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A New Six-axis Load Cell. Part I: Design

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Abstract

A new high precision six-axis load cell is presented in two companion papers. The first paper (this one) is focused on the presentation of the load cell, the conceptual design, the modelling and the embodiment design. The second paper refers to the error analysis, the construction and the experimental assessment of the performances. The new load cell is able to measure the three components of both a force and a moment acting on the load cell itself. The sensing structural element of the six-axis load cell is basically a three spoke structure. Strain gauges are conveniently located on highly stressed areas. The sensing structural element is constrained to the frame of the load cell by means of special joints conceived to avoid friction. The mechanical behaviour of the load cell is described by means of analytical equations that allow a quick preliminary design focused on the given technical specifications. A finite element model has been used to asses the mechanical behaviour of the load cell.

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Abbreviations

F :

vector of forces and moments acting at the load cell centre

Ε b :

vector of the strains \( {\varepsilon_{i\left( {1 = 1, \ldots, 6} \right)}} \)

C b :

load cell calibration matrix

T :

vector of reaction forces at the three joints N i , \( {T_{i\left( {i = 1,2,3} \right)}} \)

ΔV :

vector of the output voltages \( \Delta {V_{i\left( {i = 1, \cdots, 6} \right)}} \) at the Wheatstone bridges

K :

gauge factor

V :

excitation voltage at the Wheatstone bridges

L :

spoke length

M i :

calibration matrix

M tst :

calibration matrix (statically determined structure)

M e :

experimental calibration matrix

M thp :

calibration matrix (quasi-statically determined structure)

k a :

joint axial stiffness

k rx /k ry :

joint radial stiffnesses

k t :

joint torsional stiffness

k bt /k b :

joint bending stiffnesses

k x /k y /k z :

load cell interface stiffnesses

S :

vector of the internal forces and moments acting at the spoke tips

δ i :

displacements and rotations at a spoke tip

R t (α) :

rotation matrix

A(ω) :

Inertance—transfer function acceleration/force

h :

distance between a strain gauge and the spoke tip (see Fig. 4)

h z :

distance between a strain gauge and the three spoke structure centre (see Fig. 4)

ν :

vector of the bending moments acting on the six strain gauge bridges

ν tst :

vector of the bending moments—statically determined structure

ν thp :

vector of the bending moments—quasi-statically determined structure

χ :

load cell sensitivity matrix

χ tst :

statically determined structure sensitivity matrix

χ trj :

rigid joints model sensitivity matrix

χ trs :

rigid spokes model sensitivity matrix

δν :

error on the strain (bending) measurements

δχ :

error on the sensitivity matrix

 :

Note: bold symbol refers to a vector or a matrix

References

  1. Fischer U, Zimmermann C, Jörg W (2002) SG-Balance 192-6I calibration report. RUAG Center Aerodynamics

  2. http://www.windtunnel.soton.ac.uk/balance/balance_RJMitchell.html University of Southampton, 2010. Accessed 4/2010

  3. http://www.mmtool.com/balances.html Modern machine & tool CO., INC, Strain Gage Wind Tunnel Balances, 2010. Accessed 4/2010

  4. http://www.ruag.com RUAG Aerospace Defence Technology, Strain Gauge Balances for Wind Tunnel Measurements, 2005. Accessed 4/2010

  5. Carignan F et al (1985) Force measuring platform and load cell using strain gages to measure shear forces, Advanced Mechanical Technology, Inc., Patent US4493220

  6. Spletzer B, Marron L (2000) Information package for the simplified six-axis load cell. Sandia National Laboratories, January, e-mail: blsplet @sandia.gov

  7. BermeN (2002) Multi-component force and moment measuring platform and load transducer. Bertec Corporation, Patent US6354155

  8. Blakeborough A, Clément D, Williams MS, Woodward N (2002) Novel load cell for measuring axial force, shear force, and bending movement in large-scale structural experiments. Exp Mech 42(1):115–122

    Article  Google Scholar 

  9. Engeler P et al (1995) Multicomponent force and moment measuring arrangement. K.K. Holding AG, US5402684

  10. Meyer R et al (1998) Six axis load cell. MTS SYSTEMS CORPORATION, European Patent EP0632884B1

  11. Gobbi M, Mastinu G, Giorgetta F (2005) Sensors for measuring forces and moments with application to ground vehicle design and engineering. Proceedings ASME IMECE, paper n. IMECE2005-81143

  12. http://www.amti.biz/ Advanced Mechanical Technology, Inc, 2010. Accessed 4/2010

  13. http://www.jr3.com/ Multi-axis load cell technologies, 2010. Accessed 4/2010

  14. Yee AG, Akeel HA (1996) Six axis force sensor employing multiple shear strain gages. Fanuc USA Corporation (Elkgrove Village, IL), United States Patent 5490427

  15. Meyer RA, Olson DJ (1994) Six axis load cell. MTS Systems Corporation (Eden Prairie, MN), United States Patent 5315882

  16. Liu H, Willberg B, Meusel P (2005) Force moment sensor. Deutsches Zentrum fur Luft-und Raumfahrt e.V. (Koln, DE), United States Patent 6871552

  17. Berme N (2002) Multi-component force and moment measuring platform and load transducer. Bertec Corporation (Worthington, OH), United States Patent 6354155

  18. Meyer RA, Olson DJ (2005) Multi-axis load cell. MTS Systems Corporation (Eden Prairie, MN), United States Patent 6845675

  19. Meyer, RA, Kempainen AJ, Olson DJ (2004) Multi-axis load cell body. MTS Systems Corporation (Eden Priarie, MN), United States Patent 6769312

  20. Lerat B (1989) Transducer for bending and twisting moments. Commissariat A L’Energie Atomique, Patent US4879913

  21. Wensink H, de Boer MJ, Wiegerink RJ, Zwijze AF, Elwenspoek MC (1998) First micromachined silicon load cell for loads up to 1000 kg. Proc. SPIE ‘98, vol. 3514 Micromachined Devices and Components IV

  22. Zwijze AF, Wiegerink RJ, Lammerink TSJ, Elwenspoek MC (1998) Low creep and hysteresis load cell based on a force to liquid pressure transformation. Proceedings of the Dutch National Sensor Conference, Enschede, pp 287–292, March 2

  23. Byun Y et al (2002) Parallel type six-axes force moment measuring apparatus. Samsung Electronics Co., Ltd., Patent US6349604

  24. http://www.ati-ia.com/ ATI Industrial Automation, (Series OMEGA and GAMMA), 5/2005

  25. Mastinu G, Gobbi M (2006) Giunto elastico a cerniera sferica traslante e sensore di forze e momenti perfezionato con tale giunto. Italian Patent, ME.05.025.A, 22 May 2006

  26. Mastinu G, Gobbi M (2003) Dispositivo e metodo per la misura di forze e momenti. Italian Patent MI2003A 001500, 22 July 2003

    Google Scholar 

  27. Mastinu G, Gobbi M (2005) Device and method for measuring forces and moments. Patent WO2005015146, 17 February 2005

  28. Mastinu G, Gobbi M (2007) Elastic joint with translating spherical hinge and force and moment sensor improved by means of the said joint. International Application number PCT/IB2007/001335, 16 May 2007

  29. Gobbi M, Mastinu G, Pennati M (2007) Indoor testing of road vehicle suspensions. Meccanica 43(2):173–184. doi:10.1007/s11012-008-9119-5, 28 February 2008

    Article  Google Scholar 

  30. Cook H, Carignan FJ, White BF (1999) Multi-axis wheel transducer with angular position detector. Advanced Mechanical Technology, Inc. Patent, US5886350

  31. http://www.sendev.com/ Sensor Development Inc. (Model 77016 and 6 axis WFS), 5/2005. Accessed 4/2010

  32. http://www.michsci.com/ Michigan Scientific Corporation (model LW 12.8), 5/2005. Accessed 4/2010

  33. http://www.mts.com/ MTS Systems Corporation (SWIFT Wheel Force Transducer), 5/2005. Accessed 4/2010

  34. http://www.kistler.com Kistler (RoaDyn Wheel Force Sensor), 5/2005. Accessed 4/2010

  35. http://www.igel-ingenieure.de/ (IGel WFT Wheel Sensor), 5/2005. Accessed 4/2010

  36. Giorgetta F, Gobbi M, Mastinu G (2007) On the testing of vibration performances of road vehicle suspensions. Exp Mech 47:485–495. doi:10.1007/s11340-006-9022-8

    Article  Google Scholar 

  37. Gobbi M, Mastinu G (2004) Wheels with integrated sensors for measuring tyre forces and moments. Proc. of the AVEC 04 Symposium, Arnhem, August

  38. Gobbi M, Previati G, Guarneri P, Mastinu G (2010) A new six-axis load cell. Part II: error analysis, construction and experimental assessment of performances. Exp Mech, xxx, DOI xxx

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Authors and Affiliations

  1. Department of Mechanical Engineering, Politecnico di Milano, Via La Masa, 1, 20156, Milan, Italy

    G. Mastinu, M. Gobbi & G. Previati

Authors
  1. G. Mastinu
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  2. M. Gobbi
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  3. G. Previati
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Corresponding author

Correspondence to M. Gobbi.

Appendix

Appendix

Analytical expressions of the 18 internal forces and moments acting at the 3 spoke tips (rigid disk considered in the model):

$$ \begin{gathered} {S_{{{\underline x }_1}}} \hfill \\\left( \begin{gathered} - 1/3\left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right)\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)Ty\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right) + 2/3Fx\left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right)\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right) \hfill \\\cdot \left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right) + 1/3{\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)^3}Ty - 1/3\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)Ty\left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right) \hfill \\\end{gathered} \right) \hfill \\{\left( { - {{\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)}^2} + \left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + l} \right)}^2}}}{{kbt}}} \right)\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right) + \left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)\left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right)} \right)^{ - 1}}{\left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right)^{ - 1}} \hfill \\\end{gathered} $$
$$ \begin{gathered} {S_{{{\underline y }_1}}} \hfill \\\left( {1/3\left( { - \frac{{{L^2}}}{{EJx}} - 2\frac{{RL}}{{EJx}} - 2\frac{{R + L}}{{kb}}} \right)Tx\left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right) - 1/3Fy{{\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)}^2} + 1/3\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)Fy\left( {\frac{L}{{JpG}} + K{t^{ - 1}}} \right) + 1/3\left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right)\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)Fy} \right) \hfill \\{\left( {\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)\left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right) + \left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right) \cdot \left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right) - {{\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)}^2}} \right)^{ - 1}} \hfill \\\end{gathered} $$
$$ \begin{gathered} {S_{{{\underline z }_1}}} \hfill \\2/3FZ\left( { - {{\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)}^2} + \left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right)\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)} \right){\left( { - {{\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)}^2} + \left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right)\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right) + \left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right) + \left( {\frac{L}{{EA}} + K{a^{ - 1}}} \right)} \right)^{ - 1}} \hfill \\\end{gathered} $$
$$ \begin{gathered} {M_{{{\underline x }_1}}} \hfill \\\left( \begin{gathered} 2/3Tx\left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right)\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + Kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)\left( {\frac{L}{{EJx}} + K{b^{ - 1}}} \right) - 1/3\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)Fy\left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right) \cdot \hfill \\\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right) - 1/3\left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right)\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)Fy\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right) + 1/3Fy{\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)^3} \hfill \\\end{gathered} \right) \hfill \\\cdot {\left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right)^{ - 1}}{\left( {\left( {1/3\frac{{{L^3}}}{{EJX}} + \frac{{RL\left( {R + L} \right)}}{{EJX}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)\left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right) + \left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right)\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right) - {{\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)}^2}} \right)^{ - 1}} \hfill \\\end{gathered} $$
$$ \begin{gathered} {M_{{{\underline y }_1}}} \hfill \\\left( { - 1/3Ty{{\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)}^2} + 1/3\left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}}\frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right)\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)Ty + 1/3\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)Ty\left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right) - 1/3\left( { - \frac{{{L^2}}}{{EJy}} - 2\frac{{RL}}{{EJy}} + 2\frac{{ - R - L}}{{kbt}}} \right)Fx\left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right)} \right) \hfill \\{\left( { - {{\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)}^2} + \left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right)\left( {\frac{L}{{Ejy}} + kb{t^{ - 1}}} \right) + \left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)\left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right)} \right)^{ - 1}} \hfill \\\end{gathered} $$
$$ \begin{gathered} {M_{{{\underline z }_1}}} \hfill \\2/3Tz\left( {\left( {\frac{1}{{EJx}} + k{b^{ - 1}}} \right)\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right) - {{\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)}^2}} \right){\left( {\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)\left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right) + \left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right)\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right) - {{\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)}^2}} \right)^{ - 1}} \hfill \\\end{gathered} $$
$$ \begin{gathered} {S_{{{\underline x }_2}}} \hfill \\\left( \begin{gathered} - 1/3\left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right)\sqrt {3} \left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)Fz\left( {\frac{L}{{EA}} + ka{^{ - 1}}} \right) - 1/3Fx\left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{R + {L^2}}}{{kbt}}} \right)\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)\left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right) + 1/3{\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)^3} \hfill \\Ty - 1/3\left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right)\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)Ty\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right) - 1/3\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right) \cdot \left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)Ty\left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right) \hfill \\\end{gathered} \right) \hfill \\\left( { - {{\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)}^2} + \left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right) + \left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right) + \left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)\left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right)} \right) \cdot {\left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right)^{ - 1}} \hfill \\\end{gathered} $$
$$ \begin{aligned} & S_{{y_{{ - 2}} }} \\ & \begin{array}{*{20}l} {{{\left( {\begin{array}{*{20}c} {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3{\left( {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3\frac{{L^{3} }}{{EJr}} + \frac{{RL{\left( {R + L} \right)}}}{{EJx}} + kry^{{ - 1}} + \frac{{{\left( {R + L} \right)}^{2} }}{{kb}}} \right)}Fy{\left( {\frac{L}{{JpG}} + kt^{{ - 1}} } \right)} + 1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3{\left( {\frac{L}{{EJr}} + kb^{{ - 1}} } \right)}{\left( {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3\frac{{L^{3} }}{{EJx}} + \frac{{RL{\left( {R + L} \right)}}}{{EJx}} + kry^{{ - 1}} + \frac{{{\left( {R + L} \right)}^{2} }}{{kb}}} \right)}} \\ {Fy - 1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3Fy{\left( {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2\frac{{L^{2} }}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)}^{2} + 1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3{\left( {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2\frac{{L^{2} }}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)}Tx{\left( {\frac{L}{{JpG}} + kt^{{ - 1}} } \right)}} \\ { + 1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3{\left( {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2\frac{{L^{2} }}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)}{\sqrt 3 }{\left( {\frac{L}{{JpG}} + kt^{{ - 1}} } \right)}Tz} \\ \end{array} } \right)}} \hfill} \\ {{{\left( {{\left( {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3\frac{{L^{3} }}{{EJx}} + \frac{{RL{\left( {R + L} \right)}}}{{EJx}} + kry^{{ - 1}} + \frac{{{\left( {R + L} \right)}^{2} }}{{kb}}} \right)}{\left( {\frac{L}{{JpG}} + kt^{{ - 1}} } \right)} + {\left( {\frac{L}{{EJx}} + kb^{{ - 1}} } \right)}{\left( {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3\frac{{L^{3} }}{{EJx}} + \frac{{RL{\left( {R + L} \right)}}}{{EJx}} + kry^{{ - 1}} + \frac{{{\left( {R + L} \right)}^{2} }}{{kb}}} \right)} - {\left( {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2\frac{{L^{2} }}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)}^{2} } \right)}} \hfill} \\ \end{array} \\ \end{aligned} $$
$$ \begin{aligned} & S_{{z_{{ - 2}} }} \\ & {\left( {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3{\left( { - 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2\frac{{L^{2} }}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)}^{2} Fz - 1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3{\left( {\frac{L}{{EJy}} + kbt^{{ - 1}} } \right)}Fz{\left( {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3\frac{{L^{3} }}{{EJy}} + \frac{{RL{\left( {R + L} \right)}}}{{EJy}} + krz^{{ - 1}} + \frac{{{\left( {R - L} \right)}^{2} }}{{kbt}}} \right)} - 1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3{\sqrt 3 }{\left( { - 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2\frac{{L^{2} }}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)}^{2} Fx{\left( {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3\frac{{L^{3} }}{{EJy}} + \frac{{RL{\left( {R + L} \right)}}}{{EJy}} + krz^{{ - 1}} - \frac{{{\left( {R + L} \right)}^{2} }}{{kbt}}} \right)}} \right)}{\left( { - {\left( { - 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2\frac{{L^{2} }}{{EJy}} + \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)}^{2} + {\left( {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3\frac{{L^{3} }}{{EJy}} + \frac{{RL{\left( {R + L} \right)}}}{{EJy}} + krx^{{ - 1}} + \frac{{{\left( {R + L} \right)}^{2} }}{{kbt}}} \right)}{\left( {\frac{L}{{EJy}} + kbt^{{ - 1}} } \right)} + {\left( {\frac{L}{{EJy}} + kbt^{{ - 1}} } \right)}{\left( {\frac{L}{{EA}} + ka^{{ - 1}} } \right)}} \right)}^{{ - 1}} \\ \end{aligned} $$
$$ \begin{gathered} {M_{{x_2}}} \hfill \\\left( \begin{gathered} 1/3\left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right)\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)Fy\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right) - 1/3\sqrt {3} \left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right)Tz\left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right)\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right) - 1/3Tx\left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right)\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)\left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right) \hfill \\- 1/3\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)Fy\left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right) \cdot \left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right) + 1/3Fy{\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)^3} \hfill \\\end{gathered} \right) \hfill \\{\left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right)^{ - 1}}{\left( {\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)\left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right) + \left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right) \cdot \left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right) - {{\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)}^2}} \right)^{ - 1}} \hfill \\\end{gathered} $$
$$ \begin{gathered} {M_{{{\underline y }_2}}} \hfill \\\left( { - 1/3Ty{{\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)}^2} + 1/3\left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right)\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)Ty + 1/3\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)Ty\left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right) + 1/3\sqrt {3} \left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + - \frac{{ - R - L}}{{kbt}}} \right)\left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right)Fz + 1/3\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)Fx\left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right)} \right) \hfill \\{\left( { - {{\left( {-1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)}^2} + \left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right) \cdot \left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right) + \left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)\left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right)} \right)^{ - 1}} \hfill \\\end{gathered} $$
$$ \begin{gathered} {M_{{{\underline z }_2}}} \hfill \\\left( { - 1/3\sqrt {3} {{\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)}^2}Tx + 1/3{{\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)}^2}Tz + 1/3\left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right)\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)\sqrt {3} Tx - 1/3\left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right) \cdot \left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)Tz} \right) \hfill \\{\left( {\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)\left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right) + \left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right)\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right) - {{\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)}^2}} \right)^{ - 1}} \hfill \\\end{gathered} $$
$$ \begin{gathered} {S_{{{\underline x }_3}}} \hfill \\\left( \begin{gathered} 1/3\left( { - 1/3\frac{{{L^3}}}{{EJy}} - \frac{{RL\left( {R + L} \right)}}{{EJy}} - kr{x^{ - 1}} - \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right)\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)Ty\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + - \frac{{ - R - L}}{{kbt}}} \right) - 1/3Fx\left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right)\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right) \hfill \\\cdot \left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right) + 1/3\left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right)\sqrt {3} \left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)Fz\left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right) + 1/3{\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)^3}Ty - 1/3\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)Ty\left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right) \hfill \\\end{gathered} \right) \hfill \\\left( { - {{\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)}^2} + \left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right)\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right) + \left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right) \cdot \left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right)} \right)^{-1}{\left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right)^{ - 1}} \hfill \\\end{gathered} $$
$$ \begin{gathered} {S_{{{\underline y }_3}}} \hfill \\\left( { - 1/3\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)\sqrt {3} \left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right)Tz + 1/3\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)Fy\left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right) + 1/3\left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right)\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)Fy - 1/3Fy{{\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)}^2} + 1/3\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)Tx\left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right)} \right) \hfill \\{\left( {\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right) \cdot \left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right) + \left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right)\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right) - {{\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)}^2}} \right)^{ - 1}} \hfill \\\end{gathered} $$
$$ \begin{gathered} {S_{{{\underline z }_3}}} \hfill \\\left( { - 1/3\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)Fz\left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right) - 1/3\sqrt {3} \left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)Fx\left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right) + 1/3{{\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)}^2}Fz + 1/3\sqrt {3} {{\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)}^2}Fx} \right) \hfill \\{\left( { - {{\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)}^2} + \left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + Kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right)\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right) + \left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)\left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right)} \right)^{ - 1}} \hfill \\\end{gathered} $$
$$ \begin{gathered} {M_{{x_3}}} \hfill \\\left( \begin{gathered} - 1/3Tx\left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right)\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)\left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right) + 1/3\sqrt {3} \left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right)Tz\left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right)\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right) - 1/3\left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right)\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)Fy\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right) \hfill \\- 1/3\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)Fy\left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right) \cdot \left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{RL}}{{kb}}} \right) + 1/3Fy{\left( {1/2\frac{{{L^2}}}{{EJx}} + \frac{{FL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)^3} \hfill \\\end{gathered} \right) \hfill \\\left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right)^{-1}{\left( {\left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right)\left( {\frac{L}{{JpG}} + k{t^{ - 1}}} \right) + \left( {\frac{L}{{EJx}} + k{b^{ - 1}}} \right) \cdot \left( {1/3\frac{{{L^3}}}{{EJx}} + \frac{{RL\left( {R + L} \right)}}{{EJx}} + kr{y^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kb}}} \right) - {{\left( {1/2\frac{{{L^2}}}{{EJr}} + \frac{{RL}}{{EJr}} + \frac{{R + L}}{{kb}}} \right)}^2}} \right)^{ - 1}} \hfill \\\end{gathered} $$
$$ \begin{gathered} {M_{{{\underline y }_3}}} \hfill \\\left( {1/3\left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{RL\left( {R + L} \right)}}{{EJy}} + kr{x^{ - 1}} + \frac{{{{\left( {R + L} \right)}^2}}}{{kbt}}} \right)\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)Ty + 1/3\left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)Ty\left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right) - 1/3Ty{{\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)}^2} - 1/3\sqrt {3} \left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + - \frac{{R - L}}{{kbt}}} \right)\left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right)Fz + 1/3\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)Fx\left( {\frac{L}{{EA}} + k{a^{ - 1}}} \right)} \right) \hfill \\{\left( { - {{\left( { - 1/2\frac{{{L^2}}}{{EJy}} - \frac{{RL}}{{EJy}} + \frac{{ - R - L}}{{kbt}}} \right)}^2} + \left( {1/3\frac{{{L^3}}}{{EJy}} + \frac{{R{{\left( {R + L} \right)}}}}{{EJy}}} + krx^{-1} + \frac{{\left( {R + L} \right){^2}}}{kbt} \right) \cdot \left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right) + \left( {\frac{L}{{EJy}} + kb{t^{ - 1}}} \right)\left( {\frac{L}{{EA}} + K{a^{ - 1}}} \right)} \right)^{ - 1}} \hfill \\\end{gathered} $$
$$ \begin{aligned} & M_{{ - z_{3} }} \\ & {\left( { - 1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3{\left( {\frac{L}{{EJx}} + kb^{{ - 1}} } \right)}{\left( {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3\frac{{L^{3} }}{{EJx}} + \frac{{RL{\left( {R + L} \right)}}}{{EJx}} + kry^{{ - 1}} + \frac{{{\left( {R + L} \right)}^{2} }}{{kb}}} \right)}Tz + 1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3{\left( {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2\frac{{L^{2} }}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)}^{2} Tz + 1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3{\left( {\frac{L}{{EJx}} + kb^{{ - 1}} } \right)}{\left( {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3\frac{{L^{3} }}{{EJr}} + \frac{{RL{\left( {R + L} \right)}}}{{EJx}} + kry^{{ - 1}} + \frac{{{\left( {R + L} \right)}^{2} }}{{kb}}} \right)} \cdot {\sqrt {3Tx + 1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3} }{\sqrt 3 }{\left( {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2\frac{{L^{2} }}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)}^{2} Tx} \right)}{\left( {{\left( {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3\frac{{L^{3} }}{{EJx}} + \frac{{RL{\left( {R + L} \right)}}}{{EJx}} + kry^{{ - 1}} + \frac{{{\left( {R + L} \right)}^{2} }}{{kb}}} \right)}{\left( {\frac{L}{{JpG}} + kt^{{ - 1}} } \right)} + {\left( {\frac{L}{{JpG}} + kt^{{ - 1}} } \right)} + {\left( {\frac{L}{{EJx}} + kb^{{ - 1}} } \right)}{\left( {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3\frac{{L^{3} }}{{EJx}} + \frac{{RL{\left( {R + L} \right)}}}{{EJx}} + kry^{{ - 1}} + \frac{{{\left( {R + L} \right)}^{2} }}{{kb}}} \right)} - {\left( {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2\frac{{L^{2} }}{{EJx}} + \frac{{RL}}{{EJx}} + \frac{{R + L}}{{kb}}} \right)}^{2} } \right)}^{{ - 1}} \\ \end{aligned} $$

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Mastinu, G., Gobbi, M. & Previati, G. A New Six-axis Load Cell. Part I: Design. Exp Mech 51, 373–388 (2011). https://doi.org/10.1007/s11340-010-9355-1

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