DERIVATION OF MILES EQUATION Revision D

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1 By Tom Irvie July, DERIVATION OF MILES EQUATION Revisio D Itroductio The obective is to derive Miles equatio. This equatio gives the overall respose of a sigle-degree-of-freedom system to base excitatio where the excitatio is i the form of a radom vibratio acceleratio power spectral desity. Derivatio Cosider a sigle degree-of-freedom system m &&x k c &&y where m equals mass, c equals the viscous dampig coefficiet, ad k equals the stiffess. The absolute displacemet of the mass equals x, ad the base iput displacemet equals y. The free-body diagram is m &&x k(y-x) cy (& x&) Summatio of forces i the vertical directio, F mx && () mx && c(& y x&) k ( y x) ()

2 Substitutig the relative displacemet terms ito equatio () yields mz (&& &&) y cz& kz (3) mz && cz& kz my && () Dividig through by mass yields, By covetio, && z ( c/ m)& z ( k / m) z && y (5) ( c/ m) ( k / m) where is the atural frequecy i (radias/sec), ad is the dampig ratio. Substitutig the covetio terms ito equatio (5) yields && z z& z && y (6) Take the Fourier trasform of each side. Let Y( ) X( ) { z z z} e t & & dt { & y } e t dt & (7) { y(t) } e t dt t { x(t) } e dt Now take the Fourier trasform of the velocity term dz(t) { z(t) } e t & dt e t dt dt ()

3 Itegrate by parts { z(t) & } e t dt d{ z(t)e } t [z(t)]( )e t dt (9) { z(t) } e t dt z(t)e t & () z(t)e t dt () z(t)e t as t approaches the ± limits. () { z(t) } e t & dt () z(t)e t dt () { z(t) & } e t dt ()X( ) (3) Furthermore d z(t) { z(t) } e t & dt e t dt dt () dz(t) dz(t) { z(t) } e t dt d e t & [ ( )e t dt dt (5) dt dz(t) dz(t) { z(t) } e t dt e t & () e t dt dt (6) dt dz(t) dt as t approaches the limits. e t ± (7) dz(t) { z(t) } e t & dt () e t dt () dt 3

4 { z(t) & } e t dt ()()Z( ) & (9) { z(t) & } e t dt Z( ) & () Recall { z z z} e t & & dt { & y } e t dt & () Let the subscript A deote acceleratio. By substitutio, Z( ) ( Z( ) ) Z( ) Y A ( ) () ( ) Z( ) Y A ( ) (3) YA ( ) Z ( ) () ( ) Z A ( ) Z( ) (5) Y A ( ) Z A ( ) ( ) (6) The relative acceleratio equatio ca be expressed i terms of Fourier trasforms as ZA ( ) X A ( ) Y A ( ) (7) XA ( ) Z A ( ) Y A ( ) ()

5 Y A ( ) A ( X ) Y A ( ) (9) ( ) [ ( ) ) X ]Y A A ( ) ( (3) ( ) [ ) X ]Y A A ( ) ( (3) ( ) Multiply each side by its complex cougate [ )Y * ( ) X * ]Y A A A ( )X ][ ( A ( ) (3) ( ) ( ) [ ( ]Y * A ( )Y ( ) X * A A ( )X ) A ( ) (33) ( ) ( ) [ () ]Y * A ( )Y ( ) X * A A ( )X A ( ) (3) ( ) ( ) The Fourier trasforms are coverted ito power spectral desities usig the method show i Referece 3, where T is the duratio. lim * T X A ( )X A ( ) / T X APSD ( ) (35) limt * Y A ( )Y A ( ) / T Y APSD ( ) (36) 5

6 [ ( ) ]Y APSD ( ) X APSD ( ) (37) ( ) ( ) Equatio (A-3) ca be trasformed as a fuctio of frequecy f as follows f [f ( f ) ]ŶAPSD(f ) Xˆ APSD (f ) (3) (f f ) (f f ) Divide each side by f Xˆ APSD (f ) [ (f / f ) ]Ŷ APSD (f ) ( (f / f ) ) (f /f ) (39a) Let f / f, [ ( ) ] APSD (f ) Xˆ Ŷ APSD(f ), f / f (39b) ( ) () Defie H ( ) as H ( ) ( ) () Multiply by the complex cougate H ( ) H *( ) ( ) ( ) () Note that ( ) H * ( ) ŶA PSD (f ), f / f Xˆ A PSD (f ) H () 6

7 Rearrage equatio () as H ( ) H *( ) (3) Solve for the roots R ad R of the first deomiator. ( ) ± ( ) R,R () ± R,R (5) R, R ± (6) Solve for the roots R3 ad R of the secod deomiator. ( ) ± ( ) R3,R (7) ± R3,R () R3, R ± (9) Summary, R (5) (5) R R3 (5) (53) R (5) 7

8 Note R -R* (55) R3 R* (56) R -R* (57) Now substitute ito the deomiators. H ( ) H *( ) (5) H ( ) H *( ) ( R)( R)( R3)( R) (59) Substitute equatios (55) through (57) ito equatio (59). H ( ) H *( ) ( R)( R* )( R* )( R) (6)

9 Expad ito partial fractios. ( R)( R* )( R* )( R) α ( R) β ( R* ) λ ( R* ) σ ( R) (6) Multiply through by the deomiator o the left-had side of equatio (6). [ ] α ( R)( R* )( R* )( R) ( R) β ( R)( R* )( R* )( R) ( R* ) λ ( R)( R* )( R* )( R) ( R* ) σ ( )( )( )( ) ( R) R R* R* R (6) 9

10 [ ] α β λ σ ( R* )( R* )( R) ( R)( R* )( R) ( R)( R* )( R) ( R)( R* )( R* ) [ ] (63) [ ] α β λ σ α β λ σ ( R* )( R) ( ( R R* ) RR* )( R) ( ( R R* ) RR* )( R) ( ( R R* ) RR* )( R* ) ( 3 R R* RR* ) ( 3 ( R R R* ) ( RR* R RR* ) R R* ) ( 3 ( R R R* ) ( RR* R RR* ) R R* ) ( 3 ( R* R R* ) ( RR* RR* R* ) RR* ) (6) (65)

11 [ ] α β λ σ ( 3 R R* RR* ) ( 3 R* R R R* ) ( 3 R* R R R* ) ( 3 R R* RR* ) [ ] (66) [ α β λ σ] 3 [ Rα R* β R* λ Rσ] [ R* α R β R λ R* σ] [ RR* α R R* β R R* λ RR* σ] (67) [ ] [ α β λ σ] 3 [ Rα R* β R* λ Rσ] [ R* α R β R λ R* σ] [ R* α Rβ Rλ R* σ] RR * (6)

12 Equatio (6) ca be broke up ito four separate equatios, (69) through (7). α β λ σ (69) [ R R* β R* λ Rσ ] α (7) [ R* α R β R λ R* σ] (7) [ R * α Rβ Rλ R* σ] RR* (7) The four equatios are assembled ito matrix form. Recall R R* R* R R* R R R* R R α R β R* λ R* σ / ( RR* ) (73) (7) R R* (75) R R* (76) Substitute equatio (76) ito (73). R R* R* R* R R R* R R α R β R* λ R* σ (77)

13 3 Multiply the first row by * R ad add to the third row. σ λ β α R* R R R* R* R R* R R R* R* R (7) Scale the third row. σ λ β α R* R R R* R R* R* R (79a) Multiply the third row by - ad add to the first row. σ λ β α R* R R R* R R* R* R (79b) Multiply the first row by -R ad add to the secod row. Also multiply the first row by R* ad add to the fourth row. σ λ β α R* R R R R* R* () Multiply the third row by -R* ad add to the secod row. Also, multiply the third row by R ad add to the fourth row.

14 R* R α R β λ R* σ () The first row equatio yields α σ () The third row equatio yields λ β (3) Equatio () thus reduces to R* R R λ R* σ () Complete the solutio usig Cramer's rule. Recall R* R λ det er mi at R R* σ (5) R R* R (6) R (7) ( ) R () ( ) R (9) R* (9)

15 R * (9) ( ) R * (9) Thus, ( ) R * (93) R R* (9) [ R R* ] 6 (95) R* R λ det er mi at 6 (96) R R* σ λ 6 det er mi at R R* (97) λ 6 λ ( R* ) ( R* ) R R (9) (99) Recall, R () 5

16 λ () λ () λ (3) σ 6 R* det er miat R () σ σ [ R* ( )( R) ] 6 [ R* ( )( R) ] (5) (6) R (7) σ ( ) () 6

17 7 σ 3 (9) ( ) ( ) σ () ( ) ( ) σ () Recall, σ α () β λ (3) The complete solutio set is thus σ λ β α ()

18 The partial fractio expasio is thus H ( ) H *( ) α ( R) β ( R* ) λ ( R* ) σ ( R) (5) The overall respose & x& GRMS ca be obtaied by itegratig Xˆ A PSD(f ) across the frequecy spectrum ad the takig the square root of the area per Refereces ad. & x& ( ) GRMS f, H( ) H * ( ) ŶA PSD(f ) df (6) & x& ( ) GRMS f, f H( ) H *( ) ŶA PSD(f ) d (7) Now assume that Ŷ A PSD (f ) is a costat level represeted by the variable A. Furthermore, assume the level is costat over a ifiite frequecy rage, startig at zero. & x& ( ) GRMS f, Af H( ) H * ( ) d () {& x& ( f, ) } GRMS A f H ( ) H * ( ) d (9) Substitutig equatio (5) ito (9) yields

19 9 ( ) { } d d d d A f f, x& GRMS & ()

20 ( ) { } d d d d A f f, x& GRMS & ()

21 ( ) { } l l l l A f f, x& GRMS & () Note that [ ] ( ) ( ) π φ k exp y x l y x l (3) where φ x y arcta ()...,,, k ± ± (5) [ ] [ ] π φ k y x l y x l (6)

22 Take k. ( ) { } arcta l arcta l arcta l arcta l A f f, x& GRMS & (7) Both the real ad imagiary compoets of the atural log terms cacel out at the lower itegratio limit. The imagiary compoets of the atural log terms cacel out at the upper limit. The sum of the real compoets of the atural log terms approaches zero as the upper limit approaches ifiity.

23 3 ( ) { } arcta arcta arcta arcta A f f, x& GRMS & ()

24 ( ) { } arcta arcta arcta arcta A f f, x& GRMS & (9) Now make trigoometric substitutios.

25 5 ( ) { } { } π π π arcta arcta arcta arcta A f f, x& GRMS & (3) ( ) { } ( ) π GRMS A f, f x& & (3) ( ) { } ( ) π GRMS A f, f x& & (3) ( ) { } ( ) π A f, f x && GRMS (33) ( ) ( ) π A f, f x & GRMS & (3)

26 ( ) For., (35) For small dampig, & A f π x ( f, ) GRMS (36) Q (37) π & x GRMS ( f, Q) f Q A (3) Now cosider the realistic case where the acceleratio power spectral desity level varies with frequecy. Nevertheless, costrai the acceleratio power spectral desity level to be costat withi ± octave of the atural frequecy. A ŶAPSD(f ) (39) π & x GRMS ( f, Q) f Q ŶAPSD(f ) () Note that equatio (), the Miles equatio, may lead to error, particularly if the acceleratio power spectral desity fuctio has sigificat variatio with frequecy. Thus, the geeral method i Referece 3 is the preferred method. The geeral method is show as equatio (). N ( ) { ( } f, i ) & x GRMS ŶAPSD(fi) Δf i, i fi / f () i [ ] i i 6

27 The geeral method allows the power spectral desity to vary with frequecy. It also allows for power spectral desity iputs with fiite frequecy limits. Refereces. L. Meirovitch, Aalytical Methods i Vibratio, Macmilla, New York, W. Thomso, Theory of Vibratio with Applicatios, d Editio, Pretice Hall, New Jersey, T. Irvie, A Itroductio to the Vibratio Respose Spectrum, Vibratiodata Publicatios,

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