Bölüm 2 6 dynamic force analysis spatial

1

ME 302 DYNAMICS OF ME 302 DYNAMICS OF MACHINERYMACHINERY

Dynamic Force Analysis VI

Dr. Sadettin KAPUCU

© 2007 Sadettin Kapucu

2Gaziantep University

Kinetics of a Rigid BodyKinetics of a Rigid BodyEEquation quation oof f MMotion (otion (EOMEOM))

x

z

y

CG

r

dt

LdF

dt

Vmd CG )(

dt

Vdm CG

dt

HdM CG

dt

dmrxxrd



x

z

y

CG

r

dt

LdF

dt

Vmd CG )(

dt

Vdm CG

CzyxCG VxkVjViVdt

Vd

zyx

zyx

VVV

kji

iVV yzzy

jVV xzzx

kVV xyyx



x

z

y

CG

r

dt

LdF

dt

Vmd CG )(

dt

Vdm CG

CzyxCG VxkVjViVdt

Vd

zyx

zyx

VVV

kji

iVV yzzy

jVV xzzx

kVV xyyx

yzzyxx VVVmF

zxxzyy VVVmF

xyyxzz VVVmF



x

z

y

CG

r

dt

HdM CG

dt

dmrxxrd

yzzyxx VVVmF

zxxzyy VVVmF

xyyxzz VVVmF

zyx

zyx

rrr

kji

irr yzzy

jrr xzzx

krr xyyx

rx



x

z

y

CG

r

dt

HdM CG

dt

dmrxxrd

yzzyxx VVVmF

zxxzyy VVVmF

xyyxzz VVVmF

xyyxzxxzyzzy

zyx

rrrrrr

rrr

kji

rxxr

irr yzzy

jrr xzzx

krr xyyx

rx



x

z

y

CG

r

dt

HdM CG

dt

dmrxxrd

yzzyxx VVVmF

zxxzyy VVVmF

xyyxzz VVVmF

xyyxzxxzyzzy

zyx

rrrrrr

rrr

kji

rxxr

irrrrrr zxxzzxyyxy

jrrrrrr yzzyzxyyxx

krrrrrr yzzyyzxxzx



x

z

y

CG

r

dt

HdM CG

dt

dmrxxrd

yzzyxx VVVmF

zxxzyy VVVmF

xyyxzz VVVmF

dmrxxr dmirrrrrr zxxzzxyyxy

dmjrrrrrr yzzyzxyyxx

dmkrrrrrr yzzyyzxxzx

H



x

z

y

CG

r

dt

HdM CG

dt

dmrxxrd

yzzyxx VVVmF

zxxzyy VVVmF

xyyxzz VVVmF

dmrrrrrr

dmrrrrrr

dmrrrrrr

H

H

H

H

yzzyyzxxzx

yzzyzxyyxx

zxxzzxyyxy

z

y

x



x

z

y

CG

r

dt

HdM CG

dt

dmrxxrd

yzzyxx VVVmF

zxxzyy VVVmF

xyyxzz VVVmF

dmrrrrrr

dmrrrrrr

dmrrrrrr

H

H

H

H

yzzyyzxxzx

yzzyzxyyxx

zxxzzxyyxy

z

y

x

dmrrrrrr

dmrrrrrr

dmrrrrrr

H

H

H

H

yzyzyxzxxz

zyzzyxyxyx

zxzxzyxyyx

z

y

x

22

22

22



x

z

y

CG

r

dt

HdM CG

dt

dmrxxrd

yzzyxx VVVmF

zxxzyy VVVmF

xyyxzz VVVmF

dmrrrrdmrr

dmrrdmrrdmrr

dmrrdmrrdmrr

H

H

H

H

yxzyzyxzx

zyzzxyxyx

zxzyxyzyx

z

y

x

22

22

22

dmrrrrrr

dmrrrrrr

dmrrrrrr

H

H

H

H

yzyzyxzxxz

zyzzyxyxyx

zxzxzyxyyx

z

y

x

22

22

22

z

y

x

yxyzxz

zyzxxy

zxyxzy

z

y

x

dmrrdmrrdmrr

dmrrdmrrdmrr

dmrrdmrrdmrr

H

H

H

H

22

22

22



x

z

y

CG

r

dt

HdM CG

dt

dmrxxrd

yzzyxx VVVmF

zxxzyy VVVmF

xyyxzz VVVmF

dmrrrrdmrr

dmrrdmrrdmrr

dmrrdmrrdmrr

H

H

H

H

yxzyzyxzx

zyzzxyxyx

zxzyxyzyx

z

y

x

22

22

22

z

y

x

yxyzxz

zyzxxy

zxyxzy

z

y

x

dmrrdmrrdmrr

dmrrdmrrdmrr

dmrrdmrrdmrr

H

H

H

H

22

22

22



x

z

y

CG

r

dt

HdM CG

dt

dmrxxrd

yzzyxx VVVmF

zxxzyy VVVmF

xyyxzz VVVmF

z

y

x

yxyzxz

zyzxxy

zxyxzy

z

y

x

dmrrdmrrdmrr

dmrrdmrrdmrr

dmrrdmrrdmrr

H

H

H

H

22

22

22

z

y

x

zzzyzx

yzyyyx

xzxyxx

z

y

x

III

III

III

H

H

H

H



x

z

y

CG

r

dt

HdM CG

dt

dmrxxrd

yzzyxx VVVmF

zxxzyy VVVmF

xyyxzz VVVmF

z

y

x

zzzyzx

yzyyyx

xyxyxx

z

y

x

III

III

III

H

H

H

H

HxkHjHiHdt

Hdzyx



x

z

y

CG

r

dt

HdM CG

dt

dmrxxrd

z

y

x

zzzyzx

yzyyyx

xyxyxx

z

y

x

III

III

III

H

H

H

H

HxkHjHiHdt

Hdzyx

zyx

zyx

HHH

kji

iHH yzzy

jHH xzzx

kHH xyyx

Hx



x

z

y

CG

r

z

y

x

zzzyzx

yzyyyx

xyxyxx

z

y

x

III

III

III

H

H

H

H

HxkHjHiHdt

Hdzyx

kHHjHHiHHHx xyyxxzzxyzzy

22xyyzzyyyzzyxzxzzxyxyxxxx IIIIIIM

22xzxzzxzzxxyxzyzyyyzyxxyy IIIIIIM

22yxxyyxxxyyzzzzxyyzzyxxzz IIIIIIM



x

z

y

CG

r yzzyxx VVVmF

zxxzyy VVVmF

xyyxzz VVVmF

22xyyzzyyyzzyxzxzzxyxyxxxx IIIIIIM

22xzxzzxzzxxyxzyzyyyzyxxyy IIIIIIM

22yxxyyxxxyyzzzzxyyzzyxxzx IIIIIIM



x

z

y

CG

r

zyyyzzxxxx IIIM

zxzzxxyyyy IIIM

yxxxyyzzzz IIIM

If x,y,z are chosen such that they are principal axes than product of mass moment of inertia vanishes.

0 yzxzxy III


Gyroscopic Action in MachinesGyroscopic Action in Machines

zyyyzzxxxx IIIM

zxzzxxyyyy IIIM

yxxxyyzzzz IIIM

x

y

zp

s

Body coordinate frame

y

z

Instantaneous body coordinate frame

ts

2

2

1mrI xx 2

4

1mrII zzyy

ps

sx sinpy

y

zz

y

cospz

Magnitudes of the ws and wp are constant

0x

coscos sppy

sinsin sppz

20Gaziantep University sin21 2spz mrM

cos21 2spy mrM

0 xM


x

y

zp

s

Body coordinate frame

y

z

Instantaneous body coordinate frame

ts

2

2

1mrI xx 2

4

1mrII zzyy

ps

sx sinpy

y

zz

y

cospz

Magnitudes of the ws and wp are constant

0x

coscos sppy

sinsin sppz

zyyyzzxxxx IIIM

zxzzxxyyyy IIIM

yxxxyyzzzz IIIM


sin21 2spz mrM

cos21 2spy mrM

0 xM


x y

zp

sy

z

y

zz

y

spmr 221

M

p

sPrecession

Spin

M

o90


Gyroscopic Action in MachinesGyroscopic Action in Machinesp

s

tHH p *

s

M

H

Angular momentum

smrH 221

p

tpH

pspspot

mrIHt

H 2

2

1*lim

MH

M

Active gyroscopic couple


Gyroscopic Action in MachinesGyroscopic Action in MachinesExample 1Example 1 p

s

NmIM ps 2.9284.6*7.104*014.0

A uniform disc of 150 mm diameter has a mass of 5 kg. It is mounted centrally in bearings which maintain its axle in a horizontal plane. The disc spins about it axle with a constant speed of 1000 rpm. while the axle precesses uniformly about the vertical at 60 rpm. The directions of rotation are as shown in Figure. If the distance between the bearings is 100 mm, find the resultant reaction at each bearing due to the mass and gyroscopic effects.

x

z

y

Given:

mxCCWsradorrpmN

CCWsradorrpmNmrormmd

pp

s

1.0,/28.660

,/7.1071000,075.0150

We know that mass moment of inertia of the disc, about an axis through its centre of gravity and perpendicular to the plane of disc,

222 014.02/075.0*52/ kgmmrI Gvroscopic couple acting on the disc



s

axiszalongNmM 2.9


x

z

y

Given:

mxCCWsradorrpmN


pp

s

1.0,/28.660

,/7.1071000,075.0150

2014.0 kgmI

The direction of the reactive gyroscopic couple is shown in Figure.

NxMF 921.0/2.9/ pLet F be the force at each bearing due to the gyroscopic couple.

s

z

o90

M

The force F will act in opposite directions at the bearings as shown in Figure.

F F



s

axiszalongNmM 2.9


x

z

y

Given:

mxCCWsradorrpmN


pp

s

1.0,/28.660

,/7.1071000,075.0150

2014.0 kgmI NF 92

F F

Now let RAw and RBw be the reaction at the bearing A and B respectively due to the weight of the disc. Since the disc is mounted centrally in bearings, therefore.

NkgRR BWAW 5.2481.9*5.25.22/5

W BWRAWR

Total reaction at the bearings, therefore.

)(5.675.2492

)(5.1165.2492

downwardsNRFR

upwardsNRFR

BWB

AWA

Gyroscopic action alters the bearing forces.


Gyroscopic Action in MachinesGyroscopic Action in MachinesExample 2Example 2

A simple configuration of olive crusher is shown in figure. Radius of the crusher disc is r. Mass of the disc is m. Whole system is rotating with an angular velocity of w. Assume that there is no slip between the ground and the disc. What would be the crushing force?

rR s **

If there is no slip between the ground and disc than

R

r

r

Rs

*

s

p Precessional velocity simply will be

The disc moment of inertia will be2

2

1r

g

WI

p

M

Gyroscopic moment must be act on the system in this direction

The crushing force F is shown in the figure

F


Gyroscopic Action in MachinesGyroscopic Action in MachinesExample 2Example 2

A simple configuration of olive crusher is shown in figure. Radius of the crusher disc is r. Mass of the disc is m. Whole system is rotating with an angular velocity of w. Assume that there is no slip between the ground and the disc. What would be the crushing force?

R

rs

WRFRrg

Wps 2

2

1

p

M

The moment about O can be written as F

O

psrg

WWF 2

2

1

g

rWF

22

1

Engineering

Bölüm 2 6 dynamic force analysis spatial