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.


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About Sunny

I am an Automotive engineer by qualification. I innovate on high-tech solar and automotive applications. My hope is to engineer a more efficient smart grid-connected world.

Posted on August 11, 2015, in Uncategorized. Bookmark the permalink. Leave a comment.

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