Nervous Strain Makes You Tense, But Engineering Strain Makes You Tensor

Ok, I admit it; I love the title of this post so much I want to get it printed on a T-shirt! But that’s a job for later, right now there’s some serious work to be done.

It may surprise you to discover that I don’t just plonk down at my keyboard, take a swig of tea, and rattle off fifteen-hundred words in half-an-hour, before hitting the “submit” button and sauntering off for a glass of wine and a snooze. Nope, ninety-nine times out of a hundred my untrustworthy memory fails me and I’m forced to do some actual research in order to maintain the appearance of knowing what I’m talking about… and this is definitely one of the ninety-nine.

I put a great deal of time and effort into learning tensors and Classical Lamination Theory (CLT), but annoyingly I doubt I’ll ever be able to regurgitate it on-the-fly, at least not in a coherent manner! So I’m hitting the books again with a goal of distilling out the fundamentals required for design, hopefully without getting lost in the theoretical “long grass”. The trade-off is that some detail is going to get skimmed over, and one or two mathematicians may be harmed in the process – possibly due to falling off their chairs in horror at my blasé attitude to their craft!

So what-on-earth is a tensor?

A quick Google search turned up the following: “A mathematical object analogous to but more general than a vector, represented by an array of components that are functions of the coordinates of a space”. Phew! There’s quite a bit of assumed knowledge packed in there, which makes it a less-than-great description for our purposes, but it’s a starting point, so I’m going to run with it. To do that however, I’m going to chop it up and deal with it in parts, starting with: “A mathematical object … represented by an array of components…” which sounds strangely familiar.

Cast your mind back to the matrices we discussed in the last post. Whilst a matrix is limited to 2-dimensions (i.e. having rows and columns), and occasionally has just 1-dimension (if it is a single row or single column), a tensor may also be zero-dimension (which is just a number, called a scalar), or 3-dimensional (notionally, numbers arranged in a cube), or even “4, 5, 6, 7, 8 etc.”-dimensional (at which point a small trap-door in the back of my head opens, through which my brain attempts to squeeze and make its escape).

Avoiding the difficulty of imagining a 4-D object for a moment, I’m going to take a quick diversion into terminology. Unfortunately the word “dimension” is already in use, so tensors aren’t described as being 1-D, 2-D etc., the word “rank” is used instead, (or “degree”, or “order” –  yep, once again universal conventions are hard to come by!) So, for example, a 2-D tensor is commonly described as a “second-rank” tensor.

For our purposes a tensor is going to look exactly like a matrix – an array of numbers or symbols surrounded by square brackets. It’s worth noting however that just because a tensor can be written as a matrix, it doesn’t have to be written in matrix form and a matrix is not always a tensor. We’re simply choosing to write our tensors as a matrix, because it’s convenient way to package and manipulate the data.

A Sense of Direction

Getting back to our Googled definition, the next part notes that tensors are: “… analogous to but more general than a vector …” Any mention of vectors and you can be pretty sure that direction is going to be important, but what does, “more general that a vector” even mean?!

I previously mentioned that a zero-rank tensor is just a single number called a scalar. A scalar has no direction, only a magnitude. Speed, is a scalar (it tells you how fast you are going but not the direction); mass, temperature and absolute pressure are scalars too; they all have magnitude but no direction.

If we go up a step from a scalar and combine magnitude and direction together, you get a vector. Vectors are very common in engineering and physics, force being a good example – with the difference between tensile, compressive and shear forces being an excellent illustration of the importance of the direction component! Vectors are a first-rank tensor; they are often depicted as a line, with an arrow to indicate direction and the length denoting the magnitude. Break a vector down into coordinates and they can be represented as a single column matrix:

1st Order Tensor 3D

Figure 1 – A First-Rank Tensor (i.e. a vector)

Stepping up again from vectors we get a second-rank tensor. They don’t get a special name like scalars and vectors, they are just “second-rank tensors”, and you can represent them as a 2-D matrix. Stress and strain are examples of second-rank tensors – which we’ll be going into in more detail later. If you imagine a vector as a single arrow, then the best way I can think of to visualise a second-rank tensor is as a grouping of interrelated vectors. Here’s one way of presenting the second-rank stress tensor:

2nd Order Tensor 3D

Figure 2 – A Second-Rank Tensor (in this case representing stress)

Now we get to the mind-bending part. If we want to relate stress and strain in an isotropic material – such as most metals – we can use the stiffness (i.e. Young’s Modulus) relation and, providing the loading is simple, do a straightforward calculation:

εE = σ
(i.e. strain × stiffness = stress)

What’s not immediately obvious though, is that the above calculation is actually presenting a special case – we are treating stress and strain as scalars (or at best, vectors pointing in the same direction) when really they aren’t. As I just mentioned, stress and strain are both actually second-rank tensors, making the stiffness relationship between them (oh the horror!), a fourth-rank tensor.

Fortunately we have found a way to write fourth-rank tensors in a 2-D matrix, so there’s no need to buy special “4th dimension” paper and pencils, but they are not exactly intuitive. I may have already stretched your visualisation abilities well beyond breaking point, but if you do want to try and visualise a fourth-rank tensor, think of it as holding the mathematical relationships between two groups of interdependent vectors. (Ugh! On second thoughts, maybe don’t bother.)

Size Matters…

The final part of our Googled tensor definition stated that our array of components, “… are functions of the coordinates of a space.” By my reasoning this statement has more than one aspect to it, the first being that the number of dimensions of “space” is important. The size of the matrix required to represent a tensor depends on the number of components the tensor has, and that in turn depends on the number of spatial dimensions the tensor is working within. The rule is that the number of tensor components is equal the number of dimensions raised to the power of the tensor’s rank. A picture really helps here, so have a look at the diagram below. I skipped zero-rank tensors in the diagram as they always have only one component (any number to the power zero is equal to one):

Tensor Size

Figure 3 – The size of matrix required to describe a tensor depends on the rank and number of dimensions

I didn’t illustrate a fourth rank tensor in the above diagram, but following the same logic a fourth-rank tensor in two dimensional space will have 2⁴ = 16 components (arranged in a 4×4 matrix) and a fourth-rank tensor in three-dimensional space will have a whopping 3⁴ = 81 components (in a 9×9 matrix). Fortunately for us we can make some assumptions that cull the numbers of components in our fourth rank tensors considerably, so they won’t be totally unmanageable, but once again – that’s a subject for later!

… But It’s The Direction You Point It In That Really Counts

Now we’ve established that, when written down, tensors look like matrices, but that means we can’t tell if a matrix is a tensor or not from simply looking at it, leaving the question: “What makes a tensor a tensor?”

There is a notably tongue-in-cheek quote attributed to physicist Anthony Zee, who, on being asked a similar question by a student, “So what exactly is a Tensor?” replied:

“A tensor is something that transforms like a tensor.”

This description is annoyingly self-referencing, but it does make a valid point – transformation is what makes a tensor a tensor. Remember how I previously described a matrix as a “filing cabinet” for numbers. Well if we use that filing cabinet to store material properties or state data (e.g. stiffness, stress, strain, etc.) at a particular location and in a particular direction within a material, we can then use tensor transformation to calculate what the material properties or stress/strain state is at that same location, but in any other direction of our choosing. This is very useful indeed! (I’ll be explaining how to do it later.)

Taking a second look at our definition I think maybe the description that tensor components, “… are functions of the coordinates of a space” is back-to–front. What tensors allow us to do is change the “coordinates of space” i.e. move our reference point, and see how the thing we are examining looks from a different point-of-view.

That Wasn’t So Bad, Was It?

I’m going to pause here. We’re not done with tensors yet; we still need to cover transformation in more detail and maybe have another look at the tensor versions of stress and strain. But hopefully our: “Mathematical object analogous to but more general than a vector, represented by an array of components that are functions of the coordinates of a space.” is now slightly less of a mystery.

I’m sure you’re all familiar with the old showbiz adage insisting that you should, “Always leave them wanting more.” Unfortunately by this point I suspect you’ve already had way more than enough. Which only leaves me to hit the “submit” button and ponder which bottle of wine to open.

Cheers!

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The Matrix

Let’s be quite clear up-front, there will be no references to Keanu Reeves in this post so any sci-fi fans that have landed here by accident can move along. If however you want to analyse laminated composites it’s time to take the “red pill” and brace yourself for reality…

Before I start, however, there’s a tiny risk of confusion around the word “matrix”, so to clarify – Composites are made from a reinforcement, (usually some kind of fibre), and a binder, which holds the reinforcement in place, (usually some kind of adhesive). This binder is often formally referred to as “the matrix”, e.g. “This laminate is Carbon Fibre in an Epoxy matrix”. To be absolutely clear, this post is not about the binder/adhesive “matrix”, it’s about mathematics!

Tools of the Trade

Designing and analysing laminated composites requires some mathematical tools, of which the most fundamental is ‘matrices’ (i.e. the plural of ‘matrix’ – I don’t believe anyone says ‘matrixes’, but if you do, please submit a comment at the bottom of this post so we can all point at you and laugh… this is the internet after all!). I’ll explain why matrices are useful later, but before I do that, let’s head back to high school for a quick recap:

What are they?

Wikipedia describes a matrix as, “A rectangular array of numbers, symbols or expressions arranged in rows or columns.” I like to think of them as a ‘filing cabinet’ for numbers; a convenient way of grouping them together which also makes them easy to manipulate.

Of course for a filing cabinet to be useful, you need some way of referencing the stored data. The same is also true for a matrix, so each element in a matrix is referred to by is coordinate position based on the row and column the element is in. For example the A21­ element of matrix [A] (shown in Fig 1 below) is in the second row and first column (starting from the top left):

Matrix Fig1

Figure 1 – Numbering of matrix elements. The “A” name is arbitrary, it could just as easily be “a” or “B” or any other symbol. Subscript ‘m’ represents the total number of rows and ‘n’ the total number of columns.  

Matrix Note

Matrices can be any size and don’t need to be square (i.e. having the same number of row and columns). Single column matrices turn up quite frequently and are often presented with curly brackets, (although not here! Jones uses square brackets):

Matrix Fig2

Figure 2 – A Column Matrix

So What Can We Do With These Matrix Thingies?

Quite a lot, as it turns out. However, as I’m not attempting to write a maths text book, I’m not going to dig too deep with the following descriptions, especially as you (being sane and sensible) are unlikely to be manually number-crunching your matrices. I assume you have Microsoft Excel, (you really should), which is quite capable of doing these operations for you. So I’m just going to give you the highlights and some limitations, so if your design spreadsheet implodes you can hopefully work out why.

If you really want to know the finer details, Engineering Mathematics by K.A. Stroud covers them well, as does the Appendix of the aforementioned Mechanics of Composite Materials, and of course there’s always Google…

Getting back to matrices, let’s start with the simple – as long as two matrices are the same size you can add them or subtract them from each other. This is simply done on an element by element basis:

Matrix Fig3

Figure 3 – Matrix Addition

Multiplication can be either by a scalar (i.e. a number), in which case each element is multiplied in turn:

Matrix Fig4

Figure 4 – Matrix Multiplication

Or, two matrices can be multiplied together – but only if the number of rows in [B] is the same as the number of columns in [A]. Therefore, in the following example you can only multiply [A][B] and not [B][A]. Also note that even if [A] and [B] were both square matrices and the same size, (meaning you can multiply them in either order), [A][B] does not necessarily equal [B][A], so matrix multiplication is not commutative (unlike “normal” numbers where the order of multiplication is irrelevant):

Matrix Fig5

Figure 5 – Multiplying Matrices – The resulting [C] matrix has the same number of rows as [A] and the same number of columns as [B]

What Else Can We Do With Them?

In addition to the basic operations just described there are some additional operations particular to matrices. First up, you can transpose a matrix – denoted using a superscript T (i.e. [A]T). Transposing a matrix is a simple case of swapping the rows and columns – A12 becomes A21, A21 becomes A12 and so on; this results in the elements on the main diagonal from top left to bottom right (A11, A22, A33 etc.) remaining unchanged and all the other element positions being mirrored about this diagonal:

Matrix Fig6a

Matrix Fig6b

Figure 6 – Transposing a Matrix

The concept of transposition introduces three special matrix types. They are the symmetric matrix – which is a square matrix in which the corresponding elements on either side of the main diagonal are equal to each other; meaning for a symmetric matrix [A] = [A]T.

Matrix Fig7a

Figure 7a – A Symmetric Matrix

If all the elements on either side of the main diagonal are equal to zero you have what is called a “diagonal matrix”:

Matrix Fig7b

Figure 7b – A Diagonal Matrix

And if you have a diagonal matrix where all the elements on the main diagonal are equal to one, you have a “unit matrix”, denoted as [I]:

Matrix Fig7c

Figure 7c – A Unit Matrix

Are We There Yet?

Nearly, but there are a couple of important operations remaining: The determinant and inversion.

I’m going to skim over the determinant somewhat, not because it isn’t important, but because most of the time it isn’t important! The determinant is a way of boiling down all the elements of a square matrix into a single number. The bigger the matrix, the more involved the calculation (but of course, you can always get Excel to calculate determinants for you). As an example, for two-by-two and three-by-three matrices the calculation is as follows:

Matrix Fig8a

Figure 8a – Calculating the Determinant of a Two-by-Two Matrix, (Note the different bracket for a determinant).

Matrix Fig8b

Figure 8b – Calculating the Determinant of a Three-by-Three Matrix

Determinants may be mathematically useful, but for our purposes the critical thing is that they doesn’t end up equal to zero, because a matrix with a determinant of zero can’t be inverted, and that really is important!

Inversion is where things really start to get useful (finally!). Matrix inversion has applications for solving systems of linear equations and applying transformations which is exactly the kind of thing we’ll need to be doing later and is the reason you’ve been subjected to this lengthy and rather dry post!

An inverted matrix is notated with a superscript -1 (e.g. [A]-1) and is calculated using the following equation:

Matrix Fig9a

Figure 9a – Matrix Inversion

Matrix Fig9b

Figure 9b – Inverse Matrix Relations

I’m not going to unpack Equation 9a, as I’d have to explain the adjoint matrix  [a]T, which would take me at least another ten paragraphs, but you can immediately see why the determinant |A|must be non-zero – otherwise you get a “divide by zero” and the equation self-destructs!

Equation 9b gives a better explanation of what the inverse matrix is, when [A]-1 is multiplied with the original ‘non-inversed’ matrix [A], in either order, the result is a unit matrix (as described above).

Finally

Despite the length and heavy content, there are really only two important “take-homes” from this post. Matrices, and the ability to invert them, are a necessary tool, so you have to know what they are. But perhaps the most important point is that Excel (or any other maths software for that matter) can calculate a matrix inverse for you, so you really only need to know what might make this process fail, (and of course always “sanity check” your results, computers are never to be entirely trusted!)

Next time I’m going to get back to engineering and delve into Tensors, (where these cursed “Matrixes” can start “earning their keep”).

Composing Composites

I’m an Engineer, and whilst I’m not entirely devoid of social skills, I do suffer from a problem quite common in engineers. Simply put, things I find interesting, others (usually those unfortunate enough to find themselves sat next to me at dinner parties), don’t. This can typically be seen by my victim’s eyes glazing over, often shortly followed by them desperately seeking a means of escape, all whilst I’m busy waxing lyrical about gyroscopic loading on wind turbine bearings or some other, clearly fascinating, subject!

Why is this important? Because the problem composites have is strikingly similar to the problem of socialising with engineers: They should be exciting. Just think of all those sexy compound curves (yes, I’m talking about the composites!) or the number of times “Carbon Fibre” or “Advanced Composites” gets name-dropped in glossy marketing material. Composites are exciting, but unfortunately designing with them involves a whole load of fairly advanced maths, which, unless you are into that kind of thing, is either painfully dull or total gobbledygook.

Virtually all the content about composite design I have come across falls into one of two clear categories. Either:

(a) The Dumbed Down Version – “Composites design is so fantastically complex it requires special powers to understand so there’s no point even attempting to describe it in simple terms; just accept that it is superhuman/magic and move on”, or

(b) The Textbook Version – “Composite design is fantastically complex… And here’s how to do it in excruciating mathematical detail, (usually pitched at engineering postgraduate level)”.

Now I’m not one to shy away from a challenge, so I’m going to attempt to take the middle ground and describe composite laminate design for the non-engineer. Before we start however, a word of warning, I can’t wave my magic wand and make all the maths disappear, it really is required to do the job. What I will try and do is describe what the maths is doing in simple terms. Hopefully the end result is interesting to the non-engineer and a useful primer for those considering tackling the textbooks.

Metals are Easy!

“Why is designing in composites so hard?” is a question I have both heard, and asked myself, on more than one occasion, however I think a better question is:

“Why is designing in metal so simple?”

The answer is straightforward. Metals are, for the most part, wonderfully bland, with nice uniform properties (yes, I know there are some variations due to grain orientation and the like, but they are small – usually small enough to ignore). This uniformity means metals only have two independent engineering constants for relating stresses and strains, and they are essentially the same for all directions within the material. (Materials like this are called, “Isotropic”, which is approximately Greek for “Equal in all directions”)

“One moment”, I hear you cry, “Aren’t there usually four engineering constants?”

(I’m sure you all know these, but just in case they have momentarily slipped your mind…)

E – Elastic Modulus (i.e. Young’s Modulus); [Relates tensile stress to tensile strain]

G – Shear Modulus; [Relates shear stress to shear strain]

ν – Poisson’s Ratio [Relates axial strain to transverse strain]

K – Bulk modulus [relates change in volume to change in hydrostatic pressure]

So yes, there are four, but they aren’t independent. As long as you know two of the above you can calculate the other two because they have fixed relationships:

Engineering Constants

Composites have rather more than two independent engineering constants because they are “anisotropic”, i.e. Not equal in all directions, (laminated composites, which is what we are interested in, are actually a special subset of anisotropic materials called “Orthotropic” – which I’ll get back to in detail later).

Composites are Complicated

Now we’ve cleared that up let’s get on to composites. Unlike metals, laminated composites are not the same in all directions; in fact a laminate is made up from a stack of lamina (single layers) which are individually very directional indeed!

Lamina Fig2

Figure 1 – A Unidirectional Lamina

The simplest type of lamina is called unidirectional and has all of its fibre reinforcement aligned in one direction. Why this results in directional properties should be intuitively obvious, just imagine strands of carbon fibre all nicely aligned and cured in epoxy:

Carbon fibres are stiff. Cured epoxy in comparison has a stiffness of barely 1/20th that of the carbon fibres. But what does that mean for our lamina? If we apply a load aligned with the carbon fibres, their stiffness ensures they will carry the majority of the load and thus dominate the material properties in that direction. If on the other hand we apply a force at 90 degrees to the fibres they will carry almost none of the load, the properties of the material will be dominated by the epoxy and will only be a fraction of the stiffness.  The same idea also applies to strength. Carbon fibres are much stronger than epoxy so our lamina will carry a much greater load in alignment with the fibres. Apply a load across the fibres and the strength will be limited to that of the epoxy.

A single lamina is not a lot of use structurally, so we tend to stack them up into a Laminate:

Laminate Fig1

Figure 2 – Exploded View of a Laminate

The variation in stiffness and strength of each layer with direction makes laminated composites extremely versatile. By carefully choosing the orientation of each lamina, stiffness and strength of the laminate can be tailored to align with the expected loading, making for an extremely efficient structure. Of course it also makes composites a lot more complex to design with, there’s no free lunch!  The concepts required are not that hard to grasp, but the maths can be quite… Shall we say, “Intimidating”.

That’s enough for this post. Next time we’ll dig a little deeper…

 

Into The Vortex

Aerodynamics is a strange thing. On the one hand familiar, but also mysterious. We’ve all been outside on a windy day or stuck our hand out of a moving car window, so we’re naturally acquainted with the general effects of fast-moving air, but what the air itself is actually doing remains invisible, the realm of wind tunnels and high-powered computer simulations.

Many, many years ago I was one of those slightly annoying children who asked, “Why?”, all of the time, and growing up near a golf course, it wasn’t long before I enquired of my father, “Why are golf balls lumpy?”. Pleased to be furthering his child’s education my father confidently replied, “Son, It makes them go further”. Intrigued, and frankly somewhat suspicious, (I was old enough to know that speedy things like fast cars and aeroplanes were smooth and streamlined, and also that parents were not a reliable source of information… After all, they appeared to believe in both the tooth fairy and Santa!). So I paused, cocked my head to one side, furrowed my brow and launched my second-most-favourite question, “How?”.

Now, I don’t remember the exact response, but I’m 100% sure it didn’t involve the words, “Delaying turbulent boundary layer separation”, although there just may have been a mumbled mention of, “less drag”, immediately followed by dad disappearing behind the newspaper or going off to do something ‘urgent’ in the shed.

Why am I telling you this? Mostly because I want to talk about vortex generators, and they fall into the same category for pilots, as ‘golf ball dimples’ do for golfers: That is to say, most are familiar with them, a good portion know what they do, but far less know how they actually work.

Before we plunge headlong into the details of how vortex generators work, let’s first have a look at what they are and what they do:

VGs (to save ink/pixels I’m going to call them VGs from now on) come in many different shapes and sizes, but In their commonest form are thin, usually triangular, tabs attached perpendicular to a surface and at an angle to the oncoming airflow (see Fig.1). Invariably used in groups, when applied to aerofoils they are usually arranged in pairs along the span set-back from the leading edge.VortexGs-Fig1

Figure 1 – Typical vortex generator application

So we know what VGs look like, but what do they do? The obvious answer is, “Exactly what their name suggests”, they generate vortices. Behaving like tiny wings, each VG creates a small amount of lift perpendicular to the oncoming airflow and in the process sheds a trailing vortex downstream from its tip. This explanation is all well and good, but sadly not very enlightening, so a more practical answer is that VGs, “Fix aerodynamic problems”.

Separation Anxiety

You can be pretty confident that VGs were nowhere to be seen in the original  designs for almost every aircraft they are now attached to, in fact you can almost guarantee they were added later after something unsavory turned up during flight testing:

As far as possible, aircraft designers like the airflow to stay firmly stuck to the surface of their aeroplanes. Depending on the location, detached flow can result in a multitude of effects, from additional drag and early stall to ineffective control surfaces and stability problems. None of these traits is desirable, but unfortunately detached flows are hard to avoid. As soon as an aerodynamic body starts to narrow, such as at the rear portion of an aerofoil or fuselage the airflow wants to separate from the surface. Gentle tapering of surfaces helps, (giving familiar ‘streamlined’ shapes), but is not always practical and is ineffective at higher angles-of-attack or where surface discontinuities such as at flaps or control surfaces occur.

Hitting a Boundary

Flow separation occurs thanks to the behaviour of the air in a thin layer immediately adjacent to the aircraft’s surface (See Fig.2). Air has some viscosity – it’s not in the same league as honey, but nonetheless it possesses a degree of ‘thickness’ or internal friction. What this means is that when air flows over a surface some molecules stick to it whilst the others rub against each other as they flow past and are slowed down. This area of friction-affected air is called the boundary layer and it starts off very shallow, but thickens as the air travels further along the surface.

VortexGs-Fig2

Figure 2 – The boundary layer and flow separation

When air flows over a tapered surface such as the rear portion of an aerofoil there is a combined effect of viscosity and also an adverse pressure gradient (the pressure is lowest over the front portion of an aerofoil where most of the lift is produced and then increases as the surface tapers). In this case the air immediately adjacent to the surface experiences both viscous drag and a pressure differential trying to push it back towards the lower pressure area at the nose. This can cause the airflow at the rear of the aerofoil to turn back on itself,  reversing direction and so acting like a wedge forcing the oncoming airflow to separate from the surface.

A Quick Fix

If an aircraft design demonstrates flow separation problems the obvious solution is to tweak the aerofoil shape or re-contour the fuselage profile to solve the problem, but if the aircraft is already at the flying prototype stage, or is a one off design, significantly altering the outline of the aircraft will be expensive at best, and at worst completely impractical. This is where VGs come to the rescue. Because they are simply attached to the existing surface, aerodynamic problems can be fixed without the need for re-tooling or major structural changes.

Vortex generators work because sluggish air in the boundary layer is at the root of most separation problems. Correctly dimensioned VGs extend slightly above the boundary layer and create vortices at their tips that grab fast moving air in the free stream and mix it into the boundary layer. The now highly turbulent and energy-rich boundary layer is far more resistant to flow separation and so will follow more sharply tapered surfaces, better negotiate sharp discontinuities caused by deflected control surfaces, and resist aerodynamic stall to higher angles of attack (Fig.3).

VortexGs-Fig3

Figure 3 – Re-energising the boundary layer

No Such Thing as a Free Lunch

Of course it can’t all be good news or our aircraft would be peppered with VGs. In reality you have to pay somewhere, and for VGs that penalty is drag. Whilst they avoid the large drag increases that come with flow separation, the drag generated by VGs occurs in all flight regimes, and so adds to the total parasitic drag of the aircraft whenever it is flying – even in conditions where flow separation may not actually be a problem.

To control the drag VGs create their dimensions and positioning are critical. If VGs are well proportioned and well positioned then the bulk of the VG (around 80%) will sit inside the boundary layer and the drag penalty incurred will be modest. Make VGs too tall and unnecessary drag will result with no added benefit, too short and they simply won’t work.

In the final analysis, if VGs are taming poor stall behaviour or a loss of control authority at high angles of attack, then a modest increase in drag is a small price to pay. Similarly, curing a fuselage flow separation problem during cruising flight is almost guaranteed to give a net drag reduction and so be well worthwhile. As long as VGs are the right size and in the right place there is very little down-side to them as long as they are correctly applied. I suppose they are quite delicate, which makes them prone to damage, but that’s about it. In fact, for me at least, their biggest drawback is that whenever I see them I immediately ask myself the question, “Is that a clever piece of design, or just a band-aid solution to an unforseen problem.”

To ‘V’ or not to ‘V’

That is the question… But don’t worry, unlike Shakespeare’s Hamlet I’m not having an existential crisis, just pondering one of the mysteries of aircraft configuration: Why aren’t we all flying aircraft with V-tails?

vtail-fig1-sv11.png

Figure 1 – V-tailed ultralights are out there, like this SV11 for example

The history of aircraft design is littered with innovations which at the time of their inception were heralded as being ‘game-changing’ or even ‘revolutionary’. However, with the definite exception of the jet engine, the vast majority have failed to live up to their promises. This shouldn’t come as a surprise. Combine over-enthusiastic engineers excitedly pursuing a novel idea, with a marketing mentality keen to make grand attention-grabbing claims and it’s easy to see where the hype comes from. But whilst you can fool people, you can’t fool nature, and many a promising idea has fallen foul of the laws of physics.

I’d argue that V-tails fall into this category. On paper they have a huge amount of promise and they turn up quite regularly on UAVs and jet-fighters. However, in the ultralight and GA world they remain something of a curiosity; but why is this?

‘V’ Good

Theoretically V-tails have a lot going for them, especially in the drag department. Firstly a V-tail reduces both wetted and frontal area. The theory goes that the two diagonally mounted aerofoils of a V-tail can perform the same job as the three separate fins in a conventional empennage, but with a smaller combined area. Hey Presto – smaller area, less drag. But wait, there’s more! Because a V-tail only has two fins, there is one less intersection between surfaces and one less wing tip too, so you get a bonus reduction in interference and tip drag as well. Clearly, if you are chasing speed, a V-tail is the way to go.

Vtail Fig2 - Equivalent

Figure 2 – Scale comparison of equivalent conventional and V-tails

Next on the list of V-tail benefits is control. Conventional tails can be subject to “Rudder Lock”, a phenomenon where large yaw angles, such as those occurring during a spin, generate massive aerodynamic forces on the rudder, pinning it hard over with more force than the pilot’s legs can overcome. Obviously this is an undesirable trait, and one which should be avoided if possible (a requirement for certified aircraft and certainly recommended elsewhere!). V-tail geometry limits the aerodynamic forces on the control surfaces during a spin, providing some resistance to rudder lock.

V-tails have two other potential control benefits – based on the V configuration raising the tail surfaces relative to the fuselage. Firstly the V-tail is less exposed to ground effect, meaning it won’t suffer from the same loss of elevator effectiveness conventional horizontal tails experience when close to the ground, i.e. when flaring for landing or raising the nose for take-off. Secondly, a raised position places the centre-of-pressure of the control surfaces above the centre-of-gravity of the aircraft. The benefit here is that you get greater pitch–up authority (albeit at the cost of reduced pitch-down authority) at large control surface deflections. This is because the drag generated by the deflected control surfaces creates a supplementary pitch-up moment in addition to the primary pitch–up due to control surface lift.

Vtail Fig3 - Elevator

Figure 3 – V-tail pitch control

Vtail Fig4 - Rudder

Figure 4 – V-tail yaw control

‘V’ Bad

So that’s the good points wrapped up, but what about the bad stuff? First-up, there are some drawbacks to combining the rudder and elevator functions. In aircraft that have manual flight controls, (i.e. pretty much all ultralights), aerodynamic forces acting on the controls get fed directly back to the pilot. For pilots used to conventional aircraft this can make for some unexpected interactions between control forces, notably when large amounts of trim are applied, or when applying large amounts of ‘rudder’ input such as sideslipping for a cross-wind landing. Control forces are not the only problem, there is also potential for the controls themselves to interact, such as increased drag from large rudder inputs causing some secondary pitch-up effect.

The next problem is also control related. Mechanically combining conventional stick and rudder control inputs to give differential control surface movement for rudder, and coincident movement for elevator, requires a mechanical mixer assembly. This is not only adds weight but represents a complex mechanical linkage which is also a single point of failure for the control system, effectively putting the elevator and rudder control “eggs in one basket”. Trim can also be an issue. Providing a trim system on the pilot side of the mixer assembly is relatively straightforward, but removes the benefit of having a trim system which is independent of the primary controls. A separate trim system, on the other hand, will provide redundancy (required if an aircraft is to meet FAR Part 23), but is yet more complex and heavier to implement.

On the subject of weight, you might imagine that having less and “smaller” surfaces would produce a weight saving. Somewhat surprisingly this turns out not to be the case. Whilst there is inevitably some saving from the reduced overall surface area, each V-tail fin is doing duty as both horizontal and vertical tail and so tends to be larger in area than any single conventional tail surface. The end result is greater aerodynamic loads, which in turn require stronger and thus heavier structure, giving up significant weight benefit.

The final drawback for V-tails is adverse roll. We are all familiar with adverse yaw, the tendency for the nose of an aircraft to yaw away from the direction of bank when rolling (caused mainly by a difference in drag on each wing due to aileron deflection). The usual piloting response to adverse yaw is to apply rudder to counteract the yawing moment, but with a V-tail the act of applying rudder to counteract the yaw generates a rolling moment which tries to roll the aircraft out of the turn, i.e. adverse roll.

It’s Not Wrong, It’s Just Different

There are a few aspects of V-tails that don’t fall into the realm of advantage or disadvantage; they are just differences that need to be considered. One of these is dihedral effect. A V-tail, by definition, has a lot of dihedral and this supplements the dihedral effect of the main wing. This tends to make the aircraft more laterally stable, but also makes it more prone to Dutch Roll. With a conventional tail the solution would be to increase the directional stability by increasing the vertical tail area. This isn’t an option for a V-tail, as reducing the ‘V’ angle to give more ‘vertical’ tail area also increases the dihedral effect and so doesn’t fix the problem.  In fact the usual solution is a ‘Y-tail’ which adds a small fixed vertical tail surface to improve directional stability.

Finally, a claim often made is that V-tails are easier/cheaper to manufacture – as there is one less fin and one less control surface to build. This simplicity argument is certainly true for servo-controlled systems, but for manual systems it’s not so clear cut, having to be balanced against a control system which is significantly heavier, more complex and costly.

V. Ugly?

In summary, V-tails do have their place. If you have jet wash or water spray to avoid; or are fanatical about minimising drag, they may just be the way to go. V-tails make even more sense if your craft is unmanned or fly-by-wire – thereby avoiding the control feedback quirks. However, for an Ultralight, I don’t really see the point. As a comparison between the Sonex and Waiex demonstrates, there is no real performance or weight difference to be had between the two tails. In the end it really comes down to aesthetics, so if you like the look, why not? Just don’t expect miracles in the performance department.

Battle Fatigue

Last month we had a look at metal fatigue and why we are not immune to it despite the low number of hours the average homebuilt acquires. This month we’ll get onto more of a design footing and look at how aircraft designers tackle the fatigue problem.

Taking a Gamble

One of the key challenges when designing for fatigue is the probabilistic nature of fatigue itself. It simply isn’t possible to predict failure after an exact number of load cycles. The fact is there is a large variation in fatigue life – even for apparently identical parts. All fatigue design ultimately boils down to a gamble, albeit one with the odds heavily stacked in your favour.

If you take a collection of steel samples, all of the same dimensions and polished to the same surface finish, and expose them repeatedly to a loading equivalent to 75% of their ultimate strength, you’ll find they break after somewhere between 10,000 and 100,000 cycles. Do the same test at 55% of the ultimate strength and the samples will last somewhere between 250,000 cycles and infinity – that’s a fair amount of uncertainty!

Add to the mix real-world loading, which varies considerably in both magnitude and frequency; and then pity the poor engineer who has to answer the superficially simple question, “Will it break?”

Personally, my preferred response to the above question has always been, “Yes, eventually! How long would you like it to last?” which gets to the crux of the problem, especially when combined with, “…and how confident would you like me to be?”

Fatigue2 - Fig1 S-n Diagram

Figure 1 – A generic S-n diagram

For a more in depth explanation take a look at Figure 1. Known as an S-n diagram it plots stress against load cycles for a given material. To produce the chart a large number of identical samples are repeatedly loaded and unloaded at a variety of stress levels and the number of cycles to failure at each level is recorded, (on a logarithmic scale, which allows data from ten thousand to one hundred million cycles to be shown in one chart). With enough samples a best-fit curve can be drawn to give an idea of the average life of a sample at any given stress level.

This is useful information, but from a design point of view you don’t really want to know the stress level at which 50% of your parts have already failed! Instead some clever statistical analysis is required, producing another curve of the stress level or number of cycles at which 99% of parts can be expected to survive. In addition, for some materials an endurance limit can also be determined, giving a stress level below which no fatigue failures should occur irrespective of how many load cycles are experienced.

Armed with this data, plus the desired reliability and predicted loading, and with additional allowances made for effects such as temperature, corrosion, surface finish and stress concentrations (among others!), a designer should finally have the information required to produce an acceptably durable part… although in a commercial environment analysis alone is not considered enough, and the final judgement invariably comes down to testing.

So How Long Will It Last?

There is an old adage in motor racing that the perfectly engineered race-car should, “Break down just as it crosses the finish line”, thus demonstrating it has enough durability to finish the race, but is carrying no more weight than the absolute minimum necessary to complete the task. Aircraft designers find themselves in a similar situation; there’s no question that aircraft have to meet their design-life requirements, but any excess strength means excess weight and a corresponding loss of performance, range or payload. This quest for a happy-medium has historically led to four different approaches to fatigue design:

Design for Infinite Life – Components are designed to be stressed below their endurance limit, (sometimes called the fatigue limit), plus a margin of safety, the goal being to provide a unlimited fatigue life. In the case of materials such as aluminium, which have no clearly defined endurance limit, a limitless fatigue life is not possible so an exposure well beyond anything that could be expected in service is selected instead, effectively negating the risk of a fatigue failure.

Safe-Life Design – A finite life is deliberately included in the a component’s design after which it is required to be replaced. A suitable margin of safety is applied to the required design life of the part; the expected loading and operating conditions; and also to account for the statistical uncertainty of fatigue properties resulting in an acceptably small probability of failure during the part’s lifetime. Safe-life design results in ‘lifed’ parts; component required to be replaced during scheduled maintenance prior to reaching a specified number of hours in service.

Fail-Safe Design – Rather than attempting to avoid fatigue failures altogether, fail-safe design accepts that part failures may occur and instead focuses on making the system as a whole ‘failure tolerant’. Structures are designed with multiple redundant load paths, allowing loads to be safely transferred around a failed part without causing further damage. Of course failed parts still need to be detected and replaced, but in the meantime the aircraft will still be safe to operate, albeit with a reduced margin of safety. As an example, large aircraft skin panels are typically designed with “crack stoppers” – stiffeners directly attached to the skin, dividing it into ‘bays’. If a crack occurs in the skin it will only be able to grow as far as an adjacent stiffener, limiting the maximum crack size to a single fuselage bay. To meet the fail-safe requirement the skin in the adjacent bays is then designed to be capable of carrying the additional load incurred should a panel fail.

Damage-Tolerant Design – Extends the fail-safe design concept to include capturing fatigue failures before they occur (and thus minimising the demands on the fail-safe structure!). Much like fail-safe design it acknowledges that fatigue cracks will develop, but based on a knowledge of crack growth behaviour, and an ability to reliably detect cracks using non-destructive inspection, the intent is that failing parts will be identified and replaced before they endanger the aircraft. By calculating the predicted rate of crack growth on a component, maintenance intervals can be set such that a crack will be discovered by inspection before the parts residual strength is reduced to a dangerous level.

Which Method is Best?

None of the above approaches is inherently better than the others. Part of the skill of the designer is the ability to select the most applicable method for the task in hand, and all the above approaches all have their strengths and weaknesses.

Starting with the Infinite Life approach, the primary drawback is that it produces components are heavier than is strictly necessary. This is generally not a good thing for an aeroplane, but is certainly a practical solution for engine components such as valve springs which can see billions of load cycles in a lifetime, and which don’t lend themselves to regular inspection or frequent replacement.

Safe-life design will save weight when compared to the Infinite Life approach, but an accurate knowledge of both the loading and conditions a part will experience in service is critical if premature failures are to be avoided. Safe-life has a financial impact too, “lifed” parts either need to be economically replaceable, (although you wouldn’t guess it from the cost of overhauling a TBO expired engine!), or alternatively, forced retirement of the aircraft when the hours are up has to be accepted.

Fail-safe design should similarly save some weight when compared to an Infinite Life approach, although it inherently involves redundant structure and thus ‘extra’ weight by definition. Careful analysis on the part of the engineer is also required to ensure all single points of failure have been identified and eradicated from the design – not necessarily a simple task. For example, a wing with multiple spars may have adequate residual strength to accommodate a single spar failure, but the if accompanying reduction in wing stiffness leads to aerolastic problems like flutter, the design may appear to be fail-safe when it actually isn’t.

Finally, Damage-Tolerant Design has the greatest potential for weight saving, but comes at a heavy price in the form of analysis, testing and especially ongoing non-destructive inspections. For a commercial airliner these costs are easily justified by the lifetime savings in fuel and/or corresponding increase in payload, but this is certainly not the case for your average homebuilt.

Making The Most Of It

We’ve covered the overall design approach, but achieving the best possible fatigue life on an individual component level is just as critical; so what are the tools available to a designer to really get the best from his parts?

Firstly, surface finish counts. Surface imperfections are just tiny cracks waiting to happen, so removing them by polishing or surface grinding can massively improve fatigue life. If there are initially no cracks in a part they must form at a microscopic level before they can grow. This “crack nucleation” process can take a long time and so represents a significant opportunity to extend the total fatigue life of a part. In the same vein, residual compression forces on a part’s surface inhibit crack initiation, so even if polishing is impractical the life of a part can be usefully extended through surface treatments such as shot peening or burnishing which leave residual surface compression stresses.

Cracking is bad, but once a crack has started the battle is not entirely lost. A part can have considerable life remaining, providing the crack grows slowly and the critical crack length (beyond which a rapid failure will occur) is not too short. This is where material properties, in particular Fracture Toughness, become vital. Materials with high fracture toughness are tolerant of cracking, giving slow crack development and long critical crack length. Now, I’m not going to plunge into the details of fracture mechanics here, but it’s worth noting that this is not a simple case of selecting steel over aluminium or even selecting a particular aluminium alloy. These choices do have an impact, but simply selecting a different type of heat treatment can change the fracture toughness by more than 50%. The devil, as they say, is in the detail.

Summing Up

As a final word of warning, fatigue damage is cumulative and for the most part occurs at a microscopic level where it is not readily apparent to a visual inspection. It is a brave (or foolhardy) maintainer that uses a part beyond its stated life, even if it still looks, “As good as new”.

Fatigue2 - Fig2 Comet Window

Figure 2 – Feather edges produced by countersunk holes and large stress concentrations in the vicinity of square windows proved to be fatal flaws in the design of the DeHavilland Comet.

Chronic Fatigue

I was at an air show recently and was more than a little taken aback to overhear someone expressing a view that, “Fatigue isn’t an issue for homebuilts, they just don’t fly enough hours for it to be a problem.”

It’s easy to see where this kind of opinion comes from. Certified airframes have fatigue lives equivalent to tens, or even hundreds of thousands of flying hours, whereas most homebuilts will be lucky to collect more than couple of thousand hours in a lifetime; so they should be fine, right?

Unfortunately this belief misses the crucial point that certified aircraft are carefully designed and tested to achieve a certain design life, and even then there’s still been a few occasions when the big boys got it wrong, (Aero Commander wing spars and deHavilland Comet windows spring to mind). Designing for fatigue life is tricky, and just because an aircraft is only going to accumulate a few thousand hours certainly does not mean fatigue can be conveniently ignored. But before we look at how long your beloved homebuilt is going to last, let’s go back to the origins of fatigue.

An Age Old Problem

In the first half of the 19th century it was noticed that some railway carriage axles would fail unexpectedly after relatively short periods in service, despite the fact that they were operating at loads well below their designed and tested strength. By the 1850’s there was a growing appreciation in the engineering community that metal components exposed to cyclic loading displayed a tendency to weaken over time. They dubbed this phenomenon ‘fatigue’ as it was postulated that the material was somehow ‘tiring’ through use, and losing its strength. Systematic investigation followed, revealing that fatigue failures actually result from the progressive growth of initially microscopic cracks. These cracks develop gradually over repeated loading cycles until a part is so weakened that catastrophic failure occurs at well below the designed strength.

Three Steps to Failure

Fatigue failure occurs in three stages. Firstly a crack needs to initiate, this will typically occur at a pre-existing surface defect such as a tooling mark, an area of damage, or material defects such as a void or contamination in a casting. However even apparently defect free highly polished parts will initiate cracks eventually, triggered by tiny imperfections in the material microstructure. It’s worth noting that, for a part with good surface finish and no damage, the first 90% of the fatigue life can pass with no cracking visible to the naked eye.

Once a crack has initiated it will then go through a period of slow growth extending by a tiny amount with each load cycle. Despite being damaged a cracked part can remain serviceable in this state; provided the design loads are not exceeded and the crack is shorter than the critical length the part will not fail catastrophically. Finally, after a period of crack growth which may last months, or even years in service, the length of the crack will reach a point where the rate of growth increases exponentially, rapidly leading to final failure of the part.

This fatigue process is clearly visible when examining the failure surface of a broken part as shown in Figure 1: Fatigue Fig1

Figure 1 – A Classic Fatigue Failure

A crack has initiated from an area of damage (in this case a tool mark) and has then grown slowly over repeated loading cycles creating a fairly smooth but subtly ‘beach marked’ region similar in appearance to tree rings, finally the crack has grown large enough that the remaining material lacked the strength to carry the load and the part has failed creating a large rough area indicative of rapid fracture.

An Old Age Problem?

It should be clear by now that fatigue life is not actually about age, instead the primary criteria involved are the number and magnitude of the loading cycles. Low stress loading cycles are much less damaging than high stress ones and will cause failure to occur far more slowly. To give a couple of examples: Landing gear legs see large stress variations with each take-off and landing, which is much more arduous from a fatigue point of view, but the number of cycles will be low – maybe a few thousand in a plane’s entire lifetime. On the other hand an engine mount is exposed to constant vibration whenever the engine is running, the magnitude of the stress variation is low but the exposure is huge – even if you consider only the vibration directly due to the firing of the cylinders, a Rotax four stroke produces well over 500 million loading cycles for every 1000 hours it runs!

Smoothly does it

If you’ve built a metal aeroplane you’ll be well aware of the mantra to, “smooth edges and deburr holes”, and not without good reason. Burrs and rough edges provide a multitude of tiny ‘notches’ – sites for cracks to initiate and propagate from – which can dramatically reduce the fatigue life. But notches are not just a builders problem, from a design point of view holes, corners, and changes in thickness should always treated with suspicion, after all, they are basically blunt cracks deliberately included in the design! Now I’d challenge anyone to design an aircraft without using any holes, but placement of these features is critical and can be the difference between a part that lasts weeks and one that lasts years.

Fatigue Fig2

Figure 2 – Fatigue Cracks frequently initiate at holes

So why do notches cause problems? Firstly they cause stress concentrations – small localised areas of higher stress – and secondly, by definition, they are on the surface of the part and so are likely to already be in a high stress area – especially for parts loaded in bending.

As a side note it is this same property of notches that makes corrosion such a problem. Loss of material to corrosion obviously reduces a parts strength, but the surface damage, cracking and pitting that corrosion creates can be far more critical and have a huge impact on fatigue life.

Fatigue Fig3Figure 3 – Corrosion not only weakens the base material but provides an opportunity for cracks to develop

Material Matters

Correct material selection is vital for good fatigue performance. For some metals, such as steel there is a fatigue endurance limit – a stress level below which cracks won’t initiate or grow, providing theoretically infinite fatigue life providing stress levels are low enough. Unfortunately aluminium doesn’t display this property and even at very low stress levels fatigue failures can still theoretically occur – albeit at massive numbers of load cycles. This doesn’t make aluminium useless, but it does mean that high frequency vibration and aluminium don’t play together nicely, and probably explains why you don’t see many aluminium engine mounts.

Do we Need to Worry?

Getting back to my “Air Show Expert”, was he right? I guess time will tell. Fatigue may well be a lesser problem at our end of aviation, but we are not immune and it’s certainly not something we should simply ignore. So, with that in mind, how do designers combat the problem of fatigue? We’ll find out next time.