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Chapter#3 Met 305 2-_velocity_relative

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MET 305                                     MECHANICS OF MACHINES                          TUTORIAL. 2


Student Name & No._________________________                       Date: ____________________

Section: ________________________________                         Score: ____________________

                               MECHANISMS - II
      Velocity in mechanisms- Relative velocity method (velocity diagrams)

Objective
   a) To determine the velocity of a point on any link of a mechanism by the relative
       velocity method
   b) To determine the angular velocity of any link of a mechanism by the relative velocity

Problems

1. In a four bar chain ABCD, AD is fixed and is 150 mm long. The crank AB is 40 mm long and rotates at
120 r.p.m. clockwise, while the link CD = 80 mm oscillates about D. BC and AD are of equal length. Find
the angular velocity of link CD when angle BAD = 60°.




Answer :-Angular velocity of link CD = 4.8 rad/s
2. The crank and connecting rod of a theoretical steam engine are 0.5 m and 2 m long respectively.
The crank makes 180 r.p.m. in the clockwise direction. When it has turned 45° from the inner dead
centre position, determine: I. velocity of piston, 2. angular velocity of connecting rod, 3. velocity of
point E on the connecting rod 1.5 m from the gudgeon pin, 4. position and linear velocity of any
point G on the connecting rod which has the least velocity relative to crank shaft.




Answers: -
  Velocity of the piston = 8.15 m/s
  Angular velocity of the connecting rod AB = 3.4 rad/s
  VE = 8.5 m/s
  Crank pin = db/2 x (angular velocity of BO + angular velocity of PB) = 0.6675 m/s
  Pin cross head = dc/2 x angular velocity of PB = 0.051 m/s




Yanbu Industrial College                                                                       1/5/2013
MET 305                                     MECHANICS OF MACHINES                         TUTORIAL. 2


3. In the given mechanism, the angular velocity of the crank OA is 600 r.p.m. Determine the linear
velocity of the slider D and the angular velocity of the link BD, when the crank is inclined at an angle
of 75° to the vertical. The dimensions of various links are: OA = 28 mm; AB = 44 mm; BC 49 mm; and BD
= 46 mm. The centre distance between the centers of rotation O and C is 65 mm. The path of travel of
the slider is 11 mm below the fixed point C. The slider moves along a horizontal path and OC is
vertical.




                Answers:-
                        1.      linear velocity of the slider D = 1.6 m/s
                        2.      Angular velocity of link BD = 36.96rad/s
        4. Draw the velocity diagram of the engine mechanism shown below. Find out the
        acceleration of the slider D and the angular acceleration of link CD.
        The crank OA rotates uniformly at 180 r.p.m. in clockwise direction. The various
        lengths are: OA = 150 mm; AB = 450 mm; PB = 240 mm ; EC = 70 mm ; CD = 660 mm.




Yanbu Industrial College                                                                       1/5/2013
MET 305                                     MECHANICS OF MACHINES                TUTORIAL. 2




        Solution
        Given: NAO = 180 r.p.m., or wao = 2 pi x 180/60 = 18.85 rad/s ; OA = 150 mm = 0.15
        m; AB = 450 mm = 0.45 m ; PB = 240 mm = 0.24 m ; CD = 660 mm = 0.66 m
        We know that velocity of A with respect to O or velocity of A, = wAO xOA =18.85 x
        0.15 = 2.83 m/s (Perpendicular to OA)
        First of all draw the space diagram, to some suitable scale, as shown in Fig.(a). Now
        the velocity diagram, as shown in Fig.(b), is drawn as discussed below:




    (a) Space diagram         (b) Velocity diagram

   1. Since O and P are fixed points, therefore these points lie at one place in the velocity
   diagram. Draw vector oa perpendicular to OA, to some suitable scale, to represent the velocity
   of A with respect to O or velocity of A (i.e. vao or va), such that
   vector oa = vao = va = 2.83 m/s

   2. Since the point B moves with respect to A and also with respect to P, therefore draw vector
   ab perpendicular to AB to represent the velocity of B with respect to A i.e. vBA and from point
Yanbu Industrial College                                                             1/5/2013
MET 305                                MECHANICS OF MACHINES                      TUTORIAL. 2


   p draw vector pb perpendicular to PB to represent the velocity of B with respect to P or
   velocity of B (i.e. vbp or vb). The vectors ab and pb intersect at b.

   3. Since the point C lies on PB produced, therefore divide vector pb at c in the same ratio as C
   divides PB in the space diagram. In other words, pb/pc = PB/PC.

   4. From point c, draw vector cd perpendicular to CD to represent the velocity of D with respect
   to C and from point o draw vector od parallel to the path of motion of the slider D (which is
   vertical), to represent the velocity of D, i.e. vd.

   By measurement, we find that velocity of the slider D,vd = vector od = 2.36 m/s
   Velocity of D with respect to C, vdc = vector cd=l.2 m/s
   Velocity of B with respect to A, vba = vector ab = 1.8 m/s and
   Velocity of B with respect to P, vbp = vector pb = 1.5 m/s

   5. In a toggle mechanism shown below, the slider D is constrained to move on aa horizontal
   path. The crank OA is rotating in the counter-clockwise direction at a speed
  of 180 r.p.m. increasing at the rate of 50 rad/s2. The dimensions of the various links are as
  follows: OA = 180 mm ; CB = 240 mm ; AB = 360 mm ; and BD = 540 mm.
   For the given configuration, find 1. Velocity of slider D and angular velocity of BD, and 2.
   Acceleration of slider D and angular acceleration of BD.




Yanbu Industrial College                                                              1/5/2013
MET 305                    MECHANICS OF MACHINES   TUTORIAL .2




Yanbu Industrial College
1/5/2013

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Chapter#3 Met 305 2-_velocity_relative

  • 1. MET 305 MECHANICS OF MACHINES TUTORIAL. 2 Student Name & No._________________________ Date: ____________________ Section: ________________________________ Score: ____________________ MECHANISMS - II Velocity in mechanisms- Relative velocity method (velocity diagrams) Objective a) To determine the velocity of a point on any link of a mechanism by the relative velocity method b) To determine the angular velocity of any link of a mechanism by the relative velocity Problems 1. In a four bar chain ABCD, AD is fixed and is 150 mm long. The crank AB is 40 mm long and rotates at 120 r.p.m. clockwise, while the link CD = 80 mm oscillates about D. BC and AD are of equal length. Find the angular velocity of link CD when angle BAD = 60°. Answer :-Angular velocity of link CD = 4.8 rad/s 2. The crank and connecting rod of a theoretical steam engine are 0.5 m and 2 m long respectively. The crank makes 180 r.p.m. in the clockwise direction. When it has turned 45° from the inner dead centre position, determine: I. velocity of piston, 2. angular velocity of connecting rod, 3. velocity of point E on the connecting rod 1.5 m from the gudgeon pin, 4. position and linear velocity of any point G on the connecting rod which has the least velocity relative to crank shaft. Answers: - Velocity of the piston = 8.15 m/s Angular velocity of the connecting rod AB = 3.4 rad/s VE = 8.5 m/s Crank pin = db/2 x (angular velocity of BO + angular velocity of PB) = 0.6675 m/s Pin cross head = dc/2 x angular velocity of PB = 0.051 m/s Yanbu Industrial College 1/5/2013
  • 2. MET 305 MECHANICS OF MACHINES TUTORIAL. 2 3. In the given mechanism, the angular velocity of the crank OA is 600 r.p.m. Determine the linear velocity of the slider D and the angular velocity of the link BD, when the crank is inclined at an angle of 75° to the vertical. The dimensions of various links are: OA = 28 mm; AB = 44 mm; BC 49 mm; and BD = 46 mm. The centre distance between the centers of rotation O and C is 65 mm. The path of travel of the slider is 11 mm below the fixed point C. The slider moves along a horizontal path and OC is vertical. Answers:- 1. linear velocity of the slider D = 1.6 m/s 2. Angular velocity of link BD = 36.96rad/s 4. Draw the velocity diagram of the engine mechanism shown below. Find out the acceleration of the slider D and the angular acceleration of link CD. The crank OA rotates uniformly at 180 r.p.m. in clockwise direction. The various lengths are: OA = 150 mm; AB = 450 mm; PB = 240 mm ; EC = 70 mm ; CD = 660 mm. Yanbu Industrial College 1/5/2013
  • 3. MET 305 MECHANICS OF MACHINES TUTORIAL. 2 Solution Given: NAO = 180 r.p.m., or wao = 2 pi x 180/60 = 18.85 rad/s ; OA = 150 mm = 0.15 m; AB = 450 mm = 0.45 m ; PB = 240 mm = 0.24 m ; CD = 660 mm = 0.66 m We know that velocity of A with respect to O or velocity of A, = wAO xOA =18.85 x 0.15 = 2.83 m/s (Perpendicular to OA) First of all draw the space diagram, to some suitable scale, as shown in Fig.(a). Now the velocity diagram, as shown in Fig.(b), is drawn as discussed below: (a) Space diagram (b) Velocity diagram 1. Since O and P are fixed points, therefore these points lie at one place in the velocity diagram. Draw vector oa perpendicular to OA, to some suitable scale, to represent the velocity of A with respect to O or velocity of A (i.e. vao or va), such that vector oa = vao = va = 2.83 m/s 2. Since the point B moves with respect to A and also with respect to P, therefore draw vector ab perpendicular to AB to represent the velocity of B with respect to A i.e. vBA and from point Yanbu Industrial College 1/5/2013
  • 4. MET 305 MECHANICS OF MACHINES TUTORIAL. 2 p draw vector pb perpendicular to PB to represent the velocity of B with respect to P or velocity of B (i.e. vbp or vb). The vectors ab and pb intersect at b. 3. Since the point C lies on PB produced, therefore divide vector pb at c in the same ratio as C divides PB in the space diagram. In other words, pb/pc = PB/PC. 4. From point c, draw vector cd perpendicular to CD to represent the velocity of D with respect to C and from point o draw vector od parallel to the path of motion of the slider D (which is vertical), to represent the velocity of D, i.e. vd. By measurement, we find that velocity of the slider D,vd = vector od = 2.36 m/s Velocity of D with respect to C, vdc = vector cd=l.2 m/s Velocity of B with respect to A, vba = vector ab = 1.8 m/s and Velocity of B with respect to P, vbp = vector pb = 1.5 m/s 5. In a toggle mechanism shown below, the slider D is constrained to move on aa horizontal path. The crank OA is rotating in the counter-clockwise direction at a speed of 180 r.p.m. increasing at the rate of 50 rad/s2. The dimensions of the various links are as follows: OA = 180 mm ; CB = 240 mm ; AB = 360 mm ; and BD = 540 mm. For the given configuration, find 1. Velocity of slider D and angular velocity of BD, and 2. Acceleration of slider D and angular acceleration of BD. Yanbu Industrial College 1/5/2013
  • 5. MET 305 MECHANICS OF MACHINES TUTORIAL .2 Yanbu Industrial College 1/5/2013