GATE Mechanical Engineering (ME) 2018 Shift 1 Solved Paper

Question : 58 of 65

Marks: +1, -0

A machine of mass

m=200 {kg}

is supported on two mounts, each of stiffness

{k}=10 {kN} {/} {m}

. The machine is subjected to an external force (in

{N}

)

{F}( {t})=50 {cos} 5 {t}

. Assuming only vertical translator motion, the magnitude of the dynamic force (in

{N}

) transmitted from each mount to the ground is ___________ (correct to two decimal places).

Solution:

m=200 {kg}, k=10 {kN} {/} {m}=10000 {N} {/} {m}

K_{e q}=K+K=10000+10000=20000 {N}{/} {m}

F_{t}=50 \cos 5 t

F_{0}=50 {N}

ω=5 {rad} {/} {s}

ω_{n}=√{\;{K_{e}}/{m}}=√{\;{20000}/{200}}=√{100}=10 {rad} {/} {s}

\;{ω}/{ω_{n}}=\;{5}/{10}=\;{1}/{2}

No damping

⇒ c=0 ⇒ ξ=0

Transmissibility

∊=\;{√{1+({2 ξ ω}/{ω_{n}})^{2}}}/{√{1-(\;{ω}/{ω_{n}})^{2}\}^{2}+\{\;{2 ξ ω}/{ω_{n}}\}^{2}}}

=\;{1}/{√{(1-(\;{ω}/{ω_{n}})^{2}]^{2}}}=\;{1}/{√{[1-(\;{1}/{2})^{2}\}^{2}}}

=\;{1}/{1-\;{1}/{4}}=\;{1}/{{3}/{4}}=(\;{4}/{3})

∊=\;{F_{t}}/{F_{0}}

\;{4}/{3}=\;{F_{t}}/{50}

F_{t}=\;{200}/{3}=66.66 {N}

Force from each mount

=\;{F_{t}}/{2}=\;{66.666}/{2}=33.333 {N}

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