12. Structural damping#

As a first approximation, a large number of structures can be treated as undamped structures that, if constrained so that there’s no rigid motion allowed, can be represented by the second order system

\[\mathbf{M} \ddot{\mathbf{x}} + \mathbf{K} \mathbf{x} = \mathbf{f} \ ,\]

being mass and stiffness matrices \(\mathbf{M}\), \(\mathbf{K}\) that are positive definite, and symmetric if derived as an example from a Lagrangian formulation of the problem.

todo Add reference to Lagrange mechanics and its properties in the classical mechanics bbooks.

This kind of systems are conveniently described using modal basis, as modes (or free/natural modes of vibrations) are orthogonal w.r.t. both mass and stiffness matrix.

todo Add reference; add comment: diagonal, or diagonalizable with coincident eigenvectors.

Free response using modal basis

12.1. Small damping#

Structural small damping can be treated as a first order perturbation of the undamped system,

\[\mathbf{M} \ddot{\mathbf{u}} + \mathbf{C} \dot{\mathbf{u}} + \mathbf{K} \mathbf{u} = \mathbf{f} \ ,\]

or in Laplace domain

\[\left[ s^2 \mathbf{M} + s \mathbf{C} + \mathbf{K} \right] \mathbf{u} = \mathbf{f} \ .\]

Here two assumptions are made and justified later:

  • matrix \(\mathbf{C}\) is (semi)positive symmetric

  • matrix \(\mathbf{C}\) becomes diagonal in the modal basis, i.e. modal basis simoultaneously diagonalize mass, damping and stiffness matrices

If these two assumption holds, using the modal base collected in matrix \(\mathbf{U}\),

\[\mathbf{u} = \mathbf{U} \mathbf{q} \ ,\]

the diagonalization reads

\[\begin{split}\begin{aligned} \mathbf{U}^T \mathbf{f} & = \mathbf{U}^T \left\{ \mathbf{M} \mathbf{U} \ddot{\mathbf{q}} + \mathbf{C} \mathbf{U} \dot{\mathbf{q}} + \mathbf{K} \mathbf{U} \mathbf{q} \right\} = \\ & = \text{diag} \{ m_i \} \ddot{\mathbf{q}} + \text{diag}\{ c_i \} \dot{\mathbf{q}} + \text{diag}\{ k_i \} \mathbf{q} = \\ & = \text{diag} \{ m_i \ddot{q}_i + c_i \dot{q}_i + k_i q_i \} \ . \end{aligned}\end{split}\]

being \(m_i := \mathbf{u}_i^T \mathbf{M} \mathbf{u}_i\), \(c_i := \mathbf{u}_i^T \mathbf{C} \mathbf{u}_i\), \(k_i := \mathbf{u}_i^T \mathbf{K} \mathbf{u}_i\), the modal mass, damping and stiffness.

(Semi)definite positive damping matrix

Starting from the equations of motion

\[\mathbf{M} \ddot{\mathbf{u}} + \mathbf{C} \dot{\mathbf{u}} + \mathbf{K} \mathbf{u} = \mathbf{f} \ ,\]

the kinetic energy and the mechanical energy balance is derived (todo add references) with scalar multiplication by \(\dot{\mathbf{u}}\). For constant matrices,

\[\begin{split}\begin{aligned} \dot{\mathbf{u}}^T \mathbf{M} \ddot{\mathbf{u}} + \dot{\mathbf{u}}^T \mathbf{C} \dot{\mathbf{u}} + \dot{\mathbf{u}}^T \mathbf{K} \mathbf{u} & = \dot{\mathbf{u}}^T \mathbf{f} \\ \dfrac{d}{dt} \left[ \dfrac{1}{2} \dot{\mathbf{u}}^T \mathbf{M} \dot{\mathbf{u}} + \dfrac{1}{2} \mathbf{u}^T \mathbf{K} \mathbf{u} \right] & = \dot{\mathbf{u}}^T \mathbf{f} - \dot{\mathbf{u}}^T \mathbf{C} \dot{\mathbf{u}} \\ \dfrac{d}{dt} \left( K + V \right) & = \dot{\mathbf{u}}^T \mathbf{f} - \underbrace{\dot{\mathbf{u}}^T \mathbf{C} \dot{\mathbf{u}}}_{ D \ge 0} \ , \\ \end{aligned}\end{split}\]

having recognized \(D = \dot{\mathbf{u}}^T \mathbf{C} \dot{\mathbf{u}} \ge 0\) as the dissipation from damping, that can’t make the mechanical energy of the system \(K + V\) increase. This condition implies that \(\mathbf{C}\) is (semi)definite positive.

Diagonal damping in modal basis

Let’s write here the perturbed free damped system in Laplace domain using modal basis,

\[\left[ s^2 \mathbf{M} + s \mathbf{C} + \mathbf{K} \right] \mathbf{u} = \mathbf{0} \ ,\]

and evaluate the derivative of this relation w.r.t. a parameter \(p\) associated to the damping, and not influencing mass or stiffness properties, \(\mathbf{M}_{/p} = \mathbf{0}\), \(\mathbf{K}_{/p} = \mathbf{0}\),

\[\begin{split}\begin{aligned} \mathbf{0} & = \left\{ \left[ s^2 \mathbf{M} + s \mathbf{C} + \mathbf{K} \right] \mathbf{u} \right\}_{/p} = \\ & = \left[ (2 s \mathbf{M} + \mathbf{C} ) s_{/p} + s \mathbf{C}_{/p} \right] \mathbf{u} + \left[ s^2 \mathbf{M} + s \mathbf{C} + \mathbf{K} \right] \mathbf{u}_{/p} \ . \end{aligned}\end{split}\]

Let’s investigate the effect of small damping on the eigensolution \(\left(s_i, \mathbf{u}_i \right)\). Exploiting the symmetry of the matrices of the system (following from the assumed simultaneous diagonalization of the damping matrix \(\mathbf{C} = \mathbf{U} \text{diag}\{ c_i \} \mathbf{U}^*\)), and evaluating the dot product of the latter relation for the \(i\)-th eigensolution with the eigenvector \(\mathbf{u}_i\),

\[\begin{split}\begin{aligned} 0 & = \mathbf{u}_i^T \left\{ \left[ (2 s_i \mathbf{M} + \mathbf{C} ) s_{i/p} + s_i \mathbf{C}_{/p} \right] \mathbf{u}_i + \left[ s_i^2 \mathbf{M} + s_i \mathbf{C} + \mathbf{K} \right] \mathbf{u}_{i/p} \right\} = \\ & = \mathbf{u}_i^T \left[ (2 s_i \mathbf{M} + \mathbf{C} ) s_{i/p} + s_i \mathbf{C}_{/p} \right] \mathbf{u}_i + \mathbf{u}_{i/p}^T \underbrace{\left[ s_i^2 \mathbf{M} + s_i \mathbf{C} + \mathbf{K} \right] \mathbf{u}_i}_{=\mathbf{0}} = \\ \end{aligned}\end{split}\]

it follows that the derivative of the \(i^{th}\) eigenvalue w.r.t. the parameter \(p\) reads

\[s_{i/p} = - \dfrac{s_i \mathbf{u}_i^T \mathbf{C}_{/p} \mathbf{u}_i}{\mathbf{u}_i^T (2 s_i \mathbf{M} + \mathbf{C} ) \mathbf{u}_i} \ .\]

This derivative evaluated for the reference undamped condition \(\mathbf{C} = \mathbf{0}\) becomes

\[s_{i/p} = - \dfrac{1}{2} \dfrac{\mathbf{u}_i^T \mathbf{C}_{/p} \mathbf{u}_i}{\mathbf{u}_i^T \mathbf{M} \mathbf{u}_i} = - \dfrac{1}{2} \dfrac{\mathbf{u}_i^T \mathbf{C}_{/p} \mathbf{u}_i}{m_i} \ ,\]

having recognized the modal mass \(m_i := \mathbf{u}_i^T \mathbf{M} \mathbf{u}_i\) associated to the \(i^{th}\) mode. Now, let’s evaluate the derivative of the eigenvalue \(s_i\) w.r.t. the components of the damping matrix \(\mathbf{C}\), i.e. \(\mathbf{C}_{/C_{jk}}\) that is a matrix full of zero, except for the component \((j,k)\) equal to one,

\[s_{i/C_{jk}} = - \dfrac{1}{2} \dfrac{\mathbf{u}_i^T \mathbf{C}_{/C_{jk}} \mathbf{u}_i}{m_i} = -\dfrac{1}{2} \dfrac{u^{(i)}_j u^{(i)}_k}{m_i} \ ,\]

and the first order polynomial expansion of \(s_i\) in coefficients \(C_{jk}\) reads

\[\begin{split}\begin{aligned} s_i & = s_{i,0} + s_{i/C_{jk}} C_{jk} = \\ & = s_{i,0} - \dfrac{1}{2} \dfrac{u^{(i)}_j C_{jk} u^{(i)}_k}{m_i} = \\ & = s_{i,0} - \dfrac{1}{2} \dfrac{\mathbf{u}_i^T \mathbf{C} \mathbf{u}_i}{m_i} \ . \end{aligned}\end{split}\]

From this expression, it’s possible to deduce that the \(i^{th}\) eigenvalue of the slightly damped system differs from the \(i^{th}\) eigenvalue of the undamped system \(s_{i,0} = \mp j \omega_{i}\) of a real non-positive (as \(\mathbf{C} \ge 0\) for dissipative damping actions) term \(\Delta s_i = - \frac{1}{2} \frac{\mathbf{u}_i^T \mathbf{C} \mathbf{u}_i}{m_i} \in \mathbb{R}\), \(\Delta s_i \le 0\), depending only on the damping matrix and the \(i^{th}\) mode. This term shifts the eigenvalue \(s_i\) to the left in the complex plane, and thus makes it asymptotically stable.

As the variation \(\Delta s_i\) only depends on the \(i^{th}\) eigenvector, and not on other eigenvectors, the assumption of siultaneously diagonalizable damping matrix is consistent with the results from this assumption.

12.2. Sensitivity of eigenvalues and eigenvectors#

Link to Sensitivity of spectral decomposition, for first order equations. The sensitivity of the \(i^{th}\) eigenvalue to a general parameter reads

\[s_{i/p} = - \dfrac{s_i \mathbf{u}_i^* \mathbf{C}_{/p} \mathbf{u}_i}{\mathbf{u}_i^* (2 s_i \mathbf{M} + \mathbf{C} ) \mathbf{u}_i} \ .\]

The sensitivity of this eigenvalue to damping \(\mathbf{C}\) of the undamped system as reference condition \(\mathbf{C} = \mathbf{0}\) reads

\[s_{i/\mathbf{C}} = - \dfrac{1}{2} \dfrac{\mathbf{u}_i \otimes \mathbf{u}_i}{m_i} \ .\]

The sensitivity of the eigenvector \(\mathbf{u}_{i/p}\) can be evaluated as the solution of a linear system derived from the derivation of the eigenvalue problem w.r.t. the parameter \(p\)

\[\begin{split}\begin{aligned} \mathbf{0} & = \dfrac{d}{dp}\left\{ \left( s^2 \mathbf{M} + s \mathbf{C} + \mathbf{K} \right) \mathbf{u} \right\} = \\ & = \left( s^2 \mathbf{M} + s \mathbf{C} + \mathbf{K} \right) \mathbf{u}_{/p} + \left( 2 s s_{/p} \mathbf{M} + s_{/p} \mathbf{C} + s \mathbf{C}_{/p} \right) \mathbf{u} \ , \end{aligned}\end{split}\]

having assumed here \(\mathbf{M}_{/p} = \mathbf{0}\) and \(\mathbf{K}_{/p} = \mathbf{0}\). The linear system thus becomes

\[\left( s^2 \mathbf{M} + s \mathbf{C} + \mathbf{K} \right) \mathbf{u}_{/p} = \underbrace{- \left( 2 s s_{/p} \mathbf{M} + s_{/p} \mathbf{C} + s \mathbf{C}_{/p} \right) \mathbf{u}}_{= \mathbf{b}}\ .\]

  • This linear system is singular, as \(s\) is an eigenvalue of the system, and

    \[\left( s^2 \mathbf{M} + s \mathbf{C} + \mathbf{K} \right) \mathbf{x} = \mathbf{0} \ ,\]

    for every \(\mathbf{x}\) that is a linear combination of the eigenvectors with eigenvalue \(s\). Thus, if the linear system has a solution, it has infinite solution. The linear system has a solution if the RHS \(\mathbf{b}\) is in the range of the matrix \(\mathbf{A}(s) := s^2 \mathbf{M} + s \mathbf{C} + \mathbf{K}\).

  • If \(\mathbf{b} \in \text{R}(\mathbf{A})\) then it’s orthogonal to \(\text{K}(\mathbf{A}^*)\), and viceversa. Kernel of \(\mathbf{A}^* := s^{* 2} \mathbf{M}^* + s^* \mathbf{C}^* + \mathbf{K}^*\) is spanned by the left eigenvectors \(\mathbf{v}\) associated with the eigenvalue \(s\). Direct evaluation of the product \(\mathbf{v}^* \mathbf{b}\) proves that \(\mathbf{b} \perp \text{K}(\mathbf{A}^*)\) and thus \(\mathbf{b} \in \text{R}(\mathbf{A})\), and thus a solution exists,

    \[\begin{split}\begin{aligned} \mathbf{v}^* \mathbf{b} & = \mathbf{v}^* \left( - 2 s s_{/p} \mathbf{M} - s_{/p} \mathbf{C} - s \mathbf{C}_{/p} \right) \mathbf{u} = \\ & = \mathbf{v}^* ( - 2 s \mathbf{M} - \mathbf{C} ) \mathbf{u} s_{/p} - s \mathbf{v}^* \mathbf{C}_{/p} \mathbf{u} = \\ & = \mathbf{v}^* ( - 2 s \mathbf{M} - \mathbf{C} ) \mathbf{u} \dfrac{-s \mathbf{v}^* \mathbf{C}_{/p} \mathbf{u}}{\mathbf{v}^* ( 2 s \mathbf{M} + \mathbf{C}) \mathbf{u}} - s \mathbf{v}^* \mathbf{C}_{/p} \mathbf{u} = 0 \ . \end{aligned}\end{split}\]

  • Since a solution exists, an infinite number of solution exists. Given a solution \(\widetilde{\mathbf{u}}_{/p}\), adding a linear combination of the eigenvectors with eigenvalue \(s\) produces a solution as well,

    \[\widetilde{\mathbf{u}}_{/p} + \mathbf{U} \boldsymbol\beta \ .\]

  • In order to remove the arbitariness, it’s possible to introduce some conditions, like the orthogonality condition \(\mathbf{V}^* \mathbf{M} \mathbf{u}_{/p} ) \mathbf{0}\)

  • Writing the solution as a linear combination of eigenvectors \(\mathbf{U}\) of the linear system (are they a basis?) and explicitly discern the eigenvectors with eigenvalue \(s_i\) fro the other ones,

    \[\mathbf{u}_{/p} = \mathbf{U}_{\notin i} \boldsymbol\alpha + \mathbf{U}_i \boldsymbol\beta \ ,\]

    the linear system and the orthogonality condition \(\mathbf{V}_i^* \mathbf{M} \mathbf{U}_{\notin i} = \mathbf{0}\) give the decoupled linear system

    \[\begin{split}\begin{cases} \mathbf{V}^*_{\notin i} \left( s_i^2 \mathbf{M} + s_i \mathbf{C} + \mathbf{K} \right) \mathbf{U}_{\notin i} \boldsymbol\alpha = \mathbf{V}^*_{\notin i} \mathbf{b} \\ \mathbf{V}_i \mathbf{M} \mathbf{U}_i \boldsymbol\beta = \mathbf{0} \ . \end{cases}\end{split}\]

    Under the assumption that mass, damping and stiffness matrices are simultaneously diagnoalized, it immediately follows that this linear system is diagonal,

    \[\begin{split}\begin{cases} \text{diag} \left\{ s_i^2 m_{j \notin i} + s_i c_{j \notin i} + k_{j\notin i} \right\} \boldsymbol\alpha = \mathbf{V}_{\notin i}^* \mathbf{b}\\ \text{diag} \left\{ m_i \right\} \boldsymbol\beta = \mathbf{0} \ . \end{cases}\end{split}\]

    If mass normalization is chosen, \(\mathbf{V}^* \mathbf{M} \mathbf{U} = \mathbf{I}\), i.e. \(m_j = 1\), \(k_j = \omega_j^2 = |s_j|^2\), and \(c_i = 2 \xi_j \omega_j = - 2 \, \text{re}\{ s_j \}\), being \(s_j = \omega_j \left( - \xi_j \mp i \sqrt{ 1 - \xi^2_j } \right)\). It’s immediate to prove that

    \[s^2 - 2 s \, \text{re}\{ s_j \} + |s_j|^2 = ( s - s_j^* ) ( s - s_j ) \ ,\]

    as

    \[(s - s_j^* ) (s - s^j ) = s^2 - s (s_j + s_j^*) + s_j^* s_j = s^2 - 2 \, \text{re}\{ s_j \} + |s_j|^2 \ .\]

    The solution of the linear system thus reads

    \[\begin{split}\begin{cases} \boldsymbol\alpha = \text{diag}\left\{ \dfrac{1}{s_i^2 + 2 \xi_j \omega_j s_i + \omega_j^2} \right\} \mathbf{V}^*_{\notin i} \mathbf{b} \\ \boldsymbol\beta = \mathbf{0} \ , \end{cases}\end{split}\]

    s.t. the unique solution of the augmented problem reads

    \[\begin{split}\begin{aligned} \mathbf{u}_{i/p} & = \mathbf{U}_{\notin i} \text{diag}\left\{ \dfrac{1}{s_i^2 + 2 \xi_j \omega_j s_i + \omega_j^2} \right\} \mathbf{V}^*_{\notin i} \mathbf{b}_i = \\ & = - \mathbf{U}_{\notin i} \text{diag}\left\{ \dfrac{1}{s_i^2 + 2 \xi_j \omega_j s_i + \omega_j^2} \right\} \mathbf{V}^*_{\notin i} \left( 2 s_i s_{i/p} \mathbf{M} + s_{i/p} \mathbf{C} + s_i \mathbf{C}_{/p} \right) \mathbf{u}_i = \\ & = - s_i \mathbf{U}_{\notin i} \text{diag}\left\{ \dfrac{1}{s_i^2 + 2 \xi_j \omega_j s_i + \omega_j^2} \right\} \mathbf{V}^*_{\notin i} \mathbf{C}_{/p} \mathbf{u}_i = \\ & = \dots \end{aligned}\end{split}\]
Algebra with components

Let the matrix \(\mathbf{U}_{\notin i} = \left[ \dots | \mathbf{u}_j | \dots \right]\), and \(U_{ab}^{\notin i} = u^{(b)}_a\), \(V_{ab}^{\notin i} = v^{(b)}_a\),

\[\begin{split}\begin{aligned} u^{(i)}_{a/p} & = - s_i U^{\notin i}_{ab} \dfrac{\delta_{be}}{s_i^2 + 2 \xi_b \omega_b s_i + \omega_b^2} V^{\notin i, \ *}_{ce} C_{cd/p} u^{(i)}_d = \\ & = - u^{(b)}_a \dfrac{s_i}{s_i^2 + 2 \xi_b \omega_b s_i + \omega_b^2} v^{(b) \, *}_{c} C_{cd/p} u_d^{(i)} = \\ & = - \sum_{b \ne i} \dfrac{s_i}{s_i^2 + 2 \xi_b \omega_b s_i + \omega_b^2} \mathbf{v}_b^* \mathbf{C}_{/p} \mathbf{u}_i \, \mathbf{u}_b \end{aligned}\end{split}\]
Parameter \(\ p = C_{rs}\)

If \(p = C_{rs}\), then \(C_{cd/p} = C_{cd/C_{rs}} = \delta_{cr} \delta_{ds}\), and

\[\begin{split}\begin{aligned} u^{(i)}_{a/C_{rs}} & = - s_i U^{\notin i}_{ab} \dfrac{\delta_{be}}{s_i^2 + 2 \xi_b \omega_b s_i + \omega_b^2} V^{\notin i, \ *}_{ce} C_{cd/C_{rs}} u^{(i)}_d = \\ & = - \sum_{b \ne i} u^{(b)}_a \dfrac{s_i}{s_i^2 + 2 \xi_b \omega_b s_i + \omega_b^2} v^{(b) \, *}_{c} \delta_{cr} \delta_{ds} u_d^{(i)} = \\ & = - \sum_{b \ne i} u^{(b)}_a \dfrac{s_i}{s_i^2 + 2 \xi_b \omega_b s_i + \omega_b^2} v^{(b) \, *}_{r} u_s^{(i)} = \\ \end{aligned}\end{split}\]

is a third-order tensor, as it’s the derivative of a vector quantity w.r.t. a second-order tensor \(\mathbf{C}\)1, and its index representation has 3 non-dummy indices, namely \(a\), \(r\), \(s\).

The first-order approximation of the eigenvector reads

\[\begin{split}\begin{aligned} u^{(i)}_a & \simeq u^{(i)}_{a,0} + \sum_{r,s} C_{rs} u^{(i)}_{a/C_{rs}} = \\ & = u^{(i)}_{a,0} - \sum_{r,s} C_{rs} \sum_{b \ne i} u_a^{(b)} \dfrac{s_i}{s_i^2 + 2 \xi_b \omega_b s_i + \omega_b^2} v_r^{(b) \ *} u_s^{(i)} = \\ & = u^{(i)}_{a,0} - \sum_{b \ne i} u_a^{(b)} \dfrac{s_i}{s_i^2 + 2 \xi_b \omega_b s_i + \omega_b^2} \sum_{r,s} v_r^{(b) \ *} C_{rs} u_s^{(i)} \ , \end{aligned}\end{split}\]

or using vector formalism

\[\mathbf{u}_i = \mathbf{u}_{i,0} - \sum_{b \ne i} \dfrac{s_i}{s_i^2 + 2 \xi_b \omega_b s_i + \omega_b^2} \mathbf{v}_{b,0}^* \mathbf{C} \mathbf{u}_{i,0} \, \mathbf{u}_{i,0} \ .\]
1

Is this really a tensor? Should we recall the definition of tensor here? What’s the right name of the extension of a matrix with 3 indices?