Dynamic substructuring (DS) is an engineering tool used to model and analyse the dynamics of mechanical systems by means of its components or substructures. Using the dynamic substructuring approach one is able to analyse the dynamic behaviour of substructures separately and to later on calculate the assembled dynamics using coupling procedures. Dynamic substructuring has several advantages over the analysis of the fully assembled system: Substructures can be modelled in the domain that is most appropriate, e.g. experimentally obtained substructures can be combined with numerical models. Large and/or complex systems can be optimized on substructure level. Numerical computation load can be reduced as solving several substructures is computationally less demanding than solving one large system. Substructure models of different development groups can be shared and combined without exposing the modelling details. Dynamic substructuring is particularly tailored to simulation of mechanical vibrations, which has implications for many product aspects such as sound / acoustics, fatigue / durability, comfort and safety. Also, dynamic substructuring is applicable to any scale of size and frequency. It is therefore a widely used paradigm in industrial applications ranging from automotive and aerospace engineering to design of wind turbines and high-tech precision machinery.
History
The roots of dynamic substructuring can be found in the field of domain decomposition. In 1890 the mathematician Hermann Schwarz came up with an iterative procedure for domain decomposition which allows to solve for continuous coupled subdomains. However, many of the analytical models of coupled continuous subdomains do not have closed-form solutions, which led to discretization and approximation techniques such as the Ritz method (which is sometimes called the Rayleigh-Ritz method due to the similarity between Ritz's formulation and the Rayleigh ratio) the boundary element method (BEM) and the finite element method (FEM). These methods can be considered as "first level" domain decomposition techniques.
The finite element method proved to be the most efficient method and the invention of the microprocessor made it possible to easily solve a large variety of physical problems. In order to analyse even larger and more complex problems, methods were invented to optimize the efficiency of the discretized calculations. The first step was replacing the direct solvers by iterative solvers such as the conjugate gradient method. The lack of robustness and slow convergence of these solvers did not make them an interesting alternative in the beginning. The rise of parallel computing in the 1980s however sparked their popularity. Complex problems could now be solved by dividing the problem into subdomains, each processed by a separate processor, and solving for the interface coupling iteratively. This can be seen as a second level domain decomposition as is visualized in the figure.
The efficiency of dynamic modelling could be increased even further by reducing the complexity of the individual subdomains. This reduction of the subdomains (or substructures in the context of structural dynamics) is realized by representing substructures by means of their general responses. Expressing the separate substructures by means of their general response instead of their detailed discretization led to the so-called dynamic substructuring method. This reduction step also allowed for replacing the mathematical description of the domains by experimentally obtained information. This reduction step is also visualized by the reduction arrow in the figure.
The first dynamic substructuring methods were developed in the 1960s and were more commonly known under the name component mode synthesis (CMS). The benefits of dynamic substructuring were quickly discovered by the scientific and engineering communities and it became an important research topic in the field of structural dynamics and vibrations. Major developments followed, resulting in e.g. the classic Craig-Bampton method. The Craig-Bampton method employs static condensation (Guyan reduction) and modal truncation techniques to effectively reduce the degrees of freedom in a system.
Due to improvements in sensor and signal processing technology in the 1980s, substructuring techniques also became attractive for the experimental community. Methods dealing with structural dynamic modification were created in which coupling techniques were directly applied to measured frequency response functions (FRFs). Broad popularity of the method was gained when Jetmundsen et al. formulated the classical frequency-based substructuring (FBS) method, which laid the ground work for frequency-based dynamic substructuring. In 2006 a systematic notation was introduced by De Klerk et al. in order to simplify the difficult and elaborate notation that had been used prior. The simplification was done by means of two Boolean matrices that handle all the "bookkeeping" involved in the assembly of substructures
Domains
Dynamic substructuring can best be seen as a domain-independent toolset for assembly of component models, rather than a modelling method of its own. Generally, dynamic substructuring can be used for all domains that are well suited to simulate multiple input/multiple output behaviour. Five domains that are well suited for substructuring are summarized in the below table.
+Dynamic equations for five domainsDomainDynamic equationAdditional informationPhysical domain represent the linear(ised) mass, damping and stiffness matrix of the system. Modal domain represent the modally reduced mass, damping and stiffness matrix; is the set of modal amplitudes.Frequency domain is the impedance FRF matrix; is the admittance FRF matrix.Time domain is the IRF matrix.State-space domain are the state-space matrices; , and represent the state, input and output vector.
The physical domain concerns methods that are based on (linearised) mass, damping and stiffness matrices, typically obtained from numerical FEM modelling. Popular solutions to solve the associated system of second order differential equations are the time integration schemes of Newmark and the Hilbert-Hughes-Taylor scheme. The modal domain concerns component mode synthesis (CMS) techniques such as the Craig-Bampton, Rubin and McNeal method. These methods provide efficient modal reduction bases and assembly techniques for numerical models in the physical domain. The frequency domain is more popularly known as frequency-based substructuring (FBS). Based on the classic formulation of Jetmundsen et al. and the reformulation of De Klerk et al., it has become the most commonly used domain for substructuring, because of the ease of expressing the differential equations of a dynamical system (by means of frequency response functions, FRFs) and the convenience of implementing experimentally obtained models. The time domain refers to the recently proposed concept of impulse-based substructuring (IBS), which expresses the behaviour of a dynamic system using a set of impulse response functions (IRFs). The state-space domain, finally, refers to methods proposed by Sjövall et al. that employ system identification techniques common to control theory.
As dynamic substructuring is a domain-independent toolset, it is applicable to the dynamic equations of all domains. In order to establish substructure assembly in a particular domain, two interface conditions need to be implemented. This is explained next, followed by a few common substructuring techniques.
Interface conditions To establish substructuring coupling / decoupling in each of the above-mentioned domains, two conditions should be met: Coordinate compatibility, i.e. the connecting nodes of two substructures should have equal interface displacement. Force equilibrium, i.e. the interface forces between connecting nodes have equal magnitude and opposing sign. These are the two essential conditions that keep substructures together, hence allow to construct an assembly of multiple components. Note that the conditions are comparable with Kirchhoff's laws for electric circuits, in which case similar conditions apply to currents and voltages though/over electric components in a network; see also Mechanical–electrical analogies.
Substructure connectivity
Consider two substructures A and B as depicted in the figure. The two substructures comprise a total of six nodes; the displacements of the nodes are described by a set of Degrees of Freedom (DoFs). The DoFs of the six nodes are partitioned as follows: DoFs of the internal nodes of substructure A; DoFs of the coupling nodes of substructures A and B, i.e. interface DoFs; DoFs of the internal nodes of substructure B. Note that the denotation 1, 2 and 3 indicates the function of the nodes/DoFs rather than the total amount. Let us define the sets of DoFs for the two substructures A and B in concatenated form. The displacements and applied forces are represented by the sets and . For the purpose of substructuring, a set of interface forces is introduced which only contains non-zero entries on the interface DoFs:
The relation between dynamic displacements and applied forces of the uncoupled problem is governed by a particular dynamic equation, such as presented in the table above. The uncoupled equations of motion are augmented by extra terms/equations for compatibility and equilibrium, as discussed next.
Compatibility The compatibility condition requires that the interface DoFs have the same sign and value at both sides of the interface: . This condition can be expressed using a so-called signed Boolean matrix, denoted by . For the given example this can be expressed as:
In some cases the interface nodes of the substructures are non-conforming, e.g. when two substructures are meshed separately. In such cases a non-Boolean matrix has to be used in order to enforce a weak interface compatibility.
A second form in which the compatibility condition can be expressed is by means of coordinate substitution by a set of generalised coordinates . The set contains the unique coordinates that remain after assembly of the substructures. Every matching pair of interface DoFs is described by a single generalised coordinate, which means that the compatibility condition is automatically enforced. Expressing using gives:
Matrix is referred to as the Boolean localisation matrix. A useful relation between matrix and can be exposed by noting that compatibility should hold for any set of physical coordinates expressed by . Indeed, substituting in the equation :
Hence represents the nullspace of :
This means in practice that one only needs to define or ; the other Boolean matrix is calculated using the nullspace property.
Equilibrium The second condition that has to be satisfied for substructure assembly is the force equilibrium for matching interface forces . For the current example, this condition can be written as . Similar to the compatibility equation, the force equilibrium condition can be expressed using a Boolean matrix. Use is made of the transpose of the Boolean localisation matrix that was introduced to write compatibility:
The equations for and state that the interface forces on internal nodes are zero, hence not present. The equation for correctly establishes the force equilibrium between a matching pair of interface DoFs according to Newton's third law.
A second notation in which the equilibrium condition can be expressed is by introducing a set of Lagrange multipliers . The substitution of these Lagrange multipliers is possible as and differ only in sign, not in value. Using again the signed Boolean matrix :