Spur or Helical Gear


Symbol:
Identifier: PowerTransmission.Transmissions.Gear
Version: 3.9
File: PowerTransmission.cat
Connectors: Connectors for Gear 1:
  Mechanical Connector (linear) ctrT1x radial (tangential) support gear 1, x-component;
connectable to elements of the Linear Mechanics library
  Mechanical Connector (linear) ctrT1y radial support gear 1, y-component;
connectable to elements of the Linear Mechanics library
  Mechanical Connector (linear) ctrT1z axial support gear 1, z-component;
connectable to elements of the Linear Mechanics library
  Mechanical Connector (rotational) ctrR1 element for gear 1 can be connected here (rotation around z);
connectable to elements of the Rotational Mechanics library
  Connectors for Gear 2:
  Mechanical Connector (linear) ctrT2x radial support gear 2, x-component;
connectable to elements of the Linear Mechanics library
  Mechanical Connector (linear) ctrT2y radial support gear 2, y-component;
connectable to elements of the Linear Mechanics library
  Mechanical Connector (linear) ctrT2z axial support gear 2, z-component;
connectable to elements of the Linear Mechanics library
  Mechanical Connector (rotational) ctrR2 element for gear 2 can be connected here (rotation around z);
connectable to elements of the Rotational Mechanics library
Parameters and Results: see Table of Parameters and Result Quantitiesbelow
  • Elastischer Zahneingriff für schräg- oder geradverzahnten Stirnradgetriebestufen
  • Zahnsteifigkeit nach DIN 3990: eingriffsabhängige Steifigkeitsschwankung, Drehflankenspiel und Dämpfung
  • Alternativ starre Modellierung des Zahneingriffs und Berücksichtigung des Spiels nach den Stoßgesetzen
  • Optional Dämpfung im Verzahnungsspiel
  • Berücksichtigung von Verlusten im Zahnkontakt und in den Radlagern (Wirkungsgrade etc.)
  • Elastic gearing for helical or spur gear stages
  • Tooth stiffness according to DIN 3990: contact-dependent stiffness fluctuation, backlash and damping
  • Alternative rigid of tooth contact model and consideration of backlash according to impact laws
  • Optional damping in gear backlash
  • Consideration of losses in tooth contact and gear bearings (efficiency etc.)

Description:

The model Spur or Helical Gear stage can be used for the conversion of rotational speed and torque during the transmission process. It models the meshing of the gears, taking the # tooth stiffness and transverse backlash into account. It is suitable for modeling spur and helical gears. If required, the gear contact can be modeled as rigid.

The radial and axial support behavior can be modeled externally and thus can be rigid or elastic. Force, spring, and pre-load models are possible.

Figure 1: Coordinate systems of the Gear stage model

The gear element transforms displacements, velocities, accelerations, and forces (torques) between the red connector coordinate systems and the green inner coordinate system at the tooth contact (indicated by I). The tooth contact model is described in the chapter Tooth Contact.

Furthermore, the element is able to represent losses. The losses can be considered in the tooth contact and/or in the gear supports (at the rotational connectors).

Parameters:

Geometry

It is possible to use only constructional data for the parameterization of the gear (see Figure 1). These are in particular:

  • the Normal Modulus,
  • the Tooth Width ,
  • the Number of Teeth z1 and z2 for the two gears,
  • the Helix Angle (measured with respect to gear 1, see Figure 1),
  • the Normal Pressure Angle ,
  • and the Transverse Backlashjt.

All the mechanical parameters (stiffness, damping, overlap) can be calculated internally, if nothing else is specified. In addition each parameter (if known) can be altered manually in further parameter groups in order to precisely adjust the model to the situation to be simulated.

Tooth Contact

The tooth contact can be modeled as rigid (see below) or elastic. If no values are given, SimulationX computes the tooth stiffness by an internal approach. Otherwise you can choose between an elastic or a rigid tooth contact model. Further information for parameterization, calculation and results of the tooth contact can be found in chapter Tooth Contact.

The principal model of the gear stage is defined by the Rigid checkbox. It can behave as

  • Spring-Damper-Backlash: The Rigid checkbox is deselected. The behavior is defined by the given or computed stiffness, damping, and backlash parameters. Within the backlash the stiffness and damping are always zero.
  • Rigid End Stop: If the Rigid checkbox is selected, the model behaves as rigid. I.e., it consists of an ideal transformation with rigid connection between all connectors (kindR=="without Backlash") or an ideal transformation combined with a rigid end stop. The kind of impact can be specified by the user via the Tooth Contact parameter - in the same way as for the rigid end stop in the Linear Mechanics library. If a backlash is present, it is set via the Rotational Backlash parameter.
  • If the checkbox Rigid is selected and if the Contact enumeration kindR is switched to "without Backlash", the gear stage as a transformer (transmission). In this case, forces will only be transferred via the right tooth flanks, no matter in which direction the power is transmitted. So, the right tooth flanks transfer tensile and compressive forces (tensile forces have a negative sign). In this case, the result variables for the forces of the left tooth flanks Fbnl and Ftl are deactivated.
  • If Consideration of Stiffness Change is selected (only possible if Rigid is deselected) the element computes a variable toothing stiffness by an internal approach (see chapter Tooth Contact).

Stiffness of Toothing / Damping of Toothing

  • If Rigid is deselected, the properties dialog page Free Definition 1 provides parameters to define the elastic tooth contact model. Read more about the parameters and the calculation in the chapter Tooth Contact.

Modified Profile / Contact Ratio

This group provides further geometry parameters of the gear. If Consideration of Modified Profile is selected, it is possible to enter

  • profile offset factors x1 and x2 for every gear. Based on this the effective radii for the force and motion quantity transformation between Tooth Contact and connector coordinate systems will be computed.
  • the center distance a of the gears. Together with the tooth numbers, the effective transformation radii can be computed.
  • Furthermore, the Input of an Addendum Modification is possible. The Addendum Modification kmn in the element Gear is an absolute value. If only an relative value kmn,rel is given, the absolute value can be computed using the normal modulus mn by:
  • Finally the Total Contact Ratio can be specified. By default it is computed internally,
    but it is possible to enter a constant value for epsilon (, see chapter Tooth Contact) in order to consider different kinds of modified profiles. The total contact ratio is used for the computation of the total stiffness by the specific stiffness of one pair of teeth (see also chapter Tooth Contact). Thus, this parameter is redundant if Summated Meshing Spring Stiffness is selected for kindS on page Free Definition 1. If you decide to enter a value for the total contact ratio, the addendum modification is obsolete, because this modification is always considered within the total contact ratio value.

Losses (tabs Losses 1 and Losses 2)

There are three locations for the consideration of losses:

  • the tooth contact,
  • the gear bearing at gear 1 (ctrR1) and ...
  • the gear bearing at gear 2 (ctrR2).

Each loss location can be parametrized separately. All losses can be considered separately or in combination.
If reliable data is only available for the whole gear stage, e.g., the overall efficiency, only the parameters in the box Toothing / Gear Stage on the dialog page Losses 1 have to be entered and will determine the overall efficiency.
A detailed description of the loss models in gear stage elements can be found in the chapter Losses in Gear Stages.

Results:

  • The gear model provides the normal forces, the tangential, radial, and axial forces, deformations and velocity differences of the tooth contact for the left and the right flank (see chapter Tooth Contact).
    The tangential, radial, and axial forces are acting as pressure forces w.r.t. the green coordinate system in Figure 1 on gear 2. So, a negative radial force will move the gears apart.
  • Furthermore the Total Meshing Spring Stiffness is computed during simulation and can be observed in the result variable kbt.
  • If a spring-damper-backlash model (non-rigid model) of the tooth contact is selected, the Change of Potential Energy (the power flow into the spring in the contact) and the Power Loss (power dissipated in the damper) can be examined too.
  • If there is a change of the contact surface, there is also a change in the place and direction where the spring stiffness, damping, and the normal force act.
  • If Rigid is selected and kindR=="without Backlash", the tooth contact forces Fbnl and Ftl are inactive (see section Tooth Contact in this chapter).
  • The transmission error TE is computed by: TE = |r1*phi1| - |r2*phi2| using the operation radii of both gears r1 and r2

Periodic Steady-State Simulation:

The relative damping and Lehr's damping factor D as well as the spectral powers of the left and the right flank PSpecl and PSpecr, resp., are only taken into account in the Periodic Steady-State Simulation. These parameters are described in some detail in sections Periodic Steady-State Simulation (Parameters) and Periodic Steady-State Simulation (Results) of the chapter Tooth Contact.

Remarks:

The gear element only calculates the deformation forces in the tooth engagement. The inertia properties, i.e. the moments of rotational inertia in the gears, have to be assigned to externally connected objects.

The linear (translational) connectors ctrT1x, ctrT1y, ctrT1z, ctrT2x, ctrT2y, and ctrT2z can be used to model support properties, as well as for example a preload, which are applied to the gear stage. Some examples on how to model such situations are given in the subsection Linear connections of the Gear.

Table of Parameters and Result Quantities:

Parameters: Geometry:
  Normal Modulus mn  
  Tooth Width bw  
  Helix Angle beta  
  Transverse Backlash jt not visible for Rigid model and kindR=="Without Backlash"
  No. of Teeth, Gear 1 z1  
  No. of Teeth, Gear 2 z2  
  Normal Pressure Angle alphan  
  Tooth Contact:
  Rigid rigid Selects rigid (end stop) behavior instead of elastic behavior in
tooth contact
  Tooth Contact kindR only available, if Rigid is selected;
Possible settings: "without Backlash", "Rigid Plastic", "Rigid Elastic", and "Coefficient of Restitution"
  Coefficient of Restitution ci only available, if kindR== "Coefficient of Restitution"
  Consideration of Stiffness Change StiffChange internal approach for a variable toothing stiffness; not visible, if Rigid or Preset of Stiffness is selected
  Free Definition of Elastic Toothing (Free Definition):
  Preset of Stiffness (specific, related to common face width) stiff not available, if Rigid is selected;
switches to manual stiffness definition
  Preset of Stiffness as kindS only available if Preset of Stiffness is selected;
Possible settings: "Function over Normalized meshing length Ln", "Characteristics over Normalized meshing length Ln", "Summated meshing spring stiffness"
  Function f(Ln) kf only available, if kindS== "Function over Normalized meshing length Ln"
  Characteristics kc Curve Parameter;
only available, if kindS=="Characteristics over Normalized meshing length Ln"
  as kindK only available, if kindS=="Summated meshing spring stiffness";
Possible settings: "Constant Mean Value", "Function of the angle
of gear 1"
, "Function of the angle of gear 2"
  Summated Stiffness ksm only available, if kindK=="Constant Mean Value"
  Summated Stiffness ksc only available, if kindK=="Function of the angle of gear 1" or "Function of the angle of gear 2"
  Damping of Toothing (Free Definition 1):
  Preset of Damping damp not available, if Rigid is selected
Switches to manual damping definition
  as kindB only available if Preset of Damping is selected;
Possible settings: "Total Damping",
"Factor B for Relation to Tooth Stiffness b=B*sqrt(k)"
  Damping b only available, if kindB=="Total Damping"
  Damping Factor B only available, if kindB=="Factor B for Relation to Tooth Stiffness b=B*sqrt(k)"
  Damping in Toothing Backlash bL only available if Preset of Damping is selected
  Input of Stiffness/Damping kindD Interpretation of entered stiffness and damping values "in Meshing Direction" or "in Circumference Direction"
  Modified Profile / Contact Ratio:    
  Consideration of Modified Profiles modProf  
  Input of kindM Possible settings: "Profile Offsets" and "Center Distance";
only available if modProf=="true"
  Profile Offset Factor (Gear 1) x1 only available if kindM=="Profile Offsets"
  Profile Offset Factor (Gear 2) x2 only available if kindM=="Profile Offsets"
  Center Distance a only available if kindM==Center Distance"
  Input of Addendum Modification undercut not available if kindC=="Input"
  Addendum Modification kmn only available if undercut=="true"
  Total Contact Ratio kindC not available, if Rigid is selected;
Possible settings: "Internal Calculation" and "Input"
  Total Contact Ratio (input) epsilon only available, if kindC=="Input"
  Losses in Toothing / Gear Stage (page Losses 1 & Losses 2):    
  Description kindL Description of the friction model used in the tooth contact or for the complete gear stage.
  Efficiency eta Please pay attention to direction of the value, see also chapter Losses in Gear Stages;
It is possible to use this value for entering a degree of efficiency of the whole gear stage (incl. bearing losses);
only available if kindL=="Constant Efficiency"
  Direction of eta from Gear 2 to Gear 1 dir Measuring direction of eta;
only available if kindL=="Constant Efficiency"
  Static Friction Coefficient mu0 only available if kindL=="Friction Coefficients" or kindL=="Stribeck Curve"
  Sliding Friction Coefficient mu only available if kindL=="Friction Coefficients" or kindL=="Stribeck Curve"
  Limit Velocity vL only available if kindL=="Stribeck Curve"
  Consideration of Velocity Dependency switchV only available if kindL=="Stribeck Curve"
  Coefficient Kv only available if kindL=="Stribeck Curve"
  Exponent Av only available if kindL=="Stribeck Curve"
  Force of Static Friction Fst only available if kindL=="Friction Force Characteristic" or kindL=="Friction Forces"
  Force of Sliding Friction Fsl only available if kindL=="Friction Forces"
  Force of Sliding Friction Curve FslCurve only available if kindL=="Friction Force Characteristic"
  Losses in Gear Bearing 1    
  Description kindLB1 Description of the friction model used at the bearing of gear 1.
  Efficiency etaB1 Please pay attention to direction of the value, see also chapter Losses in Gear Stages;
only available if kindLB1=="Constant Efficiency"
  Direction of etaB1 from Gear 2 to Gear 1 dirB1 Measuring direction of etaB1;
only available if kindLB1=="Constant Efficiency"
  Static Friction Coefficient mu0B1 only available if kindLB1=="Friction Coefficients" or kindLB1=="Stribeck Curve"
  Sliding Friction Coefficient muB1 only available if kindLB1=="Friction Coefficients" or kindLB1=="Stribeck Curve"
  Friction Radius rB1 only available if kindLB1=="Friction Coefficients";
E.g., the radius of the ball bearing, measured at in the middle of the roller bearing retainer
  Limit Angular Velocity omLB1 only available if kindLB1=="Stribeck Curve"
  Consideration of Angular Velocity Dependency switchVB1 only available if kindLB1=="Stribeck Curve"
  Coefficient KvB1 only available if kindLB1=="Stribeck Curve"
  Exponent AvB1 only available if kindLB1=="Stribeck Curve"
  Static Friction Torque TstB1 only available if kindLB1=="Friction Torque Characteristic" or kindLB1=="Friction Torques"
  Sliding Friction Torque TslB1 only available if kindLB1=="Friction Torques"
  Sliding Friction Torque Curve TslB1Curve only available if kindLB1=="Friction Torque Characteristic"
  Losses in Gear Bearing 2    
  Description kindLB2 Description of the friction model used at the bearing of gear 2.
  Efficiency etaB2 Please pay attention to direction of the value, see also chapter Losses in Gear Stages;
only available if kindLB2=="Constant Efficiency"
  Direction of etaB1 from Gear 2 to Gear 1 dirB2 Measuring direction of etaB2;
only available if kindLB2=="Constant Efficiency"
  Static Friction Coefficient mu0B2 only available if kindLB2=="Friction Coefficients" or kindLB1=="Stribeck Curve"
  Sliding Friction Coefficient muB2 only available if kindLB2=="Friction Coefficients" or kindLB1=="Stribeck Curve"
  Friction Radius rB2 only available if kindLB2=="Friction Coefficients";
E.g., the radius of the ball bearing, measured at in the middle of the roller bearing retainer
  Limit Angular Velocity omLB2 only available if kindLB2=="Stribeck Curve"
  Consideration of Angular Velocity Dependency switchVB2 only available if kindLB2=="Stribeck Curve"
  Coefficient KvB2 only available if kindLB2=="Stribeck Curve"
  Exponent AvB2 only available if kindLB2=="Stribeck Curve"
  Static Friction Torque TstB2 only available if kindLB2=="Friction Torque Characteristic" or kindLB2=="Friction Torques"
  Sliding Friction Torque TslB2 only available if kindLB2=="Friction Torques"
  Sliding Friction Torque Curve TslB2Curve only available if kindLB2=="Friction Torque Characteristic"
Results: Stiffness of Toothing    
  Total Meshing Spring Stiffness kbt not available, if Rigid is selected
  Toothing Forces    
  Normal Force, Right Flank Fbnr  
  Normal Force, Left Flank Fbnl not available, if Rigid is selected and kindR=="without Backlash"
  Tangential Force, Right Flank Ftr  
  Tangential Force, Left Flank Ftl not available, if Rigid is selected and kindR=="without Backlash"
  Radial Force Fr  
  Axial Force Fa  
  Transmission Error    
  Transmission Error TE  
  Deformations and Velocity Differences of the Toothing    
  Displacement Difference, Left dxbnl  
  Displacement Difference, Right dxbnr  
  Velocity Difference, Left dvbnl  
  Velocity Difference, Right dvbnr  
  Powers    
  Change of Potential Energy Pp not available, if Rigid is selected
Power Loss Pl not available, if Rigid is selected
  Energy Loss El only available if rigid==false and backlash is taken into account
Steady-State Simulation: Preset of Damping kindDamping Only Effective in Steady-State Simulation
  Relative Damping psi only available if kindDamping=="Relative Damping"
  Damping Factor D only available if kindDamping=="Lehr's Damping Factor"
  Spectral Power of Left Flank PSpecl  
  Spectral Power of Right Flank PSpecr