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#  />   Find the transfer function between the active cart position (IPO2) as X1 and the input voltage Va Attachment 1 Attachment 2 Attachment 3 Attachment 4 Attachment 5 ATTACHMENT PREVIEW Download attachment d1.png IV. Mathematical Modeling Please complete the modeling of the dynamics. Ra La W Ktla Gearbox ia (t ) + v(t) vo (t ) = Kbom Jm Pinion Bm Gear DC Motor N Rack Figure 11: DC motor schematic diagram relating applied voltage to position of the motor pinion gear, 0g i. DC Motor Current By using Kirchhoff's Voltage Law on the circuit diagram in Figure 11, Raia(t)+ La + VBEMF(t) = VA(t) (1) ATTACHMENT PREVIEW Download attachment d2.png Here, the conslants RE and Lu represent the armature resistance and inductance+ and the variables iﬂﬂ]. and 15.13.} represent the motor current draw and the applied voltage respectively. The voltage fed hack into the circuit+ as a result of the motion of wires in a magnetic ﬁeld is called the hack electromotive force, or wag-&quot;F. and can be determined from the equation+ Vssusﬂ] = &quot;Kali; {2} where N is the gear ratio of the planetary gearbox. K3 is the motor speed coefﬁcient. and {is is the rotational velocity of the motor pinion gear. ii. DC Motor Gear Angle The sum of the moments about the rotor ofthe DC motor gives the second equation, Lydia + 3...“, = r... + r; {3} Here, I,&quot; is the rotor inertia, B,&quot; is the damping acting on the rotor+ r... and T; are the torques on the motor the to the magnetic ﬁeld and the external load applied via the interactions of the rack and motor pinion gear respectively. The equations for these torques are given by the following relations, r... = strum {431' n; = -r.: 5: {41:} where Kt is the motor torque constant provided by the manufacturer+ I; is the reaction force applied to the motor pinion gear by the rack, and 1:. is the radius of the motor pinion gear. ATTACHMENT PREVIEW Download attachment d3.png iii. IP02 Cart Position Consider the two carts with their respective free body diagrams in Figure 12. 12 X1 B2X2 K(x1 - X2) Mixi LFJ Cart By (1 1 - 12) IP02 Cart Figure 12: Linearized external and inertial forces affecting both carts The third equation can be obtained from the sum of the forces in the x-direction on the IP02 cart as, Mix, + By(x1 - x2) + Ky(x] - x2) + Bjin = fc (5) 12 ATTACHMENT PREVIEW Download attachment d4.png Here, M, is combined mass of the IP02 cart, weight and additional 'mass' due to the IP02 cabling, B, is the additional damping observed by the active cart, By is the damping caused by the linear flexible joint between the two-masses, Ky is the stiffness due to the compression spring, x, is the position of the IP02 cart in the x-direction, x2 is the position of the LFJ cart in the x-direction, fe is the external force applied to the IP02 cart at the motor pinion, and ( represents time derivatives of the respective variables. iv. LFJ Cart Position The fourth and final equation of motion can be obtained from the sum of the forces on the LFJ cart as, Maxz + By(x2 - x1) + Bzx2 + Ky(x2 - x) =0 (6) where, M2 is combined mass of the LFJ cart and weights, and B2 is the additional damping observed by the passive cart. v. Combined Transfer Function To combine these equations of motion via substitution, Eq. 3 can be rewritten to isolate fe as, JmNeg + BmNeg = Kia - fed (7a) Jmpeg - Bmy Ogt - Ktla = fc (7b) By substituting Eq. 7b into Eq. 5, and realizing that the angle of the motor pinion gear may be written as 0. = =, Eq. 5 may be rewritten as, Mix, + By(x1 - 12) + Ky(x1 - x2) + Bix = -Jmrs -Bg - Bmeg+ NKtia (8a) (Mi + / m = ) \$1 + (B, + By + Bmz 1 1 + Kyx1 - Byx2 - K,x2 = =Kia (8b) ATTACHMENT PREVIEW Download attachment d5.png Now, by assuming zero initial conditions and applying the Laplace transform, we can collapse Eq. 1, 2, 6, and 8b to form a single transfer function describing the input-output relationship between applied voltage and cart position. The resulting transfer function will be of the fifth order, and appear as follows, G(S) = X2 (s) bis+bo Va (s) ass5+as4+a3s3+a2s+ajs (9) The numerator coefficients, b, and the denominator coefficients, a;, are readily found by this formulation and dictate the open loop zeros and poles respectively. In Matlab notation, (9) is given by the following numerator, denominator coefficients: n = [106604.95, 6880865.21] d = [1.0, 14456.76, 280415.99, 7100018.02, 55372388.44, 0] 13

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