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Vibration Prediction of Bladed Disks Coupled by Friction Joints

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Abstract

The present review article addresses the vibration behavior of bladed disks encountered e.g. in aircraft engines as well as industrial gas and steam turbines. The utilization of the dissipative effects of dry friction in mechanical joints is a common means of the passive mitigation of structural vibrations caused by aeroelastic excitation mechanisms. The prediction of the vibration behavior is a scientific challenge due to (a) the strongly nonlinear contact interactions involving local sticking, sliding and liftoff, (b) the model order required to accurately describe the dynamic behavior of the assembly, and (c) the multi-disciplinary character of the problem associated with the need to account for structural mechanical as well as fluid dynamical effects. The purpose of this article is the overview and discussion the current state of the art of vibration prediction approaches. The modeling approaches in this work embrace the description of the rotating bladed disk, the contact modeling, the consideration of aeroelastic effects, appropriate model order reduction techniques and the exploitation of the rotationally periodic nature of the problem. The simulation approaches cover the direct computation of periodic, steady-state externally forced and self-excited vibrations using the high-order harmonic balance method, the formulation of the contact problem in the frequency domain, methods for the solution of the governing algebraic equations and advanced simulation approaches, including the concept of nonlinear modes.

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Notes

  1. Note that the actual values of \({}^{\,(n)}_{{\mathrm{fe}}}{\varvec{f}}_{{\mathrm{c}}}\) and \({}^{\,(n)}_{{\mathrm{fe}}}{\varvec{f}}_{{\mathrm{a}}}\) still depend on the independent displacement vectors \({}^{\,(n)}_{{\mathrm{fe}}}{\varvec{u}}\) and are, of course, generally not equal to each other.

  2. Hence, the part \({}_{\,\mathrm{fe}}{\varvec{K}}\) related to elastic forces is sometimes referred to as ‘tangent’ stiffness matrix.

  3. Sometimes also referred to as cyclic or nodal diameter coordinates.

  4. The actual Mortar method does not only involve the discretization by means of contact segments, but is commonly associated with an augmented Lagrangian formulation of the contact laws. Here, ‘Mortar-like’ only refers to the discretization using contact segments.

  5. \({\varvec{u}}\) is a vector of generalized displacements of dimension \(n_{{\mathrm{d}}}\times 1\). In particular, it can stand for \({}_{\,\mathrm{fe}}{\varvec{u}}\) or \({}_{\,\mathrm{tw}}{\varvec{u}}\).

  6. This coordinate transform may also involve reduced descriptions of subdomains in terms of component modes, see Sect. 2.3, in which case \({\varvec{u}}\) refers to generalized coordinates instead of nodal coordinates. This also includes the special case where one or both of the contacting surfaces belongs to a rigid body, where \({\varvec{B}}\) takes into account rigid body kinematics.

  7. The actual reduction depends on the discretizaton. In accordance with the notation introduced in the previous subsection, for the variant (a), the number of coupling coordinates equals the number of rows of \({\varvec{B}}\) containing non-zero elements. For variant (b), the number of coupling coordinates is \(3n_{{\mathrm{c}}}\), that is, the dimension of the vector of contact gaps \({}_{\,\mathrm{c}}{\varvec{g}}\).

  8. For more information and an illustration of such traveling waves, see ‘Appendix 1’.

  9. Note that one approach to formally derive Eq. (17) is the principle of virtual work.

  10. Since this matrix is formulated in the modal space, it is often referred to as modal aerodynamical influence matrix, and its entries are referred to as modal aerodynamical influence coefficients.

  11. Note that for assessing the aeroelastic stability, an alternative to the energy method is to carry out a fluid-structure simulation of the whole annulus, starting from an initial perturbation, and analyzing whether the vibrations grow or decay [166].

  12. As discussed in Sect. 2.1.2, this is the case if the sector-to-sector deviations of (geometrical, material and contact) properties are sufficiently small or the inter-sector coupling is sufficiently strong.

  13. For enlightening illustrations of this rule, the reader is referred to [114].

  14. It appears to be a common belief that the equations of motion can be solved exactly if piecewise linear contact laws are considered. Indeed, the set of ordinary differential equations becomes piecewise integrable. However, the transition times between the different contact states (stick, slip, liftoff) are generally not a priori known and need to be determined from the transition conditions. The latter are usually transcendental equations in the unknown transition times, rendering an exact solution impossible.

  15. In the literature, other widely used names for the method described here are the ‘Describing Function method’ and the ‘Krylov–Bogoliubov–Mitropolsky method’. Moreover, the prefixes ‘multi’ or ‘high-order’ are often used for the harmonic balance method in order to clarify the difference to the single-term variant which only considers the fundamental harmonic.

  16. Hence, this procedure is considered a pure frequency-domain method, since there is no need to switch to the time domain, in contrast to the alternating frequency–time scheme presented in the following paragraph.

  17. cf. discussion in paragraph ‘Tangential contact’ in Sect. 2.2.3.

  18. Note that the normal preload is sometimes also referred to as initial normal load, which emphasizes that the actual normal load may change due to vibrations. In fact, even the static component (or average value) of the normal load is generally influenced by the dynamic contact interactions.

  19. The task of finding such an appropriate guess is addressed in Sect. 3.4.4.

  20. In the literature, the name ‘Newton–Raphson’ method is also commonly used for the method described here.

  21. This method is sometimes referred to as the Riks method.

  22. In fact, in the case of the horizontal tangent condition, it cannot even be ensured that the traced point is a maximum, but a local minimum or a saddle node might be traced ‘by mistake’.

  23. In literature, the term Nonlinear Normal Mode (NNM) is quite common. However, the term ‘normal’ may lead to the wrong conclusion that nonlinear modes are orthogonal to each other. Apparently this term goes back to Rosenberg [137], who defined nonlinear modes as vibrations in unison, i. e. where all material points cross their equilibrium points and their extremum points simultaneously. For this type of vibration, the motions take place on so-called modal curves in the generalized displacement space which are normal to the surface of maximum potential energy [167]. However, this property is only valid for symmetric conservative systems, whereas non-trivial phase lags among the oscillations of the coordinates may exist in general. Hence the term ‘normal’ in this context is avoided in this article.

  24. In the literature, this number is also referred to as ‘harmonic index’. To avoid confusion with (temporal) harmonics in the context of frequency domain methods, this terms is avoided in this work. Moreover, the term ‘nodal diameter number’ is also common for this number.

  25. Here, ‘left’ and ‘right’ is meant with respect to the rotation axis (and, consequently, the numbering of the sectors).

  26. Note that \({\varvec{B}}^{\mathrm{T}}{\varvec{u}}= \left[ {\begin{array}{cc} {\varvec{B}}^{\mathrm{T}}\left( {\varvec{B}}^{\mathrm{T}}\right) ^{+}&{} {\varvec{B}}^{\mathrm{T}}{\varvec{N}}_{{\varvec{B}}^{\mathrm{T}}}\\ \end{array}}\right] \left[ {\begin{array}{c} {}_{\,\mathrm{c}}{\varvec{g}}\\ {\varvec{u}}_{{\mathrm{rem}}}\\ \end{array}}\right] = {}_{\,\mathrm{c}}{\varvec{g}}\), in full accordance with Eq. (6).

Abbreviations

\(N_0\) :

Initial normal load

\(g_{{\mathrm{n}},0}\) :

Initial normal gap

\({\mathcal{H}}\) :

Set of (temporal) harmonics

\(m\) :

Engine order

\(m_0\) :

Fundamental engine order

\({\mathcal{M}}\) :

Set of relevant engine orders

\(\epsilon _{{\mathrm{DL}}}\) :

Dynamic Lagrangian penalty coefficient

\(\varOmega _{{\mathrm{rot}}}\) :

Rotational speed

\(\theta\) :

Inter-blade phase angle

\({\varvec{f}}_{{\mathrm{a}}}\) :

Aerodynamical forces

\({\varvec{f}}_{{\mathrm{ae}}},{\varvec{F}}_{{\mathrm{ae}}}\) :

Aerodynamical external forces (time domain, frequency domain)

\({\varvec{f}}_{{\mathrm{ai}}}\) :

Aerodynamical interaction forces

\({\varvec{f}}_{{\mathrm{c}}},{\varvec{F}}_{{\mathrm{c}}}\) :

Global contact forces (time domain, frequency domain)

\({\varvec{g}}\) :

Contact gaps

\({\varvec{\lambda }},{\varvec{\varLambda }}\) :

Local contact forces (time domain, frequency domain)

\({\varvec{u}},{\varvec{U}}\) :

Vector of (generalized) coordinates (time domain, frequency domain)

\({\varvec{p}}\) :

Pressure

\({\varvec{B}}\) :

Interface coupling matrix

\({\varvec{D}}\) :

Damping matrix

\({\varvec{G}}_{{\mathrm{ai}}}\) :

Aerodynamic influence matrix

\({\varvec{H}}\) :

Dynamic compliance matrix

\({\varvec{I}}\) :

Identity matrix

\({\varvec{K}}\) :

Matrix of velocity proportional forces

\({\varvec{M}}\) :

Mass matrix

\({\varvec{S}}\) :

Dynamic stiffness matrix

\({\varvec{T}}\) :

Matrix of component modes

\({\varvec{W}}_{n_{{\mathrm{s}}}}\) :

Discrete Fourier matrix for \(n_{{\mathrm{s}}}\) samples

\({\varvec{\nabla }}\) :

Frequency domain derivative matrix

\(n_{{\mathrm{c}}}\) :

Number of contact points

\(n_{{\mathrm{d}}}\) :

Number of (generalized) coordinates

\(n_{{\mathrm{fe}}}\) :

Number of finite element nodal degrees of freedom

\(n_{{\mathrm{fe,s}}}\) :

… per sector

\(n_{{\mathrm{if}}}\) :

Number of interfaces

\(n_{{\mathrm{r}}}\) :

Number of component modes

\(n_{{\mathrm{s}}}\) :

Number of sectors

\((\,)_{{\mathrm{n}}}\) :

Associated to the normal direction

\((\,)_{{\mathrm{t}}}\) :

Associated to the tangential direction

\({}^{(n)}(\,)\) :

Associated to sector n

\({}_{{\mathcal{C}}}(\,)\) :

In the coordinates of the continuous contact interface

\({}_{{\mathrm{c}}}(\,)\) :

In the coordinates of the discrete contact interface

\({}_{{\mathrm{fe}}}(\,)\) :

In the physical degrees of freedom of the finite element model

\({}_{\,\mathrm{r}}(\,)\) :

In the generalized coordinates of the component modes

\({}_{\,\mathrm{tw}}(\,)\) :

In traveling wave coordinates

\({(\,)}^*\) :

Complex conjugate

\({\mathfrak {I}}\left\{ (\,) \right\}\) :

Imaginary part

\({\mathfrak {R}}\left\{ (\,) \right\}\) :

Real part

\((\,)^{+}\) :

Pseudo inverse

\((\,)^{\mathrm{H}}\) :

Hermitian transpose

\((\,)^{\mathrm{T}}\) :

Transpose

\({\varvec{N}}_{\varvec{A}}\) :

Null space of matrix \({\varvec{A}}\)

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Authors and Affiliations

  1. Institute of Aircraft Propulsion Systems, University of Stuttgart, Pfaffenwaldring 6, 70569, Stuttgart, Germany

    Malte Krack

  2. Department of Mechanical Engineering, Vibration University Technology Centre, Imperial College London, Exhibition Road, London, SW7 2AZ, UK

    Loic Salles

  3. Laboratoire de Tribologie et Dynamique des Systmes, École Centrale de Lyon, 36 Avenue Guy de Collongue, 69134, Ecully Cedex, France

    Fabrice Thouverez

Authors
  1. Malte Krack
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  2. Loic Salles
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Correspondence to Malte Krack.

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Appendices

Appendix 1: The Traveling Wave Coordinate System

In this appendix, we define the traveling wave coordinate system and illustrate the notion of traveling waves. The traveling wave coordinates are related to the physical coordinates by the (inverse) discrete Fourier transform. The transformation can be applied to any physical quantity (displacement, force, etc.). For the displacement, this transform reads

$${}_{\,\mathrm{tw}}{\varvec{u}}_j(t)= \frac{1}{\sqrt{n_{{\mathrm{s}}}}}\sum \limits _{n=0}^{n_{{\mathrm{s}}}-1}{\mathrm{e}}^{-{\mathrm{i}}\frac{2\pi j n}{n_{{\mathrm{s}}}}}\,{}^{\,(n)}_{{\mathrm{fe}}}{\varvec{u}}(t),\quad j\in \left[ 0,n_{{\mathrm{s}}}-1\right] ,$$
(39)
$${}^{\,(n)}_{{\mathrm{fe}}}{\varvec{u}}(t)= \frac{1}{\sqrt{n_{{\mathrm{s}}}}}\sum \limits _{j=0}^{n_{{\mathrm{s}}}-1}{\mathrm{e}}^{{\mathrm{i}}\frac{2\pi j n}{n_{{\mathrm{s}}}}}{}_{\,\mathrm{tw}}{\varvec{u}}_j(t),\quad n\in \left[ 0,n_{{\mathrm{s}}}-1\right] ,$$
(40)
$$\begin{aligned} {}_{\,\mathrm{tw}}{\varvec{u}}(t) &= \left( {\varvec{W}}_{n_{{\mathrm{s}}}}^{\mathrm{H}}\otimes {\varvec{I}}_{n_{{\mathrm{fe,s}}}}\right) {}_{\,\mathrm{fe}}{\varvec{u}}(t)\,\, \Leftrightarrow \nonumber \\ {}_{\,\mathrm{fe}}{\varvec{u}}(t) &= \left( {\varvec{W}}_{n_{{\mathrm{s}}}}\otimes {\varvec{I}}_{n_{{\mathrm{fe,s}}}}\right) {}_{\,\mathrm{tw}}{\varvec{u}}(t). \end{aligned}$$
(41)

Herein, \({}_{\,\mathrm{tw}}{\varvec{u}}_j\) denotes the displacement vector associated with the (spatial) wave numberFootnote 24 j, and \({}^{\mathrm{H}}\) denotes the Hermitian transpose. A congruent discretization and ordering is here assumed for each sector, such that the physical displacement vector \({}^{\,(n)}_{{\mathrm{fe}}}{\varvec{u}}\) comprises the same number of degrees of freedom, \(n_{{\mathrm{fe}},n}=n_{{\mathrm{fe,s}}}\) for each sector \(n\in \left[ 0,n_{{\mathrm{s}}}-1\right]\). The vector \({}_{\,\mathrm{tw}}{\varvec{u}}_j\) has the same number of coordinates as the physical displacement vector \({}^{\,(n)}_{{\mathrm{fe}}}{\varvec{u}}\). The more compact notation in Eq. (41) involves the Fourier matrix \({\varvec{W}}_{n_{{\mathrm{s}}}}\), defined in Eq. (47).

Remark

At this point, it is important to note that both \({}_{\,\mathrm{fe}}{\varvec{u}}(t)\) and \({}_{\,\mathrm{tw}}{\varvec{u}}(t)\) are complex-valued quantities in general. The complex arithmetic is very convenient for the mathematical derivations. Of course, eventually we are interested only in the physical part, that is, the real component.

Fig. 17
figure 17

Illustration of the spatiotemporal nature of a traveling wave: a \(q\) as a function of the sector number and the temporal phase similar to [114], b temporal phase as a bijective function of time

Full size image

To illustrate the traveling wave character of the coordinate system, regard the k-th wave component, \({}_{\,\mathrm{tw}}{\varvec{q}}_k\), of a physical quantity \({\varvec{q}}\) (displacement, force, etc.), and consider the case of an oscillation with

$${}_{\,\mathrm{tw}}{\varvec{q}}_k(t) = {\sqrt{n_{{\mathrm{s}}}}}{\varvec{Q}}{\mathrm{e}}^{{\mathrm{i}}\phi (t)}.$$
(42)

Herein, \({\varvec{Q}}\in {\mathbb {C}}^{n_{{\mathrm{fe,s}}}}\) is a complex-valued amplitude vector and \(\phi (t)\) is the phase, which is assumed to be strictly monotonous in time with \(\dot{\phi }(t)>0\) in the considered time span, see Fig. 17b. As a consequence, the relation between phase and time is bijective. All other wave components are assumed to be zero, \({}_{\,\mathrm{tw}}{\varvec{q}}_j={\mathbf{0}}\quad\forall j\in \left[ 0,n_{{\mathrm{s}}}-1\right] \backslash k\). Taking into account the transform defined in Eqs. (39)–(41), the response of sector n reads

$$\begin{aligned} {}^{(n)}{\varvec{q}}\left( \phi \left( t\right) \right) &= \frac{1}{{\sqrt{n_{{\mathrm{s}}}}}}{\mathrm{e}}^{{\mathrm{i}}\frac{2\pi kn}{n_{{\mathrm{s}}}}}{}_{\,\mathrm{tw}}{\varvec{q}}_k\left( \phi \left( t\right) \right) \nonumber \\ &={ \varvec{Q}}{\mathrm{e}}^{{\mathrm{i}}\left( \phi (t)+\frac{2\pi kn}{n_{{\mathrm{s}}}}\right) }={}^{(0)}q\left( \phi (t)+n\theta _k\right) .\end{aligned}$$
(43)

Herein, the abbreviation \(\theta _k\) is used, with

$$\theta _k=\frac{2\pi }{n_{{\mathrm{s}}}}k.$$
(44)

\(\theta _k\) is the phase lag between neighboring sectors and is hence often referred to as inter-blade phase angle (IBPA). Since the relation between phase and time is assumed as bijective, Eq. (43) defines a unique time lag \(\varDelta t\) with \({}^{(n)}{\varvec{q}}(t)={}^{(0)}q(t+\varDelta t)\). Moreover, \({\varvec{q}}(t)\) is spatially periodic; i.e.,\({}^{(n+n_{{\mathrm{s}}})}{\varvec{q}}(t) = {}^{(n)}{\varvec{q}}(t)\), which can be easily verified from Eq. (43). Hence, the spatiotemporal form of the quantity \({\varvec{q}}(t)\) can be identified as a traveling wave, discrete in space and continuous in time, as illustrated in Fig. 17a. Since \(\dot{\phi }\) is allowed to be time-dependent, the time lag \(\varDelta t\) generally varies with time t and sector number n. This means that the wave does not have to propagate with constant speed. However, the case of constant wave speed is of particular practical relevance, which coincides with a harmonic oscillation of constant angular frequency \(\varOmega , \dot{\phi }(t)=\varOmega\). In this case, the constant wave speed is \(\varOmega /k\) (in \({\hbox {rad}}\,\hbox {s}^{-1}\)).

Table 3 Correspondence between wave number k, IBPA \(\theta _k\), nature of the apparent wave form and the short name (FTW: forward traveling wave, BTW: backward traveling wave, SW: standing wave, ND: nodal diameter)
Full size table

It should be noted that Eq. (43) defines a strict backward traveling wave. However, the traveling wave nature is seen to alias relative to \(n_{{\mathrm{s}}}\), depending on the wave number k. The ranges of k and \(\theta _k\) that correspond to apparent forward and backward traveling waves (FTW and BTW, respectively), and the special cases of standing waves (SW), are given in Table 3. Herein, \(s^-_{n_{{\mathrm{s}}}}\) and \(s^+_{n_{{\mathrm{s}}}}\) depend on \(n_{{\mathrm{s}}}\), with

$$s^-_{n_{{\mathrm{s}}}}= {\left\{ \begin{array}{ll} \frac{n_{{\mathrm{s}}}}{2}-1 &{} n_{{\mathrm{s}}}\,\,{\mathrm{even}}\\ \frac{n_{{\mathrm{s}}}-1}{2} &{} n_{{\mathrm{s}}}\,\,{\mathrm{odd}} \end{array}\right. }$$
(45)
$$s^+_{n_{{\mathrm{s}}}}= {\left\{ \begin{array}{ll} \frac{n_{{\mathrm{s}}}}{2}+1 &{} n_{{\mathrm{s}}}\,\,{\mathrm{even}}\\ \frac{n_{{\mathrm{s}}}+1}{2} &{} n_{{\mathrm{s}}}\,\,{\mathrm{odd}} \end{array}\right. }$$
(46)

In this sense, the columns of the Fourier matrix \({\varvec{W}}_{n_{{\mathrm{s}}}}\) in Eq. (41) can be interpreted as discrete unit traveling waves, such that \({}_{\,\mathrm{tw}}{\varvec{u}}_k\) corresponds to a wave form with wave number k and IBPA \(\theta _k\).

Remark

It should be emphasized that wave numbers and nodal diameter numbers are only illustrative expressions for strictly mathematical concepts. The physical number of waves or nodal diameters can deviate from the mathematical one. Consider the example of a rotationally symmetric disk. We can divide the disk into a finite number of \(n_{{\mathrm{s}}}\) sectors. The highest possible wave number is then bounded by \(n_{{\mathrm{s}}}-1\) in accordance with our definition. But, of course, the disk can carry an infinite number of waves. The higher wave forms are generally not lost by the dissection into a finite number of sectors, but represent higher modes of vibration for a specific mathematical wave number.

In the above considerations, the complex-valued amplitude vector \({\varvec{Q}}\) is assumed to be constant in time. If \({\varvec{Q}}(t)\) depends on time, the strict relation \({}^{(n)}{\varvec{q}}(t)={}^{(0)}q(t+\varDelta t)\) between time and sector number is no longer satisfied. However, it still holds that \({}^{(n)}{\varvec{q}}(t) = {}^{(0)}q(t){\mathrm{e}}^{{\mathrm{i}}n\theta _k}\); that is, there is a constant phase lag between the individual sectors for a given k. The corresponding motion for time-dependent \({\varvec{Q}}(t)\) can thus be interpreted as a pseudo-traveling wave. This notion can be useful to describe vibration phenomena during run-up or run-down of a rotating machine. In this case, the oscillation frequency and the amplitudes vary with time, but excitation and vibration response might still exhibit a characteristic traveling wave form. For instance, in the case of constant acceleration, the phase \(\phi\) would be defined as \({\ddot{\phi }}(t)=\alpha\).

The discrete Fourier matrix \({\varvec{W}}_{n_{{\mathrm{s}}}}\) for a number of \(n_{{\mathrm{s}}}\) sectors (or samples in general) is defined as,

$${\varvec{W}}_{n_{{\mathrm{s}}}}= \left[ W_{ab}\right] = \left[ {\begin{array}{ccc} {\varvec{w}}_{1}&{} \cdots &{} {\varvec{w}}_{n_{{\mathrm{s}}}}\\ \end{array}}\right] ,\,\,\text {with}\, W_{ab} = w_{n_{{\mathrm{s}}}}^{\left( a-1\right) \left( b-1\right) }.$$
(47)

where \(w_{n_{{\mathrm{s}}}} = {\mathrm{e}}^{{\mathrm{i}}\frac{2\pi }{n_{{\mathrm{s}}}}}\) is the \(n_{{\mathrm{s}}}\)-th root of unity.

Appendix 2: Traveling Wave Structural Matrices

In this appendix, we describe how the structural matrices in traveling wave coordinates \({}_{\,\mathrm{tw}}{\varvec{K}}_k, {}_{\,\mathrm{tw}}{\varvec{D}}_k\) and \({}_{\,\mathrm{tw}}{\varvec{M}}_k\) can be obtained from the structural matrices of a reference sector. Suppose that a finite element model of the reference sector is given. The sector spans an angular region of \(2\pi /n_{{\mathrm{s}}}\), and can be divided into an inner volume, and left and right boundary.Footnote 25 Typically, the finite element model spans the whole sector, including left and right boundaries, making the description of the reference sector somewhat redundant. The sector’s displacement vector in physical coordinates can be permuted and partitioned as \(\left[ {\begin{array}{ccc} {\varvec{u}}_{{\mathrm{l}}}^{\mathrm{T}}&{} {\varvec{u}}_{{\mathrm{i}}}&{} {\varvec{u}}_{{\mathrm{r}}}^{\mathrm{T}}\\ \end{array}}\right] ^{\mathrm{T}}\), where \({\varvec{u}}_{{\mathrm{l}}}\) and \({\varvec{u}}_{{\mathrm{r}}}\) are degrees of freedom associated with left and right boundaries and \({\varvec{u}}_{{\mathrm{i}}}\) are the inner degrees of freedom. \({\varvec{u}}_{{\mathrm{l}}}\) and \({\varvec{u}}_{{\mathrm{r}}}\) shall have the dimension \(n_{{\mathrm{b}}}\) and \({\varvec{u}}_{{\mathrm{i}}}\) shall have the dimension \(n_{{\mathrm{i}}}\). Because of the redundancy, the total number of degrees of freedom of the sector (without constraints on left and right boundaries) is \(2n_{{\mathrm{b}}}+n_{{\mathrm{i}}}\) and exceeds the number \(n_{{\mathrm{fe,s}}}=n_{{\mathrm{fe}}}/n_{{\mathrm{s}}}\) of degrees of freedom per sector (of the full model) by the number of degrees of freedom of one boundary, \(n_{{\mathrm{b}}}\). The accordingly ordered structural matrices (in physical coordinates) without any constraints on left and right boundaries are denoted \({}_{\,\mathrm{fe}}^{(0)}{\varvec{K}}, {}_{\,\mathrm{fe}}^{(0)}{\varvec{D}}\) and \({}_{\,\mathrm{fe}}^{(0)}{\varvec{M}}\), and they take the form

$${\varvec{A}}= \left[ {\begin{array}{ccc} {\varvec{A}}_{{\mathrm{ll}}}&{} {\varvec{A}}_{{\mathrm{li}}}&{} {\mathbf{0}}\\ {\varvec{A}}_{{\mathrm{li}}}^{\mathrm{T}}&{} {\varvec{A}}_{{\mathrm{ii}}}&{}{\varvec{A}}_{{\mathrm{ri}}}^{\mathrm{T}}\\ {\mathbf{0}} &{} {\varvec{A}}_{{\mathrm{ri}}}&{} {\varvec{A}}_{{\mathrm{rr}}}\\ \end{array}}\right] ,\,\quad{\varvec{A}}\in \lbrace {}_{\,\mathrm{fe}}^{(0)}{\varvec{K}}, {}_{\,\mathrm{fe}}^{(0)}{\varvec{D}}, {}_{\,\mathrm{fe}}^{(0)}{\varvec{M}}\rbrace ,$$
(48)

where \({\varvec{A}}_{{\mathrm{ll}}}, {\varvec{A}}_{{\mathrm{rr}}}\) and \({\varvec{A}}_{{\mathrm{ii}}}\) account for the coupling within each boundary and inner volume, and the matrices \({\varvec{A}}_{{\mathrm{li}}}\) and \({\varvec{A}}_{{\mathrm{ri}}}\) account for the coupling between boundaries and inner volume. It is here assumed that left and right boundaries are disjunct, so that no coupling exists between them.

The matrices \({}_{\,\mathrm{tw}}{\varvec{A}}_k\) for each IBPA \(\theta _k\) can be obtained as [5],

$${}_{\,\mathrm{tw}}{\varvec{A}}_k= {\varvec{P}}_{k}^{\mathrm{H}}{}_{\,\mathrm{fe}}^{(0)}{\varvec{A}}{\varvec{P}}_{k},\,\quad{\varvec{A}}\in \lbrace {\varvec{K}}, {\varvec{D}}, {\varvec{M}}\rbrace ,$$
(49)

with the matrix \({\varvec{P}}_{k}\)

$${\varvec{P}}_{k}= \left[ {\begin{array}{cc} {\varvec{I}}_{n_{{\mathrm{b}}}} {\mathrm{e}}^{{\mathrm{i}}\theta _k} &{} {\mathbf{0}}\\ {\mathbf{0}} &{} {\varvec{I}}_{n_{{\mathrm{i}}}}\\ {\varvec{I}}_{n_{{\mathrm{b}}}} &{} {\mathbf{0}}\\ \end{array}}\right] .$$
(50)

The resulting matrices \({}_{\,\mathrm{tw}}{\varvec{A}}_k\) then have the proper size \(n_{{\mathrm{fe,s}}}\times n_{{\mathrm{fe,s}}}\).

It is assumed in the above formulations that left and right boundary have matching nodes and the local coordinate systems are conform after rotation by the sector angle. If this is not the case, Eq. (50) has to be adjusted by accounting for the coupling of non-conforming meshes and coordinate transformation. Note that the formal relation \({}_{\,\mathrm{tw}}{\overline{{\varvec{A}}}}_k= \left( {\varvec{W}}_{n_{{\mathrm{s}}}}^{\mathrm{H}}\otimes {\varvec{I}}_{n_{{\mathrm{fe,s}}}}\right) {\overline{\varvec{A}}}\left( {\varvec{W}}_{n_{{\mathrm{s}}}}\otimes {\varvec{I}}_{n_{{\mathrm{fe,s}}}}\right)\) could also be utilized to obtain these matrices. However, the method described in this appendix is much more efficient, since it involves only a single sector.

Appendix 3: Transformation to Relative Coordinates at the Contact Interface

In this appendix, we describe how the transformation to relative coordinates discussed in Sect. 2.3.1 is applied. To this end, the global displacement vector \({\varvec{u}}\) is expressed in terms of relative coordinates \({}_{\,\mathrm{c}}{\varvec{g}}\) at the contact interface and remaining coordinates \({\varvec{u}}_{{\mathrm{rem}}}\),

$${\varvec{u}}=\left[ {\begin{array}{cc} \left( {\varvec{B}}^{\mathrm{T}}\right) ^{+}&{} {\varvec{N}}_{{\varvec{B}}^{\mathrm{T}}}\\ \end{array}}\right] \left[ {\begin{array}{c} {}_{\,\mathrm{c}}{\varvec{g}}\\ {\varvec{u}}_{{\mathrm{rem}}}\\ \end{array}}\right] = {\varvec{L}}{\varvec{u}}^{(b)}.$$
(51)

Herein, \({}^{+}\) denotes the pseudo-inverse, and \({\varvec{N}}_{{\varvec{A}}}= {\text {Null}}\left( {\varvec{A}}\right)\) denotes the nullspace of matrix \({\varvec{A}}\), and (b) refers to the variant (b) for the definition of the coupling coordinates, as introduced in Sect. 2.3.1.Footnote 26

In general, the transformation matrix \({\varvec{L}}\) could be expensive to compute, since it involves the computation of the pseudo-inverse and the null space of the comparatively large matrix \({\varvec{B}}^{\mathrm{T}}\) (dimension \(n_{{\mathrm{d}}}\times 3n_{{\mathrm{c}}}\), where \(n_{{\mathrm{d}}}\) could be \(n_{{\mathrm{fe,s}}}\) or \(n_{{\mathrm{fe}}}\)). The computational cost can be considerably reduced by taking advantage of the local nature of the problem, i.e., by computing the sub-matrices of \({\varvec{L}}\) separately for each interface. For convenience, the coordinate vector \({\varvec{u}}\) is rearranged in such a way that the nodal DOFs associated with a particular interface are grouped together. The matrix \({\varvec{B}}\) then takes the form \({\varvec{B}}={\mathbf{bdiag}}\lbrace {\varvec{B}}_{1},\ldots ,{\varvec{B}}_{n_{{\mathrm{if}}}}\rbrace\) where \({\varvec{B}}_{n}\) is the local coupling matrix of interface n and \(n_{{\mathrm{if}}}\) is the number of interfaces. The \(n_{{\mathrm{int}}}\) interior DOFs are not associated with any of the interfaces and form the rear part of the re-ordered vector \({\varvec{u}}\), where typically \(n_{{\mathrm{int}}}\gg 3n_{{\mathrm{c}}}\). The null space associated to the interior DOFs is trivial and does not have to be computed explicitly. The matrix \({\varvec{L}}\) can then be assembled as

$${\varvec{L}}= \left[ {\begin{array}{ccccccc} \left( {\varvec{B}}_1^{\mathrm{T}}\right) ^{+}&{} \cdots &{} {\mathbf{0}} &{} {\varvec{N}}_{{\varvec{B}}_1^{\mathrm{T}}}&{} \cdots &{} {\mathbf{0}}&{} {\mathbf{0}}\\ \vdots &{} \ddots &{} \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ {\mathbf{0}} &{} \cdots &{} \left( {\varvec{B}}_{n_{{\mathrm{if}}}}^{\mathrm{T}}\right) ^{+}&{} {\mathbf{0}} &{} \cdots &{} {\varvec{N}}_{{\varvec{B}}_{n_{{\mathrm{if}}}}^{\mathrm{T}}}&{} {\mathbf{0}}\\ {\mathbf{0}} &{} \cdots &{} {\mathbf{0}} &{} {\mathbf{0}} &{} \cdots &{} {\mathbf{0}}&{} {\varvec{I}}_{n_{{\mathrm{int}}}}\\ \end{array}}\right] .$$
(52)

The coordinate transform is applied by substituting Eq. (51) into the equations of motion and left-multiplication by \({\varvec{L}}^{\mathrm{T}}\). It should be noted that the matrix \({\varvec{L}}\) defined in Eq. (52) has full rank, and, thus, Eq. (51) defines an invertible coordinate transform.

Resulting structure of the contact force vector The structure of the contact force vector depends on the variant pursued for the definition of the coupling DOFs. If the conventional variant (a) is used, the contact force vector \({\varvec{f}}_{{\mathrm{c}}}\) takes the form,

$${\varvec{f}}_{{\mathrm{c}}}^{(a)}\left[ {\varvec{u}}\right] = {\varvec{B}}{\varvec{\lambda }}\left[ {\varvec{B}}^{\mathrm{T}}{\varvec{u}}\right] ,$$
(53)

where \({\varvec{\lambda}}[\cdot]\) represents the actual contact force law formulated in terms of the contact gaps (and/or velocities). The gaps are determined by means of the transform \({\varvec{B}}^{\mathrm{T}}{\varvec{u}}\), every time when the contact force vector \({\varvec{\lambda}}\) is evaluated, and a multiplication by \({\varvec{B}}\) is formally necessary to determine the force vector \({\varvec{f}}_{{\mathrm{c}}}\) acting on the global displacement vector \({\varvec{u}}\). In the case of variant (b), this transformation is applied, once and for all, during the dynamic substructuring procedure. The contact force vector thus becomes

$$\begin{aligned} {\varvec{f}}_{{\mathrm{c}}}^{(b)}\left[ {\varvec{u}}^{(b)}\right]&= {\varvec{L}}^{\mathrm{T}}{\varvec{f}}_{{\mathrm{c}}}^{(a)}\left[ {\varvec{L}}{\varvec{u}}\right] \nonumber \\ &= {\varvec{L}}^{\mathrm{T}}{\varvec{B}}{\varvec{\lambda }}\left[ {\varvec{B}}^{\mathrm{T}}{\varvec{L}}{\varvec{u}}^{(b)}\right] \nonumber \\ &= \left[ {\begin{array}{c} {\varvec{I}}_{3n_{{\mathrm{c}}}}\\ {\mathbf{0}}\\ \end{array}}\right] {\varvec{\lambda }}\left[ \left[ {\begin{array}{cc} {\varvec{I}}_{3n_{{\mathrm{c}}}} &{} {\mathbf{0}}\\ \end{array}}\right] {\varvec{u}}^{(b)}\right] \nonumber \\ &= \left[ {\begin{array}{c} {\varvec{\lambda }}\left[ {}_{\,\mathrm{c}}{\varvec{g}}\right] \\ {\mathbf{0}}\\ \end{array}}\right] .\end{aligned}$$
(54)

Owing to the preliminary coordinate transformation, the global contact vector depends and acts on only the first \(3n_{{\mathrm{c}}}\) components of the coordinate vector. Hence, no transformation is necessary during the nonlinear dynamic analysis.

Appendix 4: Craig-Bampton and MacNeal-Rubin Method

In this appendix, explicit expressions are given for the matrix \({\varvec{T}}\) of component modes for the well-known CB and MR methods, see e.g. [28]. Consider an initial model with a hermitian, positive-definite stiffness matrix \({\varvec{K}}={\varvec{K}}^{\mathrm{H}}>{\mathbf{0}}\) and a hermitian, positive-definite mass matrix. The associated vector of coordinates \({\varvec{u}}\) is of the form \({\varvec{u}}= \left[ {\begin{array}{cc} {\varvec{u}}_{{\mathrm{ret}}}^{\mathrm{T}}&{} {\varvec{u}}_{{\mathrm{del}}}^{\mathrm{T}}\\ \end{array}}\right] ^{\mathrm{T}}\) where \({\varvec{u}}_{{\mathrm{ret}}}\) and \({\varvec{u}}_{{\mathrm{del}}}\) denote the coordinates to be retained in the reduced model and those that are (deleted and) replaced by generalized coordinates, respectively.

In the case of the CB method, the reduction basis is spanned by constraint modes and a set of fixed interface normal modes,

$${\varvec{T}}_{{\mathrm{cb}}}= \left[ {\begin{array}{cc} {\varvec{I}}_{n_{{\mathrm{ret}}}} &{} {\mathbf{0}}_{n_{{\mathrm{ret}}}\times n_{{\mathrm{del}}}} \\ {\varvec{\varPsi }}&{} {\varvec{\varPhi }}^{{\mathrm{fixed}}}\\ \end{array}}\right] .$$
(55)

The first hyper-column represents the constraint modes, which are static deformation shapes for a unit displacement applied to one of the coupling DOFs, while the remaining coupling DOFs are kept fixed,

$${\varvec{K}}_{{\mathrm{del}},\mathrm{del}}{\varvec{\varPsi }}= -{\varvec{K}}_{{\mathrm{del}},\mathrm{ret}},$$
(56)

where \({\varvec{K}}_{{\mathrm{del}},\mathrm{ret}}\) refers to the restriction of \({\varvec{K}}\) to the rows associated with \({\varvec{u}}_{{\mathrm{del}}}\) and the columns associated with \({\varvec{u}}_{{\mathrm{ret}}}\) and so on. The fixed interface normal modes, assembled in the matrix \({\varvec{\varPhi }}^{{\mathrm{fixed}}}\) in Eq. (55), are obtained from modal analysis of the system with all coupling DOFs fixed,

$$\begin{aligned} {\varvec{\varPhi }}^{{\mathrm{fixed}}} &= \left[ {\begin{array}{c} {{\varvec{\phi} }^{{\mathrm{fixed}}}_1 \cdots {\varvec{\phi} }^{{\mathrm{fixed}}}_{n_{{\mathrm{mod}}}}} \end{array}}\right] ,\nonumber \\ &\quad \left( {\varvec{K}}_{{\mathrm{del}},\mathrm{del}} - {\omega ^{{\mathrm{fixed}}}_{j}}^2 {\varvec{M}}_{{\mathrm{del}},\mathrm{del}} \right) {\varvec{\phi }}^{{\mathrm{fixed}}}_{j} = {\mathbf{0}},\nonumber \\ &\quad\omega ^{{\mathrm{fixed}}}_{1}\le \ldots \le \omega ^{{\mathrm{fixed}}}_{n_{{\mathrm{mod}}}}.\end{aligned}$$
(57)

It should be emphasized that the coupling DOFs are either the nodal or the relative DOFs at the interface, as explained in Sect. 2.3.1.

The MR method is the complement of the CB method with free interface normal modes. In the case of the MR method, the reduction basis is, thus, spanned by the residual attachment modes and a set of free interface normal mode shapes \({\varvec{\varPhi }}^{{\mathrm{free}}}\),

$${\varvec{T}}_{{\mathrm{mr}}}= \left[ {\begin{array}{cc} {\varvec{I}}_{n_{{\mathrm{ret}}}} &{} {\mathbf{0}}_{n_{{\mathrm{ret}}}\times n_{{\mathrm{del}}}} \\ {\varvec{\varDelta }}&{} {\varvec{\varPhi }}^{{\mathrm{free}}}_{{\mathrm{del}}} - {\varvec{\varDelta }}{\varvec{\varPhi }}^{{\mathrm{free}}}_{{\mathrm{ret}}} \end{array}}\right] .$$
(58)

The residual attachment modes essentially represent the static deformation shapes for a unit force applied to one of the coupling DOFs, while the remaining DOFs are not loaded.

$$\begin{aligned} {\varvec{\varDelta }} &= {\varvec{R}}_{{\mathrm{del}}}{\varvec{R}}_{{\mathrm{ret}}}^{-1},\nonumber \\ \left[ {\begin{array}{c} {\varvec{R}}_{{\mathrm{ret}}}\\ {\varvec{R}}_{{\mathrm{del}}}\\ \end{array}}\right] &= \left[ {\begin{array}{c} \left( {\varvec{K}}^{-1}\right) _{{\mathrm{ret}},{\mathrm{ret}}}\\ \left( {\varvec{K}}^{-1}\right) _{{\mathrm{del}},\mathrm{ret}}\\ \end{array}}\right] -\sum \limits _{j=1}^{n_{{\mathrm{mod}}}} \left[ {\begin{array}{c} {\varvec{\varPhi }}^{{\mathrm{free}}}_{{\mathrm{ret}}}\\ {\varvec{\varPhi }}^{{\mathrm{free}}}_{{\mathrm{del}}}\\ \end{array}}\right] \frac{{{\varvec{\varPhi }}^{{\mathrm{free}}}_{{\mathrm{ret}}}}^{\mathrm{H}}}{\omega ^{{\mathrm{free}}}_j}.\end{aligned}$$
(59)

Herein, \({\varvec{R}}\) denotes the residual static flexibility associated with loading of the retained coordinates. The presence of rigid body modes requires special attention [28]; however, this case is not further discussed in this work. The free interface normal modes are defined as

$$\begin{aligned} &{\varvec{\varPhi }}^{{\mathrm{free}}} = \left[ {\begin{array}{ccc} {\varvec{\phi }}^{{\mathrm{free}}}_{1} &{} \cdots & {\varvec{\phi }}^{{\mathrm{free}}}_{n_{{\mathrm{mod}}}}\\ \end{array}}\right] ,\nonumber \\ & \left( {\varvec{K}} - {\omega ^{{\mathrm{free}}}_{j}}^2 {\varvec{M}} \right) {\varvec{\phi }}^{{\mathrm{free}}}_{j} = {\mathbf{0}},\,\,\omega ^{{\mathrm{free}}}_{1}\le \cdots \le \omega ^{{\mathrm{free}}}_{n_{{\mathrm{mod}}}}.\end{aligned}$$
(60)

Appendix 5: Exact Condensation Procedure

In this appendix, an exact procedure is presented for the condensation of the harmonic balance equations, which takes advantage of the sparsity of the nonlinear terms. To this end, it is convenient to arrange the equations of motion in such a manner that the nonlinear force and generalized coordinates vectors have the form

$${\varvec{f}}_{{\mathrm{c}}}\left[ {\varvec{u}}\right] = \left[ {\begin{array}{c} {\varvec{\lambda }}\left[ {}_{\,\mathrm{c}}{\varvec{g}}\right] \\ {\mathbf{0}}\\ \end{array}}\right] ,\quad {\varvec{u}}= \left[ {\begin{array}{c} {}_{\,\mathrm{c}}{\varvec{g}}\\ {\varvec{u}}_{{\mathrm{rem}}}\\ \end{array}}\right] .$$
(61)

Herein, \({\varvec{f}}_{{\mathrm{c}}}\) and \({\varvec{u}}\) have the dimension \(n_{{\mathrm{d}}}\), whereas \({\varvec{\lambda }}\) and \({}_{\,\mathrm{c}}{\varvec{g}}\) have the dimension \(3n_{{\mathrm{c}}}\). We refer to \({}_{\,\mathrm{c}}{\varvec{g}}\) as nonlinear coordinates, and to \({\varvec{u}}_{{\mathrm{rem}}}\) as linear coordinates, since for given \({}_{\,\mathrm{c}}{\varvec{g}}(t)\) a linear ODE governs \({\varvec{u}}_{{\mathrm{rem}}}(t)\). The vector of nonlinear forces is considered as sparse, if \(3n_{{\mathrm{c}}}\ll n_{{\mathrm{d}}}\). This sparsity is inherited by the harmonics \({\varvec{\varLambda }}\) and the associated gradients. The extent of this sparsity depends on the choice of the generalized coordinates. If the physical coordinates \({}_{\,\mathrm{c}}{\varvec{g}}\), that describe the (relative) interface motions, are not retained, the sparsity is generally lost. In the simplest case, \({}_{\,\mathrm{c}}{\varvec{g}}\) represents the local relative deformation at the contact points.

Taking advantage of this sparsity during the numerical solution process is a common procedure in conjunction with harmonic balance, see e.g. [3, 18, 54, 77]. To this end, one condenses the set of \(n_{{\mathrm{d}}}\) nonlinear algebraic equations for each harmonic to a set of \(3n_{{\mathrm{c}}}\) equations. This can significantly reduce the number of explicit unknowns and thus reduce the computational effort required for the iterative solution process.

The procedure is exemplified for the balance of generalized displacements given in Eq. (28), but a fully analogous procedure is available for the balance of generalized forces given in Eq. (27), see e.g. [133]. To this end, Eq. (28) is split into the individual harmonics,

$${\varvec{U}}_n+ {\varvec{H}}_{n}\left( \varOmega \right) {\varvec{F}}_{{\mathrm{c,n}}}\left( {\varvec{U}}_{0},\ldots ,{\varvec{U}}_{H}\right) = {\varvec{U}}_{{\mathrm{ae}},n}\quad \forall n\in {\mathcal {H}}.$$
(62)

The matrices \({\varvec{H}}_{k}\) are also partitioned as in Eq. (61),

$${\varvec{H}}_{k}= \left[ {\begin{array}{cc} {\varvec{H}}_k^{\mathrm{nl,nl}}&{} {\varvec{H}}_k^{\mathrm{nl,l}}\\ {\varvec{H}}_k^{\mathrm{l,nl}}&{} {\varvec{H}}_k^{\mathrm{l,l}}\\ \end{array}}\right].$$
(63)

With this, the first hyper-row of Eq. (62) reads

$${\varvec{G}}_{k}+ {\varvec{H}}_k^{\mathrm{nl,nl}}(\varOmega ){\varvec{\varLambda }}_{k}\left( {\varvec{G}}_0,\ldots ,{\varvec{G}}_{n_{{\mathrm{h}}}}\right) = {\mathbf{0}}\quad\forall \,n\in {\mathcal {H}},$$
(64)

where \({\varvec{H}}_k^{\mathrm{nl,nl}}\) is a portion of the matrix \({\varvec{H}}_{k}\). Equation (64) only depends on the harmonic components \({\varvec{G}}_0,\ldots ,{\varvec{G}}_{n_{{\mathrm{h}}}}\) of the nonlinear coordinates, but not on \({\varvec{U}}_{{\mathrm{rem}},k}\), the harmonic components of the linear coordinates. It is thus sufficient to solve Eq. (64), which is of much smaller dimension than Eq. (62) if \(3n_{{\mathrm{c}}}\ll n_{{\mathrm{d}}}\). Upon solution of Eq. (64) for \({\varvec{G}}_{k}\), the remaining portion of the generalized coordinates can be determined using the explicit formulation \({\varvec{U}}_{{\mathrm{rem}},k}= -{\varvec{H}}_k^{\mathrm{l,nl}}{\varvec{\varLambda }}_{k}\). It should be noted that this dynamic condensation procedure is mathematically exact, so that it does not suffer from poor accuracy like, e.g., the static (Guyan) condensation procedure.

Note that the dynamic compliance matrix is defined as the inverse of the dynamic stiffness matrix. Computing \({\varvec{H}}_{n}\) by matrix inversion, however, would be time consuming. This is particularly true since \({\varvec{H}}_{n}\) depends on \(\varOmega\) and, thus, typically has to be re-computed in every iteration. As long as the dynamic stiffness matrix can be expressed as a polynomial in \(\varOmega\) with constant coefficient matrices, the matrix inversion can be replaced by a small number of matrix multiplications and the trivial inversion of a diagonal matrix, see e.g. [84, 133].

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Krack, M., Salles, L. & Thouverez, F. Vibration Prediction of Bladed Disks Coupled by Friction Joints. Arch Computat Methods Eng 24, 589–636 (2017). https://doi.org/10.1007/s11831-016-9183-2

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