reference Frame

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From the course by Georgia Institute of Technology

Advanced Engineering Systems in Motion: Dynamics of Three Dimensional (3D) Motion

119 оценки

Georgia Institute of Technology

Advanced Engineering Systems in Motion: Dynamics of Three Dimensional (3D) Motion

119 оценки

This course is an advanced study of bodies in motion as applied to engineering systems and structures. We will study the dynamics of rigid bodies in 3D 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 course entitled Engineering Systems in Motion: Dynamics of Particles and Bodies in 2D Motion.” 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.

From the lesson

Course Introduction; Angular Velocity; Angular Acceleration

In this section students will learn to derive the "derivative formula." We will define angular velocity for 3D motion and learn to determine and solve for the Angular Acceleration for a body.

Course Introduction5:20

Module 2: Derive the “Derivative Formula”; Define Angular Velocity for 3D Motion12:55

Module 3: Define the Properties of Angular Velocity for 3D Motion5:40

Module 4: Solve for the Angular Velocity of a body undergoing 3D Motion10:20

Module 5: Determine the Angular Acceleration for a Moving Reference Frame Relative to another Reference Frame8:11

Module 6: Solve for the Angular Acceleration for a Body expressed in a Series of Multiple Reference Frames10:30

Meet the Instructors

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

[BLANK_AUDIO].

Hi this is module 5 of Three Dimensional Dynamics.

Today's learning outcome is to determine the angular acceleration for a

moving reference frame relative to another

reference frame for three dimensional motion.

So here's my situation.

I have a body F or a reference frame F and a body B or a reference frame B.

Here's an example on an actual robot.

Here's frame F and here's frame B.

And to, I've got frame B is, ha, has an angular velocity with respect to frame F.

So I call that Omega B with respect to F.

If I want to find the angular acceleration now,

what I do is I take the time derivative.

And so I'm going to take the time derivative of B with respect to F.

But we see that B, with respect to F, is expressed in the B frame, and we want to

take the derivative back here in the F frame,

so I'm going to have to use this derivative formula.

And I'll let A, be, A is my arbitrary vector, I'll

let it be the angular velocity B with respect to A itself.

And so I get the time derivative of the angular velocity of B, with respect

to F in the F frame, is equal to the time derivative of the angular velocity of

B with respect to F in the B frame, plus the angular velocity of B with respect

to F, cross with the vector itself which is also Omega B with respect to F.

And so my question to you is, what's this term equal to?

And what you should say, a vector crossed with itself is equal to 0 and so now I

have the, the time derivative of the angular velocity

of B with respect to F in the F frame.

That's just going to be the angular acceleration of B with respect to F.

And so the angular acceleration of B with respect to F then, is equal to, the time

derivative of the angular acceleration of B with respect to F in the F frame.

But it's also equal to the ang, the time derivative of the

angular velocity of B with respect to F in the B frame.

And so we see that the angular acceleration is the same, regardless

of whether I'm looking at it from the F or the B frame.

Okay now let's extend that and let's look

at a body, with three reference frames now.

And so here's my reference frame F, here's my

reference frame B attached to this segment of the

robotic arm, and then I have, reference frame C

that's attached to this segment of the the robotic arm.

And we've, we found that, for the angular velocities, we could use, we developed, we

pr, actually proved the addition theorem, where

the angular velocity of the frame our body

C with respect to F was equal to the angular velocity of the body C with

respect to B, plus the angular velocity of the body B with respect to F.

And so now if I want to find the relationship

for accelerations, angular accelerations, I'm going to have to do what?

[BLANK_AUDIO].

And what you should say is we're going to have to take the angular, excuse me, we're

going to have to take the time derivative, so the time derivative of C with

respect to F will be the angular acceleration of C with respect to F is

equal to, now, we've got the, well this is the time derivative of C with respect

to F dot.

That's equal to the time derivative of the angular

velocity of C with respect to B, taken in the F frame.

Plus the time derivative of the angular velocity of

B with respect to F taken in the F frame.

And now, what's this term equal to?

[BLANK_AUDIO].

And what you should say is, okay, this is the time derivative of B with respect to

F and so that's just going to be the angular

acceleration of the frame B with respect to F.

Or, I can write it as alpha B with respect to F.

Now, we're going to need to be more careful with this term,

[BLANK_AUDIO].

Because for this term we've got [COUGH] excuse me,

the angular velocity of C with respect to B,

so it's a velocity expressed in the B frame

but we're taking the time derivative in the F frame.

And so I've gotta be again, more careful.

And so how would I take that time derivative?

[BLANK_AUDIO].

And what you should say is, we're going to have to use the derivative formula again,

for taking a derivative of a vector expressed

in one frame with respect to the other frame.

In this case, I'm going to let the angular velocity

C with respect to B, be my arbitrary vector A.

And so I get the time derivative of C with respect to B in

the F frame, is equal to the time derivative of C with

respect to B in the B frame, plus Omega B with respect to F.

Crossed with the vector itself, which is Omega C with respect to B.

And so my next question to you is, what's this term equal to?

Omega C with respect to B, the time

derivative of that expressed in the B frame.

And what you should say is, okay that's the time derivative

of C with respect to B taken in the B frame.

Well that's just the angular acceleration of body C or frame C with respect to B.

So this is just alpha C with respect to B.

And so now I can take all that result and I can substitute it

up in here for this term, and I get, okay, now I have the

angular acceleration of C with respect to F is equal to.

I've got now the angular acceleration of B with respect to F, plus all of this term.

So plus the angular acceleration is C with respect to B, plus

Omega B with respect to F crossed with omega C with respect to B.

And so that's my relative acceleration, excuse me, not my relative, that's

my angular acceleration, for a three dimensional body.

And so, let's look at that result again, we have the addition theorem.

Here's my, my acceleration equation, and so what you see here is, that for angular

velocities for three different frames, we could use an addition theorem.

But now for angular acceleration, well, we get this extra term here.

We, and we call that a gyroscopic term, or gyroscopic acceleration.

And we'll talk more about it in, in, in later modules.

So the addition theorem does not hold for angular acceleration.

There is this extra gyroscopic term, and so

we'll talk more about that in in future lessons.

And we'll do a problem of angular acceleration in the next module.

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