1 Modeling Domains

Simulate sound & vibration behavior prior to prototype

Dynamic Substructuring (DS) is a powerful engineering concept to model and analyze dynamic systems in terms of their components or substructures. It derives from the ancient principle of “divide and conquer” (“divide et impera”), which suggests that large problems are most effectively governed by dividing them into smaller parts and choosing the best approach for each. Dynamic Substructuring does just that: it allows to model each component using the most quality- and cost-effective technique and “substructure” its dynamics together. This way, one can simulate the sound and vibration behavior of the full assembly (the end-product), long before any physical prototype is available.

Domains

For structural dynamic simulation, several domains can be chosen depending on the component at hand, the targeted analysis, the availability of computational or experimental resources, and perhaps personal preference. The following figure shows a schematic overview of five domains commonly encountered in structural dynamic simulation: the physical, modal, frequency, time, and state-space domains. Arrows indicate some typical (but not all) conversions between the domains.

All domains implement a common system to describe the interface (or boundary) conditions, which allows a user to “hybridize” a substructuring simulation. This page introduces the basic formulations to model a linear dynamic system in each of the five domains.

The physical domain modeling approach yields a second-order differential equation based on an equilibrium between internal forces of the structure (due to inertia, viscous damping, and elasticity) and externally applied forces.

$$ \mathbf{M}\ddot{\mathbf{u}}\mathrm{(t)} + \mathbf{C}\dot{\mathbf{u}}\mathrm{(t)} + \mathbf{Ku}\mathrm{(t)} = \mathbf{f}\mathrm{(t)} $$

$$ \mathbf{M} $$, $$ \mathbf{C} $$, $$ \mathbf{K} $$  represent the linearized mass, damping and stiffness matrix of the system. These matrices are often obtained from Finite Element (FE) modeling, and are referred to as the numerical model of the structure. Furthermore, $$ \mathbf{u} $$ represents the DoFs and $$ \mathbf{f} $$ the set of externally applied loads which are time dependent $$ \mathrm{(t)} $$.

To simulate the structure’s response to a load case in the time domain, one could apply a time integration algorithm to either the full or the reduced-order system matrices. The latter is obtained by a modal reduction, which leads to a representation in the modal domain.

Modal domain

A dynamic analysis typically concentrates on the structural behavior in the lower frequency range, for which the structure deforms rather globally. Finite Element models accommodate such rapid phenomena in their solution. By discarding the solution space associated with these rapid and local phenomena, one retains a smaller solution space, suited to describe the global behavior of the structure more efficiently. This is the general idea of modal reduction.

Natural vibration modes (eigenmodes)

In the absence of damping, the eigenmodes and their corresponding eigenfrequencies $$ \mathrm{\omega} _{r} $$ can be computed iteratively by an eigensolver that seeks for a solution in the form:

$$ (\mathbf{K} – \mathrm{\mathrm{\omega}}_{r}^{2} \mathbf{M})\boldsymbol{\varphi}{_r } = \mathbf{0} \;\;\;\;\;\;\;\;\;\; {r} = 1,…,n $$

A numerical model consists of $$ \mathit{n} $$ eigensolutions, which equals the number of DoFs in the system. The eigenmodes $$ \boldsymbol{\varphi}{_r } $$ are typically stored in the columns of a mode shape matrix $$ \boldsymbol{\Phi} $$, in order of increasing eigenfrequency $$ \omega _{{r}} $$, orthogonal and mass-normalized, such that:

$$ \boldsymbol{\Phi } = \left [ \boldsymbol{\varphi }_{1}, \boldsymbol{\varphi }_{2}, …, \boldsymbol{\varphi }_{n} \right ] $$

$$ \boldsymbol{\Phi} ^{\mathrm{T}}\mathbf{ M} \boldsymbol{\Phi } = \mathbf{I} $$

$$ \boldsymbol{\Phi} ^{\mathrm{T}}\mathbf{ K} \boldsymbol{\Phi } = \mathrm{diag}\left ( \omega _{1}^{\mathrm{2}}, …, \omega _{n}^{2} \right ) $$

A special solution are the rigid body modes for which $$ \omega _r = 0 $$.

Modal reduction by truncation

By limiting the displacements $$ \mathbf{u}\mathrm{(t)} $$ to the the first $$ m \ll n $$ eigenmodes, one can drastically reduce the effective dimension of the original problem. This reduction matrix $$ \mathbf{R} = \left [ \boldsymbol{\varphi }_{1}, \boldsymbol{\varphi }_{2}, …, \boldsymbol{\varphi }_{m} \right ] $$ yields an approximation of the vector of displacements:

$$ \mathbf{u} \left ( \mathrm{t} \right ) \approx \mathbf{R}\boldsymbol{\eta }\left ( \mathrm{t} \right ) $$

The physical displacements are now replaced by a smaller set of time-dependent modal coordinates $$ \boldsymbol{\eta }\left ( \mathrm{t} \right ) $$, representing the amplitudes of the mode shapes in the columns of $$ \mathbf{R} $$. Since the reduction is an approximation, an additional residual term $$ \mathbf{r} $$ arises:

$$ \mathbf{MR}\ddot{\boldsymbol{\eta }} \mathrm{(t)} + \mathbf{CR}\dot{\boldsymbol{\eta }} \mathrm{(t)} + \mathbf{KR}\boldsymbol{{\eta }} \mathrm{(t)} = \mathbf{f}\mathrm{(t)} + \mathbf{r}\mathrm{(t)} $$

$$ \mathbf{R}^\mathrm{{T}}\mathbf{r}\mathrm{(t)} = \mathbf{0} $$

The modally reduced problem of dimension $$ m $$ reads:

$$ \mathbf{M}_{m}\ddot{\boldsymbol{\eta }}\mathrm{(t)} + \mathbf{C}_{m}\dot{\boldsymbol{\eta }}\mathrm{(t)}+ \mathbf{K}_{m}\boldsymbol{{\eta }}\mathrm{(t)} = \mathbf{R}^\mathrm{{T}}\mathbf{f}\mathrm{(t)} \;\; \left\{\begin{matrix}
\mathbf{M}_{m} = \mathbf{R}^{\mathrm{T}} \mathbf{MR}\\
\mathbf{C}_{m} = \mathbf{R}^{\mathrm{T}} \mathbf{CR}\\
\mathbf{K}_{m} = \mathbf{R}^{\mathrm{T}} \mathbf{KR}
\end{matrix}\right. $$

The matrices with subscript $$ m $$ are the reduced-order system matrices in modal coordinates.

Practical benefit:

• A projection on the space of natural vibration modes is remarkably efficient, because the resulting modal matrices are diagonal if the damping matrix is proportional.

Frequency domain

One can apply a Fourier transform and write the dynamic equilibrium as a function of the excitation frequency $$ \omega $$ . Inserting the fact that $$ \dot{\mathbf{u}} (\omega ) = i\omega \mathbf{u}(\omega) $$ and $$ \ddot{\mathbf{u}} (\omega ) = – \omega^\mathrm{{2}}\mathbf{u}(\omega) $$ allows to gather the physical domain system matrices into a single dynamic stiffness matrix $$ \mathbf{Z}(\omega) $$, depending on frequency:

$$ \mathbf{M}\ddot{\mathbf{u}} (\omega) + \mathbf{C}\dot{\mathbf{u}} (\omega) + \mathbf{K{u}} (\omega) = \mathbf{f }(\omega) $$

$$ \left [ -\omega ^2 \mathbf{M} + i \omega \mathbf{C} + \mathbf{K}\right ] \mathbf{u}(\omega) = \mathbf{f}(\omega) $$

$$ \mathbf{Z}(\omega) \mathbf{u}(\omega) = \mathbf{f}(\omega) $$

The dynamic stiffness $$ \mathbf{Z}(\omega) $$ describes the force needed to generate a unit harmonic displacement on one particular DoF, while all other DoFs are constrained. $$ \mathbf{Z}(\omega) $$ can be a sparse matrix, as it only comprises element behavior. By inversion of this matrix, the receptance matrix $$\mathbf{Y}(\omega) $$ is obtained:

$$ \mathbf{u}(\omega) = \mathbf{Y}(\omega) \mathbf{f}(\omega) \;\;\;\; \mathrm{with} \;\;\;\; \mathbf{Y}(\omega) = (\mathbf{Z(\omega)})^{-1} $$

The elements in the receptance matrix $$ \mathbf{Y}(\omega) $$ are called Frequency Response Functions (FRFs) and describe the displacement response as a result of a unit harmonic force. $$ \mathbf{Y}(\omega) $$ is generally a full matrix.

Notation:

The following table gives an overview of common terminology for relations between motion and forces.

Relation to experimental testing:

The admittance FRFs are more familiar and intuitive, as they are naturally obtained by experimental testing. A column of $$ \mathbf{Y}(\omega) $$ governs the global response as a result of an excitation at one DoF.

Practical benefits:

• Admittance FRFs employ condensed dynamic information and are therefore particularly suited for substructuring.
• To that extent, the frequency domain forms a bridge between the numerical modeling and experimental testing approach.

Time domain

By inverse Fourier transform of the admittance matrix, we obtain Impulse Response Functions (IRFs). These functions, convolved with the history of applied forces, also yield a response solution in the time domain.

$$ \mathbf{u}\mathrm{(t)} = \mathbf{Y}\mathrm{(t)} \; * \; \mathbf{f}\mathrm{(t)} = \int_{\tau =0}^{\mathrm{t}} \mathbf{Y}(\tau) \mathbf{f} (\mathrm{t}-\tau) d\tau $$

$$ \mathbf{Y}\mathrm{(t)} $$ is a matrix of IRFs, with each entry describing the displacement response to an applied impulse of unit momentum and infinitely short time.

Practical benefits:

• Concerning experimental modeling, a good approximation of an IRF is obtained from an impact hammer measurement.
• In contrast to the original physical domain, only the input/output DoFs in the system need to be considered for a simulation, as the effects of the internal dynamics are already condensed in the IRFs.

State-space domain

The state-space domain, finally, expresses the structural dynamics as a system of first-order differential equations. It employs the ABCD-quadruple, which can be obtained by transforming the physical domain system matrices (the numerical approach) or by system identification using measurement data (the experimental approach).

$$ \begin{matrix}
\dot{\mathbf{x}}\mathrm{(t)} = \mathbf{Ax}\mathrm{(t)} + \mathbf{Bv}\mathrm{(t)}\\
\mathbf{{y}}\mathrm{(t)} = \mathbf{Cx}\mathrm{(t)} + \mathbf{Dv}\mathrm{(t)}
\end{matrix} \;\;\; \mathrm{with} \;\;\; \mathbf{x}\mathrm{(t)} = \begin{bmatrix}
\mathbf{u}\mathrm{(t)}\\
\dot{\mathbf{u}}\mathrm{(t)}
\end{bmatrix}, \;\;\; \mathbf{v}\mathrm{(t)} = \mathbf{f}\mathrm{(t)} $$

Practical benefit:

• Advanced system identification methods are available within the research field of control theory and signal processing that can be very well utilized for structural dynamic identification.

Conversion between domains

This section discusses several standard techniques to convert between domains, to allow for hybrid analysis and design strategies.

Mode synthesis

Directly inverting the dynamic stiffness can be an expensive operation if the dimension size $$ n $$ is large and the frequency dependency $$ \omega $$ is evaluated at many frequency points $$ (\omega_{1}, …,\omega_{k}) $$.

An alternative way to compute FRFs is by means of mode synthesis, which uses the natural vibration modes or normal modes of the structure to build up the FRFs. This is only possible for systems with uncoupled modes, i.e. linear systems with a proportional damping matrix. Assuming mass-normalized modes, the uncoupled modal parameters read:

$$ \boldsymbol{\Phi}^{\mathrm{T}} \mathbf{M} \boldsymbol{\Phi} = \mathbf{I} $$

$$ \boldsymbol{\Phi}^{\mathrm{T}} \mathbf{C} \boldsymbol{\Phi} = 2 \; \mathrm{diag}(\zeta_{1}\omega_{1}, …,\zeta_{n}\omega_{n} ) $$

$$ \boldsymbol{\Phi}^{\mathrm{T}} \mathbf{K} \boldsymbol{\Phi} = \mathrm{diag}(\omega_{1}^2, …,\omega_{n}^2 ) $$

Admittance FRFs can be synthesized using a superposition of single-DoF modal solutions. The $$ m_{\mathrm{RB}} $$ rigid body modes (if present) need to be treated differently as they have an eigenfrequency $$ \omega_{r} = 0 $$ and no damping ratio related to frequency.

$$ \mathbf{Y}(\omega)\approx \underbrace{-\frac{1}{\omega^2}\sum_{1}^{m_{\mathrm{RB}}} \boldsymbol{\varphi} _{r} \boldsymbol{\varphi} _{r}^\mathrm{T}}_{\mathrm{rigid \;body \;modes}} \; + \; \underbrace{\sum_{m_{\mathrm{RB}}+1}^{m}\frac{\boldsymbol{\varphi} _r \boldsymbol{\varphi}_r^\mathrm{T} }{-\omega^2 + 2i\omega\zeta _r\omega_r + \omega_r^2}}_{\mathrm{vibration \; modes}} $$

It is good practice to include eigenfrequencies up to twice the highest frequency of interest in the summation, i.e. $$ \omega_m > 2\omega_K $$.
Analogously, the IRFs can be constructed from a similar mode synthesis technique.

$$ \mathbf{Y}(\mathrm{t})\approx \underbrace{\mathrm{t}\sum_{1}^{m_{\mathrm{RB}}} \boldsymbol{\varphi} _{r} \boldsymbol{\varphi} _{r}^\mathrm{T}}_{\mathrm{rigid \;body \;modes}} \; + \; \underbrace{\sum_{m_{\mathrm{RB}}+1}^{m}{\boldsymbol{\varphi} _r \boldsymbol{\varphi}_r^\mathrm{T} }\frac{e^{\lambda _{r}t}e^{\lambda _{r}^*t}}{2i\omega_\mathrm{d}}}_{\mathrm{vibration \; modes}} $$

With the parameters $$ \lambda_r $$ forming the complex poles corresponding to the vibration modes.

Practical benefits:

• Allows constructing FRFs from a smaller selection of modes $$ m<n $$ (modal truncation).
• The nominator remains constant over frequency, whereas the denominator controls the spectral part, adding a resonance frequency for each mode $$ r $$, namely for $$ \omega = \omega_r $$.

Modal parameter estimation

Inverting the mode synthesis approach can be understood as modal parameter estimation. Experimentally obtained admittance FRFs can be used to identify the vibration modes from measured response data, resulting in a series of mode shapes with associated frequency and damping parameters. Methods for modal identification of linear systems can roughly be put in the following categories:

• Frequency domain- vs. time domain-based. A frequency domain method tries to fit FRFs by introducing a finite number of vibration modes at resonance frequencies that are usually already identified. Time domain solutions may operate fully automatically and be more similar to Finite Impulse Response (FIR) techniques.
• Single-DoF fit vs. multi-DoF algorithm. Single-DoF models fit the response functions by consecutively introducing a single mode, i.e. without considering the interaction between modes. It is often preferred for systems with close resonance peaks or high damping to use a multi-DoF model that tries to identify a more extensive set of modes at once.
• Normal mode vs. complex mode model. A normal mode model assumes proportional damping and synchronous response at all DoFs. This implies that the modes are real-valued and are assumed to be superimposed (see mode synthesis). A complex mode model fits response functions using complex pairs of modes, allowing for more sophisticated damping properties.

Summary

The following table summarizes the five domains for modeling and simulation of a structure’s dynamics. The respective equations of motion all relate displacements to applied force — either directly or through some modal transformation — but vary in the representation or manifold (in the meaning of topological space) to describe the structure’s internal dynamics.

It should be noted that for a given structure, some reduction methods may be more effective than others.

Practical benefits:

• A structure with few modes and low damping can be very effectively reduced in the modal domain, but may lead to agonizingly slow simulation in the time domain due to the long decay time of the natural vibrations.
• It is therefore great to have different modeling domains at hand and benefit from the particular specialties of each of them, provided that we can combine them in a dynamically correct way.