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DYNAMICS - CASE STUDY SOLUTION


Problem Diagram
 

The velocity needs to be determined before the acceleration can be calculated.

The motion of the lift can be determined using the following conditions;

  1. Point B only moves vertically
  2. Vertical speed of point C is twice that of point B
  3. Point C and the lift have the same vertical speed
   
    Velocity of Point B


Velocity Diagram for Member AC

 

The geometric constraints of points A and B are critical in determining the member AC angular velocity. The velocity vectors for points A and B are

     vA = vAi

     vB = vBj

The relative velocity equation for bar AB is

     vB = vA + vB/A

     vB = vA + ωAB × rAB

     vBj = vAi + ωABk × rAB (cosθi + sinθj)

          = vAi + ωAB rAB (cosθj - sinθi)

Sum in both the i and j directions gives,

     j :    vB = ωABrABcosθ

     i :    vA = ωABrABsinθ

There are only two unknowns, vA and ωAB, which can be solved for, giving

     

The vertical velocities of points C and B are related by a factor of 2, giving,

     vC = 2vB = 1.33 ft/s

     
    Acceleration of Point C


Acceleration of Member AC

 

As with the velocity, the geometric constraints of points A and B are used to determine their acceleration direction,

     aA = aAi

     aB = aBj

The relative acceleration equation for bar AB is

     aB = aA + aB/A

         = aA + (aB/A)n + (aB/A)t

         = aA + ωAB × (ωAB × rAB) + (αAB × rAB)

     aBj = aAi + ωABk × (ωAB k × rAB (cosθi + sinθj))
               + αABk × rAB (cosθi + sinθj)

     aBj = aAi + ω2AB rAB (-cosθi - sinθj))
              + αAB rAB (cosθj - sinθi)

Summing in both the i and j directions gives,

     i :    0 = aA - αABrABsinθ - ω2ABrABcosθ

     j :    aB = αAB rAB cosθ - ω2ABrABsinθ

     


Motion of Platform

 

These two equations can be used to solve for the two unknowns, αAB and aB. Solving for αAB in the first equation and then substituting into the second equation gives,

     

Using numerical values give,

    

     αAB = 0.04938 rad/s2

     aB = -0.06959 ft/s2

The vertical acceleration of the lift is twice the vertical acceleration of point B,

     alift = 2aB = -0.1392 ft/s2

     
   
 
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