Overview of Free Interface Substructuring Approaches for Systems with Arbitrary Viscous Damping in Dynamic Substructuring

Abstract

Most classical substructuring methods yield great approximation accuracy if the underlying system is not damped. One approach is a fixed interface method, the Craig-Bampton method. In contrast, many other methods (e.g., MacNeal method, Rubin method, Craig-Chang method) employ free interface modes, (residual) attachment modes, and rigid body modes. None of the aforementioned methods takes any damping effects into account when performing the reduction. If damping significantly influences the dynamic behavior of the system, the approximation accuracy can be very poor. One procedure to handle arbitrarily viscously damped systems and to take damping effects into account is to transform the second-order differential equations into twice the number of first-order differential equations resulting in state-space representation of the system. Solving the corresponding eigenvalue problem allows the damped equations to be decoupled; however, complex eigenmodes and eigenvalues occur. The complex modes are used to build a reduction basis that includes damping properties.

The derivation of different Craig-Bampton substructuring methods (fixed interface) for viscously damped systems was presented in Gruber et al. (Comparison of Craig-Bampton approaches for systems with arbitrary viscous damping in dynamic substructuring). In contrast, we present here the derivation of different free interface substructuring methods for viscously damped systems in a comprehensible consistent manner. Craig and Ni suggested a method that employs complex free interface vibration modes (1989). De Kraker and van Campen give an extension of Rubin’s method for general state-space models (1996). Liu and Zheng proposed an improved component modes synthesis method for nonclassically damped systems (2008), which is an extension of Craig and Ni’s method. A detailed comparison between the different formulations will be given. Liu and Zheng’s method can be considered as a second-order extension of Craig and Ni’s method. We propose a third-order extension and a generalization to any given higher order. Moreover, a new method combining the reduction basis of Liu and Zheng’s approach with the primal assembly procedure applied by de Kraker and van Campen is proposed. The presented theory and the comparison between the methods will be illustrated in different examples.

11.1 Introduction

Dynamic substructuring techniques reduce the size of large models very efficiently. A large finite element model is divided into a number of substructures; each substructure is analyzed and reduced separately and then assembled into a low-order reduced model. This low-order reduced model approximates the original large model’s behavior. During this process, each substructure’s degrees of freedom (DOFs) are divided into internal DOFs (those not shared with any adjacent substructure) and boundary or interface DOFs (those shared with adjacent substructures and therefore forming the model’s interface DOFs) [1]. Many substructuring methods that work with second-order equations of motion have been proposed in the past [2,3,4,5,6,7,8,9]. Only the substructures’ mass and stiffness properties are commonly taken into account for the reduction by all those methods. Each substructure’s reduction basis is thereby constructed exclusively on the properties of the mass matrix and stiffness matrix, but the properties of the damping matrix are ignored. The undamped equations of motion are assumed to correctly describe the substructures’ dynamics: it is either assumed that there is no damping or that damping effects are completely negligible when building the reduction basis.

Substructuring methods working with the undamped equations of motion afford great approximation accuracy if the underlying system is damped only slightly or not at all [10]. The most popular approach is a fixed interface method, the Craig-Bampton method [2], which is based on fixed interface vibration modes and interface constraint modes. In contrast, many other methods (e.g., MacNeal method [3], Rubin method [4], Craig-Chang method [6]) employ free interface modes, (residual) attachment modes, and rigid body modes.

None of the aforementioned methods take any damping effects into account when performing the reduction. There is generally no justification for neglecting damping effects. If damping significantly influences the dynamic behavior of the system under consideration, then the approximation accuracy of these methods can be very poor. The damped equations of motion have to be taken into account to incorporate damping effects. One procedure to handle arbitrarily damped systems and to take damping effects into account is to transform the second-order differential equations into twice the number of first-order differential equations, resulting in a state-space representation of the system. Solving the corresponding eigenvalue problem allows the damped equations to be decoupled, but complex eigenmodes and eigenvalues occur. The complex modes are used to build a reduction basis that includes damping properties.

The derivation of different Craig-Bampton substructuring methods (fixed interface) for viscously damped systems is given in [11]. Thereby, Hasselman and Kaplan’s approach [12], which is an extension of the Craig-Bampton method through its employment of complex component modes, Beliveau and Soucy’s method [13], which replaces the real fixed interface normal modes of the second-order system with the corresponding complex modes of the first-order system, and an approach of de Kraker [14], which uses complex normal modes and modified static modes, were investigated [11].

In contrast, in this contribution, we wish to present the derivation of different free interface substructuring methods for viscously damped systems in a comprehensible consistent manner: Craig and Ni [15] presented a method that employs complex free interface vibration modes. De Kraker and van Campen [16] gave an extension of Rubin’s method for general state-space models. Liu and Zheng [17] proposed an improved component modes synthesis method for nonclassically damped systems, which is an extension of Craig and Ni’s method. All these methods will be briefly derived and a detailed comparison between the different formulations will be given. The presented theory and the comparison between the methods will be illustrated in different examples. Based on those insights, the following improvements will be proposed: Liu and Zheng’s method can be considered as a second-order extension of Craig and Ni’s method. We propose a third-order extension and a generalization to arbitrarily higher orders. Moreover, a new method combining the reduction basis of Liu and Zheng’s approach with the primal assembly procedure applied by de Kraker and van Campen is proposed. The approximation accuracy of these new approaches will be compared to the existing methods and illustrated in different examples.

Section 11.2 recalls the superordinate governing equations, and the terminology and notation used throughout this paper is established up. In Sect. 11.3, a literature review of existing free interface substructuring approaches for viscously damped systems is given. Starting in Sect. 11.3.1, the first-order approach of Craig and Ni is presented. Following this, Liu and Zheng’s approach is derived in Sect. 11.3.2, which is the second-order extension of the method of Craig and Ni. Section 11.3.3 recalls de Kraker and van Campen’s approach, which is an extension of Rubin’s method for general state-space models. Improvements of the existing free interface substructuring approaches are suggested in Sect. 11.4: a generalization to arbitrarily higher orders of Liu and Zheng’s method is derived in Sect. 11.4.1. Following this, a new approach combining Liu and Zheng’s reduction basis with a primal assembly procedure is demonstrated in Sect. 11.4.2. The properties of the presented methods are subsequently illustrated in detail in Sect. 11.5 using three different examples. Finally, a brief summary of findings and conclusions is given in Sect. 11.6.

11.2 Governing Equations

11.2.1 Undamped and Classically Damped Systems

Consider the equations of motion of a viscously damped linear system with m degrees of freedom (DOFs)

$$\displaystyle \begin{aligned} \boldsymbol{M} \boldsymbol{\ddot{u}} + \boldsymbol{C} \boldsymbol{\dot{u}}+ \boldsymbol{K} \boldsymbol{u} = \boldsymbol{f} {}\end{aligned} $$

(11.1)

with mass matrix M, damping matrix C, stiffness matrix K, displacement vector u, and external force vector f. Associated with the undamped system of Eq. (11.1), i.e., C = 0, is the eigenvalue problem

$$\displaystyle \begin{aligned} \left( - \omega^2_j \boldsymbol{M} + \boldsymbol{K} \right) \boldsymbol{\theta}_j = \boldsymbol{0} {} \end{aligned} $$

(11.2)

with eigenvalue $\omega ^2_j$ and corresponding eigenvector θ _j for j = 1, …, m. The eigenvectors θ _j of Eq. (11.2) are called normal modes or natural modes and form the columns of the modal matrix Θ:

(11.3)

If the modal matrix Θ is normalized with respect to the mass matrix M, then the normal modes’ orthogonality conditions give

$$\displaystyle \begin{aligned} \boldsymbol{\Theta}^T \boldsymbol{M} \boldsymbol{\Theta} = \boldsymbol{I}, \qquad\boldsymbol{\Theta}^T \boldsymbol{K} \boldsymbol{\Theta} = \boldsymbol{\Omega}^2 = \text{diag}\left(\omega^2_1, \, \omega^2_2, \, \dots, \, \omega^2_m\right) . {} \end{aligned} $$

(11.4)

Using the modal transformation

$$\displaystyle \begin{aligned} \boldsymbol{u} = \boldsymbol{\Theta} \boldsymbol{p} {} \end{aligned} $$

(11.5)

with the vector of modal coordinates p, Eq. (11.1) takes the canonical form of the damped system [18]

$$\displaystyle \begin{aligned} & \boldsymbol{\ddot{p}} + \boldsymbol{C}_{\mathrm{modal}} \; \boldsymbol{\dot{p}}+ \boldsymbol{\Omega}^2 \boldsymbol{p} = \boldsymbol{\Theta}^T \boldsymbol{f} \qquad\text{with } \qquad\boldsymbol{C}_{\mathrm{modal}}= \boldsymbol{\Theta}^T \boldsymbol{C} \boldsymbol{\Theta}. {} \end{aligned} $$

(11.6)

Mass matrix M and stiffness matrix K have been diagonalized by the modal transformation of Eq. (11.5) and the diagonal eigenvalue matrix Ω ² is given in Eq. (11.4). Matrix C _modal is referred to as generalized damping matrix or modal damping matrix [1]. Any modal coupling in a linear system occurs exclusively through damping [19]. A damped system is termed “classically damped” if it can be decoupled by the modal matrix Θ [1], i.e., C _modal is also diagonalized by the modal transformation of Eq. (11.5). A necessary and sufficient condition that the damped system can be decoupled and hence that the damping matrix C can be diagonalized by the modal matrix Θ is [20]:

$$\displaystyle \begin{aligned} \boldsymbol{C}\boldsymbol{M}^{-1}\boldsymbol{K} = \boldsymbol{K}\boldsymbol{M}^{-1}\boldsymbol{C} {} \end{aligned} $$

(11.7)

Condition (11.7) is usually not satisfied. Only under special conditions are the equations of motion completely diagonalized by the classical modal transformation [18], and these conditions appear to have little physical justification [21]. For instance, the often used but simplifying assumptions of mass proportional damping, stiffness proportional damping, Rayleigh damping [22], or modal damping fulfill condition (11.7) and diagonalize the generalized damping matrix C _modal. Nevertheless, decoupling of Eq. (11.1) is not generally possible using classical modal analysis.

11.2.2 Nonclassically Damped Systems

A procedure to handle and decouple systems that are not classically damped, i.e., systems with general viscous damping, is to transform the m second-order equations of motion (11.1) into 2m first-order equations [1, 23,24,25]. The state-space vector of dimension n = 2m is

(11.8)

with $\boldsymbol {v}(t) = \boldsymbol {\dot {u}}(t)$. Adding the m redundant equations $\boldsymbol {M}\boldsymbol {v}(t) = \boldsymbol {M}\boldsymbol {\dot {u}}(t)$ to the equations of motion (11.1), the generalized state-space symmetric form

$$\displaystyle \begin{aligned} \boldsymbol{{A}} \dot{\boldsymbol{z}} + \boldsymbol{{B}} \boldsymbol{z} = \boldsymbol{{F}} {} \end{aligned} $$

(11.9)

is obtained with

(11.10)

Associated with Eq. (11.9), which includes the damping matrix C, is the eigenvalue problem

$$\displaystyle \begin{aligned} \left( \lambda_j \boldsymbol{{A}} + \boldsymbol{{B}} \right) \boldsymbol{\phi}_{j} = \boldsymbol{0} {} \end{aligned} $$

(11.11)

with eigenvalue λ _j and corresponding state-space eigenvector ϕ _j for j = 1, …, n. Since matrices A and B are real, the n eigenvalues λ _j must either be real or they must occur in complex conjugate pairs [24]. For underdamped systems, the eigenvalues λ _j and corresponding eigenmodes ϕ _j occur in complex conjugate pairs [26] and are termed “underdamped eigenvalues” and “underdamped eigenmodes”, respectively. Since real eigenvalues indicate very high damping leading to overdamped modes, most structures have m complex conjugate pairs of eigenvalues and corresponding eigenvectors [1]. The eigenvectors ϕ _j are referred to as complex normal modes.

Similar to the undamped case in Eq. (11.3), the state-space eigenvectors ϕ _j form the complex state-space modal matrix

(11.12)

An extensive investigation of properties and relations between complex normal modes of Eq. (11.12) and real normal modes of Eq. (11.3) is given in [27]. If the complex modal matrix Φ is normalized with respect to the state-space matrix A, the orthogonality conditions of the complex normal modes give

$$\displaystyle \begin{aligned} \boldsymbol{\Phi}^T \boldsymbol{{A}} \boldsymbol{\Phi} = \boldsymbol{I}, \qquad\boldsymbol{\Phi}^T \boldsymbol{{B}} \boldsymbol{\Phi} = -\boldsymbol{\Lambda} = -\text{diag}\left( \lambda_1, \, \lambda_2, \, \dots, \, \lambda_{n} \right). {} \end{aligned} $$

(11.13)

Complex modal matrix Φ decouples the damped system if it is written in state-space format as in Eq. (11.9). Eigenvalue matrix Λ contains the eigenvalues λ _j of Eq. (11.11) for j = 1, …, n as diagonal entries. For the orthogonality conditions in Eq. (11.13), it is assumed that the system Eq. (11.9) is non-defective. The special case of defective systems is illustrated in detail in [28].

11.3 Literature Review of Free Interface Substructuring Approaches for Viscously Damped Systems

The equations of motion in state-space form of one substructure s are

(11.14)

Equation (11.14) has n ^(s) DOFs and the superscript (s) is the label of the particular substructure s. State-space matrices A ^(s) and B ^(s) are constructed according to Eq. (11.10) whereby the substructure’s mass matrix M ^(s), damping matrix C ^(s) and stiffness matrix K ^(s) are used. Vector f ^(s) corresponds to the vector of the external forces and vector g ^(s) corresponds to the vector of internal reaction forces occurring at the interface due to the connection to adjacent substructures. The substructure state-space vector z ^(s) is constructed according to Eq. (11.8) and can be divided into internal DOFs (those not shared with any adjacent substructure) and boundary DOFs (those shared with adjacent substructures and therefore forming the model’s interface DOFs):

(11.15)

The reduction is performed by using a substructure reduction matrix Φ ^(s). Thus, the n ^(s) DOFs of the substructure’s state-space vector z ^(s) are approximated by a reduced number of generalized DOFs p ^(s):

(11.16)

In the reduction according to Eq. (11.16), $\boldsymbol {\Phi }_{\mathrm{kept}}^{(s)}$ contains $n_{\mathrm{kept}}^{(s)}$ kept eigenmodes that are solutions of the substructure’s eigenvalue problem

$$\displaystyle \begin{aligned} \left( \lambda_j^{(s)} \boldsymbol{{A}}^{(s)} + \boldsymbol{{B}}^{(s)} \right) \boldsymbol{\phi}_{j}^{(s)} = \boldsymbol{0} . {} \end{aligned} $$

(11.17)

Depending on the boundary conditions of the substructure, rigid body modes $\boldsymbol {\Phi }_{r}^{(s)}$ can also occur. Using the attachment modes $\boldsymbol {\Phi }_{a}^{(s)}$, forces occurring at the interface of connected substructures are represented statically completely.

The basic idea of model order reduction is the approximation by a reduced number of modes. Therefore, the matrix $\boldsymbol {\Phi }_{\mathrm{kept}}^{(s)}$ contains only selected kept modes, where $n_{\mathrm{kept}}^{(s)} \,<\, n^{(s)}$ holds. The selection is made according to a selection criterion. Usually, the kept modes are the eigenmodes corresponding to eigenvalues with the smallest magnitude, up to a limiting frequency, which depends on the application. This strategy is used in Craig and Ni’s method [15] and in de Kraker and van Campen’s method [16]. Liu and Zheng [17] proposed selecting the kept modes according to a weighting factor, which evaluates the influence of the respective mode on the transfer function. In this paper, the first selection criterion, i.e., keeping the modes corresponding to the eigenvalues with the smallest absolute magnitude, will be applied.

The reduced equations of motion of a substructure are obtained by inserting the reduction according to Eq. (11.16) into the substructure’s equations of motion (11.14) and left multiplication with Φ ^{(s) T}. The reduced system can be written as

$$\displaystyle \begin{aligned} {\boldsymbol{A}_{\mathrm{red}}^{(s)}}{\boldsymbol{\dot p}^{(s)}}+{\boldsymbol{B}_{\mathrm{red}}^{(s)}}{\boldsymbol{p}^{(s)}}={\boldsymbol{\Phi}^{(s)}}^{T} {\boldsymbol{F}^{(s)}} {} \end{aligned} $$

(11.18)

with the reduced state-space matrices

$$\displaystyle \begin{aligned} \boldsymbol{A}_{\mathrm{red}}^{(s)}={\boldsymbol{\Phi}^{(s)}}^{T}\boldsymbol{A}^{(s)}\boldsymbol{\Phi}^{(s)} \qquad\text{and} \qquad\boldsymbol{B}_{\mathrm{red}}^{(s)}={\boldsymbol{\Phi}^{(s)}}^{T}\boldsymbol{B}^{(s)}\boldsymbol{\Phi}^{(s)} . \end{aligned} $$

(11.19)

For the reduced substructure matrices, the following dimension properties apply:

$$\displaystyle \begin{aligned} \boldsymbol{A}_{\mathrm{red}}^{(s)} \in \mathbb{C}^{{(n_{\mathrm{kept}}^{(s)}+n_{r}^{(s)}+n_{a}^{(s)})}\times{(n_{\mathrm{kept}}^{(s)}+n_{r}^{(s)}+n_{a}^{(s)})}} \quad\text{and}\quad{\boldsymbol{B}_{\mathrm{red}}^{(s)} \in \mathbb{C}^{{(n_{\mathrm{kept}}^{(s)}+n_{r}^{(s)}+n_{a}^{(s)})}\times{(n_{\mathrm{kept}}^{(s)}+n_{r}^{(s)}+n_{a}^{(s)})}}}, \end{aligned}$$

whereby, $n_{r}^{(s)}$ is the number of state-space rigid body modes and $n_{a}^{(s)}$ is the number of attachment modes of substructure s. In the following, different substructuring methods for viscously damped systems are reviewed.

11.3.1 Craig and Ni’s Method [15]

To start with, the first-order reduction according to Craig and Ni (CN) [15] is considered. Therefore, the reduction basis as proposed by Craig and Ni will be derived in Sect. 11.3.1.1, and afterwards the assembly procedure of substructures is recalled in Sect. 11.3.1.2.

11.3.1.1 Reduction Basis

Firstly, the substructures are divided into substructures without rigid body modes, and substructures with rigid body modes.

Substructures Without Rigid Body Modes

Eq. (11.16) shows that the approximation of the state-space vector z of a substructure¹ is obtained by the superposition of a dynamic part and a static part. The dynamic part is described using the kept eigenmodes Φ _kept. These are solutions of the corresponding eigenvalue problem of Eq. (11.17). The eigenmodes are orthonormalized, so that they satisfy the orthogonality conditions for complex modes

$$\displaystyle \begin{aligned} \boldsymbol{\Phi}^{T}\boldsymbol{A}\boldsymbol{\Phi}=\boldsymbol{I}\qquad \qquad\boldsymbol{\Phi}^{T}\boldsymbol{B}\boldsymbol{\Phi}=-\boldsymbol{\Lambda}. {} \end{aligned} $$

(11.20)

Here, $\boldsymbol {\Lambda } \in \mathbb {C}^{{n}\times {n}}$ corresponds to the diagonal matrix, which contains the eigenvalues (see Sect. 11.2.2). The kept eigenmodes Φ _kept are identical for all reduction methods considered in this paper. The static part is used to fully represent the connection forces that occur at the interface of adjacent substructures [10]. By considering only the static part, i.e., $\boldsymbol {\dot {z}}=\boldsymbol {0}$ in Eq. (11.14), and assuming that no external forces f = 0 occur, the static problem can be written as

(11.21)

Matrix F _a is a force matrix, which consists of unit-force vectors arranged in columns for each interface displacement DOF. The attachment modes Φ _a result by solving BΦ _a = F _a and can be symbolically represented as

$$\displaystyle \begin{aligned} \boldsymbol{\Phi}_{a}=\boldsymbol{B}^{-1}\boldsymbol{F}_{a}=\boldsymbol{G}_{e}\boldsymbol{F}_{a}. {}\end{aligned} $$

(11.22)

Here, G _e corresponds to the flexibility matrix, which for substructures without rigid body modes is equal to the inverse of the state-space stiffness matrix B. Eq. (11.22) shows that attachment modes are columns of the corresponding flexibility matrix. Based on the definition of the force matrix F _a according to Craig and Ni, the attachment modes of substructures without rigid body modes contain only entries in the displacement DOFs.

The basic idea of model order reduction is the approximation by keeping only a reduced number of modes Φ _kept. The information of these modes is already contained in the reduction basis. Therefore, the goal in the flexibility matrix is to solely consider information from the truncated modes Φ _trunc. The part of the flexibility matrix resulting from the truncated modes is called the residual flexibility matrix [1]. Using Eq. (11.20) this reads [29]

$$\displaystyle \begin{aligned} \boldsymbol{G}_{\mathrm{res}} =\boldsymbol{\Phi}_{\mathrm{trunc}}(-\boldsymbol{\Lambda}_{\mathrm{trunc}})^{-1}\boldsymbol{\Phi}_{\mathrm{trunc}}^{T}=\boldsymbol{G}_{e}+\boldsymbol{\Phi}_{\mathrm{kept}}\boldsymbol{\Lambda}_{\mathrm{kept}}^{-1}\boldsymbol{\Phi}_{\mathrm{kept}}^{T}. {}\end{aligned} $$

(11.23)

The advantage of the second representation is that only the eigenmodes corresponding to the smallest eigenvalues have to be calculated. From Eq. (11.23), the attachment modes Φ _a,CN according to Craig and Ni’s method arise thus:

$$\displaystyle \begin{aligned} \boldsymbol{\Phi}_{a,\mathrm{res}} =\boldsymbol{G}_{\mathrm{res}}\boldsymbol{F}_{a}=\boldsymbol{\Phi}_{a,\mathsf{CN}} {} \end{aligned} $$

(11.24)

For Craig and Ni’s attachment modes, $\boldsymbol {\Phi }_{a,\mathsf {CN}}\in \mathbb {C}^{{n}\times {n_{b,u}}}$ applies. Here, n _b,u corresponds to the number of interface displacement DOFs. The reduction of substructures without rigid body modes according to Craig and Ni’s method is finally done by

(11.25)

Substructures with Rigid Body Modes

In the case of substructures with rigid body modes, the substructure stiffness matrix K is singular. The substructure has m _r physical rigid body modes θ _r with zero eigenvalues, for which the following applies:

$$\displaystyle \begin{aligned} \boldsymbol{K} \boldsymbol{\theta}_{r}=\boldsymbol{0} \end{aligned} $$

(11.26)

In state-space, the occurrence of a physical rigid body mode θ _r leads to either one or two rigid body modes. For a detailed explanation, please refer to [28]. Thus, for the number n _r of rigid body modes in the state-space $m_{r}\leqslant n_{r}\leqslant 2m_{r}$ holds. Using the rigid body modes θ _r, two cases are distinguished [16, 17]:

Case 1: Cθr = 0

In this case, the rigid body motion does not cause any damping forces. The substructure is termed “defective”. In the mathematical sense, a matrix of size n × n is deemed defective if it does not have a linearly independent set of n eigenvectors [1]. The corresponding rigid body movements in the state-space result in

(11.27)

Here, ϕ _reg is referred to as regular rigid body mode and ϕ _gen is referred to as a generalized rigid body mode.

Case 2: Cθr ≠ 0

In this case, the rigid body motion causes damping forces. The substructure is non-defective. In contrast to defective matrices, matrices are called non-defective if the n-dimensional space is spanned by a complete set of independent eigenvectors [30]. Thus, there is only one regular rigid body mode ϕ _reg and no generalized rigid body mode occurs in the state-space:

(11.28)

All state-space rigid body modes can be combined in the state-space rigid body matrix Φ _r. In the case of substructures with rigid body modes, an adaptation of the procedure is necessary to determine the attachment modes according to Eq. (11.24). As a first step, a projection matrix P is defined for substructures with rigid body modes [1, 15]

$$\displaystyle \begin{aligned} \boldsymbol{P}=\boldsymbol{I}-\boldsymbol{A}\boldsymbol{\Phi}_{r}\boldsymbol{A}_{rr}^{-1}\boldsymbol{\Phi}_{r}^{T} \qquad \text{with}\quad\boldsymbol{A}_{rr}=\boldsymbol{\Phi}_{r}^{T}\boldsymbol{A}\boldsymbol{\Phi}_{r}. {} \end{aligned} $$

(11.29)

In Eq. (11.29), A _rr corresponds to the rigid body mass matrix in state-space. If an arbitrary force vector F is multiplied by the matrix P, no rigid body modes are excited by the force vector [15]. Additionally, due to the singularity of the stiffness matrix K, it is not possible to invert the state-space stiffness matrix B. Therefore, as a second step, the n _i internal DOFs z _i of substructures with rigid body modes are divided into a part z _r corresponding to the number n _r of rigid body modes and a part z _e corresponding to the number n _e of elastic modes, where n _i = n _r + n _e applies [16]

(11.30)

The rigid body parts of the stiffness matrix K are fixed, from which the constrained stiffness matrix K _c follows:

$$\displaystyle \begin{aligned} \boldsymbol{K}_c = \left[ \begin{array}{ *{2}{c} } \boldsymbol{K}_{ee} & \boldsymbol{K}_{eb}\\ \boldsymbol{K}_{be} & \boldsymbol{K}_{bb}\\ \end{array} \right] \end{aligned} $$

(11.31)

Matrix K _c is regular and can therefore be inverted. The pseudo flexibility matrix G for substructures with rigid body modes results in [15]

(11.32)

Finally, the flexibility matrix G _e for substructures with rigid body modes is defined through the combination of both steps

$$\displaystyle \begin{aligned} \boldsymbol{G}_{e}=\boldsymbol{P}^{T}\boldsymbol{G}\boldsymbol{P}. {} \end{aligned} $$

(11.33)

Analogous to substructures without rigid body modes, the residual flexibility matrix is defined as $\boldsymbol {G}_{\mathrm{res}}=\boldsymbol {G}_{e}+\boldsymbol {\Phi }_{\mathrm{kept}}\boldsymbol {\Lambda }_{\mathrm{kept}}^{-1}\boldsymbol {\Phi }_{\mathrm{kept}}^{T}$ and the residual attachment modes are obtained by

$$\displaystyle \begin{aligned} \boldsymbol{\Phi}_{a,\mathsf{CN}} =\boldsymbol{G}_{\mathrm{res}}\boldsymbol{F}_{a}. {} \end{aligned} $$

(11.34)

Here, Λ _kept contains the eigenvalues of the kept eigenmodes without rigid body modes. Thus, the reduction of substructures with rigid body modes according to Craig and Ni’s method is

(11.35)

11.3.1.2 Assembly Procedure (Elimination of Interface DOFs)

The assembly of two substructures α and β according to Craig and Ni is based on the interface displacement compatibility and the interface force equilibrium

$$\displaystyle \begin{aligned} &\boldsymbol{u}_{b}^{(\alpha)} =\boldsymbol{u}_{b}^{(\beta)}, {} \end{aligned} $$

(11.36)

$$\displaystyle \begin{aligned} &\boldsymbol{g}_{b}^{(\alpha)}+\boldsymbol{g}_{b}^{(\beta)} =\boldsymbol{0}, {} \end{aligned} $$

(11.37)

where the index b denotes the boundary DOFs. The interface forces g _b are approximated by the attachment modes. Thus, they correspond physically to the generalized attachment modes’ DOFs p _a [31]. The condition

$$\displaystyle \begin{aligned} \boldsymbol{p}_{a}^{(\alpha)}=-\boldsymbol{p}_{a}^{(\beta)} {} \end{aligned} $$

(11.38)

follows from the interface force equilibrium of Eq. (11.37) The assembly is done on the basis of the displacements. For this purpose, only the displacement part u of the approximation of the state-space vector in Eq. (11.25) is considered:

$$\displaystyle \begin{aligned} \boldsymbol{u} \approx \boldsymbol{\Phi}_{\mathrm{kept},u} \; \boldsymbol{p}_{\mathrm{kept}} + \boldsymbol{\Phi}_{a,u} \; \boldsymbol{p}_{a} {} \end{aligned} $$

(11.39)

In Eq. (11.39), Φ _kept,u corresponds to the displacement part of the kept eigenmodes Φ _kept and Φ _a,u corresponds to the displacement part of the attachment modes Φ _a. Inserting the approximation (11.39) into the displacement compatibility condition (11.36) yields

$$\displaystyle \begin{aligned} {\boldsymbol{\Phi}_{\mathrm{kept},u,b}^{(\alpha)}}{\boldsymbol{p}_{\mathrm{kept}}^{(\alpha)}}+{\boldsymbol{\Phi}_{a,u,b}^{(\alpha)}}{\boldsymbol{p}_{a}^{(\alpha)}} ={\boldsymbol{\Phi}_{\mathrm{kept},u,b}^{(\beta)}}{\boldsymbol{p}_{\mathrm{kept}}^{(\beta)}}+{\boldsymbol{\Phi}_{a,u,b}^{(\beta)}}{\boldsymbol{p}_{a}^{(\beta)}}. {} \end{aligned} $$

(11.40)

From Eq. (11.40), the following arises, by using condition (11.38) and rearranging:

$$\displaystyle \begin{aligned} {\boldsymbol{\Phi}_{a,u,b}^{(\alpha)}}{\boldsymbol{p}_{a}^{(\alpha)}}+{\boldsymbol{\Phi}_{a,u,b}^{(\beta)}}{\boldsymbol{p}_{a}^{(\alpha)}} =-{\boldsymbol{\Phi}_{\mathrm{kept},u,b}^{(\alpha)}}{\boldsymbol{p}_{\mathrm{kept}}^{(\alpha)}}+{\boldsymbol{\Phi}_{\mathrm{kept},u,b}^{(\beta)}}{\boldsymbol{p}_{\mathrm{kept}}^{(\beta)}}, \end{aligned} $$

(11.41)

which can be written in matrix notation as

(11.42)

Thus, it is possible to express the generalized attachment modes’ DOFs p _a depending on the remaining DOFs p _kept of both substructures:

(11.43)

The interface parameters are eliminated by means of

(11.44)

with

(11.45)

This finally results in the reduced overall system

$$\displaystyle \begin{aligned} \boldsymbol{A}_{\mathrm{red}}\boldsymbol{\dot p}_{\mathrm{kept}} + \boldsymbol{B}_{\mathrm{red}}\boldsymbol{p}_{\mathrm{kept}} = \boldsymbol{0} \end{aligned} $$

(11.46)

with

For the assembled reduced matrices, the following dimension properties apply:

$$\displaystyle \begin{aligned} \boldsymbol{A}_{\mathrm{red}} \in \mathbb{C}^{{(n_{\mathrm{kept}}^{(\alpha)}+n_{r}^{(\alpha)}+n_{\mathrm{kept}}^{(\beta)}+n_{r}^{(\beta)})}\times{(n_{\mathrm{kept}}^{(\alpha)}+n_{r}^{(\alpha)}+n_{\mathrm{kept}}^{(\beta)}+n_{r}^{(\beta)})}} \quad\text{and} \quad\boldsymbol{B}_{\mathrm{red}} \in \mathbb{C}^{{(n_{\mathrm{kept}}^{(\alpha)}+n_{r}^{(\alpha)}+n_{\mathrm{kept}}^{(\beta)}+n_{r}^{(\beta)})}\times{(n_{\mathrm{kept}}^{(\alpha)}+n_{r}^{(\alpha)}+n_{\mathrm{kept}}^{(\beta)}+n_{r}^{(\beta)})}} \end{aligned}$$

This shows that the interface displacement DOFs and the interface forces are completely eliminated during assembly according to Craig and Ni. This is similar to the assembly procedure of MacNeal’s method for the undamped case.

11.3.2 Liu and Zheng’s Method [17]

11.3.2.1 Reduction Basis

The second-order reduction according to Liu and Zheng (LZ) [17] represents an extension of the first-order reduction according to Craig and Ni described in the previous section. For this reason, only the differences to the procedure in Sect. 11.3.1.1 are explained below. The exact representation (no reduction) of the state-space vector z considering all eigenmodes ϕ _k is

$$\displaystyle \begin{aligned} \boldsymbol{z}=\sum_{k=1}^n \boldsymbol{\phi}_{k}p_{k} = \sum_{k=1}^{n_{\mathrm{kept}}}\boldsymbol{\phi}_{\mathrm{kept},k}p_{\mathrm{kept},k} + \sum_{k=n_{\mathrm{kept}}+1}^n\boldsymbol{\phi}_{\mathrm{trunc},k}p_{\mathrm{trunc},k}. {} \end{aligned} $$

(11.47)

The basic idea of model order reduction is the approximation by keeping only a reduced number n _kept of eigenmodes ϕ _k. Vector ϕ _kept,k in Eq. (11.47) denotes one kept eigenmode ϕ _k and vector ϕ _trunc,k denotes one truncated eigenmode. According to Craig and Ni’s reduction basis, the influence of the truncated modes on the dynamic behavior is statically approximated by means of the attachment modes. The extension in the second-order reduction according to Liu and Zheng refers to the additional determination of second-order attachment modes. In comparison to Craig and Ni’s first-order reduction, a dynamic part is retained with the second-order attachment modes. Thus, the inertia and damping effects of the truncated modes are considered in the following [17]. The kept eigenmodes Φ _kept used for the reduction are identical to the kept modes according to Craig and Ni’s reduction basis.

Substructures Without Rigid Body Modes

In the case of substructures without rigid body modes, the attachment modes according to Liu and Zheng are defined as [17]

$$\displaystyle \begin{aligned} \boldsymbol{\Phi}_{a1,\mathsf{LZ}} =\boldsymbol{Q}_{1}\boldsymbol{F}_{a}\ \qquad\text{and} \qquad\boldsymbol{\Phi}_{a2,\mathsf{LZ}} =\boldsymbol{Q}_{2}\boldsymbol{F}_{a} {} \end{aligned} $$

(11.48)

with

$$\displaystyle \begin{aligned} \boldsymbol{Q}_{1} = {\boldsymbol{B}^{-1}}+\boldsymbol{\Phi}_{\mathrm{kept}}\boldsymbol{\Lambda}_{\mathrm{kept}}^{-1}\boldsymbol{\Phi}_{\mathrm{kept}}^{T} \qquad\text{and} \qquad\boldsymbol{Q}_{2} = -{\boldsymbol{B}^{-1}}\boldsymbol{A}{\boldsymbol{B}^{-1}}+\boldsymbol{\Phi}_{\mathrm{kept}}\boldsymbol{\Lambda}_{\mathrm{kept}}^{-2}\boldsymbol{\Phi}_{\mathrm{kept}}^{T}. {} \end{aligned} $$

(11.49)

Utilizing the relation G _e = B ⁻¹, Eq. (11.49) can be rewritten as

$$\displaystyle \begin{aligned} \boldsymbol{Q}_{1} = {\boldsymbol{G}_{e}}+\boldsymbol{\Phi}_{\mathrm{kept}}\boldsymbol{\Lambda}_{\mathrm{kept}}^{-1}\boldsymbol{\Phi}_{\mathrm{kept}}^{T}=\boldsymbol{G}_{\mathrm{res}} \qquad\text{and}\qquad\boldsymbol{Q}_{2} = -{\boldsymbol{G}_{e}}\boldsymbol{A}{\boldsymbol{G}_{e}}+\boldsymbol{\Phi}_{\mathrm{kept}}\boldsymbol{\Lambda}_{\mathrm{kept}}^{-2}\boldsymbol{\Phi}_{\mathrm{kept}}^{T}. {} \end{aligned} $$

(11.50)

From Eq. (11.50), it can be seen that the first-order attachment modes Φ _a1,LZ according to Liu and Zheng are identical to the attachment modes Φ _a,CN as defined by Craig and Ni in Eq. (11.24):

$$\displaystyle \begin{aligned} \boldsymbol{\Phi}_{a1,\mathsf{LZ}}=\boldsymbol{\Phi}_{a,\mathsf{CN}} \end{aligned} $$

(11.51)

The difference between this and the reduction according to Craig and Ni is the extension of the first-order attachment modes by the second-order attachment modes Φ _a2,LZ of Eq. (11.48). For the dimension of the attachment modes according to Liu and Zheng, $\boldsymbol {\Phi }_{a1,\mathsf {LZ}}\in \mathbb {C}^{{n}\times {n_{b,u}}}$, $\boldsymbol {\Phi }_{a2,\mathsf {LZ}}\in \mathbb {C}^{{n}\times {n_{b,u}}}$ and thus $\boldsymbol {\Phi }_{a,\mathsf {LZ}}\in \mathbb {C}^{{n}\times {2n_{b,u}}}$ apply. The reduction of substructures without rigid body modes results in

(11.52)

Substructures with Rigid Body Modes

In the case of substructures with rigid body modes, the attachment modes are defined as [17]

$$\displaystyle \begin{aligned} \boldsymbol{\Phi}_{a1,\mathsf{LZ}} = \boldsymbol{G}_{\mathrm{res}}\boldsymbol{F}_{a}=\boldsymbol{\Phi}_{a,\mathsf{CN}} \qquad \qquad\boldsymbol{\Phi}_{a2,\mathsf{LZ}} = -\boldsymbol{G}_{e}\boldsymbol{A}\boldsymbol{G}_{e}\boldsymbol{F}_{a}. {} \end{aligned} $$

(11.53)

In Eq. (11.53), the matrices G _res and G _e are identical to the definition according to Craig and Ni in Eq. (11.33). In comparison to the attachment modes of substructures without rigid body modes, it can be seen that for substructures with rigid body modes, the dynamic term $\boldsymbol {\Phi }_{\mathrm{kept}}\boldsymbol {\Lambda }_{\mathrm{kept}}^{-2}\boldsymbol {\Phi }_{\mathrm{kept}}^{T}$ is not considered for the computation of second-order attachment modes Φ _a2,LZ by Liu and Zheng [17]. This results in the reduction of substructures with rigid body modes according to Liu and Zheng as

(11.54)

11.3.2.2 Assembly Procedure (Elimination of Interface DOFs)

For assembly of the reduced substructures according to Liu and Zheng, the interface displacement compatibility (11.36) and the interface force equilibrium (11.37) are extended to the state-space form [17]:

(11.55)

(11.56)

Equations (11.55) and (11.56) show that for the assembly according to Liu and Zheng an additional velocity compatibility and an impulse equilibrium at the interface are used. Thus, it is possible to eliminate all interface DOFs. From the extended equilibrium of forces in Eq. (11.56), the state-space equilibrium condition

(11.57)

follows. By inserting the approximation of Eq. (11.52) into the extended compatibility condition of Eq. (11.55), the following is obtained:

$$\displaystyle \begin{aligned} {\boldsymbol{\Phi}_{\mathrm{kept},b}^{(\alpha)}}{\boldsymbol{p}_{\mathrm{kept}}^{(\alpha)}}+{\boldsymbol{\Phi}_{a1,b}^{(\alpha)}}{\boldsymbol{p}_{a}^{(\alpha)}}+{\boldsymbol{\Phi}_{a2,b}^{(\alpha)}}{\boldsymbol{\dot p}_{a}^{(\alpha)}} = {\boldsymbol{\Phi}_{\mathrm{kept},b}^{(\beta)}}{\boldsymbol{p}_{\mathrm{kept}}^{(\beta)}}+{\boldsymbol{\Phi}_{a1,b}^{(\beta)}}{\boldsymbol{p}_{a}^{(\beta)}}+{\boldsymbol{\Phi}_{a2,b}^{(\beta)}}{\boldsymbol{\dot p}_{a}^{(\beta)}} {}\end{aligned} $$

(11.58)

From Eq. (11.58) follows under utilization of Eq. (11.60) and rearrangement

(11.59)

The concluding transformation is analogous to Craig and Ni’s assembly

(11.60)

with

(11.61)

For the assembled reduced matrices, the following dimension properties apply:

$$\displaystyle \begin{aligned} \boldsymbol{A}_{\mathrm{red}} \in \mathbb{C}^{{(n_{\mathrm{kept}}^{(\alpha)}+n_{r}^{(\alpha)}+n_{\mathrm{kept}}^{(\beta)}+n_{r}^{(\beta)})}\times{(n_{\mathrm{kept}}^{(\alpha)}+n_{r}^{(\alpha)}+n_{\mathrm{kept}}^{(\beta)}+n_{r}^{(\beta)})}} \quad\text{and}\quad\boldsymbol{B}_{\mathrm{red}} \in \mathbb{C}^{{(n_{\mathrm{kept}}^{(\alpha)}+n_{r}^{(\alpha)}+n_{\mathrm{kept}}^{(\beta)}+n_{r}^{(\beta)})}\times{(n_{\mathrm{kept}}^{(\alpha)}+n_{r}^{(\alpha)}+n_{\mathrm{kept}}^{(\beta)}+n_{r}^{(\beta)})}}. \end{aligned}$$

Thus, the size of the reduced assembled system is identical to the size of the reduced assembled system according to Craig and Ni’s method.

11.3.3 De Kraker and van Campen’s Method [16]

De Kraker and van Campen’s method (KC) [16] can be considered as an extension of Rubin’s method for the case of arbitrarily viscously damped systems.

11.3.3.1 Reduction Basis

The reduction according to de Kraker and van Campen represents another approach for the determination of attachment modes. In their approach, the force matrix is extended to the velocity boundary DOFs [16]:

(11.62)

Substructures Without Rigid Body Modes

The attachment modes of substructures without rigid body modes according to de Kraker and van Campen are defined as follows

$$\displaystyle \begin{aligned} \boldsymbol{\Phi}_{a,\mathsf{KC}} =\boldsymbol{G}_{e}\boldsymbol{F}_{a,\mathsf{KC}}. {} \end{aligned} $$

(11.63)

Since the force matrix F _a,KC consists of a displacement and a velocity part, the resulting matrix Φ _a,KC of the attachment modes according to de Kraker and van Campen can symbolically be divided into a displacement and a velocity part:

(11.64)

In Eq. (11.64), the displacement attachment modes Φ _a are identical to the attachment modes according to Craig and Ni as well as Liu and Zheng in Eq. (11.22). The velocity attachment modes $\boldsymbol {\dot \Phi }_{a}$ extend the reduction basis compared to the reduction approach according to Craig and Ni. For the attachment modes according to de Kraker and van Campen $\boldsymbol {\Phi }_{a}\in \mathbb {C}^{{n}\times {n_{b,u}}}$, $\boldsymbol {\dot \Phi }_{a,\mathsf {KC}}\in \mathbb {C}^{{n}\times {n_{b,u}}}$ and thus $\boldsymbol {\Phi }_{a,\mathsf {KC}}\in \mathbb {C}^{{n}\times {2n_{b,u}}}$ apply. Accordingly, the matrix Φ _a,KC has twice the number of columns compared to the matrix Φ _a,CN and has the same dimension as the attachment modes Φ _a,LZ according to Liu and Zheng. In addition, de Kraker only uses attachment modes and not residual attachment modes. De Kraker and van Campen’s reduction is

(11.65)

Substructures with Rigid Body Modes

For substructures with rigid body modes, de Kraker and van Campen define the pseudo flexibility matrix in a manner different from the reduction methods according to Craig and Ni as well as Liu and Zheng. From de Kraker and van Campen’s definition results [16]

(11.66)

$$\displaystyle \begin{aligned} \boldsymbol{\overline K}^{\ -1}= \left[ \begin{array}{ ll } \begin{array}{ l } \boldsymbol{0} \end{array} & \begin{array}{ *{2}{l} } \quad\boldsymbol{0} & \quad\boldsymbol{0}\\ \end{array} \\ \begin{array}{ c } \boldsymbol{0} \\ \boldsymbol{0} \\ \end{array} & \left[ \begin{array}{ *{2}{c} } \boldsymbol{K}_{ee} & \boldsymbol{K}_{eb}\\ \boldsymbol{K}_{be} & \boldsymbol{K}_{bb}\\ \end{array} \right] ^{-1}\\ \end{array} \right] \quad\text{and} \quad\boldsymbol{\overline M}^{\ -1}= \left[ \begin{array}{ ll } \begin{array}{ l } \boldsymbol{0} \end{array} & \begin{array}{ *{2}{l} } \quad\boldsymbol{0} & \quad\boldsymbol{0}\\ \end{array} \\ \begin{array}{ c } \boldsymbol{0} \\ \boldsymbol{0} \\ \end{array} & \left[ \begin{array}{ *{2}{c} } \boldsymbol{M}_{ee} & \boldsymbol{M}_{eb}\\ \boldsymbol{M}_{be} & \boldsymbol{M}_{bb}\\ \end{array} \right] ^{-1}\\ \end{array} \right]. \end{aligned}$$

In Eq. (11.66), it can be seen that parts of the mass matrix M are also fixed. The attachment modes follow thus:

$$\displaystyle \begin{aligned} \boldsymbol{\Phi}_{a,\mathsf{KC}} = \boldsymbol{G}_{e,\mathsf{KC}}\boldsymbol{F}_{a} \qquad \text{with}\qquad\boldsymbol{G}_{e,\mathsf{KC}}=\boldsymbol{P}^{T}\boldsymbol{G}_{\mathsf{KC}}\boldsymbol{P} {} \end{aligned} $$

(11.67)

The reduction of substructures with rigid body modes according to de Kraker and van Campen is

(11.68)

11.3.3.2 Assembly Procedure (Primal Assembly)

De Kraker and van Campen use a primal assembly approach for a reduction with free interface modes as proposed by Martinez et al. [32]. In their concept, the boundary displacement and velocity DOFs are to be retained in physical coordinates after the reduction. Hereby, superelements are generated out of the individual substructures, which are assembled in a primal way. The approach is based on the reduction of Eq. (11.68). By arranging the DOFs z ^(s) according to internal DOFs $\boldsymbol {z}_{i}^{(s)}$ and boundary DOFs $\boldsymbol {z}_{b}^{(s)}$, the approximation of the state-space vector can be written as

(11.69)

To consider the cases of substructures with and without rigid body modes at the same time, the rigid body modes $\boldsymbol {\Phi }_{r}^{(s)}$ are included in the matrix $\boldsymbol {\Phi }_{\mathrm{kept}}^{(s)}$ in the following if the substructure possesses rigid body modes. The goal is now to have the boundary displacement and velocity DOFs as physical coordinates $\boldsymbol {z}_{b}^{(s)}$ after the reduction. For this reason, the second line of Eq. (11.69) $\boldsymbol {z}_{b}^{(s)}=\boldsymbol {\Phi }_{\mathrm{kept},b}^{(s)}\boldsymbol {p}_{\mathrm{kept}}^{(s)}+ \boldsymbol {\Phi }_{a,b}^{(s)}\boldsymbol {p}_{a}^{(s)}$ is considered and resolved for $\boldsymbol {p}_{a}^{(s)}$:

$$\displaystyle \begin{aligned} \boldsymbol{p}_{a}^{(s)} = {\boldsymbol{\Phi}_{a,b}^{(s)}}^{-1}\boldsymbol{z}_{b}^{(s)}-{\boldsymbol{\Phi}_{a,b}^{(s)}}^{-1}\boldsymbol{\Phi}_{\mathrm{kept},b}^{(s)}\boldsymbol{p}_{\mathrm{kept}}^{(s)} {} \end{aligned} $$

(11.70)

From Eq. (11.70), the transformation matrix $\boldsymbol {\Phi }_{2}^{(s)}$ can be derived, which expresses the generalized DOFs $\boldsymbol {p}_{a}^{(s)}$ depending on the physical DOFs $\boldsymbol {z}_{b}^{(s)}$ [32]:

(11.71)

The final reduction follows from multiplying reduction matrix $\boldsymbol {\Phi }_{\mathsf {KC}}^{(s)}$ and transformation matrix $\boldsymbol {\Phi }_{2}^{(s)}$:

(11.72)

It can be seen that only the internal DOFs are transformed into generalized coordinates and that the interface displacement and velocity DOFs are present in physical coordinates. Thus, it is possible to assemble the substructures in a primal way after the transformation and superelements are created. For the assembled reduced matrices, the following dimension properties apply:

$$\displaystyle \begin{aligned} \boldsymbol{A}_{\mathrm{red}} \in \mathbb{C}^{{(n_{\mathrm{kept}}^{(\alpha)}+n_{r}^{(\alpha)}+n_{\mathrm{kept}}^{(\beta)}+n_{r}^{(\beta)}+n_{a}^{(\alpha)})}\times{(n_{\mathrm{kept}}^{(\alpha)}+n_{r}^{(\alpha)}+n_{\mathrm{kept}}^{(\beta)}+n_{r}^{(\beta)}+n_{a}^{(\alpha)})}}\quad\text{and} \end{aligned}$$

$$\displaystyle \begin{aligned} \boldsymbol{B}_{\mathrm{red}} \in \mathbb{C}^{{(n_{\mathrm{kept}}^{(\alpha)}+n_{r}^{(\alpha)}+n_{\mathrm{kept}}^{(\beta)}+n_{r}^{(\beta)}+n_{a}^{(\alpha)})}\times{(n_{\mathrm{kept}}^{(\alpha)}+n_{r}^{(\alpha)}+n_{\mathrm{kept}}^{(\beta)}+n_{r}^{(\beta)}+n_{a}^{(\alpha)})}}. \end{aligned}$$

By keeping the boundary displacement and velocity DOFs, the size of the assembled system increases with the number of attachment modes $n_{a}^{(\alpha )}$ and $n_{a}^{(\beta )}$ compared to Craig and Ni’s method and Liu and Zheng’s method.

11.4 Improvements of Free Interface Substructuring Approaches

11.4.1 Third and Higher-Order Reduction Interface Flexibility Representation

The determination of higher-order attachment modes as in the reduction according to Liu and Zheng can be extended arbitrarily. Thus, the influence of the truncated modes is represented more accurately. However, the size of the reduced system also increases.

Substructures Without Rigid Body Modes

In the case of substructures without rigid body modes, the attachment modes of higher order are generated by

$$\displaystyle \begin{aligned} \boldsymbol{\Phi}_{a,k} =\boldsymbol{Q}_{k}\boldsymbol{F}_{a} {} \end{aligned} $$

(11.73)

with

$$\displaystyle \begin{aligned} \boldsymbol{Q}_{k}=(-1)^{k-1}\boldsymbol{G}_{e}(\boldsymbol{A}\boldsymbol{G}_{e})^{k-1}+\boldsymbol{\Phi}_{\mathrm{kept}}\boldsymbol{\Lambda}_{\mathrm{kept}}^{-k}\boldsymbol{\Phi}_{\mathrm{kept}}^{T}\qquad\text{for}\qquad k = 1,\,\ldots ,\,h. {} \end{aligned}$$

Here, h corresponds to the highest order of the reduction.

Substructures with Rigid Body Modes

For substructures with rigid body modes, the attachment modes of higher order are generated by

$$\displaystyle \begin{aligned} \boldsymbol{\Phi}_{a,k} =\boldsymbol{Q}_{k}\boldsymbol{F}_{a} {} \end{aligned} $$

(11.74)

with

$$\displaystyle \begin{aligned} \boldsymbol{Q}_{k}=(-1)^{k-1}\boldsymbol{G}_{e}(\boldsymbol{A}\boldsymbol{G}_{e})^{k-1}\qquad\text{for}\qquad k = 1,\,\ldots ,\,h. {} \end{aligned}$$

Third-Order Reduction

Based on Eqs. (11.73) and (11.74), a third-order reduction (TO) will be explicitly given in the following. The third-order attachment modes for substructures without rigid body modes are defined as

$$\displaystyle \begin{aligned} \boldsymbol{\Phi}_{a3} =\left( (-1)^{2}\boldsymbol{G}_{e}(\boldsymbol{A}\boldsymbol{G}_{e})^{2}+\boldsymbol{\Phi}_{\mathrm{kept}}\boldsymbol{\Lambda}_{\mathrm{kept}}^{-3}\boldsymbol{\Phi}_{\mathrm{kept}}^{T}\right) \boldsymbol{F}_{a}, \end{aligned} $$

(11.75)

and the third-order attachment modes for substructures with rigid body modes are defined as

$$\displaystyle \begin{aligned} \boldsymbol{\Phi}_{a3} =\left( (-1)^{2}\boldsymbol{G}_{e}(\boldsymbol{A}\boldsymbol{G}_{e})^{2}\right) \boldsymbol{F}_{a}. \end{aligned} $$

(11.76)

Thus, the reduction is

(11.77)

The first and second-order attachment modes correspond to the attachment modes according to Craig/Ni and Liu/Zheng. In Eq. (11.77), $\boldsymbol {\ddot p}_{a}$ represents the third-order attachment modes’ DOFs, which can be interpreted as the second derivative of the generalized displacements DOFs p _a and can conceptually be identified as accelerations. The assembly follows the assembly procedure according to Liu and Zheng in Sect. 11.3.2.2 on the basis of the displacements and velocities. The elimination of acceleration DOFs $\boldsymbol {\ddot p}_{a}$ is not possible in state-space. Therefore, they are kept as independent DOFs and increase the size of the reduced system.

11.4.2 Combination of Liu/Zheng’s Reduction Basis and Primal Assembly Procedure

Furthermore, a combination of Liu and Zheng’s reduction basis as given in Sect. 11.3.2.1 and the primal assembly procedure as shown in Sect. 11.3.3.2 will be examined. In order to assemble the substructures in a primal fashion, the physical boundary displacement and velocity DOFs have to be retained in the reduced system. Thus, the assembled reduced system has a larger number of DOFs compared to Liu and Zheng’s assembly procedure, which eliminates all boundary DOFs. Following the reduction approach of Sect. 11.3.2.1, the assembly has to be based on the Eq. (11.52). By arranging the displacement and velocity DOFs in the state-space vector z ^(s) according to internal DOFs $\boldsymbol {z}_{i}^{(s)}$ and boundary DOFs $\boldsymbol {z}_{b}^{(s)}$, Eq. (11.52) can be partitioned:

(11.78)

For clarity, $\boldsymbol {\Phi }_{a1}^{(s)}$ and $\boldsymbol {\Phi }_{a2}^{(s)}$ are combined to and $\boldsymbol {p}_{a}^{(s)}$ and $\boldsymbol {\dot p}_{a}^{(s)}$ are combined to . In order to have the interface displacement and velocity DOFs in physical coordinates $\boldsymbol {z}_{b}^{(s)}$ after the reduction, the second row of Eq. (11.78) is solved for $\boldsymbol {p}_{a,\mathsf {LZ}}^{(s)}$:

$$\displaystyle \begin{aligned} \boldsymbol{p}_{a,\mathsf{LZ}}^{(s)}={\boldsymbol{\Phi}_{a,\mathsf{LZ},b}^{(s)}}^{-1}\boldsymbol{z}_{b}^{(s)}-{\boldsymbol{\Phi}_{a,\mathsf{LZ},b}^{(s)}}^{-1}\boldsymbol{\Phi}_{\mathrm{kept},b}^{(s)}\boldsymbol{p}_{\mathrm{kept}}^{(s)}. {} \end{aligned} $$

(11.79)

From Eq. (11.79), the transformation matrix $\boldsymbol {\Phi }_{2,\mathsf {LZ}}^{(s)}$ can be derived, which expresses the generalized DOFs $\boldsymbol {p}_{a,\mathsf {LZ}}^{(s)}$ depending on the physical DOFs $\boldsymbol {z}_{b}^{(s)}$ [32]

(11.80)

The final reduction follows from multiplying reduction matrix $\boldsymbol {\Phi }_{\mathsf {LZ}}^{(s)}$ and transformation matrix $\boldsymbol {\Phi }_{2,\mathsf {LZ}}^{(s)}$:

(11.81)

It can be seen that only the internal DOFs are transformed into generalized coordinates and the boundary displacement and velocity DOFs are present in physical coordinates. Thus, the substructures can be assembled in a primal way after the transformation and superelements are created. For the assembled reduced matrices, the following dimension properties apply:

$$\displaystyle \begin{aligned} \boldsymbol{A}_{\mathrm{red}} &\in \mathbb{C}^{{(n_{\mathrm{kept}}^{(\alpha)}+n_{r}^{(\alpha)}+n_{\mathrm{kept}}^{(\beta)}+n_{r}^{(\beta)}+n_{a1}^{(\alpha)}+n_{a2}^{(\alpha)})}\times{(n_{\mathrm{kept}}^{(\alpha)}+n_{r}^{(\alpha)}+n_{\mathrm{kept}}^{(\beta)}+n_{r}^{(\beta)}+n_{a1}^{(\alpha)}+n_{a2}^{(\alpha)})}} \qquad\text{and} \\ \boldsymbol{B}_{\mathrm{red}} &\in \mathbb{C}^{{(n_{\mathrm{kept}}^{(\alpha)}+n_{r}^{(\alpha)}+n_{\mathrm{kept}}^{(\beta)}+n_{r}^{(\beta)}+n_{a1}^{(\alpha)}+n_{a2}^{(\alpha)})}\times{(n_{\mathrm{kept}}^{(\alpha)}+n_{r}^{(\alpha)}+n_{\mathrm{kept}}^{(\beta)}+n_{r}^{(\beta)}+n_{a1}^{(\alpha)}+n_{a2}^{(\alpha)})}} \end{aligned} $$

By keeping the boundary DOFs, the size of the assembled system increases with the number of attachment modes $n_{a1}^{(\alpha )}+n_{a2}^{(\alpha )}$ and $n_{a1}^{(\beta )}+n_{a2}^{(\beta )}$ compared to Liu and Zheng’s method.

11.5 Numerical Applications

11.5.1 Beam Structure with Two Localized Dampers

To evaluate the proposed methods numerically, the clamped beam structure with two localized dampers shown in Fig. 11.1 is analyzed [16]. This example has already been used in [16]. The beam structure consists of 18 Euler-Bernoulli beam elements and only the bending vibration (no axial deformation) is considered. The total system has a length of 1.8 m and is divided into substructure α, consisting of ten beam elements and a length of 1.0 m, and substructure β, consisting of eight beam elements and a length of 0.8 m. A localized viscous damper with a damper constant c =  1.0 × 104 N s m−1 is attached to each substructure. The total system has m = 36 DOFs in the physical space and n = 72 DOFs in the state-space. Substructure α has n ^(α) = 40 DOFs in state-space and substructure β has n ^(β) = 36 DOFs in the state-space. Furthermore, substructure β has two physical rigid body modes $m_{r}^{(\beta )}=2$ (translation and rotation). In state-space representation, the physical rigid body modes lead to a single zero eigenvalue for the translational movement and to one zero eigenvalue with multiplicity two for the rotational movement. This can be traced back to the fact that the translation rigid body movement causes a damping force. On the other hand, the rotational rigid body movement is freely possible and doesn’t cause a damping force. This results in a regular rigid body mode for the translation and a regular and a generalized rigid body mode for the rotation in the state-space. Accordingly, there are $n_{r}^{(\beta )}=3$ state-space rigid body modes.

Fig. 11.1

Clamped beam structure with two localized dampers divided into two substructures [16]. The beam consists of 18 Euler-Bernoulli beam elements (Young’s modulus 2.1 × 1011 N m−2, density 7.8 × 103 kg m−3, cross-section 9.0 × 10−4 m2, moment of inertia 7.0 × 10−8 m4) and has n = 72 DOFs in state-space. The total length of the beam is 1.8 m. The length of substructure α is 1.0 m and the length of substructure β is 0.8 m. The damper constant c =  is 1.0 × 104 N s m−1

Figure 11.2 shows all 72 eigenvalues of the unreduced system in the complex plane. It can be seen that complex conjugate eigenvalues occur to a large extent . Altogether, there are 34 complex conjugate eigenvalue pairs. Thus, there are 68 complex eigenvalues with imaginary parts. In contrast, there are four real eigenvalues without imaginary parts (two of them with real part $\mathfrak{R} \left ( \lambda \right )< {-5000}\,{\mathrm{rad}\, \mathrm{s}^{-1} }$ ). This indicates very high damping of these eigenvalues [16].

Fig. 11.2

Exact eigenvalues of the coupled unreduced system of Fig. 11.1

Figure 11.3 shows the exact eigenvalues of both substructures. For substructure α, there are 19 eigenvalue pairs and 2 real eigenvalues. For substructure β, there are 16 eigenvalue pairs and one real eigenvalue as well as 3 rigid body modes with zero eigenvalue.

Fig. 11.3

Exact eigenvalues of the unreduced substructures of Fig. 11.1. (a) Substructure α. (b) Substructure β

The clamped beam is reduced and assembled using the substructuring methods described in Sects. 11.3 and 11.4. For the reduction of substructure α, $n_{\mathrm{kept}}^{(\alpha )}=20$ eigenmodes belonging to the 20 eigenvalues with the lowest absolute value are kept. These are nine complex conjugate pairs and two real eigenvalues without imaginary parts. For substructure β $n_{\mathrm{kept}}^{(\beta )}=15$, eigenmodes belonging to the 15 eigenvalues with the lowest absolute value are kept. These are seven complex conjugate pairs and one real eigenvalue without imaginary parts. Additionally, for the reduction of substructure β, the $n_{r}^{(\beta )}=3$ rigid body modes are used.

For the reduction according to Craig and Ni, both attachment modes are determined for substructure α according to Eq. (11.24) and for substructure β according to Eq. (11.34). For the reduction according to Liu and Zheng and for the reduction according to de Kraker and van Campen, the four attachment modes are determined for substructure α according to Eqs. (11.48) and (11.63), respectively, and for substructure β according to Eqs. (11.53) and (11.67), respectively. For the third order approach, six attachment modes are determined according to Eq. (11.73) for substructure α and according to Eq. (11.74) for substructure β.

After assembly, the reduced system according to Craig and Ni’s method (CN) and Liu and Zheng’s method (LZ) has n _red,CN = n _red,LZ = 38 DOFs. According to de Kraker and van Campen’s method (KC), Liu and Zheng’s reduction basis with primal assembly (LZ,KC), and the third order approach (TO), the reduced system has n _red,KC = n _red,LZ,KC = n _red,TO = 42 DOFs. Table 11.1 summarizes the number of used modes and the size of the reduced assembled systems.

Table 11.1 Modes used for reduction and resulting size of the reduced assembled system

In order to quantify the differences of the individual methods, the relative error of the real parts and imaginary parts of the eigenvalue λ _k is used in the following. The eigenvalues of the full unreduced system λ _full,k are set in relation to the eigenvalues of the assembled reduced system λ _red,k, which leads to the relative errors $\varepsilon _{\mathrm{rel},\Re ,k}$ of the real part and the relative errors ε _rel,ℑ,k of the complex parts:

$$\displaystyle \begin{aligned} \varepsilon_{\mathrm{rel},\Re,k}=\frac{\left| \Re\left( \lambda_{\mathrm{red},k}\right)-\Re\left( \lambda_{\mathrm{full},k}\right) \right|}{\Re\left( \lambda_{\mathrm{full},k}\right)}\quad\text{and}\quad\varepsilon_{\mathrm{rel},\Im,k}=\frac{\left| \Im\left( \lambda_{\mathrm{red},k}\right)-\Im\left( \lambda_{\mathrm{full},k}\right) \right|}{\Im\left( \lambda_{\mathrm{full},k}\right)} \end{aligned} $$

(11.82)

Figure 11.4 shows the relative errors of the real and imaginary parts corresponding to the 34 eigenvalues with the lowest absolute value for the various methods.For better distinguishability, only the relative errors ε _rel > 3 ⋅ 10⁻⁹ are depicted in the following figures. The relative errors of both eigenvalues belonging to a complex-conjugate pair are represented. The absence of a relative error of the imaginary part ε _rel,ℑ implies that the associated eigenvalue is purely real and has no imaginary part. Relative errors increase with increasing eigenvalue. In addition, the imaginary parts show a slightly better agreement than the real parts. In the case of purely real eigenvalues, the real parts approximate the exact eigenvalues very well. Comparing the substructuring methods of Sect. 11.3, the eigenvalues are approximated one to two orders of magnitude more accurately by using Liu and Zheng’s method and de Kraker and van Campen’s method compared to Craig and Ni’s method. Furthermore, no significant difference can be seen between Liu and Zheng’s method and de Kraker and van Campen’s method. Thus, Liu and Zheng’s method approximates the exact eigenvalues at a smaller size of the reduced system as accurately as de Kraker and van Campen’s method. Both approaches shown in Sect. 11.4 (the third order reduction in Sect. 11.4.1 and Liu and Zheng’s reduction basis with primal assembly in Sect. 11.4.2) lead to an improvement of the approximation in comparison to the aforementioned methods. The overall best approximation is achieved by the third order reduction.

Fig. 11.4

Relative error of the real and imaginary parts of the approximated eigenvalues λ _red of the clamped beam of Fig. 11.1. The relative errors of the 34 eigenvalues with the lowest absolute value are shown using Craig and Ni’s method (CN), Liu and Zheng’s method (LZ) and de Kraker and van Campen’s method (KC). Additionally, the combination of Liu and Zheng’s reduction basis and primal assembly (LZ,KC) as well as the third order reduction (TO) are shown. The number of DOFs of the assembled system according to Craig and Ni’s method as well as Liu and Zheng’s method is n _red,CN = n _red,LZ = 38 and according to de Kraker and van Campen’s method, the combination of Liu and Zheng’s reduction basis and primal assembly, as well as the third order reduction is n _red,KC = n _red,LZ,KC = n _red,TO = 42. (a) Relative error $\varepsilon _{\mathrm{rel},\Re ,k}$ of real part to eigenvalue λ _red,k. (b) Relative error ε _rel,ℑ,k of imaginary part to eigenvalue λ _red,k

Comparison Based on the Identical Size of the Reduced System

In Fig. 11.4, the comparison is based on an identical number of kept eigenmodes for the reduction in all approaches. Thus, the reduced assembled systems have a different number of remaining DOFs, depending on the assembly procedure. In contrast, the relative errors for reduced systems of the same size is considered here. In order to achieve this, different numbers of eigenmodes are kept for the methods, in such a way that the size n _red = 42 of the reduced system is the same for all methods. For Craig and Ni’s method as well as Liu and Zheng’s method, 22 eigenmodes are kept. De Kraker and van Campen’s method, Liu and Zheng’s reduction and primal assembly, as well as the third order reduction, are based on the identical number of modes as in the preceding analysis. Table 11.2 shows the modes used for the reduction and the resulting size of the reduced assembled system. Figure 11.5 shows the relative errors of the clamped beam of Fig. 11.1 based on the identical size of the reduced system. It can be seen that this leads to improved results for Craig and Ni’s method as well as Liu and Zheng’s method. However, Liu and Zheng’s reduction basis and primal assembly, as well as the third order reduction, still show better approximations of the exact eigenvalues.

Table 11.2 Modes used for reduction and resulting size of the reduced assembled system relating to Fig. 11.5

Fig. 11.5

Relative error of the real and imaginary parts of the approximated eigenvalues λ _red of the clamped beam of Fig. 11.1. The relative errors of the 34 eigenvalues with the lowest absolute value are shown using Craig and Ni’s method (CN), Liu and Zheng’s method (LZ) and de Kraker and van Campen’s method (KC). Additionally, the combination of Liu and Zheng’s reduction basis and primal assembly (LZ,KC), as well as the third order reduction (TO), are shown. The number of DOFs of the reduced assembled system is identical for all approaches n _red,CN = n _red,LZ = n _red,KC = n _red,LZ,KC = n _red,TO = 42. (a) Relative error $\varepsilon _{\mathrm{rel},\Re ,k}$ of real part to eigenvalue λ _red,k. (b) Relative error ε _rel,ℑ,k of imaginary part to eigenvalue λ _red,k

11.5.2 Damped Free-Free Beam Model

In this section, the damped free-free beam structure shown in Fig. 11.6 is analyzed [17]. This example has already been used in [17]. The beam structure consists of 20 Euler-Bernoulli beam elements and the bending vibration (no axial deformation) is considered. The total system has a length of 1.8 m and is divided into substructure α, consisting of 12 beam elements and a length of 1.08 m, and substructure β, consisting of eight beam elements and a length of 0.72 m. The damping is proportional to the stiffness for each element. For elements 1–5 and 13–16 (left parts of both substructures), a damping matrix C _e = 0.0002K _e is assumed. For elements 6–12 (right part of substructure α) the damping matrix is C _e = 0.0004K _e and for elements 17–20 (right part of substructure β) the damping matrix is C _e = 0.0001K _e. Due to the different damping per element, the damping of both the substructures and the overall system is non-proportional overall.

Fig. 11.6

Free-free beam structure divided into two substructures [17]. The beam consists of 20 Euler-Bernoulli beam elements (Young’s modulus 7.0 × 1010 N m−2, density 2.7 × 103 kg m−3, cross-section 5.0 × 10−5 m2, moment of inertia 1.04 × 10−10 m4) and has n = 84 DOFs in the state-space. The total length of the beam is 1.8 m. The length of substructure α is 1.08 m and the length of substructure β is 0.72 m. The damping for elements 1–5 and 13–16 is C _e = 0.0002K _e, for elements 6–12 C _e = 0.0004K _e and for elements 17–20 C _e = 0.0001K _e

The total system has n = 42 DOFs in the physical space and m = 84 DOFs in the state-space. Substructure α has n ^(α) = 52 DOFs in the state-space and substructure β has n ^(β) = 36 DOFs in the state-space. Furthermore, both substructures possess two physical rigid body modes ($m_{r}^{(\alpha )}=m_{r}^{(\beta )}=2$). In state-space representation, they lead to zero eigenvalues with multiplicity two for the translation as well as for the rotation for both substructures. This can be attributed to the fact that, due to the absence of local dampers, both the translational rigid body movements as well as the rotational rigid body movements do not cause any damping forces. Thus, for both substructures, one regular and one generalized rigid body mode apply with regard to the translation and also one regular and one generalized rigid body mode for the rotation result in the state-space, i.e., $n_{r}^{(\alpha )}=n_{r}^{(\beta )}=4$.

Figure 11.7 shows all 84 eigenvalues of the unreduced system in the complex plane. It can be seen that compared to the previous example in Fig. 11.2 significantly more real eigenvalues occur without imaginary parts. Altogether, there are 20 complex conjugate eigenvalue pairs. Thus, there are 40 complex eigenvalues with imaginary parts. In contrast, there are 40 real eigenvalues without imaginary parts. Additionally, four rigid body modes with zero eigenvalue occur. Furthermore, it can be seen that the maximum amount of the real parts is larger by two orders of magnitude compared to the previous example in Fig. 11.2, whereas the imaginary part is smaller by approximately two orders of magnitude.

Fig. 11.7

Exact eigenvalues of the coupled unreduced system of Fig. 11.6

Figure 11.8 shows the exact eigenvalues of the substructures. For substructure α there are 10 eigenvalue pairs and 28 real eigenvalues as well as 4 rigid body modes with zero eigenvalue. For substructure β there are 9 eigenvalue pairs and 14 real eigenvalues as well as 4 rigid body modes with zero eigenvalue.

Fig. 11.8

Exact eigenvalues of the unreduced substructures of Fig. 11.6. (a) Substructure α. (b) Substructure β

The damped free-free beam is reduced and assembled using the substructuring methods described in Sects. 11.3 and 11.4. For the reduction of substructure α, $n_{\mathrm{kept}}^{(\alpha )}=11$ eigenmodes belonging to the 11 eigenvalues with the lowest absolute value are kept. These are 5 complex conjugate pairs and one real eigenvalue without imaginary parts. For substructure β, $n_{\mathrm{kept}}^{(\beta )}=10$ eigenmodes belonging to the 10 eigenvalues with the lowest absolute value are kept. These are 5 complex conjugate pairs. Additionally, for the reduction of substructure α and substructure β the $n_{r}^{(\alpha )}=n_{r}^{(\beta )}=4$ rigid body modes are used.

For the reduction according to Craig and Ni, both attachment modes are determined according to Eq. (11.34) for substructure α and substructure β. For the reduction according to Liu and Zheng and for the reduction according to de Kraker and van Campen, four attachment modes are determined per substructure according to Eqs. (11.53) and (11.67). For the third order approach, six attachment modes are determined per substructure according to Eq. (11.74). After assembly, the reduced system according to Craig and Ni’s method (CN) and according to Liu and Zheng’s method (LZ) has n _red,CN = n _red,LZ = 29 DOFs. According to de Kraker and van Campen’s method (KC), Liu and Zheng’s reduction basis with primal assembly (LZ,KC), as well as the third order reduction (TO), the reduced systems have n _red,KC = n _red,LZ,KC = n _red,TO = 33 DOFs. Table 11.3 summarizes the number of used modes and the size of the reduced assembled system.

Table 11.3 Modes used for reduction and resulting size of the reduced assembled system

Additionally, the methods are compared based on the identical size of the reduced system. Therefore, an increased number of eigenmodes are kept for Craig and Ni’s method as well as Liu and Zheng’s method. The number of kept eigenmodes and the size of the reduced systems are given in brackets in Table 11.3.

Figures 11.9 and 11.10 show the relative errors of the real and imaginary parts belonging to the 21 eigenvalues with the lowest absolute value. In Fig. 11.9, Craig and Ni’s method and Liu and Zheng’s method have a smaller number of DOFs of the reduced system. In contrast, in Fig. 11.10, the number of kept eigenmodes for Craig and Ni’s method and Liu and Zheng’s method is increased compared to the other methods. Thus, the number of DOFs of the reduced system is identical for all approaches. The assembled system has four rigid body modes in the state-space. Therefore, no relative errors are determined for the eigenvalues 1–4.

Fig. 11.9

Relative error of the real and imaginary parts of the approximated eigenvalues λ _red of the free-free beam of Figure 11.6. The relative errors of the 21 eigenvalues with the lowest absolute value are shown using Craig and Ni’s method (CN), Liu and Zheng’s method (LZ) and de Kraker and van Campen’s method (KC). Additionally, the combination of Liu and Zheng’s reduction basis and primal assembly (LZ,KC) as well as the third order reduction (TO) are shown. The number of DOFs of the assembled system according to Craig and Ni’s method as well as Liu and Zheng’s method is n _red,CN = n _red,LZ = 29 and according to de Kraker and van Campen’s method, the combination of Liu and Zheng’s reduction basis and primal assembly, as well as the third order reduction is n _red,KC = n _red,LZ,KC = n _red,TO = 33. (a) Relative error $\varepsilon _{\mathrm{rel},\Re ,k}$ of real part to eigenvalue λ _red,k. (b) Relative error ε _rel,ℑ,k of imaginary part to eigenvalue λ _red,k

Fig. 11.10

Relative error of the real and imaginary parts of the approximated eigenvalues λ _red of the free-free beam of Figure 11.6. The relative errors of the 21 eigenvalues with the lowest absolute value are shown using Craig and Ni’s method (CN), Liu and Zheng’s method (LZ) and de Kraker and van Campen’s method (KC). Additionally, the combination of Liu and Zheng’s reduction basis and primal assembly (LZ,KC) as well as the third order reduction (TO) are shown. The number of DOFs of the reduced assembled system is identical for all approaches n _red,CN = n _red,LZ = n _red,KC = n _red,LZ,KC = n _red,TO = 33. (a) Relative error $\varepsilon _{\mathrm{rel},\Re ,k}$ of real part to eigenvalue λ _red,k. (b) Relative error ε _rel,ℑ,k of imaginary part to eigenvalue λ _red,k

Comparing the substructuring methods described in Sect. 11.3, Liu and Zheng’s method shows the most accurate results. Despite the increased size of the reduced system, de Kraker and van Campen’s method leads to larger relative errors. When considering the relative error ε _rel,ℑ of the imaginary parts of eigenvalue 5 and 6, the difference between Liu and Zheng’s method and de Kraker and van Campen’s method is four orders of magnitude. The worst approximation is obtained using Craig and Ni’s method.

Liu and Zheng’s reduction basis with primal assembly leads to an improvement in the results compared to Liu and Zheng’s method and de Kraker and van Campen’s method. Compared to Liu and Zheng’s method, the relative error according to Liu and Zheng’s reduction with primal assembly is lower by a half to one order of magnitude.

The most exact approximation is achieved by the third order reduction. In comparison to Liu and Zheng’s reduction and primal assembly, the exact eigenvalues are approximated more precisely by about one order of magnitude using the third order reduction.

11.5.3 Damped Beam Model with Four Substructures and Localized Dampers

As a concluding example, the damped beam structure with four substructures and localized dampers as shown in Fig. 11.11 is analyzed. The beam structure consists of 26 Euler-Bernoulli beam elements and the bending vibration (no axial deformation) is considered. The total system has a length of 2.6 m and is divided into substructure α, consisting of eight beam elements and a length of 0.8 m, substructure β, consisting of six beam elements and a length of 0.6 m, substructure γ consisting of five beam elements and a length of 0.5 m, and substructure δ consisting of seven beam elements and a length of 0.7 m. Two localized viscous dampers are attached to substructure β and one localized viscous damper to substructure δ, each with a damper constant of c =  1.0 × 104 N s m−1. In addition, the damping matrix per element is assumed to be proportional to the stiffness matrix. For elements 1–8 and 20–26, a damping matrix of C _e = 0.0002K _e is assumed. C _e = 0.0001K _e is prescribed for elements 9–14 and C _e = 0.0004K _e for the elements 15–19. Thus, the damping of the overall system is not proportional, due to the localized dampers on the one hand and due to the different stiffness proportional damping properties per element on the other hand. The total system has n = 52 DOFs in physical space and m = 104 DOFs in state-space. Substructure α has n ^(α) = 32 DOFs in state-space, substructure β has n ^(β) = 28 DOFs in state-space, substructure γ has n ^(γ) = 24 DOFs in state-space and substructure δ has n ^(δ) = 32 DOFs in state-space. Furthermore, substructures β, γ and δ have two physical rigid body modes ($m_{r}^{(\beta )}=m_{r}^{(\gamma )}=m_{r}^{(\delta )}=2$). For the state-space rigid body modes, different cases occur considering the three substructures β, γ and δ. For substructure β, each of the physical rigid body modes leads in state-space to a simple zero eigenvalue for the translation and for the rotation. This can be attributed to the fact that due to the two localized dampers, both the translational rigid body motion and the rotational rigid body motion cause damping forces. Considering substructure γ, there is one zero eigenvalue with multiplicity two for the translation as well as for the rotation in state-space. This follows from the fact that due to the absence of localized dampers, both the translational rigid body motion and the rotational rigid body motion do not cause any damping forces. For substructure δ, the translational rigid body motion causes a damping force, but the rotational rigid body motion is possible without causing any damping force. One single zero eigenvalue for the translation and one zero eigenvalue with multiplicity two for the rotation result for substructure δ in state-space. Thus, in state-space, the physical rigid body modes for substructure β lead to one regular state-space rigid body mode for the translation and one regular state-space rigid body mode for the rotation. For substructure γ, they lead to one regular and one generalized state-space rigid body mode for the translation and also one regular and one generalized state-space rigid body mode for the rotation. Considering substructure δ, one regular state-space rigid body mode for the translation and one regular and one generalized state-space rigid body mode for the rotation occur. Accordingly, $n_{r}^{(\beta )}=2$, $n_{r}^{(\gamma )}=4$ and $n_{r}^{(\delta )}=3$ holds for the number of state-space rigid body modes. This example represents all possible combinations of regular and generalized state-space rigid body modes.

Fig. 11.11

Clamped beam structure with three local dampers divided into four substructures. It consists of 26 Euler-Bernoulli beam elements (Young’s modulus 2.1 × 1011 N m−2, density 7.8 × 103 kg m−3, cross-section 5.0 × 10−5 m2, moment of inertia 1.04 × 10−10 m4) and has n = 104 DOFs in the state-space. The total length of the beam is 2.6 m. The length of substructure α is 0.8 m, the length of substructure β is 0.6 m, the length of substructure γ is 0.5 m and the length of substructure δ is 0.7 m. The damper constant c =  is 1.0 × 104 N s m−1. The damping for the elements 1–8 and 20–26 is C _e = 0.0002K _e, for the elements 9–14 C _e = 0.0001K _e and for the elements 15–19 C _e = 0.0004K _e

Figure 11.12 shows all 104 eigenvalues of the unreduced system in the complex plane. In total, there are 28 complex conjugate eigenvalue pairs. Thus, there are 56 complex eigenvalues with imaginary parts. In contrast, there are 48 real eigenvalues without imaginary parts. The real parts and imaginary parts are roughly in the order of magnitude of the preceding example in Fig. 11.7.

Fig. 11.12

Exact eigenvalues of the coupled unreduced system of Fig. 11.11

Figure 11.13 shows the exact eigenvalues of the substructures. For substructure α, there are 8 eigenvalue pairs and 16 real eigenvalues. For substructure β, there are 8 eigenvalue pairs and 10 real eigenvalues as well as 2 rigid body modes with zero eigenvalue. Substructure γ shows 3 eigenvalue pairs and 14 real eigenvalues as well as 4 rigid body modes with zero eigenvalue. For substructure δ, there are 7 eigenvalue pairs and 15 real eigenvalues as well as 3 rigid body modes with zero eigenvalue.

Fig. 11.13

Exact eigenvalues of the unreduced substructures of Figure 11.11. (a) Substructure β. (b) Substructure γ. (c) Substructure δ

In comparison to the beam structure in Fig. 11.1, substructure α in Fig. 11.11 and substructure α Fig. 11.1 are basically identical to one another. Substructure δ in Fig. 11.11 and substructure β in Fig. 11.1 are also basically identical to each another. The main difference between both systems is that for the beam structure in Fig. 11.11, the damping matrix of each element is additionally proportional to the stiffness matrix while for the beam structure of Fig. 11.1 viscous damping occurs only due to the two localized dampers. When considering the eigenvalues of the substructures of the two different systems in Figs. 11.3 and 11.13, this leads to a increased number of purely real eigenvalues for the substructures with stiffness proportional damping. In addition, the eigenvalues show a significantly higher real part.

The clamped beam is reduced and assembled using the substructuring methods described in Sects. 11.3 and 11.4. For the reduction of substructure α, $n_{\mathrm{kept}}^{(\alpha )}=15$ eigenmodes belonging to the 15 eigenvalues with the lowest absolute value are kept. These are 6 complex conjugate pairs and 3 real eigenvalues without imaginary parts. For substructure β, $n_{\mathrm{kept}}^{(\beta )}=10$ eigenmodes belonging to the 10 eigenvalues with the lowest absolute value are kept. These are 5 complex conjugate pairs. For substructure γ, $n_{\mathrm{kept}}^{(\gamma )}=13$ eigenmodes belonging to the 13 eigenvalues with the lowest absolute value are kept. These are 3 complex conjugate pairs and 7 real eigenvalues without imaginary parts. For substructure δ, $n_{\mathrm{kept}}^{(\delta )}=12$ eigenmodes belonging to the 12 eigenvalues with the lowest absolute value are kept. These are 5 complex conjugate pairs and 2 real eigenvalues without imaginary parts. Additionally, for the reduction of substructure β the $n_{r}^{(\beta )}=2$ rigid body modes, for substructure γ the $n_{r}^{(\gamma )}=4$ and for substructure δ the $n_{r}^{(\delta )}=3$ state-space rigid body modes are used.

For Craig and Ni’s method, two attachment modes are determined for each boundary of each substructure. For Liu and Zheng’s and for de Kraker and van Campen’s method, four attachment modes are used for each boundary. For the third order approach, six attachment modes per interface are determined according to Eqs. (11.73) and (11.74). After assembly, the reduced system has n _red,CN = n _red,LZ = 59 DOFs for Craig and Ni’s method (CN) and Liu and Zheng’s method (LZ). The reduced system has n _red,KC = n _red,LZ,KC = n _red,TO = 71 DOF for de Kraker and van Campen’s method (KC), Liu and Zheng’s reduction with primal assembly and the third order reduction (TO). Due to the increased number of interface DOFs compared to the previous examples, there is a clear difference between the approaches. Table 11.4 summarizes the number of modes used and the size of the reduced assembled systems of all methods.

Table 11.4 Modes used for reduction and resulting size of the reduced assembled system

Additionally, the methods will be compared for the identical size of the reduced system in the following. Therefore, an increased number of eigenmodes are kept for Craig and Ni’s method as well as for Liu and Zheng’s method. The number of kept eigenmodes and the size of the reduced systems are given in brackets in Table 11.4.

Figures 11.14 and 11.15 show the relative errors of the real and imaginary parts corresponding to the 39 eigenvalues with the lowest absolute value. When comparing the approaches with the identical number of remaining DOFs of the reduced system in Fig. 11.14, Liu and Zheng’s method consistently achieves more precise approximations than Craig and Ni’s methods. The difference is between five orders of magnitude for low eigenvalues and two orders of magnitude for high eigenvalues. This also holds in Fig. 11.15, where the number of kept eigenmodes for Craig and Ni’s method and Liu and Zheng’s method is increased. Liu and Zheng’s reduction basis with primal assembly as well as the third order reduction lead in both Figs. 11.14 and 11.15 to lower relative errors compared to Craig and Ni’s method, de Kraker and van Campen’s method and Liu and Zheng’s method. Overall, the third order reduction achieves the most accurate approximation.

Fig. 11.14

Relative error of the real and imaginary parts of the approximated eigenvalues λ _red of the clamped beam with four substructures of Fig. 11.11. The relative errors of the 39 eigenvalues with the lowest absolute value are shown using Craig and Ni’s method (CN), Liu and Zheng’s method (LZ) and de Kraker and van Campen’s method (KC). Additionally, the combination of Liu and Zheng’s reduction basis and primal assembly (LZ,KC) as well as the third order reduction (TO) are shown. The number of DOFs of the assembled system according to Craig and Ni’s method as well as Liu and Zheng’s method is n _red,CN = n _red,LZ = 59 and according to de Kraker and van Campen’s method, the combination of Liu and Zheng’s reduction basis and primal assembly, as well as the third order reduction is n _red,KC = n _red,LZ,KC = n _red,TO = 71. (a) Relative error $\varepsilon _{\mathrm{rel},\Re ,k}$ of real part to eigenvalue λ _red,k. (b) Relative error ε _rel,ℑ,k of imaginary part to eigenvalue λ _red,k

Fig. 11.15

Relative error of the real and imaginary parts of the approximated eigenvalues λ _red of the clamped beam with four substructures of Figure 11.11. The relative errors of the 39 eigenvalues with the lowest absolute value are shown using Craig and Ni’s method (CN), Liu and Zheng’s method (LZ) and de Kraker and van Campen’s method (KC). Additionally, the combination of Liu and Zheng’s reduction basis and primal assembly (LZ,KC) as well as the third order reduction (TO) are shown. The number of DOFs of the reduced assembled system is identical for all approaches n _red,CN = n _red,LZ = n _red,KC = n _red,LZ,KC = n _red,TO = 71. (a) Relative error $\varepsilon _{\mathrm{rel},\Re ,k}$ of real part to eigenvalue λ _red,k. (b) Relative error ε _rel,ℑ,k of imaginary part to eigenvalue λ _red,k

11.6 Conclusions

In this paper, a derivation and comparison of three existing free interface substructuring methods for viscously damped systems were provided and two new approximation approaches were proposed. The three existing investigated methods are Craig and Ni’s method, Liu and Zheng’s method, and de Kraker and van Campen’s method. Liu and Zheng’s method can be considered as a second-order extension of Craig and Ni’s method. We extended Liu and Zheng’s method to a third-order approximation and generalized it further to arbitrarily higher orders. Moreover, a new method combining the reduction basis of Liu and Zheng’s approach with the primal assembly procedure applied by de Kraker and van Campen was derived.

The new method combining Liu and Zheng’s reduction basis with primal assembly and the third-order method give the best results and are recommended for the approximation of arbitrarily viscous damped substructured systems. The third-order method in particular shows the very best results, but increases the size of the reduced system compared to Craig and Ni’s and Liu and Zheng’s method. Liu and Zheng’s reduction basis with primal assembly outperforms de Kraker and van Campen’s method, while both methods generate the same size of the reduced system.

The five methods were applied to three different beam structures. In the future, we want to apply the method to bigger problems with a larger number of DOFs. The examples used in this paper are very illustrative, allow for comparison to results in the literature and demonstrate all critical points for the application of the suggested methodology. However, they are too small for a meaningful comparison in terms of computational time. For this purpose, it is necessary to consider additional numerical examples to further examine the performance of the proposed methods.