tailieunhanh - Vibration Analysis and Control New Trends and Developments Part 3

Tham khảo tài liệu 'vibration analysis and control new trends and developments part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 40 Vibration Analysis and Control - New Trends and Developments s2 w2 Y s2Y Y k1 k2s2Y Identification of the excitation frequency w The differential equation 23 is expressed in notation of operational calculus as m1 s4Y s yk1 k2 ỉtl s2Y s Y s m2 2 U s ----------w2 Fo 2 2 a3s3 a2s2 a1 s ao 24 where ai i 0 . 3 denote unknown real constants depending on the system initial conditions. Now equation 24 is multiplied by s2 w2 leading to s4Y s2 y 1 m2 s2 w2 u w2 Ĩ0 s2 w2 3s3 a2s2 ajs ao 25 This equation is then differentiated six times with respect to s in order to eliminate the constants ai and the unknown amplitude F0. The resulting equation is then multiplied by s 6 to avoid differentiations with respect to time in time domain and next transformed into the time domain to get 11 t w2a12 t j 1 12 t w2b12 t k1 C1 t w2d1 t 26 where At t to and a11 t 2g11 t k2g12 t a12 t 2g12 t k2g13 t r 6 . .6 b12 t 2g13 t k2 At 6 Z1 J to C1 t k2g14 t k2 2g12 t d1 t k2 I At 6 u k2 2g13 t J to with g11 t 720 y 6 y 4320 y 5 At y 5400 y 4 At 2 y 2400 y 3 At 3 y to to to to f 2 A f . c 45 At 4 y 36 At 5 y At 6 y to to g12 t 360 y 6 At 2 y 480 y 5 At 3 y 180 y 4 At 4 y to to to 24 At 5 y i At 6 y to to Design of Active Vibration Absorbers Using On-Line Estimation of Parameters and Signals 41 13 t 30 6 At 4 y-12 5 At 5 y 4 At 6 y t0 t0 t0 14 t 30 ỉ 6 At 4 u-12 ỉ 5 At 5 u y 4 At 6 u t 0 t 0 to Finally solving for the excitation frequency w in 26 leads to the following on-line algebraic identifier w2 _ N1 t _ c1 t a11 t m1 a12 t k1 e D1 t a12 t m1 b12 t k1 - d1 t 27 This estimation is valid if and only if the condition D1 t _ 0 holds in a sufficiently small time interval t0 t0 Ỗ0 with Ỗ0 0. This nonsingularity condition is somewhat similar to the well-known persistent excitation property needed by most of the asymptotic identification methods Isermann Munchhof 2011 Ljung 1987 Soderstrom 1989 . In particular this obstacle can be overcome by using numerical resetting algorithms or further .