MP2002 Kinematics and Dynamics of Machinery

Tutorial 4&5

2009/2010 Sem 2

1

MP2002 Kinematics and Dynamics of Machinery

Tutorial 4&5

Q1. (Tutorial) The Figure below shows the kinematic diagram of a

six-bar mechanism. The input link O2B is rotating at a constant angular

velocity of 2 rad/s (ccw).

(i) Complete the velocity polygon on the answer sheet and find the

velocities of points D and E, and the angular velocities of link 3, link 4,

link 5 and link 6.

Contruct the velocity polygon

VC = V

B + V

CB

from the diagram VC = 22 mm/s ; V

CB = 17.5 mm/s

Use VCB

& VC to calculate the angular velocities ω

3 and ω

4:

ω3 = 0.44 rad/s (ccw) ; ω

4 = 1.46 rad/s (cw)

Use velocity image of link 4 to locate D directly on the velocity

diagram.

ω 2

= 2 rad/s (ccw)

O2B = 15 mm

BC = 40 mm

CD = 30 mm

O4C = 15 mm

O4D = 37 mm

DE = 25 mm

O6E = 15 mm

1

6

5

4

3

2

O4

E D

C

B

O6

O2

MP2002 Kinematics and Dynamics of Machinery

Tutorial 4&5

2009/2010 Sem 2

2

Q1 (cont.)

Contruct the velocity polygon

V

E = V

D + V

ED

from the diagram VE = 46 mm/s ; V

ED = 9.5 mm/s

Use VED

& VE to calculate the angular velocities ω

5 and ω

6:

ω3 = 0.38 rad/s (cw) ; ω

4 = 3.06 rad/s (cw)

(ii) Construct the required acceleration polygon on the answer sheet and

find the angular acceleration of link 4 and the acceleration of point D.

Contruct the acceleration polygon

An

C + A

t

C = A

n

B + A

t

B + A

n

CB + A

t

CB

from the diagram At

C = 40.5 mm/s

2

; At

CB = 81 mm/s

2

Use At

CB & A

t

C to calculate the angular accelerations α

3 and α

4:

α3 = 2 rad/s

2

(cw) ; α4 = 2.7 rad/s

2

(cw)

Use acceleration image of link 4 to locate D directly on the

acceleration diagram.

AD = 127 mm/s

2

__________________________________________________________________

(zero)

MP2002 Kinematics and Dynamics of Machinery

Tutorial 4&5

2009/2010 Sem 2

3

Q2. The Figure below the kinematic diagram of a six-bar mechanism.

The input link O2C is rotating at a constant angular velocity of 2 rad/s

(ccw).

(i) Complete the velocity polygon on the answer sheet and find the

sliding velocity of link 4, the velocity of point E and the angular

velocities of link 3, link 5 and link 6.

Contruct the velocity polygon

VB = V

A + V

BA

from the diagram VB = 32 mm/s ; V

BA = 16 mm/s

Use VBA

to calculate the angular velocities ω3:

ω3 = 0.64 rad/s (cw)

Contruct the velocity polygon

VD = V

C + V

DC

from the diagram VD = 54 mm/s ; V

DC = 57 mm/s

2

= 2 rad/s (ccw)

O2A = 15 mm

O2C = 30 mm

AB = 25 mm

CD = 40 mm

DE = 70 mm

O6D = 50 mm

4

E

6

2

5 3

1

C

D

A

B

O2

O6

MP2002 Kinematics and Dynamics of Machinery

Tutorial 4&5

2009/2010 Sem 2

4

Q2(cont.)

Use VDC

& VD to calculate the angular velocities ω

5 and ω

6:

ω5 = 1.43 rad/s (cw) ; ω

6 = 1.1 rad/s (cw)

Use velocity image of link 5 to locate E directly on the velocity

diagram.

VE = 92 mm/s

(ii) Complete the acceleration polygon on the answer sheet and find the

sliding acceleration of link 4, the acceleration of point E and the angular

accelerations of link 3, link 5 and link 6.

Contruct the acceleration polygon

AB = A

n

A + A

t

A + A

n

BA + A

t

BA

from the diagram AB = 23 mm/s

2

; At

BA = 51 mm/s

2

Use At

BA to calculate the angular accelerations α

3:

α3 = 2 rad/s

2

(ccw)

Contruct the acceleration polygon

An

D + A

t

D = A

n

C + A

t

C + A

n

DC + A

t

DC

from the diagram At

D = 138 mm/s

2

; At

DC = 96 mm/s

2

Use At

DC & A

t

D to calculate the angular accelerations α

5 and α

6:

α5 = 2.4 rad/s

2

(cw) ; α6 = 2.76 rad/s

2

(ccw) __________________________________________________________________

(zero)

(zero)

MP2002 Kinematics and Dynamics of Machinery

Tutorial 4&5

2009/2010 Sem 2

5

Answer Sheet MP2002

Acceleration polygon

Scale :

ω 2

= 2 rad/s (ccw)

O2A = 15 mm

O2C = 30 mm

AB = 25 mm

CD = 40 mm

DE = 70 mm

O6D = 50 mm

4

E

6

2

5 3

1

C

D

A

B

O2

O6 Velocity polygon

Scale :

MP2002 Kinematics and Dynamics of Machinery

Tutorial 4&5

2009/2010 Sem 2

6

Q3. The Figure below shows the kinematic diagram of a five-bar

mechanism. At the instant, the angular velocity and angular

acceleration of link 2 are 1 rad/s (ccw) and 1 rad/s2

(ccw) respectively.

The velocity of slider 5 is 50 mm/s to the right and its acceleration is 50

mm/s2

to the left.

(i) Complete the velocity polygon on the answer sheet and find the

velocity of point C, and the angular velocities of link 3 and link 4.

This is a 2-dof mechanism and two inputs are needed.

Contruct the velocity polygon

VC = V

B + V

CB

VC = V

D + V

CD

from the diagram VCB

= 45 mm/s ; VCD

= 107 mm/s ;

VC = 58 mm/s

Use VCB

& VCD

to calculate the angular velocities ω3 and ω

4:

ω3 = 1.5 rad/s (cw) ; ω

4 = 4.3 rad/s (ccw)

⎨

⎩

ω 2

= 1 rad/s (ccw)

α 2

= 1 rad/s2

(ccw) V

D = 50 mm/s (to the right)

AD = 50 mm/s

2

(to the left)

O2B = 50 mm

BC = 30 mm

CD = 25 mm

1

5

4

3

2

D

C

B

O2

MP2002 Kinematics and Dynamics of Machinery

Tutorial 4&5

2009/2010 Sem 2

7

Q3(cont.)

(ii) Construct the acceleration polygon on the answer sheet and find the

acceleration of point C, and the angular accelerations of link 3 and link 4.

Contruct the acceleration polygon

AC = A

n

B + A

t

B + A

n

CB + A

t

CB

AC = A

D + A

n

CD + A

t

CD

from the diagram At

CB = 475 mm/s

2

; At

CD = 300 mm/s

2

AC = 535 mm/s

2

Use At

CB & A

t

CD to calculate the angular accelerations α

3 and α

4:

α3 = 15.8 rad/s

2

(ccw) ; α4 = 12 rad/s

2

(cw)

__________________________________________________________________

Note: the normal component of acceleration depends only on velocity and

therefore its magnitude and deirction are known when performing

acceleration analysis:

An

= ω

2

r

The tangential component can only be calculated if the angular

acceleration is known:

At

= α r

__________________________________________________________________

⎨

⎩