What Is the Mechanical Advantage of a Bolt-and-Nut Power Transmission?
First, let’s understand how the power is transmitted. When we turn the head of the bolt,
its threads slide over the threads of the nut and, while sliding, displace the nut and the
bolt relative to one another. To facilitate understanding of the problem, let us analytically
unwrap the cylindrical contacting surfaces of the bolt and the nut into a plane, and only
consider one full revolution of the thread.
Here is the resulting problem depiction:
Pdr, driving force
component parallel
to the thread
bolt
nut
Fdr, applied
driving force
N, normal
component of
resulting force
Pr, reaction force
component parallel
to the thread
Ffr, full reaction
force
R, resulting axial
force for true
friction value
R max, resulting
force, zero
friction
On this picture, the grey triangle represents one revolution of the nut’s thread, and the
grey box on top of the triangle – a piece of the bolt’s thread sliding over the nut’s thread.
The incline of the thread  (marked with the small black triangles) can be estimated from
the knowledge of the thread radius and the number of threads per inch. So, the vertical
leg of the triangle will be equal to the distance between two consecutive threads and the
horizontal leg will be equal to the circumference of the threaded bolt.
Given the applied driving force Fdr, we will find the resulting driven axial force on the
bolt R. Fdr itself is equal to the force applied to the bolt’s handle times the mechanical
advantage of the handle. If the force is applied to the handle at 5.5” distance from the
center of the bolt and the radius of the bolt is 0.5”, the mechanical advantage is 11, and
Fdr is 11 times the force applied to the handle.
By applying a given force to the handle we can only drive the bolt into the nut until the
equilibrium condition is satisfied – when the driving force becomes equal to the force of
reaction. Accordingly, the components of the applied force Pdr and the reaction force Pr
parallel to the threads also become equal to one another. Their value is, hence, equal to
the Fdr times the cosine of .
For ideal sliding with no friction, the resulting driving force can be now found as P r
divided by sin(). The resulting value is indeed impressive, but hardly achievable due to
friction. In reality, the maximum force at which the bolt can be driven into the nut is
always limited by friction, bringing us to the condition of Pdr = Pr = Ffr, where Ffr is the
force of frictional resistance. Frictional force is proportional to the normal force between
the contacting bodies with a constant friction coefficient. This coefficient depends on
several factors, including the material from which the bodies are made. For example, its
value is 0.47 for steel being rubbed over aluminum.
This coefficient will be useful for the estimation of a normal force between the threads:
N = Pr / {friction coefficient}
By now, we have obtained the values of both components of the full reaction force, and
therefore, we also can find its value Ffr. Notice that the full reaction force is not vertical
in our representation, and, therefore, not parallel to the axis of the thread. Physically this
can be interpreted as follows: friction both reduces the resulting axial force and produces
the reactive frictional torque – that is the non-vertical component in our problem
representation.
To obtain only the vertical or axial component of the full resulting reaction force, we can
represent the full reaction force in a different basis. This can be conveniently done by
calculating the value of an angle marked with the white triangle  as a difference of a
straight angle and ( + ), where  is a value of a hatched angle known from the triangle
of components of the reaction forces. The sought value R is equal to Ffr times cos().