9.1: Force Output of an Extending Cylinder
- Page ID
- 116650
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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 an Extending Cylinder
When a pneumatic cylinder extends a load, the air pressure in the cap end of the cylinder exerts force on all internal surfaces. However, the only area where this pressure produces force to move the load is on the piston’s surface. The force output of the cylinder rod depends on how much pressure in the cap end there is and how large the piston area is.
How to Calculate the Force Output of an Extending Cylinder
To calculate the force output of an extending cylinder, you'll need to consider a few factors:
- Pressure: The force exerted by a cylinder is directly related to the pressure of the fluid (usually hydraulic oil or air) inside the cylinder. Pressure is typically measured in pounds per square inch (psi) or Newtons per square meter (Pascal).
- Area: The area of the piston or plunger inside the cylinder is crucial. Force equals pressure multiplied by the area over which it acts. The formula for force (F) is:
𝐹=𝑃×𝐴
Where:
F is the force (in pounds Force).
P is the pressure (in psi or pounds per square inch).
A is the Area (in square inches) of the piston or plunger.
The piston is circular, so its area can be calculated using the formula for a circle:
Area=πr^2 = pi x r^2 = 3.14 x r^2
"r" or radius is equal to half of the diameter of the circle. This is a difficult length to measure and most measure the diameter and divide by 2 to get the radius. Instead of having the extra step, let's use diameter within our equation and try to find a simpler equation to use from it.
Using diameter:
Area=(πd^2)/4 or (3.14 x d x d)/4
To simplify, let's divide 3.14 by 4.
This will result in:
Area=0.7854 x d^2 or 0.7854 x d x d
By using the simplified value of 0.7854 for the area of a circle, you can streamline the calculation process while still obtaining an accurate estimation of the force output of the extending cylinder.
Where "d" is the piston’s diameter, this will be the distance across the widest part of the piston.
This formula is essential in determining the theoretical force exerted by an extending cylinder.
Frictional losses: In real-world scenarios, frictional losses may reduce the effective force output of the cylinder. This is particularly relevant if the cylinder is lifting a load or moving against resistance.
An Example of Calculating the Extension Force of a Cylinder Given its Size and Pressure
Let's walk through an example of calculating the extension force of a cylinder given its size (diameter) and pressure.
Let's say we have a hydraulic cylinder with the following specifications:
Diameter of the piston: 4 inches
Pressure of the hydraulic fluid: 2500 psi
We'll use the simplified formula 𝐴=0.7854×𝑑^2 to calculate the area of the piston, and then use F=P×A to find the force.
Step 1: Calculate the area of the piston.
A=0.7854×(4 inches)^2 or A=0.7854×(4 inches x 4 inches)
A=0.7854×16 square inches
A≈12.5664 square inches or about 12.57 sq. in.
Step 2: Find the force exerted by the cylinder.
Given: Pressure (P) = 2500 psi
Area (A) ≈ 12.57 square inches
Solution
F=P×A
F=2500 psi×12.57 square inches
F≈31,425 pounds Force
So, the force exerted by the hydraulic cylinder, at a pressure of 2500 psi, with a piston diameter of 4 inches, is about 31, 425 pounds Force.
This calculation demonstrates how to determine the extension force of a cylinder based on its size (diameter) and the pressure of the hydraulic fluid.
Theoretical Calculation
This calculation provides a theoretical estimation of the extension force of the cylinder under ideal conditions. Here's why it's considered theoretical:
- Ideal Fluid Properties: The calculation assumes that the hydraulic fluid behaves as an ideal fluid with constant pressure throughout the system. In reality, factors like fluid viscosity, temperature variations, and flow dynamics can affect pressure distribution and thus impact the actual force output.
- Ideal Cylinder Conditions: The calculation assumes that the cylinder operates under ideal conditions, with no friction or mechanical losses. However, in real-world scenarios, friction between moving parts, seal friction, and other mechanical inefficiencies can reduce the effective force output of the cylinder.
- Uniform Force Distribution: The calculation assumes that the force is uniformly distributed across the entire surface area of the piston. In practice, uneven loading, piston misalignment, or other factors may result in non-uniform force distribution, affecting the actual force exerted.
- Neglects External Factors: The calculation does not account for external factors such as external loads, gravitational forces, or dynamic loads, which can influence the actual force required to extend or retract the cylinder.
Therefore, while the theoretical calculation provides a useful estimate of the extension force based on fundamental principles, actual operating conditions may vary, and practical considerations need to be taken into account for precise engineering design and operation. Experimental testing and validation are often necessary to verify the performance of hydraulic systems in real-world applications.