Review of Basic Mechanics and
Mechanical Systems
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Levers
• The ratio of the effort arm length to the
resistance arm length is inversely proportional
to the ratio of the resistance to the effort
required to overcome it.
ℓ
L
E
R
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Fulcrum
L=Length of effort arm ,ℓ=Length of resistance arm, R=Resistance weight or force
E=Effort
Mechanical Advantage
• The ratio of the required effort to the weight of
the object to be removed provides the
mechanical advantage of a lever.
• If Mechanical Advantage > 1, the lever has positive gain of
force.
• If Mechanical Advantage < 1, the lever has a negative gain
of force.
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Grippers
• Grippers can be considered as levers. The
following equation can be used to calculate
force and length.
• Examples:
F1
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a
F1
F2
a
b
F2
Moment of Force
• The Moment M of force about a point 0 is the
perpendicular distance from 0 to the line of
action of force F times the force F.
M=Moment , F=Force and d=distance
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Rotational Kinetic Energy
• A mass having either linear motion or rotary motion
has kinetic energy (KE). Kinetic energy of a mass ‘m’
moving at a velocity v is
• Also kinetic energy of a mass ‘m’ moving at an angular
velocity of ω is,
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Rotational Kinetic Energy
Axis of Rotation
m
r
Also
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Kinetic Energy
• Kinetic energy of a rotating body is equal to half
the sum of the value of all its particles
multiplied by the square of its angular speed.
or
The quantity
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is called Moment of Inertia of the body
Moment of Inertia: Different
Types of Bodies
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Belt Drive & Gears
• Levers are limited by angle through which they
can operate. On the other hand belt and gear
can be used on a continuous basis.
• A pair of rolling cylinders can transfer rotary
motion from one shaft to another by using a
belt or a gear.
• Mechanical advantage in a belt drive is achieved
at the expense of motion.
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Belt Drive & Gears
•The higher the output torque relative to the input
torque, the more turn of the drive pulley are
needed per turn of the driven one.
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is the radius of driving pulley and is the radius of the driven pulley.
Mechanical
Advantage
Ideal
Mechanical
Advantage
Actual
Mechanical
Advantage
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Other Important Relationships
The ratio of angular speed is the inverse of the ratio of pulley diameters.
When motor (input) pulley is smaller than the machine(output)
pulley, the machine turns at a smaller number of rpm (revolutions per minute) than the motor but the torque provided to the machine shaft is higher
than that of the motor
And
When motor pulley is larger than machine pulley, the machine speed is
higher but the available torque is lower.
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Gears
• Gear drives have the advantage over the belt
drive to transfer higher torque with less chance
of slippage. The disadvantage of using gears is
cost, weight and it requires more precision.
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Spur Gears
• Spur gears have their teeth cut parallel to
the axis of rotation.
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Image Source: http://www.dallasmusic.org/gearhead/
Helical Gears
• Helical gears have curved teeth cut in a
spiral pattern at an angle to their axis.
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Image Source: http://www.elizabethtown.kctcs.edu/members/jnail/BRX-Gears.htm
More Formulas
• Speed Change:
v1 and v2 are tangential speed of the gears, n1 and n2 are number of
teeth and ω1 and ω2 are tangential speed in radians/seconds.
• Mechanical Advantage ( Theoretical Mechanical
Advantage):
or
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Torque Advantage
Where n1= number of teeth on the driving gear and n2= number of
teeth on the driven gear.
M1 and M2= Turning Moments
v1 and v2= Tangential speed of the gear
n1 and n2= Number of teeth
ω1 and ω2= Tangential Speed in radians/seconds.
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*Note: Friction is considered to be negligible or zero.