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function [yp] = reduced2(t,y,T_a) beta = 13*(pi/180); %Helix angle (rad) speed = 1000/60; % Speed in Revs/sec Freq = 1000*20/60; %Speed divided by gear ratio % Common parameters theta_a_vec = -speed*2*pi*t; %torsional angle of driver gear (rad) e = 0; %circumferential relative displacement of the teeth (m) z = e*tan(beta); %axial tooth displacement caused by internal excitation (m) e_a = 48e-6; %circumferential displacement of the driver gear (m) e_p = 48e-6; %circumferential displacement of the driver gear (m) ks = 0.9e3 + 20*sin(2*pi*Freq*t); %Shear stiffness Cs = 0.1 + 0.001*sin(2*pi*Freq*t); %Shear damping % Driver gear m_a = 0.5; %mass of driver gear (kg) c_ay=500; %Damping of driver gear in y direction (Ns/m) c_az=500; %Damping of driver gear in z direction (Ns/m) k_ay= 8e7; %Stiffness of driver gear in y direction (N/m) k_az= 5e7; %Stiffness of driver gear in z direction (N/m) R_a = 51.19e-3; %Radius I_a = 0.5*m_a*(R_a^2); %Moment of inertia of driver gear (Kg.m3) y_ac= e_a + theta_a_vec*R_a; %circumferential displacement at the meshing point on the driver gear (m) y_pc = e_p - theta_a_vec*R_a; %circumferential displacement at the meshing point on the driven gear (m) z_a = e_a*tan(beta); z_p = e_p*tan(beta); z_ac = (z_a-y_ac)*tan(beta); %axial displacements of the meshing point on the driving gear (m) z_pc = (z_p+y_pc)*tan(beta); %axial displacements of the meshing point on the driven gear (m) yp = zeros(12,1); %vector of 12 equations % Excitation forces Fy=ks*cos(beta)*(y_ac-y_pc-e) + Cs*cos(beta)*2*R_a*speed*2*pi; %axial dynamic excitation force at the meshing point (N) Fz=ks*sin(beta)*(z_ac-z_pc-z) - Cs*sin(beta)*2*tan(beta)*R_a*speed*2*pi; %circumferential dynamic excitation force at the meshing point (N) %Time interpolation - Needed for solution of equations (It basically uses %your torque and prescribed time matrices to generate a time matrix to be %used in the ODE solver) time = 0:0.00001:0.06; Torque = interp1(time,T_a,t); Opp_Torque = 68.853; % Average torque value %Driver gear equations yp(1) = y(2); yp(2) = (-Fy - k_ay*y(1) - c_ay*y(2))/m_a; %acceleration in y (up and down) direction (m/s2) yp(3) = y(4); yp(4) = (-Fz - k_az*y(3) - c_az*y(4))/m_a; %acceleration in z (side to side) direction (m/s2) yp(5) = y(6); yp(6) = (Torque - Fy*R_a + Opp_Torque)/I_a; % angular acceleration theta_a double dot (rad/s2) % Driven gear m_p = 0.5; %mass of driven gear (kg) c_py=500; %Damping of driven gear in y direction (Ns/m) c_pz=500; %Damping of driven gear in z direction (Ns/m) k_py= 9.5e7; %Stiffness of driven gear in y direction (N/m) k_pz =9e7; %Stiffness of driven gear in z direction (N/m) R_p = 139.763e-3; %Radius I_p = 0.5*m_p*(R_p^2); %Moment of inertia of driven gear (Kg.m3) n_a = 20; % no of driver gear teeth n_p = 60; % no of driven gear teeth i = n_p/n_a; % gear ratio %Driven gear equations yp(7) = y(8); yp(8) = (-Fy - k_py*y(7) - c_py*y(8))/m_p; %acceleration in y (up and down) direction (m/s2) yp(9) = y(10); yp(10) = (-Fz - k_pz*y(9) - c_pz*y(10))/m_p; %acceleration in y (side to side) direction (m/s2) yp(11) = y(12); yp(12) = (-Torque*i - Fy*R_p + Opp_Torque*i)/I_p; % angular accelration theta_p double dot (rad/s2) end
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