Optimizing Shape of an Offset-Link Spring 1

[dtronvig]

Hi Guys,

I’m trying to optimize the shape of a flat spring, generally semicircular in plan but tapering toward both ends like a crescent, acting as an offset link, in order to get the maximum axial deflection before yield for a given axial force. I’d love to find some simple math to get me pretty close to the optimum shape before I start cutting and stretching steel. The spring is part of a latch that I’m hoping will take 1000 pounds tension before yield but will allow some elastic flex (I’m hoping for about 0.1”) to allow the latch to handle situations where the latch halves are kept apart by some obstruction (that’s a longer story).

The spring needs to fit inside the female latch housing that at this point restricts the spring to 0.3” thickness and an outside radius of about 0.8” but I suspect I’ll have to expand the housing to allow an outside radius of 1” or so and may have to compromise on the maximum deflection. The spring could be somewhat elliptical--with the midpoint pulled a little away from the axis--but it needs to be roughly semicircular because a hook from the male half of the latch comes in from the open side of the semicircle to grab one end of the spring, the other end being pinned to the female latch housing. There’s an eccentric cam on the male side of the latch that pulls the hook tight.

I’m trying to taper the cross-sectional height of the spring so that with 1000 pounds axial tension the inside surface of the spring will all be at yield stress, except for about the last 10 degrees at each end, where it has to expand for connection details. I’m designing around full-hard 301 stainless, with a yield strength of about 140,000 psi but I may have to go to something more exotic.

I think I have the math to derive the deflection from a given shape, but the math that I’ve tried to apply for inside stress around the semicircle is straightforward at mid-span but goes haywire toward the ends. I’d love to find a reasonable approximation of the optimum shape or some reasonable math for homing in on it. My dif eq is rusty but I’ll brush up if I must. It might be better still if there’s cheap modeling software I can get or good software I can access a few times for successive iterations of the design.

Thanks for any help or leads you can provide,
Drew

 

[zekeman]

Consider the semicircular leafspring of your description. By symmetry you can deal with 1/2 of it with the 1/4 circular end built-in and an end force = to P. Using the energy approach to solving this we first write the energy equation of the spring:
E=integral(M^2/2EI dS) limits 0 to pi/2.
Since you want a constant max bending stress Mc/I and c is the 1/2 thichness is constant, then M/I is constant. Therefore the energy integral is
E=1/2*M/EI*Integral(M dS)limits 0 to pi/2
where dS is the incremental length along the circle and is
r*d(theta) where theta is measured from the built-in point clockwise
Now the M is found = P*r*cos(theta)
From Castigliano, the partial derivative of the energy with respect to P is
1/2*M/EI*Integral(r^2*cos (theta)*d(theta) lim 0 to pi/2
The integral of the cosine over 0 to pi/2 is unity and the result is the deflection for the 1/2 section is simply
1/2*M*r^2/EI and for the whole semicircular section is double this or M*r^2/EI. Now we set Mc/I=100,000 for example.Substituting in the deflection eq
dL=r^2E*100,000/c
where it is recognized that minimzing c increases the deflection. For example, if the thicness is .04, then c=.02 and and r=1, dL becomes
100000*(1.0)^2/.02/30,000,000= .16 inches.
This should be a starting point for you.

 

[dtronvig]

Easy for you to say, but thanks very much, Zekeman; I'll work my way through that.