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The flow characteristic (hydraulic cylinder)

The flow characteristic (hydraulic cylinder) r a b β α r a b β T L S S a 2 a 1 γ δ α
The flow characteristic (hydraulic cylinder)

Values for calculation

$α$ $\mathrm{°}$
$β$ $\mathrm{°}$
$r$ $\mathrm{mm}$
$a$ $\mathrm{mm}$
$b$ $\mathrm{mm}$
$L_d$ $\mathrm{\%}$
$T_c$ $\mathrm{s}$
$T_d$ $\mathrm{s}$
$K_{Q-valve}[1]$
$K_{Q-valve}[2]$
$K_{Q-valve}[3]$
$K_{Q-valve}[4]$
$K_{Q-valve}[5]$
$K_{Q-valve}[6]$
$K_{Q-valve}[7]$
$K_{Q-valve}[8]$
$K_{Q-valve}[9]$
$K_{Q-valve}[10]$

Calculation

Stroke

$$S_{max}=\sqrt{\left(b-r\cdot\sin{\left(α+β\right)}\right)^2+\left(a-r\cdot\cos{\left(α+β\right)}\right)^2}-\sqrt{\left(b-r\cdot\sin{\left(α\right)}\right)^2+\left(a-r\cdot\cos{\left(α\right)}\right)^2}$$
$S_{max}=\sqrt{\left(b-r\cdot\sin{\left(α+β\right)}\right)^2+\left(a-r\cdot\cos{\left(α+β\right)}\right)^2}-\sqrt{\left(b-r\cdot\sin{\left(α\right)}\right)^2+\left(a-r\cdot\cos{\left(α\right)}\right)^2}$
Equations in LaTeX code
S_{max}=\sqrt{\left(b-r\cdot\sin{\left(α+β\right)}\right)^2+\left(a-r\cdot\cos{\left(α+β\right)}\right)^2}-\sqrt{\left(b-r\cdot\sin{\left(α\right)}\right)^2+\left(a-r\cdot\cos{\left(α\right)}\right)^2}

Distance between the axis of rotation of the valve and the axis of the hydraulic cylinder

$$L=\sqrt{a^2+b^2}$$
$L=\sqrt{a^2+b^2}$
Equations in LaTeX code
L=\sqrt{a^2+b^2}

Requirements

$$α\geq\tan^{-1}{\cfrac{b}{a}}\cdot\cfrac{180}{π}$$
$α\geq\tan^{-1}{\cfrac{b}{a}}\cdot\cfrac{180}{π}$
Equations in LaTeX code
α\geq\tan^{-1}{\cfrac{b}{a}}\cdot\cfrac{180}{π}
$$β\le\tan^{-1}{\cfrac{b}{a}}\cdot\cfrac{180}{π}+180-α$$
$β\le\tan^{-1}{\cfrac{b}{a}}\cdot\cfrac{180}{π}+180-α$
Equations in LaTeX code
β\le\tan^{-1}{\cfrac{b}{a}}\cdot\cfrac{180}{π}+180-α
$$0.8\cdot T_c\geq T_d$$
$0.8\cdot T_c\geq T_d$
Equations in LaTeX code
0.8\cdot T_c\geq T_d
Stroke from open position
The distance between the axis of the hydraulic cylinder and the axis of the eye of the hydraulic cylinder
Height $ T $
The angle between the axis of the hydraulic cylinder and the imaginary line between the axis of the closure and the pivot axis of the hydraulic cylinder
$s$ $S_S$ $T$ $γ$
$\mathrm{\%}$ $\mathrm{mm}$ $\mathrm{mm}$ $\mathrm{°}$
No data
Stroke from open position The distance between the axis of the hydraulic cylinder and the axis of the eye of the hydraulic cylinder Height $ T $ The angle between the axis of the hydraulic cylinder and the imaginary line between the axis of the closure and the pivot axis of the hydraulic cylinder
$s$ $S_S$ $T$ $γ$
$\mathrm{\%}$ $\mathrm{mm}$ $\mathrm{mm}$ $\mathrm{°}$
No data
Unrounded data
Stroke from open position;The distance between the axis of the hydraulic cylinder and the axis of the eye of the hydraulic cylinder;Height $ T $ ;The angle between the axis of the hydraulic cylinder and the imaginary line between the axis of the closure and the pivot axis of the hydraulic cylinder;
s;S_S;T;γ;
\%;mm;mm;°;

$S_S \mathrm{[mm]}$
No data
$s\mathrm{[\%]}$
The distance between the axis of the hydraulic cylinder and the axis of the eye of the hydraulic cylinder

Stroke from open position The distance between the axis of the hydraulic cylinder and the axis of the eye of the hydraulic cylinder
$s$ $S_S$
$\mathrm{\%}$ $\mathrm{mm}$
No data
Unrounded data
Stroke from open position; The distance between the axis of the hydraulic cylinder and the axis of the eye of the hydraulic cylinder;
s; S_S;
\%; mm;
$$S_S=S_{max}-\cfrac{S_{max}}{100}\cdot s+\sqrt{\left(b-r\cdot\sin{\left(α\right)}\right)^2+\left(a-r\cdot\cos{\left(α\right)}\right)^2}$$
$S_S=S_{max}-\cfrac{S_{max}}{100}\cdot s+\sqrt{\left(b-r\cdot\sin{\left(α\right)}\right)^2+\left(a-r\cdot\cos{\left(α\right)}\right)^2}$
Equations in LaTeX code
S_S=S_{max}-\cfrac{S_{max}}{100}\cdot s+\sqrt{\left(b-r\cdot\sin{\left(α\right)}\right)^2+\left(a-r\cdot\cos{\left(α\right)}\right)^2}

$T \mathrm{[mm]}$
No data
$s\mathrm{[\%]}$
Height $ T $

Stroke from open position Height $ T $
$s$ $T$
$\mathrm{\%}$ $\mathrm{mm}$
No data
Unrounded data
Stroke from open position; Height $ T $ ;
s; T;
\%; mm;
$$T=\sqrt{S_S^2-\left(\cfrac{L^2-r^2+S_S^2}{2\cdot L}\right)^2}$$
$T=\sqrt{S_S^2-\left(\cfrac{L^2-r^2+S_S^2}{2\cdot L}\right)^2}$
Equations in LaTeX code
T=\sqrt{S_S^2-\left(\cfrac{L^2-r^2+S_S^2}{2\cdot L}\right)^2}

$γ \mathrm{[°]}$
No data
$s\mathrm{[\%]}$
The angle between the axis of the hydraulic cylinder and the imaginary line between the axis of the closure and the pivot axis of the hydraulic cylinder

Stroke from open position The angle between the axis of the hydraulic cylinder and the imaginary line between the axis of the closure and the pivot axis of the hydraulic cylinder
$s$ $γ$
$\mathrm{\%}$ $\mathrm{°}$
No data
Unrounded data
Stroke from open position; The angle between the axis of the hydraulic cylinder and the imaginary line between the axis of the closure and the pivot axis of the hydraulic cylinder;
s; γ;
\%; °;
$$γ=\sin^{-1}{\cfrac{T}{S_S}}\cdot\cfrac{180}{π}$$
$γ=\sin^{-1}{\cfrac{T}{S_S}}\cdot\cfrac{180}{π}$
Equations in LaTeX code
γ=\sin^{-1}{\cfrac{T}{S_S}}\cdot\cfrac{180}{π}
Stroke from open position
Length $ a_2 $
Length $ a_1 $
The angle between the lever axis and the imaginary line between the valve axis and the pivot axis of the hydraulic cylinder
Angle rotation of the rocking motion
$s$ $a_2$ $a_1$ $δ$ $β_S$
$\mathrm{\%}$ $\mathrm{mm}$ $\mathrm{mm}$ $\mathrm{°}$ $\mathrm{°}$
No data
Stroke from open position Length $ a_2 $ Length $ a_1 $ The angle between the lever axis and the imaginary line between the valve axis and the pivot axis of the hydraulic cylinder Angle rotation of the rocking motion
$s$ $a_2$ $a_1$ $δ$ $β_S$
$\mathrm{\%}$ $\mathrm{mm}$ $\mathrm{mm}$ $\mathrm{°}$ $\mathrm{°}$
No data
Unrounded data
Stroke from open position;Length $ a_2 $ ;Length $ a_1 $ ;The angle between the lever axis and the imaginary line between the valve axis and the pivot axis of the hydraulic cylinder;Angle rotation of the rocking motion;
s;a_2;a_1;δ;β_S;
\%;mm;mm;°;°;

$a_2 \mathrm{[mm]}$
No data
$s\mathrm{[\%]}$
Length $ a_2 $

Stroke from open position Length $ a_2 $
$s$ $a_2$
$\mathrm{\%}$ $\mathrm{mm}$
No data
Unrounded data
Stroke from open position; Length $ a_2 $ ;
s; a_2;
\%; mm;
$$a_2=S_S\cdot\cos{γ}$$
$a_2=S_S\cdot\cos{γ}$
Equations in LaTeX code
a_2=S_S\cdot\cos{γ}

$a_1 \mathrm{[mm]}$
No data
$s\mathrm{[\%]}$
Length $ a_1 $

Stroke from open position Length $ a_1 $
$s$ $a_1$
$\mathrm{\%}$ $\mathrm{mm}$
No data
Unrounded data
Stroke from open position; Length $ a_1 $ ;
s; a_1;
\%; mm;
$$a_1=L-a_2$$
$a_1=L-a_2$
Equations in LaTeX code
a_1=L-a_2

$δ \mathrm{[°]}$
No data
$s\mathrm{[\%]}$
The angle between the lever axis and the imaginary line between the valve axis and the pivot axis of the hydraulic cylinder

Stroke from open position The angle between the lever axis and the imaginary line between the valve axis and the pivot axis of the hydraulic cylinder
$s$ $δ$
$\mathrm{\%}$ $\mathrm{°}$
No data
Unrounded data
Stroke from open position; The angle between the lever axis and the imaginary line between the valve axis and the pivot axis of the hydraulic cylinder;
s; δ;
\%; °;
$\text{if }\ a_1\le0$
$\text{if }\ a_1\le0$
Equations in LaTeX code
\text{if }\ a_1\le0
$$δ=90+\cos^{-1}{\cfrac{T}{r}}\cdot\cfrac{180}{π}$$
$δ=90+\cos^{-1}{\cfrac{T}{r}}\cdot\cfrac{180}{π}$
Equations in LaTeX code
δ=90+\cos^{-1}{\cfrac{T}{r}}\cdot\cfrac{180}{π}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$δ=90-\cos^{-1}{\cfrac{T}{r}}\cdot\cfrac{180}{π}$$
$δ=90-\cos^{-1}{\cfrac{T}{r}}\cdot\cfrac{180}{π}$
Equations in LaTeX code
δ=90-\cos^{-1}{\cfrac{T}{r}}\cdot\cfrac{180}{π}

$β_S \mathrm{[°]}$
No data
$s\mathrm{[\%]}$
Angle rotation of the rocking motion

Stroke from open position Angle rotation of the rocking motion
$s$ $β_S$
$\mathrm{\%}$ $\mathrm{°}$
No data
Unrounded data
Stroke from open position; Angle rotation of the rocking motion;
s; β_S;
\%; °;
$$β_S=δ_{[0]}-δ$$
$β_S=δ_{[0]}-δ$
Equations in LaTeX code
β_S=δ_{[0]}-δ
Stroke from open position
Flow coefficient $ K_{Q-valve} $
Flow coefficient $ K_{Q-hydraulic-cylinder} $
$s$ $K_{Q-valve}$ $K_{Q-hydraulic-cylinder}$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$
No data
Stroke from open position Flow coefficient $ K_{Q-valve} $ Flow coefficient $ K_{Q-hydraulic-cylinder} $
$s$ $K_{Q-valve}$ $K_{Q-hydraulic-cylinder}$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Stroke from open position;Flow coefficient $ K_{Q-valve} $ ;Flow coefficient $ K_{Q-hydraulic-cylinder} $ ;
s;K_{Q-valve};K_{Q-hydraulic-cylinder};
\%; ; ;

$K_{Q-valve} \mathrm{[-]}$
No data
$s\mathrm{[\%]}$
Flow coefficient $ K_{Q-valve} $

Stroke from open position Flow coefficient $ K_{Q-valve} $
$s$ $K_{Q-valve}$
$\mathrm{\%}$ $\mathrm{ }$
No data
Unrounded data
Stroke from open position; Flow coefficient $ K_{Q-valve} $ ;
s; K_{Q-valve};
\%; ;

$K_{Q-hydraulic-cylinder} \mathrm{[-]}$
No data
$s\mathrm{[\%]}$
Flow coefficient $ K_{Q-hydraulic-cylinder} $

Stroke from open position Flow coefficient $ K_{Q-hydraulic-cylinder} $
$s$ $K_{Q-hydraulic-cylinder}$
$\mathrm{\%}$ $\mathrm{ }$
No data
Unrounded data
Stroke from open position; Flow coefficient $ K_{Q-hydraulic-cylinder} $ ;
s; K_{Q-hydraulic-cylinder};
\%; ;
$\text{if }\ s=0$
$\text{if }\ s=0$
Equations in LaTeX code
\text{if }\ s=0
$$K_{Q-hydraulic-cylinder}=K_{Q-valve}$$
$K_{Q-hydraulic-cylinder}=K_{Q-valve}$
Equations in LaTeX code
K_{Q-hydraulic-cylinder}=K_{Q-valve}
$\text{else if }\ s=100$
$\text{else if }\ s=100$
Equations in LaTeX code
\text{else if }\ s=100
$$K_{Q-hydraulic-cylinder}=K_{Q-valve}$$
$K_{Q-hydraulic-cylinder}=K_{Q-valve}$
Equations in LaTeX code
K_{Q-hydraulic-cylinder}=K_{Q-valve}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$K_{Q-hydraulic-cylinder}=K_{Q-valve}-\cfrac{K_{Q-valve}-K_{Q-valve}[i+1]}{β/10}\cdot\left(β_S-\cfrac{β}{100}\cdot s\right)$$
$K_{Q-hydraulic-cylinder}=K_{Q-valve}-\cfrac{K_{Q-valve}-K_{Q-valve}[i+1]}{β/10}\cdot\left(β_S-\cfrac{β}{100}\cdot s\right)$
Equations in LaTeX code
K_{Q-hydraulic-cylinder}=K_{Q-valve}-\cfrac{K_{Q-valve}-K_{Q-valve}[i+1]}{β/10}\cdot\left(β_S-\cfrac{β}{100}\cdot s\right)
Time value
Percentage of hydraulic cylinder stroke at a given time
Flow coefficient
$T_s$ $S_T$ $K_Q$
$\mathrm{s}$ $\mathrm{\%}$ $\mathrm{ }$
No data
Time value Percentage of hydraulic cylinder stroke at a given time Flow coefficient
$T_s$ $S_T$ $K_Q$
$\mathrm{s}$ $\mathrm{\%}$ $\mathrm{ }$
No data
Unrounded data
Time value;Percentage of hydraulic cylinder stroke at a given time;Flow coefficient;
T_s;S_T;K_Q;
s;\%; ;

$S_T \mathrm{[\%]}$
No data
$T_s\mathrm{[s]}$
Percentage of hydraulic cylinder stroke at a given time

Time value Percentage of hydraulic cylinder stroke at a given time
$T_s$ $S_T$
$\mathrm{s}$ $\mathrm{\%}$
No data
Unrounded data
Time value; Percentage of hydraulic cylinder stroke at a given time;
T_s; S_T;
s; \%;
$\text{if }\ T_s<\left(T_c-T_d\right)$
$\text{if }\ T_s<\left(T_c-T_d\right)$
Equations in LaTeX code
\text{if }\ T_s<\left(T_c-T_d\right)
$$S_T=\cfrac{100-L_d}{T_c-T_d}\cdot T_s$$
$S_T=\cfrac{100-L_d}{T_c-T_d}\cdot T_s$
Equations in LaTeX code
S_T=\cfrac{100-L_d}{T_c-T_d}\cdot T_s
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$S_T=100-L_d+\cfrac{L_d}{T_d}\cdot\left(T_s-T_c+T_d\right)$$
$S_T=100-L_d+\cfrac{L_d}{T_d}\cdot\left(T_s-T_c+T_d\right)$
Equations in LaTeX code
S_T=100-L_d+\cfrac{L_d}{T_d}\cdot\left(T_s-T_c+T_d\right)

$K_Q \mathrm{[-]}$
No data
$T_s\mathrm{[s]}$
Flow coefficient

Time value Flow coefficient
$T_s$ $K_Q$
$\mathrm{s}$ $\mathrm{ }$
No data
Unrounded data
Time value; Flow coefficient;
T_s; K_Q;
s; ;
$\text{if }\ s=0$
$\text{if }\ s=0$
Equations in LaTeX code
\text{if }\ s=0
$$K_Q=K_{Q-hydraulic-cylinder}$$
$K_Q=K_{Q-hydraulic-cylinder}$
Equations in LaTeX code
K_Q=K_{Q-hydraulic-cylinder}
$\text{else if }\ s=100$
$\text{else if }\ s=100$
Equations in LaTeX code
\text{else if }\ s=100
$$K_Q=K_{Q-hydraulic-cylinder}$$
$K_Q=K_{Q-hydraulic-cylinder}$
Equations in LaTeX code
K_Q=K_{Q-hydraulic-cylinder}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$K_Q=K_{Q-hydraulic-cylinder}-\cfrac{K_{Q-hydraulic-cylinder}-K_{Q-hydraulic-cylinder}[i+1]}{10}\cdot\left(S_T-s\right)$$
$K_Q=K_{Q-hydraulic-cylinder}-\cfrac{K_{Q-hydraulic-cylinder}-K_{Q-hydraulic-cylinder}[i+1]}{10}\cdot\left(S_T-s\right)$
Equations in LaTeX code
K_Q=K_{Q-hydraulic-cylinder}-\cfrac{K_{Q-hydraulic-cylinder}-K_{Q-hydraulic-cylinder}[i+1]}{10}\cdot\left(S_T-s\right)
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