MODULUS OF ELASTICITY |
| The modulus of elasticity (λ) is calculated by dividing the stress by the strain: |
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| where |
λ is the modulus of elasticity (or Young's modulus), (P) |
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F is the force (N) |
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A is the cross-sectional area through which the force is applied (m²) |
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x is the extension (m) |
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l is the natural length (m) |
Example: A typical λ for rubber would be 6,9 MPa (1000 psi). For an average strength steel cord conveyor belt the modulus of elasticity would be 200 kN/mm and for an average textile conveyor belt around 7 kN/mm.
In other words: The higher the modulus the lower the elastic elongation per unit stress |
TENSION FORCE |
| The modulus of elasticity of a material can be used to calculate the tension force it exerts under a specific extension: |
|
| where |
T is the tension force |
| |
λ is the modulus of elasticity |
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A is the cross-sectional area |
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x is the extension |
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l is the length (m) |
EYTELWEIN EQUATION |
| The minimum belt tensions for transmitting the pulley peripheral forces are calculated as follows: |
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| where |
α is the length (circular measure) that the belt is wrapped around the pulley (arc of contact) |
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µ is the coefficient of friction between belt and pulley |
TAKE-UP LENGTH |
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where
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SSp is take-up length (m) |
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L is centre distance (m) |
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ε is belt elongation, elastic and permanent (%) |
| As a rough guideline, use 1,5% elongation for textile belts and 0,25% for steel cord conveyor belts. |
COEFFICIENT C |
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The coefficient C is a function of the length of the installation.
The total resistances without slope and special resistances are divided by the main resistances. |
ARRHENIUS EQUATION |
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where
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k is the rate constant of chemical reactions on the temperature |
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EA is the activation energy |
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T is the temperature (°K) |
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R is the gas constant |
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A is the prefactor (frequency factor) |
| The Arrhenius equation describes the quantitative relation between reaction velocity and temperature (as you know, the speed of chemical reactions increase with rising temperature) |
STRESS IN RUBBER |
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where
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s is the stress |
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v is the period of strain oscillation |
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d is the phase lag between stress and strain |
STRAIN IN RUBBER |
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where
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e is the strain |
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v is the period of strain oscillation |
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t is time |
|
|
| where |
E' is the storage modulus |
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s is the stress |
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e is the strain |
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d is the phase lag between stress and strain |
LOSS MODULUS |
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where
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E'' is the loss modulus |
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s is the stress |
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e is the strain |
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d is the phase lag between stress and strain |
INTERNAL FRICTION |
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| where |
tan d is the internal friction of a rubber |
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E' is the storage modulus |
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E'' is the loss modulus |
The tan d is sometimes used to determine the indentation loss of a conveyor belt cover (cf. Energy Saving Belts). E' and E'' should be as low as possible. However, there are a number of misconceptions related to specifiying E' and E''. |
LENGTH RELATED MASS FLOW (m³/h) |
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| where |
v is the belt velocity (m/s) |
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lvth is the theoretical volume flow (m³/h) |
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r is the bulk density of the conveyed material (t/m³) |
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jSt is the coefficient for determination of the volume flow |
|
|
| where |
rB0 is the braking factor related to the rated torque of all drive motors |
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hges is the overall efficiency of all transmission elements between motor and pulley shaft |
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PMerf is the total capacity of the drive motors required in a steady operating state |
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PMinst is the total installed capacity of the drive motors (N) |
MINIMUM BELT TENSION FOR BELT SAG LIMITATION (top side, loaded) |
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| where |
g is gravity (9,81 m/s²) |
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m'Li is the mass of the conveyed material, uniformly distributed across a section of the conveyor (kg/m) |
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m'G is the length related mass of the conveyor belt (kg/m) |
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IRo is the idler spacing in top run (m) |
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hrel is the maximum belt sag related to the spacing between the carry idlers (%) |
MINIMUM BELT TENSION FOR BELT SAG LIMITATION (bottom side, unloaded) |
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| where |
g is the gravity (9,81 m/s²) |
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m'G is the length related mass of the conveyor belt (kg/m) |
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IRu is the idler spacing in bottom run (m) |
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hrel is the maximum belt sag related to the spcing between the carry idlers (%) |
PRIMARY RESISTANCES IN AN EVENLY TILTED CONVEYOR |
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| where |
f is the friction factor in top and bottom run |
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L is the conveyor length (m) |
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g is the gravity acceleration (m/s²) |
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m'R is the mass of the idlers (kg/m) |
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m'L is the mass of the conveyor belt with an evenly distributed load (kg/m) |
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d is the even inclination of the conveyor (°) |
| MAXWELL MODEL |
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| where |
e is strain |
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s is stress |
| VOIGT MODEL |
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| where |
e is strain |
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s is stress |
| Used to express the relaxation behavior of polymers. |
| ROLLING RESISTANCE |
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| where |
F is resistance force |
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Crr is the dimensionless rolling resistance coefficient |
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Nf is the normal force |
| MOTIONAL RESISTANCE |
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| where |
C is length coefficient |
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f is resistance coefficient |
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L is conveyor length (m) |
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m' is length related mass (kg/m) |
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g is acceleration |
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H is conveying lift (m) |
| MINIMUM TRANSITION LENGTH (m) |
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| where |
B is belt width (mm) |
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l is troughing angle (°) |
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S is the safety factor |
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KG is the belt parameter |
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Kf1 is the troughing parameter |
| PERIPHERAL FORCE (N) |
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| where |
FH is the main resistance |
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FN is the secondary resistance |
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FS1 are the special main resistances |
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FS2 are the special secondary resistances |
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FSt are the resistances due to slope |
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| where |
PTr is the drive power (pulley) |
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v is speed (m/s) |
| |
|
| where |
C is the coefficient (main resistance factor) |
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f is the resistance coefficient |
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L is belt length (m) |
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g is acceleration (m/s²) |
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qRO is the mass of the idlers on top side (kg/m) |
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qRU is the mass of the idlers on bottom side (kg/m) |
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qB is the belt mass (kg/m) |
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qG is the mass of the conveyed material (kg/m) |
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H is the lift (m) |
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FS1 are the special main resistances |
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FS2 are the special secondary resistances |
|
|
| where |
G'' is the loss modul |
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G' is the storage modul |
|
|
| where |
qG is the conveying mass (kg/m) |
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H is the lift (m) |
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g is acceleration (m/s²) |
|
|
| where |
m'G is the length related mass of the conveyor belt (kg/m) |
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g is acceleration (m/s²) |
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b is width (mm) |
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l is troughing angle |
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l is idler length (mm) |
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B is belt width (mm) |
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Tx is drive traction |
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