Mechanical Vibrations Chapter 13

Transcription

1 Mechanical Vibrations Chapter 3 Peter Avitabile Mechanical Engineering Department University of Massachusetts Lowell Dr. Peter Avitabile

2 Random Vibrations Up until now, everything analyzed was deterministic. Other loading conditions eist that are not deterministic - random vibrations. The function below is irregular but may have some statistical character. Dr. Peter Avitabile

3 Random Vibrations Each record is called a sample - the total set is called an ensemble. If the function is evaluated at (t) and (t + τ ) and the averaged the function shows no difference then the signal is stationary. 3 Dr. Peter Avitabile

4 Time Average - Epected Values The epected value can be obtained if time averaging is performed over a long time record E[(t)] (t) (t) lim T T 0 (t)dt (3..) Mean value E[ (t)] lim T T 0 (t)dt (3..4) Variance σ lim T T 0 ( ) dt () (3..5) 4 Dr. Peter Avitabile

5 Frequency Response Function The linear input-output relationship also holds true for random signals. In the time domain, the response can be determined in terms of the impulse response function using the convolution (Duhamel) integral as (t) t 0 f ( ξ)h(t ξ)dξ (3.3.) 5 Dr. Peter Avitabile

6 Frequency Response Function For the input-output problem, a simplier approach utilizes a frequency domain & frequency response function under stationary or steady state condition For a sinusoidal ecitation, the SDOF response is or (t) k mω (t) + H( ω)fe Fe jcω jωt jωt (3.3.4) Mean Squared Response is H( ω) F 6 Dr. Peter Avitabile

7 Probability Distribution Many times it is desirable to know the probability of a certain value of a time signal. The probability density function is and the variance Gaussian and Rayleigh distributions widely used lim p() P( + ) P() 0 + σ ) p()d () (3.4.) ( (3.4.7) 7 Dr. Peter Avitabile

8 Time Correlation Functions Correlation is the measure of similatiry between two signals. By time shifting one time signal relative to another time signal, a correlation function can be obtained. 8 Dr. Peter Avitabile

9 Auto- and Cross-Correlation Function Auto-Correlation R ( τ) E[(t)(t + τ)] (t)(t + τ) R ( τ) lim T T T / T / (t)(t + τ) dt (3.5.) Cross-Correlation R y ( τ) E[(t)y(t + τ)] (t)y(t + τ) R y ( τ) lim T T T / T / (t)y(t + τ) dt (3.5.3) 9 Dr. Peter Avitabile

10 Fourier Transforms Fourier Integral is used for the transformation Fourier Transform Pair + + jπft (t) S(f )e df πft S(f ) (t)e dt (3.7.) j (3.7.) or using ω Fourier Transform Pair + j ω t (t) S( ω)e dω π S( ω) + (t)e jωt dt (3.7.3) (3.7.4) Note: Thompson uses X(f) as a linear spectrum and S(f) as a power spectrum These notes use S(f) as a linear spectrum and G(f) as a power spectrum 0 Dr. Peter Avitabile

11 Fourier Transforms Differentiation is simply (t) & + j ω t jωs( ω)e dω π which is just multiplication by jω FT FT [ (t) & ] jωft[ (t) ] [&& (t) ] ω FT[ (t) ] Dr. Peter Avitabile

12 Fourier Transforms Thus transforming the differential equation ( mω m && + c& + k f (t) + jcω + k)x( ω) F( ω) H ( ω)x( ω) F( ω) or as more commonly written X( ω) H( ω)f( ω) Note the simple multiplication rather than the convolution integral in the time domain Dr. Peter Avitabile

13 Parseval s Theorum Useful for converting time domain integration into frequency domain integration + + (t) (t)dt S (f )S (f )df + S (f )S (f )df 3 Dr. Peter Avitabile

14 Auto-Correlation Function The auto-correlation function is lim R τ + ( ) (t)(t + τ) dt T T Applying Parseval, rearranging terms, simplifying R and the inverse G ( τ) + S G + (f )S (f ) (f )e S jπft jπft (f )S (f ) R ( τ)e dτ df (f ) (3.7.9) (3.7.0) These are the Wiener-Khintchine Equations 4 Dr. Peter Avitabile

15 Cross-Correlation Function The cross-correlation function is lim T / R y( τ) (t)y(t T T T / + τ) dt Applying Parseval, rearranging terms, simplifying R y and the inverse G ( τ) y + S + (f )S Gy (f ) S jπft (f ) R ( τ)e dτ y y (f )e jπft (f )S df y (f ) (3.7.) (3.7.3) These are the Wiener-Khintchine Equations 5 Dr. Peter Avitabile

16 Fourier Response Technique The Frequency Response Function (FRF) is the input-output relation X( ω) H ( ω) F( ω) FT[(t)] FT[f (t)] (3.8.) Multiplying and dividing by the conjugate of the input force spectrum yields H( ω) S S f ( ω)s ( ω)s f f ( ω) ( ω) G G f ff ( ω) ( ω) (3.8.) S ( ω)s f ( ω) H( ω)s f ( ω)s f ( ω) G f ( ω) H( ω)g ff ( ω) 6 Dr. Peter Avitabile

17 Fourier Response Technique Schematic INPUT TIME FORCE FFT f(t) OUTPUT TIME RESPONSE y(t) IFT INPUT SPECTRUM FREQUENCY RESPONSE FUNCTION OUTPUT SPECTRUM f(j ω) h(j ω) y(j ω) 7 Dr. Peter Avitabile

18 Fourier Response Technique Using the frequency domain input-output relationships, Output Response System Characteristic X Input Forces the response due to many forces can be computed y N o i (jω) hij(jω)f j(jω) j The frequency response function is needed for this response h ij (jω) r ij,k r ij, + jω λ k jω λ k m k k 8 Dr. Peter Avitabile

19 Fourier Response Technique Frequency domain input-output schematic OUTPUT SPECTRUM y(j ω) h ij (jω) r ij,k r ij, k + jω p k jω p k m k FREQUENCY RESPONSE FUNCTION INPUT SPECTRUM f(j ω) y N o i (jω) hij( jω)f j(jω) j 9 Dr. Peter Avitabile