Q$3.$ Balancing a rotating machine

A four cylinder engine runs at $\backslash (6000\backslash$mathrm$\{$rev$\}/\backslash$mathrm$\{$min$\}\backslash )$ and the crank arrangement is $1,3,4,2$ phased $\backslash (90^\{\backslash$circ$\}\backslash )$ apart. The main moving parts are idealised to a set of rotating masses, and a set of reciprocating masses. The four rotating masses, each of $1\mathrm{.}5$ kg $,$ are at a crank radius of $60$ mm $.$ And the four identical reciprocating masses, each of $0\mathrm{.}5$ kg $,$ have effective connecting rod lengths of $150$ mm and the axial distance between each unit is $100$ mm $.$ Assume that the mass centre of the rotating assembly lies in the vertical plane midway between cylinders $2$ and $3.$

i$)$ In terms of primary and secondary forces and moments, determine the state of external balance of the engine, i$.$e$.$ calculate the primary and secondary forces and moments. Show your working using a table to calculate moments and check with a diagram showing the vector addition.

$[14$ marks$]$

ii$)$ To eliminate the primary pitching moment, balance masses are to be fitted on the crankshaft in the planes of cylinder $1$ and cylinder $4,$ at the crank radius. Find the magnitude and relative angular positions of the pair of balance masses needed.