To check the accuracy of the calculations of the EULER PC by students and teachers of the Moscow State University. M.V. Lomonosov solved a large number of different problems in theoretical mechanics. The solution of these problems showed that the accuracy obtained in the EULER, with the appropriate specification of the numerical solution parameters, is at least six significant digits (the relative error is less than 1.0e-6). This is significantly higher than the accuracy that is normally required for engineering calculations.

The results of modeling the dynamic behavior of real mechanical systems always differ from the results of field trials. These differences are explained by the following three factors:

1.Errors in the measurement of characteristics during testing. These include the errors of the measuring sensors, the error of their installation and the error in processing the recorded readings.

2. Inaccuracies in the original model data. They are associated with the inability to accurately determine the various characteristics of the design of a mechanical system that has undergone field testing. These include the inauthenticity of specifying in the model the characteristics of elastic and damping elements, errors in geometric data and other parameters.

3. Methodical errors of the model. They are associated with the assumptions and idealizations adopted in the model in comparison with the real mechanical system and with the accuracy of calculating the model.

When simulating simple mechanisms, the discrepancy with the results of field trials is usually 1-2 percent. When modeling complex systems, these discrepancies usually are much larger.


Comparison of the movement characteristics of the UAZ-3151 vehicle in the process of performing the "rearrangement" maneuver, obtained during the simulation, with the results of full-scale tests. The average discrepancy in the values of the characteristics during the movement of the car between modeling and full-scale tests is 7.4%. This discrepancy is largely due to the errors in the original model data.

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Comparison of the results of mathematical modeling of the attraction of the cathedral mountain "Cobra" with the results of full-scale tests. During the simulation, the maximum height of the train was determined after the passage of the track. The average discrepancy in the loss in height of the center of mass of the train after the passage of the path between the simulation results and full-scale tests is 0.42%.

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The results of the testing of the EULER software package for energy conservation during the motion of conservative mechanical systems are presented. In conservative mechanical systems, at any point in their motion, the sum of the potential and kinetic energies should remain constant. To test the accuracy of energy conservation in calculating the motion of conservative systems, the following examples were compiled in the EULER:

  • Example 1. One body connected by a rotational hinge, in the field of gravity.
  • Example 2. Three bodies connected by rotational hinges, in the field of gravity.
  • Example 3. An unbalanced gyroscope in the field of gravity.

In all the presented examples, in the process of motion of conservative mechanical systems, the relative error of conservation of the total energy is calculated. This error depends on the step and time of numerical integration.

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In this section, we compare the results of mathematical modeling with the EULER software package with the results of analytical solutions. As the samples under investigation, a set of known problems for each student on theoretical mechanics.

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Latest news

Euler version 12.12

A new version of the software package has been released
EULER 12.12 
The demo version 12.12.03 of the software package is available at the link

DOWNLOAD the demo version of EULER

May, 2023

Версия Euler 12.12

Выпущена новая версия программного комплекса
EULER 12.12

Май, 2023

Update of the EULER software package - version 12 has been released

We continue to actively work on improving the EULER software package.

A big update of the software package - the 12th version of EULER has been released, in which the following important changes have been made:

  • a new computational core with significantly expanded capabilities has been developed;
  • increased calculation speed;
  • implicit integration methods are implemented, the use of which reduces the calculation time of rigid systems by orders of magnitude;
  • a number of user interface improvements have been made.
Euler version 11.14

A new version of the EULER 11.14 software package has been released

December, 2020