Graphical analytical modeling of the kinematic scheme of a rock-piston pump

Two kinematic diagrams are presented, consisting of two combined toggle mechanisms and a piston pump. Kinematic calculations of the moving link parameters for both kinematic diagrams resulted in the determination of the displacement of the working and idle stroke lengths <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> of the piston as a function of the toggle mechanism swing angle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math> and the change in the toggle length and crank radius of the piston pump. The numerical value of the coefficient <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> of the average toggle mechanism slider velocity, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:mi> </mml:mi> </mml:math> 2, and the displacement of the piston stroke <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> were obtained: for a toggle-piston pump, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> <mml:mo>=</mml:mo> <mml:mi> </mml:mi> </mml:math> 1.25, and for a crank-toggle mechanism, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi> </mml:mi> </mml:math> 0.7 m. Various asymmetric phase angles were calculated for the working <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>φ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> </mml:math> and idle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>φ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> </mml:math> strokes of the slider during rotation of the toggle mechanism crank for both kinematic diagrams. The relationship between the center distance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and the position of the fixed support point <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>O</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> of the crank axis of rotation to the support point <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>O</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> of the rocker arm is obtained. The numerical values of the stroke displacement <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>D</mml:mi> </mml:mrow> </mml:msub> </mml:math> , linear velocity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>D</mml:mi> </mml:mrow> </mml:msub> </mml:math> , and acceleration <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>D</mml:mi> </mml:mrow> </mml:msub> </mml:math> of the pump piston for both kinematic diagrams of the rocker-piston pump mechanism are presented in tabular form by numerical values and in kinematic diagrams.