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Constraint motion clearly understood !!
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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
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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
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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
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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.
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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.
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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
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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
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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.
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