2. Acceleration Analysis

3. Acceleration analysis
Objective: Compute accelerations (linear and angular) of all components of a mechanism

4. Acceleration of link in pure rotation
AtPA P
APA
Length of link: p
AnPA
,   A
Magnitude of tangential component = p,
magnitude of normal component = p 2

5. Graphical acceleration analysis
AtBA B
AtB 3 A
AtA
Clockwise
acceleration of
crank 4 2
AnA 1

6. Problem definition: given the positions of the links, their angular velocities and the acceleration of the input link (link 2), find the linear accelerations of A and B and the angular accelerations of links 2 and 3.
Solution:
Find velocity of A
Solve graphically equation:
Find the angular accelerations of links 3 and 4

7. Graphical solution of equation AB=AA+ABA
AtB
AtBA 2 3 4 1
AtA
AnA B A
AtB AA
AtBA
AnBA
-AnB
Steps:
Draw AA, AnBA, -AtBA
Draw line normal to link 3 starting from
tip of –AnB
Draw line normal to link 4 starting from origin of AA
Find intersection and draw AtB and AtBA.

8. Guidelines Start from the link for which you have most information
Find the accelerations of its points
Continue with the next link, formulate and solve equation: acceleration of one end = acceleration of other end + acceleration difference
We always know the normal components of the acceleration of a point if we know the angular velocity of the link on which it lies
We always know the direction of the tangential components of the acceleration

9. Algebraic acceleration analysis2 3 4 1 B A a b c R4 R2 R1 1
Given: dimensions, positions, and velocities of links and angular
acceleration of crank, find angular accelerations of coupler
and rocker and linear accelerations of nodes A and B

10. Loop equation
Differentiate twice:
This equation means:

11. General approach for acceleration analysis
P, P’ (colocated points at some instant), P on slider, P’ on bar
Acceleration of P = Acceleration of P’ + Acceleration of P seen from observer moving with rod+Coriolis acceleration of P’

12. Coriolis acceleration
APslip: acceleration of P as seen by observer moving with rod
VPslip P
APcoriolis
AP’n
AP’t
APslip O AP

13. Coriolis acceleration
Coriolis acceleration=2Vslip
Coriolis acceleration is normal to the radius, OP, and it points towards the left of an observer moving with the slider if rotation is counterclockwise. If the rotation is clockwise it points to the right.
To find the acceleration of a point, P, moving on a rotating path: Consider a point, P’, that is fixed on the path and coincides with P at a particular instant. Find the acceleration of P’, and add the slip acceleration of P and the Coriolis acceleration of P.
AP=acceleration of P’+acceleration of P seen from observer moving with rod+Coriolis acceleration=AP’+APslip+APCoriolis

14. Application: crank-slider mechanism
B2 on link 2
B3 on link 3
These points
coincide at the instant
when the mechanism
is shown.
When 2=0, a=d-b
Unknown quantities marked in blue .
B2, B3
normal to crank
Link 2, a
3, 3, 3
2, 2 2 O2
Link 3, b d

15. Velocity analysis VB3= VB2+ VB3B2 VB3B2 // crank VB3 ┴ B2 on crank,
rocker
B2 on crank,
B3, on slider .
crank
rocker
VB2 ┴
crank O2

16. Rolling acceleration
First assume that angular acceleration, , is zero O  O  R
 (absolute) C
 (absolute) C r P
No slip condition: VP=0

17. Find accelerations of C and P
-(R-r)/r (Negative sign means that CCW rotation around center of big circle, O, results in CW rotation of disk around its own center)
VC= (R-r) (Normal to radius OC)
AnC=VC2/(R-r) (directed toward the center O)
AnP=VC2/(R-r)+ VC2 /r (also directed toward the center O)
Tangential components of acceleration of C and P are zero

18. Summary of results R AC, length VC2/(R-r) C VC, length (R-r) r
AP, length VC2/(R-r)+ VC2 /r P
VP=0

19. “MOTION OF ONE PART LEAD TO THEMOTION OF OTHER PARTS”

20. Relative-Motion Analysis :
Relative Velocity
Instantaneous Center of Zero Velocity
Relative Acceleration

21. Position
Rotating axes

22. Velocity VC = Velocity of the Collar, measured from
the X, Y, Z reference
VO = Velocity of the origin O of the x,y,z reference
measured from the X,Y,Z reference
(VC/O)xyz = relative velocity of “C with respect to O”
observer attached to the rotating x,y,z reference
= angular velocity of the x,y,z reference, measured
from the X,Y,Z reference
rC/O = relative position of “C with respect to O”

23. Coriolis acceleration
Whenever a point is moving on a path and the path is rotating, there is an extra component of the acceleration due to coupling between the motion of the point on the path and the rotation of the path. This component is called Coriolis acceleration.
First measured by the French engineer G.C. Coriolis
Important in studying the effect “force and acceleration” of earth
rotation on the rockets and long-range projectiles

25. Recall cylindrical coordinate
Rearrange
Compare !

27. Recall – Cylindrical coordinate
Coriolis acceleration

28. Thank You