Kinematics Rigid BodyKinematics Rigid BodyDedison Gasni, Ph.D

DefenitionDefenition

� Kinematics is the study of the relative motion between two or more physical bodies.

� Study includes the determination of

a. Position

b. Velocity

c. Acceleration

There are two methods to determine There are two methods to determine position, velocity and acceleration.position, velocity and acceleration.

� Graphical methods :

a. Instantaneous center methodsa. Instantaneous center methods

b. Relative motion methods

� Analytical methods

Position AnalysisPosition Analysis

Draw the position of mechanism below !

Type of MotionType of Motion

� Pure rotation : The body possesses one point (center of rotation) that has no motion with respect to the “stationary” frame of reference. All other points on the body describe arcs about that center. A reference line drawn on the body through the center changes only its angular orientation.

� Pure translation : All points on the body describe parallel � Pure translation : All points on the body describe parallel (curvilinear or rectilinear) paths. A reference line drawn on the body changes its linear position but does not change its angular orientation.

� Complex motion : A simultaneous combination of rotation and translation. Any reference line drawn on the body will change both its linear position and its angular orientation. Points on the body will travel nonparallel paths, and there will be, at every instant, a center of rotation, which will continuously change location.

Quick-Return Mechanisms

Four-Bar Linkage

VECTOR ANALYSISVECTOR ANALYSIS

Vector and ScalarVector and Scalar

Certain physical quantities such as mass or the absolute temperature at some point only have magnitude. These quantities can be represented by numbers alone, with the appropriate units, and they are called scalars.

A vector is quantity that has both direction and magnitude

Vector additionVector addition

SubtractionSubtraction

Subtraction Cont. Subtraction Cont.

PROPERTIES OF PROPERTIES OF RELATIVE MOTIONRELATIVE MOTIONRELATIVE MOTIONRELATIVE MOTION

M

NB

C

R2

R1

ø

∆ɵ

∆x∆y

∆s

P

ɵ1

ɵ2

O

R P

P’

V

ω

Velocity at P

ω

A

B

O

RA

RB

VA = RA ω

VB = RB ω

Rotation speed: rpm (n)

VA = RA ω

VB = RB ω

M

N

B

C

V

V + →∆V

R’R ∆θ

The liner acceleration

The instantaneous acceleration

O

O’

The angular acceleration

V +→

∆V

∆Vt∆Vn

∆θ

The normal acceleration : The changes of direction of linear velocity

The tangential acceleration : The changes of magnitude of linear velocity

A B

VA = 60 km/jam VB = 40 km/jam

VA -VA

-VB BVA/B = 20 km/jam VB/A = 20 km/jam

VA

VB

-VB

VA

VA/B

VB

-VA

VB/A

INSTANTANEOUS INSTANTANEOUS CENTER METHODSCENTER METHODSCENTER METHODSCENTER METHODS

Instantaneous center of velocityInstantaneous center of velocity

This concept is based on the fact that at a given instanta pair of coincident points on two links in motion willhave identical velocities relative to a fixed link and,therefore, will have zero velocity relative to each other.At this instant either link will have pure rotation relativeto the other link about the coincident points.to the other link about the coincident points.

An instantaneous center is (a) a point in both bodies, (b) a point at which the two bodies have no relative velocity, and (c) a point about which one body may be considered to rotate relative to the other body at a given instant.

Instantaneous center of velocity Instantaneous center of velocity cont.cont.

Instant center of velocities between two links is the

Instant center of velocities is a simple graphical methodfor performing velocity analysis on mechanisms. Themethod provides visual understanding on how velocityvectors are related.

Instant center of velocities between two links is the location at which two coinciding points, one on each link, have identical velocities.

The most obvious instant center of velocities, or simply the instant center (IC), between two links that are pinned to each other is the point at the center of the pin joint.

The instant center of velocities may not be located within the physical boundaries of a link.

Instantaneous Center NotationInstantaneous Center Notation

Instantaneous centers of velocity are determined foreach of the moving links relative to the fixed link. Thesystem of labeling these points is such as labeled 31 toindicate the motion of 3 relative to 1.

Instant Center at A Pin ConnectionInstant Center at A Pin Connection

2

3

4

23

34

1

2

12

14

Instant Center for A Sliding BodyInstant Center for A Sliding Body1

2A

VA

O

12 ∼

B

1

2

VB

Instant Center for A Rolling BodyInstant Center for A Rolling Body

P

VP

VO

VP/O

900

1

2

O

P

12

ω2

900900

Instant Center for A Sliding BodyInstant Center for A Sliding Body

Number of instant centers for a Number of instant centers for a mechanismmechanism� In a mechanism with n links (count the

ground as one of the links), the number of instant centers is determined as:

Kennedy’s TheoremKennedy’s Theorem

� For three independent bodies in general plane motion, Kenedy’s theorem states that the three instantaneous centers lie on a common straight line. on a common straight line.

� Three independent links (1, 2, and 3) are shown in motion relative to each other. There are three instantaneous centers (12, 13, and 23), whose instantaneous locations are to be determined.

Kennedy’s RuleKennedy’s Rule

� The three instant centers between three planar links must lie on a straight line.

� This rule does not tell us where the line � This rule does not tell us where the line is or where the centers are on that line. However, the rule can be used to find the instant centers when we consider a mechanism.

Primary instant centersPrimary instant centers

� Fixed Joint

� Moving link

� See slides above

Circle diagram methodCircle diagram method

Instant Centers of A FourInstant Centers of A Four--BarBar

Instant Centers of A SliderInstant Centers of A Slider--crankcrank