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dynamic lectrure chapter 6, Lecture notes of Mechanical Engineering

Prince Sattam bin Abdulaziz University (PSAU)Mechanical Engineering

Plane Motion of Rigid Bodies:Forces and Accelerations

Typology: Lecture notes

2018/2019

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VECTOR MECHANICS FOR ENGINEERS:
DYNAMICS
DYNAMICS
Tenth
Tenth
Edition
Edition
Ferdinand P. Beer
Ferdinand P. Beer
E. Russell Johnston, Jr.
E. Russell Johnston, Jr.
Phillip J. Cornwell
Phillip J. Cornwell
Lecture Notes:
Lecture Notes:
Brian P. Self
Brian P. Self
California Polytechnic State University
California Polytechnic State University
CHAPTER
© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
16
Plane Motion of Rigid Bodies:
Forces and Accelerations
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Download dynamic lectrure chapter 6 and more Lecture notes Mechanical Engineering in PDF only on Docsity!

VECTOR MECHANICS FOR ENGINEERS:

DYNAMICS

DYNAMICS

Tenth

Tenth

Edition

Edition

Ferdinand P. Beer

Ferdinand P. Beer

E. Russell Johnston, Jr.

E. Russell Johnston, Jr.

Phillip J. Cornwell

Phillip J. Cornwell

Lecture Notes:

Lecture Notes:

Brian P. Self

Brian P. Self

California Polytechnic State University

California Polytechnic State University

CHAPTER

Plane Motion of Rigid Bodies:

Forces and Accelerations

Vector Mechanics for Engineers: Dynamics

Vector Mechanics for Engineers: Dynamics

Contents

16 - 2

Introduction

Equations of Motion of a Rigid

Body

Angular Momentum of a Rigid

Body in Plane Motion

Plane Motion of a Rigid Body:

d’Alembert’s Principle

Axioms of the Mechanics of Rigid

Bodies

Problems Involving the Motion of a

Rigid Body

Sample Problem 16.

Sample Problem 16.

Sample Problem 16.

Sample Problem 16.

Sample Problem 16.

Constrained Plane Motion

Constrained Plane Motion:

Noncentroidal Rotation

Constrained Plane Motion:

Rolling Motion

Sample Problem 16.

Sample Problem 16.

Sample Problem 16.

Sample Problem 16.

Vector Mechanics for Engineers: Dynamics

Vector Mechanics for Engineers: Dynamics

Introduction

16 - 4

In this chapter and in Chapters 17 and 18, we will be

concerned with the kinetics of rigid bodies, i.e., relations

between the forces acting on a rigid body, the shape and mass

of the body, and the motion produced.

Our approach will be to consider rigid bodies as made of

large numbers of particles and to use the results of Chapter

14 for the motion of systems of particles. Specifically,

G G

F ma M H

and

Results of this chapter will be restricted to:

plane motion of rigid bodies, and

rigid bodies consisting of plane slabs or bodies which

are symmetrical with respect to the reference plane.

Vector Mechanics for Engineers: Dynamics

Vector Mechanics for Engineers: Dynamics

Equations of Motion for a Rigid Body

16 - 5

Consider a rigid body acted upon

by several external forces.

Assume that the body is made of

a large number of particles.

For the motion of the mass center

G of the body with respect to the

Newtonian frame Oxyz ,

F m a

For the motion of the body with

respect to the centroidal frame

Gx’y’z’ ,

G G

M H

System of external forces is

equipollent to the system

consisting of

and.

G

ma H

Vector Mechanics for Engineers: Dynamics

Vector Mechanics for Engineers: Dynamics

Plane Motion of a Rigid Body: D’Alembert’s Principle

16 - 7

F ma F ma M I 

x x y y G

Motion of a rigid body in plane motion is

completely defined by the resultant and moment

resultant about G of the external forces.

The external forces and the collective effective

forces of the slab particles are equipollent (reduce

to the same resultant and moment resultant) and

equivalent (have the same effect on the body).

The most general motion of a rigid body that is

symmetrical with respect to the reference plane

can be replaced by the sum of a translation and a

centroidal rotation.

d’Alembert’s Principle : The external forces

acting on a rigid body are equivalent to the

effective forces of the various particles forming

the body.

Vector Mechanics for Engineers: Dynamics

Vector Mechanics for Engineers: Dynamics

Axioms of the Mechanics of Rigid Bodies

16 - 8

The forces act at different points on

a rigid body but but have the same

magnitude, direction, and line of action.

F F

and

The forces produce the same moment about

any point and are therefore, equipollent

external forces.

This proves the principle of transmissibility

whereas it was previously stated as an axiom.

Vector Mechanics for Engineers: Dynamics

Vector Mechanics for Engineers: Dynamics

Free Body Diagrams and Kinetic Diagrams

12 - 10

The free body diagram is the same as you have done in statics and

in Ch 13; we will add the kinetic diagram in our dynamic analysis.

2. Draw your axis system (Cartesian, polar, path)

3. Add in applied forces (e.g., weight)

4. Replace supports with forces (e.g., tension force)

1. Isolate the body of interest (free body)

5. Draw appropriate dimensions (angles and distances)

x

y

Include your

positive z-axis

direction too

Vector Mechanics for Engineers: Dynamics

Vector Mechanics for Engineers: Dynamics

Free Body Diagrams and Kinetic Diagrams

12 - 11

Put the inertial terms for the body of interest on the kinetic diagram.

2. Draw in the mass times acceleration of the particle; if unknown,

do this in the positive direction according to your chosen axes. For

rigid bodies, also include the rotational term, I

G

1. Isolate the body of interest (free body)

 F  m a

G

 MI

Vector Mechanics for Engineers: Dynamics

Vector Mechanics for Engineers: Dynamics

Free Body Diagrams and Kinetic Diagrams

2 - 13

P

L/
L/

r

A

C

x

C

y

mg

  1. Isolate body
  2. Axes
  3. Applied forces
  4. Replace supports with forces
  5. Dimensions
  6. Kinetic diagram

G

G

x

ma

y

ma

I

C

B

x

y

Vector Mechanics for Engineers: Dynamics

Vector Mechanics for Engineers: Dynamics

Free Body Diagrams and Kinetic Diagrams

2 - 14

A drum of 4 inch radius is attached

to a disk of 8 inch radius. The

combined drum and disk had a

combined mass of 10 lbs. A cord is

attached as shown, and a force of

magnitude P=5 lbs is applied. The

coefficients of static and kinetic

friction between the wheel and

ground are 

s

= 0.25 and 

k

respectively. Draw the FBD and

KD for the wheel.

Vector Mechanics for Engineers: Dynamics

Vector Mechanics for Engineers: Dynamics

Free Body Diagrams and Kinetic Diagrams

2 - 16

The ladder AB slides down

the wall as shown. The wall

and floor are both rough.

Draw the FBD and KD for

the ladder.

Vector Mechanics for Engineers: Dynamics

Vector Mechanics for Engineers: Dynamics

Free Body Diagrams and Kinetic Diagrams

2 - 17

F

A

W

N

A

F

B

N

B

x

ma

y

ma

I

  1. Isolate body
  2. Axes
    1. Applied forces
  3. Replace supports with forces
  4. Dimensions
  5. Kinetic diagram

=

x

y

0.225 m

0.225 m

Vector Mechanics for Engineers: Dynamics

Vector Mechanics for Engineers: Dynamics

Sample Problem 16.

16 - 19

20 ft

s

ft

30

0

vx

SOLUTION:

  • Calculate the acceleration during the skidding stop

by assuming uniform acceleration.

 

2  20 ft

s

ft

0 30

2

2

0

2

0

2

a

v v a x x

  

s

ft

a  22. 5

  • Draw a free-body-diagram equation expressing the

equivalence of the external and inertial terms.

  • Apply the corresponding scalar equations.

NNW  0

A B

 

eff

y y

F F

 

 

  1. 699

  2. 2

  3. 5

  

  

  

   

g

a

W W g a

N N

F F m a

k

k

k A B

A B

 

 

eff

x x

F F

Vector Mechanics for Engineers: Dynamics

Vector Mechanics for Engineers: Dynamics

Sample Problem 16.

16 - 20

N W N W

A B

   0. 350

N NW

rear A

  1. 350

2

1

2

1

  N W

rear

 0. 175

N NW

front V

  1. 650

2

1

2

1

 

N W

front

 0. 325

   F N W

rear k rear

   0. 690 0. 175

F W

rear

 0. 122

F N   W

front k front

   0. 690 0. 325

F W

front

 0. 0. 227

  • Apply the corresponding scalar equations.

     

N W

g

W a

a

g

W

N W

W N m a

B

B

B

  1. 650

5 4

12

5 4

12

1

5 ft 12 ft 4 ft

  

 

  

 

 

eff

A A

M M