Example $7\mathrm{.}2.$ An air compressor consists of four compressor discs mounted on a steel shaft $50$ mm apart as shown in Fig. $7\mathrm{.}23.$ The shaft is simply supported at either end. Each compressor wheel mass is $18$ kg $.$ Calculate the deflection at each wheel using Macaulay's method and calculate, using the Rayleigh$-$Ritz equation, the first critical frequency of the shaft. The shaft outer diameter is $0\mathrm{.}05$ m $,$ the inner diameter is $0\mathrm{.}03$ m $.$