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For the lumped mass system shown, determine: (a) the two natural frequencies and mode shapes. (b) Normalize the mode shapes by assuming U₂ = 1. corresponding k M₂ 2m L 2₂ 12
Center of mass from A: Σmx, SL = ΣΜ 3 +0=1,2-14, XG= Let 1₁₂=1₂₁-Le= U=U₁ 210 21₁+1₂ 1=₁+ = 3 3 1₁ = 1₁ + Le = "₁+U₂ Up=1₂ For free vibration, ignore the gravity effect. ΣF, =0: 3mu=-ku-ku ΣMG=0: Lõ=-ku A [3m 0 0 where I = 2m @= 8mL² 3 4kL 3 2 SL 1 k 3m 77/ -ku 2m ( ² ) ² + m (47)² = 8m²³² 3 4kL {}+ Let u = Ucos @_t,6 = Ⓒ cos@t, 4kL 3ma²-2k 3 2k 4kL 3 8mI² 3 4 k 3m 24kL² 9 24kL² 9 3mw²-2k 4kL 3 B 2 |7₁ = 8mL² 24kL² @²² 3 9 4 k @₂ m 3 m Let U=1. Find 0. (3ma-2k)(1)- 4kL 3 4kL 3 }₂ m₂ = 1 C [UG] OO 4L 2L Or, U₂ = 3L-e. One can build the equations by means of U₂, instead.
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