Experimental Mechanics

, Volume 30, Issue 1, pp 40–48 | Cite as

Optimal design of an uncoupled six degree of freedom dynamometer

  • T. P. Quinn
  • C. D. MoteJr.


A design analysis of a class of six degree of freedom strain-gage dynamometers which are virtually uncoupled in each force and moment component is presented. The method of design is detailed and an optimal design algorithm is implemented. The dynamometer is made of six or more T-sections with thin webs and flanges called shear panel elements (SPE). Complete stress and buckling analyses are carried out for the SPE, and experiments confirm the predictions of the analyses. The optimal design method is illustrated with several case studies. A dynamometer has been built and used in laboratory and field experiments.


Mechanical Engineer Fluid Dynamics Field Experiment Optimal Design Flange 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of Symbols

b, c

dimensions of the cantilever plate in the ξ and ζ directions

d, fw,ff,tw,tf

geometry of the SPE; see Fig. 1


flexural rigidity

E, ν

Young's modulus and Poisson's ratio

\(\bar e\)

shear strain in (ξ, ζ) coordinates at the center of the cantilever plate (b/2,c/2)

\(\bar e_w \)

shear strain at the center of the web\(\bar e_w = \bar e\) withL=P, b=f w ,c=d, andt=t w


coefficients in the approximation ofu andv


external force components applied to the cage


external force applied to the cage in the Y′ direction (e.g.,Fy′=F x if Y′ is aligned with X)

Kx′, Ky′, Kz

stiffness of a SPE for forces applied to the ends of the flange in the X′, Y′, Z′, directions, respectively


torsional stiffness of the web


bending stiffness of one arm of the flange


minimum allowable design stiffness of the SPE in the Y′ direction


torsional stiffness of the SPE about the X′ axis


distance from the moment axis of the reference coordinates to the flange of the SPE; see Fig. 1


force applied at the end of the cantilever plate; see Fig. 5

\(\bar L\)

the forceL at the onset of buckling

\(\bar L_f \)

the buckling force in the flange;\(\bar L_f = \bar L\) withb=f f /2,c=d, andt=t f

\(\bar L_w \)

the buckling force in the web;\(\bar L_w = \bar L\) withb=f w ,c=d, andt=t w


external moment components applied to the cage


external moment applied to cage in the Z′ direction (e.g.,M z ′=M y if Z′ is aligned with Y)


coefficients in the approximation ofw


total external force applied to a SPE


minimum allowable stiffness ratios


the set of all possibleX


SPE's of the dynamometer in Fig. 3


yield strength


thickness of the cantilever plate


shear traction applied at the end of the cantilever plate; see Fig. 5


SPE in a six degree of freedom dynamometer

u, v

displacements of the cantilever plate along ξ and ζ

\(\bar v\)


\(\bar v_f \)

\(\bar v\) withL=1/2, b=f f /2,c=d, andt=t f

\(\bar v_w \)

\(\bar v\) withL=1, b=f w ,c=d, andt=t w


transverse displacement of the cantilever plate


coordinates at the center of the dynamometer


coordinates at the base of an SPE


[l,d,f w ,f f ,t w ,t f ]


stresses in the cantilever plate

\(\sigma '_\zeta ,\sigma '_\xi ,\sigma '_{\xi \zeta } \)

nondimensional stresses in the cantilever plate

\(\bar \sigma \)

maximum von Mises stress in the cantilever plate;\(\begin{gathered} \bar \sigma = 0 \leqslant \mathop \xi \limits^{\max } \leqslant b(\sigma _{^\xi }^2 + \sigma _\zeta ^2 - \sigma _\xi \sigma _\zeta + 3\sigma _{\xi \zeta }^2 )^{1/2} \hfill \\ 0 \leqslant \zeta \leqslant c \hfill \\ \end{gathered} \)

\(\bar \sigma _f \)

maximum von Mises stress in the flange;\(\bar \sigma _f = \sigma \) withL=P/2, b=f f/2 ,c=d andt=t f

\(\bar \sigma _w \)

maximum von Mises stress in the web\(\bar \sigma _w = \bar \sigma \) withL=P, b=f w ,c=d andt=t w


potential energy functional for the cantilever plate


local coordinate system for the cantilever plate


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beckwith, T.G., Buck, N.L. andMaragoni, R.D., Mechanical Measurements, 427–428, Addison Wesley, Reading (1982).Google Scholar
  2. 2.
    Ono, K. andHatamura, Y., “A New Design for 6-Component Force/Torque Sensors”,Mechanical Problems in Measuring Force and Mass, ed. H. Wieringa, Martinus Nijhoff, Dordrecht, 39–48 (1986).Google Scholar
  3. 3.
    Chaplin, J., Lueders, M. andZhao, Y., “Three-Point Hitch Dynamometer Design and Calibration”,Appl. Eng. in Agriculture,3 (1),10–13 (1987).Google Scholar
  4. 4.
    Watson, P.C. and Drake, S.H., “Method and Apparatus for Six Degree of Freedom Force Sensing”, U.S. Patent No. 4,094,192 (June 13, 1978).Google Scholar
  5. 5.
    Molland, A.F., “A Five-Component Strain Gauge Wind Tunnel Dynamometer”,Strain,14 (1),7–13 (1978).Google Scholar
  6. 6.
    Goodwin, R.J., “An Extended Octagonal Ring Transducer for Use in Tillage Studies”,J. of Agric. Eng. Res.,20,347–352 (1975).CrossRefGoogle Scholar
  7. 7.
    Millward, A. andRossiter, J., “The Design of a Multi-Purpose Multi-Component Strain Gauge Dynamometer”,Strain,19,27–30 (1983).Google Scholar
  8. 8.
    Regan, K. andReuber, M., “3-D Force-Sensing Robot Hand”,Proc. of 1985 ASME Int. Comp. in Eng. Conf. and Exhibition, ed. R. Raghavan andS.M. Rohde 1,ASME,New York,165–167 (Aug. 1985).Google Scholar
  9. 9.
    Girard, D., “Development of Multicomponent Force Transfer Standards by ONERA for French BNM”,Mechanical Problems in Measuring Force and Mass, ed. H. Wieringa, Martinus Nijhoff, Dordrecht, 1–12 (1986).Google Scholar
  10. 10.
    Reuber, M., Kornegay, J., Melehy, W. and Panjabi, C., “Symbolic Processing as an Aid to the Solution of Inverse Design Problems: Applications”, Proc. 2nd Int. Conf. on Inverse Design Concepts and Optimization in Eng. Sci., ed. G.S. Dulikravich (1987).Google Scholar
  11. 11.
    Hou, J.W. andTwu, S.L., “Optimum Design of Internal Strain Gage Balances: An Example of Three-Dimensional Shape Optimization”,J. Mechanisms, Transmissions, and Automation in Design,109,257–262 (June 1987).Google Scholar
  12. 12.
    Lin, C.-T. andBeadle, C.W., “The Optimal Design of Force Transducers Which Are Cross-Axis Sensitive”,Sensors and Controls for Automated Manufacturing and Robotics, eded. K.A. Stetson andL.M. Sweet ASME, New York, 179–191 (Dec. 1984).Google Scholar
  13. 13.
    Sinden, F.W. andBoie, R.A., “A Planar Capacitive Force Sensor with Six Degrees of Freedom”,Proc. 1986 IEEE Int. Conf. Robotics and Automation, IEEE Computer Society Press, Washington,3,1806–1814 (April 1986).Google Scholar
  14. 14.
    Levin, A. and Mote, C.D., Jr., “An Uncoupled Six Degree of Freedom Dynamometer”, Univ. of Calif., Dept. of Mech. Eng. Rep. (1983).Google Scholar
  15. 15.
    Kuo, C.Y., Louie, J.K. andMote, C.D. Jr., “Field Measurements in Snow Skiing Injury Research”,J. Biomech.,16 (8),609–624 (1983).CrossRefPubMedGoogle Scholar
  16. 16.
    Reddy, J.N., Applied Functional Analysis and Variational Methods in Engineering, McGraw-Hill Book Co., New York, 258–285 (1986).Google Scholar
  17. 17.
    Timoshenko, S., “Mathematical Determination of the Modulus of Elasticity”,Mech. Eng.,45 (4),259–260 (April 1923).Google Scholar
  18. 18.
    Timoshenko, S.P. andGoodier, J.N., Theory of Elasticity, McGraw-Hill Book Co., New York, 309–313 (1970).Google Scholar
  19. 19.
    Timoshenko, S.P. andGere, J.M., Theory of Elastic Stability, McGraw-Hill Book Co., New York, 348–351 (1961).Google Scholar
  20. 20.
    Gill, P.E., Murray, W. andWright, M.H., Practical Optimization, Academic Press, Inc., London, 225–233 (1981).Google Scholar
  21. 21.
    Dally, J.W. andRiley, W.F., Experimental Stress Analysis, McGraw-Hill Book Co., New York, 251–253 (1978).Google Scholar
  22. 22.
    Woinowsky-Krieger, S., “Uber die Beulsicherheit von Reckteckplatten mit querverschieblichen Rändern”,Ingenieur-Archiv,19,200–207 (1951).CrossRefGoogle Scholar

Copyright information

© Society for Experimental Mechanics, Inc. 1990

Authors and Affiliations

  • T. P. Quinn
    • 1
  • C. D. MoteJr.
    • 1
  1. 1.Department of Mechanical EngineeringUniversity of CaliforniaBerkeley

Personalised recommendations