Visualize Pascal's principle force amplification in real time. Adjust piston diameters, input force, and stroke to compute output force, pressure, flow rate, and pump power instantly.
The core principle is Pascal's Law: pressure applied to a confined fluid is transmitted undiminished. This creates the force multiplication between two connected pistons.
$$P = \frac{F_1}{A_1}= \frac{F_2}{A_2}$$Where $P$ is the fluid pressure (Pa), $F_1$ and $F_2$ are the forces on the input and output pistons (N), and $A_1$ and $A_2$ are their respective areas (m²). Since area depends on diameter $d$, the force ratio is $F_2 = F_1 \cdot (d_2/d_1)^2$.
Because the fluid is incompressible, the volume displaced by the input piston must equal the volume received by the output piston. This governs the trade-off between force and distance.
$$A_1 \cdot s_1 = A_2 \cdot s_2 \quad \text{or}\quad s_2 = s_1 \cdot \frac{A_1}{A_2}$$Here, $s_1$ and $s_2$ are the stroke lengths (m) of the input and output pistons. The flow rate $Q$ (m³/s) is the volume displaced per unit time: $Q = A_1 \cdot v_1$, where $v_1$ is the input piston speed.
Automotive Lifts: Car jacks are the most common example. A mechanic applies a small force over a long stroke on the small piston, and the large piston lifts the vehicle with massive force over a short distance. The simulator's force ratio directly models this.
Hydraulic Presses: Used in manufacturing to mold metal or crush materials. They use the same principle on an industrial scale, with precisely controlled pressure to apply tons of force for shaping or assembly.
Construction Equipment: Excavators, bulldozers, and crane arms use hydraulic cylinders to generate powerful, controlled linear motion. The operator's gentle lever movement controls high-pressure fluid to move massive loads.
Aircraft Landing Gear & Brakes: Hydraulic systems are crucial in aviation for retracting landing gear and applying brake pressure. They provide reliable, powerful actuation in a compact system, where safety depends on precise force transmission.
There are a few key points you should be especially mindful of when starting to use this simulator. First is the point that "force amplification is not infinite". While increasing the output piston size raises the output force, real hydraulic cylinders always have an upper limit called the "rated pressure". For example, if the pressure rating of the pump or hoses is 21 MPa (approx. 210 atmospheres), the maximum output force you can generate there is determined by $F_{out} = P_{max} \times A_{out}$, no matter how large you make the output piston area. Indiscriminately increasing the piston size can sometimes just be a waste of cost and space.
Next is the simulator's assumption of "incompressibility". Actual hydraulic oil is slightly compressible, and hoses also expand. In machine tools requiring ultra-precise positioning, if you don't account for this "spring constant of the oil", you can run into trouble like the machine not stopping at the intended position under load. Also, while the calculation might make it seem like force is transmitted instantly, in reality, delays occur due to oil viscosity and pipe resistance. When designing a control system for emergency stops, it's dangerous not to consider this "transmission delay".
Finally, a misconception about efficiency. The "required pump power" calculated by the simulator is a theoretical value. In reality, losses occur from various factors: mechanical losses in the pump itself, motor efficiency, pressure losses in the piping, leaks at the cylinder seals, etc. When selecting actual components, the practical wisdom is to apply a safety factor of at least 1.2 to 1.5 times this theoretical value and choose motors and pumps with some margin.
The concepts you master through hydraulic jack calculations are the gateway to "fluid mechanics". The relationship between pressure and flow rate, $P$ and $Q$, that appears here can be directly applied to flow in pipes and pump selection. For instance, the principle for calculating the required pump head (equivalent to pressure) and discharge rate (flow rate) when transferring liquid from a tank in a chemical plant is the same.
Furthermore, the trade-off between force and stroke ($A_{in}\cdot s_{in}= A_{out}\cdot s_{out}$) is essentially the same as the "principle of work" or "lever principle" found in "mechanical dynamics" or "mechanism theory". Hydraulic systems can be considered a type of "force conversion device," similar to levers, gears, and link mechanisms. The thought process for determining the relationship between required thrust and moving speed is very similar even when designing "electric actuators" that use ball screws and servo motors instead of hydraulic actuators.
Taking it a step further connects to the field of "control engineering". To move a hydraulic jack at a constant speed or stop it precisely at a specific position, you need flow control valves or pressure control valves. Changing the "input stroke" in this simulator corresponds to adjusting the valve opening to control the flow rate. The smooth, unwavering movement of a construction machine's arm is thanks to this hydraulic control technology.
Once you understand the basics of Pascal's principle, try learning "the continuity equation" and "Bernoulli's theorem". While the hydraulic jack was about "static pressure," these two become your tools when considering the "dynamic pressure" and flow velocity of oil moving through pipes. For example, if you narrow a section of hose, the flow rate remains the same but the flow velocity increases, creating a section where pressure drops (Venturi effect). Understanding this helps you see why surge pressure occurs and how to prevent cavitation.
Mathematically, you used the circle area formula $\pi d^2/4$ for the area calculation, right? This is the first step to understanding "scaling laws". Doubling the diameter makes the area four times larger (2 squared) and the volume eight times larger (2 cubed). This relationship is essential knowledge for predicting how strength and required power change when scaling up machinery. It forms the basis for estimating full-scale performance from experiments on small-scale models.
As a concrete next step, we recommend using this simulator to learn how to read "hydraulic circuit diagrams (JIS symbols)" and study how pumps, valves, and cylinders combine to form a single system. For example, understanding the difference between meter-in and meter-out circuits, or the role of an accumulator, can help you leap from simple jacks to the design thinking behind complex hydraulically-driven industrial machinery. To start, it's also interesting to watch a video of a familiar hydraulic excavator and try to guess which cylinder corresponds to which movement as you observe.