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to mass changes for low screw lead but much less sensitive when the screw lead increases. This last behavior can be explained using conclusions from the degree of coupling analyzed in the previous section. For low axial–torsional coupling, as is the case for low transmission ratios, this mode is strongly axial and behaves like a spring mass system; therefore, it is very sensitive to mass changes. However, as the screw lead increases, the axial component is more coupled with the torsional component, where the latter is less sensitive to mass changes due to the reduction imposed by the
transmission. As can be expected, this mode presents high sensitivity to the combined effects of load mass and carriage position variations.
On the other hand, the second mode in general presents low variations with respect to mass and position changes, as Table 5 shows, due to its torsional predominance. Particularly, the frequency variation increases up to 5.6% for high transmission ratio when the carriage position changes, which could be due to the
coupling between the components of this mode.
Finally, the third mode presents very low variation respect to mass changes for both transmission ratios, as Table 6 shows, due to its highly torsional predominance.
However, it presents considerable variation for different carriage positions.
6 Analysis of results
The first vibration mode is predominantly axial for both transmission ratios, with a strong displacement on the carriage position, as is noticed by the large value of φuc1 in Figs. 3aand 4a. Consequently, it presents a strong frequency shift for different operating conditions involving load mass variations, large carriage displacements, or combination of both, specially for low transmission ratios. This mode will have a strong influence on the achievable bandwidth if the control loop is closed with thedirect carriage position [13]. In this case, adaptive control strategies must be used
to achieve a high bandwidth closed loop [4, 10]. For this kind of control, a high transmission ratio may be preferred because the natural frequency value will be
less sensitive to load mass variations.
The second mode presents low sensitivity to load mass variation for both transmission ratios and just a slight frequency shift for high transmission ratios when
the carriage position changes. Therefore, a low screw lead may be preferred if the loop is closed with a rotary encoder on the motor side. In this case, just a conventional
notch filter can be used to mitigate its negative effect. In addition, with a low screw lead, the first mode will be slightly reflected on the encoder signal, as is indicated by the lower value of its angular components φθm1 in Fig. 3 with respect to the corresponding one in Fig. 4.
The third mode presents a considerable frequency shift for different carriage positions but low sensitivity to mass changes. Therefore, if the application involves
short carriage displacement, a notch filter can also be used when the loop is closed with a rotary encoder. Furthermore, as this mode has a relatively high frequency,
it is expected to be slightly excited by typical position commands or external perturbations.
On the other hand, the knowledge of the system behavior in each mode can be very useful for identification purposes, especially at high frequencies. A convenient point to perform measurements to identify the first mode is the direct carriage position, where the largest deformation of this mode takes place, as can be seen in Fig. 3. Similarly, the angular position at the end of the screw θx=L can be a suitable point of measurement to identify the second and third modes.
7 Conclusions
ball screw subsystem. This model is used to predict the behavior of the first three vibration mode shapes and their frequencies. The predominance of deformation,
torsional or axial, of each mode was studied for different transmission ratios. Furthermore, the mode frequency shift for different carriage positions and load