Transcript Dynamic Force Analysis5

ME 302 DYNAMICS OF
MACHINERY
Dynamic Force Analysis V
Dr. Sadettin KAPUCU
© 2007 Sadettin Kapucu
1
Preliminary
Kinematics of a Rigid Body
Arbitrary point in the body
Y
y
P 
x
O’
z
O
Z
Rigid body angular
velocity wrt inertial frame
Body coordinate system it
rotates at the same
angular velocity as the
body
X
Inertial Frame
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Preliminary
Kinematics of a Rigid Body
Y
y

R
O
 z
Ro

P

Position of P

r




r  pxi  py j  pz k
x
O’
x
The position of P wrt inertial
coordinate frame
  
R  Ro  r
The absolute velocity of P is
Z
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

 dR dRo dr
V


dt
dt 3 dt
Preliminary
Kinematics of a Rigid Body
  
R  Ro  r
Y
y

R
O
Z
 z
Ro
P

r
O
’
X


The absolute velocity of P is


 dR dRo dr
V


dt
dt dt



x
dr dpx 
di dpy 
dj dpz 
dk

i  px 
j  py 
k  pz
dt dt
dt dt
dt dt
dt




dr dpx  dpy  dpz 
di
dj
dk

i
j
k  px  py  pz
dt dt
dt
dt
dt
dt
dt
Becomes zero
because body is rigid

dr  
 x r
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dt
 
 xr
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Preliminary
Kinematics of a Rigid Body
Y
y

R
O
 z
Ro
P



r
  
R  Ro  r
The absolute velocity of P is
x
O
’
X
Z
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

 dR dRo dr
V


dt
dt dt

dr  
 x r
dt

dRo 
 Vo
dt
   
V  Vo   x r
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Preliminary
Kinematics of a Rigid Body
  
R  Ro  r
Y
y

R
O
 z
Ro
P
The absolute velocity of P is

   
V  Vo   x r


r
Acceleration of P wrt inertial coordinate system is
x
O
’
X
Z


 
 dV dVo d ( x r )
a


dt
dt
dt


d   dr

a  ao 
x r  x
dt
dt
    

a  ao   x r   x  x r 
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Preliminary Kinematics of a Rigid Body
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Preliminary Kinematics of a Rigid Body
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8
Kinematics of a Rigid Body
Example
The 0.8 m arm OA for a remote-control mechanism is pivoted about the horizontal x-axis of
the clevis, and the entire assembly rotates about the z-axis with a constant speed
N=60rev/min. Simultaneously the arm is being raised at the constant rate b  4 rad / s . For
the position where b=60o determine (a) angular velocity of OA, (b) the angular acceleration of
OA, (c) the velocity of point A, and (d) the acceleration of point A.
x  b  4 rad / s
z  2N / 60  2 (60) / 60  6.283 rad / s
z



x


  x  z  4i  6.283k rad / s


   
     x x
  6.283k
 


  6.283k x 4i  25.13 jrad / s 2





r  0.693 j  0.4k
i
j
  
V  x r  4
0






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
k
6.283
0 0.693 0.4


9 
 4.35i 1.60 j  2.77k m / s
Kinematics of a Rigid Body
Example
The 0.8 m arm OA for a remote-control mechanism is pivoted about the horizontal x-axis of
the clevis, and the entire assembly rotates about the z-axis with a constant speed
N=60rev/min. Simultaneously the arm is being raised at the constant rate b  4 rad / s . For
the position where b=60o determine (a) angular velocity of OA, (b) the angular acceleration of
OA, (c) the velocity of point A, and (d) the acceleration of point A.


  
  x  z  4i  6.283k rad / s
 


  6.283k x 4i  25.13 jrad / s 2



r  0.693 j  0.4k
     
a   x r   x ( x r )
   
  x r  x V

z



x








i
j
k
i
j
k

a  0 25.13 0  4
0
6.283
0 0.693 0.4  4.35 1.60 2.77



 20.11i  38.44 j  6.40k m / s 2
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Kinematics of a Rigid Body
Example
The electric motor with an attached disk is running at a constant low speed of 120
rey/mm in the direction shown. Its housing and mounting base are initially at rest.
The entire assembly is next set in rotation about the vertical Z-axis at the constant
rate N=60 rev/min with a fixed angle g of 300. Determine (a) the angular velocity
and angular acceleration of the disk, (b) the space and body cones, and (c) the
velocity and acceleration of point A at the top of the disk for the instant shown.
o  120(2 ) / 60  4 rad / s
  2 (60) / 60  2 rad / s


 

  
  o    ok  K   (cosg j  sin g k )




  o k  (cosg j  sin g k )


 ( cosg ) j  (o   sin g )k


o
o
 (2 cos30 ) j  (4  2 sin 30 )k
b  4rad / s 
  ( 3 j  5.0k )rad / s
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Kinematics of a Rigid Body
Example
The electric motor with an attached disk is running at a constant low speed of 120
rey/mm in the direction shown. Its housing and mounting base are initially at rest.
The entire assembly is next set in rotation about the vertical Z-axis at the constant
rate N=60 rev/min with a fixed angle g of 300. Determine (a) the angular velocity
and angular acceleration of the disk, (b) the space and body cones, and (c) the
velocity and acceleration of point A at the top of the disk for the instant shown.
o  120(2 ) / 60  4 rad / s
  2 (60) / 60  2 rad / s



   ( 3 j  5.0k )rad / s
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Kinematics of a Rigid Body
Example
The electric motor with an attached disk is running at a constant low speed of 120
rey/mm in the direction shown. Its housing and mounting base are initially at rest.
The entire assembly is next set in rotation about the vertical Z-axis at the constant
rate N=60 rev/min with a fixed angle g of 300. Determine (a) the angular velocity
and angular acceleration of the disk, (b) the space and body cones, and (c) the
velocity and acceleration of point A at the top of the disk for the instant shown.
o  120(2 ) / 60  4 rad / s
  2 (60) / 60  2 rad / s
   
    x





  (cosg j  sin g k ) x (cosg ) j  (o  sin g )k


2
 (o cosg   sin g cosg ) i  ( sin g cosg )i



o
 (o cosg )i  (2 )(4 ) cos30 i  68.4i rad / s

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
Kinematics of a Rigid Body
Example
The electric motor with an attached disk is running at a constant low speed of 120
rey/mm in the direction shown. Its housing and mounting base are initially at rest.
The entire assembly is next set in rotation about the vertical Z-axis at the constant
rate N=60 rev/min with a fixed angle g of 300. Determine (a) the angular velocity
and angular acceleration of the disk, (b) the space and body cones, and (c) the
velocity and acceleration of point A at the top of the disk for the instant shown.
o  120(2 ) / 60  4 rad / s
  2 (60) / 60  2 rad / s



   ( 3 j  5.0k )rad / s
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Kinematics of a Rigid Body
Example
The electric motor with an attached disk is running at a constant low speed of 120
rey/mm in the direction shown. Its housing and mounting base are initially at rest.
The entire assembly is next set in rotation about the vertical Z-axis at the constant
rate N=60 rev/min with a fixed angle g of 300. Determine (a) the angular velocity
and angular acceleration of the disk, (b) the space and body cones, and (c) the
velocity and acceleration of point A at the top of the disk for the instant shown.



r  0.125 j  0.250k



i
j
k
  

V  x r  0
3
5  0.1920 i m / s
0 0.125 0.250
         
a   x r   x ( x r )   x r   x V



 


a  68.4i x (0.125 j  0.250k )   ( 3 j  5k ) x (0.192 i )


 26.6 j  11.83k m / s 2
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b  4University
rad / s
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Preliminary
Kinematics of a Rigid Body
Body coordinate frame rotates
with this angular velocirty
Y
y
A

rA

rA
O
Z


B
x
B

rB z

Letting
F
coordinate system rotates with
this angular velocirty


X
Denote the angular velocity of the reference wrt the body frame, the angular
B
velocity of the body is related to that of the coordinate system
 
    F

B
The velocity of a point of the body may be represented by
    
VA  VB  r   x r
  
 
    
VA Gaziantep
VB  F University
x r   x r  VB  Vrel   x r
B
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Preliminary
Kinematics of a Rigid Body
Body coordinate frame rotates
with this angular velocirty
Y
y

rA
x
B
X


coordinate system rotates with
this angular velocirty
The velocity of a point of the body may be represented by
    
VA  VB  r   x r
Acceleration of a point of the body is obtained as:
Z


arel   F


B

rB z
O
A

rA
B
 
 
   
  

aA  aB  rA  2 x rA   x rA   x   x rA 
B
B
B
B

 
 



 



x rA  F x F x rA 
2 x F x rA   2 x Vrel
B
B
B
 B
B
 B
 
 
   
  
aA  aB  arel  2 x Vrel   x rA   x   x rA 
B
B

   
 Gaziantep

  

   
   
University

aA  aB  B x rA  B x B x rA   2 x B x rA    x rA   x  17
 x rA 
F
B
F
B
B
B
B
 F
 F

Kinematics of a Rigid Body
Example
The motor housing and its bracket rotate about the Z axis at the constant rate   3 rad / s
The motor shaft and disk have a constant angular velocity of spin p  8 rad / s with respect
to the motor housing in the direction shown. If g constant at 30o, determine the velocity
and acceleration of point A at the top of the disk and angular acceleration  of the disk.
    
VA  VB   x r Vrel


  3K


rB  0.350J



rA  0.300 j  0.120k
B
  


VB   x rB  3K x 0.350J


 1.05I  1.05i m / s

 







 x rA  3K x (0.300 j  0.120k )  (0.9 cos30 )i  (0.36sin 30 )i  0.599i m / s

 B



Vrel  p x rA  8 j x (0.300 j  0.120k )  0.960i m / s

 Gaziantep
 University 

VA  1.05i  0.599i  0.960i  0.689i m / s
B
18
Kinematics of a Rigid Body
Example
The motor housing and its bracket rotate about the Z axis at the constant rate   3 rad / s
The motor shaft and disk have a constant angular velocity of spin p  8 rad / s with respect
to the motor housing in the direction shown. If g constant at 30o, determine the velocity
and acceleration of point A at the top of the disk and angular acceleration  of the disk.
 
 
   
  
aA  aB  arel  2 x Vrel   x rA   x   x rA 
B
B


0
 
 




VB   x ( x rB )  3K x (3K x 0.350J )  3.15J






 3.15( j cos30  k sin 30 )  2.73 j  0.899k m / s 2


 



 x ( x rA )  3K x 3K x (0.300 j  0.120k )
B




 3K x (0.599i )  1.557 j  0.899k m / s 2




 







2 x Vrel  2(3K) x 0.960i  5.76J  5.76( j cos30  k sin 30 )  4.99 j  2.88k m / s2






 
arel  p x ( p x rA )  8 j x (8 j x (0.300 j  0.120k ))  7.68k m / s 2
B



aA  0.703 j  8.086k m / s 2 aA  0.7032  8.0862  8.12 m / s2

  


  

     x   3K x (3K  8 j )   0  (24cos30 )i  20.8i rad / s 2

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