A harmonic balance-based method to predict nonlinear forced response and temperature rise of dry friction systems including frictional heat transfer

Abstract

Dry friction dampers in turbomachinery not only decrease the vibration level, but also generate frictional heat. This thermal process may cause a significant temperature rise at the contact interface, producing thermal expansion and altering tribological properties subsequently. These effects in turn can change structural dynamics. Besides, the temperature rise may also cause the material melting and ablation, leading to damper failure. Hence, the structural dynamics and the thermal process in dry friction systems are interacting. The thermomechanical coupling should be included in analyses. In this paper, a novel numerical method, namely Dry Friction Thermo-Mechanical Coupling Response Prediction (DFTMCP), is proposed. Based on the multi-harmonic balance method, the DFTMCP can synchronously predict the nonlinear forced response and interface temperature in the steady state. This method is under the framework of steady heat transfer assumption, and a dimensionless number is proposed to determine the rationality of the assumption. To guarantee efficiency and convergence, an ad hoc model reduction technique for the nonlinear thermomechanical coupling problem and the corresponding analytical Jacobian matrix, which are also the highlights of the work, are implemented. The former reduces the dimension of the governing equations by over 98.6% while the latter makes the time cost drop over 37 times. By using the proposed numerical method, two essential coupling factors, the thermoelastic deformation and the friction coefficient variation at the interface, are considered and discussed quantitatively for the influence on the forced response through the finite element model of a blade with a flat underplatform damper in engineering. A convergence analysis has been performed to validate the correctness of the simulation results. Results show that under the specific rotational speed, the average temperature at the contact surface rises by 415 °C, and the maximum local temperature increases to 1229 °C, which is close to the melting point. Ignoring the thermomechanical coupling effect leads to a 19.0% misprediction of the optimal centrifugal force and a 21.4% underestimation of the resonant peak.

Graphical abstract

Appendices

Appendix 1

The transient temperature part of the $i{\text{th}}$ component in the temperature vector ${\tilde{\mathbf{\theta }}}^{k}$ for the ${0}^{\mathrm{th}}$ harmonic:

$$ \tilde{\theta }_{i}^{0} (t) = \frac{{\tilde{q}_{i}^{0} }}{{\lambda_{{{\uptheta }i}} }}(1 - {\text{e}}^{{ - \lambda_{{{\uptheta }i}} t}} ) $$

(45)

The transient temperature part of the $i{\text{th}}$ component in the temperature vector ${\tilde{\mathbf{\theta }}}^{k}$ for other harmonics:

$$ \begin{array}{*{20}c} {\tilde{\theta }_{i}^{n} (t) = \tilde{q}_{i}^{n} \frac{{\lambda_{{{\uptheta }i}} }}{{(n\omega )^{2} + \lambda_{{{\uptheta }i}}^{2} }}{\text{e}}^{{ - \lambda_{{{\uptheta }i}} t}} + \tilde{q}_{i}^{n} \frac{{(n\omega )^{2} }}{{(n\omega )^{2} + \lambda_{{{\uptheta }i}}^{2} }}\sqrt {\left( {\frac{1}{n\omega }} \right)^{2} + \left( {\frac{{\lambda_{{{\uptheta }i}} }}{{(n\omega )^{2} }}} \right)} {\text{e}}^{ - jn\omega t} } \\ {n = 1,2, \cdots ,N_{{\text{h}}} } \\ \end{array} $$

(46)

Appendix 2

Supposing M and N are the invertible matrices with $m \times m$ and $n \times n$ dimensions. P is the $m \times n$ matrix. The inverse of the integral matrix can be expressed through the block matrix calculation as:

$$ \left[ {\begin{array}{*{20}c} {\mathbf{M}} & {\mathbf{P}} \\ {\mathbf{0}} & {\mathbf{N}} \\ \end{array} } \right]^{ - 1} = \left[ {\begin{array}{*{20}c} {{\mathbf{M}}^{ - 1} } & { - {\mathbf{M}}^{ - 1} {\mathbf{PN}}^{ - 1} } \\ {\mathbf{0}} & {{\mathbf{N}}^{ - 1} } \\ \end{array} } \right] $$

(47)

The condensation process can be expressed as

$$ \begin{array}{*{20}c} {\left[ {\begin{array}{*{20}c} {\mathbf{A}} & {\mathbf{B}} \\ {\mathbf{C}} & {\mathbf{D}} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {{\mathbf{x}}_{{\text{o}}} } \\ {{\mathbf{x}}_{{\text{s}}} } \\ \end{array} } \right\} + \left\{ {\begin{array}{*{20}c} {{\mathbf{F}}_{{\text{nl,o}}} } \\ {{\mathbf{F}}_{{\text{nl,s}}} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {{\mathbf{F}}_{{\text{e,o}}} } \\ {{\mathbf{F}}_{{\text{e,s}}} } \\ \end{array} } \right\}} \\ {\begin{array}{*{20}c} \Rightarrow & {\left( {{\mathbf{D - CA}}^{ - 1} {\mathbf{B}}} \right)} \\ \end{array} {\mathbf{x}}_{{\text{s}}} + {\mathbf{CA}}^{ - 1} ({\mathbf{F}}_{{\text{e,o}}} - {\mathbf{F}}_{{\text{nl,o}}} ) - ({\mathbf{F}}_{{\text{e,s}}} - {\mathbf{F}}_{{\text{nl,s}}} ) = {\mathbf{0}}} \\ \end{array} $$

(48)

in which A, B, C and D are the static or dynamic stiffness matrix, x is the response vector. ${\mathbf{F}}_{{{\text{nl}}}}$ and ${\mathbf{F}}_{{{\text{nl}}}}$ are the nonlinear force vector and the excitation force vector, respectively. The subscripts o and s refer to the reduced DOFs and the retained DOFs, respectively. Substituting Eqs. (47) and (48) into (19), one can derive the reduced thermomechanical coupling equations:

$$ \begin{gathered} \left( {\left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{{\mathbf{u}}}^{{{\mathbf{nl}}}} } & {{\mathbf{K}}_{{{\mathbf{u\uptheta }}}}^{{{\mathbf{nl}}}} } \\ {\mathbf{0}} & {{\mathbf{K}}_{{\mathbf{\uptheta }}}^{{{\mathbf{nl}}}} } \\ \end{array} } \right] - \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{{\mathbf{u}}}^{{{\mathbf{nll}}}} } & {{\mathbf{K}}_{{{\mathbf{u\uptheta }}}}^{{{\mathbf{nll}}}} } \\ {\mathbf{0}} & {{\mathbf{K}}_{{\mathbf{\uptheta }}}^{{{\mathbf{nll}}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{{\mathbf{u}}}^{{\mathbf{l}}} } & {{\mathbf{K}}_{{{\mathbf{u\uptheta }}}}^{{\mathbf{l}}} } \\ {\mathbf{0}} & {{\mathbf{K}}_{{\mathbf{\uptheta }}}^{{\mathbf{l}}} } \\ \end{array} } \right]^{{ - {\mathbf{1}}}} \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{{\mathbf{u}}}^{{{\mathbf{lnl}}}} } & {{\mathbf{K}}_{{{\mathbf{u\uptheta }}}}^{{{\mathbf{lnl}}}} } \\ {\mathbf{0}} & {{\mathbf{K}}_{{\mathbf{\uptheta }}}^{{{\mathbf{lnl}}}} } \\ \end{array} } \right]} \right)\left\{ {\begin{array}{*{20}c} {{\mathbf{U}}_{{{\mathbf{nl}}}}^{{\mathbf{0}}} } \\ {{\mathbf{\uptheta }}_{{{\mathbf{nl}}}} } \\ \end{array} } \right\} \hfill \\ \quad \quad \quad \quad \quad \quad \quad + \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{{\mathbf{u}}}^{{{\mathbf{nll}}}} } & {{\mathbf{K}}_{{{\mathbf{u\uptheta }}}}^{{{\mathbf{nll}}}} } \\ {\mathbf{0}} & {{\mathbf{K}}_{{\mathbf{\uptheta }}}^{{{\mathbf{nll}}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{{\mathbf{u}}}^{{\mathbf{l}}} } & {{\mathbf{K}}_{{{\mathbf{u\uptheta }}}}^{{\mathbf{l}}} } \\ {\mathbf{0}} & {{\mathbf{K}}_{{\mathbf{\uptheta }}}^{{\mathbf{l}}} } \\ \end{array} } \right]^{{ - {\mathbf{1}}}} \left\{ {\begin{array}{*{20}c} {{\mathbf{F}}_{{\mathbf{l}}}^{{\mathbf{0}}} } \\ {\mathbf{0}} \\ \end{array} } \right\}{\mathbf{ = }}\left\{ {\begin{array}{*{20}c} { - {\mathbf{F}}_{{{\mathbf{nl}}}}^{{\mathbf{0}}} } \\ {{\mathbf{q}}_{{{\mathbf{nl}}}} } \\ \end{array} } \right\} \hfill \\ \end{gathered} $$

(49)