Bepaling van de stijfheidsmatrix voor rechthoekige kokers. Ir. F.E. Veldpaus TG

Bepaling van de stijfheidsmatrix voor rechthoekige kokers. Ir. F.E. Veldpaus TG3 64-28

L - 6-

-7-4 C 8 9

-ti- (%= -4 (x=- p,

'/ I y= f u. @e. u - U.

-/y - o o o o o a,

-20 - P x=o

-23-4A -44 426 I

4 a, o b o o C i * I I o I /II -a,/ -a,/ -4

A- -/ J2 f -k f o o f o o a f o 1 -P -JA 0 0 0 o o I.P o o o o o f o U o / o o o o -02 / o o d o o o f -2 o o o o r P o 8 8 Q Q o

ur h! h 9 a A ~n o \ h o LU a h a h Y ka I Wh 6 2 cc i a, * I x 3;s a h : I -/i-

I: L -44-

-.+a--

P= o o I o

- 4 7-

procedure St(E, nu, bl, b2, tl, $2, Pn, 1, E, singular); value E, nu, bl, b2, tl, t2, Fn, 1; e E, nu, bï, b2, tl, t2, Fn, 1; wrax S; ga, singular; 1_1- i, 5, k; real G, an, al, 232, a3, c, r2, s4, S2j alfa, gamma, cl, c2, dl, d2, d3, ch, sh, f1 && I array p[lm array A, B11:6, 1.61; (2 + 2 x nu); c-4 x E/((I - nu & 2) x (bî/t1 /1\ 3 + ~,/t2 /f\ 3)); an := bi x b2; al :=4~~an/E\2x(blXtlt.bMt2~3X~n)/3;a2 22, f3, 24; := 4 X G X ayi X (bl X t2 i- b2 X t1); a3 := 4 X G X an X (- bl X t2.c b2 X tl ); r2 := a2 x c/(a x (a2 /I\ 2 - a3 II\ 2)); s4 C= c/al; s2 :a sgrt((s4); alfa :=r sqrt((s2 c r2)/2); grna := sqrt((s2 - r2)/2); cl := exp(alfa X 1); c2 := l/el; dl := sln(gwxm X l);d2 := cos(gamna /E: fl :=- Ch X dl; f2 t= Ch X d2j 23 := - sh 9( d25 f4 := sh X dl; 1.)) ch := (cl + c2)/2; sh := (el - c2)/2; A[1, li:=alif&4 + gammf2; AC1, 21:=a1ia~f5 - grnafl; ACl, 3lc=aliaxf2 - gaxamaxî4; AC1, 4J:=aLfaxfl c gm~mxf3; AC1, 51:=a3; All, 61:=û; d3 := - 2 X r2 X a3/(s4 X a2); dl := r2 X d3; d2. := 2 x aiá-a x gma x d3; AC2, li:=dlxf? -+ d2 X f3; AC2, 21:~ dl X f2 - d2 X 24; AC2, 31:=dl X f3 -* d2 X flj A[2, 4l:=dl X Y% + d.2 X f2; A[2, 61:=1; ALS, 51:=a2 X I; d3 := alfa x gamma; dl := (&Ma h 4-6 x d3 /I\ 2 + gmm /2\ 4)/s4; d2 := 4. x d3 x (alfa /2\ 2 - gamm /2\ 2)/s4; ALS, 1 I:=dl X fl i- d2 X f3; AC3, 2I:=dl X f2-82 X f4; AC3, 3I:=dl X f3 - d2 X flj A[3, 4];=dl X f4 t- d2 X 12; -, ii-l]:4[19i+ i]; AC5, i]:s--a[2, i]; A[5, i+1]:=+a[2, icl];, ii-1 1 :=+A/S,i+l] end; d1 := - al X r2; d2 := -2X al X d3; Bil, 1I:=dl X PI + d2 X 13; B[1, 21:zdl X f2 - d2 X f4; BCl, 3I:=dl X f.3 -ai d2 3( BCI, 51:=; BC1, 61:=;. for i := 1 step 1 until 6 do B[2, i1 :=; B[2, 51:=+ (a2 /1\ 2 - a3 /I\ 2)) d1 := -a1 X alfa =fa $-2-3 *á gmmm /ta 2); d2 := -al X gamma, X (3 x alfa /2\ 2 - gamm 4 2); flj B[1, 41a=d1 X f4 i- d2 X i29 BC3, l]:=dl X i4 + d2 X f2; BC3, 2]:=dl X fg - d2 X á-1; BCS, 3]:=dl X f2 - I d2 X fkj BL3, 4]:=dl X fl -I- d2 X P3; B13, SI:=; BC3, 61:=0; - for i := 1 2 5 $0 begin B[4, BC ; BE, i+l]:-b[l, i-1-11; Ed?, i]:-j3[2, i]; Uí5, i+i~:--cb[2, ic11; BC6, i]:-b[3, i]; BC6, Icî ]:=+BcS,icl1 end; -.-ii CHU ~ECWSITIülV( 6, A, A, p,cjtuci, 2,,-12, singular); CWUTINWR>E(6, A, p, A); - for i := 1 1 until 6 do for j := 1 % 1 until 6 do begin SCi, 1 -- P -7 *- ; for k := 1 until 6 -d m. m, jt:= S[i,p jl c B[i, k] X Alk, j]. end end bepaling stijfheidsmatrix S bij torsie; - _I_

PJ fo b P Li c. P n vl \. Ç u -F+ u i i X X r? Y F- -k L i% Fa, ìv X x M w u. 1% c. Fo fl 2 x Y Ç -t E X +b fv ir..e tt gr; x

- d.0 -

1 2* I Eocedure S'IYFHEIDSGHUT~DElV(E, au, bl, b2, tl, t2, Fn, a); value E, nu, bl, b2, tl, t2, Fnj real E, nu, bl, b2 tl, t2, Fn; arrez a; 4, alfa, beta- deze volgorde opgeborgen in array a); real an, cl, ~ 2 an ; := bl 3s: b2; c2 := 2 '36; E: X an/(l i- nu); cl := e2 X bl X t2) CS := e2 X b2 X tlj := 4 X E X m. X a;yi X ((BI X bl + b2 Ix '~2)/3 I- &I); al21 := c'i +- c2; a[3] := - cl i- 62; _I begin comment bepaling vaua al, a23, c, r /1\ 2, s al41 := 4 X E/((I - nu X nu) X (bl/tl /1\ 3 + b2/l2 4 3)); d51 := ac21 X a[4]/(2 X (al21 +- acu1) X (al21 -- ac31)); a[61 := a[4l/a[il; CI := sqrt(ac6j)/2; e2 := al51/2; al71 := qrt(c1 i- ei?); a[$] := sqrt(c1 - c2) I_ end bepaling interessante stijfheidsgrootheden voor een rechthoekige koker; rocedure Fl(aJ-fa, beta, x, fi); value alfa, beta, x; real alfa, betel, x; array fi; :egin coment bepaling van de hulpfuncties fil, --- -- - - real el, c2, c3, c4; e2 := exp(alfa X abs(x)); e3 := l/c2; c1 := (e2 + c3)/2; e2 :=: (e2- c3)/2; if x < Q then c2:=-c2; E= beta X x; c3 := sin(c4); 64 := CS(~~); fii.11 := ci x c3; Tic21 := cl x c4.i fic31 := c2 x c4; fir41 := e2 x e3 end bepaling fig procedure MATRIX Al (1, x, a, fi, A); value x; real 1, x; _arre;y a, fi, A; v begin coment bepaling van de matrix Al uit c = Au; I real dl, d2, d3, d4, d5, d6; d1 := a[7]; d2 := ar81; d4 := - 2 X a[51 X a[3j/(al6] X a[21); d3 := ac5.1 X d4; dk := 2 36 dl X d2 X d4; d5 := - 1 + 2 x al51 x ac5i/al61; d6 := 4 X dl x d2 X ac'21/algj; All, 11 := d1 X fir41 + d2 X fic21; All, 21 := dl X ail33 - d2 X ficî]; A[1, 31 := dl X fi[2] - d2 X ficli.1; All, 41 := dl X fi[l] + d2 X fie31; All, 51 := a[3l/(a[2] X 1); A[l, 61 := ; A[2, 11 := d3 X PiCl] + d4 X fi131; 21 := d3 X ail21 - d4 51 fic4l; A12, 31 := d3 X fic31 - d4 X fill]; A[2, 41 := d3 x flik] -c d4 x fic2l; A[2, 51 := - x/l; A[2, 61 := I; AC3, 11 := d.j X ficí1 i- d6 X fic31; AC3, 21 := d5 X fic21 - d6 X fic41; AC3, 31 := d5 X ril31 - d6 x fic11; A[', 41 := d5 x fi[k] i- d6 X file]; Als, 51 := A[3, 61 := ; I_ end bepaling vm de hulpmatrix Al; "..- -e--

procedure MATRIX A(1, begin integer i, for i := 1 - - _I_ for j := 1, 2 do -- - --._U a, fi, A, singular); value 1; real 1; arrax a, fi, A,; label singular; [1:6]; EyIA31NfX Al(l, - 1, a, fi, A); i i- I] := - A[I, i i- I]; AC4 + j, i] := - All c J, i]; Al4 4-3, i.t 'I] := AC1 c j, i c 11 end end; CRoUTDEC~MPSITIN(6, A, A, p, true, 2,u-12, singular); CHUUTX~QCSE(~, A, p, A) _L_ end bepaling van de hulpmatrix A; 1-1 procedure MATRIX BI(1, a, fi, B); v real 1; begin comment bepaling van de matrix 13 uit f = Bc; integer i; real dl, d2, d3, d4, d5, d6; - aa fi, B; dl := ac11 n53; d2 := 2 X al11 X af71 X ac81; d3 := (a[21 - ac31) X (a[21 + a[']); d5 := -I- ac11 x act/l 9(; (al71 x ac71-3 x ai81 3< a[83); d6 := -t. ac11 x ac81 x (3 x ar71 x al71 - ac81 x ac81); B[1, 11 := + dl X fic11 + d2 X fils]; B[1, 21 := + dl X fi621 - d2 X fic41; B[1, 31 := i- dl X fi[3] - d2 X fi[1 J; B[î, 41 := c dî x fic41 + d2 x fil21; BCi, 51 := ; BLI, 63 := ; for i := 1 1 until 4 do BC2, i] := ; B[2, 51 := - d3/(a[2] X 1); B[2, 61 ;= ;, TE- ' 11 := + d5 X.mT c % X fi[2]; B[3, 21 := + d5 X fi[3] - d6 X fill]; BC3, 31 := -t. d5 x fir21 - a6 X fic41; 1313, 41 := + d5 x fill] c d6 x SiC3J; BC3, 51 :a ; BC3, 61 := ; end bepaling hulpmatrix 13; rocedure MTHIX B(1, a, fi, B); value 1; real 1; array a, fi, B; I for i := i stt;d 2 until 5 do begin intege? i, j; UTmx 81(1, a, Fi, B); begin 814, i] := - B[I, i]; B[4, i c IE= - B[1, i c 11;.m"i] i- BC3, i]; B[6, i i- 11 := - B[3, i + 13. end; for i := 1 _II_ until 6 i& Bi?, i1 := ; BC5, 51 := B[2, 51; for i := 1 sts 1 until 3 -- do for j := 1 1 -until 1 6 do -[i, J] := - Blij 51 P end bepaling van matrix B; - 1_.

procedure KETVE(l, a, Qe, singular); v value 1; sg li a? Qe; P label sin,gu:lar; begin Comment bepaling van de stijfhcidsutrix bij torsie VEW rechthoekige koker met exact Vlasov-E3lement; fic1:4l AL1 :6, 1 :61, Blï :6, 1 :61; j MATHIX A(l, a, fi, A, singular); MAYNX B(1, a, fi, 23); e[i>*:= G r i1 := QeLi, jl Qe[j, 6 do for j := i 1 uti1 6 do - P k-?=-f-- 1 - I_ until 5 do QelisLT] end; I _I end bepaling va de stijfheidsmtrix Qe bij torsie van kokcrrs(exact); := &ieli, j] + B[i, kj >< ALk, j];

@af u4 (i. *r p -' II xn, f o x F X *. P u b.