This improvement in system performance without the added cost was the basis of wanting to simulate the viscous inertial damper in software. In electronics, there is a concept of Norton to Thevenin Equivalent circuits. In this transformation, a current source (or torque in the mechanical system) and the impedance representing the motor and load inertias, and shaft spring constant, can be converted to a voltage (mechanical velocity) with the impedance in series. When an additional load is then added, the current through the load (torque coupled to the mechanical damper) can be calculated as the voltage (velocity) divided by the sum of the motor/shaft/load series impedance and the added damper impedance (Thevenin equivalent circuit). In software, we can (real-time) simulate the torque that would be needed to accelerate the damper inertia to the motor velocity, given the measured motor velocity. The torque so estimated can then be subtracted from the commanded torque to the motor (from the rest of the control loop) so that the motion of the motor with the synthetic (simulated) viscous inertial damper closely approximates that of the motor and load with the physical inertial damper attached. This simulated damper gives the same improvements in gain and phase margins of the system as would the physical inertial damper but without the size and cost disadvantages. Of course, nothing is quite free. The stepwise output of a rotary encoder and the time lag involved in processing reduce some of the margins and require a bit more complexity, but in many cases, the approximation is very good and the improvements are substantial. In the previous article, we showed that a 100:1 inertial mismatch resulted in significant peaking at resonance (motor inertia Jm= 1e-5 kg-m2, Ks= 100 Nm/radian ≥ Ls=1/K = 10-2 radian/Nm, load inertia J1= 1e-3 kg-m2): The damper inertia was selected as three times the motor inertia Jd=3e-5 kg-m2, and the damping constant of the viscous oil was adjusted in the simulation to give a nice overall damping with Bd = 20N/Rad/sec. The resulting system of the motor and load and damper improved the phase margin just above the resonance from -90 (for the velocity, and -180 for position) by about 120 degrees! It also reduced the peaking from 64 dB to 30.4 dB (gain of 1631 to a gain of 33.2) at resonance. We still have a phase margin of 40 degrees at 1000Hz, so the bandwidth of the system can be significantly improved. In the system modeling, we take the voltage (motor speed) of the motor with 100 times the load inertia, and we divide it by the impedance of the (motor + inertia) plus the damper. The current (torque) transmitted by the viscous coupling in the damper is the same in this topology (which is an electrical model, as inertia is always modeled as a capacitor connected to a ground node) as in the parallel version with the current (torque source). This model is not physically realizable in the mechanical design but gives an easy method to calculate the torque absorbed by the mechanical damper attached to the motor. Note: The voltage source labeled as Vm in the series circuit model is the Thevenin equivalent voltage representing the speed of the motor/shaft/load without the damper present.

The model works as the voltage source is modeled as a zero impedance, so the damper is essentially connected across Jm. Although this model is not realizable in a mechanical system, it does allow us to easily calculate the torque that a physical damper would incur if attached. While this model makes it easy to calculate the damper torque, it may be harder to picture how it works. Another way to think about this model is to have the motor velocity (modeled as a voltage) drive the damper (Rd, Cd) with the same velocity (voltage) as the motor model. The torque (modeled as current) drawn by the damper is measured through Rd. This torque is subtracted from the original control torque It. The resultant net torque to the motor/shaft/load is identical to the torque that reaches the motor/shaft/load when a physical damper is in the system. Not surprisingly, the resulting system then produces the same response using the synthetic damper as it did with the physical damper! In an actual system, this damping torque calculation would be done by measuring the actual motor position as the input, estimating the velocity, and calculating the equivalent synthetic inertial damper torque term. This damper torque term is then subtracted from the commanded torque (after some scaling for torque units used and for motor torque constant) to result in a very helpful improvement in the system dynamics. The actual system has some additional filtering terms to reduce the effects of encoder resolution with its stepwise output. These calculations are done in real time with minimized delay and are performed in the time domain, so they are a little more complicated than the simple impedance calculations in the spreadsheet, but they produce a very similar response.