Abstract
The dynamic behavior of a servomechanism must be studied as a whole without distinction between its parts, in order to assess correctly the overall performance of the system. For this reason, in many cases of practical interest, it may be useful to analyze the dynamics of a mechatronic device using a mathematical model and a computer simulation software, in order to verify the consequences arising from the modification of a particular parameter of the system. Following this methodological approach, this paper proposes a model for the dynamic analysis of a typical industrial servomechanism characterized by elasticity and backlash in the transmission. The model has been implemented into a software application and it can be used to test, from a theoretical point of view, the dynamic behavior of the device under user-defined operating conditions. The numerical simulations, which can be performed in a short time, may help the designer to optimize the dynamic performance of the servomechanism without performing experimental activities. On the basis of the simulation results, he can then evaluate, for example, whether the presence of the backlash still allows to obtain acceptable acceleration values. The software also allows to verify the effects resulting from different settings of the position regulator, and therefore, it can be helpful to assist the designer during the calibration procedures.
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Abbreviations
- T :
-
Kinetic energy
- U :
-
Potential energy due to joint elasticity
- D :
-
Rayleigh dissipation function
- W :
-
Work done by external forces/torques
- \(\alpha _{\mathrm{ref}}\) :
-
Motor shaft rotation (reference command)
- \(\alpha \) :
-
Motor shaft rotation
- \(\beta \) :
-
Rotation of the output shaft
- \(\gamma \) :
-
Screw rotation
- \(\varepsilon \) :
-
Relative rotation
- x :
-
Carriage displacement
- e :
-
Position error (rad)
- \(e_\mathrm{s}\) :
-
Position error (step)
- y :
-
Output signal of the PID controller
- \(J_\mathrm{m}\) :
-
Moment of inertia of the motor
- \(J_\mathrm{v}\) :
-
Moment of inertia of the screw
- M :
-
Carriage mass
- z :
-
Gear ratio
- p :
-
Screw pitch
- k :
-
Joint stiffness
- c :
-
Joint damping constant
- b :
-
Backlash half amplitude
- \(\tau _\mathrm{m}\) :
-
Motor torque
- \(F_\mathrm{r}\) :
-
Resistant force
- \(k_\mathrm{m}\) :
-
Torque/back e.m.f. constant of the DC motor
- R :
-
Armature resistance
- L :
-
Armature inductance
- I :
-
Armature current
- \(V_\mathrm{m}\) :
-
Armature voltage
- \(V_{\mathrm{max}}\) :
-
Maximum armature voltage
- \(V_{\mathrm{out}}\) :
-
Output voltage of the D/A converter
- n :
-
Bit number of the D/A converter
- \(\varDelta V\) :
-
Output voltage range of the D/A converter
- \(K_{\mathrm{dac}}\) :
-
D/A converter resolution
- N :
-
Number of encoder step
- \(K_{\mathrm{enc}}\) :
-
Encoder resolution
- \(K_\mathrm{a}\) :
-
Voltage amplifier gain
- \(K_0\) :
-
Global constant
- \(K_\mathrm{p}\) :
-
Proportional gain
- \(K_\mathrm{i}\) :
-
Integral gain
- \(K_\mathrm{d}\) :
-
Derivative gain
- h :
-
Carriage total displacement
- \(T_0\) :
-
Motion time
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Incerti, G. Modeling and simulation of a position-controlled servo-axis with elasticity and backlash in the transmission. Arch Appl Mech 87, 633–645 (2017). https://doi.org/10.1007/s00419-016-1213-x
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DOI: https://doi.org/10.1007/s00419-016-1213-x