-
KINEMATICS OF PARTICLES
CONSTRAINED MOTION OF CONNECTED
PARTICLES
-
Sometimes the motions of particles are interrelated because of
the constraints imposed by interconnecting members. In such cases,
it is neccesary to account for these constraints in order to
determine the respective motions of the particles.
-
Consider first the very simple system of two interconnected
paricles A and B. It should be quite evident by inspection that the
horizontal motion of A is twice the motion of B. We can illustrate
this by using the method of analysis
Reference line
which applies to more complex situations where the results
cannot be easily obtained by inspection.
-
bryr
xL 12 2
2
The motion of B is clearly the same as that of the center of its
pulley, so we establish position coordinates y and x measured from
a convenient fixed datum. The total length of the cable is
Reference line
Here, L, r2, r1, and b are constant
-
BA aaoryx 2020
The velocity and acceleration constraint equations indicate
that, for the coordinates selected, the velocity of A must have a
sign which is opposite to that of the velocity of B, and similarly
for the accelerations. The constraint equations are valid for
motion of the system in either direction. We emphasize that is
positive to the left and that is positive down.
The first and second time derivatives of the equation give:
BA vvoryx 2020
Reference line
xvA yvB
-
Because the results do not depend on the lengths or pulley
radii, we should be able to analyze the motion without considering
them.
Reference line
This system is said to have one degree of freedom since only one
variable, either x or y is needed to specify the positions of all
parts of the system.
-
Reference line
2. cable
1. cable
The system with two degrees of freedom is shown here. The
positions of the lower cylinder and pulley C depend on the separate
specifications of the two coordinates yA and yB.
-
DADA
DA
yyyy
yyL
2020
21
constant
DADA aaandvv 2020
constant DCCB yyyyL2
1. Cable length:
2. Cable length:
The lengths of the cables attached to cylinders A and B can be
written, respectively, as
Reference line
2. cable 1. cable
DCBDCB
DCBDCB
aaaoryyy
vvvoryyy
2020
2020
-
Eliminating the terms in and gives
042
042
042
042
CBA
CBA
CBA
CBA
aaa
oryyy
vvv
oryyy
Reference line
2. cable 1. cable
Dy Dy
It is clearly impossible for the signs of all three terms to be
positive simultaneously. So, for example, if both A and B have
downward (positive) velocities, then C will have an upward
(negative) velocity.
-
1.(2/219) Determine the relationship which governs the
velocities of the four cylinders. Express all velocities as
positive down. How many degrees of freedom are there?
-
2. (2/221) Neglect the diameters of the small pulleys and
establish the relationship between the velocity of A and the
velocity of B for a given value of y.
-
3. (2/225) The power winches on the industrial scaffold enable
it to be raised or lowered. For rotation in the sense indicated,
the scaffold is being raised. If each drum has a diameter of 200 mm
and turns at the rate of 40 rev/min, determine the upward velocity
v of the scaffold.
-
4. (2/228) Collars A and B slide along the fixed right-angle
rods and are connected by a cord of length L. Determine the
acceleration ax of collar B as a function of y if collar A is given
a constant upward velocity vA.
-
5. (2/230) Under the action of force P, the constant
acceleration of block B is 3 m/s2 to the right. At the instant when
the velocity of B is 2 m/s to the right, determine the velocity of
B relative to A, the acceleration of B relative to A and the
absolute velocity of point C of the cable.
sD
