Complex machines are made up of moving parts such
as levers, gears, cams, cranks, springs, belts, and
wheels. Machines deliver a certain type of movement to a desired
location from an input force applied somewhere else. Some
machines simply convert one type of motion to another type, such as
rotary to linear. While there is a seemingly endless variety
of machines, they are all based upon simple machine elements. The elements
discussed here include inclined planes, levers, wheels and axles,
pulleys, and screws.
It is important to remember that all machines are
limited in their efficiency by friction. No machine is 100 percent efficient
in its efforts, so the mechanical advantage gained will require additional
energy to accomplish the task. For more information on friction, see this
module
The Inclined Plane
An
inclined plane decreases the force
required to raise an object a given height by
increasing the distance over which that force must be
applied, see
Figure 1. Imagine lifting something twice your weight to a 4
foot high shelf. Now imagine rolling the same mass up a gently
sloping surface. The latter would be much easier. Inclined
planes are commonly put to use in cutting devices and often
two inclined planes are put back to back to form a wedge. In a
wedge, forward movement is converted into a parting movement
acting perpendicular to the face of the blade. A zipper is simply a combination of two
lower wedges for closing and an upper wedge for opening, as shown in
Figure 2.
The Screw
The
screw is basically an
inclined plane (see
Figure 3) wrapped around a
cylinder. In an inclined plane, a linear force in the
horizontal plane is converted to a vertical "lifting"
force. With a screw, a rotary force in the horizontal plane is
converted to a vertical "lifting" force.
When a wood screw is turned, the threads of
the screw push up on the wood. A reaction force from the wood
pushes back down on the screw threads and in this way the
screw moves down even though the force of turning the screw is
in the horizontal plane. Screws are known for high friction,
which is why they are used to hold things together. A worm gear is sometimes used in machines, but they also have high friction that can waste considerable power.
Levers
A lever has three points of
interest: the fulcrum, the load, and the effort applied to the lever. The
fulcrum is the point around which the lever
pivots rotationally. The load is what we wish to
manipulate with the lever, and the load is described by its position relative to the fulcrum, and the force (magnitude and direction) it exerts at that point.
The effort is also a force that has a magnitude and
a direction, and a position with respect to the fulcrum. A lever
is used to change the direction of movement, and to
trade the magnitude of the effort for the distance over which
the effort is applied.
As shown in
Figure 4, there are
three different classes of levers defined by the relative
positions of the fulcrum, effort, and load. A
first
class lever has the fulcrum positioned between the
effort and the load. Examples of first class levers include: a
balance, a crow bar, and scissors. In a
second class
lever the load is placed between the fulcrum and the
effort. Examples of second class levers include: a
wheelbarrow, a bottle opener, and a nutcracker.
Third
class levers place the effort between the fulcrum and
the load. Examples of a third class lever are a hammer, a
fishing rod, and tweezers. Most machines that employ levers
use a combination of several levers, often of different
classes.
The Wheel and Axle
Both levers and the inclined plane lower the force required for a task at the price of having to apply that force over a longer distance.
With wheels and
axles the same is true: a poweful force and movement of the axle is converted
to a greater movement, but less force, at the circumference of the wheel. In a circular geometry,
torque is a more useful concept than force and distance. You can learn more about torque
here. The
wheel and axle can be thought of as simply a
circular lever, as shown in
Figure 5. Many common
items rely on the wheel and axle such as the screwdriver, the
steering wheel, the wrench, and the faucet.
Gears and Belts
A wheel and axle assembly becomes especially
useful when combined with gears and belts. Gears can be used to change the
direction or speed of movement, but changing the speed of
rotation inversely affects the force transmitted. A small
gear meshed with a larger gear will turn faster, but with
less force. There are four basic types of gears: spur gears,
rack and pinion gears, bevel gears, and worm gears.
Spur gears are probably the type of gear that
most people picture when they hear the word. The two wheels
are in the same plane (the axles are parallel). With
rack and pinion gears there is one wheel and
one rack, a flat toothed bar that converts the rotary motion
into linear motion. Bevel gears are also known
as pinion and crown or pinion and ring gears. In bevel
gears, two wheels intermesh at an angle changing the
direction of rotation (the axles are not parallel); the
speed and force may also be modified, if desired.
Worm gears involve one wheel gear (a pinion)
and one shaft with a screw thread wrapped around it. Worm
gears change the direction of motion as well as the speed
and force. Belts work in the same manner as
spur gears except that they do not change the direction of
motion.
In both gears and belts, the speed and force is altered
by the size of the two interacting
wheels. In any pair, the bigger wheel always rotates more
slowly, but with more force. On both the big and the small gear, the linear
velocity at the point of contact for the wheels is equal. If
it was unequal and one gear were spinning faster than the
other at the point of contact then it would rip the teeth
right off of the other gear. As the circumference of the
larger gear is greater, a point on the outside of the larger
gear must cover a greater distance than a point on the
smaller gear to complete a revolution. Therefore the smaller
gear must complete more revolutions than the larger gear in
the same time span. (It's rotating more quickly.)
The force
applied to the outer surface of each wheel must also be
equal otherwise one of them would be accelerating more
rapidly than the other, and again the teeth of the other
wheel would break. However, the forces
the forces applied to
the axles are not equal. Returning to the
concept of levers, we know that the distance from the fulcrum at which the
force is applied effects the force applied at another point, and a wheel and
axle works like a lever. Equal forces are being applied to
each wheel, but on the larger wheel that force is being
applied at a greater distance from the axle. Thus, for the larger wheel,
the force on the axle is greater than the force on the axle
of the smaller wheel.
Cams and Cranks
Both cams and cranks are useful when a
repetitive motion is desired.
Cams make rotary
motion a little more interesting by essentially moving the
axle off-center. Cams are often used in conjunction with a
rod. One end of the rod is held flush against the cam by a
spring. As the cam rotates the rod remains stationary until
the "bump" of the cam pushes the rod away from the cam's
axle. Cams can be used to
create either a linear repetitive motion such as the one
illustrated in
Figure 6, or a repetitive
rotational motion such as
using a cam and a rubber band.
Cranks convert rotary motion into
a piston-like linear motion. The best examples of cranks in
action are the drive mechanism for a steam locomotive and the
automobile engine crankshaft. In a crank, the wheel rotates
about a centered axle, while an arm is attached to the wheel
with an off-centered peg. This arm is attached to a rod fixed
in a linear path. A crank will cause the rod to move back and
forth. If instead the rod is pushed back and forth, it will cause
the crank to turn. On the other hand, cams can move their
rods, but rods cannot move the cams.
Pulleys
Pulleys can be used to simply
change the direction of an applied force or to provide a
force/distance tradeoff in addition to a directional change,
as shown in
Figure 7. Pulleys are very flexible
because they use ropes or chain to transfer force rather than a rigid
object such as a rod. Ropes can be routed through
virtually any path. They are able to abruptly change
directions in three-dimensions without consequence, except, of
course, additional friction. Ropes can
be wrapped around a motor's shaft and either wound up or let
out as the motor turns.
Figure 8 illustrates how a
compound pulley 'trades' force for distance through an
action/reaction force pair. In a double pulley, as the rope
passes over the pulley the force is transmitted entirely but
the direction has changed. The effort is now pulling up on the
left side of the bottom pulley. Now, for a moment forget that
the end of the rope is tied to the bottom of the top
pulley. The mechanics are the same if the rope is fixed to the
ceiling. The important thing is that the end of the rope is
immobile. The effort is once again transmitted entirely as the
rope passes over the bottom pulley and there is a direction
change. The end of the rope is attached to the ceiling so the
rope is pulling down on the ceiling with the force of the
effort (and half of the force of the load). We assume that the
ceiling holds up, so this must mean that there is a force
balancing out this downward force. The ceiling pulls up on the
rope as a reaction force. This upward force is equal to the
effort and now there is an upward force on the right side of
the bottom pulley. From the perspective of a free-body diagram
the compound pulley system could be replaced by tying two
ropes to the load and pulling up on each with a force equal to
the effort.
The disadvantages of pulleys, in contrast to
machines that use rigid objects to transfer force, are
slipping and stretching. A rope will permanently stretch under
tension, which may affect the future performance of a
device. If a line becomes slack, then the operation of a
machine may change entirely. Also, ropes will slip and stick
along pulley wheels just like belts. One solution to the
problems associated with rope is to use chain. Chain is
pliable like rope, and is able to transfer force through many
direction changes, but the chain links are inflexible in
tension, so that the chain will not stretch. Chains may also
be made to fit on gears so that slipping is avoided.