Introduction DCT

Transcription

1 Introduction... NASTRAN model and analytical model.... NASTRAN model.... Analytical model Comparison of NASTRAN and analytical model...7 Transformation to time domain...9. Displacement to velocity and acceleration...9. Mirroring....3 Inverse Fourier transformation....4 Tail Furter improvements Interpolation Adding zero Hz...7 Conclusion and recommendations...8 References... List of symols... Appendix A Derivation of te analytical model parameters... Appendix B Fourier transformation...3 B. Basics...3 B. Discrete Fourier Transformation...3 B.3 Fast Fourier transform...4 B.4 Properties of te Fourier transform...6 B.5 Symmetry...6 B.6 Error sources in Fourier transform...7 B.6. Signal leaage...7 B.6. Aliasing...8 Appendix C Convolution...3 C. Basic teory of convolution...3 C. Properties of te convolution product...3 Appendix D Testing metods...3 D. Transfer function estimate metod...3 D. Analytical transfer function determination...33 Appendix E Deconvolution...35 Entire or partial reproduction of te content of tis pulication in any form, witout preliminary autorization y letter y DAF is illegal, except for restrictions registered y law. Tis proiition also includes entire or partial rewriting of te pulication.

2 Introduction Te drive-off eavior is an important caracteristic for a truc. It influences ot te comfort and te performance of te truc. Te drive-off eavior of a truc depends mainly on te control strategy of te driveline (engine, clutc and gear ox), te caracteristics of te clutc facing and te dynamic eavior of te veicle structure (drive safts, cassis, cain suspension). In order to get te drive-off eavior on a ig level, optimization and tuning of te control strategy is a necessary part of te process. Tis optimization and tuning is mainly done on a real veicle, wic taes a lot of time and effort. In order to speed up and improve te process of optimizing and tuning and to e ale to anticipate on future veicles canges, it sould e possile to predict and calculate te drive off eavior up front. A necessary part of te simulation models to do so is, esides te control strategy, te caracteristic of te veicle dynamics. Witin DAF, NASTRAN models of te total veicle dynamics are availale. Recently tese models ave een extended wit a model of te driveline, wic includes a clutc plate, gearox drive saft and alf-safts. Wit tese NASTRAN models any aritrary transfer function can e determined. Tese transfer functions are all determined in te frequency domain. Because te drive-off eavior is mainly a transient penomenon, simulations in te time domain wit MATLAB/SIMULINK software are very useful. Terefore te availale NASTRAN model transfer functions will e transformed from te frequency domain into te time domain. In tis report te metod of translating a FRF into a time domain impulse response is discussed. First an overview of te NASTRAN model is sown, togeter wit an overview of an analytical model tat is used to approximate te NASTRAN driveline model. Next te necessary steps to get a proper transformation of te frequency domain transfer function to te time domain will e sown. Te transformation routine tat will e developed will e a general usale tool. Finally a conclusion and recommendations will e given. DCT 6-7

3 NASTRAN model and analytical model In order to get an optimal drive-off eavior, an optimization is necessary. Te optimization of te drive-off eavior is up to now mainly done y adusting/tuning te veicle and control strategy parameters during tests on a real veicle. Tis routine taes a lot of effort and time. Terefore tere is a need to mae te drive off eavior more predictale y means of simulations, wic will speed up and improve tis process/routine. For tis ind of simulations te veicle/driveline dynamics are a maor part of te simulation model. Witin DAF te software pacage NASTRAN is used to model te veicle eavior/dynamics. Te NASTRAN simulations are mainly performed in te frequency domain and te output consist of frequency response functions (FRF) Te manoeuvring and drive-off of a veicle are mainly transient penomena. Terefore a simulation of tis eavior in te time-domain using is more useful. Terefore te output of te NASTRAN model (FRF s) will e transformed from frequency to time domain. Tis transformation will e discussed in capter. In tis capter a description of te NASTRAN veicle model will e given. Next a asic analytical model will e sown. Tis model can e used to approximate and verify some transfer functions provided y te NASTRAN model. It will e used to descrie te time domain transformation in capter.. NASTRAN model To investigate te veicle eavior of te veicle, as was descried efore a NASTRAN model is used. Te NASTRAN model used was originally designed to perform fatigue and comfort analyses. Te original NASTRAN model is sown elow. Figure.: Original NASTRAN model used for fatigue and comfort analyses. Te original NASTRAN model as een expanded wit a model tat descries te driveline. Te driveline consists of a clutc plate, gearox, drive saft, cardanic oints, a differential gear and alf-safts. Te complete model can e used to determine te transfer functions etween te clutc torque and te veicle and driveline displacements. Tese transfer functions are a necessary part in te control loop to e optimized. Furtermore te influence of parameters lie gear ratios and saft stiffness on te veicle eavior can e investigated. Te NASTRAN driveline model is sown in te next figure. DCT 6-7

4 Figure.: NASTRAN driveline model wit all its components Te NASTRAN model determines te transfer functions y solving te general equation: ( t) Dqɺ ( t) K q( t) f ( t) M qɺ (.) Were q ( t) is te displacement column, ( t) damping matrix and K te stiffness matrix. Te general equation (.) is solved using: f de force column, M de mass matrix, D te q [ Ω M ΩD K] F a H F a (.) Ωt Ωt Were q( t) q e and f ( t) F a e [ Ω ] M ΩD H K is called te transfer function matrix or te matrix of frequency response functions (FRF matrix). All equations were found in [].. Analytical model Te drive-eavior is also determined y te caracteristics of te veicle driveline. Te driveline can e seen as a two rotating mass spring-damper system. A simple analytical model of te two mass spring-damper system can e used to approximate and verify some of te NASTRAN veicle model transfer functions. In tis analytical model te first mass represents te clutc plate and te primary part of te gearox. Te second mass represents te veicle wit its mass. Te stiffness and te damping stand for te caracteristics of te drive saft and te two alf-safts. Te metod of determining te values of,, and can e found in appendix A. Te two mass spring-damper system is sown in te next figure. 3 DCT 6-7

5 DCT Figure.3: Scematic overview of te analytical two mass spring damper system Te next equations can e derived for te standard two mass spring-damper system. M ɺ ɺ ɺɺ ɺɺ (.3) Using t e and t M e M Tis equation can also e written as: M (.4) Te equation for te first disc can e written as: ( ) ( ) M (.5) And te equation for second disc: ( ) ( ) (.6) ( ) ( ) (.7) (.8) Now te equation for te first disc can e rewritten y sustituting equation (.8) ( ) M (.9) So te transfer function etween te excitation moment and te displacement of te first disc is: M

6 H d ( ) (.) ( ) Using ɺ te transfer function etween excitation moment and rotational velocity of te first disc can e determined: H v ( ) (.) Next, using ɺ te transfer function etween te excitation moment and te acceleration of te first disc can e written as: H a ( ) ( ) (.) In te figure elow an example of te transfer functions for te first disc of te two mass spring-damper system same system is sown. Te natural frequencies are indicated, were n is an anti-resonance and n a resonance. Te static gains for te acceleration transfer function are indicated in te figure as well. displacement transfer function - n / H H -4-6 n / velocity transfer function n / - -4 n / n angle(h) angle(h) acceleration transfer function H / - /() angle(h) Figure.4: Transfer functions for te first disc of te two mass spring-damper system. 5 DCT 6-7

7 DCT Next te transfer functions for te second mass are determined. Equation (.5) can e rewritten into: ( ) ( ) M (.3) ( ) M (.4) Sustituting (.4) in to equation (.6) gives: ( ) M (.5) So te transfer function etween te excitation moment and te displacement of te second disc is: ( ) ( ) H d (.6) Using ɺ te transfer function etween excitation moment and rotation velocity of te second disc can e determined. ( ) H v (.7) Next, using ɺ ɺ te transfer function etween te excitation moment and te acceleration of te first disc can e written as: ( ) ( ) H a (.8) In te next figure te transfer functions of te second disc on displacement, velocity and acceleration level are sown.

8 H displacement transfer function n / H velocity transfer function n / angle(h) angle(h) acceleration transfer function n / H - -4 /() angle(h) Figure.5: Transfer functions for te second disc.3 Comparison of NASTRAN and analytical model To compare te analytical model wit te NASTRAN model, te transfer function etween clutc torque and te velocities of te first mass (clutc) and te second mass (end of alfsaft) are sown in te figure elow for ot te models. For te analytical model te parameters in appendix A were used. Te transfer functions were calculated for tird gear. H velocity transfer function first mass analytical model Nastran model H velocity transfer function second mass analytical model Nastran model angle(h) 5-5 angle(h) Figure.6: Comparison of te analytical model wit te NASTRAN model, sowing te transfer functions etween clutc torque and te velocity of te first mass (clutc) and te velocity of te second mass (end of alf-saft) for tird gear. 7 DCT 6-7

9 For te first mass, te analytical model coincides well wit te NASTRAN model, as can e seen in figure.6. For te motion of te first mass, wic is in fact te clutc, te simple two mass spring damper system is a good approximation for te far more complicated NASTRAN model. For te second mass, te approximation of te NASTRAN model y te analytical model is less accurate. Te approximation of te complete veicle structure y ust one simple mass is not accurate. Furtermore for ot te first and second mass only one degree of freedom is taen into account. Te eavior of te veicle (tus te second mass) cannot e descried using ust one degree of freedom (for te first mass one degree of freedom is correct ecause tere is ust one). Terefore te conclusion can e drawn tat in te investigation of te drive-off eavior te NASTRAN model cannot e totally replaced y te analytical model. Parameters lie te vertical cain acceleration, wic are important parameters in te veicle eavior, cannot e determined wit te analytical model. Terefore te NASTRAN model will still ave to e used and te NASTRAN model transfer functions will ave to e transformed in order to e ale to use in MATLAB/SIMULINK. 8 DCT 6-7

10 Transformation to time domain Te drive-off eavior is mainly a transient penomenon and furtermore influenced y te control strategy. Software lie MATLAB/SIMULINK is terefore very useful to investigate te drive-off prolems. In order to use tis software te NATRAN model output first as to e transformed from te frequency domain to time domain. Tis can e done in several ways. Maing a fit of te frequency domain transfer function is a metod tat is often used. Te fit is ten implemented in MATLAB/SIMULINK (using an s-loc). Disadvantage of using a fit is tat it will practically always contain a small error. Wen te veicle layout is canged a new fit as to e made. Te influence of te cange in layout on te veicle drive-off eavior will e difficult to determine. Te cange in drive-off eavior could also e a result of an error in te fit tat was used to approximate te frequency domain transfer function. Terefore anoter metod will e used; te Fourier transformation metod. Wit tis metod te transfer function or frequency response function tat was acquired using te NASTRAN veicle model will e converted to an impulse response function in te time domain. Te asic idea of te Fourier transformation is tat any function f ( x) can e formed as a summation of a series of cosine and sine terms of increasing frequency. Tis means tat any space or time varying data can e transformed into te frequency domain and vice versa. More information on te Fourier transform teory can e found in appendix B. In order to get a proper time domain transform, te following steps ave to e followed:. Displacement to velocity and acceleration Te output of te NASTRAN model is a transfer function tat descries te systems eavior on displacement level. Tis means tat all outputs are expressed as displacements. In te researc tat is done in order to investigate te drive-off eavior te relation etween te clutc velocity and te clutc torque is an important parameter. To investigate tis parameter te transfer function on displacement level is converted to a transfer function on velocity level. Using equation (B.4): df( u) πuf ( u) (.) du Wit u te frequency f in Hz and F te NASTRAN transfer function H ( f ) tis gives dh df ( f ) ( f ) πfh (.) In te next figure a typical NASTRAN model output transfer function on velocity level is sown. 9 DCT 6-7

11 Figure.: transfer function etween clutc torque and clutc speed Furtermore te vertical cain acceleration is an important output concerning te drive off comfort. Terefore te NASTRAN model transfer function etween clutc torque and cain displacement as to e converted to a transfer function etween clutc torque and cain acceleration. Using equation (B.5): H df ( f ) d ( πf ) H ( f ) (.3) In te figure elow te NASTRAN model transfer functions etween clutc torque and te vertical cain acceleration are sown for te first seven gears. Figure.: Transfer function etween clutc torque and vertical cain acceleration DCT 6-7

12 . Mirroring Wen a signal consists of real parts only, te Fourier transform of te signal is symmetrical wit respect to a folding frequency (see appendix B.5). Tis means tat Re( f ) Re( f ) and Im( f ) Im( f )) wit Re ( f ) te real part of H ( f ) and Im( f )) te imaginary part of H ( f ). Tis concept can also e used te oter way around. In order to ave an inverse Fourier transform signal tat is real, te original signal as to e symmetric. Tus, efore te NASTRAN model can e transformed to te time domain te NASTRAN transfer function first as to e made symmetrical. Tis can e done y folding te original transfer function data vector H ( f ) around a folding frequency f fold (see figure B.3). Te folding frequency is cosen to e te imum frequency of te NASTRAN model. Tis is done in order to eep te information in te new symmetric transfer function te same as in te original transfer function. Furtermore te mirrored part of te original transfer function H ( f ) is equal to te complex conugated of H ( f ). Tis means tat te real part of H ( f ) is line symmetrical and te imaginary part of H ( f ) is point symmetrical wit respect to te folding frequency (see B.5). H f can e made symmetrical in MATLAB using: Te transfer function ( ) H s ( f ) ([ real( H ), H ( end ), fliplr( real( H ( : end )))] [, imag( H ( : end )),, fliplr( imag( H ( : end )))]) Wit H ( f ) H te original NASTRAN model output transfer function (.4) Te zero for te imaginary part and H ( end ) for te real part indicates te values of ( f ) H at te folding frequency. Tis value is cosen to e zero for te imaginary part (oterwise te metod will not wor) and H ( end ) for te real part so tat it does not influence te transfer function. In te figure elow an example is sown of a transfer function and te corresponding symmetric transfer function transfer function symmetric transfer function H frequency frequency angle(h) angle(h) H frequency frequency Figure.3: Transfer function and te corresponding symmetric transfer function DCT 6-7

13 .3 Inverse Fourier transformation Now tat te NASTRAN model transfer function is made symmetric, it can e transformed from te frequency domain to te time domain. Tis can e done in MATLAB using te inverse fast Fourier transform commando ifft. ( t) ifft( H ( f )), wit ( f ) s H s te symmetric transfer function (.5) Tis transformation will result in an impulse response function. In te figure elow te impulse response functions for te end of te rigt alf-saft are sown on displacement, velocity and acceleration level. Te eavior of te end of te rigt (or left) alf-saft is te same as te eavior of te second mass of te analytical model. 5.5 x -3 displacement impulse response function 4 x -3 velocity impulse response function (t) (t) time [s] acceleration impulse response function time [s] (t) time [s] Figure.4: Impulse response functions descriing on displacement, velocity and acceleration level for te end of te rigt alf-saft Te impulse response function tat will e used to investigate te clutc udder eavior as to e a finite impulse response. Tis means tat te impulse response function as to e damped out completely at te end of te time ase. Te finite impulse response function as reaced a constant value at te end of te time ase, wic is maintained till infinity. As can e seen in figure.4, te impulse response function on acceleration level is te only impulse response function tat is finite. Tis is ecause te system is not constrained in space. Wen te system is suected to an impulse, te resulting displacement and velocity will eep varying till infinity, ut te acceleration will e finite. Terefore, te acceleration impulse response function for tis particular transfer function is te only function tat can e used. DCT 6-7

14 Because te clutc facing properties are depending on slip speed, te acceleration impulse response function as to e transformed to a velocity impulse response function. Terefore, an integration step as to e applied. In MATLAB/SIMULINK tis can e done y adding an integrator to te model..4 Tail In te figure elow a closer view on te acceleration impulse response function is given time [s] Figure.5: Acceleration impulse response function, descriing te eavior of te clutc plate Te figure on te rigt side sows te close-up of te tail. As can e seen in te figure, te impulse response does not totally damp out in time. Tere is a little tail at te end of te impulse response function. Tis is not realistic and does not correspond wit wat will appen in practice. Furtermore tis penomenon will give prolems wen time response signals will e determined using tis impulse response function. In order to determine te influence of te tail a test is performed. In tis test wite noise is applied to te impulse response and te transfer function of te impulse response filter is determined using transfer function estimate. Tis transfer function is ten compared to te original transfer function. Te testing metod tat was used is descried in appendix D. In te figure elow te test of te impulse response function is sown. 4 ode plot original frf versus frf determined wit impulse response original frf impulse response frf H angle(h) Figure.6: Testing te impulse response function 3 DCT 6-7

15 Figure.6 sows tat te transfer function of te tested impulse response function does not coincide wit te original transfer function. Especially in te low frequency area te magnitude and pase of te tested impulse response function differ from te original. Te prolem lies in te egin conditions of te filter tat was used to test te impulse response function. Te filter uses convolution to determine te output signal (see appendix C). Te wite noise input signal is convoluted wit te impulse response function. In tis convolution te impulse response function is used as a periodic repeating time signal. Tis is illustrated in te next figure. periodic repeating of impulse reponse function (t) time [s] Figure.7: Periodic repeating of impulse response function Te idea of convolution is tat te output signal at a certain point is equal to te multiplication of te input signal wit te value of te impulse response at tat certain point summed wit te multiplications of te all te previous input signals wit te previous values of te impulse response function (see appendix E). Terefore te output signal values at te eginning of a period are also determined y te values of te output signal of te previous period. In te testing metod tat was used to determine figure.6 only one period was used. Terefore te influence of te tail (wic would e te end of te previous period, see figure.7) on te first output values is not taen into account. In oters words, te egin conditions are taen equal to zero, wic is not te case ecause of te occurrence of te tail. In order to tae te egin conditions into account te testing metod as to e canged. Now a wite noise input signal consisting of multiple periods is applied to te impulse response function. Te output signal will terefore also consist of multiple periods. For te transfer function estimate only one period of te input and output signal is used, ut not te first period. Tis period will contain te influence of te previous period, tus te influence of te tail. Te transfer function corresponding to te impulse response function using tis metod is sown in te figure.8, togeter wit te original transfer function. 4 DCT 6-7

16 4 ode plot original frf versus frf determined wit impulse response original frf impulse response frf H angle(h) Figure.8: Testing te impulse response function wit egin conditions As can e seen in figure.8, te transfer function determined wit te impulse response coincides well wit te original transfer function. Te impulse response function wit te tail is terefore a good representation of te dynamic eavior of te system. However, te occurrence of te tail causes te prolem tat te egin conditions of te impulse response are not equal to zero. Terefore te transient pase of te output signal using te impulse response function is not correct. Te ig disadvantage of te tail is terefore tat te impulse response function can only e used for steady state situations or for multiple period input signals. Note tat in te real world, te egin conditions will always e equal to zero. Te system cannot react and move efore an excitation is applied to te system. Anoter test to cec te influence of te tail is an analytical transfer function determination. Tis metod is descried in appendix D.. Wit tis metod te influence tat cutting of te tail at te dynamic eavior is investigated. Te result is te same as for te wite noise metod; te impulse response wit te tail is a good representation, te impulse response witout te tail is not. Furtermore in tis test steady state signals are used to determine te transfer function. Tis metod again indicates tat te impulse response is a good representation, ut only for steady state situations. Te cause of te tail lies in te limited frequency range tat is analyzed in te NASTRAN model. Te transfer functions are only determined up to a imum frequency, f. Te NASTRAN transfer function can e seen as a multiplication of te infinite transfer function wit a window function. Tis concept is illustrated in te figure next using te analytical model. infinite model window function calculated finite model H.8 H angle(h) w angle(h) Figure.9: Infinite transfer function (left), window function and te calculated finite model ( NASTRAN model) 5 DCT 6-7

17 In te frequency domain, te infinite transfer function is multiplied wit te window function to otain te calculated finite transfer function. In time domain tis is equal to a convolution of te impulse response function of te infinite transfer function wit te impulse response of te window function (equation C.4). In te figure elow te impulse response functions of te infinite transfer function, te window function and te finite transfer function are sown. impulse response infinite transfer function. impulse reponse window function.4 impulse response finite transfer function time [s] time [s] time [s] Figure.: Impulse response functions of te infinite transfer function, te window function and te finite transfer function As can e seen in figure., te impulse response function of te window function causes te tail at te end of te impulse response of te finite transfer function. Furtermore also te first part of te impulse response is influenced y te window function. Tis can also e seen in figure.: te amplitudes of te infinite and finite impulse responses are not te same. Te size of te tail is determined y te amount of information tat is in te model ust efore te imum frequency were te model is cut off. Te occurrence of (anti-) resonance peas and te presence of a slope in te transfer function ust efore te imum frequency will increase te size of te tail. Te size tail will terefore increase for increasing gears (te resonance pea moves towards te imum frequency). In practice, only te finite NASTRAN model transfer function and te window function are nown. To reconstruct te original infinite transfer function and its corresponding impulse response function a deconvolution as een performed (see appendix E). Wit tis deconvolution an attempt as een done to filter out te influence of te window function in te calculated NASTRAN transfer function. Te results of tis deconvolution, sown in appendix E, are unsatisfying. Wit te deconvolution metod tat was used, it is not possile to remove te tail witout canging te dynamic eavior represented y te impulse response function. Terefore te calculated impulse response function can still only e used for steady state situations or for periodic input signals. Te representation of te transient pase using te impulse response function wit te tail is not correct..5 Furter improvements Tere are a few metods left to improve te results of te Fourier transformation if necessary..5. Interpolation Te time ase of te impulse response function is linearly proportional to te frequency step of te original transfer function, as is sown in equation (B.8). If te impulse response function is not yet damped out at te end of te time ase, interpolation can e used. Wen te original transfer function is interpolated, more data points can e created. Te frequency step ten ecomes smaller, creating a larger time ase for te impulse response function. 6 DCT 6-7

18 Decreasing te frequency step in te original NASTRAN model could also do tis ut tis would increase te calculation time NATRAN drastically. Terefore, interpolation of te NASTRAN transfer function is a more suitale solution..5. Adding zero Hz Wen an analytical model is used te zero Hz frequency point information can e determined easily. For NATRAN models tis is more complicated. NATRAN uses te inverse of te transfer function matrix to calculate te transfer function, as was sown in paragrap.. Tis metod will not wor for a frequency of zero. Tis prolem can e avoided y coosing a very small starting frequency for te NASTRAN model, wic is in fact a simulation of te zero frequency point. Tis metod is especially attractive to use wen a lot of information aout te dynamic eavior of te analyzed system is situated at low frequencies. 7 DCT 6-7

19 Conclusion and recommendations Te drive-off eavior is an important caracteristic for a truc. In order to get te drive-off eavior on a ig level, optimization is necessary. Tis optimization was mainly done on a real veicle, wic taes a lot of time and effort. In order to speed up and improve te process of optimizing and tuning and to e ale to anticipate on future veicles canges, it sould e possile to predict and calculate te drive off eavior up front. A necessary part of te simulation models to do so is, esides te control strategy, te caracteristic of te veicle dynamics. Te NASTRAN model outputs are transfer functions, wic are all in te frequency domain. Because te drive-off eavior is mainly a transient penomenon and furtermore influenced y te control strategy of te driveline, a simulation of te drive-off eavior is more useful. In order to e ale to do tese time domain simulations, te NASTRAN model transfer function ave to e transformed from te frequency domain to time domain. Tis is done using Fourier Transformation were te frequency domain transfer function will e transformed into a time domain impulse response. First a description of te NASTRAN model tat was used was given togeter wit an analytical model tat was used to approximate some of te NASTRAN model output transfer functions. Te analytical model can e used to approximate te transfer function etween te clutc torque and te clutc motion. Next te steps used for te transformation from te frequency domain to te time domain using Fourier transformation ave een descried. Furtermore te determined impulse response as een tested in order to cec te correctness. Te conclusion tat can e drawn after following tis procedure: Te analytical model can e used to approximate te transfer function etween clutc torque and clutc motion. Te approximation of te transfer function etween clutc toque and te motion of te alf-saft using te analytical model is less accurate and terefore not useful. Because te analytical model can only e used to determine a small part of te transfer functions te NASTRAN model will still ave to e used. Te impulse response function tat was determined for te use in time domain can only e used for steady state situations or for periodic input signals. Tis prolem is caused y te occurrence of a tail at te end of te impulse response. Te tail is caused y te limited frequency range tat was analysed in te NASTRAN model. In fact te real transfer function is multiplied wit a window function. In order to filter out te influence of tis window function a deconvolution as een performed. Te results of tis deconvolution owever are unsatisfying. Using tese findings te next recommendations are done: Cec te possiility to determine te impulse response function wit te NATRAN model y applying an impulse as input instead of a frequency excitation. Te prolem wit te tail will ten proaly not occur. Furtermore te transient eaviour will surely e taen into account. Increase te imum frequency for te NASTRAN model calculation. Tis will decrease te effect of windowing and tus te size of te tail. Disadvantage of tis metod is te increasing calculation time tat is needed. 8 DCT 6-7

20 Loo for oter possile metods for deconvolution. Wit a well-operating deconvolution metod, te effect of windowing could possily e filtered out. 9 DCT 6-7

21 References [] Bram de raer & Dic H. van Campen Mecanical Virations April [] Yerin Yoo Tutorial on Fourier Teory Internet paper, Marc [3] Prof David Heeger Signals, linear systems and convolution Internet paper, Septemer DCT 6-7

22 List of symols T eng engine torque [Nm] eng engine speed [rad/s] T c clutc torque [Nm] clutc speed [rad/s] c desired engine speed [rad/s] eng,d s c distance etween clutc plates [mm] M mass matrix D damping matrix K stiffness matrix. H frequency domain transfer function rotational displacement [rad] ɺ rotational speed [rad/s] ɺ ɺ rotational acceleration [rad/s ] inertia first mass of analytical model [g.m ] inertia second mass of analytical model [g.m ] stiffness analytical model [N.m] damping analytical model [N.m.s] imaginary unit frequency [rad/s] f ( t) impulse response function H s symmetric transfer function N numer of samples f frequency step T end time for impulse response function t time step for impulse response function f sample frequency s f folding frequency used for symmetry fold f imum frequency for frequency domain transfer function u ( t) time domain input signal y t time domain output signal ( ) DCT 6-7

23 Appendix A Derivation of te analytical model parameters Te values of,, and of te two mass spring damper system can e derived using te next input: prim inertia input axle gearox [g.m ] sec inertia output axle gearox [g.m ] m veicle mass [g] v R dynamic tyre radius [m] dyn i g gear ratio gearox gear dependent [-] i a gear ratio rear axle [-] inertia rear weels [g.m ] K K B B weel ds s ds s stiffness drive saft [N.m] stiffness alf-saft [N.m] damping drive saft [N.m.s] damping alf-saft [N.m.s] dyn weel ig ia mvr (A.) prim (A.) i sec g Ks K s (A.3) ia K ds (A.4) K s Kds K ds (A.5) i g B B s (A.6) i s a Bds (A.7) Bs Bds Te summation of dampers (A.7) is incorrect, ut in tis case were te damping of te drive saft is relatively large, te error will e minimal B (A.8) i ds g DCT 6-7

24 Appendix B Fourier transformation In order to transform te transfer function provided y te NASTRAN model from te frequency domain to te time domain a Fourier transformation is used. Te asic ideas of te Fourier transformation will e discussed in tis paragrap. B. Basics Te asic idea of te Fourier transformation is tat any function ( x) f can e formed as a summation of a series of cosine and sine terms of increasing frequency. Tis means tat any space or time varying data can e transformed can e transformed into te frequency domain and vice versa [5]. For a one-dimensional function f ( x) te Fourier transform is defined y: F πux ( u) f ( x) e dx (B.) And te inverse Fourier transform: f πux ( x) F( u) e du (B.) Were and u is called te frequency variale. Using Euler s Formula: θ e i cosθ sinθ (B.3) Equation. can e rewritten into; F ( u) f ( x)( πux sin πux ) cos dx (B.4) In tis equation te summation of sine and cosine terms is clearly visile. B. Discrete Fourier Transformation f is not continuous ut discrete. Te transfer function provided y te NASTRAN model for example is a discrete function. Te difference etween a continuous function and a discrete function is sown in figure B.. In practice, te function ( x) 3 DCT 6-7

25 continuous signal discrete signal x(t) time [s] - 5 Figure B.: Continuous function versus discrete function Terefore te Fourier transform is canged into te Discrete Fourier Transform [5] F N x ( u) f ( x) e πux N (B.5) And te inverse Discrete Fourier transform f N N u ( x) F( u) e πux N (B.6) Were N is te numer of samples of f ( x) or ( u) Te discrete Fourier transform taes ( N ) F. O time to process for N samples. B.3 Fast Fourier transform A proper decomposition of equation.5 can mae te numer of multiplications and addition operations proportional to N log ( N ) instead of N [8]. Tere are numer of different ways tat tis algoritm can e implemented. Te most common used algoritm is te Cooley-Tuey algoritm. In MATLAB te Fast Fourier transform Y of a vector x can e calculated using Y fft(x) And te inverse Fast Fourier transform y ifft(x) 4 DCT 6-7

26 In te figure elow an example is given of a transfer function in frequency domain and te corresponding transfer function in time domain. Te time domain function is called te impulse response function of te system. transfer function in frequency domain transfer function in time domain t H(f) (t) f frequency f time T Figure B.: A transfer function in frequency domain and te corresponding transfer function in time domain. Wen te original function consists of N sampled points: f f / N (B.7) T (B.8) f T t, wit f s te sample frequency N f s (B.9) N f / f f (B.) t T / f f s In case te impulse response function was acquired using te mirrored frequency domain transfer function: Nimpulse N frf (B.) Nimpulse N frf fs f (B.) T T 5 DCT 6-7

27 B.4 Properties of te Fourier transform Linearity: Te Fourier transform is a linear operation so tat te Fourier transform of te sum of two functions is given y te sum of te individual Fourier transforms. Terefore: { af ( x) g( x) } af( u) G(u) I (B.3) Complex conugate: Te Fourier transform of te Complex Conugate of a function is given y: * * I ( f ( x) ) F ( u) were f * ( x) is te complex conugate of ( x) Forward and inverse: ( F( u) ) f ( x) f. (B.4) I (B.5) Differentials: Te Fourier transform of te derivative of a function is given y ( x) df I πuf ( u) (B.6) dx And te second derivative is given y ( x) d f I ( πu ) F( u) (B.7) dx B.5 Symmetry In practice for most cases a sampled signal consists of real function values only. In tat case it can e easily sown tat tere is an additional property for te Fourier transform wic states [5]: X N * N [ p] X [ p], were * X is te complex conugate of X. (B.8) Tus, for real values of te sampled signal, te Fourier transform as a special symmetry wit f N respect to te frequency-line f fold, also called te folding frequency. Consequently, of all te N complex numers X only alf contain essential information. Te teory of symmetry is sown in te next figure. 6 DCT 6-7

28 Re[c n ] f fold f s / n Im[c n ] Figure B.3: Discrete frequency spectrum of complex Fourier series n B.6 Error sources in Fourier transform B.6. Signal leaage, wic is defined for infinite time, only a part is used wile sampling. In tis process, called windowing, only te data points from to T in time are taen into account. Tere are two situations were te windowing effect does not ave any influence: From a signal x ( t) In case of a transient signal (for example an impulse response) wic is (practically) damped out efore te end of te window In case of a periodic signal wen te window-lengt t is exactly a multiple of te period time If te windowed part is not exactly a multiple of te asic period of te armonic signal, discontinuities will arise on te edges of te window. Tis is illustrated in figure.4. Tis penomenon can e avoided using a different window function in stead of te rectangular window function. An example is te Hanning-window, wic reduces te edge discontinuities consideraly. In case of transient signals signal leaage can e avoided y multiplying te response wit a so-called exponential window. Tis window forces te response signal to e practically zero efore te end of te measurement time. However tis metod introduces some additional (numerical) damping in te response. If a system is udged on damping level using tese measurements tis additional damping sould e taen into account. 7 DCT 6-7

29 x(t) T time [s] Figure B.4: Time domain illustration of signal leaage B.6. Aliasing Aliasing is an effect wic is closely related to te sampling of te original (analogous) signal on te discrete time points n T. Te sampling frequency f s is defined as: [ Hz] t i f s (B.9) T A sampled signal can e written as a so-called pulse train wit variale intensity: ( t) [ T x( t) ] ( t n T ) x B δ (B.) Using [] tis relation can e transformed wit Fourier transformation into: B s ( f ) X ( f n f ) X (B.) Te Fourier transform of te sampled signal can e seen as te infinite sum of te exact Fourier transform eac time sifted over multiples of f. N situations can e distinguised: Situation : f f / s In te interval [ f f f / ] te function X B ( f ) is exactly equal to ( f ) s / s X. 8 DCT 6-7

30 So te approximation ( f ) X B is perfect in tis interval. Tis situation is sown in figure.5. X (f) X(f) -f f Situation : f > f / Tis situation is sown in figure.6 -f s f / f s s f s s Figure B.5: Situation witout Aliasing X (f) X(f) -f f -f s f s / f s f s Figure B.6: Situation wit aliasing Te asic Fourier transform is indicated y te tin solid line and te sifted functions f n X B f is indicated y te tic solid line. X ( f s ) y dotted Lines. Te final result ( ) In te interval [ f s / f f s / ] tis function ( f ) original X ( f ). Tis clearly sows te Aliasing effect. X B is not coinciding anymore wit te To avoid te Aliasing effect tere is one action tat sould always e taen care of: f f s / f, or f T > fold > (B.) Te frequency / T is called te Nyquist frequency. In case of a nown f te FFT parameters ave to e cosen suc tat tis criterion is fulfilled. 9 DCT 6-7

31 Appendix C Convolution C. Basic teory of convolution Wen te transfer function provided y te NASTRAN model is transferred from te frequency domain to te time domain, te result will e an impulse response function. Tis function descries te way te system reacts to a unit impulse. ust lie te NATRAN model transfer function te impulse response function descries te dynamic eaviour of te system. Once te impulse response function is determined, te system response to any input signal can e predicted. In order to mae tis prediction a convolution product is needed u(t) Linear System (t) y(t) Figure C.: A linear system Te output y(t) can e determined y applying a convolution product of te input u(t) and te systems impulse response function (t) [9]. y ( t) u( t) (t), were is te sign used for te convolution product. (C.) Te concept of convolution is sown in figure.7. Wen te impulse response function of a system is nown, te output can e determined. First te input signal u(t) is transformed into te sum of a set of impulses. Te response to eac input pulse can e calculated using te impulse response function (t). Te output signal y(t) is equal to te sum of te responses of all input pulse in te input signal u(t). Figure C.: Caracterizing a linear system using its impulse response. 3 DCT 6-7

32 Te convolution integral is defined as: ( t) u( t) t u( s) ( t s) y ( ) ds (C.) And te discrete form: ( ) u( ) ( ) y (C.3) signal x(t) t s signal (t) Figure C.3: Grapical presentation of te convolution integral Te lengt of te output vector y in discrete form: lengt ( y( t) ) lengt( u( t) ) lengt( ( t) ) t C. Properties of te convolution product Some general properties of te convolution product Commutative: x y y x x y z x y z (C.4) x z y z x y Associative: ( ) ( ) Distriutive: ( ) ( ) ( ) z Some properties of te convolution product wit respect to te Fourier transform. x x ( t) y( t) I X ( f ) Y ( f ) ( t) y( t) I X ( f ) Y ( f ) I ( f ) Y ( f ) x( t) y( t) I ( f ) Y ( f ) x( t) y( t) X X (C.5) 3 DCT 6-7

33 Appendix D Testing metods In order to validate te correctness of te impulse response tat was determined two testing metods are used. D. Transfer function estimate metod Te first metod is te transfer function estimate metod. In order to use tis metod, te impulse response function first as to e implemented in a MATLAB or MATLAB/SIMULINK model. Tis can e done y using te filter function. In MATLAB te filter commando is used, in MATLAB/SIMULINK a Finite Impulse Response filter loc (FIR) is used. Now te impulse response function can e compared to te original transfer function y applying a wite noise signal to te impulse response filter. Te SIMULINK model were wite noise is used to test te impulse response function is sown in te figure elow. Figure D.: MATLAB/SIMULINK model used to test te impulse response function In MATLAB te same routine can e done using te following commandos. x randn(size()); a zeros(size()); a() ; y filter(, a, x) wit te impulse response function (t). taing te egin conditions into account: creating input signal wit multiple periods (in tis case 5): ne 5; xe []; for i:ne, xe [xe x]; end xe [xe x]; 3 DCT 6-7

34 ye filter(, a, xe) y ye((ne*lengt()):(ne*lengt()lengt())); Te impulse response filter can now e analyzed and compared in te frequency domain to te original frequency domain transfer function y using te Transfer function estimate (tfe) commando. [ H, f ] tfe( x, y, nfft, fs) Were: H estimated transfer function f corresponding frequency vector x wite noise input signal nfft numer of points used to estimate te transfer function fs sampling frequency of te original model D. Analytical transfer function determination One of te disadvantages of te transfer function estimate metod is tat a possile occurrence of te tail will not e taen into account, unless multiple-period input signals are used, as was explained efore. Te tfe function does not tae te egin conditions into account. Tis is undesired wen te impulse response function as to e ceced. Terefore also anoter metod is used to cec te functionality of te acquired impulse response function. In tis metod a sine signals wit varying frequencies are put on te FIR filter in stead of wite noise. Te magnitude and te pase of te transfer function are ten determined analytically. Te magnitude can e determined y dividing te imum amplitude of te output signal y te imum amplitude of te input signal. Te imum amplitude is determined in te steady state area, after te transient penomenon in te first seconds of te output signal. To determine te pase te imum amplitude of te input and output signal are scaled to one and ten te two signals are summed. Te imum amplitude of te resulting signal can e used as a standard for te pase. Te relation etween te amplitude of te summed input and output signal and te pase are sown in figure D.. To determine te sign of te pase, te sign of ot te input and te output signal are ceced. Wen te output signals sign canges efore te sign of te input signal does, te pase will e positive and vice versa. Te main disadvantage of tis metod is tat less transfer function information can e acquired compared to te transfer function estimate metod. 33 DCT 6-7

35 pase [rad] amplitude(input output) Figure D.: Relation etween te imum amplitude of te summed scaled input and output signal and te pase. In te figure elow an example is sown of te analytical transfer function determination metod. In tis example te influence of te tail is sown. In te left figure te tail is taen into account, in te rigt figure te tail is cut of. transfer function wit tail transfer function witout tail H H angle(h) 5 angle(h) Figure D.3: Bode plot of te NASTRAN transfer function versus te transfer function acquired using te impulse response function wit (left) and witout (rigt) te tail. 34 DCT 6-7

36 Appendix E Deconvolution As was explained efore, te calculated impulse response as a little tail at te end. Tis tail was caused y te limited frequency area tat was analyzed in NASTRAN. Te calculated NASTRAN transfer function is a multiplication of te infinite transfer function wit a window function (see paragrap.4). In time domain tis is equal to a convolution of te impulse response of te infinite model wit te impulse response of te window function. In tis case, te impulse response of te finite model and te window function are nown and te impulse response of te infinite model is unnown. In order to determine tis infinite model deconvolution is applied. Note tat te result will in fact not e a real infinite model, ut a model wit a longer frequency range. Te lengt of te frequency range is determined y te lengt of te window function, wic can e cosen freely. First te convolution equation (C.) is written in matrix form n un u un 3 n n 3 n 3 n 3 3 y y y3 y n or H u y (E.) In tis case i is te impulse response coefficient of te window function, u i te impulse response coefficient of te infinite model and y i te impulse response coefficient of te finite model (NASTRAN model). In order to decrease te calculation time, only a part of te window function impulse response is used. Terefore first te zero-frequency component of te impulse response function is sifted to te center of te spectrum. Tis can e done using fftsift in MATLAB. Next a part at te eginning and te end of te impulse response function is disposed. Tis part can e disposed ecause te impulse response is almost equal to zero. Tis is illustrated in te next figure..5 sifted impulse response..5. (t) disposed used disposed time [s] Figure E.: Sifted impulse response 35 DCT 6-7

37 Te same routine is applied to te impulse response function of te NASTRAN model. Equation (E.) can now e rewritten into: un u un y y y (E.) Tis matrix form equation cannot e solved. Tere are n unnown variales of u and only n equations. In order to e ale to solve tis prolem te size of H as to e reduced from ( n ) n to n n. Tis can e done in several ways. Here, two ways are sown:. use only te rigt alf side of te matrix H. te matrix ten canges to: (E.3) In tis way no egin conditions are taen into account.. use te center part of te matrix H. te matrix ten canges into: (E.4) Now a part of te egin conditions are taen into account. In order to determine U, te impulse response of te infinite model, te inverse of H as to e determined. U H Y (E.5) Because H is a large sparse singular matrix, singular value decomposition is used to determine te inverse of H. In MATLAB: [ U S, V ] svds( H, ), (E.6) H V (:,: ) * ( diag( / diag( S( :,: ) ))) * U (:,: ) ( )' (E.7) 36 DCT 6-7

38 Wit te numer of largest singular values and associated singular vectors of te matrix H, U te n ortonormal columns, S te diagonal and V te n ortonormal columns. U * S * V ' is te closest ran approximation to H. In order to determine te optimal value for, te numer of singular values, te so-called L-curve plot is used. In tis plot te norm of te solution is plotted against te norm of te error for different values of. Te optimal value for lies at te point were ot te norm of te error and te solution are small. Tis is illustrated in te next figure. L-curve Uest optimum H.Uest - Y' Figure E.: L-curve used to determine te optimal numer of singular values Now, ot equation (E.3) and equation (E.4) are used to calculate te impulse response functions of te infinite model. Tis calculation was done using te analytical model. Te window function is cosen to e suc tat te infinite model will e four times longer ten te original model. Te impulse responses are now calculated using te deconvolution metod. Next te actual impulse response function of te infinite model is determined y increasing imum frequency of te analytical model. Te impulse response functions are ten tested y applying wite noise (see appendix D) and compared wit te original transfer function. In te next figure te calculated impulse response for te infinite model using te rigt side part of te window impulse response matrix (equation (E.3)) is sown togeter wit te real impulse response of te infinite model. Te tird gear was used for te calculation impulse response wit deconvolution using rigt side part wit deconvolution real 4 ode plot input frf versus numerical experiment (t) H [db] original tf tf infinite impulse response tf finite impulse response angle(h) [deg] time [s] - - Figure E.3: Impulse response calculated wit deconvolution (left) and tested using wite noise 37 DCT 6-7