DC Motor with Backlash

1 Standard Modelica Library Components

model DCMotorCircuit Modelica.Mechanics.Rotational.Components.Inertia inertia1(J = 1.0, w(fixed = true), phi(fixed = true)); Modelica.Electrical.Analog.Basic.Inductor inductor1(L = 1.0, i(fixed = true)); Modelica.Electrical.Analog.Basic.Resistor resistor1(R = 1.0); Modelica.Electrical.Analog.Basic.Ground ground1; Modelica.Electrical.Analog.Basic.EMF emf1(k = 1.0); Modelica.Blocks.Sources.Step step1; Modelica.Electrical.Analog.Sources.SignalVoltage signalVoltage1; equation connect(step1.y, signalVoltage1.v); connect(signalVoltage1.n, resistor1.p); connect(resistor1.n, inductor1.p); connect(signalVoltage1.p, ground1.p); connect(ground1.p, emf1.n); connect(inductor1.n, emf1.p); connect(emf1.flange, inertia1.flange_a); end DCMotorCircuit;

1.1 Simulation of DCMotorCircuit

loadModel(Modelica)
simulate( DCMotorCircuit, stopTime=10 )
plot( inertia1.w )

2 Modelica.Mechanics.Rotational.ElastoBacklash

2.1 Description

This element consists of a backlash element connected in series to a spring and damper element which are connected in parallel. The spring constant shall be non-zero, otherwise the component cannot be used.
In combination with components IdealGear, the ElastoBacklash model can be used to model a gear box with backlash, elasticity and damping.
The parameters are:

Name Default Description

b 0 Total backlash [rad]

c 1.e5S pring constant (c > 0 required) [N.m/rad]

phi_rel0 0 Unstretched spring angle [rad]

d 0 Damping constant [N.m.s/rad]

2.2 Source Code

model DCMotorCircuitBacklash Modelica.Mechanics.Rotational.Components.Inertia inertia1(J=1), inertia2(J=6, w(fixed = true), phi(fixed = true)); Modelica.Electrical.Analog.Basic.Inductor inductor1(L = 1.0, i(fixed = true)); Modelica.Electrical.Analog.Basic.Resistor resistor1(R = 1.0); Modelica.Electrical.Analog.Basic.Ground ground1; Modelica.Electrical.Analog.Basic.EMF emf1(k = 1.0); Modelica.Blocks.Sources.Step step1; Modelica.Electrical.Analog.Sources.SignalVoltage signalVoltage1; Modelica.Mechanics.Rotational.Components.ElastoBacklash elastoBacklash( b=3, c=10, d=0.8, w_rel(fixed = true), phi_rel(fixed = true)); equation connect(step1.y, signalVoltage1.v); connect(signalVoltage1.n, resistor1.p); connect(resistor1.n, inductor1.p); connect(signalVoltage1.p, ground1.p); connect(ground1.p, emf1.n); connect(inductor1.n, emf1.p); //connect(emf1.flange_b, inertia1.flange_a); connect(emf1.flange,inertia1.flange_a); connect(inertia1.flange_b, elastoBacklash.flange_a); connect(elastoBacklash.flange_b,inertia2.flange_a); end DCMotorCircuitBacklash;

2.3 Simulation of DCMotorCircuitBackClash

loadModel(Modelica)
simulate( DCMotorCircuitBacklash, stopTime=25 )
plot(elastoBacklash.w_rel)

plot( elastoBacklash.phi_rel )

plot( { inertia1.w, inertia2.w } )

For comparison we define a model with less mass in inertia2 (J=1), no damping d, and weaker string

constant c, to show more even more dramatic backlash phenomena

model DCMotorCircuitBacklash2 = DCMotorCircuitBacklash( inertia1(J=1), inertia2(J=1), elastoBacklash( b=1, c=1e5, d=0 ) );
simulate( DCMotorCircuitBacklash2, stopTime=25 )
plot( { inertia1.w, inertia2.w } )

2.4 SimpleElastoBacklash

Most properties of the elastic transmission with backlash, both continuous and discrete, are represented in the partial model SimpleElastoBacklash below. This is a typical hybrid mode switching model, with the finite state automaton represented by the equations for the Boolean variables backward, forward and slack and the conditional equation for the torque tau having different right hand side expressions in the three operationg modes.

partial model SimpleElastoBacklash Boolean backward, slack, forward; // Mode variables parameter Real b "Size of backlash region"; parameter Real c = 1.e5 "Spring constant (c>0), N.m/rad"; Flange_a flange_a "(left) driving flange - connector"; Flange_b flange_b "(right) driven flange - connector"; parameter Real phi_rel0 = 0 "Angle when spring exerts no torque"; Real phi_rel "Relative rotation angle betw. flanges"; Real phi_dev "Angle deviation from zero-torque pos"; Real tau "Torque between flanges"; equation phi_rel = flange_b.phi - flange_a.phi; phi_dev = phi_rel - phi_rel0; backward = phi_rel < -b/2; // Backward angle gives torque tau<0 forward = phi_rel > b/2; // Forward angle gives torque tau>0 slack = not (backward or forward); // Slack angle gives no torque tau = if forward then // Forward angle gives c*(phi_dev - b/2) // positive driving torque else (if backward then // Backward angle gives c*(phi_dev + b/2) // negative braking torque else // Slack gives 0); // zero torque end SimpleElastoBacklash;

The ElastoBacklash model consists of a backlash element connected in series to a spring and damper element, which are connected in parallel (Figure 13-90). This is a slight extension compared to the previous SimpleElastoBacklash model since we have introduced a damping element with a damping constant d. The damping element contributes to the torque tau equation by the expression d*w_rel that is nonzero provided d is nonzero. The spring constant must be nonzero, otherwise the model cannot be used. Another extension compared to the simple model is that a constant bEps for minimum backlash has been introduced. The backlash behavior of the model is available only if abs(b) > bEps, since if b is too small or zero, numeric problems with too many generated state events will occur. The following ElastoBacklash model is from the Modelica standard library version MSL3.2.1, Mechanics.Rotational.Components package, but slightly reformatted to increase readability.

model ElastoBacklash "Backlash connected in series to linear spring and damper (backlash is modeled with elasticity)" parameter SI.RotationalSpringConstant c(final min=Modelica.Constants. small, start=1.0e5) "Spring constant (c > 0 required)"; parameter SI.RotationalDampingConstant d(final min=0, start=0) "Damping constant"; parameter SI.Angle b(final min=0) = 0 "Total backlash"; parameter SI.Angle phi_rel0=0 "Unstretched spring angle"; extends Modelica.Mechanics.Rotational. Interfaces.PartialCompliantWithRelativeStates; extends Modelica.Thermal.HeatTransfer.Interfaces. PartialElementaryConditionalHeatPortWithoutT; protected final parameter SI.Angle bMax=b/2 "Backlash in range bMin <= phi_rel - phi_rel0 <= bMax"; final parameter SI.Angle bMin=-bMax "Backlash in range bMin <= phi_rel - phi_rel0 <= bMax"; SI.Torque tau_c; SI.Torque tau_d; SI.Angle phi_diff = phi_rel - phi_rel0; // A minimum backlash bEps is defined in order to avoid an infinite // number of state events if backlash b is set to zero. constant SI.Angle bEps=1e-10 "Minimum backlash"; equation if initial() then tau_c = if phi_diff > 1.5*bMax then c*(phi_diff - bMax) else if phi_diff < 1.5*bMin then c*(phi_diff - bMin) else (c/3)*phi_diff; tau_d = d*w_rel; tau = tau_c + tau_d; lossPower = tau_d*w_rel; else if abs(b) <= bEps then // Less than minimum backlash to avoid tau_c = c*phi_diff; // infinite number of state events tau_d = d*w_rel; tau = tau_c + tau_d; elseif phi_diff > bMax then // Forward angle gives positive torque tau_c = c*(phi_diff - bMax); tau_d = d*w_rel; tau = smooth(0, noEvent(if tau_c + tau_d <= 0 then 0 else tau_c + min(tau_c,tau_d))); elseif phi_diff < bMin then // Backward angle gives negative torque tau_c = c*(phi_diff - bMin); tau_d = d*w_rel; tau = smooth(0, noEvent(if tau_c + tau_d >= 0 then 0 else tau_c + max(tau_c,tau_d))); else tau_c = 0; // Slack gives zero torque tau_d = 0; tau = 0; end if; end if; end ElastoBacklash;