Motor Sizing

Transmission Selection: Sizing Transmission Selection: The Velocity Accuracy is almost load-independent and completely motor-independent: it does depend only on the position sensor and regulator tuning The reaction times of a brushless can be as low as few ms: the limits is now the mechanics attached to it A resonance can grow, due to load elasticity that cause oscillations that behave as high-noise vibrations (the motor over-heat in this case is due to accelerations-decelerations) Solutions: higher load stiffness or lower response time (lower gains with lower performances…)

Transmission selection (cont’d) Sizing Transmission selection (cont’d) rotation-rotation : Belt (cinghia dentata): low cost, band-width<10Hz Gearbox (riduttori): high cost, low backlash is necessary <10-15 arcmin Ball-screw (vite senza fine): efficiency very low at high speeds, high static friction (attrito di primo distacco): not good for varying speed applications rotation-linear: Belt (cinghia dentata): low cost Ball-screw (vite senza fine): good if speed <1m/s Pinion-rack (pignone-cremagliera): big backlashes, band-width <5Hz Metallic tape: high band-width but with limited loads

Transmission selection (cont’d) Sizing Transmission selection (cont’d) Power applications (mandrels, traction, avvolgitori): not relevant kinematic performances, motor cost is important. It’s useful a transmission with a reduction-adaptation stage (that allows for smaller motor). Positioning applications (position profiles): high kinematic performances. With high reduction ratio, the load inertia become less important, but the motor inertia becomes more important. The optimal ratio (with respect to the motor’s torque) is the one with same load inertia (to the rotor’s side) and rotor inertia (inertial ratio), or the direct-drive solution.

PLANETARY GEARBOXES solar wheel planetary wheel internal crown Basic parts of an epicyclical gearbox stage: solar wheel (or pinion) planetary wheel internal crown

When the planetary rollers turn they describe a geometrical profile called epicycloid, here shown:

Parts of an epicyclical gearbox with one stage pignone Carcassa con dentatura interna Planetari (3*120°) Albero di uscita Albero di ingresso

Scomponiamo un riduttore nelle sue parti…. Cannotto calettatore Cuscinetto Albero d’uscita con chiavetta 1° stadio 2° stadio Guarnizione Guarnizione Albero d’uscita con chiavetta Cuscinetto Cuscinetto 2° stadio 1° stadio Cannotto calettatore :

Parts of an epicyclical gearbox with two stages Planetary carrier of the second stage (output flange) ring gear = housing (fix) Ring gear of first stage is connected with the planetary carrier of the second stage (rotating with output) pinion of the first stage (input) Planetary gears 3x120°

ANGULAR BACKLASH / ACCURACY 1 to 10 arcmin in gearboxes for servos

Transmission selection (cont’d) Sizing Transmission selection (cont’d) Indeed: J = JM + JL/N2 ; C = J dw/dt with: qM = N qL And I look for the minimum of CM = (N JM + JL/N) d qL /dt deriving with respect to N and equaling to 0 we get: N2 = JL / JM that is: JL/N2 = JM but ATTENTION: this “inertia weighing” is for determining optimal N only; but if I can reduce JL/N2 and/or JM working directly on them, still having them different, I have to do it! It is not convenient to increase the rotor’s inertia!

Rotor-Load Inertia Ratio Sizing Il classico criterio di progetto per servoazionamenti prescrive di scegliere un insieme motore + trasmissione tale da ottenere un’inerzia del carico (ridotta all’albero motore) circa uguale all’inerzia del rotore stesso o di poco superiore. Tale impostazione è legata a problematiche distinte: 1. scegliere il rapporto N di trasmissione ottimale (derivata rispetto ad N) per massimizzare l’accelerazione del carico, una volta fissati il motore ed il carico 2. ottenere una sufficiente insensibilità delle prestazioni al variare delle condizioni operative (“robustezza del controllore” ottenuta appunto trovando il minimo e quindi con derivata che cresce lentamente da entrambi i lati) 3. ottimizzare le prestazioni dinamiche dell’asse controllato (avere deformazioni modeste tra sensore e carico) E’ chiaro che, a fronte di condizioni dinamiche, l’adozione di un motore con grande inerzia ne ridurrà gli effetti negativi, ma in generale ciò renderà anche più difficile e costoso raggiungere prestazioni elevate già nelle condizioni nominali. Una possibilità alternativa, è invece quella di cercare di stimare la variazione del comportamento dinamico istante per istante e modificare di conseguenza guadagni o strategie di controllo (ad esempio col feed-forward).

Rotor-Load Inertia Ratio Sizing Rotor-Load Inertia Ratio Bassa inerzia Jmot+carico Vel.mot. Coppia Log w = 2pf Log Log

Rotor-Load Inertia Ratio Sizing Rotor-Load Inertia Ratio Nel caso rotativo vi è di solito una risonanza (in generale a frequenze abbastanza basse) dove motore e carico si muovono in direzione opposte, deformando il collegamento motore-carico. Sopra la frequenza di risonanza la risposta in frequenza continua a scendere lungo una linea che rappresenta l’inerzia del solo rotore, fino alle frequenze più elevate dove si vede il limite della banda passante dell’anello di corrente (circa 1 kHz) dove potrebbero essere visibili le risonanze legate a deformazione della struttura che collega motore e sensore. L’adozione di motori lineari permette di superare le limitazioni legate alla presenza delle catene cinematiche tradizionali (cedevolezza tra motore e sensore di posizione, attrito variabile lungo la corsa, inerzia, …) e sviluppare quindi applicazioni con elevate prestazioni dinamiche. Per poter raggiungere tali prestazioni è però necessario mantenere sotto controllo diversi aspetti, di natura elettrica, elettronica e meccanica. Un utile strumento per fare ciò è costituito dall’analisi meccatronica, che permette, tramite modelli che uniscono una descrizione della struttura meccanica, tipicamente ottenuta tramite modelli agli elementi finiti, ad una descrizione dell’azionamento, di stimare i principali indici prestazionali e verificare se, durante il funzionamento, insorgano limitazioni legate ai specifici componenti scelti, quali l’azionamento ed i sensori di posizione.

Transmission selection (cont’d) Sizing Transmission selection (cont’d) The gearbox cost is typically similar to the motor’s cost, thus the optimal motor dimensioning is not the optimal system dimensioning, considering the gearbox problems. Every gearbox ratio N2 > JL / JM is wrong because it reduces the bandwidth and increases the motor’s energy consumption. The ideal case is thus WITHOUT GEARBOX An important exception is when JL >> JM so that the inertia rotor cannot compensate the load mechanical resonances. Cymex SW tool link

Transmission selection (cont’d) Sizing Transmission selection (cont’d) Long and thin motors have low inertia: high accelerations are possible Short and thick motors have high torsional stiffness: OK with high inertia loads Torsional Stiffness : S = p D4 78.5 109 /(32L) Resonance Frequency: F = SQR(S/JL)/(2p) Shaft diameter Steel shaft length

Servo-System Sizing: 1) Draw velocity/time diagram through the cycle 2) Port inertias and loads to the rotor’s side 3) Calculate accelerations and torques 4) Add 2) e 3) 5) Calculate: torque RMS, velocity RMS, average torque, max.torque duration, torque at the max.speed, max.torque

Servo System Sizing (cont’d) Max torque verification: The max torque required from the motor cannot be bigger than the motor’s peak torque (di targa). Motor temperature verification: With the motor thermal model, the torque over the nominal torque have to be maintained for intervals below 0.3s with 1.0s cycle, typically. The excessive temperature is the ONLY good reason to increase the motor’s size DC bus voltage verification: Vmax-speed < Emin

Servo System Sizing (cont’d) Drive size verification: Imax < Cpeak motor / Kt (for not burning the motor) IRMS > CRMS / Kt (for profile following) Imax > Cpeak profile / Kt (for profile following) Power-on (in-rush) verification: At the power-on the electronics absorbs more than double current for 0.3s for charging internal capacitors In-Rush Resistors Motion-Book SW tool link

Needed Mechanical Inputs for Motion Book: Position or Speed or Torque Profiles vs. time or machine degrees Inertias and/or Masses and Distances from rotation axis If present: Spring K const. If known: Frictions (static, viscous, …)

Electro-Magnetic Compatibility Sizing Electro-Magnetic Compatibility (EMC): The PWM has square waves with edges at 5000V/ms that in turn generates frequency spectrum that, coupling with cable and winding capacitances, produce radio emissions at 10KHz-100MHz. Useful hints: Ground the motor, drive and shield and keep this ground separate from signal ground Power and signal separated, with shielded cables, with 90deg intersection Ferrites where necessary

EMC: Frequency spectrum example Sizing EMC: Frequency spectrum example

Trouble-Shooting in First Rig Installation Sizing Trouble-Shooting in First Rig Installation Insufficient Motor Torque: increase current limit or motor size Motor get hot: increase motor size Motor is noisy: lower Kp and/or Kd of the PID Motor is unstable: lower Kp and/or Kd of the PID Motor follows the cycle but is noisy: reduce mechanical backlashes and/or lower Kd Motor doesn’t turn or does it intermittently: wrong U,V,W, cabling or feedback cabling

Trouble-Shooting in first rig installation (cont’d) Sizing Trouble-Shooting in first rig installation (cont’d) Motor is noisy where other motors are not: better load torsional stiffness or lower Kp of the PID Motor is slow to react: higher Kp and Kd of the PID Motor does not reach required velocity: motor need to be rewound for higher speed (but with lower Kt) It’s impossible to reach required accuracy without going into instability: better mechanics: no backlashes, belts and, if possible, migrate to a direct-drive; the position sensor could also be improved (sin-cos) Error at regime is too high: increase Ki of the PID