Abstract
What happens between starting the simulation and saving the results is for many a closed book. Most simulation programs offer the user the possibility to generate various calculation results completely without the knowledge of numerical analysis. A few basic concepts are discussed below.
The book of nature is written in the language of mathematics.
Galileo Galilei
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Notes
- 1.
- 2.
In the literature, one also finds it under the Euler-Cauchy method. But, because in most simulation programs only the Swiss mathematician Euler is mentioned, I will focus on him.
- 3.
At this point, I refrain from giving the derivation. The books about numerical analysis are full of them, however, one should at least understand what it means when a growth of errors is described with O(h²).
- 4.
In this context, the term flawless is often mentioned too. However, this term only describes the absence of error reports from the simulation program and has nothing to do with the erroneous discussed in Chap. 3.
- 5.
The most popular step size will probably be 1 ms. Because the system behavior is classically measured to up to 5 Hz in vehicle dynamics, this leads to at least 200 measuring points per oscillation cycle.
- 6.
Further variants exist in which the weighting factors are chosen differently.
- 7.
Predetermined by the step size h.
- 8.
In this method, cubic splines are used too. However, the neighboring intervals are considered in the calculation of the derivatives at the intervals’ transitions.
- 9.
Of course the values are unequal zero, otherwise they could be directly joined together.
- 10.
Since the sinus is just a shifted cosine, it will not be mentioned any further here. However, it can be used without worry by the supporters of this function.
References
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Hibbeler, R.C.: Technische Mechanik 3 – Dynamik, Pearson Studium, Munich, 2012
Mayr, M.: Technische Mechanik, Hanser Verlag, Munich, 2012
Munz, C. D. and Westermann, Th.: Numerische Behandlung gewöhnlicher und partieller Differenzialgleichungen, Springer, Heidelberg, 2012
Rill, G. and Schaeffer, Th.: Grundlagen und Methodik der Mehrkörpersimulation, Vieweg+Teubner, Wiesbaden, 2010
Strehmel, K., Weiner, R. and Podhaisky, H.: Numerik gewöhnlicher Differentialgleichungen – Nichtsteife, steife und differential-algebraische Gleichungen, Springer, Wiesbaden, 2012
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Adamski, D. (2021). Numerical Analysis: The Problem with the Beginning. In: Simulation in Chassis Technology . Springer, Wiesbaden. https://doi.org/10.1007/978-3-658-30678-6_4
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DOI: https://doi.org/10.1007/978-3-658-30678-6_4
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