Dynamic Response of Mix Systems

Abstract

The dynamic models are complex and highly nonlinear; thus, finding an automatic machine's time response requires numerical solutions. There are several forms of finding the time response of complex systems, in particular when combining different components. This chapter presents the dynamic response of automatic machines widely used in industrial applications. The dynamic equations were derived in previous chapters, and the solution is found using Simulink™. Examples include power supply mechanisms by electric motors, gear transmissions, belt transmissions, ball screw systems, and pneumatic applications. It also includes special codes for simulating the nonlinear terms of the gear mesh stiffness, ball bearing stiffness, and the bearing.

Appendix

6.1.1 Simulation Parameters

The simulation parameters for the system composed by a DC motor, a nonlinear transmission and elastic band are:

% %   Parameters Electric Motor (Series) % V=240;% Volts Rr=31; % Rotor's resistance Ohms Rs=57; % Stator's resistance Ohms (857 for a parallel motor) Lr=0.00001; % Rotor's inductance H Ls=0.00001; % Stator's Inductance H Ke=0.8;% Voltage constant Kt=0.8;% Torque constant C0m=0.01;% Friction coefficient motor Jm=2.188e-4;% Rotor's inertia Kg-m^2 % %   Parameters Gearbox % Np=27; Ng=41; relcon=3.7; rp=0.072; % Pinion Radius (m) rg=rp*41/27; % Gear Radius (m) Jp=0.0059616; % Moment of Inertia, pinion (Kg-m^2) Jg=0.007523; % Moment of Inertia, gear (Kg-m^2 mp=2.3; % Pinion mass (Kg) mg=3.6; % Gear mass (Kg) KT=193917.5; % Teeth stifnness (N/m) Kp=1e6; % Pinion support stiffness Kg=1e6; % Gear support stiffness Fe=0.0045/1000; % Manufacturing errors ex=0.001/1000; % Unbalance T=12; % Output torque (N-m) % %   Bearing parameters % Nb1=27; % Total number of roller elements alpha= 19; % Contact angle alpha= acos(alpha*pi/180); Nbol1=3; % Number of rollers under contact d1=5/1000; % Roller diamater D1=27/1000; % Pitch diameter Nbol2=3; d2=13/1000; D2=35/1000; Fb1=0.01/1000; % Defect % %   Parameters Ball Screw % C0t=0.0001;% Firction coefficient screw-nut psc=5/1000; % Screw's pitch [m] dt=25/1000; % Screw diameter [m] Lt=40/1000; % Screw length [m] Ktx=0.6*pi*210000e6*dt*Lt/4; % Screw stiffness eq. 5.38 and 5.39 mt=7850*pi*dt^2*Lt; % Screw mass [kg] Jt=mt*(dt/2)^2/2; %Screw and nut inertia Kg-m^2 % %   Elastic band % a2=20;% Coefficient for the EA modulus (manufacturing reference) rpol=91.7/1000; % Pulley pitch diameter (mm) part 112864 Dodge Catalog page PT11-5 mbel=0.35/2.2; %Belt mass [kg] cbel=0.02; % Estimated damping coefficient wbel=20/1000; % Belt's with tbel=2.75/1000; % Belt's thickness Lbel=40/1000; %Belt's span

The simulation parameters for the pneumatic systems are:

%    %   Piston parametres % diam=16/1000; % [m] drod=0/1000; % [m] Le=30/1000; % Equivalent length for mass estimation[m] L=42/1000; % piston stroke [m] A1=3.141592654*diam^2/4; % Nominal area A2=A1-3.141592654*drod^2/4; % Rod area A1L=1e-6; % A1/(2*L); A2L=1e-6; %A2/(2*L); V01=A1*L/100; V02=A2*L/100; % %   Dynamic parameters % mp=7850*((3.141592654*diam^2/4)*Le); % [kg] bp=-0.0028*(1000*diam)^2+1.0114*1000*diam-3.0261; % [kg-s/m] %bp=100*bp; FS=-0.0042*(1000*diam)^2+1.1442*1000*diam-8.5498; % Static coefficient [N] FD=-0.0019*(1000*diam)^2+0.5535*1000*diam-4.1324; % Dynamic coefficient [N] FNL=FD; % Nonlinear force % %   Gas properties % k=1.4; R=232; T=293; % Grados Kelvin % %   Parámetros de la válvula % KV=sqrt(2*k/(R*T*(k-1)))*0.259; AOi1=(1*1.2e-3)/(1000*60); AOi2=(1*1.2e-3)/(1000*60); AOo1=(1*1.2e-3)/(1000*60); AOo2=(1*1.2e-3)/(1000*60); % %   Coefficients and dimensions % bv=7; % Viscous coefficient [Ns/m] Kv=1000000/2.7; % Spring constant dv=0.00775; % Orifice diameter mv=0.1; % Mass [kg] y0=0.000001; % Spring preloading % %   Electric parameters % Lv0=0.001; % Inductance [H] Rv=2; % Resistance [Ohm] galf=0.05/1000; % Slinding gap FNLv=FD; % Componente no lineal % %   Operating pressure % Patm=101325; % Pas PLin=1.0342e7; % Pas=1500 psi % %   External load % Fext=300; % [N] % %   Pick and Place Cam's Dimensions % Rcam=150/1000; % [m] mgp=3;% Gripper's mass [kg] re=L/pi; % Gear radius [m] Fmu=1000; % Friction force along the sliding [N] Lcam=100/1000;% [m]