Question

Problem 2 Determine the transfer function 01(s)/M(s) for the shaft-gear mechanical system in the figure, where 1(s) and Ms) are the Laplace transforms of the angle 01(t) and of the moment m(t). Use the time-domain mathematical model of this system. Known are J1, ki, J2, c, k2, Ni and N2. N. 1000 0,m 0 000 N Problem 3 By using the transfer function 1(s)/Ms), determined in Problem 2, calculate and plot 01(t) using the step input command of MATLAB. Known are N1 = 48, N2 = 36, J1 = 0.0018 kg: m2, J2 = 0.0012 kg.m2, c = 70 Nimis, kı = 165 Nim, k2 = 100 Nim, m= 18

Answer 1

Problem 2 Let's start by drawing the FBD of the two gears: 2 ez Vm(t) MR2 Mc Mk2 1 Mki 01 »MRI In these diagrams we can see all the forces acting on the 2 gears: m(t) External moment Mk, = k101 =→ Moment on gear 1 due to spring kı Mk2 = k202 → Moment on gear 2 due to spring k2 Mc = c.2 → Moment on gear 2 due to damperc MR → Reaction moment between the gear 1 → Reaction moment between the gear 2 MR2 Apply the rotational equivalent of Newton's Second law on gear 1:

Jjë= MR, - k 01 Take Laplace transformation (Assume all initial conditions equal to 0): J15201(s) = MR, (s) – kq@(s) ... (1) Apply the rotational equivalent of Newton's Second law on gear 2: jzën = m(t) – MR2 - c), - k_02 Take Laplace transformation (Assume all initial conditions equal to 0): 12s2O2(s) = M(s) – Mr(s) – csOz(s) – k2O2(s) ... (2) Also, the relations between the angular displacements and the reactions torques on the2 gears is: MR2 N2 02 N Mp4 Substituting N1 02 -01 N. And N2 Mai Mrz N Into equation (2): J2s2O2(s) = M(s) – Mr(s) – CSO2 (s) – k2Ⓡ2(s) (N2 M(S) - k2 N Multiply both sides by 1: N2 2 N 2 12520,(s) N1 M(s) – Mg(s) - cs N2 M 0,6-) 620,68...(3) Adding the corresponding LHS and RHS terms of equations (1) and (3): Ni -M(S) (1 +))s*0,69) m -6)**,0,(9) –k, 0,6) (+4)=+*+03+266 MO - CS (s) Ni m(s) .. = N2 Multiply both sides by N2 N1 N2 :( + 12) 52 Ni + cs + N.N2 k2 + ki)0,() = M(3) N

Rearranging the terms: 1 (s) M(s) (N2 N1 N2 I1 + N212) 52 + cs + N1 N2 - kzt N2 Ni ki Problem 3 Run the following MATLAB script: % Given Data: N1=48; N2=36; Jl=0.0018; J2=0.0012; c=70; k1=165; k2=100; % System transfer function: num=[1]; den= [ (N2/N1) *J1+ (N1/N2) *J2, N1*c/N2, N1*k2/N2+N2 *k1/N1]; sys=tf(num, den); % Plot the step response: step (sys) ylabel('\theta (t) (rad)') xlabel('m(t)') Step Response * 10-3 4 3.5 3 2.5 0(t) (rad) 2 1.5 1 0.5 0 0 0.5 2.5 3 3.5 1.5 2 m(t) (seconds)