Mechanisms
Linkages
1.
The elliptic trammel is an instrument for drawing ellipses. Please prove that the
point P describes an ellipse.
2.
Please draw the equivalent linkages for following mechanisms.
3.
(a)
(b)
For a four-bar linkage, link 2 is the input link and link 4 output link. Please prove
the min and max transmission angles occurring at θ 2 = 0 o and θ 4 = 180 o ,
respectively.
4. For a four-bar linkage, the lengths of links are r1 = 100 mm, r2 = 60 mm,
r3 = 90 mm and r4 = 90 mm. Please calculate the two limit positions of the
mechanism.
5. Please design an offset slider-crank mechanism with stroke=20mm and
offset= 10 3 mm and time ratio=13/11.
6. Design a quick-return mechanism with 20mm stroke; the working stroke is to
take 0.3 sec and the return stroke 0.1sec.
7. This question concerns the classification of four-bar linkages. Link lengths: r2 ,
driver crank; r3 , coupler; r4 , driven link; r2 =100mm, r3 =200mm,
r4 =300mm. Find the ranges of values for r1 if the linkage is as follows:
(a) Grashof mechanism
(b) Crank-rocker mechanism
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(c) Drag-link mechanism
(d) Double-rocker mechanism
(e) Change-point mechanism
(f) Triple-rocker mechanism
8. Please draw the skeleton sketch and explain the function of the following
mechanisms.
(a) Toggle mechanism
(b) Universal joint
(c) Oldham coupling
(d) Scotch yoke
(e) Parallel mechanism (f) Quick-return mechanism
9. List the number of links, the numbers of various joints and calculate the degree of
freedom for the following mechanisms.
2
Kinematics
1.
For a four-bar linkage, link 2 is the input link and link 4 output link. Please derive
the position equation of link 4.
2. For a four-bar linkage, link 2 is the input link and link 4 output link. The position
analysis is finished and the velocity of link 2 is known. Please derive the velocity
equation of link 4.
3. Please state Kennedy’s theorem and prove it.
4. Please locate all the instant centers of the following four-bar linkage and calculate
the output velocity of link 4 when the input velocity is ω 2 = 10 rad/sec ccw. Please
use instant-center method and the link lengths measured in the figure.
3
2
4
1
5.
P.4
P.5
Please locate all the instant centers of above slider-crank mechanism and
calculate the output velocity of link 4 when the input velocity is ω 2 = 10 rad/sec
6.
ccw. Please use instant-center method and the link lengths measured in the figure.
Please calculate the output velocity of link 6 when the input velocity
is ω 2 = 10 rad/sec ccw. Please use instant-center method and the link lengths
measured in the figure.
3
7.
Please locate all instant centers for following mechanisms.
8.
For the six-bar mechanism shown below, please use graphical method to find the
positions of links when θ 2 = 30 o . Use the lengths measured in the figure.
9.
For the six-bar mechanism shown above. When the position analysis is finished
and input velocity ω 2 is known, please derive the velocity equations of all links
in matrix form.
10. For the inverted slider-crank mechanism shown below, please calculate the
position, velocity and acceleration of the output link 4 with θ 2 = 120 o ,
ω2 =100rpm ccw and α 2 =0. O2 O4 = 3.0cm and PO2 =7.0cm.
4
Cams
1. For a given displacement diagram, how to reduce the pressure angle of the cam?
2. Please state the definition of undercutting. How to solve the problem of
undercutting?
3. A cycloidal cam with a flat-faced follower has a rise of 20mm in an angle of
β = 75 o . Base-circle radius Rb is 80mm. Radius R g of the cutter is 50mm. For
values of θ of 30 o and 60 o calculate the cam profile. Also calculate the
location of the cutter to produce the cam profile.
4.
The total displacement h of the follower occurs as the cam rotates an angle β . For
any angle of cam rotation θ , the displacement is s.
Please derive the equation of
4-5-6-7 polynomial curve.
⎡
θ
θ
θ
θ ⎤
s = h ⎢35( )4 − 84( )5 + 70( )6 − 20( )7 ⎥
β
β
β
β ⎦
⎣
5. A disk cam is to give its follower a rise through travel h during a cam rotation α .
This motion is preceded by a dwell and is also followed by a dwell. Determine the
polynomial motion that will satisfy all displacement, velocity, and acceleration
conditions at the beginning and end of the travel.
6. Please explain following nomenclature for cam.
(a) Base circle
(b) Pitch curve
(c) Pressure angle
5
7.
A disk cam with an offset translating roller follower is shown below. Please
derive the equations of the cam profile and cutter location.
8.
What is the definition of positive-drive cam or conjugate cam? Which are
positive-drive cams in the following figures.
6
9.
For a disk cam with a translating radial roller follower, the follower is to move
outward 10mm with cycloidal motion during 180 o of cam rotation, is to dwell
for the next 60 o , and is to return with constant velocity motion during the last
120 o of cam rotation. The roller radius is 5mm, the base circle 30mm and the
cam speed is 18rpm. The 360 o of cam rotation is divided into twelve equal
divisions.
(a) Plot the displacement and velocity curves in the following figures.
(b) Plot the cam profile.
Displacement
10
8
6
4
2
0
0
90
180
270
360
0
90
180
270
360
10
5
Velocity
0
-5
-10
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Gears
1.
Please explain following nomenclature for gear.
(a) Base circle
(b) Pitch circle
(c) Pressure angle
(d) Theory of gearing
(e) Contact ratio
(f) Module
(g) Backlash
(h) Circular pitch
2.
A spur gear having 32 teeth and a pressure angle of 20 o has a module of 6.5
and an addendum of 5.5. Determine the number of teeth on the smallest pinion
that will mesh with the gear without inference.
3. A pair of spur gears has 16 and 24 teeth, a module of 6.5mm, and a 20 o
pressure angle.
(a) Determine the center distance
(b) If the center distance is increased 3mm, find the resulting pressure.
4.
Two 20 o gear have a diametral pitch of 4. The pinion has 28 teeth, while the
gear has 56 teeth. Determine the center distance for an actual pressure of 20 o .
What is the actual pressure if the center distance is increased by 0.2in?
5.
Two 25 o full-depth spur gears have a velocity ratio of 1/3. The diametral pitch
is 5 and the pinion has 20 teeth. Determine the contact ratio (number of teeth in
contact) for the gears.
Two standard 20 o full-depth gears have a module of 8 mm. The larger gear has
30 teeth, while the pinion has 15 teeth. Will the gear interfere with the pinion?
7. Suppose input 1 turns at 120rpm ccw and input 2 turns at 360 rpm cw, determine
the speed and direction of rotation of the output shaft.
6.
8
8.
For the planetary train shown below, let tooth numbers be N S =40, N P =20 and
N R =80. Find planet carrier speed if the sun gear rotates ccw at 100rpm and the
ring gear cw at 300 rpm.
9. Calculate the speed reduction ratio
ω5
for the simple planetary train with sun
ω4
gear 2 fixed.
10. Calculate the speed reduction ratio
ω2
for the following mechanism. Link 2 is
ω6
the input and link 6 output.
(b)
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11. A conventional three-speed automotive transmission is show below. Gear 1 is
driven by the engine. Gears 4, 5, 6, and 7 rotate as a unit. Gear 8 is an idler that
meshes with 7. These gears are always in motion when the shaft to the engine is
motion. Gears 2 and 3 can slide axially on the splined shaft to the driving wheels.
In the first speed, Gear 3 is shifted to the left to mesh with gear 6. For the
second speed, Gear 2 is shifted to the right to mesh with gear 5. For the third
speed, gear 2 is shifted to engage its clutch teeth with those of gear 1. For reverse,
Gear 3 is shifted to the right to mesh with idler 8. Please calculate the speed
reductions for all speeds.
12. A conventional automotive differential mechanism is show below. Please derive
the arm speed in terms of the speeds of left axle and right axle. Also explain
how does it work?
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