International Edition---engineering Mechanics: Statics, 4th Edition
International Edition---engineering Mechanics: Statics, 4th Edition
4th Edition
ISBN: 9781305501607
Author: Andrew Pytel And Jaan Kiusalaas
Publisher: CENGAGE L
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Chapter 10, Problem 10.31P
To determine

The couple C0 required to hold the mechanism in equilibrium.

Expert Solution & Answer
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Answer to Problem 10.31P

The couple C0 required to hold the mechanism in equilibrium is 9.468Nm.

Explanation of Solution

Given:

For equilibrium the value of θ is 25°.

The value of force Q is 200 N.

Concept used:

To determine the couple C0, apply a virtual angular displacement at point A.

Draw the kinematic diagram for the mechanism as shown below:

  International Edition---engineering Mechanics: Statics, 4th Edition, Chapter 10, Problem 10.31P

The instantaneous center of the system is at point D after theapplication of virtual angular displacement at point A the bar AB rotates about point A and bar BC will rotate about point D.

From above figure, by help of geometry angle α can be calculated as follows:

  sinα50=sinθ40sinα=50θ40

Simplify further for α.

  α=sin1(1.25θ) ...... (1)

Here, α is the angle made by link BC with horizontal and θ is the angle made by link AB with horizontal.

Write the expression for the distance between points A and C.

  dAC=50cosθ+40cosα......(2)

Here, dAC is the distance between the points A and C.

Write the expression for the distance between point C and D.

  dCD=dACtanθ......(3)

Here, dCD is the distance between the points C and D.

Write the expression for the distance between point A and D.

  dAD=dACcosθ......(4)

Here, dAD is the distance between the points A and D.

Write the expression for the distance between point B and D.

  dBD=dADdAB......(5)

Here, dBD is the distance between the points B and D.

Write the expression for displacement of point B fro bar AB.

  δrB=50δθ ...... (6)

Here, δrB is the displacement of point B and δθ is the virtual angular displacement at point A.

Write the expression for displacement of point B for bar BC.

  δrB=dBDδθBC ...... (7)

Here, δθBC is the angular displacement for bar BC about point D.

Write the expression for displacement of point C for bar BC.

  δrC=dCDδθBC ...... (8)

Here, δrC is the displacement of point C.

Write the expression for net virtual work done on the system.

  δU=C0δθQδrC ...... (9)

Here, δU is the net virtual work, C0 is the applied couple and Q is the applied force.

Calculation:

Substitute 25° for θ in equation (1).

  α=sin1(1.25( 25°))α=31.89°

Substitute 25° for θ and 31.89° for α in equation (2).

  dAC=50cos(25°)+40cos(31.89°)=79.28 mm

Substitute 25° for θ and 79.28mm for dAC in equation (3).

  dCD=79.28mm(tan25°)=36.97mm

Substitute 25° for θ and 79.28mm for dAC in equation (4).

  dAD=79.28 mmcos25°=87.48mm

Substitute 87.48mm for dAD and 50mm for dAB in equation (5).

  dBD=87.4850=37.48mm

Substitute 50δθ for δrB and 37.48mm for dBD in equation (7).

  50δθ=(37.48)δθBCδθBC=1.334δθ

Substitute 1.334δθ for δθBC and 36.97mm for dCD in equation (8).

  δrC=36.97(1.334δθ)=49.32δθ

Substitute 200 N for Q, 0 for δU and 49.32δθ for δrC in equation (9).

  0=C0δθ200(49.32δθ)0=(C09864)δθC0=9468Nmm( 1 m 1000 mm)C0=9.468Nm

The couple C0 required to hold the mechanism in equilibrium is 9.468Nm.

Conclusion:

Thus, the couple C0 required to hold the mechanism in equilibrium is 9.468Nm.

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Chapter 10 Solutions

International Edition---engineering Mechanics: Statics, 4th Edition

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