real-time mbs formulations: towards virtual engineering

178 J. Cuadrado, M. Gonzalez, R. Gutierrez, M.A. Naya
stop is attained. In this way, the bending modes of the chassis are excited. The described maneuver is
illustrated in Figure 7.
To obtain the motion of the car, four tri-axial
accelerometers have been attached to the chassis at the four
corners of its base. Accelerations provided by the sensors
in the motion direction have been integrated twice in order
to obtain the position and velocity of the chassis at all
times. On the other hand, four Wheatstone bridges have
been installed at four points of the chassis to measure
bending stresses. They are half-bridges, with an upper and
a lower band each, to measure the normal bending stresses
on the steel tubes surfaces. The sixteen captured signals –
twelve accelerations and four strains– were recorded by a
laptop PC after passing through a data-adquisition board..
The actual maneuver of 15 s has been simulated by
implementation of the proposed formulation, with a fixed time-step of 0.01 s, the same sample period
used for data adquisition during the experimental test, where a low-pass filter was applied with a
cutoff frequency of 10 Hz to avoid noise from engine vibration.
4.1 Accuracy
Figure 7. The maneuver
The comparison between measured and calculated bending moments at a certain point of the
chassis is shown in Figure 8. At the view of the results, three time intervals can be distinguished.
The first one is the [0,5] interval. During this period of time, the car is at rest, but some peaks can
be detected at about 2 s. They are explained because, at that time, the gear was set on drive position.
This kind of engine loads is not considered by the model
since engine torque is derived from acceleration
information. Therefore, if the car is at rest, no engine
torque is introduced in the model. This fact justifies the
discrepancies shown between measurements and
calculations during this time interval.
The second time interval is [5, 10]. During this five
seconds, the car accelerates, goes over the curb, and
brakes until reaching the rest again. Good correlation is
found during the accelerating and braking phases. The
high peaks due to the falling action are not perfectly
reproduced, although their intensities are reasonably
Figure 8. Comparison of bending moments
captured. An explanation for such peak discrepancy can
be the appearance of percussional forces in the actual
prototype, due to the presence of clearances in some joints which have not been modeled.
The third and final time interval is [10, 15], when the vehicle is again at rest. Some amount of error
in the measurements can be detected in the figures, since the bending moments should return to zero,
given that the initial and final positions of the car are exactly the same. However, the calculations
come back correctly to zero.
4.2 Efficiency and robustness
The proposed formulation has shown an efficient behavior in the analysis of this case-study. The
method was implemented on Fortran language, and the CPU time spent to run this 15-secondsimulation
was 48 s, that is, only a factor of 3 to real-time performance. The average number of
Newton-Raphson iterations to attain convergence into each time-step of 0.01 s was equal to 2, needing
a maximum of 3 iterations at the more difficult points.