Dual Arm Lever Systems Espresso Machine - The Lever Force and Work Effort Approximation

In some espresso machines, the Lever movement in Arc and pivot is driving a linear movement of the piston is a complex calculation. But the approximation and simplification on the calculation is present here: 

Assumptions:

The lever length is 26cm and the hand holding effective length is 20 cm from the pivot point.

The approximated and effective value of the hand position on the lever and the travel distance relative to the piston movement calculation:

Radius ($r$): $20\text{ cm}$

Arc: $100^\circ$ ($120^\circ$ to $20^\circ$)

Effective Hand Travel ($D_{eff}$): $28.73\text{ cm}$ (calculated via the $\sin\theta$ efficiency integral)

Piston Stroke: $6.0\text{ cm}$

Resulting NLR: $28.73 / 6.0 = \mathbf{4.79:1}$

Calculated Force for 9 Bar ($176.7\text{ kgf}$ load): $\frac{176.7}{4.79} \approx \mathbf{36.9 \text{ kgf}}$

Note on Methodology: Kinematic Approximations

The following performance metrics are time-averaged effective values. Because these machines utilize a handle attached to a rotating arc, the instantaneous mechanical advantage is non-linear and varies as a function of the handle's angular position ($\theta$).

To provide a practical benchmark, the values below are derived using:

1. Non-linear to Linear Simplification: The variable mechanical advantage is integrated across the effective operating arc ($100^\circ$ of travel).
2. Circular Efficiency Modeling: The force transfer is modeled using the average value of $\sin(\theta)$ over the displacement window, approximating the continuous variation of the input force vector relative to the vertical piston load.
3. Dynamic Force Approximation: These figures represent the mean manual effort required across the stroke to maintain a constant $9\text{ bars}$ of pressure, effectively smoothing out the non-linear "force-peak" typical of circular-drive manual lever linkages.