axial defflection studies of ring shaped force

Sudhir Kumar et al. / International Journal of Engineering Science and Technology (IJEST)

AXIAL DEFFLECTION STUDIES OF RING SHAPED FORCE TRANSDUCER: A REVIEW SUDHIR KUMAR Department of Mechanical Engineering, Krishna Engineering College, Ghaziabad, Uttar Pradesh, 201007, INDIA

WAKKAR ALI Department of Mechanical Engineering, Al Falah School of Engineering & Technology, Faridabad, Haryana, 121004, INDIA

ANIL KUMAR Mass Standard, National Physical Laboratory (CSIR) New Delhi, 110012, INDIA

HARISH KUMAR Force & Hardness Standard, National Physical Laboratory (CSIR), New Delhi, 110012, INDIA

Abstract: The ring shaped force transducers are widely used in practice and are available in varying capacities from few hundred newtons to mega newtons. The present paper discusses the deflection studies of the ring shaped force transducers under action of axial forces. Various methods leading to the measurement of deflection have been discussed and compared here. Keywords: force transducer, deflection, castigliano’s theorem, bending of curved bars Symbols b width of the cross section of ring (mm) t thickness of cross section of ring (mm) inner diameter of ring (mm) di outer diameter of ring (mm) do R mean radius (mm) F applied force (N) E young’s modulus of elasticity (N/m2) U strain energy (J) moment due to force at A (Nm) Mx M moment due to force at any position, say x(Nm)  angle of segment of ring (radian)  deflection of the ring (mm) factor that depends upon the cross section h2 1. INTRODUCTION The force transducers are commonly used in various engineering applications. They are used in weighing scales, verification of material testing machines, thrust measurement of rockets etc. Wide ranges of force transducers are available in the capacities ranging form few hundred newtons to mega newtons depending upon the suitability. But ring shaped force transducers are very common due to ease in manufacturing. The deflection is a measure of the force applied. A number of researchers have studied the ring shaped force transducers in past and various analytical expressions have been suggested on the basis of various theories. The paper attempts to summarize the findings of past research by making an intensive literature review. The deflection of the ring

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shaped force transducers is mainly analytically derived from the bending moment and strain energy. The deflection measured analytically has been derived and discussed here [1-5]. 2. ANALYTICAL STUDY The ring shaped force transducer has been analyzed under action of two axial forces (Figure 1) and has been modeled as a ring. Taking the advantage of symmetry, a quarter may be selected for the analytical study and the deflection of the quarter may be measured. The net deflection of the ring may be computed further. The free body diagram suggests that the moment M is statistically indeterminate, and there is no rotation, the strain energy U for this quadrant of the ring is due only to the bending moment M in the ring. According the castigliano’s theorem [1], F/2 F

Ri

θ

R0

Figure 1 Idealizing Circular Ring Shaped Force Proving Instrument

U 0 M 2 ( M ds ) U  2 EI

(1) (2)

Where, θ defines any section of the beam and strain energy.

( M Rd ) U  2 EI 2

Hence,

(3)

And, the radial deflection at B is calculated from the Castigliano’s theorem as,

i 

U ,.....(i  1, 2,...n) F

(4)

For any section, the bending moment may be find as,

FR (1  cos  ) 2 1 1 Mx  FR (  ) 2  FR 2 Mx  ( )(1  ) 2  3  FR    2   –      EI   4 M  Mx 

(5) (6) (7) (8)

Net deflection for a ring cross section is as follow,

 

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Using, the theory for bending of curved bars [6], Moment M, at any section may be,

M

FR 3 FR  1  sin  2 2  R  h  2

(10)

 FR  R 2 2 ( 2 )( )  sin   2 2  R h  

(11)

M Deflection of the ring,  /2

 2 0

 Rsin  FR 3 FR    Rd  1 sin  Rd      EI    R 2  h 2  2    FR 3   2  R2    2   EI  4    R  h 2      

(12)

Where, for rectangular cross section,

h2 

R3  2R  D  2 log e  R t  2R  D 

(13)

3. RESULTS & DISCUSSIONS Using the analytical expressions as discussed are compared for a number of ring shaped force transducers of varying dimensions and capacities by computing the deflection. Table 1 has been showing the deflections computed and it is found that the deflection computed by both approaches are in close agreement and may be used for deflection studies of the ring shaped force transducers [7].

Sl. No.

P

do

di

1 2 3 4 5 6 7 8 9 10

3000 5000 5000 5000 5000 10000 20000 20000 100000 100000

190.0 162.0 170.0 166.0 168.0 162.5 192.0 191.0 180.0 190.0

175.0 148.0 155.0 151.0 152.0 146.5 172.0 171.0 145.0 156.0

R

t

b t/R

91.25 7.5 25.0 77.50 7.0 24.5 81.25 7.5 24.0 79.25 7.5 25.0 80.00 8.0 23.0 77.25 8.0 23.5 91.00 10.0 45.0 90.50 10.0 50.0 81.25 17.5 48.5 86.50 17.0 50.0

A

I

E

h^2



Eq. (9) Eq. (13) 0.08 187.50 878.91 210000 6.193 1.842 1.834 0.09 171.50 700.29 210000 5.171 2.361 2.352 0.09 180.00 843.75 210000 5.884 2.258 2.250 0.09 187.50 878.91 210000 5.826 2.011 2.004 0.10 184.00 981.33 210000 6.496 1.853 1.847 0.10 188.00 1002.67 210000 6.418 3.266 3.257 0.11 450.00 3750.00 210000 9.842 2.855 2.848 0.11 500.00 4166.67 210000 9.826 2.527 2.521 0.22 848.75 21660.81 210000 26.894 1.759 1.776 0.20 850.00 20470.83 210000 25.576 2.246 2.262

Deviation % 0.43 0.38 0.36 0.35 0.31 0.29 0.24 0.23 0.99 0.71

Table 1 Deflection computation & Comparison

4. CONCLUSIONS The present study discusses the different approaches for deflection studies of ring shaped force transducers. The deflection has been measured on theories based on castigliano’s theorem and theory for bending of curved bars. The deflection measured is in close agreement for a number of ring shaped force transducers and hence justifies the both theories for deflection studies of ring shaped force transducers. The use of present study may be extended to the design of ring shaped force transducers by studying the effect of different variables over the deflection of the force transducer.

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REFERENCES [1] Josue Njock Libii, Design, Analysis and Testing of a Force Sensor for use in Teaching and Research, World Transaction on Engineering and Technology Education, 2006, Vol. 5, no. 1, pp- 175-178.

[2] Sedat Karabay, Analysis of Drill Dynamometer with Octagonal Ring type Transducers for Monitoring of Cutting Forces in Drilling and Allied Processes, Mater Des, 2007, Vol. 28, pp- 673-685.

[3] M J O’Dogherty, The Design of Octagonal Ring Dynamometers, J. agric. Engng Res., 1996, Vol. 63, pp- 9-18. [4] M A Rehman, S Rehman, A Computer Program for Designing Circular Proving Rings of Uniform Strength, Journal of the Institution of Engineers (India) Mechanical Engineering Division, 2007, Vol. 88, pp- 3-7.

[5] A Bray, the Role of Stress Analysis in the Design of Force Standard Transducers, Experimental Mechanics, 1981, Vol. 21, no. 1, pp1-20.

[6] R K Rajput, Strength of Materials, S Chand & Sons, New Delhi, India, 2008. [7] Kumar H, Sharma C, Gupta S, Deflection Studies of a Force Transducer, 10th International Conference on Research & Development in Mechanical Industry (RaDMI 2010), September 16-19, 2010, Donji Milanovac, Serbia, pp. 300-303.

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