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Receptance Coupling

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Abstract

In Chaps. 19 we discussed discrete and continuous beam models that can be used to describe the behavior of vibrating systems. We also detailed experimental techniques that we can use to identify these models. In this chapter, we will introduce an approach to combine models or measurements of individual components in order to predict the assembly’s frequency response function (FRF). This method is referred to as receptance coupling [1]; recall from Sect. 7.1 that a receptance is a type of FRF.

Keywords

  • Receptance
  • Substructure
  • Compatibility condition
  • Equilibrium condition
  • Rigid coupling
  • Flexible coupling
  • Displacement-to-force receptance
  • Displacement-to-couple receptance
  • Rotation-to-force receptance
  • Rotation-to-couple receptance
  • Modal analysis
  • Complex matrix inversion

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Fig. 10.1
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Notes

  1. 1.

    We will follow this lower case/upper case notation to differentiate between component and assembly coordinates throughout the chapter.

  2. 2.

    We will not consider axial or torsional vibrations in this analysis.

  3. 3.

    Compliance is the inverse of stiffness.

References

  1. Bishop R, Johnson D (1960) The mechanics of vibration. Cambridge University Press, Cambridge

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  2. Weaver W Jr, Timoshenko S, Young D (1990) Vibration problems in engineering, 5th edn. Wiley, New York

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  3. Burns T, Schmitz T (2004) Receptance coupling study of tool-length dependent dynamic absorber effect. In: Proceedings of American Society of Mechanical Engineers International Mechanical Engineering Congress and Exposition, IMECE2004-60081, Anaheim, CA

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  4. Burns T, Schmitz T (2005) A study of linear joint and tool models in spindle-holder-tool receptance coupling. In: Proceedings of 2005 American Society of Mechanical Engineers International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2005-85275, Long Beach, CA

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  5. Park S, Altintas Y, Movahhedy M (2003) Receptance coupling for end mills. Int J Mach Tools Manuf 43:889–896

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  6. Schmitz T, Powell K, Won D, Duncan GS, Sawyer WG, Ziegert J (2007) Shrink fit tool holder connection stiffness/damping modeling for frequency response prediction in milling. Int J Mach Tools Manuf 47(9):1368–1380

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  7. Duncan GS, Tummond M, Schmitz T (2005) An investigation of the dynamic absorber effect in high-speed machining. Int J Mach Tools Manuf 45:497–507

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Exercises

Exercises

  1. 1.

    Determine the direct frequency response function, \( \frac{X_2}{F_2} \), for the two degree of freedom system shown in Fig. P10.1 using receptance coupling. Express your final result as a function of m, c, k, and the excitation frequency, ω. You may assume a harmonic forcing function, F2eiωt, is applied to coordinate X2.

  2. 2.

    Determine the direct frequency response function, \( \frac{X_1}{F_1} \), for the two degree of freedom system shown in Fig. P10.2 using receptance coupling. Express your final result as a function of m, c, k, and the excitation frequency, ω. You may assume a harmonic forcing function, F1eiωt, is applied to coordinate X1.

  3. 3.

    Use receptance coupling to rigidly join two free-free beams and find the free-free assembly’s displacement-to-force tip receptance. Both steel cylinders are described by the following parameters: 12.7 mm diameter, 100 mm length, 200 GPa elastic modulus, and 7800 kg/m3 density. Assume a solid damping factor of 0.0015. Once you have determined the assembly response, verify your result against the displacement-to-force tip receptance for a 12.7 mm diameter, 200 mm long free-free steel cylinder with the same material properties. Select a frequency range that encompasses the first three bending modes and display your results as the magnitude (in m/N) versus frequency (in Hz) using a semi-logarithmic scale.

  4. 4.

    Plot the displacement-to-force tip receptance for a sintered carbide cylinder with free-free boundary conditions. The beam is described by the following parameters: 19 mm diameter, 150 mm length, 550 GPa elastic modulus, and 15,000 kg/m3 density. Assume a solid damping factor of 0.002. Select a frequency range that encompasses the first three bending modes and display your results as magnitude (m/N) vs. frequency (Hz) in a semi-logarithmic format.

  5. 5.

    Determine the fixed-free displacement-to-force tip receptance for a sintered carbide cylinder by coupling the free-free receptances to a rigid wall (with zero receptances). The beam is described by the following parameters: 19 mm diameter, 150 mm length, 550 GPa elastic modulus, and 15,000 kg/m3 density. Assume a solid damping factor of 0.002. Select a frequency range that encompasses the first two bending modes and display your results as magnitude (m/N) vs. frequency (Hz) in a semi-logarithmic format. Verify your result by comparing it to the displacement-to-force tip receptance for a fixed-free beam with the same dimensions and material properties.

  6. 6.

    For a rigid coupling between two component coordinates x1a and x1b, the compatibility condition is _____________.

  7. 7.

    For a flexible coupling (spring stiffness k) between two component coordinates x1a and x1b, the compatibility condition is _____________. An external force is applied to the assembly at coordinate X1a.

  8. 8.

    For a flexible-damped coupling (spring stiffness k and damping coefficient c) between two component coordinates x1a and x1b, the compatibility condition is _____________. An external force is applied to the assembly at coordinate X1a.

  9. 9.

    What are the units for the rotation-to-couple receptance, pij, used to describe the transverse vibration of beams?

  10. 10.

    What are the (identical) units for the displacement-to-couple, lij, and rotation-to-force, nij, receptances used to describe the transverse vibration of beams?

Fig. P10.1
figure 30

Two degree of freedom assembly

Fig. P10.2
figure 31

Flexible damped coupling of mass (I) to spring-mass-damper (II) to form the two degree of freedom assembly III

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Schmitz, T.L., Smith, K.S. (2021). Receptance Coupling. In: Mechanical Vibrations. Springer, Cham. https://doi.org/10.1007/978-3-030-52344-2_10

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  • DOI: https://doi.org/10.1007/978-3-030-52344-2_10

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