Introduction Rotor - 2
- 1. Presentation 2 Rotor System Modal Analysis Oil Turbo Compressor Equipment Spring 2021
- 2. vibration 𝜏𝑦 𝐹𝑦 Y 𝐹𝑧 𝜏𝑧 Z X 𝐹𝑥 𝜏𝑥 Transitional Torsional X Y z 𝜃𝑥 𝜃𝑧 𝜃𝑦 Axial Lateral (bending, transverse and flexural) Mechanical Vibration
- 3. 𝑀𝑞 + 𝐶𝑞 + 𝐾𝑞 = 𝐹 𝐹 = 𝑓 𝜏 = 𝑓𝑜𝑟𝑐𝑒 𝑡𝑜𝑟𝑞𝑢𝑒 𝑞 = 𝑥 𝑦 𝑧 𝜃𝑥 𝜃𝑦 𝜃𝑧 𝐹 = 0 𝐹 ≠ 0 Free - vibration Forced - vibration Modal Analysis Eigen Vector Eigen Value Transient Harmonic Random Degree of freedom Newton’s Law (2nd)
- 4. What is the benefit of Eigenvalue? What is the benefit of Eigenvector? • Frequency • Critical speed • Transitional and rotational vibration(rotary machine) • Mode shape (mechanical behavior) • Maximum and minimum displacement • Effects of degree of freedom • Effects of boundary conditions
- 5. Eigenvalue and Eigenvector 𝐾 − 𝑀𝑤2 𝜆 = 0 𝑀𝑞 + 𝐶𝑞 + 𝐾𝑞 = 0 𝑤2 Eigen value Eigen vector 𝑤 ≅ 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑤 = 𝐾 𝑀 𝜆 Mode shape Mass Stiffness
- 6. How to calculate K and M matrices? 𝑉 = 1 2 𝜎𝜀𝑑𝑉 𝑈 = 1 2 𝜎𝜀𝑑𝑉 𝑈 = 1 2 𝑞𝑇 𝐾𝑞 Stiffness Mass Potential Energy Kinetic Energy 𝑈 = 1 2 𝑞𝑇𝑀𝑞
- 7. Node and Element Concepts 𝑁6 𝑁5 𝑁4 𝑁3 𝑁2 𝑁1 x y z 𝜃𝑧 𝜃𝑦 𝜃𝑥
- 8. Critical Speed (rpm) Type 2504.4 𝜃𝑥 2578.26 𝜃𝑦 2594.34 𝜃𝑧 11271.6 𝜃𝑥 11443.2 𝜃𝑦 30756.6 𝜃𝑧 50842.2 𝜃𝑥 51268.8 𝜃𝑦 66960 𝜃𝑧 67188 𝜃𝑥 𝑲 = ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ x y z 𝜽𝒚 𝜽𝒛 𝜽𝒙 x y z 𝜽𝒙 𝜽𝒛 𝜽𝒚 𝐾 = ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ ∎ 𝜃𝑥 𝜃𝑦 𝜃𝑧 𝜃𝑥 𝜃𝑦 𝜃𝑧 How to know types of direction angular velocity?
- 10. Shaft 1 6 9 8 7 3 2 4 5
- 11. Natural Frequency Rotor (Hz) Critical Speed (rpm) 41.74 2504.4 42.971 2578.26 43.239 2594.34 187.86 11271.6 190.72 11443.2 512.61 30756.6 847.37 50842.2 854.48 51268.8 1116 66960 1119 67188
- 12. Campbell Diagram
- 13. Thank you