Formulario

Pagina 1

First approach Beam equation: 𝑤(𝑥,𝑡)=[𝐴cos(𝛾𝑥)+𝐵sin(𝛾𝑥)+𝐶cosh(𝛾𝑥)+𝐷sinh(𝛾𝑥)]cos(𝜔𝑡+𝜑) 𝛾4= 𝑚𝜔2 𝐸𝐽 String equation: 𝑤(𝑥,𝑡)=[𝐴cos(𝛾𝑥)+𝐵sin(𝛾𝑥)]cos(𝜔𝑡+𝜑) 𝛾= 𝜔 𝑐 𝑐=√𝑆 𝑚 Rotation for small displacements (angle): sin(𝜗)~𝜗≅ 𝜕𝑤 𝜕𝑥 Momentum: 𝑀=𝐸𝐽 𝜕2𝑤 𝜕𝑥2 Shear: 𝑇=𝐸𝐽 𝜕3𝑤 𝜕𝑥3 • When we cut a beam, we must highlight both the shear 𝑇 and the momentum 𝑀 • When we cut a string, we only have to highlight the tension 𝑆, no rotation (the shear will be the tension multiplied per the angle) • When we have to beams, we have to impose the coherence of both the displacement 𝑤 and the rotation 𝜕𝑤 𝜕𝑥 • When we have one beam and a string, we only have to impose the coherence of the displacement 𝑤 Elastic force due to a spring: 𝐹𝑒𝑙=𝑘𝑤 Force due to a spring and an imposed motion: 𝐹𝑒𝑙=𝑘(𝑤−𝑧) Torsional force due to a torsional spring: 𝐹𝑘𝑇=𝑘𝑇 𝜕𝑤 𝜕𝑥 Force due to a concentrated mass: 𝐹𝑐=𝑚𝑐 𝜕2𝑤 𝜕𝑡2 or 𝐹𝑐=𝑚𝑐𝑧̈ Momentum due to a concentrated mass: 𝑀𝑐=𝐽𝑐 𝜕3𝑤 𝜕𝑥𝜕𝑡2 Equation of the imposed motion: 𝑧=𝑍0𝑒𝑖Ω𝑡=𝑍0cos(𝜔𝑡+𝜑) 𝑧̇=−𝑍0𝜔sin(𝜔𝑡+𝜑) 𝑧̈=−𝑍0𝜔2cos(𝜔𝑡+𝜑) Modal superposition approach Displacement expression number 1: 𝑤1(𝑥1,𝑡)=Φ1 T(𝑥1)𝑞(𝑡) Displacement expression number 2: 𝑤2(𝑥2,𝑡)=Φ2 T(𝑥2)𝑞(𝑡) Mode of shape related to the first displacement expression: Φ1(𝑥1)= { Φ1 (1)(𝑥1) Φ1 (2)(𝑥1) Φ1 (3)(𝑥1) ⋮ Φ1 (𝑁)(𝑥1)} Mode of shape related to the second displacement expression: Φ2(𝑥2)= { Φ2 (1)(𝑥2) Φ2 (2)(𝑥2) Φ2 (3)(𝑥2) ⋮ Φ2 (𝑁)(𝑥2)} Modal coordinate: q(𝑡)= { 𝑞1(𝑡) 𝑞2(𝑡) 𝑞3(𝑡) ⋮ 𝑞𝑁(𝑡)} Lagrange equation: 𝑑 𝑑𝑡(𝜕𝐸𝑐 𝜕q̇) 𝑇 −(𝜕𝐸𝑐 𝜕q) 𝑇 +(𝜕𝐷 𝜕q̇) 𝑇 +(𝜕𝑉 𝜕q) 𝑇 =(𝛿𝐿 𝛿q) 𝑇 Kinetic energy of a beam: 𝐸𝑘,𝑏𝑒𝑎𝑚= 1 2∫ 𝑚𝑣𝑏𝑒𝑎𝑚 2 𝑑𝑥 𝐿 0 Beam velocity: 𝑣𝑏𝑒𝑎𝑚 = 𝜕𝑤 𝜕𝑡 sin′(𝑥)=cos(𝑥) 𝑐𝑜𝑠′(𝑥)=−sin(𝑥) cosh′(𝑥)=sinh(𝑥) sinh′(𝑥)=cosh(𝑥) Kinetic energy due to the motion of a concentrated mass: 𝐸𝑘,𝑚𝑐=𝑚𝑐𝑣𝑐2 Velocity of the concentrated mass: 𝑣𝑐= 𝜕𝑤…