/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 71 A viscous clutch is to be made f... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 71

A viscous clutch is to be made from a pair of closely spaced parallel disks enclosing a thin layer of viscous liquid. Develop algebraic expressions for the torque and the power transmitted by the disk pair, in terms of liquid viscosity, \(\mu\) disk radius, \(R,\) disk spacing, \(a,\) and the angular speeds: \(\omega_{i}\) of the input disk and \(\omega_{o}\) of the output disk. Also develop expressions for the slip ratio, \(s=\Delta \omega / \omega_{i},\) in terms of \(\omega_{i}\) and the torque transmitted. Determine the efficiency, \(\eta,\) in terms of the slip ratio.

Short Answer

Expert verified
The derived equations are: Torque \(T = \mu *\frac{{\omega_{i} - \omega_{o}}}{a} * \pi * R^{3}\), Power \(P = T * \omega_{o}\), Slip Ratio \(s = \frac{{\omega_{i} - \omega_{o}}}{\omega_{i}}\), and Efficiency \(\eta = 1 - s\). These equations help in understanding the physics behind the operation of a viscous clutch system.

Step by step solution

01

Equation for Torque

The momentum transfer between the disks is based on the viscous shear force exerted by the liquid. This viscous torque can be computed using the formula \(T = \mu *\frac{{\omega_{i} - \omega_{o}}} {a} * \pi * R^{3}\), where \(\mu\) is the viscosity of the liquid, \(\omega_{i}\) and \(\omega_{o}\) are the angular velocities of the input and output disks respectively, \(a\) is the disk spacing, and \(R\) is the disk radius.
02

Equation for Power

The power transmitted by the disks can be calculated using the formula \(P = T * \omega_{o}\), where \(T\) is the torque and \(\omega_{o}\) is the angular velocity of the output disk.
03

Equation for Slip Ratio

The slip ratio is simply the change in angular velocity relative to the input angular velocity. Therefore, it can be expressed as \(s = \frac{{\Delta \omega}}{\omega_{i}} = \frac{{\omega_{i} - \omega_{o}}}{\omega_{i}}\).
04

Equation for Efficiency

\( \eta = \frac{{\omega_{o}}}{\omega_{i}} = 1 - s \), where \(s\) is the slip ratio, \(\omega_{i}\) is the angular velocity of the input disk and \(\omega_{o}\) is the angular velocity of the output disk. It represents the proportion of power effectively delivered to the output. The utilised power is less than the supplied power due to the viscous slip between the disks.

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