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"" }}{PARA 258 "" 0 "" {TEXT -1 16 "The Beam Problem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 257 58 "We shall consider the problem of the def lection of a beam " }}{PARA 0 "" 0 "" {TEXT 258 62 "imbedded in a wall . The differential equation takes the form:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "E*I* D(D(D(D(y(x))))) = w(x );" "6#/*(%\"EG\"\"\"%\"IGF&-%\"DG6#-F)6#-F)6#-F)6#-%\"yG6#%\"xGF&-%\" wG6#F4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 42 " where the weight function over a point " }{TEXT 259 1 "x" } {TEXT -1 1 " " }}{PARA 261 "" 0 "" {TEXT -1 26 "on the beam is given b y w(" }{TEXT 260 1 "x" }{TEXT -1 8 ") and w(" }{TEXT 271 1 "x" }{TEXT -1 2 ") " }{TEXT 272 4 "= 0 " }{TEXT -1 4 "for " }{TEXT 273 5 "x > c" }{TEXT -1 3 ". " }}{PARA 276 "" 0 "" {TEXT -1 0 "" }}{PARA 275 "" 0 " " {TEXT -1 25 "The intial conditions are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 3 "a) " }{TEXT 261 9 " y(0) = 0" } {TEXT -1 8 ", and " }{TEXT 262 9 "y'(0) = 0" }{TEXT -1 50 " where th e beam is fixed in the wall at the point " }{TEXT 263 5 "x = 0" } {TEXT -1 1 "." }}{PARA 266 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT -1 30 "We also have at the free end " }{TEXT 264 5 "x= 2c" } {TEXT -1 2 ": " }}{PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 3 "b) " }{TEXT 265 10 "y''(2c)=0," }{TEXT -1 5 " and " } {TEXT 266 13 " y'''(2c)= 0." }}{PARA 264 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 5 "Let " }{TEXT 267 1 "A" }{TEXT -1 8 " denote \+ " }{TEXT 268 6 "y''(0)" }{TEXT -1 5 " and " }{TEXT 269 1 "B" }{TEXT -1 8 " denote " }{TEXT 270 7 "y'''(0)" }{TEXT -1 2 ". " }}{PARA 270 " " 0 "" {TEXT -1 37 "Using the Laplace transform we obtain" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "E*I*s^4*Y(s) - s*A - B = W(s)" "6#/,(**%\"EG\"\"\"% \"IGF'%\"sG\"\"%-%\"YG6#F)F'F'*&F)F'%\"AGF'!\"\"%\"BGF0-%\"WG6#F)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 271 "" 0 "" {TEXT -1 3 "or " }} {PARA 272 "" 0 "" {TEXT -1 0 "" }}{PARA 273 "" 0 "" {XPPEDIT 18 0 "y(x ) = (L^(-1))*( W(s)/(E*I*s^4) +A/(E*I*s^3) + B/(E*I*s^4))" "6#/-%\"yG 6#%\"xG*&)%\"LG,$\"\"\"!\"\"F,,(*&-%\"WG6#%\"sGF,*(%\"EGF,%\"IGF,F3\" \"%F-F,*&%\"AGF,*(F5F,F6F,F3\"\"$F-F,*&%\"BGF,*(F5F,F6F,F3F7F-F,F," }} {PARA 0 "" 0 "" {TEXT 256 0 "" }}{PARA 274 "" 0 "" {TEXT -1 23 "We obt ain the following" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "y(x) = A*x^2/(2*E*I ) + B*x^3/(6*E*I) + int(w(u)*(u -t)^3, u=0..t)/6" "6#/-%\"yG6#%\"xG,(* (%\"AG\"\"\"*$F'\"\"#F+*(F-F+%\"EGF+%\"IGF+!\"\"F+*(%\"BGF+*$F'\"\"$F+ *(\"\"'F+F/F+F0F+F1F+*&-%$intG6$*&-%\"wG6#%\"uGF+*$,&F@F+%\"tGF1F5F+/F @;\"\"!FCF+F7F1F+" }}{PARA 280 "" 0 "" {TEXT -1 0 "" }}{PARA 278 "" 0 "" {TEXT -1 5 "where" }{TEXT 276 6 " t < c" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 277 "" 0 "" {TEXT -1 35 "Solve this problem using Maple for " }{TEXT 274 4 "c = " }{TEXT -1 9 "1 and w(" }{TEXT 275 1 "x" }{TEXT -1 5 ") = 1" } {TEXT 277 5 " - x " }{TEXT -1 4 "with" }{TEXT 278 7 " E I=" }{TEXT -1 1 "1" }{TEXT 279 1 "." }}{PARA 284 "" 0 "" {TEXT -1 78 "Note: after you have solved the integral you must check the inital conditions " } }{PARA 285 "" 0 "" {TEXT -1 38 "on the free end of the beam to obtain \+ " }{TEXT 280 1 "A" }{TEXT -1 5 " and " }{TEXT 281 1 "B" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 281 "" 0 "" {TEXT -1 38 "Use Maple to solve the circuit p roblem" }}{PARA 282 "" 0 "" {TEXT -1 0 "" }}{PARA 279 "" 0 "" {XPPEDIT 18 0 "D(D(q(t))) + Pi^2 q(t) = f(t)" "6#/,&-%\"DG6#-F&6#-%\"q G6#%\"tG\"\"\"*&%#PiG\"\"#-F+6#F-F.F.-%\"fG6#F-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 283 "" 0 "" {TEXT -1 66 "where q(0)= 0, q'(0) =0 and f(t) is the square wave with period 2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{MARK "0 1 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }