Robert,
I'm forwarding this to the list, because your points make intuitive sense,
and I imagine other people are also wondering about the little-torque-meters
question.
The fact that little torque meters along the length of a torque wrench
would not read the same value is somewhat difficult to explain since it's
not really intuitive. I'll try to explain why this is, and then it should
be clear why handle extensions will cause a beam type torque wrench to read
innacurately.
While the torque required to turn a bolt
or raise a child on a see saw doesn't change, the bending moment along
the beam varies linearly with respect to the distance from the pivot axis.
So if you were to place little torque meters along a torque wrench, they
would read the maximum torque value at the vertical bolt axis, but nothing
at the point where you apply the linear force with your hand.
Using a handle extension does two things to the torque wrench.
First, the extension applies a uniform bending moment to the torque wrench
shaft, and secondly, it applies a linear force at the handle in the direction
of push. A theory of translation from Engineering Statics states that
an applied bending moment could be applied anywhere on a structure and have
the same effect. So the bending moment applied by the handle extension
is constant over the length of the wrench.
However, the linear force imparts a bending moment according to the
relation Moment = Force x radius. I'll construct an example with a torque
wrench of length 2 feet, and an applied handle force of 40 lb.
To analyze this, we'll create a graph. Plot bending moment on the y-axis
and distance along wrench on the x-axis, with the vertical bolt axis at the
left hand side of the x-axis. This is a bending moment diagram, and measures
the bending moment as you move along the wrench length. Evaluating this
expression at the vertical bolt axis, (LHS of graph), the result would be
40 lb x 2 ft, or 80 lb-ft. This is expected. Evaluating this expression
at the handle (RHS of graph) yields 40 lb x 0 in = zero lb-ft. The
actual bending moment present in the metal of the torque wrench handle due
to applied linear force is zero. This is not totally intuitive.
The bending moment at 1/2 distance on the wrench would be 40 lb x 1 ft or
40 lb-ft. So on the left side of the graph, our y-value is 80 lb-ft, at
midway along the x-axis the value is 40 lb-ft and at x= 2 feet, the y-value
is zero. If we continued doing these calculations at intermediate points
along x, the result would be a line sloping down to zero at the handle of
the wrench.
A real world model to show that bending moment is maximum at the vertical
pivot axis is to take a long wooden dowel and mount it in your bench vise.
If you apply a linear force at the free end of the dowel, major deformation
takes place closest to the vise. This is where fracture will occur.
This happens according to the bending moment diagram.
Now, if you apply a bending moment at the free end of the dowel, being careful
not to apply any linear loading, you will get the dowel in an arc, and
eventually into a circle, if the thing doesn't snap first. This demonstrates
the even translation of bending moment across the entire length of the beam,
and models what happens when you use a handle extension. As is evident now,
the deformed shape due to an applied force and the deformed shape due to
applied bending moment are different.
Realize that a beam type torque wrench measures deflection of the wrench,
while a clicker type measures torque at the head. Just as a dowel in the
vise would deflect acording to the type of load applied to it, the torque
wrench does the same. The linear force dowel model deflects according to
a polynomial expression, while the bending-moment-applied dowel model deflects
according to a circle.
A properly designed beam type torque wrench will have a pivot in the
handle, and this makes it difficult to apply a bending moment directly from
your fist to the wrench. This ensures accuracy of the reading.
Now for the other point, it really doesn't matter where you hold a clicker
type wrench; holding it in from the handle would produce a steeper slope on the
bending moment diagram, and you would have to push harder because your r value
is slightly less, but bending moment would be accurately read since it's value
is obtained from the head of the wrench (Left Hand Side of the bending moment
diagram.)
But, a beam type torque wrench relies on deflection and it is read at the
handle. So if you hold a beam type wrench at say, halfway along it's length,
even if you weren't applying any bending moment, the reading would be wrong.
This is of course due to the fact that only the 1st half of the beam is
deflecting, while the second half is absolutely straight.
I hope I didn't go on too long. Please respond.
Greg Meboe Meboe@wsuvm1.csc.wsu.edu
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