9.2: Force Output of a Retracting Cylinder
- Page ID
- 116651
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Calculating the Force Output of a Retracting Cylinder
When retracting, air pressure acts on the rod end of the cylinder. However, unlike in extension, the force is exerted only on the piston’s area not occupied by the rod. This net area is called the "annular area" or "donut area." The retraction force is always lower than the extension force at the same pressure, as the pressure acts over a smaller area.
The formula for retraction force modifies the extension force formula to account for the rod’s area. When accounting for the rod's area, the area is now called the Annular Area.
Force of Retraction = Pressure x Annular Area
To find the Annular Area, the area of the rod must be subtracted from the area of the piston.
Annular Area = Piston Area - Rod Area
Let's calculate the retraction force using this approach:
Let's say we have a hydraulic cylinder with the following specifications:
Diameter of the piston (Dp): 4 inches
Diameter of the rod (Dr): 2 inches
Pressure of the hydraulic fluid on the retracting side: 2500 psi
Step 1: Calculate the Annular Area.
Piston Area = 0.7854 x Dp x Dp = 0.7854 x 4 inches x 4 inches ≈ 12.5664 square inches or about 12.57 sq. in.
Rod Area = 0.7854 x Dr x Dr = 0.7854 x 2 inches x 2 inches ≈ 3.1416 square inches or about 3.14 sq. in.
Annular Area = 12.57 sq. in. - 3.14 sq. in. ≈ 9.43 square inches
Step 2: Find the force exerted during retraction. Given: Pressure (𝑃P) = 2500 psi (on the retracting side), Annular Area (𝐴annular) ≈ 9.4248 square inches
Force = Pressure x Annular Area
𝐹=2500 psi×9.43 square inches
𝐹≈23,575 pounds
So, the force exerted by the hydraulic cylinder during retraction, with a pressure of 2500 psi on the retracting side and a piston diameter of 4 inches and rod diameter of 2 inches, is approximately 23,575 pounds, considering the annular area.
When compared to the extension force (31,425 pounds Force), the retraction force is lower (23,575 pounds Force) because the pressure acts over a smaller area due to the presence of the rod.