{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 "" -1 256 "" 0 1 0 0 0 0 0 1 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 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 14 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 "MyWork" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 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 "MyWork" -1 257 1 {CSTYLE "" -1 -1 "Ti mes" 1 14 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 "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 14 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 24 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" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 24 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 "Normal" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 18 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 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }} {PARA 259 "" 0 "" {TEXT 257 21 "MECHANICAL VIBRATIONS" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 11 "Example 1.-" }} {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 358 " A m ass weighing 4 lb stretches a spring 2 in. Suppose that the mass is di splaced an additional 6 in. in the positive direction and then release d. The mass is in a medium that exerts a viscous resistance of - 6 lb \+ when the mass has a velocity of 3ft/sec. Under the assumptions discuss ed in this section, formulate the IVP that governs the motion of the m ass." }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 126 "Before formulating the IVP, let's see what we need. The equation \+ for the mass-spring system without external force is given by" }} {PARA 0 "" 0 "" {TEXT 258 1 " " }}{PARA 0 "" 0 "" {TEXT 267 67 " \+ " }{TEXT 268 20 "mu'' = - cu' - ku ," }{TEXT 266 5 " " }}{PARA 0 "" 0 "" {TEXT 269 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 22 " \+ " }}{PARA 0 "" 0 "" {TEXT 261 68 " \+ " }{TEXT 262 21 "mu'' + cu' + ku = 0, \n" }{TEXT 263 220 "\nwhere m is the mass, c is the damping coeffic ient, and k is the spring constant. The derivation of this equation of motion is based on several assumptions, including the following:\n \n the amplitude, u(t) of the m" }{TEXT 260 0 "" }{TEXT 259 129 "oti on is small;\n\n the damping force is proportional to the velocity; \n\n the spring force is proportional to the displacement.\n" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "MS:=m*diff(u(t),t$2)+c*diff(u(t),t)+k*u(t)=0;" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 69 "So we need \+ to determine \"k\", \"m\", and c out of the information given." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "k:= 24; m:= 1/8; c:= 2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "MS:=m*diff(u(t),t$2)+k*u(t)+c*diff(u(t),t)=0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "General solutio n:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(MS,u(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Initial conditions:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "u0:=1/ 2; v0:=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "dsolve(\{MS, u (0)=u0,D(u)(0)=v0\},u(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "S:=rhs(dsolve(\{MS, u(0)=u0,D(u)(0)=v0\},u(t)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "u1:=unapply(S,t);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "plot(u1(t), t=0..0.8);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 57 "Let's study now the effects on the solution behavior for " }{TEXT 270 28 "different degre es of damping" }{TEXT 271 1 ":" }}{PARA 0 "" 0 "" {TEXT 265 97 "Consid er the spring-mass system equation with m = 1, k = 1, and c (damping c oefficient) variable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "m:=1; k:=1; unassign('c');" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "MS:=m*diff(u(t),t$2)+c*diff( u(t),t)+k*u(t)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve (MS,u(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Define the initial conditions as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "init1:= u(0 )=1, D(u)(0)=1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "First case: Overdamped Motion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "c:=3;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "MS;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(MS,u(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Using Initial Conditions \+ and defining a function " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "u1aux:=dsolve(\{MS, init1\},u(t)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "u1:=unapply(rhs(u1aux), t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(u1(t,c), t=0..1 0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Second Case: Critically Damped Motion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "c:=2; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "MS;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(MS,u(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Using Initial Conditions and defin ing a function " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "u2aux:=dsolve(\{MS, init1\},u(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "u2:=unapply(rhs(u2aux),t);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(u2(t), t=0..10);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Th ird Case: Damped Motion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "c:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "MS;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsol ve(MS,u(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 49 "Using Initial Conditions and defining a function " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "u3aux:=dsolve(\{MS, init1\},u(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "u3:=unapply(rhs(u3aux),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(u3(t), t=0..10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Fourth Case: Smaller Damp ing" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "c:=1/10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 " MS;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(MS,u(t));" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "U sing Initial Conditions and defining a function " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "u4aux:=dsol ve(\{MS, init1\},u(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " u4:=unapply(rhs(u4aux),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(u4(t), t=0..30);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 59 "Superposition of the above solutions for \+ different damping:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot(\{u1(t),u2(t),u3(t),u4(t)\}, t=0..15); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Animation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "First we need to obta in the solution as a function of the damping coefficient \"c\" and the time \"t\" for the initial conditions given" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Using Initial Condition and defining a function " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "unassign('c');" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(MS,u( t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "uaux:= rhs(dsolve( \{MS,init1\},u(t)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ud: =unapply(uaux,t,c);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "animate( ud(t,c),t=0..30,c=1/10..3, numpoints=100,frames=20);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 48 "3-D plot of damping motion with variable \+ damping" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot3d(ud(t,c), t=0..30, c=0..1, numpoints=500, axes= boxed);" }}}}{MARK "0 12 2" 212 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }