Role of finite element analysis in improving hysteresis error of force transducers

Abstract

Finite element analysis has been proved a vital factor for design studies of mechanical elements. The present paper discusses the role of finite element analysis in finding the locations of strain gauges so that the performance of the force transducers may be improved. Two ring-shaped force transducers of capacities 20 and 50 kN have been developed and the strain gauges have been placed suitably at the different locations according to the stress–strain distributions. Both force transducers have been metrologically characterized according to the calibration procedure based on standard ISO 376-2004 and IS 4169-1988 (re-affirmed 2003). The metrological results have ruled that both the force transducers have significant variations in the relative hysteresis deviations, while the differences between other factors like relative deviations because of repeatability error, reproducibility error, resolution error and zero error are not significant and thereby improve metrological properties. The results have been discussed and presented here.

Keywords

Force transducer hysteresis error uncertainty of measurement

Introduction

Various types of force transducers have been in use over the decades depending upon their suitability to the applications. Presently, strain gauged force transducers are widely used and available in the range from few Newtons to mega-Newtons. The strain gauged force transducers employ strain gauges fixed over the sensing element of the force transducer according to the Wheatstone bridge configuration. As the force is applied to the force transducer, the sensing element becomes stressed and the stress has been determined by the strain gauges. The Wheatstone bridge becomes unbalanced because of changes in resistance of each strain gauge and an electrical output is obtained. Using a suitable data acquisition system, the electrical output is converted to mV/V divisions and units of force (Dhawan et al., 2004).

The force transducers may be various shapes, but ring-shaped force transducers are very commonly used. Their simplicity in manufacturing has been encouraging for large-scale use. Despite this, design studies of ring-shaped force transducers have not been found in organized forms. Various expressions have been derived by various researchers for stress–strain and deflection under force in the past. For more rigorous validation of analytical methods, various researchers have used finite element analysis in the design and development of force transducers. Finite element analysis has been able to provide findings in the form of stress–strain and deflection plots of the force transducers. The stress–strain distribution may be used to determine suitable locations for implanting the strain gauges. Two force transducers of capacities 20 and 50 kN have been developed for the measurement of forces in tension mode. Both the force transducers have been strain gauged in two different configurations (Bray, 1981; Chen et al., 2007; Diddens et al., 1995; Karaby, 2007; Libii, 2004; O’Dogherty, 1996; Rehman and Rehman, 2007).

The metrological characterization of the force transducers has been done using a 50-kN dead weight force machine with the uncertainty of force realized at ±0.003% (k=2). The 50-kN dead weight force machine has been already discussed (Jain et al., 2005; Kumar et al., 2011a). The force transducers have been calibrated according to the calibration procedure based on the ISO 376-2004 and IS 4169-1988 (re-affirmed 2003) standards. The metrological characterization reveals that the relative hysteresis deviations have been significantly improved in the 50-kN force transducer (configuration 2), whereas the remaining factors like relative deviations are repeatability error, reproducibility error, zero error, interpolation error and resolution error (IS 4169-1988; ISO 376-2004).

Finite element analysis

The ring-shaped force transducers have earlier been studied using finite element analysis. The force transducers used in the present study are ABAQUS 6.7.2. A 20-kN force transducer of material EN 24, Poission ratio 0.3 with inner radius 86 mm, outer radius 96 mm and width 45 mm has been studied with the help of finite element analysis. A quarter of the force transducer has been selected because of symmetry of the force transducer modelled as a ring to either axis. A three-dimensional solid continuum eight-node element with reduced integration is assumed and analysis is of the linear type, as the geometric order of elements is taken to be linear. The mesh size is 5 and about 560 elements are there in the quarter of the ring selected. Elements are selected from the standard library of the software and the maximum deviation factor for curvature control is 0.1. The force (concentrated/point load) is applied in a compression mode using ABAQUS for studying the stress–strain pattern at the centre. One end of the quarter of the force transducer has been fixed, as no rotation is permitted and the other end free, from which the force derives, is applied (Kumar et al., 2011b). A suitable procedure for finite element analysis of the quarter of the ring has been adopted and the stress–strain patterns have been evaluated. The findings of the finite element analysis may be summarized in form of stress–strain patterns.

Figures 1 –4 show the stress–strain distribution of the quarter of the force proving instrument when it is exposed to an external compressive force. Maximum stress–strain occurs at the upper top point where the axial external force is applied. From the point of external force application along the periphery, the stress–strain tends to decrease. At an angle of about 40°, the stress is minimum and it continues to increase along the periphery for rest of the quarter of the force proving instrument (Kumar and Jain, 2009; Kumar et al., 2011b).

Figure 1

Stress distributions in quarter of force transducer.

Figure 2

Stress distributions for quarter of force transducer different forces.

Figure 3

Strain distributions in quarter of force transducer.

Figure 4

Strain distributions for quarter of force transducer at different forces.

Also, the maximum stress/strain of the quarter of the ring by finite element analysis is neglected because, where the force has been applied, the stress/strain is abruptly high because of the assumption of the application of forces as a point/concentrated force, which does not actually happen in practice (St. Venant’s principle; DuQuesnay, 2002).

Strain gauge implantation

The strain gauges have been implanted according to two different configurations (Figures 5 and 6). In configuration 1, the strain gauges are mounted at an angle of 90° from the axis on either side. In configuration 2, two strain gauges are mounted at 90° to the axis on either side of the vertical axis and the remaining two at 40° to the inner surface because of the low stress–strain there. A balanced Wheatstone bridge has been constructed by mounting the four strain gauges over the force transducer. Before the strain gauges are implanted, the machined elements are normalized. Strain gauges (foil type, 5-mm size) have been mounted by flattening the surface with a roughness of within a few microns. The strain gauges used have the resistance of 120±0.4 Ω and the nominal gauge factor is 2.1. As the force is applied, the Wheatstone bridge becomes unbalanced and the electrical output reflects the force applied.

Figure 5

Configuration 1.

Figure 6

Configuration 2.

Calibration procedure

Force transducers of 20 and 50 kN capacity have been studied for metrological performance. A 50-kN dead weight force machine has been used for calibration of the force transducers according to standard ISO 376-2004. The calibration of the force transducers has been developed according to ISO 376-2004 and IS 4169-1988 (re-affirmed 2003) standards.

Calibration according to ISO 376-2004

Digital indicator was switched on for 30 min to warm up and is stabilized for no load output before the start of calibration, no load output was noted (before taring) and the calibration signal was noted.

Before application of calibration forces, the force transducer was preloaded three times to its maximum capacity and kept at full load for 90 s.

Calibration of the force transducer has been done in tension mode.

Calibration was carried out by applying two series of calibration forces in ascending order from 10% to 100% in steps of 10% at the initial position, considered 0°.

Two series of calibration forces have been applied at rotation positions of 120° and 240°.

The force transducer was subjected to a full load once for 90 s each time before starting calibration to a new position.

Between loadings, readings corresponding to no load at the force transducer, after waiting at least 30 s for a return to zero, were noted.

Uncertainty of measurement of the force transducer involves relative deviations because of zero error, repeatability error, reproducibility error, resolution error, interpolation error and uncertainty of measurement of force related to the force machine.

Metrological characterization has been summarized (Figures 7 and 8).

Figure 7

Hysteresis error and uncertainty of measurement of 20-kN force transducer (ISO 376-2004).

Figure 8

Hysteresis error and uncertainty of measurement of 50-kN force transducer (ISO 376-2004).

Calibration according to IS 4169-1988 (re-affirmed 2003)

The digital indicator was switched on for 30 min to warm up and is stabilized for no load output before the start of calibration, no load output was noted (before taring) and the calibration signal was noted.

Before application of the calibration forces, the force transducer was preloaded three times to its maximum capacity and kept at full load for 90 s.

Calibration of the force transducer has been done in tension mode.

Calibration was carried out by applying one series of calibration forces in ascending order from 10% to 100% in steps of 10% at the initial position, considered 0°.

Two series of calibration forces have been applied at rotation positions 120° and 240°.

The force transducer was subjected to a full load once for 90 s each time before starting calibration to a new position.

Between loadings, readings corresponding to no load at the force transducer, after waiting at least 30 s for a return to zero, were noted.

Uncertainty of measurement of the force transducer involves relative deviations because of zero error, repeatability error, resolution error, interpolation error and uncertainty of measurement of force related to the force machine.

Metrological characterization of the force transducers has been summarized (Figures 9 and 10).

Figure 9

Hysteresis error and uncertainty of measurement of 20-kN force transducer (IS 4169-1988).

Figure 10

Hysteresis error and uncertainty of measurement of 50-kN force transducer (IS 4169-1988).

The relative combined standard uncertainty w _{c (tra)} and the relative expanded uncertainty W _(tra) for k=2 (k is the coverage factor and for k=2, the confidence level is 95%) will be calculated by the following Equations (1) and (2). Other terms represent the relative variances of relative deviations of repeatability error (rep), reproducibility error (rpr), interpolation error (int), hysteresis error (hys), zero error (zer) and resolution error (res).

w_{c (tra)} = {[w_{(rep)}^{2} + w_{(rpr)}^{2} + w_{(int)}^{2} + w_{(hys)}^{2} + w_{(zer)}^{2} + w_{(res)}^{2}]}^{1 / 2}

(1)

W_{(tra)} = k w_{c (tra)}

(2)

The relative uncertainty of calibration W shall be determined by the Equation (3), considering the best measurement capability (bmc) of the force standard machine.

W = {[W_{(tra)}^{2} + W_{(bmc)}^{2}]}^{1 / 2}

(3)

Results and discussion

The calibration results have been tabulated in Tables 1 –4. The uncertainty of measurement of the force transducers involves the relative deviations because of repeatability error, reproducibility error, resolution error, zero error, interpolation error, hysteresis error and uncertainty of force machine. The metrological results shows that the values of zero error, repeatability error, reproducibility error, resolution error and interpolation error are comparatively closer, and the difference in the uncertainty of both the force transducers is primarily related to the differences in the hysteresis error of both transducers (Figures 7 –10). The relative deviation because of hysteresis error is the deviation of the readings in ascending order to that of the descending order of the force transducer for given forces and is reported in terms of percentage. This has been an important parameter for metrological investigations of the force transducer and generally ring-shaped force transducers have poor hysteresis properties. Hence, different strain gauge arrangements have been selected to investigate the effect of locations of strain gauges. The difference in the hysteresis error is significant and may be attributed to the different strain gauge arrangement for both the force transducers. The stress is minimum at a 40° angle from the vertical axis on either side to the axis for quarter of the ring. This may reduce the sensitivity of the force transducer to some extent, but may improve the metrological performance by improving the relative deviation related to hysteresis error.

Table 1

The 20-kN force transducer as per ISO 376-2004

Force	Zero deviation	Relative repeatability	Relative reproducibility	Relative resolution	Relative interpolation	Relative reversibility	W _bmc	W(k=2)

kN	fo	b	c	fr	fc	v
2	0.0028	−0.0111	−0.0251	−0.0028	−0.0034	0.2744	0.003	0.16
4	0.0028	−0.0084	−0.0112	−0.0014	0.0024	0.2610	0.003	0.15
6	0.0028	−0.0056	−0.0093	−0.0009	0.0018	0.2322	0.003	0.13
8	0.0028	−0.0056	−0.0070	−0.0007	0.0002	0.1934	0.003	0.11
10	0.0028	−0.0051	−0.0062	−0.0006	0.0001	0.1564	0.003	0.09
12	0.0028	−0.0033	−0.0042	−0.0005	−0.0009	0.1188	0.003	0.07
14	0.0028	−0.0040	−0.0048	−0.0004	−0.0008	0.0799	0.003	0.05
16	0.0028	−0.0046	−0.0060	−0.0004	0.0008	0.0540	0.003	0.03
18	0.0028	−0.0047	−0.0063	−0.0003	0.0006	0.0256	0.003	0.02
20	0.0028	−0.0040	−0.0085	−0.0003	−0.0005	0.0000	0.003	0.01

Table 2

The 50-kN force transducer as per ISO 376-2004

Force	Zero deviation	Relative repeatability	Relative reproducibility	Relative resolution	Relative interpolation	Relative reversibility	W _bmc	W(k=2)

kN	fo	b	c	fr	fc	v
5	0.0098	−0.0046	−0.0278	−0.0046	0.0029	0.1204	0.003	0.07
10	0.0098	−0.0046	−0.0162	−0.0023	−0.0017	0.1853	0.003	0.11
15	0.0098	−0.0031	−0.0155	−0.0015	−0.0004	0.1901	0.003	0.11
20	0.0098	−0.0035	−0.0186	−0.0012	−0.0003	0.1670	0.003	0.10
25	0.0098	−0.0028	−0.0158	−0.0009	0.0000	0.1439	0.003	0.08
30	0.0098	−0.0031	−0.0193	−0.0008	0.0002	0.1087	0.003	0.06
35	0.0098	−0.0033	−0.0173	−0.0007	0.0010	0.0737	0.003	0.04
40	0.0098	−0.0023	−0.0186	−0.0006	0.0007	0.0424	0.003	0.03
45	0.0098	−0.0026	−0.0145	−0.0005	−0.0023	0.0137	0.003	0.01
50	0.0098	−0.0019	−0.0154	−0.0005	0.0010	0.0000	0.003	0.01

Table 3

The 20-kN force transducer as per IS 4169-1988 (reaffirmed 2003)

Force	Zero deviation	Relative repeatability	Relative resolution	Relative interpolation	Relative reversibility	W _bmc	W(k=2)

kN	fo	b	fr	fc	v
2	0.0066	−0.0195	−0.0028	−0.0016	0.3259	0.003	0.19
4	0.0066	−0.0167	−0.0014	−0.0007	0.2973	0.003	0.17
6	0.0066	−0.0140	−0.0009	0.0022	0.2647	0.003	0.15
8	0.0066	−0.0161	−0.0007	0.0009	0.2213	0.003	0.13
10	0.0066	−0.0185	−0.0006	−0.0016	0.1802	0.003	0.10
12	0.0066	−0.0211	−0.0005	−0.0004	0.1401	0.003	0.08
14	0.0066	−0.0241	−0.0004	0.0001	0.0974	0.003	0.06
16	0.0066	−0.0282	−0.0004	−0.0004	0.0635	0.003	0.04
18	0.0066	−0.0286	−0.0003	0.0013	0.0349	0.003	0.03
20	0.0066	−0.0269	−0.0003	−0.0006	0.0000	0.003	0.02

Table 4

The 50-kN force transducer as per IS 4169-1988 (reaffirmed 2003)

Force	Zero Deviation	Rel Repeatability	Rel Resolution	Rel Interpolation	Rel Reversibility	W _bmc	W(k=2)

kN	fo	b	fr	fc	v
5	0.0103	−0.0278	−0.0046	0.0183	0.1342	0.003	0.08
10	0.0103	−0.0162	−0.0023	−0.0198	0.1830	0.003	0.11
15	0.0103	−0.0185	−0.0015	0.0048	0.1715	0.003	0.10
20	0.0103	−0.0151	−0.0012	0.0013	0.1530	0.003	0.09
25	0.0103	−0.0130	−0.0009	0.0014	0.1281	0.003	0.07
30	0.0103	−0.0162	−0.0008	0.0064	0.0928	0.003	0.06
35	0.0103	−0.0139	−0.0007	0.0004	0.0584	0.003	0.04
40	0.0103	−0.0163	−0.0006	−0.0013	0.0256	0.003	0.02
45	0.0103	−0.0119	−0.0005	−0.0035	0.0015	0.003	0.01
50	0.0103	−0.0135	−0.0005	0.0005	0.0000	0.003	0.01

Conclusions

The present paper attempts to study the effect of different strain gauge arrangement for stress–strain measurement in a force transducer. The locations of the strain gauges have been found by stress–strain distributions obtained through finite element analysis as a tool. The two force transducers developed distinctly show the wide variation in the hysteresis error, which is the primary reason for higher uncertainty of the force transducer. Hence, if the hysteresis error may be significantly improved for the force transducer, the uncertainty of measurement may be improved too.

Footnotes

Acknowledgements

The authors express their sincere thanks to Prof. R. C. Budhani, Director, National Physical Laboratory, New Delhi, India; Dr A K. Bandyopadhyay, Head, Apex Level Standards and Industrial Metrology Group, National Physical Laboratory, New Delhi, India; and Prof. Nupur Prakash, Director, Indira Gandhi Institute of Technology, Delhi, India. The authors are also grateful to Mr Anil Kumar, Head, Mass Standard Group, National Physical Laboratory, New Delhi, India, for his valuable suggestions and support in the present study.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

References

Bray

(1981) Role of stress analysis in the design of force standard transducers. Experimental Mechanics 21(1): 1–20.

Chen

Peng

(2007) Finite element analysis of ring strain sensor. Sensor and Actuators A – Physical 139: 66–69.

Dhawan

Jain

Titus

SSK

Kumar

(2004) Design, development & characterization of 5 kN force gauge. MAPAN – Journal of Metrology Society of India 19(1–2): 61–67.

Diddens

Reynaerts

Van Brussel

(1995) Design of a ring shaped three axis micro force/torque sensor. Sensor and Actuators A – Physical 46–47: 225–232.

DuQuesnay

(2002) Mech 422 – stress and strain analysis. http://www.scribd.com/doc/36649223/notes-422 (accessed 31 August 2010).

IS 4169-1988 (re-affirmed 2003). Method for calibration of force proving instruments used for verification of uniaxial testing machines.

ISO 376-2004. Metallic materials – Calibration of force proving instruments used for verification of uniaxial testing machines.

Jain

Dhawan

Kumar

(2005) Realization of force scale up to 50 kN through dead weight force. MAPAN – Journal of Metrology Society of India 20(3): 249–257.

Karaby

(2007) Design criteria for electro-mechanical transducers and arrangements for measurement of strains due to metal cutting forces acting on the dynamometers. Materials & Design 28: 496–506.

10.

Kumar

Jain

(2009) Evaluation of axial sensitivity of circular ring shaped force proving ring using finite element analysis (FEM). 7th International Conference on Advances in Metrology (Admet 2009), National Physical Laboratory, New Delhi, 18–20 February.

11.

Kumar

Yadav

(2011a) Improved performance of 50 kN dead weight force machine using automation as a tool. Measurement Science Review 11(2): 67–70.

12.

Kumar

Sharma

Kumar

(2011b) Design and development of precision force transducers. Journal of Scientific & Industrial Research 70(7): 519–524.

13.

Libii

(2004) Design analysis and testing of a force sensor for use in teaching and research. World Transaction on Engineering and Technology Education 5(1): 175–178.

14.

O’Dogherty

(1996) The design octagonal ring dynamometers. J Agricultural Engineering Research 63: 9–18.

15.

Rehman

(2007) A computer program for designing circular proving rings of uniform strength. Journal of the Institution of Engineers (India) Mechanical Engineering Division 88: 3–7.