Visvesvaraya Industrial and technology museum in Bangalore, India is an amazing place. Takes you back in time to engineering marvels of the 20th century. Also more importantly one can see the connect between ideas and the inventions that came out. The layout and design of the museum is beauty enough to visit it once, even if you are not an avid science fan.
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Sometimes one tries so hard looking for inspiration. Inventors of the past have shown us the way already. Working in adverse conditions to achieve such great marvels. One really wonders what they could have achieved in this age.
It’s been almost a week of work at the workshop and the metal structures are finally taking shape. Along the way, I had spent many hours consulting with experienced mechanical engineers on whether the structure would hold the weight I wanted them to. It was mind-boggling the amount of data they had on their finger-tips. In addition to this, I spent countless hours drawing-up CAD models, performing FEM analysis and filling-up my notebook with stress calculations. Do I trust the theoretical calculations on paper or the empirical advice of an experienced engineer? Would they add-up in the end?
This is a problem most of the engineers face. Its cheap to buy a notebook, pen, a book and workout an engineering problem from a college dorm or an office or a home. What’s not cheap is building the full-sized contraption in a humid, dusty workshop and then testing if your assumptions were correct. Would India ease this transition from chalk-board to machine-shop for budding inventors?
But the joy of working-out complex models on paper is incredible. Dr. Amar Bose had worked out the theory behind ‘noise-cancelling headphones’ on a flight between 2 cities. ‘Wearable device’ nevertheless, but extremely complex mathematics behind the electronics. I guess it did add-up in the end for Bose corp.
I had just finished registration of my company a few days back. Normally company registration in India is done in 3 steps:
1. Digital signature verification
2. Director Identification Number
3. VAT registration
Thanks to steps taken by the central government, now registration of a start-up/company has become digital. This is a big shift from the usual filing of paper-applications at the local Company Registrar office. In the past, budding company founders were taken advantage of by brokers/middlemen and clerks. Sometimes they even fell victim to corruption (so I am told by my Dad). Now that this avenue of earning has stopped for these people, they have resorted to scamming innocent Indians ( Company Incorporation VPP scam in India ).
I received a V.P.P (Value Payable Post) delivered by India Post on a Saturday. It looked like it had something to do with the Director Identification Number registration and I had no-one to consult on a Saturday (stupid me). I knew from past experiences not to return packages to India Post because it ended-up in a bad shape. So, I paid the postman Rs.998. To my surprise I am told by my auditor on the next Monday that it is a commonly recurring scam that occurs throughout India. I was angry and had to go through 4 clerks to finally get some answer. I was shocked to learn the Post office couldn’t even track the money order being sent back to the sender.
W/reg to past experiences:
I build and innovate on high tech devices. I had sourced some testing equipment from Australia and upon receiving it by India Post, was asked to pay a customs duty of Rs.950. I refused because customs didn’t have a relative Indian device to compare to and hence charge me such a hefty amount. Unable to have my customs query answered, I headed-off to the local post office to collect it. I found it below a heap of parcels with the original packing ripped-off and just cardboard + torn bubble-wrap for protection. I protested only to be threatened by the postman that if I didn’t pay then and there, he would send it back to customs. I paid the amount.
India is not yet a manufacturing economy but people have taken the first steps already. Government is still lagging behind and we still face problems when it comes to sourcing raw-material, equipment and tools. In my case, I source 90% of my equipment from abroad because India still doesn’t make such machinery. For us ‘lead time’ and ‘cost’ are highly critical parameters and our survival depends on ‘ease-of-sourcing’.
What can you do to avoid getting scammed?
Question everyone and look for accountability. The hierarchical structure of departments makes accountability a big problem. ‘Make in India’ should also be looked at from a startup point of view. Manufacturing is a follow-up of innovation and RnD.
New blog post: Relating the minimum potential energy principle to FEM.
Why systems of nature assume states with the lowest energies is a question that even automotive engineers have to contemplate with. The minimum potential energy principle is used in Finite Element structural Analysis. Read on further to learn about shape functions and the basic concepts in FEM.
Why systems of nature assume states with the lowest energies is a question that even automotive engineers have to contemplate with. The minimum potential energy principle is used in Finite Element structural Analysis.
Pierre de Fermat’s least-time principle for light states that light will travel through an optical system in such a way as to pass from starting to ending point in the least amount of time. Maybe the light has a mind of its own?
Example no.2 Tautochrone curve is the curve on which a ball sliding without friction, under the influence of gravity, takes the same exact amount of time no matter which point you place it on the curve. (Wiki) This is also a consequence of the minimum potential energy principle.
What is the mathematical formulation that describes this principle of least action?
When an external force is applied on a body and the body deforms, the energy of the body is said to have reached a higher level. This higher energy level is termed as strain energy by mechanical engineers. Note that there exists two types of strain energies:
- Elastic strain energy
- Inelastic strain energy
When a bar is loaded upto a certain point B on the load-displacement curve and the force then slowly removed (static load implies no inertial effects), a certain part of the elasticity is still retained in the bar. This is referred to as ‘elastic strain energy’.
As far I understood it, in structural analysis, any body that is elastic is said to have potential energy.
– equation 1
U- strain energy
W- potential energy of loading
W comes from the principle of ”conservation of energy” and is usually of 3 types:
- Body forces a.k.a distributed forces. It is the force per unit Volume, just like the self-weight of a bar under gravity.
- Traction forces are frictional forces.
- Concentrated point loads.
W has to be subtracted from equation 1 because this part of work potential contributes no longer to the potential energy of a system. Just like, you drop a book and it has no more potential energy.
There are many ways to express strain energy, a few examples are shown below:
Strain energy U =
Strain energy U =
The second equation form above is called the quadratic matrix form. q1,q2 are the displacements at the nodes.
What do you observe from the above figure?
It is the fact that I haven’t used the ‘method of sections’ to break-up the system and perform the force balance. This is the primary advantage of using the PMPE. It requires no force balance. One arrives at the familiar FEM equation F=KX just by using the strain energy equations.
There is one disadvantage of the above method though. You get to calculate the stiffness matrix of the system from the above method, but how accurate is this stiffness matrix? Notice that the deflections of the elements(spring-mass) are calculated only at the nodal points. What about the deflections happening within the element?
I have always found myself hitting a virtual brick learning wall when encountered with the theory of shape functions. As a student, one first encounters shape functions when reading into in the Rayleigh-Ritz method.
As pointed out in the previous section, the disadvantage of finding the stiffness matrix using the PMPE discretization approach is that deflections are computed only at the nodes. Hence, if one wanted to arrive at a more accurate solution, the number of finite element discretizations would have to be a very large amount. One can make life simpler using something known as shape functions. These are basically mathematical analytical functions that vary between 0 and 1. Why they vary between 0 and 1 is a result of the local coordinate system one chooses (phi from the diagram below). The advantage of using a local coordinate system is that irrespective of what x1 and x2 is, your shape functions would have the same form/shape. (refer to a FEM textbook for the equations on the local coordinate system phi, which in-turn is a function of x.) Equations of the shape functions N are basically functions of phi.
So now we have a way of representing the deflections in an element not only at its nodal points, but also at any point in between. Notice that the shape function N1 has the value of 0, exactly where the shape function N2 has the value of 1. This pattern, you can observe even for quadratic shape functions also.
Notice that the above element is a 2node element. In cases of a 3 node element, the shape functions take the form of a quadratic function i.e N1, N2, N3.
The split is made-up exactly where the tractional force T acts. The areas of the 2 sections can be calculated as averages.
In the news:
1. Air bag makers eye boost from new India road safety rules (Click for the article)
New blog on the mathematics of vehicle crash analysis and its connection to differential equations (DEs)
New blog on the mathematics of vehicle crash analysis and its connection to differential equations (DEs)
Second order DEs in vehicle crash analysis.
The following excerpt I take from the impeccably written book ‘Differential Equations with applications and written notes’ by George F.Simmons.
‘The use of complex numbers in the mathematics of electric circuit problems was pioneered by the mathematician, inventor and electrical engineer Steinmetz. He was employed by General Electric Company and quickly became the greatest of electrical engineers. Steinmetz solved problems of mass production of electric motors using mathematics.One day, a huge new generator at Henry Ford’s River Rouge plant had gone Kaput. His electrical engineers were unable to locate the problem. In comes Steinmetz with a piece of chalk, few sheets of paper and starts scribbling down calculations while listening to the generator for 2 whole days….doesn’t change a single part in between. He descends finally, asking the engineers to take out 16 windings from the field coil. Generator works and Steinmetz submits a bill of $ 10,000. Ford respectfully asks for a itemized statement. Steimetz replies as follows: ” Making chalk mark on generator $1. Knowing where to make mark $9,999. Total due $10,000”. ‘ Haha…..
- Second order differential equations (DEs).
- Applications of second order DEs in vehicle crash and other fields.
Most engineers come across the second order differential equation during their undergraduate years. Some of the common 2nd order DEs'(Differential Equations) in engineering are the following below, often abbreviated under MSD (Mass Spring Damper) systems:
MSD-like systems in engineering occur in various fields, for example:
- Animal running (ref: Running springs: speed and animal size)
- RLC Circuits
- Robotics (PD control) and lastly,
- Automotive (Cruise Control, car suspension etc.)
Solving linear or non-linear homogeneous equations is actually kinda easy, once you memorize the formulas and the various forms in which they appear. Its more important to understand why most of the solutions are of the form of an exponential or a sine/cosine form. Its because the exponential or the sine/cosine maintain their function form on double differentiation (2nd order). So, in conclusion, the two main behaviors that one observes from the solutions of 2nd order differential equations are 1. Trigonometric 2. Exponential 3. Combination of the two. This is probably the single important insight I gained about 2nd order DEs during my Bachelor’s degree. Solutions of differential equations are usually split into two terms:
- Complementary function
- Particular integral.
The particular integral results from the forcing part and is the steady-state part of the solution. The complementary function shows the process by which the mass reaches the steady-state solution i.e it is the transient part of the solution. Transient response implies that this term of the solution dies-out as .
When it comes to vehicle crash modeling and analysis, the analysis part is categorized into two broad categories, namely:
- Lumped parameter analysis
- Finite Element Analysis
I can explain ‘lumped parameters’ by a simple example. If you remember some of those early classes from electrical circuits, you would remember the fact that almost all conducting wires used in circuits carry some electrical resistance with it. A circuit might utilize tens of thousands of wires and it would tough assigning a resistance for each and every wire. So instead, we group all the resistances from the wires and model it using a single ‘lumped parameter’ i.e. a single resistor in the circuit. This is the most common example, although lumped parameters are also used in crash analysis to reduce the computation time. Why that is the case, you can make-up your mind after reading the explanation in the below topic.
Creating a mathematical model for vehicle crash-
Second order differential equations are also used for modeling vehicle crash situations. This is possible by making a few assumptions, for example, considering the 2 vehicles crashing into each other as 2 rigid masses and their crumple zones as a system of springs and dampers. This can be visualized from the first diagram in the pic below. This model is called as the ‘Kelvin model’ of vehicle crash analysis. I will iterate again, lumped parameter analysis does seem like a gross generalization of reality, but its worth the effort after validation from real-world tests. This is because of the use of continuous DEs and hence way faster computing times. The solution here is comprised only of the transient response part as there is no forcing function. The main aim from the above Kelvin model is to determine the crumple zone stiffness, crumple zone damping and the natural frequency of vibration. After solving the DEs in Matlab, one can obtain the graph of the relative deceleration of the vehicle 1 (mass1) w.r.t mass 2. From this, the absolute deceleration of the 2 vehicles can be separately calculated.
Using parameters in DEs to quantify solutions:
The topic of validation is also relatively new to me. But, I can put down in writing a few things I learned here and there. Obviously, it makes no sense just to chalk-up a spring-mass-damper system and derive the analytical solutions and then graph it up. The next stage is to now associate the parameters of the second order DE described above with real world parameters. This part seems the hardest to me. Normally, we would attach a mass to a spring and a dashpot, attach an displacement transducer to the mass and hit it with a impulse hammer. The readings of the transducer are then plotted with a graph against time, zero disp. being the point of attachment. The set of discrete displacements are then used to calculate parameters such as ‘period of oscillation’, ‘damping ratio’, ‘angular frequency’ etc. A brief description of how the parameters are obtained can be seen from the Figure 3 (at the end).
Vehicle crash validation method:
Vehicle crashes involve the displacement of the front-end (vehicle crush). Once the test data on the crash test is received, usually a force-displacement curve from the transducers, our next step is to setup parameters to match the graphical solutions of the second order DEs to the test curves. The forces during a crash are replaced by using pulses in our mathematical models. This can also be done by using a detailed FEM model of the crash vehicle and matching the intrusion to the test vehicle by varying the magnitudes of crash pulses.
Observe from the below figure the different stages of vehicle crash modeled from DEs. Look at the connections between maximum vehicle crush and velocity. Look at how the signage of the acceleration changes at the end of the restitution phase. These are parameters one can use to quantify the Kelvin model.
What do we infer from the graphical solution? The magnitude of the maximum displacement of vehicle crush, maximum deceleration and the resp. times during the entire collision phase.
Figure 3: Parameters used to quantify the analytical solutions of a MSD system.
NEWS ON VEHICLE CRASH:
Hallo! Time for some math. Fundamentals of mathematics are very interesting. This is because of the way these ideas came to life. Take for example the ‘Pythagoras theorem’. When you look at it at first glance, you see only a formula. A formula made up of letters with the number 2 on top of each of them. But, the idea of the theorem is something more deep. Most of the basic mathematics that you will learn in school stems from calculations on a farming field. Pythagoras theorem is no exception. Look at these notes I’ve written below.
Pythagoras theorem originated as a relation between the areas of the squares on the triangle. Take any Pythagorean triple and you will observe this phenomenon of the areas. You could go out on a beach and try drawing squares such that you have a right-angled triangle forming on the inside. Measure the sides of the squares and calculate the areas by hand. You will see that the theorem holds true. But, what you did on the beach is an empirical proof. Theorems in mathematics are not empirical, rather they have a greater meaning. Its called deduction. I can’t explain it in words. You will have to feel it.
The proof of the pythagoras theorem I performed above is a geometrical proof. The proof stems from the fact that a square can be made by joining 2 right angled triangles together.
Far more powerful is the algebraic proof that I wrote above. Now comes the most important part of today’s lesson. Remember when I told you that the Pythagoras theorem started-off as a relation between the areas of the squares? The algebraic proof shows you that the Pythagoras theorem can also be used to define the length of a line between 2 points.
Pythaogoreans were a sect existing during the time of Greek mathematics. To put time into perspective, the Indus valley civilization existed some 2000 years Before Christ was born. The Babylonians also existed some 2000 B.C. Greek civilization was around 700 B.C. Pythagoras lived at around 500 B.C. When the Pythagoreans found that the length of the side opposite the right angle(with 1 and 1 as its shorter sides) cannot be measured accurately, they did some horrible things. Thus, began a new puzzle in mathematics called the Irrational numbers.
How do you prove that a number is irrational?