Design Project $\backslash$#$1($Formal Report is due $10/25)$; Team Members:

pdf file submission to Canvas $($see the Formal Report Format; include this cover page.$)$ Grade:

Whirling $\backslash$& Vibration Analysis of a Loaded Stepped Shaft: $($Deflection and the Critical Frequency$)$: Material: Steel $1040,$ annealed $($Density: $\backslash (\backslash$rho$=7845\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ))$; Dimensions in mm$.$

Table A shows the FEA results for the load$-$induced deflections $($the shaft mass is ignored$).$

Table B shows deflections at the centers of the shaft segments due to the shaft mass only. Determine:

$1.$ Use Rayleigh's Method to investigate vibration of the shaft for two different Cases:

Case I $-$ Load$-$induced vibration: Table A shows deflections due to Force and Moment.

Case II $-$Vibration due to the combined loads and the shaft mass. Hint: use superposition of deflections due to the applied loads and the shaft mass $($Create a third Table showing deflections of all nodes: the circled nodes and the centers of segments $($the empty circles, $"$o$").$ Use Hand Calculations to show sample superposition calculations, e$.$g$.,$ deflection at the Gear on the right.

$2.$ For each Case: select a safe motor frequency $\backslash ((\backslash$mathrm$\{$HZ$\})\backslash )$ to operate the shaft.

Summary $-$ Complete the Table:

Table A$-$ Load$-$induced Deflections:

Node numbers are Circled in the Figure.

Table B$-$ Deflections due to the shaft mass only. The nodes are centers of the segments, $\text{'}$o$\text{'}$

The segment weight is assumed as a concentrated load at the center of the segment, the empty circle, $\text{'}$o$\text{'}.$

Case II requires Superposition $($see the in$-$class Examples$).$

$*$ Hint: You may use MATLAB to expedite calculations for each Case.