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# List of formulae, terms, definitions

 Circumferencial force Fu [N] Lifting force FH [N] Torque M [Nm] Belt length LB [mm] Power P [W] Span length L1, L2 [mm] Mass to be moved m [kg] Number of belt teeth zB Mass of linear slide mL [kg] Number of pulley teeth z Mass of timing belt mB [kg] Number of meshing teeth ze Mass of pulley mZ [kg] Pitch circle diameter d0 [mm] Mass of tension roller mS [kg] Crown diameter dK [mm] reduced mass mred [kg] Tension roller diameter ds [mm] specific weight ρ [kg/dm3] Bore d [mm] Acceleration a [m/s2] Belt width b [mm] Acceleration due to gravity g [m/s2] Pre-tension distance Δl [mm] Speed v [m/s] Specific elasticity cspez [N] Rotational speed n [min-1] Elasticity c [N/mm] Angular speed ω [s-1] Centre distance sA Positioning deviation Δs [mm] Usefull linear distance sL Positioning range Ps [mm] total distance of travel sges Specific tooth force Ftspez [N] Acceleration distance sB [mm] Admissible tensile load FTzul [N] Braking distance s'B [mm] Pre-tension force FTV [N] Inherent frequency fe [s-1] max. span force FTmax [N] Excitation frequency f0 [s-1] Centre load FA [N] Travel time with v=const. tv [s] Shaft force Fw [N] Overall time tges [s] Frictional force FR [N] Overall distance sges [mm]

Apply all equations with the dimensions mentioned here.

#### Calculation Circumferencial force Torque Power

#### Value to be calculated

circumferencial force FU [N]
torque M [Nm]
power P [kW]
diameter d0 [mm] Angular speed

linear and rotary
motion Rotational speed Speed / peripheral speed Acceleration time (braking time) Acceleration distance (braking time) Travel time when v = const. Travel distance when v = const. Overall time Gesamtstrecke

(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 FU [N]

1. The acceleration force FB is necessary to accelerate the linear drive with mass m e.g. from the stand still to the limit speed v.
2. The lifting force FH is necessary with a movement direction opposite to the acceleration due to gravity. With horizontal linear movement is FH = 0.
3. 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 FR = 0.

(formula 4)

#### Calculation

mL [kg] Mass of the linear slide to be moved
mB [kg] Mass of the timing belt (belt weight, see Technical Data)
mZred [kg] reduced mass of pulley(s)
mSred [kg] reduced mass of tension roller(s)

#### Calculation value

mass to be moved m [kg]

(formula 5)

m = mL + mB + mZred + mSre

The mass of a pulley and/or tension roller is calculated in relation to: Mass of the pulley mZ [kg]
Mass of the tension roller mS [kg]

The reduced mass mred 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 mZred [kg]
red. Mass of the tension roller mSred [kg]

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

(formula 7)

FV ≥ FU

Pre-tension force FV [N]

The highes span forces Fmax are to be expected within the tight span side, when both pre-tension force FV (static) and circumferencial force FU (dynamic) acting together.

(formula 8)

Fmax = FV + FU

maximum span force in the beltmax [N]

The permitted tensile load FTzul has to show safety factors to the max. occurring span force Fmax in the timing belt.
(FTzul see Technical Data).

(formula 9)

FTzul ≥ Fmax

permitted span force FTzul [N]

The static axis load FAsta act within the stand still or under no-load conditions.
FAdyn is a value depending on the effective circumferencial force.

(formula 10)

FAstat = 2 * FV

#### Calculation #### Calculation value

pre-tension distance Δl [mm]

The tensioning station can be mounted at any position on the timing belt.
Values for cspez 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 L1 and L2.
That means: Each individual position of the linear bed has its own elasticity.
The elasticity shows a minimum cmin, when L1 and L2 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 fe [s-1]

If necessary, check linear drives with regard to the occurrence of excitation frequencies f0 in the drive pulley assembly which are close to the natural frequency fe.
For technical structures, avoid compatibility of fe = f0< (resonance).
Note: In linear drives, the natural frequency fe is in general considerably higher than the excitation frequency f0 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 mL 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 FU 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.