Re: [Units] Incorrect definitions for torque, newton_meter

From: Albert Dvornik (bert-boost_at_[hidden])
Date: 2022-08-30 13:13:46

Hi, Matthias!

On 08/26/2022 12:35 PM EDT Family Schabel <family_at_[hidden]> wrote:
> Maybe this link will clarify:
> 
> https://physics.stackexchange.com/questions/37881/why-is-torque-not-measured-in-joules

> There’s also a discussion of this issue in Wikipedia:
> https://en.wikipedia.org/wiki/Radian

Thank you for your reply and the links, and of course also for your
work on Boost.Units, which I find very useful. It's good to know that
the current behavior is intentional. I had not previously encountered
the idea of expressing torque in N m/rad, so I took a while to think
it through. Another summary of the topic that I found helpful was
presented in two 2016 papers by Quincey and by Quincey and Brown:

    https://doi.org/10.1088/0026-1394/53/2/840
    https://doi.org/10.1088/0026-1394/53/3/998

I apologize in advance for my terrible math "typesetting."

To summarize: Boost.Units treats angles as base quantities, rather
than dimensionless, so it can for example distinguish between radians
and degrees. Given that, the formula for the work done by a torque on
a rotating body

    W = \integral \vec{\tau} d\vec{\theta}

implies that the torque should be expressed in N m/rad. This is not
dimensionally compatible with some other formulas for torque,
e.g. from applying a force at some distance from an axis of rotation,
traditionally written as follows, with torque in N m:

    \vec{\tau} = \vec{r} \cross \vec{F}

Since a units library such as Boost.Units needs to be consistent, it
must choose a way to reconcile the two, and can't get away with
hand-waving that radians are sometimes a base unit and sometimes just a
funny way of spelling a dimensionless constant.

> Boost Units explicitly keeps track of angular displacement for this
> (and similar reasons). The unit of torque really should not be
> energy (J or N.m) because a torque is a force applied tangentially,
> so, to compute the amount of energy required to exert a torque one
> needs to take the product of the force with the angular
> displacement.

I agree that choosing N m/rad as the units of torque seems reasonable,
if non-traditional, and I don't expect to get anyone to change that
decision now. (I don't think this was a *necessary* choice, though;
using N m would also have been reasonable.  I may be biased by the
fact that what I usually work with are force-based calculations in
static scenarios where there is no angular displacement, and so no
work being done.)

But the two equations above are still not dimensionally consistent,
and as Quincey & Brown point out, to make them consistent you have to
change the traditional torque-distance-force equation by adding a
dimensionless factor \eta = 1/radian, introduced by Torrens:

    \vec{\tau} = \eta \vec{r} \cross \vec{F}

I think this introduces a documentation problem for Boost.Units. I
don't know exactly how many users will be completely blindsided by
this, but my team at work and I were definitely included, and I think
this situation will be pretty common at least in the US. These users
may run into compilation issues they don't understand, view this as a
result of some deficiency in Boost.Units, and implement undesirable
workarounds (as an example, my team created local specializations of
boost::units::multiply_typeof_helper that "made things work").

To avoid this, I think the documentation should describe the choices
made by Boost.Units with respect to torque, moment of inertia and
angular momentum, and the implications of those choices. If the
maintainers agree, I'd be happy to draft some patches for review on
Github. It would probably also be beneficial to define \eta = 1/radian
for users to use.

> The nomenclature is a bit confusing because, as you note,
> newton_meter is not the same as newton*meter - the convention is
> that torque is measured in newton meters, but it is really newton
> meters per radian as encoded in the library.

I'm really perplexed by this statement. The Boost.Units code base
supports the general idea that angles should be treated as a base
unit. If so, then surely N m/rad and N m are different units! Blurring
this difference will increase user confusion about how to correctly
use the library. Saying that the units are N m/rad but that we choose
to label this newton_meter makes no sense to me.

(This gets extra confusing for our code base at work, which includes
calculations using torsional stiffness, i.e. torque per unit angular
displacement. If boost::units::si::newton_meter is N m/rad, what
should we call the unit of torsional stiffness (N m/rad^2 here, but
N m/rad "traditionally")? newton_meter_per_radian, for consistency
with newton_meter, but really confusing given its actual definition?
Or something else, inconsistent with the other units?)

The right choice would be to define newton_meter_per_radian as a unit
of torque, and in my opinion also to deprecate the existing definition
of newton_meter.

Thank you,
Albert