Abstract
A novel micro-slip friction modeling approach based on real pressure distribution is presented for simulating the interface friction behavior of blades with dovetail joints in aircraft engines. Considering the influence of rotational speed and stagger angle on the true pressure distribution of the dovetail joint, an asymmetric pressure density function was established and applied to the Iwan friction model. The pressure distribution at the tenon joint interface is obtained through finite element modeling. And the pressure distribution function is obtained through dimensionality reduction and data fitting. Under the framework of the Iwan model, a new tangential friction displacement relationship at the tenon interface was obtained by changing the critical slip force density function. On this basis, a dynamic model of the dovetail blade was established and the contact characteristics of the interface were analyzed. The results indicate that under the influence of uneven pressure distribution, the width and length directions of the contact interface exhibit different friction behaviors, and the nonlinear response of the blades is affected.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
The datasets generated during and analysed during the current study are not publicly available, but are available from the corresponding author on reasonable request.
References
Shangguan, B., Xu, Z.L., Wu, Q.L., Zhou, X.D., Cao, S.H.: Forced Response Prediction of Blades with Loosely Assembled Dovetail Attachment by HBM. ASME Turbo Expo 2009: Power for Land, Sea, and Air, Florida, USA, GT2009-59778, pp 421–428
She, H.X., Li, C.F., Tang, Q.S., Wen, B.C.: Nonlinear vibration analysis of a rotating disk-beam system subjected to dry friction. Shock Vib. (2020). https://doi.org/10.1155/2020/7604174
Li, C.F., Liu, X.W., Tang, Q.S., Chen, Z.L.: Modeling and nonlinear dynamics analysis of a rotating beam with dry friction support boundary conditions. J. Sound Vib. 498, 115978 (2021)
Zucca, S., Firrone, C.M., Gola, M.M.: Numerical assessment of friction damping at turbine blade root joints by simultaneous calculation of the static and dynamic contact loads. Nonlinear Dyn. 67(3), 1943–1955 (2012)
Yuan, J., Schwingshackl, C., Wong, C., Salles, L.: On an improved adaptive reduced-order model for the computation of steady-state vibrations in large-scale non-conservative systems with friction joints. Nonlinear Dyn. 103(4), 3283–3300 (2020)
Lemoine, E., Nélias, D., Thouverez, F., Vincent, C.: Influence of fretting wear on bladed disks dynamic analysis. Tribol. Int. 145, 106148 (2020)
Allara, M.: A model for the characterization of friction contacts in turbine blades. J. Sound Vib. 320(3), 527–544 (2009)
Luo, Y., Jiang, X., Wang, Y.: Modeling of microslip friction and its application in the analysis of underplatform damper. Int. J. Aeronaut. Space Sci. 19(2), 388–398 (2018)
Iwan, W.D.: A distributed-element model for hysteresis and its steady-state dynamic response. J. Appl. Mech. 33(4), 893–900 (1966)
Dahl, P.R.: Solid friction damping of mechanical vibrations. AIAA J. 14(12), 1675–1682 (1976)
de Wit, C.C., Olsson, H., Astrom, K.J., Lischinsky, P.: A new model for control of systems with friction. IEEE Trans. Autom. Control 40(3), 419–425 (1995)
Song, Y., Hartwigsen, C.J., McFarland, D.M., Vakakis, A.F., Bergman, L.A.: Simulation of dynamics of beam structures with bolted joints using adjusted Iwan beam elements. J. Sound Vib. 273(1–2), 249–276 (2004)
Rajaei, M., Ahmadian, H.: Development of generalized Iwan model to simulate frictional contacts with variable normal loads. Appl. Math. Model. 38(15–16), 4006–4018 (2014)
Segalman, D.J.: A four-parameter iwan model for lap-type joints. J. Appl. Mech. 72, 752–760 (2005)
Li, Y.K., Hao, Z.M.: A six-parameter Iwan model and its application. Mech. Syst. Signal Process. 68, 354–365 (2015)
Ahmadian, H., Rajaei, M.: Identification of Iwan distribution density function in frictional contacts. J. Sound Vib. 333(15), 3382–3393 (2014)
Li, D.W., Xu, C., Liu, T., Gola, M., Wen, L.H.: A modified IWAN model for micro-slip in the context of dampers for turbine blade dynamics. Mech. Syst. Signal Process. 121, 14–30 (2019)
Li, D.W., Botto, D., Xu, C., Gola, M.: A new approach for the determination of the Iwan density function in modeling friction contact. Int. J. Mech. Sci. 180, 105671 (2020)
Li, D.W., Xu, C., Kang, J.H., Zhang, Z.S.: Modeling tangential friction based on contact pressure distribution for predicting dynamic responses of bolted joint structures. Nonlinear Dyn. 101(12), 255–269 (2020)
Sinha, S.K., Turner, K.E.: Natural frequencies of a pre-twisted blade in a centrifugal force field. J. Sound Vib. 330(11), 2655–2681 (2011)
Yang, S.M., Tsao, S.M.: Dynamics of a pretwisted blade under nonconstant rotating speed. Comput. Struct. 62(4), 643–651 (1997)
Guo, X.M., Ni, K.X., Ma, H., Zeng, J., Wang, Z.L., Wen, B.C.: Dynamic response analysis of shrouded blades under impact-friction considering the influence of passive blade vibration. J. Sound Vib. 503, 116112 (2021)
Yagci, B., Filiz, S., Romero, L.L., Ozdoganlar, O.B.: A spectral-Tchebychev technique for solving linear and nonlinear beam equations. J. Sound Vib. 321(1–2), 375–404 (2009)
Anandavel, K., Prakash, R.V.: Effect of three-dimensional loading on macroscopic fretting aspects of an aero-engine blade–disc dovetail interface. Tribol. Int. 44(11), 1544–1555 (2011)
Ma, H., Wang, D., Tai, X.Y., Wen, B.C.: Vibration response analysis of blade-disk dovetail structure under blade tip rubbing condition. J. Vib. Control 23(2), 252–271 (2015)
Liu, X.W., Li, C.F., Lyu, Z.P., Zhang, G.B.: An approach of micro-slip modeling based on asymmetric contact pressure distribution and its application in nonlinear boundary beam system. Int. J. Non-Linear Mech. 154, 104425 (2023)
Acknowledgements
This project is supported by the National Natural Science Foundation of China (No. 52075086) and the Fundamental Research Funds for the Central Universities (No. N2203021).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
1.1 Detailed expressions of some equations
-
1.
Vibrational vector
$$ {\mathbf{U}}\left(\upeta \right) = {\mathbf{V}}\left(\upeta \right) = {\mathbf{W}}\left(\upeta \right) = {{\varvec{\upphi}}}_{{v}} \left(\upeta \right) = {{\varvec{\upphi}}}_{{w}} \left(\upeta \right) = \left[ {\begin{array}{*{20}c} {T_{1}^{ * } \left( \eta \right)} & {T_{2}^{ * } \left( \eta \right)} & \cdots & {T_{{N_{p} }}^{ * } \left( \eta \right)} \\ \end{array} } \right]^{{\text{T}}} $$(36) -
2.
Generalized displacement vector:
$$ {\mathbf{q}} = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}l} {{\mathbf{q}}_{\rm{u}} } & {{\mathbf{q}}_{{v}} } \\ \end{array} } & {{\mathbf{q}}_{{w}} } & {{\mathbf{q}}_{{{\varphi }_{{v}} }} } & {{\mathbf{q}}_{{{\varphi }_{{w}} }} } \\ \end{array} } \right]^{{\text{T}}} $$(37)where
$$ \begin{aligned} {\mathbf{q}}_{\rm{u}} \left( \text{t} \right) = \left[ {\begin{array}{*{20}l} {U_{1} q_{u1} \left( t \right)} & {U_{2} q_{u2} \left( t \right)} & \cdots & {U_{{N_{p} }} q_{{uN_{p} }} \left( t \right)} \\ \end{array} } \right]^{{\text{T}}} \\ {\mathbf{q}}_{{v}} \left( \text{t} \right) = \left[ {\begin{array}{*{20}l} {V_{1} q_{v1} \left( t \right)} & {V_{2} q_{v2} \left( t \right)} & \cdots & {V_{{N_{p} }} q_{{vN_{p} }} \left( t \right)} \\ \end{array} } \right]^{{\text{T}}} \\ {\mathbf{q}}_{{w}} \left( \text{t} \right) = \left[ {\begin{array}{*{20}l} {W_{1} q_{w1} \left( t \right)} & {W_{2} q_{w2} \left( t \right)} & \cdots & {W_{{N_{p} }} q_{{wN_{p} }} \left( t \right)} \\ \end{array} } \right]^{{\text{T}}} \\ {\mathbf{q}}_{{{\varphi }_{{v}} }} \left( \text{t} \right) = \left[ {\begin{array}{*{20}l} {\phi_{v1} q_{{\varphi_{v} 1}} \left( t \right)} & {\phi_{v2} q_{{\varphi_{v} 2}} \left( t \right)} & \cdots & {\phi_{{vN_{p} }} q_{{\varphi_{v} N_{p} }} \left( t \right)} \\ \end{array} } \right]^{{\text{T}}} \\ {\mathbf{q}}_{{{\varphi }_{{w}} }} \left( \text{t} \right) = \left[ {\begin{array}{*{20}l} {\phi_{w1} q_{{\varphi_{w} 1}} \left( t \right)} & {\phi_{w2} q_{{\varphi_{w} 2}} \left( t \right)} & \cdots & {\phi_{{wN_{p} }} q_{{\varphi_{w} N_{p} }} \left( t \right)} \\ \end{array} } \right]^{{\text{T}}} \\ \end{aligned} $$ -
3.
Mass matrix:
$$ {\mathbf{M}} = \left[ {\begin{array}{*{20}c} {{\mathbf{M}}_{{{11}}} } & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {} & {{\mathbf{M}}_{22} } & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {} & {} & {{\mathbf{M}}_{33} } & {\mathbf{0}} & {\mathbf{0}} \\ {} & {} & {} & {{\mathbf{M}}_{44} } & {\mathbf{0}} \\ {{\text{sym}}} & {} & {} & {} & {{\mathbf{M}}_{55} } \\ \end{array} } \right] $$(38) -
4.
Elements in the mass matrix:
$$ \begin{aligned} \begin{array}{*{20}c} {{\mathbf{M}}_{11} = \rho AL\int_{0}^{1} {{\mathbf{U}}\left(\upeta \right){\mathbf{U}}\left(\upeta \right)^{{\text{T}}} {\text{d}}\eta } } & {{\mathbf{M}}_{22} = \rho AL\int_{0}^{1} {{\mathbf{V}}\left(\upeta \right){\mathbf{V}}\left(\upeta \right)^{{\text{T}}} {\text{d}}\eta } } \\ {{\mathbf{ M}}_{{{33}}} = \rho AL\int_{0}^{1} {{\mathbf{W}}\left(\upeta \right){\mathbf{W}}\left(\upeta \right)^{{\text{T}}} {\text{d}}\eta } } & {{\mathbf{M}}_{{{44}}} = \rho I_{y} L\int_{0}^{1} {{{\varvec{\upphi}}}_{v} \left(\upeta \right){{\varvec{\upphi}}}_{v} \left(\upeta \right)^{{\text{T}}} } {\text{d}}\eta } \\ \end{array} \hfill \\ {\mathbf{ M}}_{{{55}}} = \rho I_{z} L\int_{0}^{1} {{{\varvec{\upphi}}}_{w} \left(\upeta \right){{\varvec{\upphi}}}_{w} \left(\upeta \right)^{{\text{T}}} } {\text{d}}\eta \hfill \\ \end{aligned} $$(39) -
5.
Structural stiffness matrix:
$$ {\mathbf{K}}_{{\text{b}}} = \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{{{11}}} } & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{K}}_{22} } & {{\mathbf{K}}_{23} } & {{\mathbf{K}}_{24} } & {{\mathbf{K}}_{25} } \\ {\mathbf{0}} & {{\mathbf{K}}_{32} } & {{\mathbf{K}}_{33} } & {{\mathbf{K}}_{34} } & {{\mathbf{K}}_{35} } \\ {\mathbf{0}} & {{\mathbf{K}}_{42} } & {{\mathbf{K}}_{43} } & {{\mathbf{K}}_{44} } & {{\mathbf{K}}_{45} } \\ {\mathbf{0}} & {{\mathbf{K}}_{52} } & {{\mathbf{K}}_{53} } & {{\mathbf{K}}_{54} } & {{\mathbf{K}}_{55} } \\ \end{array} } \right] $$(40) -
6.
Elements in the structural stiffness matrix:
$$\begin{aligned} \textbf{K}_{{11}} & = \text{EA}\frac{1}{L}\int_{0}^{1} {U^{\prime}\left( \eta \right)U^{\prime}\left( \eta \right)^{\rm{T}} \text{d}\eta } \\ \textbf{K}_{{22}} & = \kappa \text{GA}\frac{1}{L}\int_{0}^{1} {V^{\prime}\left( \eta \right)V^{\prime}\left( \eta \right)^{\rm{T}} \text{d}}\eta + \kappa \text{GAL}\gamma ^{{\prime}{2}} \int_{0}^{1} {V\left( \eta \right)V\left( \eta \right)^{\rm{T}} \text{d}}\eta \\ \textbf{K}_{{23}} & = - \kappa \text{GA}\gamma ^{\prime}\int_{0}^{1} {V^{\prime}\left( \eta \right)W\left( \eta \right)^{\rm{T}} \text{d}}\eta + \kappa \text{GA}\gamma ^{\prime}\int_{0}^{1} {V\left( \eta \right)W^{\prime}\left( \eta \right)^{\rm{T}} \text{d}}\eta \\ \textbf{K}_{{24}} & = \kappa \text{GAL}\gamma ^{\prime}\int_{0}^{1} {V\left( \eta \right)\phi _{v} \left( \eta \right)^{\rm{T}} \text{d}}\eta \\ \textbf{K}_{{25}} & = - \kappa \text{GA}\int_{0}^{1} {V^{\prime}\left( \eta \right)\phi _{w} \left( \eta \right)^{\rm{T}} \text{d}}\eta \\ \textbf{K}_{{32}} & = - \kappa \text{GA}\gamma ^{\prime}\int_{0}^{1} {W\left( \eta \right)V^{\prime}\left( \eta \right)^{\rm{T}} \text{d}}\eta + \kappa \text{GA}\gamma ^{\prime}\int_{0}^{1} {W^{\prime}\left( \eta \right)V\left( \eta \right)^{\rm{T}} \text{d}}\eta \\ \textbf{K}_{{33}} & = \kappa \text{GAL}\gamma ^{{\prime}{2}} \int_{0}^{1} {W\left( \eta \right)W\left( \eta \right)^{\rm{T}} \text{d}}\eta + \kappa \text{GA}\frac{1}{L}\int_{0}^{1} {W^{\prime}\left( \eta \right)W^{\prime}\left( \eta \right)^{\rm{T}} \text{d}}\eta \\ \textbf{K}_{{34}} & = \kappa \text{GA}\int_{0}^{1} {W^{\prime}\left( \eta \right)\phi _{v} \left( \eta \right)^{\rm{T}} \text{d}}\eta \\ \textbf{K}_{{35}} &= \kappa \text{GAL}\gamma ^{\prime}\int_{0}^{1} {W\left( \eta \right)\phi _{w} \left( \eta \right)^{\rm{T}} \text{d}}\eta \\ \textbf{K}_{{42}} & = \kappa \text{GAL}\gamma ^{\prime}\int_{0}^{1} {\phi _{v} \left( \eta \right)V\left( \eta \right)^{\rm{T}} \text{d}}\eta \\ \textbf{K}_{{43}} & = \kappa \text{GA}\int_{0}^{1} {\phi _{v} \left( \eta \right)W^{\prime}\left( \eta \right)^{\rm{T}} \text{d}}\eta \\ \textbf{K}_{{44}} & = \kappa \text{GAL}\int_{0}^{1} {\phi _{v} \left( \eta \right)\phi _{v} \left( \eta \right)^{\rm{T}} \text{d}}\eta + \text{EI}_{y} \frac{1}{L}\int_{0}^{1} {\phi _{v} ^{\prime } \left( \eta \right)\phi _{v} ^{\prime } \left( \eta \right)^{\rm{T}} \text{d}\eta } + \text{EI}_{z} L\gamma ^{{\prime}{2}} \int_{0}^{1} {\phi _{v} \left( \eta \right)\phi _{v} \left( \eta \right)^{\rm{T}} \text{d}\eta } \\ \textbf{K}_{{45}} & = - \text{EI}_{y} \gamma ^{\prime}\int_{0}^{1} {\phi _{v} ^{\prime } \left( \eta \right)\phi _{w} \left( \eta \right)^{\rm{T}} \text{d}\eta } + \text{EI}_{z} \gamma ^{\prime}\int_{0}^{1} {\phi _{v} \left( \eta \right)\phi _{w} ^{\prime } \left( \eta \right)^{\rm{T}} \text{d}\eta } \\ \textbf{K}_{{52}} & = - \kappa \text{GA}\int_{0}^{1} {\phi _{w} \left( \eta \right)V^{\prime}\left( \eta \right)^{\rm{T}} \text{d}}\eta \\ \textbf{K}_{{53}} & = \kappa \text{GAL}\gamma ^{\prime}\int_{0}^{1} {\phi _{w} \left( \eta \right)W\left( \eta \right)^{\rm{T}} \text{d}}\eta \\ \textbf{K}_{{54}} & = - \text{EI}_{y} \gamma ^{\prime}\int_{0}^{1} {\phi _{w} \left( \eta \right)\phi _{v} ^{\prime } \left( \eta \right)^{\rm{T}} \text{d}\eta } + \text{EI}_{z} \gamma ^{\prime}\int_{0}^{1} {\phi _{w} ^{\prime } \left( \eta \right)\phi _{v} \left( \eta \right)^{\rm{T}} \text{d}\eta } \\ \textbf{K}_{{55}} & = \kappa \text{GAL}\int_{0}^{1} {\phi _{w} \left( \eta \right)\phi _{w} \left( \eta \right)^{\rm{T}} \text{d}}\eta + \text{EI}_{z} \frac{1}{L}\int_{0}^{1} {\phi _{w} ^{\prime } \left( \eta \right)\phi _{w} ^{\prime } \left( \eta \right)^{\rm{T}} \text{d}\eta } + \text{EI}_{y} L\gamma ^{{\prime}{2}} \int_{0}^{1} {\phi _{w} \left( \eta \right)\phi _{w} \left( \eta \right)^{\rm{T}} \text{d}\eta } \\ \end{aligned}$$(41) -
7.
Centrifugal rigid matrix element:
$$ {\mathbf{K}}_{{\text{c}}} = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {} & {{\mathbf{K}}_{\rm{c}}^{\rm{v}} } & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {} & {} & {{\mathbf{K}}_{{\text{c}}}^{\rm{w}} } & {\mathbf{0}} & {\mathbf{0}} \\ {} & {} & {} & {\mathbf{0}} & {\mathbf{0}} \\ {{\text{sym}}} & {} & {} & {} & {\mathbf{0}} \\ \end{array} } \right] $$(42) -
8.
Element in the centrifugal stiffness matrix:
$$ \begin{array}{*{20}c} {{\mathbf{K}}_{{\text{c}}}^{\rm{v}} = \frac{1}{L}\int_{0}^{1} {f_{{\text{c}}} {\mathbf{V}}^{\prime } \left(\upeta \right){\mathbf{V}}^{\prime } \left(\upeta \right)^{{\text{T}}} {\text{d}}\eta } } & {{\mathbf{K}}_{{\text{c}}}^{\rm{w}} = \frac{1}{L}\int_{0}^{1} {f_{{\text{c}}} {\mathbf{W}}^{\prime } \left(\upeta \right){\mathbf{W}}^{\prime } \left(\upeta \right)^{{\text{T}}} {\text{d}}\eta } } \\ \end{array} $$(43) -
9.
Elastic stiffness matrix:
$$ {\mathbf{K}}_{{{\text{spr}}}} = \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{{{\text{spr}}}}^{\rm{u}} } & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {} & {{\mathbf{K}}_{{{\text{spr}}}}^{\rm{v}} } & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {} & {} & {{\mathbf{K}}_{{{\text{spr}}}}^{\rm{w}} } & {\mathbf{0}} & {\mathbf{0}} \\ {} & {} & {} & {{\mathbf{K}}_{{{\text{spr}}}}^{{{\varphi }_{{v}} }} } & {\mathbf{0}} \\ {{\text{sym}}} & {} & {} & {} & {{\mathbf{K}}_{{{\text{spr}}}}^{{{\varphi }_{{w}} }} } \\ \end{array} } \right] $$(44) -
10.
Element in the elastic stiffness matrix:
$$ \begin{aligned} \begin{array}{*{20}c} {{\mathbf{K}}_{{{\text{spr}}}}^{\rm{u}} = k_{u} {\mathbf{U}}\left( 0 \right){\mathbf{U}}\left( 0 \right)^{{\text{T}}} } & {\begin{array}{*{20}c} {{\mathbf{K}}_{{{\text{spr}}}}^{\rm{v}} = k_{v} {\mathbf{V}}\left( 0 \right){\mathbf{V}}\left( 0 \right)^{{\text{T}}} } & {{\mathbf{K}}_{{{\text{spr}}}}^{\rm{w}} = k_{w} {\mathbf{W}}\left( 0 \right){\mathbf{W}}\left( 0 \right)^{{\text{T}}} } \\ \end{array} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {{\mathbf{K}}_{{{\text{spr}}}}^{{{\varphi }_{{v}} }} = k_{{v{\text{t}}}} {{\varvec{\upphi}}}_{{v}} \left( 0 \right){{\varvec{\upphi}}}_{{v}} \left( 0 \right)^{{\text{T}}} } & {{\mathbf{K}}_{{{\text{spr}}}}^{{{\varphi }_{{w}} }} = k_{{w{\text{t}}}} {{\varvec{\upphi}}}_{{w}} \left( 0 \right){{\varvec{\upphi}}}_{{w}} \left( 0 \right)^{{\text{T}}} } \\ \end{array} \hfill \\ \end{aligned} $$ -
11.
Spin softening matrix:
$$ {\mathbf{K}}_{{\text{s}}} = \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{{\text{s}}}^{\rm{u}} } & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{K}}_{{\text{s}}}^{\rm{v}} } & {{\mathbf{K}}_{{\text{s}}}^{{\rm{vw}}} } & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{K}}_{{\text{s}}}^{{\rm{wv}}} } & {{\mathbf{K}}_{{\text{s}}}^{\rm{w}} } & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right] $$(45) -
12.
Element in the spin softening matrix:
$$ \begin{aligned} {\mathbf{K}}_{{\text{s}}}^{\rm{u}} &= - \rho AL\Omega^{2} \int_{0}^{1} {{\mathbf{U}}\left(\upeta \right){\mathbf{U}}\left(\upeta \right)^{{\text{T}}} {\text{d}}\eta } \hfill \\ {\mathbf{K}}_{{\text{s}}}^{\rm{v}} &= - \rho AL\Omega^{2} \int_{0}^{1} {\cos^{2} \gamma \left(\upeta \right){\mathbf{V}}\left(\upeta \right){\mathbf{V}}\left(\upeta \right)^{{\text{T}}} {\text{d}}\eta } \hfill \\ {\mathbf{K}}_{{\text{s}}}^{{\rm{vw}}} &= \frac{1}{2}\rho AL\Omega^{2} \int_{0}^{1} {\sin 2\gamma \left(\upeta \right){\mathbf{V}}\left(\upeta \right){\mathbf{W}}\left(\upeta \right)^{{\text{T}}} {\text{d}}\eta } \hfill \\ {\mathbf{K}}_{{\text{s}}}^{\rm{w}} &= - \rho AL\Omega^{2} \int_{0}^{1} {\sin^{2} \gamma \left(\upeta \right){\mathbf{W}}\left(\upeta \right){\mathbf{W}}\left(\upeta \right)^{{\text{T}}} {\text{d}}\eta } \hfill \\ {\mathbf{K}}_{{\text{s}}}^{{\rm{wv}}} &= \frac{1}{2}\rho AL\Omega^{2} \int_{0}^{1} {\sin 2\gamma \left(\upeta \right){\mathbf{W}}\left(\upeta \right){\mathbf{V}}\left(\upeta \right)^{{\text{T}}} {\text{d}}\eta } \hfill \\ \end{aligned} $$(46) -
13.
Coriolis force matrix:
$$ {\mathbf{G}} = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {{\mathbf{G}}_{{1{2}}} } & {{\mathbf{G}}_{13} } & {\mathbf{0}} & {\mathbf{0}} \\ {{\mathbf{G}}_{{{21}}} } & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {{\mathbf{G}}_{31} } & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right] $$(47) -
14.
Element in the Coriolis force matrix:
$$ \begin{aligned} & {{\mathbf{G}}_{{1{2}}} { = } - 2\rho AL\Omega \int_{0}^{1} {\cos \gamma \left( \eta \right){\mathbf{U}}\left(\upeta \right){\mathbf{V}}\left(\upeta \right)^{{\text{T}}} {\text{d}}\eta } } \\ & {{\mathbf{G}}_{13} { = 2}\rho AL\Omega \int_{0}^{1} {\sin \gamma \left( \eta \right){\mathbf{U}}\left(\upeta \right){\mathbf{W}}\left(\upeta \right)^{{\text{T}}} {\text{d}}\eta } } \\ & {{\mathbf{G}}_{21} = 2\rho AL\Omega \int_{0}^{1} {\cos \gamma \left( \eta \right){\mathbf{V}}\left(\upeta \right){\mathbf{U}}\left(\upeta \right)^{{\text{T}}} {\text{d}}\eta } } \\ & {{\mathbf{G}}_{{{3}1}} = - 2\rho AL\Omega \int_{0}^{1} {\sin \gamma \left( \eta \right){\mathbf{W}}\left(\upeta \right){\mathbf{U}}\left(\upeta \right)^{{\text{T}}} {\text{d}}\eta } } \\ \end{aligned}$$(48) -
15.
Excitation force vector:
$$ {\mathbf{F}} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {{\mathbf{F}}_{u} } & {{\mathbf{F}}_{v} } \\ \end{array} } & {{\mathbf{F}}_{w} } & {{\mathbf{F}}_{{\varphi_{v} }} } & {{\mathbf{F}}_{{\varphi_{w} }} } \\ \end{array} } \right]^{{\text{T}}} $$(49)where
$$ \begin{aligned} {\mathbf{F}}_{\rm{u}} &= {\mathbf{0}} \hfill \\ {\mathbf{F}}_{{v}} &= Lf_{{\text{e}}} \int_{0}^{{1}} {\cos \gamma \left( \eta \right){\mathbf{V}}(\upeta )} {\text{d}}\eta + F_{{_{{{\text{nl}}}} }}^{v} {\mathbf{V}}\left( 0 \right) \hfill \\ {\mathbf{F}}_{{w}} & = Lf_{{\text{e}}} \int_{0}^{{1}} {\sin \gamma \left( \eta \right){\mathbf{W}}(\upeta )} {\text{d}}\eta + F_{{_{{{\text{nl}}}} }}^{w} {\mathbf{W}}\left( 0 \right) \hfill \\ {\mathbf{F}}_{{{\varphi }_{{v}} }} & = {\mathbf{0}} \hfill \\ {\mathbf{F}}_{{{\varphi }_{{w}} }} & = {\mathbf{0}} \hfill \\ \end{aligned} $$(50)
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Li, Cf., Zhang, Gb., Liu, Xw. et al. Nonlinear dynamics analysis of pre-twisted blade with dovetail joint based on a two-dimensional micro-slip friction model. Nonlinear Dyn 111, 18837–18859 (2023). https://doi.org/10.1007/s11071-023-08825-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-023-08825-9