{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 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Lis t Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "L ist Subitem" 14 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 3 12 0 0 0 0 270 0 }} {SECT 0 {EXCHG {PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 41 "ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 41 "Unit 29 -- Applica tion: Nonlinear Springs" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 " " 0 "" {URLLINK 17 "Prof. Douglas B. Meade" 4 "http://www.math.sc.edu/ ~meade/" "" }}{PARA 257 "" 0 "" {URLLINK 17 "Industrial Mathematics In stitute" 4 "http://www.math.sc.edu/~IMI/" "" }}{PARA 258 "" 0 "" {URLLINK 17 "Department of Mathematics" 4 "http://www.math.sc.edu/" " " }}{PARA 259 "" 0 "" {URLLINK 17 "University of South Carolina" 4 "ht tp://www.sc.edu/" "" }}{PARA 260 "" 0 "" {TEXT -1 19 "Columbia, SC 292 08\n" }}{PARA 262 "" 0 "" {TEXT -1 7 "URL: " }{URLLINK 17 "http://ww w.math.sc.edu/~meade/" 4 "http://www.math.sc.edu/~meade/" "" }}{PARA 263 "" 0 "" {TEXT -1 25 "E-mail: meade@math.sc.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 38 "Copyright \251 2001 b y Douglas B. Meade" }}{PARA 265 "" 0 "" {TEXT -1 19 "All rights reserv ed" }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 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 29" } }{EXCHG {PARA 14 "" 0 "" {HYPERLNK 17 "29.A" 1 "" "29.A" }{TEXT -1 52 " Model for a General Spring-Mass System with Damping" }}{PARA 270 "" 0 "" {HYPERLNK 17 "29.A-1" 1 "" "29.A-1" }{TEXT -1 14 " Linear Spring " }}{PARA 270 "" 0 "" {HYPERLNK 17 "29.A-2" 1 "" "29.A-2" }{TEXT -1 12 " Soft Spring" }}{PARA 270 "" 0 "" {HYPERLNK 17 "29.A-3" 1 "" "29.A -3" }{TEXT -1 12 " Hard Spring" }}}{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( DEtools ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with( linalg \+ ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( student ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "29.A" {TEXT -1 56 "29.A Model for a General Spring-Mass System with Damping " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 335 "Consider the spring-mass syst em in which a weight is attached to one end of a massless spring and t he other end is mounted to the ceiling. When the weight is pulled down wards a short distance and released (from rest), the spring-mass syste m oscillates up and down. The motion is completely determined by Newto n's Second Law of Motion: " }{XPPEDIT 18 0 "m * Diff( y, t$2 ) = F" "6 #/*&%\"mG\"\"\"-%%DiffG6$%\"yG-%\"$G6$%\"tG\"\"#F&%\"FG" }{TEXT -1 8 " , where " }{XPPEDIT 18 0 "F" "6#%\"FG" }{TEXT -1 38 " is the total for ce on the system and " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 73 " is \+ the displacement of the mass from the spring's unstretched position, \+ " }{TEXT 256 4 "i.e." }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y" "6#%\"yG" } {TEXT -1 47 " < 0 corresponds to compressing the spring and " } {XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 286 " > 0 to an extension of the spring. Note that the displacement must remain with certain bounds to remain physically realistic. If the displacement becomes too large, t he spring will break in two. The limit on the negative displacement oc curs when the coils of the spring bunch together." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 183 "Models of this system di ffer in the way the forces included. For this discussion, the gravitat ional force, frictional force, and spring force are included. The grav itational force is " }{XPPEDIT 18 0 "-m*g" "6#,$*&%\"mG\"\"\"%\"gGF&! \"\"" }{TEXT -1 204 ". The frictional force is assumed to be proportio nal to velocity and acts in the direction opposite to the direction of motion. Thus, the second-order ODE for the displacement of the spring -mass system is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "F[gravity] := - m * g;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "F[friction] := - c * diff( y(t), t );" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "F[spring] := S( y(t) );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 69 "ode1 := m * diff( y(t), t$2 ) = F[spring] + F[ friction] + F[gravity];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "The corresponding first-order system is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "U := [ y(t), v(t) ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f := (y,v) -> v:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "g := (y,v) -> ( S(y) - c*v ) / m - g:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "sys1 := equate( diff( U, t ), [ f( y(t), v(t) ), g( y (t), v(t) ) ] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "The general condition for an equil ibrium solution is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "q1 := solve( eval( sys1, \{ y(t)=y0, v(t)=v 0 \} ), \{ y0, v0 \} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "The most note-worthy propert y of this result is that the equilibrium solutions are independent of \+ the damping force." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "29.A-1" {TEXT -1 20 "29.A-1 Linear Spring" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Hooke' s Law assumes the spring force is linear:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "SS[linear] := y -> - k*y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "with " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 10 " > 0 (and " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 5 " >0, " } {XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 10 " > 0, and " }{XPPEDIT 18 0 " g" "6#%\"gG" }{TEXT -1 6 " > 0)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 59 "There is only one equilibrium solution to the linear system" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "S := SS[linear]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "equil[linear] := q1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "for X0 in [equil[linear]] do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " b := eval( [ f(y0,v0), g(y0,v0) ], X0 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 " A := eval( evalm(jacobian( [f(y0,v0),g( y0,v0)], [y0,v0] )), X0 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " lin sys1 := equate( diff( U, t ), evalm( A &* U + b ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " print( `-------------------------------------- -----------------------` );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " pr int( `Linearization at [y0,v0]`=eval([y0,v0],X0) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " print( A = evalm( A ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " print( lambda = [eigenvals( A )] );" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 75 " print( `------------------------------------ -------------------------` );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "It is not difficult to see that there are three c ases to consider for these eigenvalues:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 12 "Case 1: 0 < " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 4 " < " }{XPPEDIT 18 0 "2*sqrt(m*k)" "6#*& \"\"#\"\"\"-%%sqrtG6#*&%\"mGF%%\"kGF%F%" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "The eigenvalues are complex conjugates w ith negative real part. Hence the equilibrium is a stable spiral. This is the underdamped case considered in " }{HYPERLNK 17 "Unit 28, Secti on A-1" 1 "unit28.mws" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "Case 2: " }{XPPEDIT 18 0 "c=2*sqrt(m*k)" "6#/%\" cG*&\"\"#\"\"\"-%%sqrtG6#*&%\"mGF'%\"kGF'F'" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "With a repeated negative eigenvalue, the equilibrium is a stable node (sink). This is the critically damped ca se discussed in " }{HYPERLNK 17 "Unit 28, Section A-2" 1 "unit28.mws" "28.A-2" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "Case 3: " }{XPPEDIT 18 0 "c " "6#%\"cG" }{TEXT -1 3 " > " }{XPPEDIT 18 0 "2*sqrt(m*k)" "6#*&\"\"# \"\"\"-%%sqrtG6#*&%\"mGF%%\"kGF%F%" }{TEXT -1 1 " " }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 133 "Both eigenvalues are negative so the equilibrium \+ is a stable node (sink). This is the overdamped case discussed in grea ter detail in " }{HYPERLNK 17 "Unit 28, Section A-3" 1 "unit28.mws" "2 8.A-3" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Thus, in all cases, the linear \+ spring-mass system returns to its equilibrium position." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "This linear spring model, " } {TEXT 258 4 "i.e." }{TEXT -1 150 ", Hooke's Law, is valid only for \"s mall\" displacements. Beyond these limits, the linear model ceases to \+ provide a good model of the spring's response." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "Nonlinear spring forces c an have wider applicability. For a nonlinear spring with spring force \+ " }{XPPEDIT 18 0 "S(y)" "6#-%\"SG6#%\"yG" }{TEXT -1 33 ", the stiffnes s of the spring is " }{XPPEDIT 18 0 "Diff( S(y), y )" "6#-%%DiffG6$-% \"SG6#%\"yGF)" }{TEXT -1 221 ". If the stiffness decreases as a functi on of displacement, the spring is said to be soft. A hard spring is ch aracterized by a stiffness increases as a function of displacement. Si mple models for soft and hard springs use" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "SS[soft] := y -> -k* y + alpha*y^3;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "SS[hard] := y -> \+ -k*y - alpha*y^3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "(Note: In all models, " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 1 " " }{TEXT 257 0 "" }{TEXT -1 8 "> 0 an d " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 6 " > 0.)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "29 .A-2" {TEXT -1 18 "29.A-2 Soft Spring" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "Because of the complexities of the general solution to a cubic equation, this discussion is restricted to a specific example." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "param1 := \{ m=1, g=98/10, c=2/10, k=10, alpha=2/10 \}:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "S := SS[soft]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sys[soft] := eval( sys1, param1 ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 47 "equil[soft] := allvalues( eval( q1, param1 ) ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf( equil[soft] ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Thus, there are three equilibria. Linearization is u sed to classify the equilibria" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "for X0 in [equil[soft]] do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " b := eval( [ f(y0,v0), g(y0,v0) ], X0 union param1 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 " A := ev al( evalm(jacobian( [f(y0,v0),g(y0,v0)], [y0,v0] )), X0 union param1 ) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " linsys1 := equate( diff( U, \+ t ), evalm( A &* U + b ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " pr int( `-------------------------------------------------------------` ) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 " print( `Linearization at [y0 ,v0]`=eval([y0,v0],X0 union param1) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " print( A = evalm( A ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " print( lambda = [eigenvals( A )] );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " print( `------------------------------------------- ------------------` );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 61 "Based on the eigenvalues, ( -1, 0 ) is a stable spiral \+ and ( " }{XPPEDIT 18 0 "(1+sqrt(197))/2" "6#*&,&\"\"\"F%-%%sqrtG6#\"$( >F%F%\"\"#!\"\"" }{TEXT -1 12 ", 0 ) and ( " }{XPPEDIT 18 0 "(1-sqrt(1 97))/2" "6#*&,&\"\"\"F%-%%sqrtG6#\"$(>!\"\"F%\"\"#F*" }{TEXT -1 24 ", \+ 0 ) are saddle points." }}{PARA 0 "" 0 "" {TEXT -1 171 "To conclude th e discussion of soft springs construct a plot containing the equilibri um solutions, representative solution curves, and the direction field \+ for this example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 66 "Pequil := pointplot( [seq( eval( [y0,v0], E \+ ), E=[equil[soft]] )]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 " \+ color=BLUE, symbolsize=18 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "icA := [ [ y(0)=-5, v(0)=-4 ], [ y(0)=-8, v(0)=8] ]; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "PsolA := DEplot( sys[soft], [ y (t), v(t) ], t=-0.1..10, icA," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " \+ stepsize=0.05, arrows=NONE ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "icB := [ [ y(0)=-6, v(0)=-4], [ y(0)=-9, v(0)=8] ];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "PsolB := DEplot( sys[soft], \+ [ y(t), v(t) ], t=-0.2..0.5, icB," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " stepsize=0.05, arrows=NONE ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "icC := [ [ y(0)=-10, v(0)=25] ];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "PsolC := DEplot( sys[soft], [ y(t), v(t) \+ ], t=-0.1..2.5, icC," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " \+ stepsize=0.1, arrows=NONE ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "icD := [ [ y(0)=-10, v(0)=30] ];" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 61 "PsolD := DEplot( sys[soft], [ y(t), v(t) ], t=-0.1. .1.1, icD," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " step size=0.1, arrows=NONE ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "icE := [ [ y(0)=10, v(0)=-10] ];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "PsolE := DEplot( sys[soft], [ y(t), v(t) ], t=-0.1..0.4, icE," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " stepsize=0.05, arr ows=NONE ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "icF := [ [ y (0)=10, v(0)=-15] ];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "PsolF := DE plot( sys[soft], [ y(t), v(t) ], t=-0.2..1.8, icF," }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 45 " stepsize=0.1, arrows=NONE ):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Pdfield := DEplot( sys[soft] , [ y(t), v(t) ], t=0..1," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " \+ y=-15..15, v=-40..40, dirgrid=[40,40] ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "display( Pequil, PsolA, PsolB, Psol C, PsolD, PsolE, PsolF, Pdfield, view=[-15..15, -40..40 ] );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "Note that the solution curves that become unbounded ceas e to be physically realistic when they exceed the spring's ability to \+ compress or extend." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "29.A-3" {TEXT -1 18 "29.A-3 Hard Spring" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Repea ting the analysis for a hard spring, the equilibrium solutions are" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "S := SS[hard]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sys[hard] : = eval( sys1, param1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 " q2 := evalf( allvalues( eval( q1, param1 ) ) );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Note t hat only one of these solutions is real; the complex-valued solutions \+ are not physical." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 42 "equil[hard] := op(remove( has, [q2], I ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 43 "Linearization about this equilibrium yields" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "for X0 in [equil[hard]] do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " b := e val( [ f(y0,v0), g(y0,v0) ], X0 union param1 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 " A := eval( evalm(jacobian( [f(y0,v0),g(y0,v0)], [y0 ,v0] )), X0 union param1 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " li nsys1 := equate( diff( U, t ), evalm( A &* U + b ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " print( `-------------------------------------- -----------------------` );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 " pr int( `Linearization at [y0,v0]`=eval([y0,v0],X0 union param1) );" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " print( A = evalm( A ) );" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " print( lambda = [eigenvals( A )] \+ );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " print( `------------------- ------------------------------------------` );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Therefore, the sole equilibri um is a stable spiral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 170 "To conclude the discussion of hard springs construc t a plot containing the equilibrium solution, representative solution \+ curves, and the direction field for this example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Pequil := p ointplot( [seq( eval( [y0,v0], E ), E=[equil[hard]] )]," }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 49 " color=BLUE, symbolsize=18 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "icA := [ [ y(0)=3*j, v(0)=0 ] $ j=-3..0 ];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "PsolA := \+ DEplot( sys[hard], [ y(t), v(t) ], t=0..10, icA," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 " stepsize=0.025, arrows=NONE ):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Pdfield := DEplot( sys[hard] , [ y(t), v(t) ], t=0..1," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " \+ y=-15..15, v=-40..40, dirgrid=[40,40] ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "display( Pequil, PsolA, Pdfield, vi ew=[-15..15, -40..40 ] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 178 "Now, all solutions appear t o remain bounded and ultimately are attracted to the equilibrium solut ion. This illustrates a key qualitative difference between soft and ha rd springs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "[Back 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 }