How the calculator works

1. What is calculated

For a lid or flap of given size and material, the calculator finds standard gas springs (force F1, stroke, length) and their two mounting points: A on the cabinet and B on the lid. It then shows, at every opening angle, whether a released lid closes, opens or stays where it is, how much force your hand needs, and how the lid moves when you let go.

2. Geometry: one coordinate for the whole motion

Every lid on the site moves with one degree of freedom: one number q (the opening angle) fixes the position of every part. For each part the model knows its position x(q) and how fast it moves per unit of q, the velocity coefficient dx/dq. A single hinge, a lift fitting with two parallel arms and a bi-fold front with a guide are all described this way.

The spring runs from A to B. Its length L(q) and its rate of change dL/dq follow from the positions of A and B. For a single hinge, |dL/dq| is exactly the perpendicular distance from the hinge to the spring line: the lever arm of the spring.

3. Statics by virtual work

Rather than writing force and moment balances for every part, the calculator uses the principle of virtual work: in equilibrium, the work of all forces over a small motion dq adds up to zero. This gives one equation per angle, whatever the mechanism:

Q(q) = n · F · κ · dL/dq − Σ mᵢ · g · dyᵢ/dq

Here n is the number of springs, F the spring force, κ = +1 for a push spring and −1 for a pull spring, mᵢ the masses (lid, extra weight), g = 9.80665 m/s² and yᵢ their heights. Q is the net turning effect on the lid. Q > 0 opens it, Q < 0 closes it.

Consequence: where the spring line passes through the hinge, dL/dq = 0 and the spring has no effect at all (dead centre). The calculator warns about this.

4. The gas spring

A gas spring holds nitrogen under pressure. Its nominal force F1 is the force when nearly fully extended at 20 °C. Pushed in, the gas is compressed and the force rises by the progression k = F2/F1, typically 1.2–1.6. The force along the stroke follows the isothermal gas law:

F(s) = F1 / (1 − c·s), c = (1 − 1/k) / S

s is the position in the stroke S, measured from the extended end. Seal friction makes the force about 10 % of F1 lower while extending than while being pushed in: at rest, a spring can deliver any force inside this band. The force scales with absolute temperature (about 3.4 % per 10 °C), and catalogue forces have a tolerance of ±5–10 %. The calculator checks the weakest and the strongest case.

5. Closes, holds or opens

At each angle, a released lid holds if the force needed to cancel its weight lies inside the spring's friction band, widened by the friction of the hinges. Above the band it closes, below it it opens. This is how the coloured zones in the drawing are made.

Hinge friction is modelled as a pin bearing: a torque μ·r·|R| with μ ≈ 0.25, pin radius r and R the load on the hinge, plus the set torque of friction hinges if chosen.

6. Choosing the springs

• Open by itself from an angle: the force is sized so the lid just balances at that angle, including hinge friction.
• Stay where I leave it: the force that makes the lid hold over the widest range of angles.
• Stay open when fully opened: the lid stays against its stop, with a margin.

The force is then rounded to standard catalogue steps and a standard stroke and length are chosen. Mounting point A is moved along the spring line until a standard length fits. "Suggest best positions" searches the allowed mounting zones for points that meet the goal with the lowest hand force and the least sensitivity to mounting errors; every layout is also checked with A and B moved by ±3 mm.

7. Special mechanisms

Lift fitting: two equal, parallel arms make a parallelogram, so the front only moves up without turning. Its weight acts like a pendulum of arm length r: Q = −m·g·r·sin q. A mechanical tension spring from a point straight below the arm pivot to a crank opposite the arm can cancel this at every height, when

n · k · a · b = m · g · r

(spring rate k, distances a and b from the pivot). The calculator sizes such springs by least squares and reports any remaining imbalance.

Bi-fold front: the upper panel hinges at the top; the lower panel hangs from it and its bottom edge runs in a guide, which fixes its angle ψ by h₁·sin q = h₂·sin ψ.

8. Motion after letting go

"Let go" integrates Lagrange's equation of motion with the same forces:

M(q)·q̈ + ½·M′(q)·q̇² = Q(q, q̇)

M is the inertia of the moving parts along q. The gas spring's oil damping slows the motion (stronger near the end of the stroke), friction opposes it, and the lid stops where static friction can hold it. The result is the time to open or close and the speed of the edge at the stop: gentle up to 0.3 m/s, firm up to 0.7 m/s, hard above.

9. Assumptions and limits

• Lids are rigid panels with their weight at the centre; the weight is estimated from the material unless you enter it.
• Springs and arms are massless; motion is planar (side view).
• Gas spring data are typical catalogue values; check the selected spring with the supplier.
• Results are indicative and do not replace a test of the first lid, especially where a falling lid could hurt someone.

10. How the results are checked

Every physics result is tested automatically against an independent result, on every change to the code:

• virtual work against moment balances about the hinge (200 random geometries);
• gravity against the derivative of potential energy; velocities against finite differences;
• the gas spring against Boyle's law; work integrals against energy changes;
• the four-bar against Freudenstein's equation; the lift fitting's balance against k·a·b = m·g·r;
• the dynamics against the period of a physical pendulum and energy conservation;
• mirror and translation invariance, and round trips through the sizing.

References

1. Norton, R. L. Design of Machinery. McGraw-Hill.
2. Uicker, J. J., Pennock, G. R., Shigley, J. E. Theory of Machines and Mechanisms. Oxford University Press.
3. Haug, E. J. Computer-Aided Kinematics and Dynamics of Mechanical Systems, Vol. I. Allyn and Bacon, 1989.
4. Nikravesh, P. E. Computer-Aided Analysis of Mechanical Systems. Prentice Hall, 1988.
5. Meriam, J. L., Kraige, L. G. Engineering Mechanics: Statics and Dynamics. Wiley.
6. Freudenstein, F. "Approximate synthesis of four-bar linkages." Transactions of the ASME 77, 1955.
7. Budynas, R. G., Nisbett, J. K. Shigley's Mechanical Engineering Design. McGraw-Hill.
8. Herder, J. L. Energy-free systems: statically balanced spring mechanisms. PhD thesis, TU Delft, 2001.
9. Hairer, E., Nørsett, S. P., Wanner, G. Solving Ordinary Differential Equations I. Springer.
10. Lawson, C. L., Hanson, R. J. Solving Least Squares Problems. Prentice-Hall, 1974.
11. BIPM. The International System of Units (SI), 9th edition, 2019.
12. USDA Forest Products Laboratory. Wood Handbook, FPL-GTR-190, 2010.
13. Technical handbooks of gas spring manufacturers (force notation F1–F4, progression, friction, temperature).