8.13
Courroies plates : Résolution de problèmes
Les courroies plates sont essentielles dans de nombreuses applications industrielles car elles aident à transmettre la puissance d'une poulie à une au…
Une courroie plate s’enroule autour de deux poulies, A et B, avec des rayons de 30 cm et 10 cm, respectivement.
L’angle entre la courroie et l’horizontale est de 20 degrés au niveau des poulies.
La poulie B tourne dans le sens des aiguilles d’une montre et entraîne la poulie A, provoquant la tension T2 à une extrémité de la courroie et la tension T1 à l’autre.
Étant donné que la tension maximale admissible T2 est de 1000 N et que le coefficient de frottement statique entre la courroie et les poulies est de 0,4, quel est le moment maximal sur la poulie A ?
L'angle de contact entre la ceinture et la surface, calculé à partir de la géométrie du système, est de 140 degrés.
L’angle de contact ceinture-surface en radians et le coefficient de frottement statique sont substitués dans l’expression des tensions de courroie pour obtenir T1.
Lorsque la poulie B tourne dans le sens des aiguilles d’une montre, une différence de tension créée à la poulie B génère un moment à la poulie A.
Un schéma de corps libre pour la poulie A est dessiné et la condition d’équilibre du moment est appliquée.
Les valeurs de rayon et de tension sont substituées pour obtenir le moment maximum.
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Q1: How does the belt-to-surface contact angle affect tension calculations in a flat belt system?
The belt-to-surface contact angle, measured in radians, is a critical input for determining belt tensions using the friction relationship. In this problem, the contact angle is 140 degrees, calculated from the system's geometry where the belt wraps around pulleys with different radii. This angle, combined with the coefficient of static friction of 0.4, is substituted into the tension expression to find T1 from the maximum allowable T2 of 1000 N.
Q2: What is the relationship between tension difference and moment generation in pulley systems?
When pulley B rotates clockwise, it creates a tension difference between the two ends of the belt—T2 minus T1. This tension difference acts at different radii on pulley A, generating a moment. The moment is calculated by multiplying the tension difference by the radius of pulley A, which is 30 cm. This moment represents the power transmission capability of the belt system.
Q3: How do you apply moment equilibrium to find the maximum moment on a pulley?
A free-body diagram of pulley A is drawn showing both tension forces acting at the pulley's radius. Moment equilibrium requires that the net moment equals zero at static conditions. By substituting the calculated tension values and the pulley radius into the moment equation, the maximum moment is determined. In this case, the maximum moment on pulley A is 186.921 N·m.
Q4: Why is the coefficient of static friction important in belt tension analysis?
The coefficient of static friction of 0.4 determines how much tension can be transmitted between the belt and pulley surfaces without slipping. This value is substituted into the belt tension relationship along with the contact angle to calculate T1 from the maximum allowable T2. A higher friction coefficient would allow greater tension transmission, while a lower coefficient would reduce it.
Q5: What role does pulley radius play in calculating the maximum moment?
Pulley radius directly determines the moment arm for the tension forces. Pulley A has a radius of 30 cm, while pulley B has 10 cm. The larger radius of pulley A means that the same tension difference produces a greater moment. The maximum moment is calculated by multiplying the tension difference by pulley A's radius of 0.3 m.
Q6: How does the geometry of the belt system determine the contact angle?
The belt-to-surface contact angle is calculated from the system's geometry, including the radii of both pulleys and the angle between the belt and horizontal at the pulleys. In this problem, with pulley radii of 30 cm and 10 cm and a 20-degree angle to the horizontal, the resulting contact angle is 140 degrees. This geometric relationship is essential for accurate tension calculations.
Q7: What is the calculated tension T1 when T2 reaches its maximum allowable value?
When the maximum allowable tension T2 is 1000 N, the calculated value of T1 is 376.93 N. This T1 value is determined by substituting the belt-to-surface contact angle in radians and the coefficient of static friction of 0.4 into the friction-based tension relationship. The tension difference of 623.07 N between T2 and T1 then generates the maximum moment on pulley A.