_jh_ex1e_r002a

_jh_ex1e_r002a.wxmx

Solution Three-command method K123 (Numerical metod)

Метод трьох команд К123 (Чисельний метод)

Визначити силу гідростатичного тиску на секторний затвор MN .
Радіус R = 1 м, ширина B = 2 м, висота затвора H = R/2 = 0.5 м.

Figure 1:D:\j_Iam_TEMP\_jh\_jh_ex1e1_job1.jpg
Diagram

Diagram

(%i1)

kill(all);

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

(%i6)

R:1;B:2;ro:1000;g:9.81;H:R/2;S:300;

\[\]\[\tag{%o1} 1\]

\[\]\[\tag{%o2} 2\]

\[\]\[\tag{%o3} 1000\]

\[\]\[\tag{%o4} 9.81\]

\[\]\[\tag{%o5} \frac{1}{2}\]

\[\]\[\tag{%o6} 300\]

(%i8)

fB(h,R):=sqrt(R··2−((R−h)··2));fB(0.5,R)−fB(0.49,R);

\[\]\[\tag{%o7} \mathop{fB}\left( h\mathop{,}R\right) \mathop{:=}\sqrt{{{R}^{2}}\mathop{-}{{\left( R\mathop{-}h\right) }^{2}}}\]

\[\]\[\tag{%o8} 0.005851002863323074\]

P_x

P_x - Horizontal projection of the force vector (Горизонтальна проекція вектору сили)

(%i9)

Pi_x(h):=ro·g·(R−h)·B;

\[\]\[\tag{%o9} \mathop{Pi\_ x}(h)\mathop{:=}\ensuremath{\mathrm{ro}} g\, \left( R\mathop{-}h\right) B\]

Figure 2:D:\j_Iam_TEMP\_jh\_jh_ex1e1_jobPx.jpg
Diagram

Diagram

(%i13)

n:1000;Px:0;dh:H/n;h:0;

\[\]\[\tag{%o10} 1000\]

\[\]\[\tag{%o11} 0\]

\[\]\[\tag{%o12} \frac{1}{2000}\]

\[\]\[\tag{%o13} 0\]

(%i14)

for i: 1 while h < H do (Pi:Pi_x(h)·dh,Px:Px+Pi,h:h+dh)$

(%i15)

Px;

\[\]\[\tag{%o15} 7359.952500000002\]

(%i16)

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

\[\]\[\tag{%o16} 7357.5\]

(%i20)

n:1000;mPx:0;dh:H/n;h:0;

\[\]\[\tag{%o17} 1000\]

\[\]\[\tag{%o18} 0\]

\[\]\[\tag{%o19} \frac{1}{2000}\]

\[\]\[\tag{%o20} 0\]

(%i21)

for i: 1 while h < H do (mPi:Pi_x(h)·(R−h)·dh,mPx:mPx+mPi,h:h+dh)$

(%i22)

mPx;

\[\]\[\tag{%o22} 5726.179158750008\]

(%i23)

mPx_test:integrate(Pi_x(hi)·(R−hi),hi,0,H);

\[\]\[\tag{%o23} 5722.5\]

(%i27)

n:1000;_mPx:0;dh:H/n;h:0;

\[\]\[\tag{%o24} 1000\]

\[\]\[\tag{%o25} 0\]

\[\]\[\tag{%o26} \frac{1}{2000}\]

\[\]\[\tag{%o27} 0\]

(%i28)

for i: 1 while h < H do (_mPi:Pi_x(h)·h·dh,_mPx:_mPx+_mPi,h:h+dh)$

(%i29)

_mPx;

\[\]\[\tag{%o29} 1633.773341249999\]

(%i30)

mPx_test:integrate(Pi_x(hi)·hi,hi,0,H);

\[\]\[\tag{%o30} 1635.0\]

(%i32)

h_D:mPx/Px;mPx_test/Px_test;

\[\]\[\tag{%o31} 0.7780184938353891\]

\[\]\[\tag{%o32} 0.2222222222222222\]

(%i34)

h_D_:_mPx/Px;mPx_test/Px_test;

\[\]\[\tag{%o33} 0.2219815061646116\]

\[\]\[\tag{%o34} 0.2222222222222222\]

P_z

P_z - Vertical projection of the force vector (Вертикальна проекція вектору сили)

Figure 3:D:\j_Iam_TEMP\_jh\_jh_ex1e1_job_Pz.jpg
Diagram

Diagram

(%i39)

R:1;B:2;ro:1000;g:9.81;H:R/2;

\[\]\[\tag{%o35} 1\]

\[\]\[\tag{%o36} 2\]

\[\]\[\tag{%o37} 1000\]

\[\]\[\tag{%o38} 9.81\]

\[\]\[\tag{%o39} \frac{1}{2}\]

(%i40)

fB(i,dh,R):=sqrt(R··2−((R−i·dh)··2));

\[\]\[\tag{%o40} \mathop{fB}\left( i\mathop{,}\ensuremath{\mathrm{dh}}\mathop{,}R\right) \mathop{:=}\sqrt{{{R}^{2}}\mathop{-}{{\left( R\mathop{-}i\, \ensuremath{\mathrm{dh}}\right) }^{2}}}\]

(%i45)

n:1000;h:0;Pz:0;dh:H/n;R;

\[\]\[\tag{%o41} 1000\]

\[\]\[\tag{%o42} 0\]

\[\]\[\tag{%o43} 0\]

\[\]\[\tag{%o44} \frac{1}{2000}\]

\[\]\[\tag{%o45} 1\]

(%i46)

pi_x(h):=ro·g·(R−h);

\[\]\[\tag{%o46} \mathop{pi\_ x}(h)\mathop{:=}\ensuremath{\mathrm{ro}} g\, \left( R\mathop{-}h\right) \]

(%i49)

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

\[\]\[\tag{%o47} \frac{1}{2000}\]

\[\]\[\tag{%o48} \mathop{fdb}\left( i\mathop{,}\ensuremath{\mathrm{dh}}\mathop{,}R\right) \mathop{:=}\frac{\mathop{fB}\left( i\mathop{+}1\mathop{,}\ensuremath{\mathrm{dh}}\mathop{,}R\right) \mathop{-}\mathop{fB}\left( i\mathop{,}\ensuremath{\mathrm{dh}}\mathop{,}R\right) }{\ensuremath{\mathrm{dh}}}\]

\[\]\[\tag{%o49} 26.18270860938711\]

(%i50)

for i: 0 while h < H do (db:fdb(i,dh,R),Pi:pi_x(h)·db·dh·B,Pz:Pz+Pi,h:h+dh)$

(%i53)

Pz,numer;Pz_test:14520.86;Rel_ERROR:(100/Pz_test)·(Pz_test−Pz,numer);

\[\]\[\tag{%o51} 14525.046184039022\]

\[\]\[\tag{%o52} 14520.86\]

\[\]\[\tag{%o53} 0.006886644454942751 false\]

(%i57)

n:1000;h:0;mPz:0;dh:H/n;

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

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

\[\]\[\tag{%o56} 0\]

\[\]\[\tag{%o57} \frac{1}{2000}\]

(%i58)

for i: 0 while h <= H do(db:fdb(i,dh,R),mPi:pi_x(h)·db·dh·B·fB(i,dh,R),mPz:mPz+mPi,h:h+dh);

\[\]\[\tag{%o58} \ensuremath{\mathrm{done}}\]

(%i60)

mPz,numer;mPz_test:5722.5;

\[\]\[\tag{%o59} 5702.602389493752\]

\[\]\[\tag{%o60} 5722.5\]

(%i61)

Rel_ERROR:(100/mPz_test)·(mPz_test−mPz,numer);

\[\]\[\tag{%o61} 0.01747487986020096 false\]

(%i62)

x_C:mPz/Pz,numer;

\[\]\[\tag{%o62} 0.39260476815282747\]

Angle (radians, degrees) Кут (радіани, градуси)

(%i64)

phi_rad:atan(Pz/Px),numer;phi_gr:phi_rad·(180/%pi),numer;

\[\]\[\tag{%o63} 1.1017969815811555\]

\[\]\[\tag{%o64} 63.12831692485351\]

Answer

Figure 4:D:\j_Iam_TEMP\_jh\_jh_ex1e1_all.jpg
Diagram

Diagram

(%i69)

Px,numer;Pz,numer;phi_gr;h_D;x_C;

\[\]\[\tag{%o65} 7359.952500000002\]

\[\]\[\tag{%o66} 14525.046184039022\]

\[\]\[\tag{%o67} 63.12831692485351\]

\[\]\[\tag{%o68} 0.7780184938353891\]

\[\]\[\tag{%o69} 0.39260476815282747\]

Sector drawing 300x300 px, scale (Малюнок сектору 300х300 пх , масштаб)

(%i73)

Px/100,numer;Pz/100,numer;h_D·S,numer;x_C·S,numer;

\[\]\[\tag{%o70} 73.59952500000003\]

\[\]\[\tag{%o71} 145.25046184039022\]

\[\]\[\tag{%o72} 233.40554815061674\]

\[\]\[\tag{%o73} 117.78143044584824\]

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The source of this Maxima session can be downloaded here.