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dynamic lecture chapter 5, Lecture notes of Mechanical Engineering

Prince Sattam bin Abdulaziz University (PSAU)Mechanical Engineering

Kinematics of Rigid Bodies lecture

Typology: Lecture notes

2018/2019

Uploaded on 05/09/2019

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VECTOR MECHANICS FOR ENGINEERS:

DYNAMICS

Tenth

Edition

Ferdinand P. Beer

E. Russell Johnston, Jr.

Phillip J. Cornwell

Lecture Notes:

Brian P. Self

California Polytechnic State University

CHAPTER

Kinematics of

Rigid Bodies

Discover Lecture notes of Mechanical Engineering Prince Sattam bin Abdulaziz University (PSAU)

Partial preview of the text

Download dynamic lecture chapter 5 and more Lecture notes Mechanical Engineering in PDF only on Docsity!

VECTOR MECHANICS FOR ENGINEERS:

DYNAMICS DYNAMICS

TenthTenth

EditionEdition

Ferdinand P. BeerFerdinand P. Beer

E. Russell Johnston, Jr.E. Russell Johnston, Jr.

Phillip J. CornwellPhillip J. Cornwell

Lecture Notes:Lecture Notes:

Brian P. SelfBrian P. Self

California Polytechnic State UniversityCalifornia Polytechnic State University

CHAPTER

Kinematics of

Rigid Bodies

Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics

15 - 2

Introduction

Translation

Rotation About a Fixed Axis: Velocity

Rotation About a Fixed Axis:

Acceleration

Rotation About a Fixed Axis:

Representative Slab

Equations Defining the Rotation of a

Rigid Body About a Fixed Axis

Sample Problem 5.

General Plane Motion

Absolute and Relative Velocity in Plane

Motion

Sample Problem 15.

Instantaneous Center of Rotation in

Plane Motion

Sample Problem 15.

Absolute and Relative Acceleration in

Plane Motion

Analysis of Plane Motion in Terms of a

Parameter

Sample Problem 15.

Rate of Change With Respect to a

Rotating Frame

Coriolis Acceleration

Sample Problem 15.

Motion About a Fixed Point

General Motion

Sample Problem 15.

Three Dimensional Motion. Coriolis

Acceleration

Frame of Reference in General Motion

Sample Problem 15.

Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics

Applications

15 - 4

How can we determine the velocity of the tip of a turbine blade?

Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics

Applications

2 - 5

Planetary gear systems are used to get high reduction ratios

with minimum weight and space. How can we design the

correct gear ratios?

Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics

Introduction

15 - 7

(^) Kinematics of rigid bodies: relations between

time and the positions, velocities, and

accelerations of the particles forming a rigid

body.

(^) Classification of rigid body motions:
- (^) general motion
- (^) motion about a fixed point
- (^) general plane motion
- (^) rotation about a fixed axis
  - (^) curvilinear translation
  - (^) rectilinear translation
- (^) translation:

Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics

Translation

15 - 8

(^) Consider rigid body in translation:
- (^) direction of any straight line inside the

body is constant,

(^) all particles forming the body move in

parallel lines.

(^) For any two particles in the body,

B A B A

r r r

(^) Differentiating with respect to time,

B A

B A B A A

v v

r r r r

All particles have the same velocity.

B A

B A B A A

a a

r r r r

(^) Differentiating with respect to time again,

All particles have the same acceleration.

Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics

Concept Quiz

15 - 10

What is the direction of the velocity

of point A on the turbine blade?

A

a) →

b) ←

c) ↑

d) ↓

v    r

  

x

y

ˆ (^) ˆ

v   k  Li



v  L  j



L

Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics

Rotation About a Fixed Axis. Acceleration

15 - 11

(^) Differentiating to determine the acceleration,

 

r v

d r r

d v a

^ 

k k k

angular ac celeration

radialacceleration component

tangentialacceleration component

a r r

(^) Acceleration of P is combination of two

vectors,

Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics

Concept Quiz

15 - 13

What is the direction of the normal

acceleration of point A on the

turbine blade?

A

a) →

b) ←

c) ↑

d) ↓

a   r

 

x

y

2 ˆ ( ) n

a    Li



2 ˆ

a  L  i



L

Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics

Equations Defining the Rotation of a Rigid Body About a Fixed Axis

15 - 14

(^) Motion of a rigid body rotating around a fixed axis

is often specified by the type of angular

acceleration.

d dt

(^) Recall or
(^) Uniform Rotation,  = 0:

     t

(^) Uniformly Accelerated Rotation,  = constant:

  0

1 0 0

t t

Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics

Sample Problem 5.

15 - 16

SOLUTION:

(^) The tangential velocity and acceleration of D are equal to the

velocity and acceleration of C.

   

 

4 rad s

12 in. s

0 0

  



  

v r

v v

D C



   

 

  2 3 rad s

9 in. s

  



  

a r

a a

D t

D t C



 

(^) Apply the relations for uniformly accelerated rotation to

determine velocity and angular position of pulley after 2 s.

4 rad s  3 rad s  2 s 10 rad s

     t   

     

14 rad

4 rad s 2 s 3 rad s 2 s

2 2

(^21)

1 0

   t   t  

  numberof revs

2 rad

1 rev 14 rad  



 



 



N N ^2.^23 rev

  

 5 in. 14 rad

5 in. 10 rad s

  

 





y r

v r

70 in.

50 in. s

 

 



Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics

Sample Problem 5.

15 - 17

(^) Evaluate the initial tangential and normal acceleration

components of D.

    9 in.s D (^) t C

a a

 

    

2 2 2 0    3 in. 4 rad s  48 in s D (^) n D a r

       

2 2 9 in. s 48 in. s D (^) t D n a a

 

Magnitude and direction of the total acceleration,

   

2 2

 9  48

  D D t D n

a a a

2  48. 8 in. s D

 

tan



D t

D n



  79. 4 

Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics

Group Problem Solving

15 - 19

A series of small machine components

being moved by a conveyor belt pass over

a 6-in.-radius idler pulley. At the instant

shown, the velocity of point A is 15 in./s to

the left and its acceleration is 9 in./s

2 to the

right. Determine ( a ) the angular velocity

and angular acceleration of the idler

pulley, ( b ) the total acceleration of the

machine

component at B.

SOLUTION:

(^) Using the linear velocity and

accelerations, calculate the angular

velocity and acceleration.

(^) Using the angular velocity,

determine the normal acceleration.

(^) Determine the total acceleration

using the tangential and normal

acceleration components of B.

Vector Mechanics for Engineers: Dynamics Vector Mechanics for Engineers: Dynamics

Group Problem Solving

15 - 20

v= 15 in/s a t

= 9 in/s

2 Find the angular velocity of the idler

pulley using the linear velocity at B.

15 in./s (6 in.)

v r 



 2.50 rad/s

2 9 in./s (6 in.)

a r 

2  1.500 rad/s

Find the angular velocity of the idler

pulley using the linear velocity at B.

Find the normal acceleration of point B****.

2 (6 in.)(2.5 rad/s)

a  r 



2 37.5 in./s n

a 

What is the direction of

the normal acceleration

of point B****?

Downwards, towards

the center

dynamic lecture chapter 5, Lecture notes of Mechanical Engineering

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Partial preview of the text

Download dynamic lecture chapter 5 and more Lecture notes Mechanical Engineering in PDF only on Docsity!

VECTOR MECHANICS FOR ENGINEERS:

DYNAMICS DYNAMICS

Contents

Applications

How can we determine the velocity of the tip of a turbine blade?

Applications

Planetary gear systems are used to get high reduction ratios

with minimum weight and space. How can we design the

correct gear ratios?

Introduction

Translation

Concept Quiz

What is the direction of the velocity

of point A on the turbine blade?

A

a) →

b) ←

c) ↑

d) ↓

x

y

L

Rotation About a Fixed Axis. Acceleration

^ 

Concept Quiz

What is the direction of the normal

acceleration of point A on the

turbine blade?

A

a) →

b) ←

c) ↑

d) ↓

x

y

L

     t

Sample Problem 5.

   t   t  

Sample Problem 5.

Group Problem Solving

SOLUTION:

Group Problem Solving

a r 