My masters explored the use of unspooling tape springs in a polygonal geometry as a potential form of self deploying structure. Upon designing and making a test model and assorted necessary paraphernalia (winding rig, release system) the dynamics exposed an interesting and not previously explained degeneration to an unstable unspooling mode. I derived a model to calculate forces and velocities of the deploying structure to try and pin down what might cause the deployment instability, by applying Lagrangian mechanics to a complicated single D.O.F. system and solving with symbolic computation on matlab. I also built a separate stationary test rig to isolate this particular behaviour. The results from this rig showed a variation in onset of instability exactly in line with the existence of a critical threshold of force applied longitudinally to the transition region at point of unspooling. The project got a first and a prize, and for anyone with a baffling amount of time on their hands, you can download it here.
Background: Deployable Structures?
Structures are special designs used to support loads (like bridges, or buildings) hold shapes (like dams or roofs) and so much more. Deployable structures are a subset of these. When deployed, they function normally, like a standard structure. However, they also have the key ability to move, or to use the jargon ‘capable of large geometric changes of shape’. Common examples of these are the umbrella, which transforms from a narrow furled rod to a large canopy, a reclining chair, a convertible car’s top, up to the enormous movable stadium roofs.
Deployable structures are useful because they enable the structure to be easily reconfigured to acheive other aims, such as transportability or ease of use. Imagine the annoyance of having to carry an open umbrella round with you all the time – you wouldn’t bother!
They are particularly important in space, where everything a satellite needs has to be packed into a tiny space where both weight and space are at a premium, so being able to deploy solar panels or antenna once successfully launched is imperative.
Such structures are therefore ‘mission critical’. People might be mildly annoyed when their umbrella broke, but what about when the British Mars Lander Beagle was a complete failure because the solar panels failed to deploy properly. This means that the analysis and design of deployable structures is an interesting and growing area.
Other uses clever people have come up with include structures based on origami that can go from flat to a useful shape such as a heart stent, or ‘metamaterials’ where materials are built up from a smaller lattice meaning new and interesting properties can be built in, like negative poissons ratio or negative stiffness.
Self Deploying Structures
Deployable structures are capable of enormous transformations, but do require something to make this happen, such as a motor. This, and its dependence on a power source, mean additional complexity, weight and ultimately points of failure. This is why there is work on ‘self-deploying’ structures that use elastic (springy) elements to drive their own deployment comes in.
This complicates design slightly – is there enough force to make it happen? How does it get applied? What geometry does there need to be? It also puts a big focus on the intermediate states between deployed and stowed, where the spring driving everything has to be pulling at all times.
Further, it turns the problem into one of dynamics, not just statics. The driving force of the springs causes the structure to accelerate, but the faster the structure moves the faster energy is released from the springs, causing more acceleration.
These dynamic forces must also be considered, to make sure it doesn’t tear itself apart before it can be of any use!
The Tape Spring
My work used tape springs – such as the snap on reflective bracelets or the steel tape of a tape measure. These have two modes:
- Uncoiled- straight, with a transverse curvature
- Coiled up – storing energy
When uncoiled, they make for good structural elements, the transverse curvature making for a light and stiff structure.
Depending on the manufacturing, one or both of these configurations can be stable. If they are built so that the coiled up state isn’t stable, a coiled spring will unwind to straight, using the stored energy to drive deployment. This deployment can only happen in some ways, and is very prone to upset and going wrong. My work investigated the stability of this deployment in different ways and applied it to trying to get a 2D net geometry to work!
Investigating Tape Spring Structure Geometires
Previous work with tape springs has used them to make structures that deploy in one direction, called booms, or how they can unwind from around a central hub.
Part of my work looked at investigating and testing whether a viable structure could be made by using several wound tape springs connected point to point in a net.
Testing
To test, I built such a structure. The spools were 3d printed so a precise inertia value could be known for them. it was found that they required serrations around the edge to grip each other when bound up, to prevent them spinning in situ.
The intended use of the structure was in space and in 0 apparent gravity. To mimic this, the device was hung from very long, fine thread, meaning they could translate with very little change in height (costheta = 1 for small theta). As a multiply spring loaded device, it turned out to require the building of a special jig to independently wind each spool without letting any of them go. By swapping the hanging pins with hex type equivalent, they could interface with ratchet spanners sliding on carriages in grooves thereby allowing the wind.
The final problem was a reliable release mechanism that was strong enough to restrain the wound system but would release quickly and reliably get out of the way, and be timable with a high speed camera. This was accomplished simply with a thick rubber band, cut on demand by a piece of fuse wire when 5 Amps were put through it.
The first attempt! Early Results
The early stages of deployment were all as predicted, with each spool unwinding outwards propagating the structure. However, most deployments would see the start of an instability, characterised by a loop leading the spool.
Instead of the spool unwinding, at some point the deployment would start feeding tape out ahead, rather than the spool travelling out.
Clearly, this is an issue ; it means that deployment is less predictable and could often result in a total failure to finish deployment properly.
Analysing the Geometry
Clearly, this is not a superificial system to analyse. It is hard to work out what forces are applied where and even what the path of the motion would be.
The model was accomplished by defining a series of vectors and applying Lagrangian mechanics.
A simplified descripiton is: The location of the spools is defined as vector Z, moving in a rotating reference frame with angular velocity $\Omega$. THe amount the spool has rotated is defined as $\theta$. They have mass m and inertia I
The tape is analysed in 2 parts. That which is still wrapped around the spool (of lenghrh $r(\theta_0 – \theta)$ is centred on and moves with the spool so is analysed with it. The length unwound from the spool is defined relative to the spool with another pair of vectors, angled relative to Z by a complicated expression.
The second diagram shows how the length of vector Z is directly related to $\theta$
After some more long winded maths, expressions for kinetic energy (derived by differentiating the position vectors to get velocities) and potential energy (the amount of energy left in the wound tapes).
At this point, for brevity and comprehension’s sake, it becomes necessary to neglect the KE of the deployed tape as this is a long winded expression. Further simplification is allowed bythe rotational symmetry of the system, which means that masses and energies cancel out by that factor.
$\Omega$, the rotation of the reference frame, is found by applying convervation of angular momentum, and substituted back in.
By applying this simplification, MATLAB’s symbolic computation was used to apply Lagrange to the system. When not neglecting the tape KE, the resulting expression is 5 pages long, with most of the terms small.
WHen making that simplification, the resulting expression is shown below, which is clearly not tractable for a human solution.
This doesn’t have an analytic solution (surprise surprise!) but can be solved by the Matlab ODE solver using Runge-Kutta. Forces can be found by applying free body diagrams with the calculated accelerations
This shows how increasing the number of spools can dramatically reduce the steady state acceleration of the system.
Isolating the instability – Tape Spring Dynamics
This part had nothing to do with deployable structures but instead isolated the mechanism and cause of breakdown in unspooling tape spring dynamics.
This particular behaviour had not been previously studied in depth. Linear applications of unspooling tape springs use rollers and guides to prevent it departing from stable deployment, rather than understand what causes it.
The onset of instability requires the tape coming off the spool to have enough slack, and it also requires the input of enough energy to form a second hinge. This means that it is probably a mixture of a static and dynamic effects.
To test this, I built another test rig to explore the onset, and how it changes with different starting conditions.
It featured a spool spinning on light polymer bearings to minimise friction. The tape coming off at B then exits, running across the pin at C to point it in the desired direction. The force on the tape from the pin is what equilibrates the torque applied by the tape that spins the spool. The base plate was configured to allow multiple tapes to unwind in different directions simultaneously.
Theinertia of the spool can be changed by sliding the position of the bolts on the beam on the top (labelled A) which act as point masses and therefore have inertia of $I = m k^2$ where k is the radial distance of the bolts.
Onset of instability
This rig is formed of a stationary spool shooting out a tape. This is very similar to the spools in the deploying net above, but with a reference frame centred on and moving with the spool. Therefore, many of the same affects should be seen.
By filming with a high speed camera (the tape shot at up to 10m/s) the exact means of breakdown can be seen. The first image shows normal deployment. At some point and for some reason, the point of transition on the tape between curled around the spool and straight moves past the spool, and it is connected to the spool with a large loose loop. Formation of such a loop requires quite a lot of energy, as this is a highly distorted state. At some point, there is enough slack for the loop to ‘snap’ and form another hinge, dropping back to a lower energy state. By this point, the link between the spools rotation and the translation of the tape is completely broken. Breakdown progresses further until a complete mess is had, shown in the last figure.
Spooling Dynamics
As the tape comes off the spool, it’s overall inertia of the system remains constant, because the rotation rate, £\dot{\theta}$ is linked to the linear velocity by $v=r \theta$, where r is the radius of the spool. As the overall inertia is constant, and a constant amount of energy is released per unit rotation, the system should exhibit a constant acceleration and therefore a quadratic characteristic when deployment is plotted against time.
The test rig showed this near perfectly, with runs at each inertia setting being fit to a quadratic with an $R^2$ of no less than 0.9997. This means that we can have a high degree of confidence that measurements regarding the onset of chaotic behaviour will also be correct.
Onset of Instability
The test rig was run with the bolts set to give varying inertias, and the point of onset of instability was measured, by finding the frame in the high speed footage.
The graph on the right shows that unstable behaviour occurs much further out when the spool has a higher inertia.
The counterintuitive result of this is, due to the longer deployment, higher inertia spools end up spinning faster before breakdown than do lower inertia spools.
By drawing a free body diagram around the spool, cutting the tape at the root and finding the force that must be applied there to equilibrate the static and dynamic forces, it can be shown that breakdown in each instance occurs when this force reaches a certain value – when the product of the mass of the deployed tape multiplied by the system acceleration reaches a critical value.
Copyright © 2024 Seb Dickson