Unit 4 Dynamics of Machines formula
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The document presents 6 key formulae relating to forced vibration. Formula 1 gives the magnification factor or amplitude relation for forced vibration as a function of excitation frequency and damping ratio. Formula 2 gives the phase lag between displacement and force vectors. Formula 3 gives the non-dimensional amplitude relation for vibration due to unbalance as a function of excitation frequency and damping ratio. Formula 4 gives the non-dimensional displacement transmission relation for forced vibrations due to base excitation. Formula 5 gives the relation for force or displacement transmissibility. Formula 6 gives the relation for phase lag between motion of the mass and motion of the support or between applied and transmitted force.
![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)](https://image.slidesharecdn.com/unit-4domformula-150816080211-lva1-app6892/85/Unit-4-Dynamics-of-Machines-formula-1-320.jpg)
Unit 4 Dynamics of Machines formula
- 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)