Important formulae used for solving problems from Fourth Unit Dynamics of Machines

1. Important Formulae - UNIT IV - Forced Vibration
1. Whatis the relationfor magnificationfactororamplituderelationin the case of a forced
vibration?
𝑴.𝑭.=
𝑨
𝑿 𝟎
=
𝟏
√[ 𝟏− (
𝝎
𝝎 𝒏
)
𝟐
]
𝟐
+ [𝟐𝝃 (
𝝎
𝝎 𝒏
)]
𝟐
… …(𝟏)
2. Whatis the relationfor phase lagbetween displacementandforce vector?
𝑷𝒉𝒂𝒔𝒆 𝒍𝒂𝒈, 𝝓 = 𝐭𝐚𝐧−𝟏 {
𝟐𝝃(
𝝎
𝝎 𝒏
)
[ 𝟏− (
𝝎
𝝎 𝒏
)
𝟐
]
} … …(𝟐)
3. Whatis the non-dimensional relationforamplitudeofvibrationdueto rotating/reciprocating
unbalance?
𝑨
(
𝒎 𝟎
𝒎
𝒆)
=
(
𝝎
𝝎 𝒏
)
𝟐
√[ 𝟏− (
𝝎
𝝎 𝒏
)
𝟐
]
𝟐
+ [𝟐𝝃 (
𝝎
𝝎 𝒏
)]
𝟐
… …(𝟑)
4. Whatis the non-dimensional relationforamplitudeofforcedvibrationsdueto base
excitation?
𝑫𝒊𝒔𝒑𝒍𝒂𝒄𝒆𝒎𝒆𝒏𝒕 𝒕𝒓𝒂𝒏𝒔𝒎𝒊𝒔𝒔𝒊𝒐𝒏,
𝑨
𝒀
=
√ 𝟏 + [𝟐𝝃 (
𝝎
𝝎 𝒏
)]
𝟐
√[ 𝟏 − (
𝝎
𝝎 𝒏
)
𝟐
]
𝟐
+ [𝟐𝝃 (
𝝎
𝝎 𝒏
)]
𝟐
…… (𝟒)
5. Whatis the relationfor force transmissibilityordisplacementtransmissibility?
𝑻𝒓𝒂𝒏𝒔𝒎𝒊𝒔𝒔𝒊𝒃𝒊𝒍𝒊𝒕𝒚 𝒓𝒂𝒕𝒊𝒐, 𝜺 =
𝑭 𝑻
𝑭 𝟎
=
√ 𝟏 + [𝟐𝝃 (
𝝎
𝝎 𝒏
)]
𝟐
√[ 𝟏 − (
𝝎
𝝎 𝒏
)
𝟐
]
𝟐
+ [𝟐𝝃 (
𝝎
𝝎 𝒏
)]
𝟐
…… (𝟓)
6. Whatis the relationfor phase lagbetween motionofmass (A) and motionofsupport (Y) (or)
FT and F0?
𝑷𝒉𝒂𝒔𝒆 𝒍𝒂𝒈, 𝝋 = 𝝓 − 𝜶 = 𝐭𝐚𝐧−𝟏 {
𝟐𝝃(
𝝎
𝝎 𝒏
)
[ 𝟏− (
𝝎
𝝎 𝒏
)
𝟐
]
} − 𝐭𝐚𝐧−𝟏 𝟐𝝃 (
𝝎
𝝎 𝒏
) …… (𝟔)