16-07-2013, 04:57 PM
Free Body Diagrams of Gear Trains
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Abstract
Many senior students in the author’s machine elements course have difficulties in drawing a
FBD (Free Body Diagram) correctly, which is the first step in force and stress analysis of a
mechanical system. One of the challenges to those students is that even though the principles to
draw a FBD are the same for every mechanical system (A FBD is a sketch of a mechanical
system cut free of its surroundings to shows all the external forces acting on the system), each
problem seems different.
Textbooks provide limited examples, especially in the case of gear sets and gear trains when
three dimensional FBDs are necessary. For students who have not mustered basic concepts in the
FBD, they need additional sample problems and concise rules to enhance their understanding. To
this end, the author has developed a course note with simple rules to determine the direction of
gear force components (radial, tangential, and axial) with extensive two and three dimensional
FBDs of gear sets and gear trains.
Background
Gear sets are used to transmit rotary motion and power from one shaft to another. The
magnitudes and directions of the tangential, radial and axial components of gear forces are
important because they act on the shafts that the gears are mounted on and contribute to the
forces acting on the bearings that support the shafts. Since the conditions of static equilibrium
will be used to determine bearing reactions, correct directions for the gear forces acting on a
shaft must be established.
Since machine design textbooks [l,2,3,4] typically include equations to determine the
magnitudes of the gear forces, the material presented here is for determination of only the gear
force directions.
Force on a spur gear
When two spur gears are meshed shown in Fig. 1, and the left one is the driver, the contact point
moves along a line as the gears rotate, as shown in Fig. 2. The line of action is sometimes called
the pressure line. The force pushing the driven gear is shown in Fig. 3, and will always be along
this line of action. The type of force is bearing (pushing) force, applying pressure to the mating
tooth. From the principal of force transmissibility in statics, we know that any point along the
line of action will still create the same torque about the gear. Therefore, we can use a fixed
contact point (pitch point) to simplify the representation of gear force during the engagement, as
in Fig. 4. The pitch point is the intersection of the pressure line and the center line of gears. The
angle between the line tangent to both pitch circles and the line of action is called the pressure
angle.
FBD of a helical gear set
Spur and helical gears are used to connect two shafts that are parallel to each other. Spur gears
are teeth parallel to the shaft, while helical gears are with teeth at an angle to the axis, as shown
in Fig., 9. The force on a helical gear has three components: radial, tangential and axial. The
force is still normal to the gear tooth, at an elevated angle of φ. Because the teeth are at a helix
angle to the gear axis, the gear force has an axial component.
The directions of Ft and Fr are determined the same way as in the spur gear. The direction of Fa
will be decided by that of Ft. This is because Fa and Ft are on the same side of the gear tooth
since they are components of the gear force F, which is a compressive force, pushing the tooth in
the normal direction.
FBD of a worm gear set
Worm gears are used to provide right-angle connection of two non-intersecting shafts, as shown
if Fig. 12. The worm resembles a screw, and the worm gear is a helical gear with the top of the
teeth concaved to provide maximum contact between the worm and worm gear. High gear ratios
can be achieved with worm gear sets, but at the expense of efficiency. The directions of the gear
forces can then be determined with the following rules.
The direction of Fr is determined the same way as in the spur gear. The direction of Fwt is
determined the same way as in the spur gear. That of FGt is not as straight-forward, as FGt and
Fwt are in different directions and of different magnitudes, unlike other types of gears discussed
so far. Because the two gears are at a right angle, the direction of FGa and Fwt are equal and
opposite.