sounds to me like the belt has stretched slightly and dried on the long journey. My timing belt, fan belt and air-con belt was changed last may, however, since then, i havent done a long journey like this with my air-con switched on, on the motorway. As already explained in the section Belt speeds, the belt strains ε and the belt speeds v are directly interrelated. Mathematically this can be expressed as follows: Thus, the elastic slip S can also be determined by the power loss ΔP with respect to the power P i at the input pulley: The elastic slip influences the belt speed and thus the power but not the circumferential force or the torque!

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The reduction in peripheral speed between the driving pulley and the driven pulley is directly relatet to a loss in power, because a decrease of the circumferential speed v at a transmitting circumferential force F c means a direct decrease in power according to the P=F c⋅v. The entire wrap area φ can therefore be divided into two zones. In the co-called sliding zoneφ’ a relative motion takes place between belt and pulley. With the acting sliding friction, this zone ensures the transmission of the circumferential force. In the remaining adhesion zone, the belt adheres to the pulley without a relative motion and without force transmission.i also found that the problem of the noise tends to happen 'more' when my air-conditioning is switched on, rather than just the cool air in the cabin. Equation (\ref{4367}) also shows that if no circumferential force is transmitted (F c=0) there is no sliding zone (ln(1)=0!) but only an adhesion zone. With the transmission of a circumferential force, however, a sliding zone is created which increases with increasing circumferential force. As a result, the elastic slip also increases.The exact relationship between elastic slip S and circumferential force F c to be transmitted is to be derived in the following sections. S =\frac{\Delta v}{v_i} = \frac{v_i-v_o}{v_i} = \frac{v_t-v_s}{v_t} = \frac{(1+\epsilon_t) – (1+\epsilon_s)}{1+\epsilon_t} = \frac{\epsilon_t-\epsilon_s}{1+\epsilon_t} \\[5px]

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As can be seen from the animation below, the belt is running more and more ahead due to the increasing stretching on the driven pulley. This means that the speed of an imaginary point on the belt is always slightly higher than the peripheral speed of the pulley. This corresponds to the relative motion between belt and pulley described above. The relative motion on the pulleys, which is always present due to the elasticity of the belt, is called elastic slip (partial relative motion between belt and pulley)! Such (elastic) stretching or shrinking processes of the belt on the pulleys, which inevitably lead to relative motion, are also called elastic slip. From the belt’s point of view, elastic slip should always be kept as low as possible, otherwise enormous belt wear will occur due to the strong relative motion. The surfaces of pulleys must therefore not be too rough, as one might misleadingly assume due to the increased static friction on rough surfaces! The higher the circumferential forces to be transmitted and the more elastic the belt is, the higher the elastic slip! Note that the relative motion around the pulley is constantly increasing as the belt increases its speed more and more in accordance with the increasing elongation (condition of continuity!), but the pulley has a constant circumferential speed. This means that the belt speed and the peripheral speed of the pulley are only equal when the belt runs onto the driven pulley, otherwise the belt speed will be higher or the pulley speed lower.The figure below shows schematically the distribution of the speed along the belt according to the animation above. Figure: Speed distribution along the belt

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So if the driving pulley generally moves faster than the belt and the driven pulley is slower, then the circumferential speeds of the pulleys are obviously no longer identical (this would only be the case with a complete inelastic belt).This ultimately results in a loss of circumferential speed between the drive pulley (rotating faster than the belt) and the output pulley (rotating slower than the belt). S = \frac{\epsilon_t-\epsilon_s}{1+\epsilon_t} = \frac{\frac{F_t}{E \cdot A}-\frac{F_s}{E\cdot A}}{1+\frac{F_t}{E\cdot A}} = \frac{F_t-F_s}{E \cdot A+F_t}\\[5px] In addition to elastic slip, which is due to the elasticity of the belt, the belt can also slip completely over the entire driven pulley in the event of overload. This is then referred to as sliding slip.Note that every belt has a certain elasticity and therefore always results in elastic slip, whereby sliding slip should always be avoided.

The belt adapts to the different speeds by elastic slip on the pulleys! Circumferential speed of the pulleys The higher the circumferential forces to be transmitted and the more elastic the belt is (e.g. low Young’s modulus!), the greater the elastic slip. With the increased elastic slip, the sliding zone also includes a larger portion of the wrap angle. Figure: Increase in elastic slip with increase in circumferential force Since the circumferential speeds can also be expressed by the rotational speeds n and pulley diameters d (v=π⋅d⋅n), the elastic slip can also be determined as follows: With the definition of the elastic slip S as the ratio of speed loss Δv and circumferential speed of the input pulley v i, the following formula applies v o: circumferential speed of the output pulley): The sliding angle φ’ relevant for power transmission can also be expressed by the circumferential force F c to be transmitted. Thus follows with F c=F t-F s or F t=F c+F s: