DESIGN OF MACHINERY - 6th Ed SOLUTION MANUAL 2-1-1
PROBLEM 2-1
Statement: Find three (or other number as assigned) of the following common devices. Sketch careful
kinematic diagrams and find their total degrees o f freed om.
a. An automobile hood hinge mechanism
b. An automobile hatchback lift mechanism
c. An electric can opener
d. A folding ironing board
e. A folding card table
f. A folding beach chair
g. A baby swing
h. A folding baby walker
i. A fancy corkscrew as shown in Figure P2-9
j. A windshield wiper mechanism
k. A dump-truck dump mechanism
l. A trash truck dumpster mechanism
m. A pickup tailgate mechanism
n. An automobile jack
o. A collapsible auto radio antenna
Solution: See Mathcad file P0201.
Equation 2.1c is used to calculate the mobility (DOF) of each of the models below.
a. An automobile hood hinge mechanism.
The hood (3) is linked to the body (1) through two rocker links (2 and 4).
Number of links L 4 HOOD
3
Number of full joints J1 4 2 4
1
Number of half joints J2 0
BODY
3 ( L 1 ) 2 J1 J2
1
b. An automobile hatchback lift mechanism.
The hatch (2) is pivoted on the body (1) and is linked to the body by the lift arm, which can be modeled as
two links (3 and 4) connected through a translating slider joint.
,DESIGN OF MACHINERY - 6th Ed SOLUTION MANUAL 2-1-2
Number of links L 4 HATCH
Number of full joints J1 4 2
3
1
Number of half joints J2 0
4
M 3 ( L 1 ) 2 J1 J2 1
BODY
M1
c. An electric can opener has 2 DOF.
d. A folding ironing board.
The board (1) itself has one pivot (full) joint and one pin-in-slot sliding (half) joint. The two legs (2 and 3)
have a common pivot. One leg connects to the pivot joint on the board and the other to the slider joint.
Number of links L 3
1
Number of full joints J1 2
3 2
Number of half joints J2 1
M 3 ( L 1 ) 2 J1 J2
M1
e. A folding card table has 7 DOF: One for each leg, 2 for location in xy space, and one for angular orientation.
f. A folding beach chair.
The seat (3) and the arms (6) are ternary links. The seat is linked to the front leg(2), the back (5) and a
coupling link (4). The arms are linked to the front leg (2), the rear leg (1), and the back (5). Links 1, 2, 4, and
5 are binary links. The analysis below is appropriate when the chair is not fully opened. When fully opened,
one or more links are prevented from moving by a stop. Subtract 1 DOF when forced against the stop.
Number of links L 6 5
6
Number of full joints J1 7
4
1 2
Number of half joints J2 0 3
M 3 ( L 1 ) 2 J1 J2
M1
g. A baby swing has 4 DOF: One for the angular orientation of the swing with respect to the frame, and 3 for the
location and orientation of th e frame wit h respect to a 2-D frame.
,DESIGN OF MACHINERY - 6th Ed SOLUTION MANUAL 2-1-3
h. A folding baby walker has 4 DOF: One for the degree to which it is unfolded, and 3 for the location and
orientation of t he walker with respect to a 2-D frame.
i. A fancy corkscrew has 2 DOF: The screw can be rotated and the arms rotate to translate the screw.
j. A windshield wiper mechanism has 1 DOF: The position of the wiper blades is defined by a single input.
k. A dump-truck dump mechanism has 1 DOF: The angle of the dump body is determined by the length of the
hydraulic cylinder that links it to the body of the truck.
l. A trash truck dumpster mechanism has 2 DOF: These are generally a rot ation and a translation.
m. A pickup tailgate mechanism has 1 DOF:
n. An automobile jack has 4 DOF: One is the height of the jack and the other 3 are the position and orientation
of the jack with respect to a 2-D frame.
o. A collapsible auto radio antenna has as many DOF as there are sections, less one.
, DESIGN OF MACHINERY - 6th Ed. SOLUTION MANUAL 2-2-1
PROBLEM 2-2
Statement: How many DOF do you have in your wrist and hand combined?
Solution: See Mathcad file P0202.
1. Holding the palm of the hand level and facing toward the floor, the hand can be rotated about an axis
through the wrist that is parallel to the floor (and perpendicular to the forearm axis) and one
perpendicular to the floor (2 DOF). The wrist can rotate about the forearm axis (1 DOF).
2. Each finger (and thumb) can rotate up and down and side-to-side about the first joint. Additionally, each
finger can rotate about each of the two remaining joints for a total of 4 DOF for each finger (and thumb).
3. Adding all DOF, the total is
Wrist 1
Hand 2
Thumb 4
Fingers 4x4 16
TOTAL 23
PROBLEM 2-1
Statement: Find three (or other number as assigned) of the following common devices. Sketch careful
kinematic diagrams and find their total degrees o f freed om.
a. An automobile hood hinge mechanism
b. An automobile hatchback lift mechanism
c. An electric can opener
d. A folding ironing board
e. A folding card table
f. A folding beach chair
g. A baby swing
h. A folding baby walker
i. A fancy corkscrew as shown in Figure P2-9
j. A windshield wiper mechanism
k. A dump-truck dump mechanism
l. A trash truck dumpster mechanism
m. A pickup tailgate mechanism
n. An automobile jack
o. A collapsible auto radio antenna
Solution: See Mathcad file P0201.
Equation 2.1c is used to calculate the mobility (DOF) of each of the models below.
a. An automobile hood hinge mechanism.
The hood (3) is linked to the body (1) through two rocker links (2 and 4).
Number of links L 4 HOOD
3
Number of full joints J1 4 2 4
1
Number of half joints J2 0
BODY
3 ( L 1 ) 2 J1 J2
1
b. An automobile hatchback lift mechanism.
The hatch (2) is pivoted on the body (1) and is linked to the body by the lift arm, which can be modeled as
two links (3 and 4) connected through a translating slider joint.
,DESIGN OF MACHINERY - 6th Ed SOLUTION MANUAL 2-1-2
Number of links L 4 HATCH
Number of full joints J1 4 2
3
1
Number of half joints J2 0
4
M 3 ( L 1 ) 2 J1 J2 1
BODY
M1
c. An electric can opener has 2 DOF.
d. A folding ironing board.
The board (1) itself has one pivot (full) joint and one pin-in-slot sliding (half) joint. The two legs (2 and 3)
have a common pivot. One leg connects to the pivot joint on the board and the other to the slider joint.
Number of links L 3
1
Number of full joints J1 2
3 2
Number of half joints J2 1
M 3 ( L 1 ) 2 J1 J2
M1
e. A folding card table has 7 DOF: One for each leg, 2 for location in xy space, and one for angular orientation.
f. A folding beach chair.
The seat (3) and the arms (6) are ternary links. The seat is linked to the front leg(2), the back (5) and a
coupling link (4). The arms are linked to the front leg (2), the rear leg (1), and the back (5). Links 1, 2, 4, and
5 are binary links. The analysis below is appropriate when the chair is not fully opened. When fully opened,
one or more links are prevented from moving by a stop. Subtract 1 DOF when forced against the stop.
Number of links L 6 5
6
Number of full joints J1 7
4
1 2
Number of half joints J2 0 3
M 3 ( L 1 ) 2 J1 J2
M1
g. A baby swing has 4 DOF: One for the angular orientation of the swing with respect to the frame, and 3 for the
location and orientation of th e frame wit h respect to a 2-D frame.
,DESIGN OF MACHINERY - 6th Ed SOLUTION MANUAL 2-1-3
h. A folding baby walker has 4 DOF: One for the degree to which it is unfolded, and 3 for the location and
orientation of t he walker with respect to a 2-D frame.
i. A fancy corkscrew has 2 DOF: The screw can be rotated and the arms rotate to translate the screw.
j. A windshield wiper mechanism has 1 DOF: The position of the wiper blades is defined by a single input.
k. A dump-truck dump mechanism has 1 DOF: The angle of the dump body is determined by the length of the
hydraulic cylinder that links it to the body of the truck.
l. A trash truck dumpster mechanism has 2 DOF: These are generally a rot ation and a translation.
m. A pickup tailgate mechanism has 1 DOF:
n. An automobile jack has 4 DOF: One is the height of the jack and the other 3 are the position and orientation
of the jack with respect to a 2-D frame.
o. A collapsible auto radio antenna has as many DOF as there are sections, less one.
, DESIGN OF MACHINERY - 6th Ed. SOLUTION MANUAL 2-2-1
PROBLEM 2-2
Statement: How many DOF do you have in your wrist and hand combined?
Solution: See Mathcad file P0202.
1. Holding the palm of the hand level and facing toward the floor, the hand can be rotated about an axis
through the wrist that is parallel to the floor (and perpendicular to the forearm axis) and one
perpendicular to the floor (2 DOF). The wrist can rotate about the forearm axis (1 DOF).
2. Each finger (and thumb) can rotate up and down and side-to-side about the first joint. Additionally, each
finger can rotate about each of the two remaining joints for a total of 4 DOF for each finger (and thumb).
3. Adding all DOF, the total is
Wrist 1
Hand 2
Thumb 4
Fingers 4x4 16
TOTAL 23
