Kinematics velocity analysis unit 2 lesson 7

velocity analysis

1. 1io1071 Construeren/inleiding
ME 33 -Theory of Machines
Lesson 7
Velocity Analysis -Analytical
and Graphical Methods

2. 2io1071 Construeren/inleiding
ME 33 -Theory of Machines
Velocity analysis…
➢Velocity analysis involves determining
“how fast” certain points on the links of
a mechanism are traveling.
➢Velocity is important because it
associates the movement of a point on a
mechanism with time. Often the timing
in a machine is critical.
➢Examples …
▪mechanism that feeds packing material
into a crate must operate in sequence
with the conveyor that indexes the crates.
▪A windshield wiper mechanism
operating on high speed must sweep the
wiper across the glass at least 45 times
every minute.

3. 3io1071 Construeren/inleiding
ME 33 -Theory of Machines
➢Two common velocity analysis
procedures …
▪the relative velocity
method, and
▪the instantaneous center
method.
•Linear velocity, v
•Angular velocity, w
Linear velocity,V, of a point is
the linear displacement (s) of
that point per unit time..
Angular velocity,w, of a link is
the angular displacement of
that link per unit time..
??????=lim
Δ??????→0
????????????
????????????dt
d
t

w
0
lim
→
=
Velocity analysis…

4. 4io1071 Construeren/inleiding
ME 33 -Theory of Machines
Linear Velocity of Rectilinear Points
➢A pointcan move along either a
straight or curved path. Many links
are constrained to straight-line, or
rectilinear, motion.
➢For points that are attached to a link
that is restricted to rectilinear
motion, the equation below can be
used to calculate the magnitude of
the velocity.
➢The orientation of the linear velocity
vector is simply in the direction of
motion, which is usually obvious.dt
dR
v
t0
lim
→
=
Velocity analysis…

5. 5io1071 Construeren/inleiding
ME 33 -Theory of Machines
Linear Velocity of a General Point
I.This Figure illustrates the velocity of two
points on a link. The velocities of points A and
B are denoted as V
Aand V
B, respectively.
II.Although they are on the same link, both
these points can have different linear
velocities. Points that are farther from the
pivot travel faster.
III.The velocity of point A, V
A, is directed along
the path that point A is moving at this instant–
–that is, tangent to an arc centered at O,
which is also perpendicular to link OA. In
casual terms, if point A were to break away
from link 2 at this instant, point A would travel
in the direction of its linear velocity.
O
Velocity analysis…

6. 6io1071 Construeren/inleiding
ME 33 -Theory of Machines
➢Several points on a link can have
drastically different linear velocities. This is
especially true as the link simply rotates
about a fixed point, as shown in the
Figure.
➢In general, the motion of a link can be
rather complex as it moves (translates)
and spins (rotates).
➢Any motion, however complex, can be
viewed as a combinationof a straight-line
movement and a rotational movement.
➢Fully describing the motion of a link can
consist of identification of the linear
motion of one point and the rotational
motion of the link.
O
➢Relationship
between linear and
angular velocitieswrv=
Velocity analysis…

7. 7io1071 Construeren/inleiding
ME 33 -Theory of Machines
➢In kinematic analysis, all the motions are
measured relative to some reference
axes or planes.
➢Usually, the earth is taken to be a fixed
reference plane, and all such motions
relative to it are termed as absolute
motion.
➢When motion of a body is measured with
respect to another body, in motion or
steady state, it is called relative motion.
➢In mechanism, the motion of a link can
be measured with respect to fixed points
as well as moving points on the links.
Velocity analysis…

8. 8io1071 Construeren/inleiding
ME 33 -Theory of Machines
➢In general terms, velocity (v)is the
derivative of the displacement
function (s) with respect to time, and
acceleration (a) is the derivative of
the velocity function. Mathematically, dt
vd
a
dt
sd
v
)(
;
)(
==
Procedure
1.Develop a displacement function,
s as a function of time
2.Differentiate the function with
respect to time to obtain velocity.
3.Differentiate the resulting
velocity function to obtain
acceleration )cos(35200 tts w−=
Example
➢Suppose the following
displacement function has been
developed based on the
dimensions and a given
configuration of mechanism
➢Differentiate to obtain velocity)cos(35
))sin(35200()(
)sin(35200
))cos(35200()(
2
ta
dt
td
dt
vd
a
tv
dt
ttd
dt
sd
v
ww
ww
ww
w
=
+
==
+=
−
==
Velocity analysis…

9. 9io1071 Construeren/inleiding
ME 33 -Theory of Machines
Relative Velocity
➢The difference between the
motionof two points is
termed relative motion.
➢The speeds measured in
relationship to a stationary
radar unit define absolute
motion.
➢Relative velocity is a term
used when the velocity of one
object is related to that of
another reference object,
which can also be moving.
Notations
➢V
A= absolute velocity of point A
➢V
B= absolute velocity of point B
➢V
B/A= relative velocity of point B with
respect to A = velocity of point B “as
observed” from point A).ABAB
ABAB
vvv
or
vvv
/
/
+=
−=
Mathematically
Velocity analysis…

10. 10io1071 Construeren/inleiding
ME 33 -Theory of Machines
Relative Velocity
➢In order to use the vector equations,
vector polygons must be prepared in
accordance with the equations.
➢Relative velocity between two points
on a link is useful in determining
velocity characteristics of the link.
➢Specifically, the relative velocity of
any two points on a link can be used
to determine the angular velocity of
that link.
➢Assuming that points A, B, and C lay
on a link, the angular velocity can be
stated asCA
CA
CB
CB
BA
BA
r
v
r
v
r
v
/
/
/
/
/
/
===w
➢when v
B/Ais directed up
and to the left, the
angular velocity of the
link is counterclockwise.
Conversely, when v
B/Ais
directed down and to
the right, the angular
velocity of the link is
clockwise.
Velocity analysis…

11. 11io1071 Construeren/inleiding
ME 33 -Theory of Machines
Example I
The Figure below shows a kinematic
diagram of a cargo lift mechanism for a
delivery truck. At this instant, point A
has a velocity of 12 in./s in the direction
shown, and point B has a velocity of
10.4 in./s, also in the direction shown.
Determine the angular velocity of the
lower link and the relative velocity of
point B relative to point A. Length of
link 2 is 24 in.
Solutionsrad
in
sin
r
v
BA
BA
/5.0
24
/12
/
/
===w
➢Angular velocity of link 2
ccwsin
vvvvv
BBBABA
/.6
30cos..2
022
/
=
−+=
➢Relative velocity of B
(Graphical or analytical
methods can be used)
Vector polygon
Velocity analysis…

12. 12io1071 Construeren/inleiding
ME 33 -Theory of Machines
Example II (Homework)
The Figure below shows a kinematic
diagram of a four bar mechanism.
Determine the angular velocity of link 4.BCBC vvv
/+=
➢Decides on an appropriate
relative velocity equationsrad
r
v
srad
r
v
CB
BC
CD
C
/25.0
)400(
)101(
/36.4
180
784
/
/
3
4
===
===
w
w
➢Calculate angular velocities
➢Vector polygonsmmrv
B /754)120(
30
)60(
22 ===

w
➢Velocity of link 2, v
B
Solution
➢Measure velocity v
Cand v
C/Bon
the polygon (v
C=784mm/s; (v
C/B=101mm/s)
Velocity analysis…

13. 13io1071 Construeren/inleiding
ME 33 -Theory of Machines
Procedure (summary)...
➢Decides on an appropriate
relative velocity equation
➢Label/identify known
parameters
➢Draw vector polygon to
scale
➢Measure unknown/desired
parameters on the vector
polygon
➢Calculate the desired
velocities
Velocity analysis…

14. 14io1071 Construeren/inleiding
ME 33 -Theory of Machines
End…