Abstract
Mass measurement using relay feedback of velocity and restoring force compensation is investigated for determining the mass of an object under weightless conditions. In the measurement system, the velocity of the object is fed back through a relay with hysteresis and the force acting on the object is switched from a positive value to a negative value when the velocity reaches a positive threshold and vice versa. As a result, a limit cycle is induced in the measurement system and the mass is estimated based on the period of the limit cycle. In addition, restoring force compensation with a spring is introduced to avoid the drift of the trajectory. This compensation makes the static equilibrium state unique. However, the trajectory still drifts slightly. It causes some error in measurement when a simple formula of estimating mass is applied. To eliminate such an error, a new formula is derived to estimate the mass independently of the position of the trajectory that is determined by the switching positions in the relay actions. When the switching positions deflect from the origin at which the spring is in the natural length, the trajectory is not at the center and becomes asymmetric. It is analytically shown that the period of the limit cycle is minimum when the switching positions are at the origin. It indicates that mass is overestimated with the simple estimation formula when the trajectory is not at the center. The validity of the modified formula and the analytical results are confirmed experimentally.
Highlights
The determination of the mass of an object is one of the oldest measurement techniques.1,2 At the very early stages of trade, it was necessary to find out the mass of an object
It is analytically shown that the period of the limit cycle is minimum when the switching positions are at the origin
It is to be mentioned that the solution trajectory of the system described by equations [9] and [10] can be derived for any given value of x0. It indicates that the actual trajectory solution is not unique; if a switching position varies from the previous one due to some disturbance, a new trajectory will start from the varied position
Summary
The determination of the mass of an object is one of the oldest measurement techniques. At the very early stages of trade, it was necessary to find out the mass of an object. Equation [8] indicates that the mass of the object is estimated from the period of the limit cycle T. In the original measurement system described in the previous sections, the velocity of the object is solely fed back and there is no restoring force. It is to be mentioned that the solution trajectory of the system described by equations [9] and [10] can be derived for any given value of x0 It indicates that the actual trajectory solution is not unique; if a switching position varies from the previous one due to some disturbance, a new trajectory will start from the varied position.
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