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Kinematics of Rigid Bodies

DINAMIK BMCM 1713

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ContentsContentsIntroductionTranslationRotation About a Fixed Axis: VelocityRotation About a Fixed Axis: AccelerationRotation About a Fixed Axis: Representative SlabEquations Defining the Rotation of a Rigid Body About a Fixed AxisSample Problem 5.1General Plane MotionAbsolute and Relative Velocity in Plane MotionSample Problem 15.2Sample Problem 15.3Instantaneous Center of Rotation in Plane MotionSample Problem 15.4Sample Problem 15.5

Absolute and Relative Acceleration in Plane MotionAnalysis of Plane Motion in Terms of a ParameterSample Problem 15.6Sample Problem 15.7Sample Problem 15.8Rate of Change With Respect to a Rotating

FrameCoriolis AccelerationSample Problem 15.9Sample Problem 15.10Motion About a Fixed PointGeneral MotionSample Problem 15.11Three Dimensional Motion. Coriolis

AccelerationFrame of Reference in General MotionSample Problem 15.15

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IntroductionIntroduction• 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:

- rotation about a fixed axis – moving circle centre on the same axis (axis of rotation)

• curvilinear translation – the rigid body moving along curvilinear path

• rectilinear translation – the rigid body moving along straight path

- translation:

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- general motion

- motion about a fixed point

IntroductionIntroduction

- general plane motion – other types of plane motion. Any plane motion which is neither a rotation nor a transalation or combination of both motion.

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TranslationTranslation• 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,

ABAB rrr

• Differentiating with respect to time,

AB

AABAB

vv

rrrr

All particles have the same velocity.

• rB/A must have constant direction and magnitude

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TranslationTranslation

““When a rigid body in translation, When a rigid body in translation, all the points of the body have the all the points of the body have the same velocity and same acceleration same velocity and same acceleration at any given instant”at any given instant”

AB

AABAB

aa

rrrr

• Differentiating with respect to time again,

All particles have the same acceleration.

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Rotation About a Fixed Axis. Rotation About a Fixed Axis. VelocityVelocity

• Consider rotation of rigid body about a fixed axis AA’

• Let P as a point of the body , r as position vector with respect to a fixed frame of reference.

• Position of P can be define by θ called as angular coordinate. θ Defined positive when viewed counterclockwise from A.

• Unit use degree (°) @ revolution (rev)

1 rev = 2π = 360°

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Rotation About a Fixed Axis. Rotation About a Fixed Axis. VelocityVelocity

sinsinlim0

rt

rdt

dsv

t

• Velocity vector of the particle P is tangent to the path with magnitude

• ∆s :length of arc described by P when the body rotates through ∆θ

dtrdv

dtdsv

sinrBPs

locityangular vekkrdt

rdv

• Divided both sided with ∆t

• The same result is obtained if we drew along AA’ a vector and formed vector ω x rk

• Direction can be determined using right hand rule from sense of rotation body

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Rotation About a Fixed Axis. Rotation About a Fixed Axis. AccelerationAcceleration

• Differentiating to determine the acceleration,

vrdt

ddt

rdr

dt

d

rdt

d

dt

vda

•

kkk

celerationangular acdt

d

componenton accelerati radial

componenton accelerati l tangentia

r

r

rra

• Acceleration of P is combination of two vectors,

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Rotation About a Fixed Axis. Rotation About a Fixed Axis. Representative SlabRepresentative Slab

• Consider the motion of a representative slab in a plane perpendicular to the axis of rotation.

• Choose x,y plane as reference plane. Assume z-axis pointing out of paper

• Acceleration of any point P of the slab,

rrk

rra

2

• Resolving the acceleration into tangential and normal components,

22 rara

rarka

nn

tt

• Velocity of any point P of the slab,

rv

rkrv

Direction can be obtained by rotating r, 90° in sense of rotation slab

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

• Motion of a rigid body rotating around a fixed axis is often specified by the type of angular acceleration.

• Can be expressed as a known function of t. (usually not!)

d

d

dt

d

dt

d

ddt

dt

d

2

2

or• Recall

• Uniform Rotation, = 0:

t 0

• Uniformly Accelerated Rotation, = constant:

020

2

221

00

0

2

tt

t

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Sample Problem 5.1Sample Problem 5.1

Cable C has a constant acceleration of 9 in/s2 and an initial velocity of 12 in/s, both directed to the right.

Determine (a) the number of revolutions of the pulley in 2 s, (b) the velocity and change in position of the load B after 2 s, and (c) the acceleration of the point D on the rim of the inner pulley at t = 0.

SOLUTION:

• Due to the action of the cable, the tangential velocity and acceleration of D are equal to the velocity and acceleration of C. Calculate the initial angular velocity and acceleration.

• Apply the relations for uniformly accelerated rotation to determine the velocity and angular position of the pulley after 2 s.

• Evaluate the initial tangential and normal acceleration components of D.

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Sample Problem 5.1Sample Problem 5.1SOLUTION:• The tangential velocity and acceleration of D are equal to the

velocity and acceleration of C.

srad4

3

12

sin.12

00

00

00

r

v

rv

vv

D

D

CD

2srad33

9

sin.9

r

a

ra

aa

tD

tD

CtD

• Apply the relations for uniformly accelerated rotation to determine velocity and angular position of pulley after 2 s.

srad10s 2srad3srad4 20 t

rad 14

s 2srad3s 2srad4 22212

21

0

tt

revs ofnumber rad 2

rev 1rad 14

N rev23.2N

rad 14in. 5

srad10in. 5

ry

rv

B

B

in. 70

sin.50

B

B

y

v

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Sample Problem 5.1Sample Problem 5.1• Evaluate the initial tangential and normal acceleration

components of D.

sin.9CtD aa

2220 sin48srad4in. 3 DnD ra

22 sin.48sin.9 nDtD aa

Magnitude and direction of the total acceleration,

22

22

489

nDtDD aaa

2sin.8.48Da

9

48

tan

tD

nD

a

a

4.79

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General Plane MotionGeneral Plane Motion

• General plane motion is neither a translation nor a rotation.

• General plane motion can be considered as the sum of a translation and rotation.

• Displacement of particles A and B to A2 and B2 can be divided into two parts: - translation to A2 and- rotation of about A2 to B2

1B1B

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Absolute and Relative Velocity in Absolute and Relative Velocity in Plane MotionPlane Motion

• Any plane motion can be replaced by a translation of an arbitrary reference point A and a simultaneous rotation about A.

ABAB vvv

rvrkv ABABAB

ABAB rkvv

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Absolute and Relative Velocity in Absolute and Relative Velocity in Plane MotionPlane Motion

• Assuming that the velocity vA of end A is known, wish to determine the velocity vB of end B and the angular velocity in terms of vA, l, and .

• The direction of vB and vB/A are known. Complete the velocity diagram.

tan

tan

AB

A

B

vv

v

v

cos

cos

l

v

l

v

v

v

A

A

AB

A

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Absolute and Relative Velocity in Absolute and Relative Velocity in Plane MotionPlane Motion

• Selecting point B as the reference point and solving for the velocity vA of end A and the angular velocity leads to an equivalent velocity triangle.

• vA/B has the same magnitude but opposite sense of vB/A. The sense of the relative velocity is dependent on the choice of reference point.

• Angular velocity of the rod in its rotation about B is the same as its rotation about A. Angular velocity is not dependent on the choice of reference point.

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Sample Problem 15.3Sample Problem 15.3

The crank AB has a constant clockwise angular velocity of 2000 rpm.

For the crank position indicated, determine (a) the angular velocity of the connecting rod BD, and (b) the velocity of the piston P.

SOLUTION:

• Will determine the absolute velocity of point D with

BDBD vvv

• The velocity is obtained from the given crank rotation data.

Bv

• The directions of the absolute velocity and the relative velocity are determined from the problem geometry.

Dv

BDv

• The unknowns in the vector expression are the velocity magnitudeswhich may be determined from the corresponding vector triangle.

BDD vv and

• The angular velocity of the connecting rod is calculated from .BDv

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Sample Problem 15.3Sample Problem 15.3SOLUTION:

• Will determine the absolute velocity of point D with

BDBD vvv

• The velocity is obtained from the crank rotation data. Bv

srad 4.209in.3

srad 4.209rev

rad2

s60

min

min

rev2000

ABB

AB

ABv

The velocity direction is as shown.

• The direction of the absolute velocity is horizontal. The direction of the relative velocity is perpendicular to BD. Compute the angle between the horizontal and the connecting rod from the law of sines.

Dv

BDv

95.13in.3

sin

in.8

40sin

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Sample Problem 15.3Sample Problem 15.3

• Determine the velocity magnitudes from the vector triangle.

BDD vv and

BDBD vvv

sin76.05

sin.3.628

50sin95.53sinBDD vv

sin.9.495

sft6.43sin.4.523

BD

D

v

v

srad 0.62in. 8

sin.9.495

l

v

lv

BDBD

BDBD

sft6.43 DP vv

kBD

srad 0.62