A direct method for updating piezoelectric smart structural models based on measured modal data

1.   Introduction

Throughout this paper, $\mathbb{R}^{m\times n}$, $\mathbb{OR}^{n\times n}$ and $\mathbb{SR}^{n\times n}$ denote the sets of all $m \times n$ real matrices, all $n \times n$ orthogonal matrices and all $n \times n$ symmetric matrices, respectively. $A^{\top}$ and tr$(A)$ stand for the transpose and the trace of a matrix $A$, respectively. $I_n$ denotes the identity matrix of size $n$.

Piezoelectric smart materials are a class of materials with piezoelectric effect. Due to the development of smart materials and structures, these materials are endowed with strong vitality. Piezoelectric smart materials can rapidly transform pressure, vibration into electrical signals, or electrical signals into vibration signals, that is, the piezoelectric elements can be used as both sensors and actuators, which realizes the unity of the sensing elements and the action elements, and make them widely used in engineering. For example, piezoelectric materials are applied in active vibration control ^[1,2], distributed dynamic measurement ^[3] and structural health monitoring ^[4,5,6], etc.

By using the finite element techniques, the global equation of motion for the undamped piezoelectric smart structure system with $n$ degrees of freedom can be written as ^[7]:

The matrices $M_{A}, K_{A},$ and the vectors ${{{\boldsymbol{v}}}}, {{{\boldsymbol{f}}}}$ are of the form

where $\tilde{M}_{uu}\in \mathbb{SR}^{n_{u}\times n_{u}}$ is the structural mass matrix, $\tilde{K}_{uu}\in \mathbb{SR}^{n_{u}\times n_{u}}$ is the structural stiffness matrix, $\tilde{K}_{u \phi}\in \mathbb{R}^{n_{u}\times n_{\phi}} \ (n_{u}+n_{\phi} = n)$ is the piezoelectric coupling matrix, $\tilde{K}_{\phi \phi}\in \mathbb{SR}^{n_\phi\times n_\phi}$ is the dielectric stiffness matrix, ${\boldsymbol{u}} \in \mathbb{R}^{n_{u}}$ denotes the mechanical displacement vector, ${\boldsymbol{\phi}}\in \mathbb{R}^{n_{\phi}}$ denotes the electrical potential vector, ${\boldsymbol{f}}_{u}\in\mathbb{R}^{n_{u}}$ denotes the mechanical external force vector and ${\boldsymbol{f}}_{\phi}\in \mathbb{R}^{n_{\phi}}$ denotes the external electric charge vector. It is known that the vibration of the mathematical model $(1.1)$ is characterized by eigenvalues and eigenvectors of the following generalized inverse eigenvalue problem:

In general, owning to the difficulty in accurately determining some structural parameters, the unreasonable coupling simplification ^[8] and the mathematical description error of geometry and boundary conditions ^[9], the piezoelectric smart structure model established by finite element techniques may not truly describe the actual characteristics of the structure. Therefore, we need to update the model by applying the measured modal data such that the updated model can better reflect the physical structure and the measured results. Mathematically, the problem of updating piezoelectric smart structure model can be formulated as the following problem.

Problem IEP. Let $\Omega = \mbox{diag}(\omega_{1}, ..., \omega_{p})\in \mathbb{R}^{p \times p}$ and $Z = [Z_{1}^\top, Z_{2}^\top]^\top\in \mathbb{R}^{n \times p}$ be the measured eigenvalue and eigenvector matrices, where $Z_{1}\in \mathbb{R}^{n_u \times p}$, $Z_{2}\in \mathbb{R}^{n_\phi \times p}$ and $\mbox{rank}(Z_{1}) = p.$ Find matrices $M = \left[ \begin{array}{cc} {M}_{uu} & 0 \\ 0 & 0 \\ \end{array} \right], K = \left[ \begin{array}{cc} K_{uu} & K_{u \phi} \\ K_{u \phi}^\top & K_{\phi \phi} \\ \end{array} \right]\in \mathbb{SR}^{n\times n}$ such that

It is well known that the numerical model is a "good" representation of the structure, we hope to find a model that is closest to the original model. Thus, we should further consider the following best approximate problem:

Problem BAP. Given matrices $M_{A}, K_{A}\in \mathbb{SR}^{n\times n}.$ Find $(\hat{M}, \hat{K})\in \mathbb{\kappa_{S}}$ such that

where ${\|\cdot \|}$ is the Frobenius norm and $\mathbb{\kappa_{S}}$ is the solution set of Problem IEP.

Fish and Chen ^[10] developed the solution procedures for large-scale transient analysis of piezocomposites by using the representative volume element-based multilevel method. Xu and Koko ^[11] presented a general purpose design scheme of actively controlled piezoelectric smart structures by finite element modal analysis. More recently, Zhao and Liao ^[12] solved the updating problem of undamped piezoelectric smart structure systems with no-spillover and derived a set of parametric solutions. Nevertheless, the problem of BAP seems rarely to be discussed in the literatures, which motivates us to provide a numerical method to solve problems IEP and BAP. By applying the generalized singular value decomposition(GSVD) of a matrix pair, the expression of the general solution of Problem IEP is derived when the solvability conditions are satisfied and the best approximate solution of Problem BAP is obtained. Finally, two numerical examples are given to verify the correctness of the results.

2.   The solution to Problem IEP

Note that $\mbox{rank}(Z_{1}) = p$, then the GSVD ^[13,14] of the matrix pair $[Z_{1}^\top, Z_{2}^\top]$ is of the following form:

where $N \in \mathbb{R}^{p \times p}$ is nonsingular, and

and

with

By (2.1), Eq (1.4) can be equivalently written as

that is,

Let

Thus, Eqs (2.4)-(2.6) are equivalent to

Comparing two sides of (2.12)-(2.14) , we have

In summary, we can state the follwing theorem.

Theorem 2.1. Suppose that $\Omega = {\rm{diag}}(\omega_{1}, ..., \omega_{p})\in \mathbb{R}^{p \times p}$ and $Z = [Z_{1}^\top, Z_{2}^\top]^\top\in \mathbb{R}^{n \times p}$ are the measured eigenvalue and eigenvector matrices, where $Z_{1}\in \mathbb{R}^{n_u \times p}$, $Z_{2}\in \mathbb{R}^{n_\phi \times p}$ and ${\rm{rank}}(Z_{1}) = p.$ Let the GSVD of the matrix pair $[Z_{1}^\top, Z_{2}^\top]$ be given by (2.1). Then Problem IEP is solvable if and only if

In this case, the solution set of Problem IEP can be expressed as

where

$M_{h3} (h = 1, 2),$ $L_{3k} (k = 1, 2, 3),$ $G_{12}, G_{13}, G_{23}$ are arbitrary matrices, and $M_{33}, F_{33},$ $G_{kk} (k = 1, 2, 3)$ are arbitrary symmetric matrices; and $M_{21},$ $F_{12}, F_{13}, F_{23},$ $L_{hk} (h = 1, 2, k = 1, 2, 3)$ and $M_{kk}, F_{kk} (k = 1, 2)$ are given by (2.15) – (2.21).

3.   The solution to Problem BAP

In order to solve Problem BAP, the following lemma is needed.

Lemma 3.1. Let $A, \ B\in \mathbb{SR}^{s\times s},$ $C\in \mathbb{R}^{s\times s},$ and $\Theta = {\rm{diag}}(\theta_1, \cdots, \theta_s)\in \mathbb{R}^{s\times s}, \ \Delta = {\rm{diag}}(\delta_1, \cdots, \delta_s)\in \mathbb{R}^{s\times s}$ satisfy $\theta_i^2+\delta_i^2 = 1, \ i = 1, \cdots, s.$ Then

if and only if

Proof. Let $A = [a_{ij}], \ B = [b_{ij}], \ C = [c_{ij}]\in \mathbb{R}^{s\times s}$, and $G_{22} = [g_{ij}]\in \mathbb{R}^{s\times s}$. Then

Now we minimize the quantities

By direct calculation, we have the minimizers

By rewriting (3.2) in matrix form, we can get (3.1) . □

It is easy to verify that $\mathbb{\kappa_{S}}$ is a closed convex subset of $\mathbb{R}^{n\times n}\times \mathbb{R}^{n\times n}.$ From the best approximate theorem ^[15], we know that there exists a unique solution $(\hat{M}, \hat{K})\in \mathbb{\kappa_{S}}$ to Problem BAP. For the given matrices ${M}_{A}, {K}_{A}\in \mathbb{SR}^{n\times n},$ write

Then

Therefore, $\|M-{M}_{A}\|^{{2}} +\|K-{K}_{A}\|^{2} = \min$ if and only if

From (3.7), we have

Thus,

Clearly, $f(M_{13}, M_{23}, L_{32}) = \min$ if and only if

When $\frac{\partial f(M_{13}, M_{23}, L_{32})}{\partial M_{13}} = 0,$ we arrive at

where

When $\frac{\partial f(M_{13}, M_{23}, L_{32})}{\partial M_{23}} = 0,$ we obtain

where

When $\frac{\partial f(M_{13}, M_{23}, L_{32})}{\partial L_{32}} = 0,$ we get

where

Substituting (3.13) into (3.11) leads to

where

Substituting (3.13) and (3.14) into (3.12), we have

From (3.8), we have

Consequently,

when $\frac{\partial f(G_{12})}{\partial G_{12}} = 0$, we conclude that

From (3.10), we have

Thus,

when $\frac{\partial f(G_{23})}{\partial G_{23}} = 0$, we can get

Solving the minimization problem $f(G_{22})$ by using Lemma 3.1, we have

Theorem 3.1. Given matrices ${M}_{A}, {K}_{A}\in \mathbb{SR}^{n\times n}.$ If the solvability conditions of (2.22) are satisfied, then the solution of Problem BAP is

where

and $M_{13}, \ M_{23}, \ L_{32}, \ G_{12}, \ G_{22},$ $G_{23}$ are given by (3.14), (3.15), (3.13), (3.6), (3.18), (3.17), respectively.

4.   Numerical examples

According to Theorems 2.1 and 3.1, we can describe a numerical algorithm to solve Problem BAP.

Example 4.1. Let $n = 10, \ p = 3$, and the matrices $\Omega, \ Z, \ M_{A}$ and $K_{A}$ be given by

It is easy to verify that the conditions (2.22) hold. By applying Algorithm 1, we can obtain the unique solution $(\hat{M}, \hat{K})$ of Problem BAP as follows:

and

which implies that $\hat{M}Z \Omega = \hat{K}Z$ reproduces the desired eigenvalues and eigenvectors.

Example 4.2. Let $n = 8, \ p = 3$, and the matrices $\Omega, \ Z, \ M_{A}$ and $K_{A}$ be given by

It can easily be seen that the conditions (2.22) hold. By applying Algorithm 1, we can obtain the unique solution $(\hat{M}, \hat{K})$ of Problem BAP as follows:

and

Observe that the prescribed eigenvalues and eigenvectors have been embedded in the new model $\hat{M}Z \Omega = \hat{K}Z$.

5.   Conclusions

A direct updating method for the piezoelectric smart structural models has been established by applying the generalized singular value decomposition. This method makes use of the constrained minimization theory to formulate the minimization error function such that the resulting changes to mass and stiffness matrices are a minimum. The updated model can accurately reproduce the measured eigenstructure data. The approach was verified by two numerical examples and the reasonable results were obtained.

Use of AI tools declaration

The authors declare that we have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare that there is no conflict of interest.

Acknowledgments

The authors are grateful to three referees for their careful reading of the manuscript and making useful comments and suggestions which greatly improved the original presentation.