Multiple torsion spring problem 1

[jbendercp]

Will try to do my best at explaining, you can see the less-than-helpful graphic attached as well.

I have a linkage of 4 segments joined end-to-end in a straight line. Rotation at each of the three joint locations is controlled by torsion springs (all three of same spring constant). A single cable passes along all the members at a fixed distance away from the joint centers and applies the tension to cause the same moment at each joint location (similar to how tension on fishing line bends the fishing pole). I am trying to solve for the amount of cable tension required to cause each joint to rotate a specific amount of degrees.

Aaaaannnd unfortunately I'm having a complete mental lapse. I am assuming that, in this case, the 3 springs are acting as if in series (because the total cable displacement is evenly divided amongst the three springs), and therefore the Keq would be (K1*K2*K3) / (K1+K2+K3). Am I doing this right? Or am I completely off base?

Thanks!
J

 

[hydtools]

I don't see how putting tension on the cable parallel to the rods will cause any movement or rotation of the rods. Fishing rods bend because the tension is at some angle to the axis of the rod shaft. Pulling parallel through the eyelets will not bend the rod shaft. If the cable is attached near the rod joint, then I would expect the cable stretch due to tension would cause the joint to rotate proportional to the cable stretch.

Ted

 

[jbendercp]

Sorry, my diagram is admittedly less than helpful. The cable causes rotation in the segments because eyelets hold the cable (which is fastened at the far end) a perpendicular distance 'R' away from the joint center. Thus the tension force 'F' creates a moment about the joints due the force 'F' acting about the lever arm 'R'.

I have a working model on my desk so I know it works in this fashion, just not sure if the math can be simplified using springs in serial or if the situation is much more complex.

 

[Jboggs]

You might do yourself some good by sketching the assembly in a tensioned state, just to show the angular displacements and related geometry. IF I am understanding this correctly (a very big IF), you are trying to create an equal angular displacement in each segment. One reason I suggested you draw it in a deflected state is that you will realize this fact: in the straight condition, the length of cable between each segment will equal the length between rod joints, but after you deflect it, the length of cable between each segment will be slightly less than the length between rod joints. I don't see how you can make that happen if you have prepositioned eyelets at fixed points along the cable.

 

[jbendercp]

I think my picture is poorly drawn to illustrate the situation. Eyelets are not right at the joints themselves, but set a bit off from the joint center. Thus, as the segments rotate (angle becomes less between segments), the eyelets from each segment are actually moving closer together (thus allowing for the shortening of the cable.

Working off my phone at the moment but i'll try to get a better sketch up when i'm at the computer.

Otherwise, imagine if each segment only had one eyelet square in the middle of it. as the segments rotate the eyelets would move closer together, eventually touching each other if allowed to complete full 180 degrees of rotation. This would allow for cable shortening. My 'eyelets' are just set closer to the joint center(s) to reduce amount of gapping between segments and cable, and reduce cable excursion necessary to rotate segments.

Again. really sorry my descriptions are falling short as I know this physically works with a working model. Just feel my explanation is letting me down in accurately explaining setup. I'll see if I can get a better pic or sketch up at some point.

 

[jbendercp]

Okay, hopefully this diagram is a bit more helpful (see attached). Also drew it at a slightly tensioned state to (maybe) help with the visualization. Again, original question is regarding calculation of the amount of cable tension force (F) to generate, a given degree of rotation amongst the segments, specifically whether or not the torsion springs could be simplified using the spring-in-series formula, or whether the situation is inherently more complex.

 

[3DDave]

The series calculation looks OK if they are similar. One way to look at this is the energy method. You store an amount of energy for each amount of bend; that energy comes from putting tension into the cable and pulling though a distance. Since it's a continuous function, the actual amount of energy will be found via integration.

The multiplication of joints will see a multiplication of the amount of cable pulled, not the tension required. Since the tension is a function of angle, but the amount is a linear function of the number of joints, then more joints just means more cable, acting just like series tension springs.

 

[jbendercp]

Thanks 3DDave, this is exactly what I was looking for.

In the end, just looking to size my cable and power source that generates the tension. Hoping to use some simplified model & assumptions (e.g assuming eyelets are close enough to joint centers to keep lever arm essentially constant, even though some gapping and lengthening of lever arm will occur) to get a 'worst case' estimate, which should ensure adequate choices for both.

Glad to here the series assumption holds water, this certainly makes my life much easier. Thanks for the help!

 

[tbuelna]

I appreciate that your sketch may be a simplified representation of the mechanism. But one issue I see that may likely have a significant effect on your analysis is how the (helical?) torsion springs are constrained, how they are connected to the segments, and their specific design characteristics (wire size, coil diameter, etc). If the helical wire coil is not constrained radially, the end coils may tend to deform radially as well as winding about the center axis.

Elasticity in the cable and effects of friction at several eyelet contacts in series will make your analysis more complicated.

 

[jbendercp]

Good points. Manufacturer provides details for radial constraint, max dowel diameter, and related torsional spring constant, planning on using these for analysis. While idealized, assuming (correctly, i hope) end result will be close enough for my needs.

Cable stretch and friction are more my worries w/ how big of a factor they may play:

Glad you brought up elasticity. I was planning on using heavy fishing line but can only get monofilament here, so after your comment did some testing and got 10%-20% stretch - yikes! Will probably switch to something like bike control cable that has an elastic stretch in ranges I can tolerate.

May end up having to throw cable friction into the equation, but in a reasonable manner that doesn't cost me more in time than the cost of a few trial-and-error iterations.

 

[BUGGAR]

Out of curiosity, I looked at this with interconnected hydraulic cylinders pushing on the outside of the links. I think it's similar?

What if you varied a cylinder size? Would this be akin to lengthening one of your string connectors? This is the finger gripping mechanism that I haven't studied yet.

 

[gruntguru]

Moment applied at each hinge is string tension times perpendicular distance from string to hinge-axis. See attachment for FBD and calc. Theta = articulation angle, F = string tension, all other symbols are geometry constants.

je suis charlie