State Space Model of Structural Dynamics#

When an multiple degree-of-freedom (MDOF) system is subject to multiple support excitation, such as in the figure above, the displacement vector is extended to include the support DOF. An equation of motion is derived as follows.

Begin by forming a partitioned equation of dynamic equilibrium for all the DOF:

Partitioned Equation of Dynamic Equilibrium#

\[\begin{split}\begin{bmatrix} \mathbf{m} & \mathbf{m}_{g} \\ \mathbf{m}^T_{g} & \mathbf{m}_{gg} \end{bmatrix} \begin{bmatrix} \mathbf{\ddot{u}}^{t}_{f} \\ \mathbf{\ddot{u}}_{g} \end{bmatrix} + \begin{bmatrix} \mathbf{c} & \mathbf{c}_{g} \\ \mathbf{c}^T_{g} & \mathbf{c}_{gg} \end{bmatrix} \begin{bmatrix} \mathbf{\dot{u}}^{t}_{f} \\ \mathbf{\dot{u}}_{g} \end{bmatrix} + \begin{bmatrix} \mathbf{k} & \mathbf{k}_{g} \\ \mathbf{k}^T_{g} & \mathbf{k}_{gg} \end{bmatrix} \begin{bmatrix} \mathbf{u}^{t}_{f} \\ \mathbf{u}_{g} \end{bmatrix} = \begin{bmatrix} \mathbf{0} \\ \mathbf{p}_{g} \end{bmatrix}\end{split}\]

where the subscript \(g\) indicates support DOF, the subscript \(f\) indicates structural DOF, and the superscript \(t\) indicates the total of quasi-static (\(\mathbf{u}^{s}_{f}\), due to static application of support displacements) and dynamic (\(\mathbf{u}_{f}\), evaluated by dynamic analysis) structural displacements.

Taking the first half of the partitioned equilibrium, separating the structural displacements (\(\mathbf{u}^{t}_{f}=\mathbf{u}^{s}_{f}+\mathbf{u}_{f}\)), and moving all \(\mathbf{u}_{g}\) and \(\mathbf{u}^{s}_{f}\) terms to the right side,

\[\mathbf{m}\mathbf{\ddot{u}}_{f} + \mathbf{c}\mathbf{\dot{u}}_{f} + \mathbf{k}\mathbf{u}_{f} = -(\mathbf{m}\mathbf{\ddot{u}}^{s}_{f}+\mathbf{m}_{g}\mathbf{\ddot{u}}_{g}) -(\mathbf{c}\mathbf{\dot{u}}^{s}_{f}+\mathbf{c}_{g}\mathbf{\dot{u}}_{g}) -(\mathbf{k}\mathbf{u}^{s}_{f}+\mathbf{k}_{g}\mathbf{u}_{g})\]

The term \((\mathbf{k}\mathbf{u}^{s}_{f}+\mathbf{k}_{g}\mathbf{u}_{g})=\mathbf{0}\) due to static equilibrium, allowing the term to be dropped and giving \(\mathbf{u}^{s}_{f} = \mathbf{-k}^{-1}\mathbf{k}_{g}\mathbf{u}_{g} = \mathbf{\iota u}_{g}\); the term \((\mathbf{c}\mathbf{\dot{u}}^{s}_{f}+\mathbf{c}_{g}\mathbf{\dot{u}}_{g})\) is dropped because it is usually small relative to the inertia term; and the term \(\mathbf{m}_{g}\mathbf{\ddot{u}}_{g}\) is dropped because mass is usually neglected at supports.

The equilibrium equation thus simplifies.

Equation of Motion#

\[\begin{split}\begin{aligned} \mathbf{M\ddot{u}}_{f}(t) + \mathbf{Z\dot{u}}_{f}(t) + \mathbf{Ku}_{f}(t) &= -\mathbf{M\iota}\mathbf{\ddot{u}}_{g}(t) \\ \mathbf{m}\mathbf{\ddot{u}}_{f} + \mathbf{c}\mathbf{\dot{u}}_{f} + \mathbf{k}\mathbf{u}_{f} &= -\mathbf{m}\mathbf{\iota}\mathbf{\ddot{u}}_{g} \end{aligned}\end{split}\]

Hence, the following equation presents the continuous linear time-invariant (LTI) state-space representation of a structural system.

Continuous LTI State-Space Representation#

\[\begin{split}\begin{aligned} \mathbf{\dot{x}} &= \mathbf{A}_{c}\mathbf{x} + \mathbf{B}_{c}\mathbf{u} \\ \begin{bmatrix} \mathbf{\dot{u}}_{f}(t) \\ \mathbf{\ddot{u}}_{f}(t) \end{bmatrix} &= \begin{bmatrix} \mathbf{0} & \mathbf{I} \\ -\mathbf{M}^{-1}\mathbf{K} & -\mathbf{M}^{-1}\mathbf{Z} \end{bmatrix} \begin{bmatrix} \mathbf{u}_{f}(t) \\ \mathbf{\dot{u}}_{f}(t) \end{bmatrix} + \begin{bmatrix} \mathbf{0} \\ -\mathbf{\iota} \end{bmatrix} \mathbf{\ddot{u}}_{g}(t) \\ \\ \mathbf{y} &= \mathbf{Cx} + \mathbf{Du} \\ \mathbf{\ddot{u}}_{f}(t) &= \begin{bmatrix} -\mathbf{M}^{-1}\mathbf{K} & -\mathbf{M}^{-1}\mathbf{Z} \end{bmatrix} \begin{bmatrix} \mathbf{u}_{f}(t) \\ \mathbf{\dot{u}}_{f}(t) \end{bmatrix} + \begin{bmatrix} -\mathbf{\iota} \end{bmatrix} \mathbf{\ddot{u}}_{g}(t) \end{aligned}\end{split}\]

In order to move from the continuous to the discrete case, the coefficients \(\mathbf{A}_{c}\) and \(\mathbf{B}_{c}\) are transformed by solving the first-order differential equation with the signal’s value held constant between time steps (“zero-order hold method”). The coefficients \(\mathbf{C}\) and \(\mathbf{D}\) are unchanged. The results are shown in the following equation.

Discrete LTI State-Space Representation#

\[\begin{split}\begin{aligned} \mathbf{x}_{k+1} &= \mathbf{Ax}_{k} + \mathbf{Bu}_{k} \\ \mathbf{y}_{k} &= \mathbf{Cx}_{k} + \mathbf{Du}_{k} \\ \end{aligned}\end{split}\]

\[\mathbf{x}_{k} = \mathbf{x}(k\Delta t), \hspace{1cm} \mathbf{u}_{k} = \mathbf{u}(k\Delta t), \hspace{1cm} \mathbf{y}_{k} = \mathbf{y}(k\Delta t)\]

\[\mathbf{A} = e^{\mathbf{A}_{c}\Delta t}, \hspace{1cm} \mathbf{B} = \int_{0}^{\Delta t}{e^{\mathbf{A}_{c}\tau}}\mathbf{B}_{c}d\tau\]

where:

\[\begin{split}\begin{aligned} \mathbf{A} & \text{: discrete state transition matrix} \\ \mathbf{B} & \text{: discrete input influence matrix} \\ \mathbf{C} & \text{: output influence matrix} \\ \mathbf{D} & \text{: direct transmission or feed-through matrix} \end{aligned}\end{split}\]