_jh_ex2a

D:\j_Iam_TEMP\_jh\_jh_ex2a.wxmx

Умова

Визначити силу гідростатичного тиску на бокову грань "MN" параболоїдального каналу висотою Hmn = 1 метр та шириною B = 3 метри. Рівень води у каналі H = 2 метри.
Аналітичні та чисельні розрахунки методом K123.

(%i1)

kill(all);

\[\]\[\tag{%o0} \ensuremath{\mathrm{done}}\]

(%i6)

ro:1000;g:9.81;H:2;B:3;Hmn:1;n:1000;

\[\]\[\tag{%o1} 1000\]

\[\]\[\tag{%o2} 9.81\]

\[\]\[\tag{%o3} 2\]

\[\]\[\tag{%o4} 3\]

\[\]\[\tag{%o5} 1\]

\[\]\[\tag{%o6} 1000\]

(%i11)

fh(x):=x··2;fHab(x):=Hmn;fH(x):=H;Fx(h):=sqrt(h);diff(Fx(h),h);

\[\]\[\tag{%o7} \mathop{fh}(x)\mathop{:=}{{x}^{2}}\]

\[\]\[\tag{%o8} \mathop{fHab}(x)\mathop{:=}\ensuremath{\mathrm{Hmn}}\]

\[\]\[\tag{%o9} \mathop{fH}(x)\mathop{:=}H\]

\[\]\[\tag{%o10} \mathop{Fx}(h)\mathop{:=}\sqrt{h}\]

\[\]\[\tag{%o11} \frac{1}{2 \sqrt{h}}\]

(%i12)

dh(h):=1/(2·sqrt(h));

\[\]\[\tag{%o12} \mathop{dh}(h)\mathop{:=}\frac{1}{2 \sqrt{h}}\]

(%i13)

plot2d([fh(x),fHab(x),fH(x)],[x,0,2]);

\[\]\[\tag{%o13} false\]

Figure 1:D:\j_Iam_TEMP\_jh\_jh_ex2.png
Diagram

(%i14)

plot2d(Fx(h),[h,0,H]);

\[\]\[\tag{%o14} false\]

Figure 2:D:\j_Iam_TEMP\_jh\_jh_ex2a.png
Diagram

(%i15)

fp(h):=ro·g·(H−h);

\[\]\[\tag{%o15} \mathop{fp}(h)\mathop{:=}\ensuremath{\mathrm{ro}} g\, \left( H\mathop{-}h\right) \]

P_x - горизонтальна проекція сили тиску

(%i16)

P_x:integrate(fp(h)·B,h,0,Hmn);

\[\]\[\tag{%o16} 44145.0\]

(%i17)

mP_x:integrate(fp(h)·(H−h)·B·h,h,0,Hmn);

\[\]\[\tag{%o17} 26977.5\]

(%i18)

mP_x_down:integrate(fp(h)·B·h,h,0,Hmn);

\[\]\[\tag{%o18} 19620.0\]

(%i19)

h_D:mP_x/P_x;

\[\]\[\tag{%o19} 0.6111111111111112\]

(%i20)

h_D_:mP_x_down/P_x;

\[\]\[\tag{%o20} 0.4444444444444444\]

(%i21)

h_D+h_D_;

\[\]\[\tag{%o21} 1.0555555555555556\]

Figure 3:D:\j_Iam_TEMP\_jh\_jh_ex2_P_x.jpg
Diagram

P_x - чисельний алгоритм методу К123

(%i24)

P_x_sum:0;dh:Hmn/n;h:0;

\[\]\[\tag{%o22} 0\]

\[\]\[\tag{%o23} \frac{1}{1000}\]

\[\]\[\tag{%o24} 0\]

(%i25)

Pi_x(h):=ro·g·(H−h)·B;

\[\]\[\tag{%o25} \mathop{Pi\_ x}(h)\mathop{:=}\ensuremath{\mathrm{ro}} g\, \left( H\mathop{-}h\right) B\]

(%i27)

for i:1 while h < Hmn do (Pi:Pi_x(h)·dh,P_x_sum:P_x_sum+Pi,h:h+dh);P_x_:P_x_sum;

\[\]\[\tag{%o26} \ensuremath{\mathrm{done}}\]

\[\]\[\tag{%o27} 44159.71499999999\]

(%i28)

Px_test:integrate(Pi_x(hi),hi,0,Hmn);

\[\]\[\tag{%o28} 44145.0\]

(%i31)

mP_x_sum:0;dh:Hmn/n;h:0;

\[\]\[\tag{%o29} 0\]

\[\]\[\tag{%o30} \frac{1}{1000}\]

\[\]\[\tag{%o31} 0\]

(%i33)

for i:1 while h < Hmn do (Pi:Pi_x(h)·dh·h,mP_x_sum:mP_x_sum+Pi,h:h+dh);mP_x_:mP_x_sum;

\[\]\[\tag{%o32} \ensuremath{\mathrm{done}}\]

\[\]\[\tag{%o33} 19605.280095000024\]

(%i34)

kill(h);

\[\]\[\tag{%o34} \ensuremath{\mathrm{done}}\]

(%i35)

mPx_test:integrate(Pi_x(h)·(h),h,0,Hmn);

\[\]\[\tag{%o35} 19620.0\]

(%i37)

h_D:mP_x_sum/P_x_sum;h_D_test:mPx_test/Px_test;

\[\]\[\tag{%o36} 0.44396301232922425\]

\[\]\[\tag{%o37} 0.4444444444444444\]

(%i38)

Rel_ERROR:(100/h_D_test)·(h_D_test−h_D),numer;

\[\]\[\tag{%o38} 0.10832222592453838\]

P_z - вертикальна проекція сили тиску

(%i39)

kill(h);

\[\]\[\tag{%o39} \ensuremath{\mathrm{done}}\]

(%i44)

ro:1000;g:9.81;H:2;B:3;Hmn:1;

\[\]\[\tag{%o40} 1000\]

\[\]\[\tag{%o41} 9.81\]

\[\]\[\tag{%o42} 2\]

\[\]\[\tag{%o43} 3\]

\[\]\[\tag{%o44} 1\]

(%i45)

P_z:integrate(fp(h)·B·dh(h),h,0,Hmn);

\[\]\[\tag{%o45} 49050.0\]

(%i46)

mP_z:integrate(fp(h)·B·dh(h)·(Fx(h)),h,0,Hmn);

\[\]\[\tag{%o46} 22072.5\]

(%i47)

x_C:mP_z/P_z;

\[\]\[\tag{%o47} 0.45\]

Figure 4:D:\j_Iam_TEMP\_jh\_jh_ex2_P_z.jpg
Diagram

P_z - чисельний алгоритм методу К123

(%i52)

ro:1000;g:9.81;H:2;B:3;Hmn:1;

\[\]\[\tag{%o48} 1000\]

\[\]\[\tag{%o49} 9.81\]

\[\]\[\tag{%o50} 2\]

\[\]\[\tag{%o51} 3\]

\[\]\[\tag{%o52} 1\]

(%i55)

P_z_sum:0;dh:Hmn/n;h:0;

\[\]\[\tag{%o53} 0\]

\[\]\[\tag{%o54} \frac{1}{1000}\]

\[\]\[\tag{%o55} 0\]

coordinate - x {for h=i*dh x=sqrt(h)}

(%i57)

fB(i,dh):=sqrt(i·dh); fB(1,dh),numer;

\[\]\[\tag{%o56} \mathop{fB}\left( i\mathop{,}\ensuremath{\mathrm{dh}}\right) \mathop{:=}\sqrt{i\, \ensuremath{\mathrm{dh}}}\]

\[\]\[\tag{%o57} 0.03162277660168379\]

(%i59)

Pi_x(h):=ro·g·(H−h)·B; Pi_x(0.2);

\[\]\[\tag{%o58} \mathop{Pi\_ x}(h)\mathop{:=}\ensuremath{\mathrm{ro}} g\, \left( H\mathop{-}h\right) B\]

\[\]\[\tag{%o59} 52974.0\]

(%i61)

fdb(i,dh):=(fB((i+1),dh)−fB((i),dh))/dh;fdb(1000,dh),numer;

\[\]\[\tag{%o60} \mathop{fdb}\left( i\mathop{,}\ensuremath{\mathrm{dh}}\right) \mathop{:=}\frac{\mathop{fB}\left( i\mathop{+}1\mathop{,}\ensuremath{\mathrm{dh}}\right) \mathop{-}\mathop{fB}\left( i\mathop{,}\ensuremath{\mathrm{dh}}\right) }{\ensuremath{\mathrm{dh}}}\]

\[\]\[\tag{%o61} 0.4998750624609638\]

(%i62)

for i: 0 while h < Hmn do (db:fdb(i,dh),Pi:Pi_x(h)·db·dh,P_z_sum:P_z_sum+Pi,h:h+dh)$

(%i64)

P_z_:P_z_sum,numer;h;

\[\]\[\tag{%o63} 49064.52275520602\]

\[\]\[\tag{%o64} 1\]

FOR MOMENT P_z coordinate - x {for h=i*dh x=sqrt(h)}

(%i67)

mP_z_sum:0;dh:Hmn/n;h:0;

\[\]\[\tag{%o65} 0\]

\[\]\[\tag{%o66} \frac{1}{1000}\]

\[\]\[\tag{%o67} 0\]

(%i68)

for i: 0 while h < Hmn do (db:fdb(i,dh),Pi:Pi_x(h)·db·dh·fB(i,dh),mP_z_sum:mP_z_sum+Pi,h:h+dh)$

(%i70)

mP_z_:mP_z_sum,numer;h;

\[\]\[\tag{%o69} 22003.335303355238\]

\[\]\[\tag{%o70} 1\]

(%i72)

x_C_:mP_z_/P_z_,numer; x_C;

\[\]\[\tag{%o71} 0.44845713496765977\]

\[\]\[\tag{%o72} 0.45\]

(%i73)

Rel_ERROR:(100/x_C)·(x_C−x_C_),numer;

\[\]\[\tag{%o73} 0.3428588960756102\]

P - Сила гідростатичного тиску

(%i74)

P:sqrt(P_x··2+P_z··2);

\[\]\[\tag{%o74} 65990.02595089655\]

(%i75)

phi_rad:atan(P_z/P_x),numer;

\[\]\[\tag{%o75} 0.83798122500839\]

(%i76)

phi_grad:atan(P_z/P_x)·(180/%pi),numer;

\[\]\[\tag{%o76} 48.01278750418334\]

(%i77)

d:tan(P_z/P_x);

\[\]\[\tag{%o77} 2.0199703317182265\]

Plot2d

(%i78)

fh_D(x):=h_D_;

\[\]\[\tag{%o78} \mathop{fh\_ D}(x)\mathop{:=}\ensuremath{\mathrm{h\_ D\_ }}\]

(%i79)

plot2d([fh(x),fHab(x),fH(x),fh_D(x),[discrete,[0.44,0.44],[0,1]],
       [discrete,[0.44],[0.45]]],[x,0,1.5],
   [legend, "parabola","Top_box", "Water", "P_x", "P_z", "point"],
   [style, [lines, 5,5], lines, [lines, 3,1], lines, lines, [points, 3,2]],
   [point_type, circle]);

\[\]\[\tag{%o79} false\]

Figure 5:D:\j_Iam_TEMP\_jh\_jh_ex2c.png
Diagram

eq:y=x_C;plot2d([fh(x),fHab(x),fH(x),fh_D(x),eq],[x,0,2],[y,0,4]);

Answer

(%i85)

P_x;P_z;P;h_D_;x_C;phi_grad;

\[\]\[\tag{%o80} 44145.0\]

\[\]\[\tag{%o81} 49050.0\]

\[\]\[\tag{%o82} 65990.02595089655\]

\[\]\[\tag{%o83} 0.4444444444444444\]

\[\]\[\tag{%o84} 0.45\]

\[\]\[\tag{%o85} 48.01278750418334\]

Figure 6:D:\j_Iam_TEMP\_jh\_jh_ex2.jpg
Diagram

Scale screen 300 px

(%i88)

P_x/300;P_z/300;P/300;

\[\]\[\tag{%o86} 147.15\]

\[\]\[\tag{%o87} 163.5\]

\[\]\[\tag{%o88} 219.9667531696552\]

(%i89)

P_:sqrt(P_x_··2+P_z_··2);

\[\]\[\tag{%o89} 66010.66445717201\]

REL_ERRORS

(%i90)

Rel_ERROR_P_x:(100/P_x)·(P_x−P_x_),numer;

\[\]\[\tag{%o90} \mathop{-}0.033333333333308936\]

(%i91)

Rel_ERROR_mP_x_:(100/mP_x_down)·(mP_x_down−mP_x_),numer;

\[\]\[\tag{%o91} 0.0750249999998791\]

(%i92)

Rel_ERROR_mP_z:(100/mP_z)·(mP_z−mP_z_),numer;

\[\]\[\tag{%o92} 0.31335234633486214\]

(%i93)

Rel_ERROR_P_z:(100/P_z)·(P_z−P_z_),numer;

\[\]\[\tag{%o93} \mathop{-}0.029608063620829527\]

(%i94)

Rel_ERROR_P:(100/P)·(P−P_),numer;

\[\]\[\tag{%o94} \mathop{-}0.031275190421675786\]

Created with wxMaxima.

The source of this Maxima session can be downloaded here.