Circumferencial force | F_{u} | [N] | Lifting force | F_{H} | [N] | |

Torque | M | [Nm] | Belt length | L_{B} | [mm] | |

Power | P | [W] | Span length | L_{1}, L_{2} | [mm] | |

Mass to be moved | m | [kg] | Number of belt teeth | z_{B} | ||

Mass of linear slide | m_{L} | [kg] | Number of pulley teeth | z | ||

Mass of timing belt | m_{B} | [kg] | Number of meshing teeth | z_{e}_{} | ||

Mass of pulley | m_{Z} | [kg] | Pitch circle diameter | d_{0} | [mm] | |

Mass of tension roller | m_{S} | [kg] | Crown diameter | d_{K} | [mm] | |

reduced mass | m_{red} | [kg] | Tension roller diameter | d_{s} | [mm] | |

specific weight | ρ | [kg/dm^{3}] | Bore | d | [mm] | |

Acceleration | a | [m/s^{2}] | Belt width | b | [mm] | |

Acceleration due to gravity | g | [m/s^{2}] | Pre-tension distance | Δl | [mm] | |

Speed | v | [m/s] | Specific elasticity | c_{spez}_{} | [N] | |

Rotational speed | n | [min^{-1}] | Elasticity | c | [N/mm] | |

Angular speed | ω | [s^{-1}] | ||||

Centre distance | s_{A}_{} | Positioning deviation | Δs | [mm] | ||

Usefull linear distance | s_{L} | Positioning range | P_{s} | [mm] | ||

total distance of travel | s_{ges} | |||||

Specific tooth force | F_{tspez} | [N] | Acceleration distance | s_{B} | [mm] | |

Admissible tensile load | F_{Tzul} | [N] | Braking distance | s'_{B} | [mm] | |

Pre-tension force | F_{TV}_{} | [N] | Inherent frequency | f_{e}_{}_{} | [s^{-1}] | |

max. span force | F_{Tmax} | [N] | Excitation frequency | f_{0} | [s^{-1}] | |

Centre load | F_{A}_{} | [N] | Travel time with v=const. | t_{v} | [s] | |

Shaft force | F_{w}_{} | [N] | Overall time | t_{ges} | [s] | |

Frictional force | F_{R} | [N] | Overall distance | s_{ges} | [mm] |

Apply all equations with the dimensions mentioned here.

#### Calculation

#### Value to be calculated

circumferencial force F_{U} [N]

torque M [Nm]

power P [kW]

diameter d_{0} [mm]

linear and rotary

motion

(formula 3)

FU = Acceleration force (1.) + lifting force (2.) + friction force (3.)

= m * a + m * g + m * µ

required circumferencial force at the drive pulley F_{U} [N]

- The acceleration force F
_{B}is necessary to accelerate the linear drive with mass m e.g. from the stand still to the limit speed v. - The lifting force F
_{H}is necessary with a movement direction opposite to the acceleration due to gravity. With horizontal linear movement is F_{H}= 0. - A friction force is required when opposite to the moving direction a force is taking effect, e.g. friction force. Can the frictional drags be neglected is F
_{R}= 0.

(formula 4)

#### Calculation

m_{L} [kg] Mass of the linear slide to be moved

m_{B} [kg] Mass of the timing belt (belt weight, see Technical Data)

m_{Zred} [kg] reduced mass of pulley(s)

m_{Sred} [kg] reduced mass of tension roller(s)

#### Calculation value

mass to be moved m [kg]

(formula 5)

m = m_{L} + m_{B} + m_{Zred} + m_{Sre}

The mass of a pulley and/or tension roller is calculated in relation to:

Mass of the pulley m_{Z} [kg]

Mass of the tension roller m_{S} [kg]

The reduced mass m_{red} of a pulley and/or tension roller is an equivalent mass with equal load bearing to the effective line of the timing belt, the same as the rotational solid to the rotational axis.

(formula 6)

red. Mass of the pulley m_{Zred} [kg]

red. Mass of the tension roller m_{Sred} [kg]

A linear drive is pre-tensioned correctly, when under maximum effective circumferencial force F_{Umax} (from acceleration and braking) the slack span side of the belt stayes tight. A minimum pre-tension force is to be considered:

(formula 7)

F_{V} ≥ F_{U}

Pre-tension force F_{V} [N]

The highes span forces F_{max} are to be expected within the tight span side, when both pre-tension force F_{V} (static) and circumferencial force F_{U} (dynamic) acting together.

(formula 8)

F_{max} = F_{V} + F_{U}

maximum span force in the belt_{max} [N]

The permitted tensile load F_{Tzul} has to show safety factors to the max. occurring span force F_{max} in the timing belt.

(F_{Tzul} see Technical Data).

(formula 9)

F_{Tzul} ≥ F_{max}

permitted span force F_{Tzul} [N]

The static axis load F_{Asta} act within the stand still or under no-load conditions.

F_{Adyn} is a value depending on the effective circumferencial force.

(formula 10)

F_{Astat} = 2 * F_{V}

Axis load [N]

#### Calculation

#### Calculation value

pre-tension distance Δl [mm]

The tensioning station can be mounted at any position on the timing belt.

Values for c_{spez} see Technical Data.

(formula 11)

Elasticity c [N/mm]

Linear systems show a variable elasticity. The elasticity behaviour of the linear slide and/or linear bed depends on the length proportion L_{1} and L_{2}.

That means: Each individual position of the linear bed has its own elasticity.

The elasticity shows a minimum c_{min}, when L_{1} and L_{2} are equal in length.

For this case the following relation is valid:

(formula 12)

Is an external force acting on a linear slide a positioning deviation s results from the relation:

(formula 13)

Positioning deviation Δs [mm]

Under the effect of a triggered force, a mass connected to the timing belt (elasticity/mass system) assumes a damped natural vibration.

(formula 14)

Natural frequency f_{e} [s^{-1}]

If necessary, check linear drives with regard to the occurrence of excitation frequencies f_{0} in the drive pulley assembly which are close to the natural frequency f_{e}.

For technical structures, avoid compatibility of f_{e} = f_{0}< (resonance).

Note: In linear drives, the natural frequency f_{e} is in general considerably higher than the excitation frequency f_{0} of the drive, in which case no resonance is to be expected . We recommend a special examination, if necessary, where stepping motors are used. Measures in the event of resonance: Increase the stiffness of the timing belt by choosing a larger belt width.

#### How to proceed

The above mentioned equations can be used to comprehensively compute BRECO linear drives. The type of the individual examinations depends on the task. If necessary, request technical support from our sales outlets.

#### General kinematics

If the movement sequence of the linear drive has to be timed, we recommend to proceed in accordance with the linear movement values of the equations (3).

#### Coarse design according to mass and acceleration

Generally, the mass of the linear slide m_{L} and the acceleration a represent the decisive values for the design of linear drives. The selection diagram, the belt type and timing belt width can be determined, based on mass and acceleration shown on the page Determination of type of belt and belt width.

In conjunction with the coarse design, we recommend to adopt the pulley dimensions (as a provisional measure). Note the permissible minimum number of teeth or minimum diameters.

#### Drive pulley assembly

The required circumferential force F_{U} in the drive pulley assembly has to be determined according to equation (4). By provisionally assuming the pulley size, it is possible to calculate the attendant drive torque M according to equation (2) for the drive pulley assembly. In how far the calculated torque M can be harmonised with the torque sequence of the motor, depends on the type and selection of the drive motor. The selection of the motor also depends on the desired servo and positioning tasks. Once the drive motor has been decided upon, the actual torque sequence of the motor has to be taken into consideration for the further precise design of the timing belt.