Signal GeneratorPowerAmplifier2XVs+15V-15VAdjustable SpringAir BearingsF = K imVoice coilMotorLDVTScopeShaftVax, vMF = K im+emRmVa+iFe = K vmmMassachusetts Institute of Technology Department of Mechanical Engineering 2.003 Modeling Dynamics and Control I Spring 2005, Prelab 4 Assigned 3/10/05, due at start of lab during week 3/14/05 Figure 1: Schematic of Lab setup For Lab 4, you will be examining the step response of the spring/mass/damper system studied in Lab 3. For this lab, we need to generate a new system model since we are using the voice coil motor to excite the system. Figure 2: Circuit of voice coil motor driven by the amplifier voltage (Va) In general, dc motors (a voice coil motor is a single phase dc motor that produces linear motion) produce a force that is proportional to the current driven through the motor Fm = Kmi (1) where Km is the motor force constant with units of N/A for linear systems. A dc motor can be driven either by a current or voltage source. For this lab you will be driving the system with a voltage source (the power amplifier). The circuit diagram for this system is shown in Figure 2. In this circuit, Va is the voltage at the output of the power amplifier, Rm is the electrical resistance of the motor coils, and x is motor displacement (For our purposes, we can2.003 Prelab 4 Assigned 3/10/05 ignore the coil inductance and capacitance). // The actuator introduces a new variable em called the back electromotive force (back emf), which is the em the voltage generated in the motor coils when the coils are moved through a magnetic field. It can be shown that em = Kmx˙ (2) [J ] [J/s] =[W ]=[N · m]Units: [V ] =≡ [C] [C/s] [A] [A s]· Summing the voltages around the circuit of Figure 2 yields −Va + imRm + em = 0, or (3) Va = iRm + em = iRm + Kmx˙ (4) Solving equation (4) for i yields x˙Va − Kmi = (5)Rm Substituting i into (1) allows us to determine the voice coil force Fm which acts on the mass as x˙ K2 Fm = Kmi = Km Va − Km = Km Va − m x˙ (6)R R R The force from the motor contains both a term proportional to the system in-put Va and the negative of a term proportional to the velocity of the coil/shaft system. The K2 m/R term can be interpreted as an equivalent damping con-stant beq with units of N s/m. As you saw in Lab 3, the back emf is not the only source of damping in the voice coil motor. Another source of damping is the currents flowing in the aluminum cup supporting the motor coils. Since aluminium is conductive, the same physics (Faraday’s law of induction) that apply to moving the copper motor coils through a magnetic field apply to the aluminium cup. Moving the cup through the motor’s magnetic field causes a voltage potential in the cup. Since the cup is a continuous part, the voltage induces a current (typically termed eddy current) to flow around the cup. The induced current then interacts with the magnetic field to create a force on the cup (Ampere’s law). It is possible to analytically calculate a damping constant associated with the eddy currents in the cup but for present purposes we will use the damping constant measured in lab (beddy ≈ 5 Ns/m). Including the eddy current damping in our motor model yields Km � K2 � Fm = Va − m + beddy x˙ (7)R R 22.003 Prelab 4 Assigned 3/10/05 Note that because the amplifier acts as voltage source, the damping in the system is the same as with the actuator coil shorted. The amplifier is con-figured with a gain of 2, which means that Va = 2Vs. Since non-idealities can distort the initial movement of a system, one commonly used metric to avoid these non-idealities is the 10-90% rise time. For underdamped systems the rise time is approximated by 0.8 + 2.5ζ tr (10 − 90%) ≈ ωn for 0 ≤ ζ ≤ 1. (8) Problems 1. Given the motor force characteristics (equation 7), derive the differ-ential equation relating the voltage input from the signal generator vs to the shaft position x for this system. Be sure to include the power amplifier voltage gain of 2V/V. Write your expression in terms of the system variables. 2. It can be difficult to measure the 2% settling time of a system in the lab. A more commonly used metric is the 5% settling time. Determine the relationship between the system parameters and the 5% settling time (ie. ts(5%) = aτ where a is a constant). 3. Given that Km = 7.1 [N] [A] R = 5.5 [Ω] m = 0.85 [kg] k = 0.132 l3 [N] [m] Analytically calculate the step response for spring lengths of 50, 100, and 150 mm. That is, derive an expression for the response x(t) to a 1V olt step in Vs, for the three spring lengths. (ie. determine x(t) for each spring length). Use MATLAB to plot each of these time responses. 4. For each response in 3 , graphically determine tr (10 − 90%), tp, Mp, and ts(5%). 5. Calculate tr (10 − 90%), tp, Mp, and ts(5%) directly from the differential equation using the formulas presented in the supplemental notes on 1rst and 2nd order systems. Compare these to your graphically determined values.