Review of Dynamic-Force Correction Method for a Force Transducer with the Levitation Mass Method

In this paper, the measurement and correction metho d of the dynamic error of a force transducer are reviewe d. The dynamic force, an impact load in this paper, is app lied to an S-shaped strain-gauge force transducer is measured with using the Levitation Mass Method (LMM). In the LMM, the force acting on the force transducer is measure d as a reaction force acting on a moving part of an aerost atic linear bearing which is collided with the force tra nsducer. The force acting on the moving part is measured as an inertial force of the moving part. Finally, the ine rtial force is calculated by the mass and the acceleration of the moving part. As a result of comparing the forces measured by the transducer and by the LMM, it is shown that the dyn amic error can be estimated by the second time derivativ e of the output of the transducer. Therefore, the dynamic er ror of the force transducer can be corrected by using the outp ut data from the transducer itself, without using the LMM.


Introduction
Force transducers are widely used in industrial and research areas.For static force calibration for the transducers, some methods that use the static standard force are established.However, there are no standard methods for dynamic force calibration.This shows that it is very difficult to estimate the dynamic error of force transducers and correct the error.Some researchers have proposed methods for evaluating the dynamic error of force transducers.Kumme proposed a method in which an oscillation force is applied to a force transducer. (1,2)In this method, a mass is attached to the sensing part of the transducer, and the transducer with the mass is vertically attached to a shaker.The oscillation force applied to the transducer is the inertial force of the mass. In this method, only the Doppler frequency shift of the laser light reflected on the mass is accurately measured, using an optical interferometer.All the other quantities such as velocity, acceleration, displacement, and inertial force are then calculated from the frequency obtained.7) In this paper, we review the method for measuring dynamic force applied to a force transducer and the method for correcting the difference between the applied force and the force measured by the transducer.

Experimental Setup
Figure 1 shows a schematic diagram of the experimental setup for measuring the dynamic response of the force transducers.The transducer being tested is fixed on a base.An aerostatic linear bearing is used to achieve linear motion with sufficiently small friction acting on the mass, i.e., the moving part of the bearing. (8)The moving part with a corner cube prism (CC) and an extension rod is given its initial velocity by hand, and the moving part then collides with the sensing point of the force transducer.
A rubber block of sufficiently small mass is attached to the sensing point to adjust the steepness of the impact.The velocity of the mass v 1 and the velocity of the sensing point of the transducer v 2 are measured using two optical interferometers, called interferometer1 and interferometer2, respectively.The force acting on the sensing point is calculated as reaction force acting on the moving part by the equation of motion, F = ma, where m is the mass and a is the acceleration of the moving part.
The mass of the moving part, including the CC and the extension rod, M1, is 2.653 kg.The mass of the contact point of the transducer, including the CC and their base plate, M2, is 0.082 kg.
A Zeeman-type two-frequency He-Ne laser was used as the light source of the interferometers.The fundamental frequency difference of the laser, f rest , was measured by a photo (PD), PD0, as the beat frequency of the two frequencies.Two beat frequencies, f beat1 and f beat2 , between the fundamental frequency and frequencies modulated by Doppler effects at the moving part and at the sensing point of the transducer were measured by PD1 and PD2.
The force measured by the transducer, F trans , was calculated using the output signal of the transducer V trans that was stored in a dynamic strain recorder (DSR; model: DC-204R, Tokyo Sokki Kenkyujo) with its static calibration result.F trans is compared with the force measured as the inertial force of the mass, F mass .
F mass is calculated by the following formulas.
) ( The acceleration of contact point a2 is calculated by a similar way, ) ( To perform simultaneous measurements, three frequency counters and the DSR were triggered by a single signal that originated from a light switch composed of a laser diode (LD) and a PD.F trans shows the electric response of the force transducer under the impact load.The difference between F trans and F mass is derived mainly from the difference between the static and the dynamic characteristics of the transducer.The root mean square (RMS) value of the difference between F trans and F mass during the collision, 0 ms < t < 14 ms, was approximately 6.2 N.

Results and Discussion
Figure 3 shows the relationship between the acceleration of the sensing part of the transducer a 2 and F diff .The plot shows a linear relationship between a 2 and F diff .The solid line in Fig. 3 shows a regression line with the following equation: As shown in Fig. 3, the difference between the force measured by using the LMM and the force measured by the transducer is linear to the acceleration of the sensing part of the transducer because the dynamic error was caused by the inertial force of the transducer itself.Therefore, if we can get the acceleration of the transducer, we can correct the dynamic error.According to the principle of the strain gauge transducer, the transducer measures displacement of the sensing point based on the Hooke's law.Accordingly, the second time derivative of the output of the transducer should have linear relationship with the acceleration, farther more, with the dynamic error, as following equation: where C is a parameter for dynamic error estimation.The parameter C is obtained from the slope of the regression line determined by a least-square method, which resulted in C = -2.49× 10 -7 s 2 in this experiment.
Figure 4 shows the difference between F trans and F mass and the inertial forces of the transducer that is estimated using the parameter C, F reg .The measurement error, F diff , is almost the same as the result of the regression analysis, F reg .From this result, the corrected force can be calculated using  The RMS values of F diff and F res appear to be almost square functions of the maximum force F trans, max .In Fig. 5, the solid and dashed curves represent the regression curves of the RMS values of F diff and F res , respectively.The RMS values of the error between F diff and its regression value and that of the error between F res and its regression value are 1.34 × 10 -1 N and 3.58 × 10 -1 N, respectively.These results show the potential for estimating the RMS values of the F diff and F res from the F trans , max .

Figure 2 -Fig. 1 .
Figure 2-5 shows the results of an impact test.Fig. 2 shows the force measured by the force transducer F trans , the force acting on the moving part F mass , and their difference F diff , calculated by mass trans diff F F F − = .(7)

ForceFig. 2 .-Fig. 3 .
Fig. 2. Force measured with the transducer F trans and the force measured with the interferometer F mass , and the difference between the two forces, F diff .