Module 17: Review Particle Kinetics; Newton’s Laws for Particles; and Euler’s 1st Law for Bodies

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

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Georgia Institute of Technology

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

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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.

Из урока

Eulerian Angles; Eulerian Angles Rotation Matrices; Angular Momentum in 3D; Inertial Properties of 3D Bodies

In this section students will learn about Eulerian Angles rotation matrices, angular momentum in 3D, and intertial properties of 3D bodies.

Module 13: Eulerian Angles for 3D Rotational Motion8:54

Module 14: Angular Velocity of Bodies in 3D Motion using Eulerian Angles6:23

Module 15: Derive Rotational Transformation Matrices6:14

Module 16: Solve a Problem Using Rotational Transformation Matrices7:08

Module 17: Review Particle Kinetics; Newton’s Laws for Particles; and Euler’s 1st Law for Bodies10:05

Module 18: Review the Definition of Angular Momentum; and Euler’s 2nd Law for Bodies7:35

Module 19: Angular Momentum for Bodies in 3D Motion12:27

Module 20: Review Mass Moments of Inertia and Products of Inertia; Inertial Property Matrix11:09

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

[BLANK_AUDIO].

This is module 17 of three dimensional dynamics, and I'd like

to begin with a recap of where we're at in the course.

And so we have actually finished the kinematics

portion of our study for three dimensional dynamics.

Kinematics again are the geometrical aspects of the

motion, relating things like position, velocity, acceleration, and time.

And so we did angular velocity and angular acceleration of bodies in 3D.

We did velocity accelerations, in moving reference frames.

We looked at Eulerian angles, and then we finished up with, rotation matrices.

And now we're going to move into kinema, kinetics, where we look at the

forces acting on a body and relate it to that motion of the body.

And so today's m, learning outcome is to

review particle kinetics and Newton's laws for particles.

And review Euler's first law for motion of the mass center of bodies.

And now I've covered these actually in my two dimensional course.

And so we're going to take a look back for the remainder of this module at,

the development that I did in that 2-D

course for particle kinetics and Newton's laws for particles.

And then we're going to go on to Euler's first

law for the motion of mass centers of bodies.

And then we're eventually going to move into and extend this for 3-D motion.

So, kinetics of particles and mass centers of particles.

So far we've looked at the kinematics

of particles and systems of particles, and that

was the geometrical aspects of the motion,

relating things like position, velocity, acceleration, and time.

Now, kinetics relates the forces acting on a particle or a

body to the changes in the motion of that particle or body.

And so here's, as a re, as a

review from my earlier courses, here's Newton's First Law.

Remember now, Newton's laws apply to particles.

And so the First Law says that if I have a balance of forces on a particle,

then the particle is at rest or remains in constant velocity, okay?

And so what I want you to do now is to write

down, write out in your own words what's Newton's Second Law for particles.

What you've should of come up with is that Newton's

Second Law says that the time rate of change of

the linear momentum or the momentum of a particle is

equal to the sum of the forces acting on the particle.

So no longer do we have a balance of

forces, we don't have some of the forces equal zero.

But we have some change in the momentum.

And we're going to be talking about in this course, classical dynamics,

where there are no relativistic effects, and so we have constant mass.

And I can take the mass, which is constant, outside

of the, the derivative and I get mass times acceleration.

So this is your famous F equals MA for particles.

And then finally write down in your own words what Newton's third law is.

Okay, so Newton's third law says if I have

two particles, the forces between those particles are equal

in magnitude, opposite in direction, and along the same

line of action, or colinear, so here's my notation.

This is the force on particle one, due to particle

two, is equal to negative, because it's in the opposite direction.

The force on particle two, by particle one.

So, equal and opposite forces.

So now, let's extend these laws for particles, to Euler's Laws for bodies now.

We want to look at bodies as a whole or systems of particles and then onto bodies.

And so here I have a, a body composed of capital N particles.

Where capital N is a large number.

I've only drawn a few of these particles.

But I could have millions of these little particles in this body.

And so that's a system of particles there is a bunch

of external forces acting on this system of particles or body

so I'm saying that f sub i is the net external

force acting on particle i and here's particle i right here.

I've labeled it m sub i because it has a mass m sub i, here's particle j.

So I would say there are i through j, i through n particles in the body.

So a total of n particles.

In addition to the external forces felt by particle i, the i particle

also feels these internal forces from all the other particles in the body.

And so, now you can write the Newton's

Second Law for the ith particle, and so for

the ith, try and do it on your own and come back and see how you did.

And so what we have here is the external forces

acting on particle i, plus the internal forces acting on particle

i, or the total forces acting on particle i, equals the

mass times the acceleration of particle i by Newton's second law.

And then we can, for the entire system of particles

or body, we can sum up all of those equations.

So we're going to sum N equations over N particles and now I have, I put the

summation sign on here in addition to all of these individual particles.

We sum them up from i equals 1 to N for the total number of particles in our body.

So my next question to you is, let's look at this second term,

and can you tell me, can you simplify this second term for me?

And after you think about that for awhile, come on back.

Okay, so what you should have realized is that by

Newton's third law this term is equal to zero because if

I go through my indices from I equals one and

J equals one, I have F one two, F two one.

Remember they cancel out, F one three, F three one, they cancel out.

F two three and F three two cancel out and so if I sum up over the

entire body, because each of those internal forces

are equal and opposite, they end up zeroing out.

And so all I'm left with is the total summation of forces, external forces on

our system of particles is equal to

the total of these, particles mass times acceleration.

And so here again is, is, is that same equation.

And now I've drawn my my system of particles or body here again.

And Ive got, and I'm looking at it from an inertial reference frame where I'll call

that point O, and the origin of which I'm looking at it from, the body from.

And I've got a position vector R to each of the particles.

This is a position vector from O to the Ith particle, and then I also

have written the position vector from O to the mass center of the, the body.

And so since I've shown the position vector from O to the

ith particle, I can sum up over the entire, body again and,

and pull the double integral out because we know that for the acceleration,

it's the second derivative of the position and so, these are equivalent.

Once I've done that let's look at the definition of center of mass.

If I sum the mass of each of the particles

times their position vector, so I go to this particle,

and this particle, and this particle, sum them all up

and then I divide by the total mass of the body.

That by definition of the mass center will give me a vector from O

to C, the mass center of the systems of, of particle or the body.

Okay, and I can rearrange that.

And so I have m times R OC, instead of a summation sign.

If I change this to a continuous body, so I have an infinite number

of particles, then it becomes the integral of the position vectors over

each little piece of mass or differential piece of mass in the body.

Okay.

We can now use this definition and put in for the summation of mR.

For a continuous body, is equal to r, m-r-o-c, or the interval of, R dm.

And, so, this is two forms for a continuous body I can write.

And then I come up with what's called Euler's, first law.

If I take the second derivative, remember, mass is constant.

We're talking about classical dynamics.

And so the second derivative of the vector from O to

the mass center is just the acceleration of the mass center.

So, what this says is for a system of particles or

body, the sum of the external forces of the acting on that

system of particles or body is equal to the mass times

a very particular point, mass times the acceleration of the mass center.

And so you can see Euler's first law for

bodies is completely analogous to Newton's second law for particles.

For particle we don't have to have the subscript of C here, but for

the body we have to specify the

particular point that we're talking about the acceleration.

And so we, this is the equation that

describes the motion of the mass center of bodies.

So that was a look back at the development of the particle kinetics Newton's

Laws for particles and Euler's First Law of motion in the mass center of bodies.

And we'll continue on to look at Euler's Second Law for

the motion of bodies and review that, in the next module.

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