Design News, February 2013

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M S S E A CE E IC L Control Electronics Digital Control Systems mechatronics Mechanical CAD Electronic Systems Electromechanics TU RI NG Software RAPHY ROG XE S M FA Y E C Mechanical Systems S T U M AT E R I ALS PR O AERO SP AC Control Systems SE FRESH IDEAS ON INTEGRATING MECHANICAL SYSTEMS, ELECTRONICS, CONTROL SYSTEMS AND SOFTWARE TIVE MO G N FE DE MECHATRONICS IN DESIGN N TO D I AU M S CO A N S CT NSU MER PRODU Inertia Mismatch: Fact or Fiction? Is this often-quoted concept an excuse for inadequate system modeling? SERVO SYSTEMS may cause servo-system stability compliance, and the choice of the be direct-drive or geared problems? This is a system inertia ratio and transmission ratio systems. For geared systems, (e.g., gear, belt, or lead screw) and the question and to answer this the choice of motor also design of the feedback control system question we must examine involves a choice of gear are done based on a system analysis the frequency response plot ratio. Figure 1 shows a and not by trusting poorly defined for a compliantly coupled rules of thumb. DN motor connected to a load motor and load, as shown through an ideal (i.e., no in Figure 3 (N = 1, BM friction, no backlash, and no = 0, BL = 0). The anticompliance) gear train with Kevin C. Craig, Ph.D., resonance frequency gear ratio N. The equation Robert C. Greenheck ωAR always occurs before of motion for this oneChair in Engineering the resonance frequency degree-of-freedom system Design & Professor ωR. At a low JL/JM ratio, is shown in terms of the of Mechanical the resonance and antiload angular velocity, ωL. Engineering, College resonance frequencies are The inertia ratio is the ratio of Engineering, close to each other at a Figure 1: Motor connected to load by an of the load inertia to the Marquette University. high frequency. As JL/JM ideal gear train. motor inertia, JL/JM. What increases, both the antiis the optimum value of this ratio for a resonance and resonance frequency particular value of N? If we ignore the decrease, with the anti-resonance motor and load damping, BM and BL, frequency decreasing at a faster rate. respectively, and the load torque, TL, the For a given JL, to increase the resoinertia ratio that maximizes the power nance frequency, either increase the transferred from motor to load is N2, shaft stiffness, KS, or decrease the i.e., the reflected motor inertia equals motor inertia. As KS increases, both the load inertia. When friction and load ωR and ωAR increase. The smaller torques and system compliance (e.g., the inertia ratio, the less compliance coupling or timing-belt compliance) will affect the system. are significant, the selection of an optiAll mechanical systems have mum inertia ratio is less straightforward. Figure 2 shows this more general case. Why do deviations from the so-called ideal inertia ratio of N2, called inertia mismatch, particularly in compliantly Figure 3: Frequency response of a compliantly Figure 2: Motor connected to load by an ideal gear train coupled systems, often with compliance. coupled motor-load (N=1). Design News | FEBRUARY 2013 | www.d esign n –28–

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