{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 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 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "bars.mws 12/24/97" }} {PARA 0 "" 0 "" {TEXT -1 72 "vibrations of bars, starting with free-fr ee bars, even and odd solutions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 76 "Then go to clamped-clamped solutions, and finally to clamped-free solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart;with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "s_even:=fsolve(tan(x)=-tanh(x), x,0 ..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "s_odd:=fsolve(tan( x)=tanh(x), x,0..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "s2: =fsolve(cos(x)*cosh(x) = 1,x,5..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "s2/s_odd;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(s_even/(3*Pi/4));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "L:=2; xe:=x*s_even*2/L; xo:=x*s_odd*2/L;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 51 "ff_even:=cos(xe)/cos(s_even)+cosh(xe)/cosh(s_e ven);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot(ff_even,x=-L/ 2..L/2,title=`free-free 1st even`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "ff_odd:=sin(xo)/sin(s_odd)+sinh(xo)/sinh(s_odd);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot(ff_odd,x=-1..1,title=`f ree-free 1st odd`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Now for cl amped-clamped solutions, which must vanish, along with their slope at \+ +/- L/2" }}{PARA 0 "" 0 "" {TEXT -1 88 "Note that they have the same f requencies as free-free solutions, but not the same shape." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "cc_even:=cos(xe)/cos(s_even)-cosh(xe)/cosh( s_even);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot(cc_even,x= -L/2..L/2,title=`clamped-clamped (1st even)`);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 48 "cc_odd:=sin(xo)/sin(s_odd)-sinh(xo)/sinh(s_odd );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot(cc_odd,x=-L/2..L /2,title=`clamped-clamped 1st odd`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "kL_cf:=fsolve(cos(x)*cosh(x)=-1,x,0..4); #kL clamped- free" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalf(kL_cf*2/Pi); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "xp:=x/L*kL_cf; # xp = k x" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "clamped-free wave function - satisifes y(0)=y'(0), and y''(L)=0=y'''(L)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "ycf:=(cos(xp)-cosh(xp))/(cos(kL_cf)+cosh(kL_cf))-( sin(xp)-sinh(xp))/(sin(kL_cf)+sinh(kL_cf));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(ycf,x=0..L,title=`clamped-free(lowest)`);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "kL_cf:=fsolve(cos(x)*cosh(x)=-1,x,4..6); #kL (2) clam ped-free" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "xp:=x/L*kL_cf; \+ # xp = kx (2nd solution)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "ycf:=(cos(xp)-cosh(xp))/(cos(kL_cf)+cosh(kL_cf))-(sin(xp)-sinh(xp))/( sin(kL_cf)+sinh(kL_cf));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot(yc f,x=0..L,title=`clamped-free (next-lowest)`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "24 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }