Second order DEs in vehicle crash analysis

New blog on the mathematics of vehicle crash analysis and its connection to differential equations (DEs)

Second order DEs in vehicle crash analysis.


Second order DEs in vehicle crash analysis

New blog on the mathematics of vehicle crash analysis and its connection to differential equations (DEs)
Second order DEs in vehicle crash analysis.

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:

m\ddot{x}+c\dot{x}+kx=F(x)  – 1

L\ddot{I}+R\dot{I}+\frac{1}{C}I=\dot{E}   – 2

MSD-like systems in engineering occur in various fields, for example:

  1. Animal running (ref: Running springs: speed and animal size)
  2. RLC Circuits
  3. Robotics (PD control) and lastly,
  4. 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:

  1. Complementary function
  2. 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 t\rightarrow \infty .

Lumped parameters-

When it comes to vehicle crash modeling and analysis, the analysis part is categorized into two broad categories, namely:

  1. Lumped parameter analysis
  2. Finite Element Analysis
(refer: Lumped parameter analysis-example case)

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. Image für Crashaufprall D.g           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).

real vs test

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.


Cashless payment for vehicle crashes in India

Ford F150 gets mixed crash results

PoPP3- Pythagoras theorem [Ved1]

For Ved.

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?

Pen on Paper Pictures[PoPP 2]-Dampers

Pen on Paper Pictures[PoPP 2]-Dampers.

Pen on Paper Pictures[PoPP 2]-Dampers

What’s PoPP?

I am creating PoPP because of the risks involved when one posts pictures on blogs (copyright infringement maybe..) and also because it feels good to put Pen on Paper, especially if your an engineer and have to perform calculations on software. Some of these topics might sound complicated and nerdy but most of them comprise of stuff we use in our everyday lives. Understanding them could help you choose a better car or get better grades in University or just pass time. I’m writing today’s blog on dampers and Dr.Amar Bose.


Vehicle dampers and Dr. Amar Bose


Comfort and safety for the passengers.


Dampers and their behaviour

Dampers and their behaviour

Introduction of the topic:

Lets assume that a vehicle body is rigidly connected to the wheels (no suspension). If your car travelling at 40 kph encounters a pot-hole, what do you think would happen? As a passenger, you’d feel a disturbance in the form of a upward jerk. If you were driving, your head would most likely dash against the steering wheel. Inorder to avoid this, you want the wheel to have some ‘travel’. With the spring between the wheel and the body, the vehicle is no longer a complete rigid system. The spring absorbs most of the kinetic energy generated as a result of the disturbance.

As suspension springs are designed according for the maximum load case (4 passengers+luggage), our spring absorbs more than enough of the kinetic energy and this usually makes the car bounce up and down.

Look at what happens to this car without a rear damper and see how it bounces up and down. That’s not so comfy and an Automotive engineer would always want the rear stable.

Description of the PoPP:

Dampers are an additional mechanical component that stop the springs from bouncing too much i.e. they perform 1 function and that is absorbing the kinetic energy of the springs. Hence, the damping force is proportional to the vertical velocity of the wheel or the body and this is not so intuitive. Look at the equation I’ve written on paper. Notice that the resisting force provided by the damper depends on two terms, namely a ‘valve constant k’ and the ‘vertical velocity v’.

Vehicles are complicated. They show a particular behaviour when you drive straight (accelerating/braking), and behave differently when turning(left/right/skid). For example, when you brake, the car performs a ‘brake dive’. Brake dives are classified as a low speed disturbance and you would like in such instances that the front dampers avoid ‘pitching of the car. A car driving over a series of speed breakers is classified as a high speed disturbance. Unbelievable as it may seem, you want the damper to change behaviour according to the situation and speed of the disturbance (like a chameleon changes colour). This brings me to the great man Dr.Amar Bose. Bose corp., during the mid 80s researched mechanical spring-damper systems and found that such a system would always have its limits to adapt to different road conditions. (fluid flow causes lag)

I have always been fascinated by MIT Professors and their knowledge of applied engineering. But, after reading a short bio of Dr.Amar Bose, I came to the conclusion that all great thinkers have a child-like curiosity.

Such was that extent of Bose’s curiosity, that he invented noise-cancelling headphones during a single flight trip from city A to city B (who cares about airspace…..he did it). Read somewhere that he wrote down the equations during the flight and the moment he landed ordered the engineers to start working on a prototype. Inspiring boss!

The graphs that you see at the bottom of the paper show the difference between passive, semi-active and active dampers. Depending on whether your on ‘Sports Mode’ or ‘Sedan Mode’, the valves in the damper behave a certain way thereby making the damper stiffer or softer. Its most important to remember that dampers behave differently during spring compression than during spring expansion. This is because rebound damping has to just dissipate the stored spring energy. Compression damping has to control spring compression and also at the same time absorb some of the bump force itself. I’m still trying to get to grips with this difference.

State-of-the-art examples:

Spring-Damper systems are incredibly complicated. For example, as an engineer you even have to take into account the differences in behaviour between your rear susp. system and front susp. system.

Front suspension frequency and rear suspenion freq

Front suspension frequency and rear suspenion freq

This is why most modern cars today use Semi-active and Active damper systems. You want the suspension to react as fast as possible and push against the disturbance.


An ideal car behaves similarly regardless of whether your turning or braking/accelr. Unfortunately, the two actions can’t co-exist with a mechanical suspension. Simply impossible. It took some geniuses at Bose corp. to create something out of nothing- Electromagnetic suspensions. That’s the thing about these eccentric CEOs (Musk,Bose). They somehow have to ability to look beyond the standards that exist. Its like a Mathematician ignoring the axioms that exist and formulating a hypothetical equation just for the fun of it. Let us try to think beyong what exists in the books.

Websites to read further:

1. Dr. Amar Bose bio-





Pen on Paper Pictures [PoPP 1]- Vehicle body structure

Pen on Paper Pictures [PoPP 1]- Vehicle body structure.

Pen on Paper Pictures [PoPP 1]- Vehicle body structure

What’s PoPP?

I am creating PoPP because of the risks involved when one posts pictures on blogs (copyright infringement maybe..) and also because it feels good to put Pen on Paper, especially if your an engineer and have to perform calculations on software. Some of these topics might sound complicated and nerdy but most of them comprise of stuff we use in our everyday lives. Understanding them could help you choose a better car or get better grades in University or just pass time. I’m writing today’s blog on vehicle body structures because I recently finished my Thesis in the very same field….so kind-of an expert.


Vehicle body structure and its behaviour during a crash.


Saves passenger’s lives during a crash.

PoPP 1:

Pen on Paper drawing of a vehicle body architecture

Pen on Paper drawing of a vehicle body architecture

Introduction of the topic

Back in the middle part of the 20th century, an engineer by the name of Béla Barényi started work at Daimler-Benz and invented the new field of ‘Passive safety’.  Mr.Barényi was a prolific inventor with more than 2500 patents but the most popular among them were the ‘Safety Cell’ and the ‘Collapsable steering column’. Most engineers like inventing stuff, but to do it with an ideal of ‘saving lives’ is why I always look at Barényi as my hero. When you see a modern vehicle crash, you might observe that everything apart from the passenger cell is crushed to oblivion. This might give you the impression that the vehicle is not safe. But, this is where the paradox lies- The more Kinetic energy you absorb during a crash, the safer are my occupants. Imagine this, your vehicle hits the wall at 50 kph and comes to an abrupt stop. Do you think the passengers would also come to an abrupt stop? No! They keep moving at 50 kph. This is why we wear seat belts. Just saying….. :). So the aim is to slow-down the rate at which the passengers hit the airbag and also make sure the steering column collapses.

Buying a car ? Today’s cars come equipped with ‘Active safety’ systems. Google it and you’ll see that some of really nifty.

Description of the PoPP

Vehicles are usually conceptualised from the inside-out. What do I mean by this? First, I need to decide how many passengers my vehicle would be seating. (Or even weirder, how many wheels would my vehicle need to have?). Where is my engine going to be positioned? Where would I want the trunk/cargo space to lie? So you see, there are a ton of stuff that one needs to decide before it comes to the stage of Art-drawing contests. Automotive engineers need to decide on the passenger space inside the vehicle depending on the size of its occupants. Take a look at the drawing again. See that I’ve labelled the ‘Manikins’ (engineering lingo for dummys) with ‘95%male’ and ‘5% female’. 95% male denotes that the dummy is heavier and taller than 95% of human males. Notice that the rear seats seat only people of size 5% female (roughly 5 feet in height). A common example of this type of ‘package’ is the Audi TT.

Most of the frontal impact energy during a crash is taken-up by the frontal longitudinal members. These are the 2 longitudinal beams in the engine compartment. These beams aren’t desingned to break but are designed to ‘crumple’ (like paper)…well this actually depends on the material, for example, CFRP can absorb energy by breaking. But, you probably get the idea by now. Modern vehicles have such complex frontal crash absorbing structures that it almost looks like a maze. Electric vehicles are easier to design for taking-up a crash, because of the absence of complex mechanical parts in the front. An Internal Combustion Engine on the front is basically a solid block and one wouldn’t want it jutting into the passenger compartment during a crash. So, for an ICE vehicle, the vehicle body looks a bit different.

Again, just a simple googling of the words ‘crumple zone’ will open-up a whole new idea of how vehicles dissipate forces during a crash.

State-of-the-art examples

Earlier generation vehicles dissipated frontal crash forces along the side structures, like door beams for example. This was not a good philosophy because it resulted in the A-pillar (see pic) buckling and the roof structure would eventually give-way. So, crash force dissipation became a big topic in the late 20th century. Take a look at some of the youtube crash-test videos and look at how an older vehicle behaves in comparison to a modern vehicle during a crash ( That didn’t look good, did it? Notice how the A-pillar buckled and the roof followed along with it? That means you have a bad frontal crash structure. But lets go easy on the engineers from the 50s, shall we? Who knows….in 60 years time, the current Tesla Model S might seem unsafe. The Tesla Model S body structure is my most favourite. Its a conventional build but they put-in a few extra beams in the front (mostly for driving dynamics) but some also for crash absorption. The modern Honda cars have re-defined the idea of ‘crash compatibility’ i.e. making sure crashes between vehicles of different sizes dont result in serious injuries. A heavier car certainly has an advantage in terms of safety of its occupants as compared to a smaller one. This is the reason why smaller cars like the Daimler Smart have a super stiff passenger cell. Another example: Mazda 3 uses Skyactiv technology and also has a ‘front crash prevention’ system….and the list of cars goes on.


Once one gets really into this topic, it is funny how fast one forgets that the passengers are the most important part of the vehicle. One starts thinking of ways to protect the passenger cell but then completely forgets about the passengers inside…haha. So, if your an engineer reading this and are part of the crash design team, hats-off to you because your already on the path to making the world safer. 2 billion ICE cars and a few million electric cars…imagine the change you could make as a mere design engineer.

Websites to read further:

1. Book H-point-The Fundamentals of Car Design & Packaging by S.Macey, R.Gilles and G.Wardle.

2.    (awesome blog on the evolution of vehicles and their technologies).

3. Wikipedia Béla Barényi for inspiration.

4. Youtube crash test videos. (good passtime too).




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