Module 30: Derive Angular Momentum for a Rigid Body in 2D Planar Motion; Define Moments of Inertia and Products of Inertia

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Engineering Systems in Motion: Dynamics of Particles and Bodies in 2D Motion

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Georgia Institute of Technology

Engineering Systems in Motion: Dynamics of Particles and Bodies in 2D Motion

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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.

Из урока

Planar (2D) Rigid Body Kinetics I

In this section students will learn about planar (2D) rigid body kinetics, translation, moment of momentum - angular momentum, and equations of motion.

Module 28: Develop the Kinetic Equatoin for a Body Translating in 2D Planar Motion5:51

Module 29: Solve a Problem for a Body Translating in 2D Planar Motion11:39

Module 30: Derive Angular Momentum for a Rigid Body in 2D Planar Motion; Define Moments of Inertia and Products of Inertia8:03

Module 31: Explain and Determine Mass Moments of Inertia and Products of Inertia7:27

Module 32: Derive the Equations of Motion for a Rigid Body in 2D Planar Motion8:53

Module 33: Solve a Problem for Motion of a Rigid Body in 2D Planar Motion16:06

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

[BLANK_AUDIO]

Hi and welcome to Module 30 of Two-Dimensional Dynamics.

For today's learning outcomes we're going to

define or derive actually, angular momentum about

any point in a rigid body, for a rigid body in 2-D planar motion.

And we're also going to define something called

Moments of Inertia and Products of Inertia.

And so, first I want to recall from an earlier

module when we talked about angular momentum and develop

it for a, a rigid body in 2-D planar motion.

And so, if we had a particle or a system of

particles, each of those particles has a momentum m times v.

And if I take and I cross r, a position vector with that m times v that becomes

the moment of momentum, and we call that the

angular momentum, and we give it the symbol H.

Now if let's extend that to a rigid body and so we're going to

take an infinite number of particles and make this into a, a rigid body.

Or, and with now we have around some point P a position vector crossed with a

differential piece of mass d m with its

velocity and we integrate it over the entire body.

And that becomes our definition of the angular

momentum, about point P. And so here's where we left off.

And now let's restrict that so far I haven't restricted the motion at all.

It's been three dimensional but I can now restrict it to two dimensional motion.

And about some Z axis, and so, here is my body, okay?

It's just the generic body.

It's going to be rotating around this Z axis, okay?

So it's going to only be in a pla, planar motion.

So, will look at a point Q in a reference plane.

And any point that lies along a axis parallel to the Z axis at that point

Q will have the same velocity, if the

body is moving in planar, rigid body, two-dimensional motion.

And I'll show you that with a demo in a second.

But dm will now be any point in the body. And Q is in the plane of motion

and it has the corresponding x and y coordinates of the Q point.

And so, we call Q a companion point to all of these points

that are along the line that is parallel to the, to the Z axis.

And so let's go ahead before I go onto the next

point and go over to a demo here. So here is my generic

body, I've taken it, deflated basketball so

it's just some blob and we're rotating it now around

the Z axis. And when I rotate it around the Z axis in

the plane there is a point Q. And every point that's

parallel to the Z axis at the point will have the same

linear velocity as it goes around the Z axis.

So, back to my theory here. I've got

the angular momentum about P and I put in for R now,

the coordinates from P to any point. Okay?

Even though that point will have the same angular velocity here or here.

And so I cross that with my velocity.

Instead of crossing it with the velocity here, it's going

to have the same velocity as the velocity at Q, okay?

All right, so I've got the angular momentum defined

as R cross vQ, where R is the position vector

to any point on the body and it's integrated over the entire body.

And now we're going to use relative velocity

kinematics to find out what that velocity is

for this point or any point along the

line, including point Q, the comp, companion point.

And so we've got R crossed with okay, v of

Q, we're going between two points on the same body.

And so v of Q will be v of P plus omega.

How the body is, is, the angular velocity of the body

about the K axis crossed with however far we go in

the X axis and however far we go in the Y

axis and that'll give us our, the velocity of point Q.

Again, the same as the velocity of this point

out here, and every point in, on, on the body

that's parallel to that Z axis and we do that

over and over and over again, for the entire body.

Okay, so if I do those cross products, I'm going

to pull this term out and move it over here.

Okay?

So I've got the integral of R dm.

I'm taking the dm with it and I'm breaking it into a second integral.

The integral of R crossed with this velocity of Q, excuse me not the

velocity of Q but a omega cross x plus y. All right.

Now this R, the integral of Rdm is the mass times the position vector

from the point P to the mass center by the definition of the mass center.

And we end up with this.

And now if you this right hand side if you take

everything that's in the integral. And we can use the

vector identity that we've used before to rearrange that.

And so this is this term B is going to be the omega

k and x plus y or xi plus yj is going to be C.

I can go ahead and do that mathematics and this is the result I end up with.

And so if I substitute that result back into my

expression for the angular momentum, this is what I get.

And, now we're going to define these integrals.

This integral over the body of x squared plus y squared dm is going

to be called the mass moment of intertia about the Z axis through point P.

This minus the integral over the body of

xz dm, is called the product of inertia with respect to the X and Z axis.

Through point P and this is also a product of inertia term

with respect to the Y and the Z axis through point P.

And so I'll use that nomenclature these mass moments of inertia and

products of inertia to simplify the equation, and it looks like this.

And so this is the angular momentum. Well we, this is one of our objectives

for today's lesson.

The angular momentum about any point for a body in 2-D planar motion.

If we take a special point, if we say that P is either the mass center, if P is

the mass center, the distance from R to P to C will cause this term to go to zero.

Or if P had zero velocity, that makes this velocity turns goes to zero.

In either of those two cases, the angular momentum

simplifies down to this.

And in the future modules we'll talk physically about what the

meaning is of these mass moments of inertia and products inertia.

But you can see here that there is a little

bit of an analogy remember what the linear momentum was.

Linear momentum is mass times velocity and the angular

momentum, although it's a little bit more complicated here.

The angular momentum is given the symbol H instead of L.

The velocity now is at an angular velocity instead of a linear velocity.

And instead of using M we're going to have

these mass moments of inertia and products of inertia.

Which are going to include information about the

mass and the geometry of the body.

And we'll look at those in more depth in future modules.

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