is called the 3dB bandwidth. (d)(6pts) The Dryden model MIL-F-8785C for simulating turbulence in relation to the short period mode is

Size: px

Start display at page:

Download "is called the 3dB bandwidth. (d)(6pts) The Dryden model MIL-F-8785C for simulating turbulence in relation to the short period mode is"

Valentine Atkins
5 years ago
Views:

1 Homeork 6 AERE355 Fall 7 Due /7 (M) SOLUTION PROBLEM (3pt The transfer function of a first order Lo Pass Filter (LPF) is frequency rane [, ] is called the filter 3dB bandidth () (a)(8pt The dc (ie very lo frequency) ain is ( i) computation to determine hy the interval [, ] H( ( s / H ives lo H( i) lo() db is called the 3dB bandidth here the ) Use the same / Solution: H ( i ) ives H( i ) Hence: lo H( i ) lo / lo() 3 db i The interval is the rane of frequencies such that the manitude of is ith 3dB of its static ain [, ] H ( i) (b)(6pt The bode(h) command as used to obtain the Frequency Response Function (FRF) shon at riht for rad/s For an input u( t) sin(t) use the information in Fiure (b) to estimate the values of M and expression for the output y ( t) M sin(t ) Solution: M () () db db M o ( ) 83 45rad (c)(pt The term lo pass filter is due to the fact that hen an input is supplied to H (, the output is a filtered version of the input Specifically, loer frequency (meanderin) components are alloed to pass freely, hile hiher frequency (rapidly jitterin) components are suppressed To visualize this, use the lsim command to complete the code at (c) in relation to a measured hite noise input, This is an array of numbers randomly u(t) sampled from a normal distribution havin mean and standard deviation Overlay plots of the input and output, and discuss ho they differ Solution: [See (c)] It is clear that the output is far less jittery than the input Fiure (b) FRF for H( ith Fiure (c) Plots of input and output (d)(6pt The Dryden model MIL-F-8785C for simulatin turbulence in relation to the short period mode is / 6 8 ( / 4b) H ( The input to this model is hite noise ith standard deviation Obtain the / 3 V L (4b / V ) s expression for the -3 db bandidth of this shapin filter Then describe hat the involved parameters are, and ho they control the bandidth Solution: V / b This is proportional to the plane speed V, and is inversely proportional to the in span b 4

2 PROBLEM (35pt The dynamics of the short period response to ind can be approximated by: q M Z M Z M M q Z q M M q q () Consider the Boein 747 (@ M=9) ith the folloin information: u=87; =37; Za=-34365; Ma=-68; Mad=-43; Mq=-396; Zde=86; Mde=-4; (a)(pt The dynamical system () is a -input/-output system Write a Matlab proram that uses the command sstf to obtain the to transfer functions of the system () associated ith only the input Solution: [See (a)] H ( ) 453 s 44 ( s 9335s 774 and (t) s 77 s 9335s 774 ( ) H q ( (b)(6pt Use the Matlab bode command to obtain plots the to FRF s associated ith the transfer functions in (a) Solution: [See (b)] Manitude (db) Phase (de) Phase (de) Manitude (db) SortPeriod FRF for a(t) in Response to (t) Frequency (rad/ Short Period FRF for q(t) in Response to (t) - Frequency (rad/ ( Fiure (b) Plot of the short period mode FRF s ) ( ) ( i) and H q ( i) H (c)(pt Give the system characteristic polynomial, and use it to obtain numerical values for (i) (iv) d, and (v) Solution: p ( s 9335s 774 s n n, (ii) n, (iii), ns Hence: (i) n 9335/ 4667 ; (ii) n r / s ; (iii) 4667/ ; (iv) 48 r s ; (v) / 43 sec d n / n (d)(9pt Use the Matlab command step to obtain a plots of the to step responses to a sharp ede ust Use the data cursor in relation to each response to estimate (i) d, and (ii) 4 [The data cursor information should be included in your plots] Finally, compare these numbers to those associated ith your ansers in (c) Solution: [See (d)]

3 3 Amplitude Amplitude 4 x -4 System: Ha Time (second: 8 Amplitude: 3 3 Short Period a(t) in Response to unit step (t) System: Ha Time (second: 97 Amplitude: 35 System: Ha Time (second: Amplitude: 84e-5 Time (second x Short Period q(t) in Response to unit step q (t) System: Hq Time (second: 54 Amplitude: -5 System: Hq Time (second: 498 Amplitude: -36 System: Hq Time (second: Amplitude: Time (second Fiure (d) Plots of the short period mode (t) and q(t) responses to a sharp ede (ie unit step) ust, (t) In relation to In relation to (t) Results from (c): : T (379 8) 5sec / T 5r s & 4 97sec T d d d / (498 54) 488sec / T 88r & 4 849sec q(t) : s d d d / 48 r s & sec d / Comment: The results from both plots compare favorably to the results in (c), iven that they ere obtained by eyeballin :)

4 4 PROBLEM 3(35pt As a continuation of Problem, in this problem e consider the short period mode responses to von Karmon turbulence (t) From p8 of Nelson, the vertical ind turbulence spatial psd is iven by: 8 8 (339L ) (339 ) L L 3 L ( ) 3 / 6 (654*) (339L ) (339 L ) [*The rihtmost approximate equality is reasonable, and is needed to simplify the folloin parts] In vie of the fact that ( ) [see (655) on p9], the temporal psd is iven by: L 8 (339L 3 (339L ) a ) c ( b ) here e have, for convenience, defined a 339L ; b a 8/ 3 ; c L / (3) (a)(5pt Define the hite noise psd From (3) sho that the expression for Solution: i) ( ) i) Then (65) on p8 is: ( ) ( ) i) c[ b( i)] in relation to the rihtmost expression is iven by i) [ a( i)] c ( b ) c[ b( i)] c[ b( i)] c[ b( i)] i) i) Hence, i) a [ a( i)] [ a( i)] [ a( i)] c( bs ) 4 (b)(5pt From (a) the turbulence shapin filter is G ( For L ft and u 87fps use the bode ( as ) command to obtain a plot of this filter FRF Solution: [See 3(b)] Manitude (db) Vertical Von Karmon Turbulence Shapin Filter Phase (de) Frequency (rad/ Fiure 3(b) Shapin filter for the von Karmon vertical turbulence psd (c)(5pt The shapin filter (3) for the turbulence is a lo-pass filter From your plot in (b) find the ratio of the poer density at rad/sec divided by the enery at rad/sec Give your anser both as a difference in ddb, as ell as an actual ratio Solution: The psd at rad/sec is -db relative to the enery at rad/sec Since lo ( r), the ratio is r

5 5 4 (d)(7pt Use the Matlab command * normrnd(,,,) for /(L ) 57( 4 ), to simulate the hite noise that ill serve as the input to the shapin filter Then use the command lsim(sys,, tvec) to compute the filter output Your time array should be tvec should use the increment, and here 339L is the shapin filter time constant Plot both the input (hite noise) and output (turbulence) time series Solution: [See 3(d)] dt / Fiure 3(d) Plots of hite noise input and of simulated turbulence, (t) (e)(7pt Run your simulated turbulence in (d) throuh your -input / -output short period transfer function to obtain plots of the simulated and (t) Solution: [See 3(e)] q(t) responses Fiure 3(e) Short period mode (t) and q(t) responses to (t) H( (f)(6pt The block diaram related to the above is shon at the riht It reveals the (t) ( t) (t) poer of the Laplace transform Specifically, it shos that q( t) A( H( W ( Q( Hence, as an extension of the discussion in (b) e obtain the folloin expression for the psds S ( ) for (t) and q (t) : H( i) i) S ( ) Use the bode command in relation to the transfer funtion q H( in order to arrive at plots of these to psds Solution: [See 3(f)] Fiure 3(f) Plots of theoretical psd s for (t) and q (t)

6 Matlab Code %PROGRAM NAME: h6m %PROBLEM : %(b): =; s=tf('s'); H=/((s/) + ); fiure() bode(h) title('frf for H ith _B_W= r/s') % %(c): rn('shuffle') %This ill ensure that no to students have the same numbers n=; u=normrnd(,,n,); dt=; t=:n-; t=dt*t'; y=lsim(h,u,t); fiure() plot(t,[u,y]) title('plots of Input and Output') xlabel('time (sec)') %==================================================== %PROBLEM %Simulate Boein 747 response to von Karmon ind turbulence % Plane Information: u=87; =37; Za=-34365;Ma=-68;Mad=-43;Mq=-396;Zde=86;Mde=-4; % A, B, C & D Matrices: A=[Za/u, ; Ma+Mad*Za/u,Mad+Mq]; B=[-Za/u^,;-Ma/u^,-Mq]; C=eye(); D=zeros(,); %(a): % Creation of state space system & transfer functions related to : Hss=ss(A,B,C,D); [Hn,Hd]=sstf(A,B,C,D,); %Transfer functions re: ; Ha=tf(Hn(,:),Hd) Hq=tf(Hn(,:),Hd) %(b): fiure() vec=lospace(-,,); subplot(,,),bode(ha,vec) title('short Period Bode Plot for a(t) in Response to (t)') subplot(,,),bode(hq,vec) title('short Period Bode Plot for q(t) in Response to (t)') %(d): fiure() subplot(,,),step(ha) title('short Period a(t) in Response to unit step _(t)') subplot(,,),step(hq) title('short Period q(t) in Response to unit step q_(t)') %=========================================================== %PROBLEM 3 %(b): % Turbulence Information: L=^4; si=pi/(*l); s=(si**l)/pi; tau=339*l/u; =/tau; Hnum=s*[tau*(8/3)^5, ]; Hden=[tau^, *tau, ]; H=tf(Hnum, Hden); fiure(3) bode(h) title('vertical Von Karmon Turbulence Shapin Filter Bode Plot') %(d): %TURBULENCE SIMULATION: n=; n=sqrt(si)*randn(n,); A=-; B=; C=; D=; 6

7 dt=tau/; t=:n-; t=dt*t'; sysu=ss(a,b,c,d); =lsim(sysu,n,t); fiure(3) subplot(,,), plot(t,n) title('white noise input to the shapin filter') subplot(,,),plot(t,) title('von Karmon _(t) turbulence') %(e): %SIMULATION OF PHUGOID RESPONSE TO TURBULENCE: a=lsim(ha,,t); q=lsim(hq,,t); fiure(33) subplot(,,),plot(t,a) title('a(t) Response to _(t)') subplot(,,),plot(t,q) title('q(t) Response to _(t)') %(f): % Compute Theoretical psd's for a(t) and q(t): vec=lospace(-,,); HHa=sqrt(si)*H*Ha; [M,P]=bode(HHa,vec); psd_a = squeeze(m); psd_adb=*lo(psd_a); HHq=sqrt(si)*H*Hq; [M,P]=bode(HHq,vec); psd_q = squeeze(m); psd_qdb=*lo(psd_q); fiure(34) semilox(vec',[psd_adb, psd_qdb]) leend('psd for a(t)','psd for q(t)') xlabel('frequency (rad/sec)') ylabel('db') title('theoretical PSDs for a(t) and q(t)') 7

Homework 6 AERE355 Fall 2017 Due 11/27 (M) Name. is called the. 3dB bandwidth.

Homework 6 AERE355 Fall 2017 Due 11/27 (M) Name. is called the. 3dB bandwidth. 1 Homeork 6 AERE355 Fall 17 Due 11/7 (M) Name PROBLEM 1(3pt The transfer function of a first order Lo Pass Filter (LPF) is frequency rane [, BW ] is called the filter 3dB bandidth (BW) (a)(8pt The dc (ie