Gregory L. Hansen wrote:

In article <1128455970.752360.190540@g43g2000cwa.googlegroups.com>,
<timwryan@gmail.com> wrote:

Here is the problem.

Bodies A, B, and C are arrayed in a triangle. A and B are connected by
a spring with spring constant K and length r. B and C are also
connected by a spring with the same spring constant K and same length
r. Let the angle ABC be theta radians. Angle ABC has a bending spring
constant k_bend. All potentials are harmonic.

So, what is the effective spring constant between A and C?

This problem has me stumped.

Thanks a lot.

Who would give an insane spring problem like this one?

It seems like a strange homework problem, but not a strange
real-life problem. Perhaps I want to know how to build spring
systems with any desired spring constant out of a stock collection
of springs.

First we have to make sure all our assumptions are straight. We always
start with a diagram. Slightly rotated for ASCII purposes...

K
A-------B
|
k_bend | K
|
C

I suppose the effective spring constant between A and C means the force
per unit displacement if we tied strings to A and C and pulled them apart.
That is, the force will act directly through the line connecting A and C.

Yes. I envision a piece of spring steel mounted at the
corner ABC, pre-bent to the desired equilibrium angle.

Not explicitly stated in the problem, but probably reasonable, is to
suppose that the springs on AB and BC are in telescoping tubes that keep
the springs straight, and B is hinged.

If AB and BC are connected by rigid rods and B is hinged with a spring
constant k_bend, can you calculate the effective spring constant between A
and C?

If AB and BC are connected by springs inside telescoping tubes and B is
locked into position at some angle theta, can you calculate the effective
spring constant between A and C?

Couldn't you do it all in one step? Stretch the line
AC by some amount dx. This will cause a change in lines
AB and BC by an amount which depends on theta. It will
also change theta by an amount d(theta).

As a result of those three stretches, there will be a
total restoring force. The ratio between force and dx
is the effective spring constant.

Of course because of the dependence on theta, I suspect that
Hooke's law doesn't apply for large deformations (dF/dx is
not a constant), so the idea of an "effective spring constant"
only works if dx is small compared to, say, r.

Thanks for the response. Here is my attempt at a diagram. Recall that
the distance |AB| is equal to the distance |BC| and the spring constant
k is the same for each of these two springs.

k k_bend
A----------B
` |
` |
` | k
` |
` C

I believe I still need to know how to transform the "angular" or
"torsional" spring constant k_bend into a function of the linear
displacement along the line AC, in order to derive an expression for
the total potential as a function of the linear displacement along AC.
Really, I am trying to understand AC as an oscillator. That is, I would
like to get a handle on how the magnitude of the distance between AC
fluctuates, with specific interest in the potential energy of such
oscillations.