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The Levitation Mass Method
Measuring force by its definition.

The Levitation Mass Method (LMM) is a precision force measurement method proposed by the author. A force is applied to a mass M levitated with almost no friction by an air bearing, and the motion of the mass is measured with a laser interferometer. The force F is obtained from the very definition of force, the product of mass and acceleration, F = Ma. Its greatest strength is that time-varying forces can be measured without a force transducer.

Figure: Principle of the LMM. A force F is applied to a rigid object (mass M) levitated by an air bearing, and its motion is measured with an optical interferometer

Key points

1. Force from its definition. Force is the product of mass and acceleration; the inertial force of the levitated mass serves directly as the reference force.
2. Only frequency is measured. During measurement, only the motion-induced Doppler-shift frequency of the laser beam is recorded. Velocity, position, acceleration and force are calculated numerically afterwards, so all quantities are precisely synchronised.
3. Filling the gap in dynamic calibration. No method had been established for calibrating force transducers against time-varying forces. The LMM supplies impact, oscillation and step reference forces.
4. Born from space. The starting point was a ground test setup for a "space balance" that measures mass under weightless conditions.
5. The calibration setup becomes a material tester. Replace the transducer with a material and you have a tester that uses neither a force transducer nor a position sensor, for micro-forces, viscoelasticity, friction, and even the response of human fingers and arms.
6. Still evolving. Self-correction of dynamic error using the second derivative of sensor output (2025) and noise-robust frequency estimation (2026).

Where to start

• New readers: the principle (Chapter 1) and why it is needed (Chapter 2).
• Space and aerospace: the space balance (Chapter 3).
• Force measurement and calibration: dynamic calibration (Chapter 4) and dynamic error correction (Chapter 5).
• Materials, mechanics and biomechanics: air-bearing friction (Ch. 6), material testing (Ch. 7), biomechanics and impact (Ch. 8).
• Measurement techniques: frequency estimation and interferometer improvements (Chapter 9).
• Researchers: list of papers (Chapter 10).

Contents

• 1. Principle: Air bearing, optical interferometer, and F = Ma
• 2. Why It Is Needed: There was no proper way to measure time-varying forces
• 3. Space Balance: The LMM grew out of an instrument for measuring astronauts' mass under weightlessness
• 4. Dynamic Calibration: Supplying impact, oscillation and step forces based on the definition of force
• 5. Dynamic Error Correction: Estimating the transducer's own inertial force and self-correcting its dynamic error
• 6. Air-Bearing Friction: Measuring the small friction of a "frictionless" bearing
• 7. Material Testing and Micro-Force: Replace the transducer in the calibration setup with a material, and it becomes a material tester
• 8. Biomechanics and Impact: From human fingers, arms and palms to a ball dropping onto water
• 9. Measurement Techniques: Frequency estimation with a digitiser, a pendulum mechanism, and a dual-beat interferometer
• 10. Papers: Latest papers (with links) and the main papers on the LMM
• 11. Contact: Contact for research and collaboration

Next: 1. Principle →

The Levitation Mass Method: precision force measurement by the definition of force | Yusaku Fujii (Gunma University)