# An analytical calculation of the Jacobian matrix for 3D friction contact model applied to turbine blade shroud contact

M. Afzal, I. Lopez Arteaga, L. Kari

15 Citaties (Scopus)

### Uittreksel

An analytical expression is formulated to compute the Jacobian matrix for 3D friction contact modeling that efficiently evaluates the matrix while computing the friction contact forces in the time domain by means of the alternate frequency time domain approach. The developed expression is successfully used for the calculation of the friction damping on a turbine blade with shroud contact interface having an arbitrary 3D relative displacement. The analytical expression drastically reduces the computation time of the Jacobian matrix with respect to the classical finite difference method, with many points at the contact interface. Therefore, it also significantly reduces the overall computation time for the solution of the equations of motion, since the formulation of the Jacobian matrix is the most time consuming step in solving the large set of nonlinear algebraic equations when a finite difference approach is employed. The equations of motion are formulated in the frequency domain using the multiharmonic balance method to accurately capture the nonlinear contact forces and displacements. Moreover, the equations of motion of the full turbine blade model are reduced to a single sector model by exploiting the concept of cyclic symmetry boundary condition for a periodic structure. Implementation of the developed scheme in solving the equations of motion is proved to be effective and significant reduction in time is achieved without loss of accuracy.
Originele taal-2 Engels 204-217 Computers and Structures 177 https://doi.org/10.1016/j.compstruc.2016.08.014 Gepubliceerd - 2016

### Vingerafdruk

Jacobian matrices
Jacobian matrix
Equations of motion
Equations of Motion
Friction
Turbines
Contact
Contact Force
Frequency Domain
Time Domain
Finite Difference
Nonlinear Algebraic Equations
Periodic structures
Periodic Structures
Nonlinear equations
Finite difference method
Large Set
Model

### Citeer dit

title = "An analytical calculation of the Jacobian matrix for 3D friction contact model applied to turbine blade shroud contact",
abstract = "An analytical expression is formulated to compute the Jacobian matrix for 3D friction contact modeling that efficiently evaluates the matrix while computing the friction contact forces in the time domain by means of the alternate frequency time domain approach. The developed expression is successfully used for the calculation of the friction damping on a turbine blade with shroud contact interface having an arbitrary 3D relative displacement. The analytical expression drastically reduces the computation time of the Jacobian matrix with respect to the classical finite difference method, with many points at the contact interface. Therefore, it also significantly reduces the overall computation time for the solution of the equations of motion, since the formulation of the Jacobian matrix is the most time consuming step in solving the large set of nonlinear algebraic equations when a finite difference approach is employed. The equations of motion are formulated in the frequency domain using the multiharmonic balance method to accurately capture the nonlinear contact forces and displacements. Moreover, the equations of motion of the full turbine blade model are reduced to a single sector model by exploiting the concept of cyclic symmetry boundary condition for a periodic structure. Implementation of the developed scheme in solving the equations of motion is proved to be effective and significant reduction in time is achieved without loss of accuracy.",
author = "M. Afzal and {Lopez Arteaga}, I. and L. Kari",
year = "2016",
doi = "10.1016/j.compstruc.2016.08.014",
language = "English",
volume = "177",
pages = "204--217",
journal = "Computers and Structures",
issn = "0045-7949",
publisher = "Elsevier",

}

In: Computers and Structures, Vol. 177, 2016, blz. 204-217.

TY - JOUR

T1 - An analytical calculation of the Jacobian matrix for 3D friction contact model applied to turbine blade shroud contact

AU - Afzal, M.

AU - Lopez Arteaga, I.

AU - Kari, L.

PY - 2016

Y1 - 2016

N2 - An analytical expression is formulated to compute the Jacobian matrix for 3D friction contact modeling that efficiently evaluates the matrix while computing the friction contact forces in the time domain by means of the alternate frequency time domain approach. The developed expression is successfully used for the calculation of the friction damping on a turbine blade with shroud contact interface having an arbitrary 3D relative displacement. The analytical expression drastically reduces the computation time of the Jacobian matrix with respect to the classical finite difference method, with many points at the contact interface. Therefore, it also significantly reduces the overall computation time for the solution of the equations of motion, since the formulation of the Jacobian matrix is the most time consuming step in solving the large set of nonlinear algebraic equations when a finite difference approach is employed. The equations of motion are formulated in the frequency domain using the multiharmonic balance method to accurately capture the nonlinear contact forces and displacements. Moreover, the equations of motion of the full turbine blade model are reduced to a single sector model by exploiting the concept of cyclic symmetry boundary condition for a periodic structure. Implementation of the developed scheme in solving the equations of motion is proved to be effective and significant reduction in time is achieved without loss of accuracy.

AB - An analytical expression is formulated to compute the Jacobian matrix for 3D friction contact modeling that efficiently evaluates the matrix while computing the friction contact forces in the time domain by means of the alternate frequency time domain approach. The developed expression is successfully used for the calculation of the friction damping on a turbine blade with shroud contact interface having an arbitrary 3D relative displacement. The analytical expression drastically reduces the computation time of the Jacobian matrix with respect to the classical finite difference method, with many points at the contact interface. Therefore, it also significantly reduces the overall computation time for the solution of the equations of motion, since the formulation of the Jacobian matrix is the most time consuming step in solving the large set of nonlinear algebraic equations when a finite difference approach is employed. The equations of motion are formulated in the frequency domain using the multiharmonic balance method to accurately capture the nonlinear contact forces and displacements. Moreover, the equations of motion of the full turbine blade model are reduced to a single sector model by exploiting the concept of cyclic symmetry boundary condition for a periodic structure. Implementation of the developed scheme in solving the equations of motion is proved to be effective and significant reduction in time is achieved without loss of accuracy.

U2 - 10.1016/j.compstruc.2016.08.014

DO - 10.1016/j.compstruc.2016.08.014

M3 - Article

VL - 177

SP - 204

EP - 217

JO - Computers and Structures

JF - Computers and Structures

SN - 0045-7949

ER -