P1a.m₁ = m₂ m₂ = 2mk₁=k₂k₂ = 2k[m₂μ₂ = −k₂ (u₂-u₂)[m₁ü₁ = −k₂(µ₁ − u₂)-₁₁[m₂ï₁ + (k₂ +k₂)µ₁ −k₂u₂ = 0[m₂²₂ −k₂U₁ +k₂U₂ = 0Setting u₁ = U₁ cos at, u₂ = U₂ cos cot[_m₂w²U₁ + (k₂ +k₂)U₁ - k₂U₂ = 0|_m₂w²U₂ −k₂U₁ + k₂U₂ = 0[(3k-mw³)U₁-2kU₂ = 0[-kU₁ + (k-ma²)U₂ = 0(3k-ma²) -2k160²30/-0=-k(k-ma²)(3k-ma²)(k-ma²) - 2k² = 0(mw² ) ² − 4k (mw² )+k² = 0Dividing by m² and solve for ²(~³²)³² - 4 ( # )(~³²) + ( # ) ³ - 0mmkm|«² = (2 ± √3)P1b.[(3k-ma²)U₁-2kU₂ = 0[-kU₁ + (k-mw³)U₂ = 0By setting U₂=1,m=20kg, k=1000 N/m|U₁ =2k(3k-ma²)(note: The 2nd eqn will givethe same answer){0.73205)77(¹)=at a = 3.66[-2.73205)1at a = 13.66[u₂ (t))[[u₁₂ (1))[U₁cos(art-a₂)]- [7²) 7(²2) ] {U₂ cos(art-a₂))77With u₁(0)1,u₂ (0)=-1,₁ (0) = 0,u₂ (0) = 0,determine U₁,U₁,a₁ and ₂ For the 2 DOF system shown with m₁ = m, m₂ = 2m and k₁=k, k₂ = 2k, determine: (a) The naturalfrequencies and the corresponding mode shapes. (b) The response of the system, u₁(t) and u₂(t).For this part, assume m = 20 kg, k = 1000 kg, and initial conditions:101 = 1, 02 = -1; 1=0,02 = 0. (c) Plot u, (t) and u₂(t) of part b on the same graph.иции[mi] Tui (t)K=m₂ Tuz(t)

Answered: Image /qna-images/answer/79c5116a-8aa1-458e-adb2-893b712cd6e0.jpg

For the 2 DOF system shown with m₁ = m, m₂ = 2m and k₁=k, k₂ = 2k, determine: (a) The natural frequencies and the corresponding mode shapes. (b) The response of the system, u₁(t) and u₂(t). For this part, assume m = 20 kg, k = 1000 kg, and initial conditions: 101 = 1, 02 = -1; 1=0,02 = 0. (c) Plot u, (t) and u₂(t) of part b on the same graph. иции [mi] Tui (t) K= m₂ Tuz(t)

For the 2 DOF system shown with m₁ = m, m₂ = 2m and k₁=k, k₂ = 2k, determine: (a) The natural frequencies and the corresponding mode shapes. (b) The response of the system, u₁(t) and u₂(t). For this part, assume m = 20 kg, k = 1000 kg, and initial conditions: 101 = 1, 02 = -1; 1=0,02 = 0. (c) Plot u, (t) and u₂(t) of part b on the same graph. иции [mi] Tui (t) K= m₂ Tuz(t)

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter12: Coupled Oscillations

Section: Chapter Questions

Problem 12.3P

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