Axisymmetric thermoelastic contact vibration between a viscoelastic half-space and a rotating spherical punch

Abstract

The axisymmetric thermoelastic contact vibration of a rigid rotating spherical punch against a homogeneous half-space is investigated by considering dynamic deformation and frictional heat. Complex-value modulus is introduced to characterize the hysteretic damping of viscoelastic materials. Under the assumption of sufficiently small oscillating force, the dynamic contact pressure is derived with the unknown disturbance variable of the contact radius by employing the perturbation method. Dynamic normal displacement is obtained in the form of complex integral by using Hankel integral transformation. Dynamic contact stiffness (DCS) factor considering the frictional heat is discussed under the approximate dynamic boundary conditions. Illustrative examples are presented to clarify the influences of the rotational speed, friction coefficient, thermal expansion coefficient, thermal conductivity coefficient, contact radius, and damping ratio on the DCS factor and surface temperature. The obtained results indicate that the friction coefficient and rotational speed can increase the DCS and surface temperature in a certain frequency range.

Appendix A

The expression of ${\bf{T}}_{1} \left( {s,z} \right)$ appearing in Eq. (42) is:

$${\varvec{T}}_{1} \left( {s,z} \right) = \left[ {T_{1i}^{a} \left( {s,z} \right),T_{2i}^{a} \left( {s,z} \right),T_{3i}^{a} \left( {s,z} \right),T_{4i}^{a} \left( {s,z} \right)} \right]^{\rm T} ,\;i = 1,2,$$

(A.1)

where

$$T_{1i}^{a} = e^{{m_{i} z}} ,$$

(A.2)

$$T_{2i}^{a} = a_{i} \left( s \right)e^{{m_{i} z}} ,$$

(A.3)

$$T_{3i}^{a} = \left[ {\lambda s + \left( {\lambda + 2\mu } \right)a_{i} \left( s \right)m_{i} } \right]e^{{m_{i} z}} ,$$

(A.4)

$$T_{4i}^{a} = \left[ {\mu m_{i} - \mu sa_{i} \left( s \right)} \right]e^{{m_{i} z}} .$$

(A.5)

The expression of ${\bf{T}}_{2} \left( {s,z} \right)$ appearing in Eq. (42) is:

$${\varvec{T}}_{2} \left( {s,z} \right) = \left[ {T_{1}^{b} \left( {s,z} \right),T_{2}^{b} \left( {s,z} \right),T_{3}^{b} \left( {s,z} \right),T_{4}^{b} \left( {s,z} \right)} \right]^{\rm T} ,$$

(A.6)

where

$$T_{1}^{b} = F_{1} \left( s \right)e^{sz} ,$$

(A.7)

$$T_{2}^{b} = F_{2} \left( s \right)e^{sz} ,$$

(A.8)

$$T_{3}^{b} = \left[ {\lambda sF_{1} \left( s \right) + \left( {\lambda + 2\mu } \right)sF_{2} \left( s \right) - \left( {3\lambda + 2\mu } \right)\alpha_{T} } \right]e^{sz} ,$$

(A.9)

$$T_{4}^{b} = \left[ { - \mu sF_{1} \left( s \right) - \mu sF_{2} \left( s \right)} \right]e^{sz} .$$

(A.10)