Rotational Systems, Gears, and DC Servo Motors

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1 Rotational Systems Rotational Systems, Gears, and DC Servo Motors Rotational systems behave exactly like translational systems, except that The state (angle) is denoted with rather than x (position) Inertia is called J (rather than M) Friction is called D (rather than B) θ Otherwise, the symbols, circuit equivalent, and analysis for rotational systems is exactly like translational systems. Z = R (resistor) Z = Ls (inductor) Z = 1/Cs (capacitor) Electrical Circuits I = 1 Z V Inertia Translational Systems F M X F = Ms 2 X Friction B F = BsX Spring K F = KX Inertia Rotational Systems T Q J T = Js 2 θ Friction D T = Dsθ Spring K T = Kθ JSG 1 October 5, 216

2 Example: Draw the circuit equivalent and write the equations of motion for the following rotational system. T Q2 Q3 K1 K2 K3 J1 J2 J3 D1 D2 D3 Inertia - Spring Rotational System Solution: Note that each term is an admittance. Draw the circuit equivalent. There are three masses, which result in three voltages nodes (each angle corresponds to a voltage node). Each element corresponds to an impedance. K1 Q2 K2 A3 T J1 s 2 D1 s J2 s 2 J3 s 2 D2 s D3 s K3 The voltage node equations are then: Circuit Equivalent ( s 2 + D 1 s + K 1)θ 1 (K 1 )θ 2 = T (J 2 s 2 + D 2 s + K 1 + K 2)θ 2 (K 1 )θ 1 (K 2 )θ 2 = (J 3 s 2 + D 3 s + K 2 + K 3)θ 3 (K 2 )θ 2 = Placing this in state-space form is identical to what we did with mass-spring systems. Solve for the highest derivative: s 2 θ 1 = D 1 s + K 1 θ 1 + K 1 θ T s 2 θ 2 = D 2s J 2 + K 1+K 2 J 2 θ 2 + K 1 J 2 θ 1 + K 2 J 2 θ 2 s 2 θ 3 = D 3s J 3 + K 2+K 3 J 3 θ 3 + K 2 J 3 θ 2 Defining the states to be the angles and the derivatives, in matrix form this is JSG 2 October 5, 216

3 Gears NDSU Rotational Systems, Gears, and DC Servo Motors ECE 461 sθ 1 sθ 2 sθ 3... s 2 θ 1 s 2 θ 2 s 2 θ 3 = D. 1 K 1 K 1 K 1 K 1 K 2 K 2. D 2 K 2 K 2 K 3. D 3 J 3 J 2 θ 1 θ 2 θ 3... sθ 1 sθ 2 sθ T Gears are like transformers: They convert one speed to another as the gear ratio, They convert one impedance to another as the gear ratio squared. The governing equation for a gear is that the distance traveled at the interface must be the same for both gears: distance = rθ d 1 = r 1 θ 1 d 2 = r 2 θ 2 d 1 = d 2 or r 1 θ 1 = r 2 θ 2 θ 1 = r 2 r 1 θ 2 JSG 3 October 5, 216

4 Gear 2 Gear 1 F2 r2 r1 Q2 F1 Two gears, where gear 1 turns gear 2 The torques are also related by the turn ration. Torque is: T 1 = F 1 r 1 T 2 = F 2 r 2 At the tip, each force has an equal and opposite force: resulting in F 1 = F 2 T 1 r 1 = T 2 r 2 T 1 = r 1 r 2 T 2 Admittance is the ratio T 1 = Ms 2 θ 1 Relative to the other side of the gear: or r 1 r 2 T 2 = (Ms 2 ) r 2 r 1 θ 2 T 2 = r 2 r 1 2(Ms 2 )θ 2 Impedances go through a gear as the turn ratio squared JSG 4 October 5, 216

5 Example: T K1 1 : 5 = 5 Q2 Q3 K3 J1 J3 D1 D3 Q2 J2 K2 D2 1 : 5 Q2 = 5 Q3 i) Draw the circuit equivalent with the gears K1 K2 Q2 1 : 5 1 : 5 A3 T J1 s 2 D1 s J2 s 2 J3 s 2 D2 s D3 s K3 Circuit Equivalent for Rotational Gear System ii) Take out the gears by scaling the impedance by the turn ratio squared Y new where going to where coming from 2 Yold K1 Q2'.4 K2 Q3'.4 J2 s 2.16 J3 s 2.16 D3 s T J1 s 2 D1 s.4 D2 s.16 K3 x 1/25 x 1/625 JSG 5 October 5, 216

6 iii) Write the equations of motion (i.e. write the voltage node equations, noting that all terms are impedance). (note: is used rather than as a reminder that the node voltages are scaled by the gear ratio.) φ θ ( s 2 + D 1 s + K 1)φ 1 (K 1 )φ 2 = T J s 2 + D s + K 1 + K φ 2 (K 1 )φ 1 K φ 3 = J s 2 + D s + K 1 + K φ 3 K φ 2 = From this point onward, it's the same as before. DC Servo Motors A DC Servo Motor is a type of motor which is often used with rotational systems. It has the advantage that Torque is proportional to current (making it easy to apply torques to a rotational system) Spins when a constant voltage is applied (operating as a motor), or Provides a constant voltage when spun (operating as a DC generator - a.k.a. a dynamo) A DC servo motor is a 2-lead motor which spins when a constant voltage is applied (motor) or produces a constant voltage when spun (generator) (image from Internally, a DC motor has A permanent magnet (field winding) on the outside, An electromagnet on the inside (armature), and A commutator which switches the polarity of the current (voltage) so that the torque remains in the same direction as the motor spins. A DC servo motor is essentially an electromagnet, with the following circuit equivalent: JSG 6 October 5, 216

7 Ra Las Ia Q Va Kt*dQ/dt + - KtIa Js 2 Ds Circuit equivalent for a DC servo motor Kt is a constant for the motor. It describes how the motor works as a generator, creating a back voltage proportional to speed E a = K t dθ dt = K t ω as well as a motor where the torque is proportional to the armature current T = K t I a It isn't obvious, but Kt is the same for both equations. To see this, note that power out = power in P in = P out E a I a = Tω Substituting for Ea and T: (K t1 ω)i a = (K t2 I a)ω K t1 = K t2 Here you can see that the two constants are the same. The equations for this are then or V a = K t(sq) + (R a + L a s)i a K t I a = (Js 2 + Ds)Q Solving gives Q = 1 s K t 2 (Js+D)(L as+r a)+k t if the output is angle (Q) or ω= dq dt = if the output is speed. K t 2 (Js+D)(L as+r a)+k t V a V a JSG 7 October 5, 216

8 Example: Determine the transfer function for a DC servo motor with the following specifications: ( K t =.3Nm/A Terminal resistance = 1.6 (Ra) Armature inductance = 4.1mH (La) Rotor Inertia =.8 oz-in/sec/sec ugh - english units Damping Contant =.25 oz-in/krpm double ugh Torque Constant = 13.7 oz-in/amp Peak current = 34 amps Max operating speed = 6 rpm You need to translate to metric: Kt: 13.7 oz in amp 1m 39.4in D:.25 oz in min krev J:.8 oz in sec 2 1lb 16oz.454kg lb 9.8N kg =.967 N m A 1Nm 141.7oz in Nm 141.7oz in 6 sec min krev 1rev kg 9.8N = kg m rev 2πrad = Nm s 2 rad/sec Plugging in numbers: Q = s(s s+63 2 ) V = 1.31(63) 2 s(s j598)(s j598) V This means... If you apply a constant +12V to the motor, the motor spins at a speed of rad/sec. (1.31*12) It takes about 2ms for the motor to get up to speed (Ts = 4/196) The motor will overshoot it's final speed (123 rad/sec) by 35% (damping ratio =.3119) JSG 8 October 5, 216

9 Gears and Wheels: It isn't obvious, but a wheel is a type of gear or transformer. The circuit symbol for a gear 5 : 1 Q2 means 5 θ 1 = 1 θ 2 A wheel converts rotational motion to translational motion as where x = rθ x is the displacement in meters r is the radius of the wheel and θ is the angle in radians. Example: Find the dynamics of the previous motor if it is driving a cart with a mass of 2kg through a wheel with a diameter of 3cm (.3m) X 2kg Q First, draw the circuit equivalent La s Ra Ia Q r : 1 x Va + - Kt W + - Kt Ia Js2 Bs 2kg Remove the gear by translating the 2kg mass to the motor angle x = rθ 2kg r 2 1 =.18 kg m 2 The net inertia is then JSG 9 October 5, 216

10 J motor + J mass = = The motor dynamics are then θ= 1361 s(s+3.273)(s+387) V a The motor dynamics have a complex pole - which might seem odd. Motors are intended to be used to drive something, however. If you drive a cart with a mass of 2kg through wheels with a radius of 3cm, the complex poles shift to { , -387 }. The complex poles will shift (and probably become real) when connected to a load. JSG 1 October 5, 216

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