{VERSION 4 0 "IBM INTEL NT" "4.0" } {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 "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {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 6 6 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 4 4 1 0 1 0 2 2 0 1 } {PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 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 "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Unit 1: Ordinary Different ial Equations - Part 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Chapter 5: Second-Order Differential Equations" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Section 5 .11: resonance" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Copyright" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Copyright * 2001 by Addison Wesley Longman, Inc. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 302 "All \+ rights reserved. No part of this publication may be reproduced, store d in a retrieval system, or transmitted, in any form or by any means, \+ electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United \+ States of America." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Initializations" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Reson ance" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 336 "Resonance can be charcterized as that physical phenomenon wher eby Mother Nature seems to over react to a lesser, but periodic, stimu lus. This over reaction to a small stimulus is called a temper tantru m in a child, \"having a fit\" in an adult, and \"resonance\" in Mothe r Nature. The response is simply out of proportion to the stimulus." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 347 "Of cou rse, we are just creating imagery to focus attention. However, anyone who has seen the four-minute film clip of the Tacoma Narrows Bridge, \+ Tacoma, Washington, twisting in the wind and collapsing in 1940 would \+ need no additional metaphors for resonance, the phenomenon responsible for such a ruinous end to a magnificent work of engineering." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 310 "Resonanc e is quantized within the driven damped oscillator. A linear, damped \+ oscillator is driven with a sinusoidal input having a variable angular frequency. The response of the system is observed as a function of t he driving input frequency. The frequency for which the response is g reatest is called the " }{TEXT 261 18 "resonant frequency" }{TEXT -1 126 ", a frequency at which the magnitude of the system's response can be far larger than the magnitude of the input driving force." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 402 "We disti nguish two cases of resonance. If damping is assumed to be zero, the \+ system is a fiction, since even at supercooled temperatures, friction \+ does not disappear entirely. The resonance model in which no damping \+ term appears has characteristics that just cannot be found in nature s ince a frictionless system is just not found in nature. For this reas on, this model is called \"unreal resonance.\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The more realistic model \+ for resonance includes frictional damping, and is called \"real resona nce.\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "Unreal Resonance" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "The undamped undamped oscillat or with mass " }{XPPEDIT 18 0 "m=1" "6#/%\"mG\"\"\"" }{TEXT -1 21 " an d spring constant " }{XPPEDIT 18 0 "k=16" "6#/%\"kG\"#;" }{TEXT -1 66 ", driven by a periodic force of amplitude 1 and angular frequency " } {XPPEDIT 18 0 "omega=4" "6#/%&omegaG\"\"%" }{TEXT -1 41 ", is modeled \+ by the differential equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`y''`+16*y=cos(4*t)" "6#/,& %$y''G\"\"\"*&\"#;F&%\"yGF&F&-%$cosG6#*&\"\"%F&%\"tGF&" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "that \+ is, by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "q := diff(y(t),t,t) + 16*y(t) = cos(4*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "If this system goes into motion with the inert initial conditio ns " }{XPPEDIT 18 0 "y(0)=`y'`*``(0)" "6#/-%\"yG6#\"\"!*&%#y'G\"\"\"-% !G6#F'F*" }{TEXT -1 59 " = 0, we can describe the resulting motions by the solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Y := rhs(dsolve(\{q, y(0) = 0, D(y)(0) = 0\},y(t )));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "graphed as the solid curve in the following figur e (Figure 5.13)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "plot([t/8,-t/8,Y],t=0..10,color=[red,red,blac k], linestyle=[2,2,1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 296 "The oscillations of the syst em grow without bound, getting ever larger and larger. Of course, thi s is not what happens in reality where some frictional damping always \+ exists. In this model with no damping, the envelope of the oscillatio ns is linear, as shown by the dotted lines in Figure 5.13." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 214 "Fortunately, thi s unreal resonance is never seen in nature since nothing in nature tak es place without frictional losses. But to see why the model predicts these unbounded motions, obtain the characteristic roots " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 3 " = " }{TEXT 256 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4*i" "6#*&\"\"%\"\"\"%\"iGF%" }{TEXT -1 33 " \+ from the characteristic equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "lambda^2+16=0" "6#/,& *$%'lambdaG\"\"#\"\"\"\"#;F(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The latent roots from the driving term " }{XPPEDIT 18 0 "r(t)" "6#-%\"rG6#%\"tG" }{TEXT -1 16 " are themselves " }{TEXT 262 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4*i " "6#*&\"\"%\"\"\"%\"iGF%" }{TEXT -1 42 ", so the particular solution \+ will contain " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "t*(a*cos(4*t)+b*sin(4*t))" "6#*&%\"tG\"\"\",&*&%\"aGF%- %$cosG6#*&\"\"%F%F$F%F%F%*&%\"bGF%-%$sinG6#*&F-F%F$F%F%F%F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Driving the system at its natural (angular) frequency " }{XPPEDIT 18 0 "omega=4" "6#/%&omegaG\"\"%" }{TEXT -1 67 " means the trig terms \+ in the particular solution are multiplied by " }{XPPEDIT 18 0 "t" "6#% \"tG" }{TEXT -1 158 ". Physically, this correspondence means every im posed push lines up exactly with the natural peak in the motion, and t he oscillations increase without bound." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Let us implement this investigation \+ in Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "To begin, obtain the homogeneous and particular solutions from the general solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "q2 := dsolve(q,y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 146 "The particul ar solution survives when the arbitrary constants in the homogeneous s olution are set equal to zero. Thus, the particular solution is" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "yp := subs(_C1=0, _C2=0, rhs(q2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "The homogeneo us solution contains the same term (" }{XPPEDIT 18 0 "cos(4*t)" "6#-%$ cosG6#*&\"\"%\"\"\"%\"tGF(" }{TEXT -1 478 ") that is being used to dri ve the system. The driving angular frequency exactly matches the natu ral frequency as determined by the characteristic roots. The homogene ous solution captures the \"natural\" response of the system, the ocil latory motion whose (angular) frequency is 4. Driving the system at t he same rate in the absence of damping means no energy is lost, and ev ery push lines up exactly with the peak of the motion. Hence, the osc illations increase without bound." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 86 "The natural frequency of the undamped sys tem is determined by the homogeneous equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m*`y' '`+k*y=0" "6#/,&*&%\"mG\"\"\"%$y''GF'F'*&%\"kGF'%\"yGF'F'\"\"!" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The characteristic equation is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "m*lambda^2+ k=0" "6#/,&*&%\"mG\"\"\"*$%'lambdaG\"\"#F'F'%\"kGF'\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "so \+ the with characteristic roots are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG " }{TEXT -1 3 " = " }{TEXT 263 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "s qrt(k/m)*i" "6#*&-%%sqrtG6#*&%\"kG\"\"\"%\"mG!\"\"F)%\"iGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "These roots are complex, so the fundamental set will now be" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "\{cos(sqrt(k/m)*t),sin(sqrt(k/m)*t)\}" "6#<$-%$cosG6#*& -%%sqrtG6#*&%\"kG\"\"\"%\"mG!\"\"F-%\"tGF--%$sinG6#*&-F)6#*&F,F-F.F/F- F0F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 9 "Clearly, " }{XPPEDIT 18 0 "omega[N]=sqrt(k/m)" "6#/&%&om egaG6#%\"NG-%%sqrtG6#*&%\"kG\"\"\"%\"mG!\"\"" }{TEXT -1 278 " becomes \+ the natural (angular) frequency since that is the frequency this syste m must necessarily exhibit if no external driving forces act on it. D riving the undamped system at its natural (angular) frequency causes i t to resonate, a resonance that predicts unbounded motions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "However, in rea l life, there is always damping. Hence, we next consider the same sys tem (" }{TEXT 258 1 "m" }{TEXT -1 6 " = 1, " }{TEXT 259 1 "k" }{TEXT -1 237 " = 16) but add some damping to explore the case of real resona nce. We will drive the system with a cosine term for which the freque ncy is not specified and seek to study the response of the system as a function of the driving frequency." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "Real Resonance" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Consider the damped oscillator for which the mass, damping, and spring constant are " }{XPPEDIT 18 0 "m= 1, b=4,k=16" "6%/%\"mG\"\"\"/%\"bG\"\"%/%\"kG\"#;" }{TEXT -1 36 ", and for which the driving term is " }{XPPEDIT 18 0 "cos(omega*t)" "6#-%$c osG6#*&%&omegaG\"\"\"%\"tGF(" }{TEXT -1 55 ". A model for this system is the differential equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`y''`+4*`y'`+16*y=cos (omega*t)" "6#/,(%$y''G\"\"\"*&\"\"%F&%#y'GF&F&*&\"#;F&%\"yGF&F&-%$cos G6#*&%&omegaGF&%\"tGF&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "q3 := diff(y(t),t,t) + 4*diff(y(t),t) + 16*y(t) = cos(omega*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "with chara cteristic equation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "lambda^2+4*lambda+16=0" "6#/,(*$%'la mbdaG\"\"#\"\"\"*&\"\"%F(F&F(F(\"#;F(\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "and characteristi c roots " }{XPPEDIT 18 0 "lambda=-2" "6#/%'lambdaG,$\"\"#!\"\"" } {TEXT -1 1 " " }{TEXT 264 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*sqrt (3)*i" "6#*(\"\"#\"\"\"-%%sqrtG6#\"\"$F%%\"iGF%" }{TEXT -1 9 ". Since " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "\{exp(-2*t)*cos(2*sqrt(3)*t),exp(-2*t)*sin(2*sqrt(3)*t )\}" "6#<$*&-%$expG6#,$*&\"\"#\"\"\"%\"tGF+!\"\"F+-%$cosG6#*(F*F+-%%sq rtG6#\"\"$F+F,F+F+*&-F&6#,$*&F*F+F,F+F-F+-%$sinG6#*(F*F+-F36#F5F+F,F+F +" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "is then a fundamental set, the natural (angular) frequenc y for the system is " }{XPPEDIT 18 0 "omega[N]=2*sqrt(3)" "6#/&%&omega G6#%\"NG*&\"\"#\"\"\"-%%sqrtG6#\"\"$F*" }{TEXT -1 32 " and the homogen eous solution is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y[h]=exp(-2*t)*(c[1]*cos(2*sqrt(3)*t)+c [2]*siin(2*sqrt(3)*t)" "6#/&%\"yG6#%\"hG*&-%$expG6#,$*&\"\"#\"\"\"%\"t GF/!\"\"F/,&*&&%\"cG6#F/F/-%$cosG6#*(F.F/-%%sqrtG6#\"\"$F/F0F/F/F/*&&F 56#F.F/-%%siinG6#*(F.F/-F<6#F>F/F0F/F/F/F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "We can obtain th ese results in Maple as follows. To obtain the homogeneous solution, \+ we find the characteristic equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "q4 := lambda^2 + 4*lambda \+ + 16 = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "and its (characteristic) roots" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "q 5 := solve(q4, lambda);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Complex exponential solutions are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "q6 := exp(q5[1]*t);\nq7 := exp(q5[2]*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "and Euler's formulas let us write these complex exponentials as " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "q8 := evalc(q6);\nq9 := evalc(q7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "T aking linear combinations of these complex solutions yields other solu tions that are real." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "y1 := (q8 + q9)/2;\ny2 := (q8 - q9)/(2*I) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "The homogeneous solution is then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "yh := colle ct(A*y1 + B*y2, exp(-2*t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "At steady-state when the t erm " }{XPPEDIT 18 0 "exp(-2*t)" "6#-%$expG6#,$*&\"\"#\"\"\"%\"tGF)!\" \"" }{TEXT -1 209 " has done its work and effectively become zero, the homogeneous solution hardly contributes to the general solution. The steady-state solution is essentially the particular solution which we write in the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "yp := A*cos(omega*t - phi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "The amplitude A and the phase angle " }{XPPEDIT 18 0 "phi" "6#% $phiG" }{TEXT -1 111 " are determined by the method of Undetermined Co efficients. Substituting into the differential equation yields" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "q10 := eval(subs(y(t) = yp, q3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "This must be \+ an identity in " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 38 ", so we se ek values of the parameters " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 55 " which will ma ke this so. Thus, in Maple, we calculate" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "q11 := solve(identit y(expand(q10),t),\{A,phi\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "We are really only inte rested in the amplitude of the motion. We therefore extract just the \+ positive value of " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := subs(q11[1],A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "and graph the amplitude fu nction as a function of " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 24 ", obtaining Figure 5.14." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(f, omega = 0..5);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "For some value of the driving frequency " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 130 ", the magnitude of the steady-stat e's amplitude peaks. The driving frequency for which this amplitude i s a maximum is called the " }{TEXT 257 8 "resonant" }{TEXT -1 62 " fre quency which can be found exactly by solving the equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f'`*``(omega)=0" "6#/*&%#f'G\"\"\"-%!G6#%&omegaGF&\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "Thus, in Maple we obtain" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "q12 := solve(diff(f,omega),omega);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "and the reson ant frequency is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "omega[R] = q12[2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "The natura l frequency of this system is found in the homogeneous solution:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "yh;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The natural frequency is " }{XPPEDIT 18 0 "2*sqrt(3)" "6#*&\"\"#\"\"\"-%%sqrtG6#\"\"$F%" }{TEXT -1 45 ", sligh tly larger than the resonant frequeny " }{XPPEDIT 18 0 "2*sqrt(2)" "6# *&\"\"#\"\"\"-%%sqrtG6#F$F%" }{TEXT -1 198 ". For unreal (undamped) r esonance, the natural frequency and the resonant frequency are the sam e. For real (damped) resonance the resonant frequency is slightly sma ller than the natural frequency." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "Steady-State Solution as a Functi on of " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "As a function of " } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "omega " "6#%&omegaG" }{TEXT -1 28 ", the particular solution is" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y[p]=cos(omega*t)-arctan(4*omega/(16-omega^2)))/sqrt(omega^4-16*ome ga^2+256)" "6#*&/&%\"yG6#%\"pG,&-%$cosG6#*&%&omegaG\"\"\"%\"tGF/F/-%'a rctanG6#*(\"\"%F/F.F/,&\"#;F/*$F.\"\"#!\"\"F:F:F/-%%sqrtG6#,(*$F.F5F/* &F7F/*$F.F9F/F:\"$c#F/F:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "which we obtain in Maple as follow s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "The steady-state solution as a function of the driving frequency " } {XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 11 " is simply " } {XPPEDIT 18 0 "y[p](t)" "6#-&%\"yG6#%\"pG6#%\"tG" }{TEXT -1 60 ". To \+ obtain it, we first get the following expressions for " }{XPPEDIT 18 0 "cos(phi)" "6#-%$cosG6#%$phiG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "s in(phi)" "6#-%$sinG6#%$phiG" }{TEXT -1 13 " in terms of " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "C := simplify(subs(q11[1],co s(phi)));\nS := simplify(subs(q11[1],sin(phi)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Then, the particular solution in terms of " }{XPPEDIT 18 0 "omega" "6#%&ome gaG" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "YP := subs(A=f, cos(phi)=C, sin(phi )=S, expand(yp));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "The following figure, Figure 5.15, shows graph of three steady-state solutions for" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "omega =3/2,omega=omega[R]" "6$/%&omegaG*&\"\"$\"\"\"\"\"#!\"\"/F$&F$6#%\"RG " }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2*sqrt(2)" "6#*&\"\"#\"\"\"-%%sqrt G6#F$F%" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "omega=4" "6#/%&omegaG\" \"%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 112 "as the red, black, and green curves, respectively. Th is figure therefore shows the variation in amplitude with " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "plot([subs(omega=3/ 2,YP), subs(omega=2*sqrt(2),YP), subs(omega=4,YP)], t=0..2*Pi, color=[ red,black,green]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "The following animation is a more dynamic representation of the variation of amplitude with driving fre quency." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "animate(YP,t=0..2*Pi, omega=0..6, frames=60, color=bl ack);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "Dependence of \+ Resonance on Damping" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "It is important to quantify the resonant frequ ency's dependence on damping. Hence, let the damped oscillator above \+ now have variable damping " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 60 ", so that the system is modeled by the differential equation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`y''`+b*`y'`+16*y=cos(omega*t)" "6#/,(%$y''G\"\"\"*&%\" bGF&%#y'GF&F&*&\"#;F&%\"yGF&F&-%$cosG6#*&%&omegaGF&%\"tGF&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "t hat is, by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 64 "q13 := diff(y(t),t,t) + b*diff(y(t),t) + 16*y(t) = \+ cos(omega*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "As before, we want the system's s teady-state response to the driving term, from which we extract the va lue of the driving frequency " }{XPPEDIT 18 0 "omega" "6#%&omegaG" } {TEXT -1 83 " maximizing the amplitude of the response. Again taking \+ the particular solution as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "yp;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "the parameter s A and " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 103 " are determin ed in Maple via the method of Undetermined Coefficients which starts w ith the substitution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "q14 := eval(subs(y(t) = yp, q13));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Since this must be an identity in " }{XPPEDIT 18 0 "t" "6 #%\"tG" }{TEXT -1 8 ", we use" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "q15 := solve(identity(expand (q14),t),\{A,phi\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "to determine the appropriate par ameters. Since we really want just the amplitude of the steady-state \+ solution, we extract the positive value of A from the solution set in \+ q15." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := subs(q15[1],A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Again interes ted in how " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 76 ", the amplitud e of the steady-state response, varies with driving frequency " } {XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 15 ", we give the " }} {PARA 0 "" 0 "" {TEXT -1 20 "damping coefficient " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 12 " the values " }{XPPEDIT 18 0 "1,2,`...`,5" "6& \"\"\"\"\"#%$...G\"\"&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "for k from 1 to 5 do\n f||k := subs(b = k, f);\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "and plot the resulting r esonance curves in the following figure, Figure 5.16." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot([f| |(1..5)], omega=0..6, color = [red,blue,green,magenta,black]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "The red curve corresponds to " }{XPPEDIT 18 0 "b=1" "6#/% \"bG\"\"\"" }{TEXT -1 245 ". The more the damping, the smaller the ma ximum peak at resonance, and the less the damping, the greater the res onant peak. Also, if the damping is great enough there is no resonanc e peak at all. Hence, sufficient damping precludes resonance." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "The foll owing figure, Figure 5.17, is another way of looking at the dependence of the response amplitude " }{XPPEDIT 18 0 "A(b,omega) " "6#-%\"AG6$% \"bG%&omegaG" }{TEXT -1 60 " on both damping and input driving frequen cy. The function " }{XPPEDIT 18 0 "A(b,omega)" "6#-%\"AG6$%\"bG%&omeg aG" }{TEXT -1 34 " is plotted as a surface over the " }{XPPEDIT 18 0 " b" "6#%\"bG" }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 28 "-plane. The plane sections " }{XPPEDIT 18 0 "b=constant" "6#/%\"bG%)constant G" }{TEXT -1 38 " are the curves seen in Figure 5.16. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot3d (f, b = 1..5, omega = 0..6,axes=boxed, style=patchnogrid);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Both Figures 5.16 and 5.17 suggest that as the driving frequenc y " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 104 " is increased w ell past the resonant frequency, the amplitude of the steady-state res ponse goes to zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 51 "Analytically, this is shown by the following limit." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Limit(f, omega=infinity) = limit(f,omega=infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 294 "Physically, this means that for high enough driving frequency, the system cannot react to the rapidity of the swings being imposed o n it and the system becomes \"paralysed.\" So if you have a motor sha king itself to pieces because of resonance, either add damping or inc rease the driving speed!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 99 "Finally, we obtain an expression for the resonant \+ frequency as a function of the damping parameter " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 61 ". This expression is determined by solving th e equation f '(" }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 10 ") = 0 for " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "q 16 := solve(diff(f,omega) = 0, omega);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The resonant \+ frequency is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "omega[R] = q16[2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "Formulae for the dependence of the resonant frequency on all three parameters of m ass, damping, and spring constant are obtained below." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Dependence of Resonance on " } {XPPEDIT 18 0 "m, b" "6$%\"mG%\"bG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "k" "6#%\"kG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 195 "Complete formulae for the dependence of the resonant \+ frequency on mass, damping, and spring constant can be obtained by fin ding the steady-state (particular) solution of the differential equati on" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "m*`y''`+b*`y'`+k*y=cos(omega*t)" "6#/,(*&%\"mG\"\"\" %$y''GF'F'*&%\"bGF'%#y'GF'F'*&%\"kGF'%\"yGF'F'-%$cosG6#*&%&omegaGF'%\" tGF'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "k := 'k':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "q17:= m*diff(y(t),t,t) + b*diff(y(t),t) + k*y(t) = cos(omega*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Assuming the particular solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "yp;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "the differential equation yields" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "q18 := eval(subs(y(t) = yp, q17));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "from which th e matching principle yields" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "q19 := solve(identity(expand (q18),t),\{A,phi\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "The positive amplitude of this st eady-state solution is given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := subs(q19[1],A);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "from which we obtain the maximizing value of " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 16 " by solving f '(" }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 10 ") = 0 for " }{XPPEDIT 18 0 "om ega" "6#%&omegaG" }{TEXT -1 8 ". Thus," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "q20 := solve(diff(f,om ega) = 0, omega);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "The resonant frequency is" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "omega[R] = q20[2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "which can be written as" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "omega[R]=sqrt(k/m-b^2/2/m^2)" "6#/&%&omegaG6#%\"RG-%%sq rtG6#,&*&%\"kG\"\"\"%\"mG!\"\"F.*(%\"bG\"\"#F3F0*$F/F3F0F0" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt((2*m*k-b^2)/2/m^2)" "6#-%%sqrtG6#*(,&*(\" \"#\"\"\"%\"mGF*%\"kGF*F**$%\"bGF)!\"\"F*F)F/*$F+F)F/" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The f irst form for " }{XPPEDIT 18 0 "omega[R]" "6#&%&omegaG6#%\"RG" }{TEXT -1 72 " is more prevalent in texts, but the second makes it easier to \+ see that " }{XPPEDIT 18 0 "b^2<2*m*k" "6#2*$%\"bG\"\"#*(F&\"\"\"%\"mGF (%\"kGF(" }{TEXT -1 28 " is necessary for resonance." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Resonance Wrap-Up" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 194 "Here are som e questions about resonance, and oscillations in damped spring-mass sy stems. Answering them will help clarify the connections, formulas, an d insights generated throughout this unit." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Questions" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 65 "Can overd amped and critically damped systems be made to resonate?" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 49 "Can every under damped system be made to resonate?" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 71 "What is the threshold of damping abov e which there can be no resonance?" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 81 "What is the threshold of damping abov e which there are no transient oscillations?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "To answer these questions, we rec all formulas for the natural frequency and the resonant frequency. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "The qu adratic formula for the characteristic roots of the characteristic equ ation gives the natural frequency:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {XPPEDIT 18 0 "lambda = -b/(2*m);" "6#/%'lambdaG,$*& %\"bG\"\"\"*&\"\"#F(%\"mGF(!\"\"F," }{TEXT -1 2 " " }{TEXT 265 1 "+" }{TEXT -1 2 " " }{XPPEDIT 18 0 "sqrt(b^2-4*m*k)/(2*m);" "6#*&-%%sqrtG 6#,&*$%\"bG\"\"#\"\"\"*(\"\"%F+%\"mGF+%\"kGF+!\"\"F+*&F*F+F.F+F0" } {TEXT -1 11 " => " }{XPPEDIT 18 0 "omega[N] = sqrt(k/m-b^2/(4*m ^2));" "6#/&%&omegaG6#%\"NG-%%sqrtG6#,&*&%\"kG\"\"\"%\"mG!\"\"F.*&%\"b G\"\"#*&\"\"%F.*$F/F3F.F0F0" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "From " }{XPPEDIT 18 0 "b^2-4*m*k " "6#,&*$%\"bG\"\"#\"\"\"*(\"\"%F'%\"mGF'%\"kGF'!\"\"" }{TEXT -1 103 " , the discriminant of the characteristic equation, we can classify the motion as a function of damping:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {XPPEDIT 18 0 "PIECEWISE([b^2 = 4*m*k, `Critically d amped`],[b^2*`>`*4*m*k, Overdamped],[b^2 < 4*m*k, Underdamped]);" "6#- %*PIECEWISEG6%7$/*$%\"bG\"\"#*(\"\"%\"\"\"%\"mGF-%\"kGF-%2Critically~d ampedG7$*,F)F*%\">GF-F,F-F.F-F/F-%+OverdampedG7$2*$F)F**(F,F-F.F-F/F-% ,UnderdampedG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The resonant frequency is" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "omega[R]=sqrt(k/m-b^2/2/m^2)" "6#/&%&om egaG6#%\"RG-%%sqrtG6#,&*&%\"kG\"\"\"%\"mG!\"\"F.*(%\"bG\"\"#F3F0*$F/F3 F0F0" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt((2*m*k-b^2)/2/m^2)" "6#-% %sqrtG6#*(,&*(\"\"#\"\"\"%\"mGF*%\"kGF*F**$%\"bGF)!\"\"F*F)F/*$F+F)F/ " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "from which we draw the two inferences:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "0 underdamped " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "0 resonance " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The following p lot (Figure 5.18) summarizes all this data on a " }{XPPEDIT 18 0 "b^2 " "6#*$%\"bG\"\"#" }{TEXT -1 1 "-" }{TEXT 260 11 "number line" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 742 "p1 := plot([t,0,t=0..6],thickness=3, color=black):\n p2 := plot(\{[[0,0],[0,1/2]],[[2,0],[2,1/2]],[[4,0],[4,1/2]]\}, style= line, color=black,thickness=3):\np3 := plot(\{[[0,0],[0,-.5]],[[4,0],[ 4,-.5]]\}, color=black,linestyle=2):\np4 := textplot(\{[0,.7,`0`],[2,. 7,`2mk`],[4,.7,`4mk`]\}, font=[TIMES,BOLD,14]):\np5 := textplot([6.3,0 ,`b`],font=[TIMES,BOLD,14]):\np6 := textplot([6.4,.2,`2`]):\np7 := tex tplot([2,-.3,`<----- underdamped ----->`], font=[TIMES,ROMAN,12]):\np8 := textplot([5.5,-.3,`<-- overdamped -->`], font=[TIMES,ROMAN,12]):\n p9 := textplot(\{[4,-.7,`critically`],[4,-1,`damped`]\}, font=[TIMES,R OMAN,12]):\np10 := textplot([1,.3,`resonance`], font=[TIMES,ROMAN,12]) :\ndisplay([p||(1..10)],axes=none,scaling=constrained, title = `Figure 5.18`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Answers" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "* Overdamped, critically damped , and \"half\" the underdamped systems cannot be made to resonate. In particular," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "0 underdamped with resonance possible" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "2*m*k<=b^2 " "6#1*(\"\"#\"\"\"%\"mGF&%\"kGF&*$%\"bGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``<4*m*k" "6#2%!G*(\"\"%\"\"\"%\"mGF'%\"kGF'" }{TEXT -1 47 " = => underdamped but resonance not possible " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We quote \"half\" because syste ms for which " }{XPPEDIT 18 0 "b^2<4*m*k" "6#2*$%\"bG\"\"#*(\"\"%\"\" \"%\"mGF)%\"kGF)" }{TEXT -1 47 " are all underdamped, but only those f or which " }{XPPEDIT 18 0 "b^2<2*m*k" "6#2*$%\"bG\"\"#*(F&\"\"\"%\"mGF (%\"kGF(" }{TEXT -1 275 " have a positive resonant frequency. In Figu re 5.18 the underdamped systems for which resonance is possible appear s to be \"half\" of the underdamped systems. Thus, the answer to the \+ first question is \"no, overdamped and critically damped systems canno t be made to resonate.\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 99 "* The answer to the second question is \"no, not \+ every underdamped system can be made to resonate.\"" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "* The answer to the th ird question is \"" }{XPPEDIT 18 0 "b^2=2*m*k" "6#/*$%\"bG\"\"#*(F&\" \"\"%\"mGF(%\"kGF(" }{TEXT -1 67 " is the threshold above which a syst em cannot be made to resonate.\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 43 "* And the answer to the last question is \"" }{XPPEDIT 18 0 "b^2=4*m*k" "6#/*$%\"bG\"\"#*(\"\"%\"\"\"%\"mGF)% \"kGF)" }{TEXT -1 68 " is the threshold above which a system cannot be made to oscillate.\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "7" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }