Tensegrity: Model Files

I have decided to make available a number of viewable 3D model files from the most recent articles relating to Tensegrity Symmetry.

The link will open a page on Google Docs where the file can be downloaded. Each file created is a 3D PDF which can be viewed in the free Adobe Acrobat Reader. The 3D PDF file format allows the user to interact with the model.

Tensegrity 3v Pars Tetra RS




Tensegrity Cube RS




Tensegrity Tetrahedron RS




Tensegrity T-Prism RS



Tensegrity Octahedron RS




Tensegrity Skew Prism Arch



NOTE: “RS” denotes models inclusive of Axis/Rotational Symmetry elements; in each case the brown struts are the actual tensegrity compression struts, the green and purple struts are virtual elements representing the transitional states and the blue struts are virtual elements to represent the axis of symmetry.

When you “activate” a 3D PDF the file opens with a toolbar across the top of the screen. Some of these files may have been processed as perspective views, so to change this to orthographic just select the toolbar button as highlighted above.
Other toolbar features include lighting setups, rotational and preset views, model hierarchy menu and CAD model presentation options.
If you have any questions about use of these files, viewing options or if you have problems accessing the link, then please drop me a line via the Contact page.

The Google Docs is a temporary location until I find something more permanent.

Tensegrity: T-Prism Geometrix

Considering my current research, it was inevitable that I would have to revisit my earlier theories on the construction of the tensegrity T-Prism and have a look at this structure for rotational symmetry.

http://spatialgeometrix.blogspot.com/2010/09/placeholder-geometry.html

My earlier thoughts on the development of this type of tensegrity still hold true, but building a model based on rotational symmetry has greater appeal and simplicity.

 

 

 

If we consider again the parameters of transition of the morphing from one state to another we will see that the green triangle form transitions to a single linear form as shown on the left image.

The right image (above) shows the relationship of the green triangular form to the blue struts representing the axis of symmetry. These lines intersect on the midpoints of the green struts.

Now we introduce the T-Prism or 3-Prism structure to this assembly and align it accordingly.

The brown struts are located at the midpoint with the blue axis struts resulting in a balanced arrangement.

To create a tapered T-prism move the intersection point of the brown struts towards one end. In each case the brown struts remain perpendicular to the axis of the blue struts.

The last image shows the parallel relationship of all three planes, assuming a plane connecting the triangular arrangement at the top and bottom of the T-prism.

 

 

The concept of rotational symmetry as discussed in the recent posts here is something I am quite excited about.

I would be interested to receive comments on my ideas and hopefully enter into some discussion with others about tensegrity and rotational symmetry.

So please drop me a line at hught2008@gmail.com

Tensegrity: Cube Geometrix

Rotational/Geometric Symmetry; Slowly progressing thru the range of tensegrity structures I have so far modelled it seemed appropriate to now have a look at the cube!

The tensegrity cube featured here is affectionately referred to by Bob Burkhardt as a Zig-Zag cube, somewhat differentiating it from the Orthogonal variant (not shown). This structure in contrast to its cousin has better stability and not quite so “jiggly”. Thus the reason why I am doing this one and not its cousin!

When I started to look for the geometric symmetry in this model I was puzzled as to how this formation could translate from the green cube to the star linear format (purple) as shown in the next 2 images.

 

In combination (above left) the translation formats do little to clarify the situation.

However when I introduced the cube itself into this geometry model I created a third construction set as a reference to represent the axis of geometric symmetry.

The blue axis of symmetry shown below has a common centre point passing through the centre of each strut. Thus we have the rotational axis about which the struts will maintain centre locality regardless of the dimensional configuration.

Having created the blue axis it was simply a case of re-introducing the cube and aligning the midpoints of each brown strut with the associated blue reference axis. The image (above left) shows the axis in combination with the cube as a dimetric view and the image on the right shows this in plan.

This image shows everything in combination, quite an overwhelming array of struts!
It would seem in principle that for the majority of “standard” tensegrity structural forms we can derive geometric relationships and symmetry.

The cube, axis of symmetry and star linears are proxy objects, modelled for illustration purposes only. You can see clearly that the axis of symmetry passes through the centre of the brown struts as well as the vertex and centres of the proxy cube elements (green).

Knowing the symmetry and underlying geometry we can define the locations of the actual tensegrity members which is useful as an aid to building a physical tensegrity model or even a 3D CAD model.

Tensegrity: 3V Pars Tetra

Following on from my last post I decided to revisit some of my earlier models to see how they may also exhibit rotational axis symmetry similar to the truncated tetrahedron.
In this case I selected the 3v Pars Tetra mentioned in one of my first postings:

http://spatialgeometrix.blogspot.com/2010/09/tetrahedron-tensegrity.html

 
I took the original model (left) and inserted a cross axis model; similar to before (blue); aligning the centre line axis with the midpoint of the brown struts in each direction and as you can see we can again show rotational symmetry.

We can also show that if we draw a connecting line between the end points of matching pairs of brown struts that this line should pass thru the centre of the assembly.

Tensegrity: Tetrahedron Geometrix

Introduction to Tensegrity Symmetry: This post has been updated to reflect changes to my blog and was essentially my first study of geometric symmetry.

Bob Burkhardt presented the mathematical formulae to develop this tetrahedron and Marcela Pars also covered this on his website.

It occurred to me that this type of structure must follow some basic rules, so I started looking at proportional geometric relationships for a tetrahedron based on a fixed length of compression strut.

At first my attention was focussed on identifying proportional relationships for the end triangular components for each of the tetrahedron size variants I had already built. I created a circle based on the 3 connection points in each case and then analysed the results. After a great deal of study there was no immediate proportional relationships that could be easily identified.

So I started again this time working from the extremes of tetrahedron displacement. The process is actually quite difficult to explain but I will make an attempt and perhaps follow through later with a more detailed report.

A Tetrahedron in its most basic form is essentially a polyhedron or pyramid shape (above left). The creation of the truncated tetrahedron as we know it is the transition when we morph from the platonic form to a linear form (above right). To better understand this morphing I would refer you to http://complexity.xozzox.de/tensegrity.html, where you can see via the Java applet how the transition works.

The green and blue “struts” are visual reference elements for purposes of clarification and not struts in the sense of being physical components. The blue struts are representative of the linear form which coincidentally also represents an axis had it been projected from the centre of the polyhedron edges to the centroid.

The image above left shows the extreme parameters of the transition from pyramid to linear form. In the image on the right I have introduced a tensegrity tetrahedron to illustrate how the struts splay out creating four triangular truncations. For clarity I have omitted the tendons.

After drawing numerous circles and construction geometries the entire study was becoming very complex and the myriad of lines I had was quite overwhelming, which may explain why I initially missed the most interesting aspect of a tensegrity tetrahedron.

At this stage I had created another tetrahedron with a dimensional variance on the tendons, but still using the same size of strut and placed it into my model..

This is where it started to get interesting as at last a pattern was beginning to emerge!

If you look closely at this image you can see that in each case the brown struts intersect exactly at the midpoint with the blue axis struts (representing the linear extreme parameter).

Essentially the blue struts define an axis of rotational symmetry.

To me my findings are very significant by showing a specific common relationship that ties the struts to a known alignment. I should also note that in each case the brown struts are perpendicular to the blue struts. Please note; it may appear that the same is true for the intersections between the brown struts and the green struts but in fact that is not the case.

So simply by knowing the base geometry and the end linear form geometry we can determine any intermediate location of the struts from the graphical perspective, which ultimately simplifies the modelling; both virtual and physical.

Just to give you some idea of how complex this study got I captured the following image:

Coincidentally this image also helps to show the strut relationships…the centre intersections and also the perpendicular aspects.

I am not sure if it is very clear, but you may be able to make out the circle perimeter line drawn through the end points of the brown and blue struts that I used to graphically check perpendicularity.

Having now studied in detail the geometry of the T-Prism and the Tetrahedron I am sure that other tensegrity structures may also exhibit some form of physical consistency such that the development of same is much easier to construct and ultimately calculate.

Update: Feb 2012. As this was my first study of Tensegrity Symmetry I have left the post as I wrote it at that time. The process of discovery is typical for a study of this nature concluding that the brown compression struts retain a midpoint relationship perpendicular to a projected axis from the polyhedron to the centroid. The transition to a tensegrity structure maintains this relationship as each strut revolves around this axis.

Thus the term axis/rotational symmetry evolved. At this time having now gained a better understanding of the potential underlying planar geometry pertinent to tensegrity structures I was tempted to rewrite this article but sometimes it is important to record how these discoveries came about.

Tensegrity: Wind Turbine

Is this the worlds first Tensegrity Wind Turbine??
This model combines the membrane Tetrahedron in my last post with the Skylon mast, also described earlier. The big question is “does it work?”

Obviously a model of this nature demands a lot more analysis before it even becomes a viable working design. The mast for example would probably not be sufficient in its current form to be an effective support for the mechanisms required to generate power and I don't have the necessary tools to do a proper design analysis.
However as a simple structure to be moved by the wind the combination I believe would work rather well. I am convinced that there is potential for membrane construction techniques for tensegrity structures and is something I intend to explore further.
Consider for a moment the nature of the membranes in this example. As it turns out they exhibit double curvature properties which is an ideal surface for acting in multi directional airflow situations.


This surface modulation could be enhanced even more by redefining the geometry of the tetrahedron.
hught2008@gmail.com

Tensegrity: Membranes

Having now developed a number of tensegrity structures, based on stick and string or strut and tension designs I decided to have a look at options whereby we replace the tension components with membranes or diaphragms.

I took a fairly common structure known as a truncated tetrahedron and explored a number of variations using membranes.
I should note that the Cad program I use; Soildworks; is not very good at depicting true generation of stressed membranes so the edge curvature is approximated in each case just to give some idea of what it may look like.

Eventually I will try and get hold of an evaluation copy of a membrane design program to test this out to gain a better perspective.

The image on the left assumes a planar arrangement following the route of three continuous tendons and the line of the main connecting strut. I have opted to include a centre connection point on the main strut which resulted in applying a bending force on the strut and needed to be countered by inclusion of the small internal triangular membrane.
The image on the right is a much simpler arrangement essentially following the perimeter of each perceived face of the tetrahedron. As all the points are interconnected there is no requirement to close the triangular end openings.
Kenneth Snelson has a detailed article on triangulated tension networks which could be adapted to develop membrane arrangements.
http://kennethsnelson.net/2010/12-triangulated-tension-networks/
I have no immediate plans to build any of these membrane structures at this time, but again the composite construction technique would be ideal for doing this.

Tensegrity: Geometrix

Tensegrity geometry involves fairly complex mathematics and it is the work of Robert William Burkhardt,that I continually refer to. A detailed account of his mathematical research and design theory is presented in the publication: A Practical Guide to Tensegrity Design.

In this publication he starts with the most basic of Tensegrity forms the T-Prism also known as 3-Prism.
 
These are just some of the images; captured from his publication; that illustrate his theories.
The mathematics are well documented but are too complex to discuss in detail here. He also covers how to adapt the formulae depending on what parameters are sought or known.

A recommended read for anyone interested in Tensegrity structures.
I would like to explore an alternative geometrix theory for this basic form of Tensegrity. If we consider a number of factors relating to this form, we find that for all compressive struts being equal in length the top triangular plane is parallel to the bottom triangular plane. We also know that for any given situation the rotational symmetry; that is the rotation about the centre axis (z-axis in above); has limitations due to the length of the struts assuming a fixed point at the base.

Essentially if we take the static vertical arrangement as per the first illustration above and rotate the top triangular form we reach this limitation at approx 180 degrees or the perceived point of intersection of all 3 struts.
 
These illustrations are from my own work with the first one showing the final 3 Prism arrangement and the second showing the theoretical graphical geometry for determining the location and size of the vertical tension members.
What I have done is create 2 circles defining the perimeter transcribed by the rotational movement assuming a fixed point at the base. The top circle is located at the strut length fully vertical with the lower one at the elevation on the point of maximum rotation. The path of movement from the vertical position to the maximum rotational symmetry achievable is shown as the Radial Path.

The radial path is determined by plotting a series of points at intervals tracking the motion of the strut from the vertical position. This path is transcribed on the perimeter surface of the 2 projected circles. This path passes beyond the minimum distance between the rotated strut and the eventual base point that would be connected by a tension strut.
So using the measuring capabilities of Solidworks it is simply a case of selecting the Radial Path and the associated connecting point at the base to determine the minimum dimension and thus we derive the ideal tension member length.
My findings at this stage are theoretical and I still need to prove the results with further experimentation. In this case I compared my theory with the model created using a dataset conceived by Robert William Burkhardt and there was a variation of only 0.04mm.
This variation may be attributed to the spline created from the series of points defining the radial path. The interpolation of this spline is dependant on the number of points (6) selected and consequently proportionally accurate.

Tensegrity Modelling: Design Considerations

Tensegrity structures are distinguished by the way forces are distributed within them.
The members of a tensegrity structure are either in tension or in compression. For the structures I have so far researched; that have been done by others; the tensile members are usually cables or rods, while the compression members are sections of tubing, pipe or rods.

For the tension members I have decided to work with mono line rated at 30lb or 14kg which satisfies my initial criteria for tension capacity.

Compression Members:

This is my primary focus at this time, as I am keen to evolve the definition of what constitutes an acceptable compression strut for the purpose of tensegrity model making. Something that opens up more opportunity for individual expression and provide additional options. It is very tempting to replicate the fine work of artists like Bruce Hamilton, who has developed a series of extraordinary and very elegant models, but that would not satisfy my curiosity for exploration.

What I doing though is looking at compound construction techniques that utilise materials which normally in isolation are ineffective but in conjunction with other material work well together to form strong compression struts.

Design Considerations:

In engineering, buckling is a failure mode characterized by a sudden failure of a structural member subjected to high compressive stress, where the actual compressive stress at the point of failure is less than the ultimate compressive stresses that the material is capable of withstanding. This mode of failure is also described as failure due to elastic instability. source: Wikipedia

Some brief notes on bending due to compression that is an inherent consideration in respect of developing compound compression struts.

Compression members, such as columns, are mainly subjected to axial forces. The failure of a short compression member resulting from the compression axial force looks like:

 

 

However, when a compression member becomes longer, the role of the geometry and stiffness becomes more and more important. For a long (slender) column, buckling occurs well before the normal stress reaches the strength of the column material.

In the case of Tensegrity models the struts replace the column in this instance. The tendency to bend under stress can be managed with compound materials and hence satisfy the conditions for structural strength.

Thus the purpose of my research is simply to explore these options.

For more specific details on bending under stress and slenderness ratio with some very good graphics on this subject I would suggest:

http://sites.google.com/site/simeonlapinbleu/tensegrity

3d Cad: Tetrahedron Tensegrity 3v.

This is my first 3D Cad Tensegrity model, technically described as a 3v Pars Tetra Tensegrity.

This design is actually not from the Bucky Fuller stable or indeed from Kenneth Snelson, instead it is devised by a very clever fellow called Marcelo Pars, who has done extensive work with tensegrity structures. I only recently came across Marcelos’ work which has opened a whole new understanding of the subject.

The design is based on a data-set created by Bob Burkhardt, who has studied in detail the mathematics of tensegrity structures.

3D CAD is an ideal medium to explore these structures enabling a degree of accuracy that physical models cant achieve. This environment also provides additional visualization tools that can dissect the models to further examine the construction and gain insights into the underlying geometric symmetry.

For all new tensegrity structures I would first develop the design in 3d Cad using where possible available data-sets from various sources. Occasionally where no data-set exists I would simply develop the design from visual inspection.

To fully comprehend the structural form it is absolutely imperative to attain a high degree of accuracy working to approximations will not suffice. It is therefore not unusual to have to work to 4 decimal places to achieve a fully constrained accurate model.

Working with datasets is an ideal starting point but in my experience still requires additional micro adjustment to fully constrain the final model.

Normally the process involves starting with building the struts and tendons as individual parts and laying out a rough shape of the form as an assembly and then constraining each of the individual struts to each node point on the tendons.

However it is very easy for this whole thing to become chaotic as you constrain the struts and occasionally the structure may become unstable as the Cad program will take the shortest route to connect the elements. To negate this and provide some control it is always best to ‘ground’ one component prior to connecting another keeping the items as close as possible to each other before constraining.

When the assembly is complete it is not unusual to find that the model exhibits some flexibility which technically should not be there. Its not because the model is not fully constrained its just that dimensionally struts or tendons are not nearly accurate enough to fully define the model. This is where it gets tedious...in order to achieve the final "fixed" result we need to select a tendon part or strut and micro adjust the dimension of this part to finalise the structure. I should note that selection of a "part" in an assembly will apply the adjustment to all the parts with the same properties (which is exactly what we want).

Once completed we can then begin to explore the model for geometric consistencies and symmetry. 

To prove geometric symmetry and relationships it is necessary to build a number of dimensional variations of the same structure and overlay these in one assembly. In the blog entries to follow I have often demonstrated this principle by showing dimensional variations of one model in the context of proving an observation.

For more information please contact me at hught2008@gmail.com
Bob Burkhardt website: http://bobwb.tripod.com/synergetics/photos/pars.html
Marcelo Pars: http://www.tensegriteit.nl/

Updated: 11th February 2012