{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 14 5 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "List Subitem" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 3 12 1 0 2 2 270 5 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 41 "ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 72 "Unit 35 -- Applica tion: Transfer Functions and Frequency Response Curves" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {URLLINK 17 "Prof. Douglas B. M eade" 4 "http://www.math.sc.edu/~meade/" "" }}{PARA 256 "" 0 "" {URLLINK 17 "Industrial Mathematics Institute" 4 "http://www.math.sc.e du/~IMI/" "" }}{PARA 256 "" 0 "" {URLLINK 17 "Department of Mathematic s" 4 "http://www.math.sc.edu/" "" }}{PARA 256 "" 0 "" {URLLINK 17 "Uni versity of South Carolina" 4 "http://www.sc.edu/" "" }}{PARA 256 "" 0 "" {TEXT -1 19 "Columbia, SC 29208\n" }}{PARA 256 "" 0 "" {TEXT -1 7 " URL: " }{URLLINK 17 "http://www.math.sc.edu/~meade/" 4 "http://www.m ath.sc.edu/~meade/" "" }}{PARA 256 "" 0 "" {TEXT -1 25 "E-mail: meade@ math.sc.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 38 "Copyright \251 2001 by Douglas B. Meade" }}{PARA 256 " " 0 "" {TEXT -1 19 "All rights reserved" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 67 "-------------------------------- -----------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 18 "Outline of Unit 35" }}{PARA 14 "" 0 "" {HYPERLNK 17 "35 .A" 1 "" "35.A" }{TEXT -1 19 " Transfer Functions" }}{PARA 257 "" 0 " " {HYPERLNK 17 "35.A-1" 1 "" "35.A-1" }{TEXT -1 13 " Definition 1" }} {PARA 257 "" 0 "" {HYPERLNK 17 "35.A-2" 1 "" "35.A-2" }{TEXT -1 34 " C omputation of Transfer Functions" }}{PARA 257 "" 0 "" {HYPERLNK 17 "35 .A-3" 1 "" "35.A-3" }{TEXT -1 13 " Definition 2" }}{PARA 14 "" 0 "" {HYPERLNK 17 "35.B" 1 "" "35.B" }{TEXT -1 26 " Frequency Responce Curv es" }}{PARA 257 "" 0 "" {HYPERLNK 17 "35.B-1" 1 "" "35.B-1" }{TEXT -1 27 " Introduction to Bode Plots" }}{PARA 257 "" 0 "" {HYPERLNK 17 "35. B-2" 1 "" "35.B-2" }{TEXT -1 10 " Example 1" }}{PARA 257 "" 0 "" {HYPERLNK 17 "35.B-3" 1 "" "35.B-3" }{TEXT -1 10 " Example 2" }}{PARA 257 "" 0 "" {HYPERLNK 17 "35.B-4" 1 "" "35.B-4" }{TEXT -1 10 " Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 " " 0 "" {TEXT -1 14 "Initialization" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( D Etools ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with( linalg ):" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "with( inttrans ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "alias( X(t)=laplace(x(t),t,s) ):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "35.A" {TEXT -1 23 "35.A Transfer Functions" }}{SECT 0 {PARA 4 "" 0 "35.A-1" {TEXT -1 19 "35.A-1 Definition 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 232 "The tr ansfer function for a linear ODE is the Laplace transform of the funda mental solution to the ODE. While this definition applies to an ODE an y order, only second-order linear ODEs will be discussed here. In part icular, consider" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ode1 := m*diff( x(t), t$2 ) + b*diff( x(t), t ) + k*x(t) = g(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "The fundamental solution can be ob tained as the solution for the IVP with " }{XPPEDIT 18 0 "g(t)=delta(t )" "6#/-%\"gG6#%\"tG-%&deltaG6#F'" }{TEXT -1 36 " and homogeneous init ial conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "ic1 := x(0)=0, D(x)(0)=0;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "35.A-2" {TEXT -1 40 "35.A-2 Computation of Transfer Functions" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 96 "The definition suggests finding the transfer function f rom the fundamental solution of the ODE :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "q1 := dsolve( \{ eva l(ode1,g(t)=Dirac(t)), ic1 \}, x(t), method=laplace );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "q2 := laplace( rhs(q1), t, s );" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "H1 := 1/expand(simplify(1/q 2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 161 "However, it is generally more efficient to simpl y take the Laplace transform of the IVP and solve for the Laplace tran sform of the fundamental solution. That is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "H2 := solve( eval ( laplace( eval(ode1,g(t)=Dirac(t)), t, s ), \{ic1\} ), X(t) );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "These results are obviously equivalent." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 4 "" 0 "35.A-3" {TEXT -1 19 "35.A-3 Definition 2" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Note that since " }{XPPEDIT 18 0 " L( delta(t) )" "6#-%\"LG6#-%&deltaG6#%\"tG" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "exp(-0*s)" "6#-%$expG6#,$*&\"\"!\"\"\"%\"sGF)!\"\"" } {TEXT -1 209 " = 1, an equilvalent definition of the transfer function is the ratio of the Laplace transform of the solution (with homogeneo us initial conditions) and the Laplace transform of the forcing functi on. That is, " }{XPPEDIT 18 0 "H(s)" "6#-%\"HG6#%\"sG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "L( x(t) ) / L( g(t) )" "6#*&-%\"LG6#-%\"xG6#%\"tG\" \"\"-F%6#-%\"gG6#F*!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "X(s) / G(s )" "6#*&-%\"XG6#%\"sG\"\"\"-%\"GG6#F'!\"\"" }{TEXT -1 3 " . " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 3 "" 0 " 35.B" {TEXT -1 30 "35.B Frequency Response Curves" }}{SECT 0 {PARA 4 " " 0 "35.B-1" {TEXT -1 33 "35.B-1 Introduction to Bode Plots" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "The frequency response curves, or Bode p lots, are plots of the magnitude and argument (in degrees) of the tran sfer function evaluated at " }{XPPEDIT 18 0 "s=i*omega" "6#/%\"sG*&%\" iG\"\"\"%&omegaGF'" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "i=sqrt(-1)" "6#/% \"iG-%%sqrtG6#,$\"\"\"!\"\"" }{TEXT -1 32 ") as functions of the frequ ency " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 47 ". That is, if the transfer function is H, then " }{XPPEDIT 18 0 "M(omega) = abs( H( i*omega) )" "6#/-%\"MG6#%&omegaG-%$absG6#-%\"HG6#*&%\"iG\"\"\"F'F0" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "N(omega) = arg( H(i*omega) )" "6#/- %\"NG6#%&omegaG-%$argG6#-%\"HG6#*&%\"iG\"\"\"F'F0" }{TEXT -1 164 ". Th e functions M and N correspond to the amplitude and phase shift, respe ctively, of the steady-state solution to the linear ODE with sinusoida l forcing function, " }{XPPEDIT 18 0 "g(t) = sin( omega*t )" "6#/-%\"g G6#%\"tG-%$sinG6#*&%&omegaG\"\"\"F'F-" }{TEXT -1 8 ", i.e., " } {XPPEDIT 18 0 "x( t ) = M(omega) * sin( omega*t + N(omega) )" "6#/-%\" xG6#%\"tG*&-%\"MG6#%&omegaG\"\"\"-%$sinG6#,&*&F,F-F'F-F--%\"NG6#F,F-F- " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "For example, when" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "H := unapply( H1, s );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "the amplitude and argument (in degrees) of the steady-sta te solution with frequency " }{XPPEDIT 18 0 "omega" "6#%&omegaG" } {TEXT -1 4 " are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "M := unapply( evalc( abs( H(I*omega) ) ), (m, b,k) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "N := unapply( ev alc( argument( H(I*omega) ) ) * 180/Pi, (m,b,k) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The quantity " }{XPPEDIT 18 0 "20*log[10](M(omega))" "6#*&\"#?\"\"\"-&%$l ogG6#\"#56#-%\"MG6#%&omegaGF%" }{TEXT -1 169 " is the gain, in decibel s (dB). Bode plots typically display the gain (in decibels) and phase \+ angle (in degrees) versus the angular frequency on a logarithmic scale for " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }}{SECT 0 {PARA 4 "" 0 "35.B-2" {TEXT -1 16 "35.B-2 Example 1" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "For example, when " }{XPPEDIT 18 0 "m=1" "6#/%\"mG\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b=1/4" "6#/% \"bG*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "k=1" "6 #/%\"kG\"\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "semilogplot( 20*log[10](M(1, 1/4,2)), omega=0.1..10," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " \+ axes=framed, labels=[omega,\"gain (dB)\"]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " labeldirections=[HORIZONTAL,VERTICAL] ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 52 "semilogplot( N(1,1/4,2), omega=0.1..10, axes=f ramed," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " labels=[omeg a,\"phase shift (degrees)\"]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " \+ labeldirections=[HORIZONTAL,VERTICAL] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "35.B-3" {TEXT -1 16 "35.B-3 Example 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 197 "One im portant use of the Bode plot for the magnitude is to determine the res onant frequency. The resonant frequency for a system occurs when the m agnitude of the steady-state response is maximized." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "dM := dif f( M(m,b,k), omega );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "q3 := solve( dM=0, omega );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "res_freq := omega = op(select( w->evalb(eval(w,\{m=1.,b=0,k=1.\})> 0), [q3] ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Note that for the undamped system, the re sonant frequency is found to be" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "simplify( eval( res_freq, b= 0 ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "as expected." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "35.B-4" {TEXT -1 16 "35.B-4 Example 3" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 253 "Another use of Bode plots is to investigate the behavior of a system as one of the parameters changes. To illustrate, the gain and phase angle are investigated as the mass and spring constant are \+ held fixed and the damping coefficient decreases to zero." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Consider the dampe d spring-mass system with " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 6 " = 1, " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 44 " = 4, and damping c oefficients selected from" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "damp_vals := [2^(-i) $ i=-3..8, 0]; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Also, chose an interval of positive frequencies to u se as the domain for the Bode plots." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "freq_range := 0.1..10;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 71 "The animated Bode plots of the magnitude for each dampi ng coeffcient is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "display( [seq( semilogplot( 20*log[10](M(1,b, 4)), omega=freq_range," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 " \+ axes=framed, labels=[omega,\"gain (dB)\"]," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 " labeldi rections=[HORIZONTAL,VERTICAL] )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 " b=damp_vals )], insequence=true );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Note that there is no (interior) maximum amplitude in the first two frames of this animation (" }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 9 " = 8 and " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 108 " = 4). Th is is consistent with the fact that a damped spring-mass system has a \+ resonant frequency only when " }{XPPEDIT 18 0 "b^2" "6#*$%\"bG\"\"#" } {TEXT -1 3 " < " }{XPPEDIT 18 0 "4*m*k" "6#*(\"\"%\"\"\"%\"mGF%%\"kGF% " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The animated Bode plots for the phase angle for each damp ing coefficient is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "display( [seq( semilogplot( N(1,b,4), omega =freq_range," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 " \+ axes=framed, labels=[omega,\"phase shift (degrees)\"]," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 " labeldi rections=[HORIZONTAL,VERTICAL] )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 " b=damp_vals )], insequence=true );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Note the jump in the phase shift when " }{XPPEDIT 18 0 "b=0" "6 #/%\"bG\"\"!" }{TEXT -1 76 ". The argument jumps to 180 degrees instea d of -180 degrees because Maple's " }{HYPERLNK 17 "arctangent" 2 "arct an" "" }{TEXT -1 30 " function returns values in ( " }{XPPEDIT 18 0 "- Pi" "6#,$%#PiG!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Pi" "6#%#PiG" } {TEXT -1 3 " ]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "And, now, the two Bode plots with all 13 frames superimpo sed on the same set of axes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "semilogplot( [seq( 20*log[10 ](M(1,b,4)), b=damp_vals )]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " \+ omega=freq_range, axes=framed," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " labels=[omega,\"gain (dB)\"]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " labeldirections=[HORIZONTAL, VERTICAL] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "semilogplot( [seq( N(1,b,4), b=damp _vals )]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " omega=fre q_range, axes=framed," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " \+ labels=[omega,\"phase shift (degrees)\"]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 " labeldirections=[HORIZONTAL,VERTICAL] ); \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "[Bac k to " }{HYPERLNK 17 "ODE Powertool Table of Contents" 1 "unit00.mws" "" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }