**4.1 Deadbeat mode behavior as** *k***<sup>1</sup>** ! **1**

As *k*<sup>1</sup> approaches 1, the spectral branches are moving upward and toward the imaginary axis (**Figures 6** and **7**). As a result of this motion, eigenvalues approach the imaginary axis at different rates depending on whether *k*<sup>1</sup> approaches 1 from above or below.

As follows from **Table 3**, the real parts of the eigenvalues decrease steadily as *k*<sup>1</sup> ! 1þ, to a point where the eigenvalue becomes a deadbeat mode. An increase in the number of deadbeat modes can be seen as *k*<sup>1</sup> ! 1þ, which is in agreement with Statement (3) of Theorem 1. One can see from **Table 3** that there are pairs of modes such that the distance between them tends to zero as *k*<sup>1</sup> ! 1þ. (Compare modes no.5 and no.7 for <sup>∣</sup>*k*<sup>1</sup> � <sup>1</sup><sup>∣</sup> <sup>¼</sup> <sup>10</sup>�4, modes no.4 and no.7 for <sup>∣</sup>*k*<sup>1</sup> � <sup>1</sup><sup>∣</sup> <sup>¼</sup> <sup>10</sup>�6, modes no.5 and no.8 for <sup>∣</sup>*k*<sup>1</sup> � <sup>1</sup><sup>∣</sup> <sup>¼</sup> <sup>10</sup>�8, and modes no.4 and no.7 for <sup>∣</sup>*k*<sup>1</sup> � <sup>1</sup><sup>∣</sup> <sup>¼</sup> <sup>10</sup>�10). Such behavior indicates convergence of the two simple deadbeat modes to a double mode, which is consistent with Theorem 2.

Analyzing **Table 4**, one can see that the eigenvalues get closer to the imaginary axis as *k*<sup>1</sup> ! 1�. However the rate at which their real parts approach zero is

**Figure 6.** *Eigenvalues with* ∣Re*λ*∣ . 10 *as k*<sup>1</sup> ! 1þ*.*

If *f* ¼ *f ξ*<sup>1</sup> ð Þ*; f ξ*<sup>2</sup> ½ � ð Þ*;* …*; f*ð Þ *ξ<sup>N</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.86940*

*<sup>D</sup>*<sup>11</sup> ¼ �*DNN* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>ð Þ *<sup>N</sup>* � <sup>1</sup> <sup>2</sup>

**5.1 Discretization of** L*<sup>k</sup>***1***,k***<sup>2</sup>**

and

D L*<sup>k</sup>*1*,k*<sup>2</sup> ð Þ¼fð Þ *u*0*; u*<sup>1</sup>

where <sup>H</sup> <sup>¼</sup> *<sup>H</sup>*<sup>2</sup>

*u*0ð Þ*ξ* ≈ ∑ *N k*¼1

**143**

**Φ** ¼ ½ � Φ1*;* Φ2*;* …*;* Φ*<sup>N</sup>*

4*EI*ð Þ1 *u*″

k k *<sup>U</sup>* <sup>2</sup> <sup>H</sup> <sup>¼</sup> <sup>1</sup> 4 Z <sup>1</sup> �1

<sup>0</sup>ð Þ� �1*;* <sup>1</sup> *<sup>L</sup>*<sup>2</sup>

*<sup>T</sup>* and *<sup>g</sup>* <sup>¼</sup> *<sup>f</sup>*

� � for *<sup>j</sup>* 6¼ *<sup>k</sup>:*

Rescaling the independent variable *x* as *ξ* ¼ 2*x* � 1, we rewrite the operator and

*∂*2

*<sup>∂</sup>ξ*<sup>2</sup> *EI*ð Þ*<sup>ξ</sup> <sup>∂</sup>*<sup>2</sup>

<sup>0</sup>ð Þ*<sup>ξ</sup>* � �<sup>0</sup>

ð Þ �1*;* 1 , equipped with the norm

*N k*¼1

<sup>0</sup>ð Þ*<sup>ξ</sup>* � � � � 2

We approximate the action of L*<sup>k</sup>*1*,k*<sup>2</sup> on the finite-dimensional subspace H*<sup>N</sup>* ⊂ H

*g* ¼ *Df*, where *D* is the Chebyshev derivative matrix with the elements

*Spectral Analysis and Numerical Investigation of a Flexible Structure with…*

*Dj,k* <sup>¼</sup> *cj*ð Þ �<sup>1</sup> *<sup>j</sup>*þ*<sup>k</sup>*

its domain, representations (14) and (15), in the form

L*<sup>k</sup>*1*,k*<sup>2</sup> ¼ �*i*

*ck ξ<sup>j</sup>* � *ξ<sup>k</sup>*

� <sup>16</sup> *ρ ξ*ð Þ

*<sup>T</sup>* <sup>∈</sup> <sup>H</sup> : *<sup>u</sup>*<sup>0</sup> <sup>∈</sup> *<sup>H</sup>*<sup>4</sup>ð Þ �1*;* <sup>1</sup> *, u*<sup>1</sup> <sup>∈</sup> *<sup>H</sup>*<sup>2</sup>

16*EI*ð Þ*ξ u*″

of polynomials of degree at most ð Þ *N* � 1 . Using the CGL grid and the cardinal

Let **Φ** and **Θ** be *N*-dim vectors and **Ψ** be a 2*N*-dim vector defined by

*T,* **<sup>Θ</sup>** <sup>¼</sup> ½ � <sup>Θ</sup>1*;* <sup>Θ</sup>2*;* …*;* <sup>Θ</sup>*<sup>N</sup>*

�<sup>16</sup> *EI ρ*

where *IN*�*<sup>N</sup>* is the *N* � *N* identity matrix and *D* is the derivative matrix (58).

Let *L* be the finite-dimensional approximation of the differential operator L*<sup>k</sup>*1*,k*<sup>2</sup> .

**0** *IN*�*<sup>N</sup>*

3

*D*<sup>4</sup> **0**

functions, we substitute for *u*<sup>0</sup> and *u*<sup>1</sup> their truncated expansions:

Φ*kψk*ð Þ*ξ ,* Φ*<sup>k</sup>* ¼ *u*<sup>0</sup> *ξ<sup>k</sup>* ð Þ*, u*1ð Þ*ξ* ≈ ∑

The discretized operator *L* induced by L*<sup>k</sup>*1*,k*<sup>2</sup> can be given by

*L* ¼ �*i*

2 4

<sup>0</sup>ð Þ¼� 1 *k*1*u*1ð Þ1 *;* 4 *EI*ð Þ*ξ u*″

2 4

<sup>6</sup> *, Dkk* ¼ � *<sup>ξ</sup><sup>k</sup>*

<sup>0</sup> *ξ*<sup>1</sup> ð Þ*; f*

2 1 � *<sup>ξ</sup>*<sup>2</sup> *k*

0 1

*∂ξ*<sup>2</sup> � � <sup>0</sup>

> � � � *ξ*¼1

<sup>þ</sup> *ρ ξ*ð Þj j *<sup>u</sup>*1ð Þ*<sup>ξ</sup>* <sup>2</sup> h i*dξ:* (61)

<sup>0</sup> *ξ*<sup>2</sup> ð Þ*;* …*; f*

� � for 1 , *<sup>k</sup>* , *N,*

3

<sup>0</sup>ð Þ �1*;* 1 ; *u*1ð Þ¼ �1 *u*<sup>0</sup>

<sup>1</sup>ð1Þg*,*

Θ*kψk*ð Þ*ξ ,* Θ*<sup>k</sup>* ¼ *u*<sup>1</sup> *ξ<sup>k</sup>* ð Þ*:* (62)

**Θ**

5*,* (64)

� �*:* (63)

*T,* **<sup>Ψ</sup>** <sup>¼</sup> **<sup>Φ</sup>**

¼ *k*2*u*<sup>0</sup>

5*,* (59)

� �*<sup>T</sup>*

0 ð Þ *ξ<sup>N</sup>* , then

(58)

<sup>1</sup>ð Þ¼ �1 0;

(60)

**Figure 7.** *Eigenvalues with* ∣Re*λ*∣ . 10 *as k*<sup>1</sup> ! 1�*.*

significantly lower than in the case *<sup>k</sup>*<sup>1</sup> ! <sup>1</sup>þ. Even at *<sup>k</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> � <sup>10</sup>�10, the eigenvalue closest to the imaginary axis has a real part of about 10�4, which means that it is not a deadbeat mode (see Statement (1) of Theorem 1).

The eigenvalues near the imaginary axis approach the same double deadbeat modes in both cases when *k*<sup>1</sup> ! 1� (see Statement (2) of Theorem 1). In conclusion, one can claim that the eigenvalues are indeed approaching the imaginary axis; however, the rate of this approach is different for *k*<sup>1</sup> ! 1� and *k*<sup>1</sup> ! 1þ. In the former case, an eigenvalue's distance from the imaginary axis decreases very slowly; in the latter case, the eigenvalues quickly "jump" on the imaginary axis and turn into deadbeat modes.

#### **5. Outline of the numerical scheme**

To carry out the numerical analysis of the differential operator L*<sup>k</sup>*1*,k*<sup>2</sup> , we use the Chebyshev collocation method and cardinal functions [14–16].

Recall that the *N*th Chebyshev polynomial of the first kind is defined by

$$T\_N(\xi) = \cos N\theta, \quad -1 \le \xi \le 1 \qquad \text{where} \qquad \xi = \cos \theta, \quad \theta \in [0, \pi]. \tag{55}$$

The cardinal functions, ψ*k*ð Þ*ξ* , and the Chebyshev-Gauss-Lobatto (CGL) grid points f g *ξ<sup>k</sup>* are defined as follows:

$$\varphi\_k(\xi) = (-1)^k \frac{(1 - \xi^2)T'\_{N-1}(\xi)}{c\_k (N-1)^2 (\xi - \xi\_k)}, \qquad \xi\_k = \cos \frac{(k-1)\pi}{N-1}, \quad \text{for} \quad 1 \le k \le N,\tag{56}$$

where coefficients *ck* are such that *c*<sup>1</sup> ¼ *cN* ¼ 2 and *ck* ¼ 1 for 1 , *k* , *N*. The main property of cardinal functions is *ψ<sup>k</sup> ξ<sup>j</sup>* <sup>¼</sup> *<sup>δ</sup>kj* (using the Kronecker delta). The family f g *ψ<sup>k</sup> N <sup>k</sup>*¼<sup>1</sup> forms a basis in the space of polynomials of degree ð Þ *N* � 1 , i.e., if *f* is such polynomial, then *f* and *f* <sup>0</sup> can be written in the forms

$$f(\xi) = \sum\_{k=1}^{N} f(\xi\_k) \boldsymbol{\mu}\_k(\xi) \qquad \text{and} \qquad f'(\xi) = \sum\_{k=1}^{N} f'(\xi\_k) \boldsymbol{\mu}\_k(\xi). \tag{57}$$

*Spectral Analysis and Numerical Investigation of a Flexible Structure with… DOI: http://dx.doi.org/10.5772/intechopen.86940*

If *f* ¼ *f ξ*<sup>1</sup> ð Þ*; f ξ*<sup>2</sup> ½ � ð Þ*;* …*; f*ð Þ *ξ<sup>N</sup> <sup>T</sup>* and *<sup>g</sup>* <sup>¼</sup> *<sup>f</sup>* <sup>0</sup> *ξ*<sup>1</sup> ð Þ*; f* <sup>0</sup> *ξ*<sup>2</sup> ð Þ*;* …*; f* 0 ð Þ *ξ<sup>N</sup>* � �*<sup>T</sup>* , then *g* ¼ *Df*, where *D* is the Chebyshev derivative matrix with the elements

$$D\_{11} = -D\_{NN} = \frac{1 + 2(N - 1)^2}{6}, \qquad D\_{kk} = -\frac{\xi\_k}{2\left(1 - \xi\_k^2\right)} \quad \text{for} \quad 1 \le k \le N,\tag{58}$$

$$D\_{j,k} = \frac{c\_j (-1)^{j+k}}{c\_k \left(\xi\_j - \xi\_k\right)} \quad \text{for} \quad j \ne k.$$

### **5.1 Discretization of** L*<sup>k</sup>***1***,k***<sup>2</sup>**

Rescaling the independent variable *x* as *ξ* ¼ 2*x* � 1, we rewrite the operator and its domain, representations (14) and (15), in the form

$$\mathcal{L}\_{k\_1,k\_2} = -i \begin{bmatrix} 0 & 1 \\ -\frac{16}{\rho(\xi)} \frac{\partial^2}{\partial \xi^2} \left( EI(\xi) \frac{\partial^2}{\partial \xi^2} \right) & 0 \end{bmatrix},\tag{59}$$

and

significantly lower than in the case *<sup>k</sup>*<sup>1</sup> ! <sup>1</sup>þ. Even at *<sup>k</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> � <sup>10</sup>�10, the eigenvalue closest to the imaginary axis has a real part of about 10�4, which means that it is not

The eigenvalues near the imaginary axis approach the same double deadbeat modes in both cases when *k*<sup>1</sup> ! 1� (see Statement (2) of Theorem 1). In conclusion, one can claim that the eigenvalues are indeed approaching the imaginary axis; however, the rate of this approach is different for *k*<sup>1</sup> ! 1� and *k*<sup>1</sup> ! 1þ. In the former case, an eigenvalue's distance from the imaginary axis decreases very slowly; in the latter case, the eigenvalues quickly "jump" on the imaginary axis and turn

To carry out the numerical analysis of the differential operator L*<sup>k</sup>*1*,k*<sup>2</sup> , we use the

*TN*ð Þ¼ *ξ* cos *Nθ,* � 1≤ *ξ*≤ 1 where *ξ* ¼ cos *θ, θ* ∈½ � 0*; π :* (55)

*<sup>N</sup>* � <sup>1</sup> *,* for 1<sup>≤</sup> *<sup>k</sup>*<sup>≤</sup> *N,*

¼ *δkj* (using the Kronecker delta). The

<sup>0</sup> *ξ<sup>k</sup>* ð Þ*ψk*ð Þ*ξ :* (57)

(56)

The cardinal functions, ψ*k*ð Þ*ξ* , and the Chebyshev-Gauss-Lobatto (CGL) grid

Recall that the *N*th Chebyshev polynomial of the first kind is defined by

*ck*ð Þ *<sup>N</sup>* � <sup>1</sup> <sup>2</sup> *<sup>ξ</sup>* � *<sup>ξ</sup><sup>k</sup>* ð Þ*, <sup>ξ</sup><sup>k</sup>* <sup>¼</sup> cos ð Þ *<sup>k</sup>* � <sup>1</sup> *<sup>π</sup>*

*f ξ<sup>k</sup>* ð Þ*ψk*ð Þ*ξ* and *f*

where coefficients *ck* are such that *c*<sup>1</sup> ¼ *cN* ¼ 2 and *ck* ¼ 1 for 1 , *k* , *N*. The

*<sup>k</sup>*¼<sup>1</sup> forms a basis in the space of polynomials of degree ð Þ *N* � 1 , i.e., if *f*

0 ð Þ¼ *ξ* ∑ *N k*¼1 *f*

<sup>0</sup> can be written in the forms

Chebyshev collocation method and cardinal functions [14–16].

*<sup>N</sup>*�<sup>1</sup>ð Þ*<sup>ξ</sup>*

a deadbeat mode (see Statement (1) of Theorem 1).

**5. Outline of the numerical scheme**

points f g *ξ<sup>k</sup>* are defined as follows:

*<sup>ψ</sup>k*ð Þ¼ � *<sup>ξ</sup>* ð Þ<sup>1</sup> *<sup>k</sup>* <sup>1</sup> � *<sup>ξ</sup>*<sup>2</sup> *<sup>T</sup>*<sup>0</sup>

family f g *ψ<sup>k</sup>*

**142**

*N*

is such polynomial, then *f* and *f*

*f*ð Þ¼ *ξ* ∑ *N k*¼1

main property of cardinal functions is *ψ<sup>k</sup> ξ<sup>j</sup>*

into deadbeat modes.

*Eigenvalues with* ∣Re*λ*∣ . 10 *as k*<sup>1</sup> ! 1�*.*

**Figure 7.**

*Functional Calculus*

$$\mathcal{D}(\mathcal{L}\_{k\_1,k\_2}) = \{ (u\_0, u\_1)^T \in \mathcal{H} \, : \, u\_0 \in H^4(-1, 1), \,\, u\_1 \in H\_0^2(-1, 1); \,\, u\_1(-1) = u\_1'(-1) = 0;$$

$$4EI(\mathbf{1})u\_0'(\mathbf{1}) = -k\_1 u\_1(\mathbf{1}), \,\, 4\{EI(\boldsymbol{\xi})u\_0^"(\boldsymbol{\xi})\big|\_{\boldsymbol{\xi}=1}^\prime = k\_2 u\_1'(\mathbf{1})\},\,\, \, \, \, \, \, \},\,\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,$$

where <sup>H</sup> <sup>¼</sup> *<sup>H</sup>*<sup>2</sup> <sup>0</sup>ð Þ� �1*;* <sup>1</sup> *<sup>L</sup>*<sup>2</sup> ð Þ �1*;* 1 , equipped with the norm

$$\left\|\left\|U\right\|\right\|\_{\mathcal{H}}^2 = \frac{1}{4} \int\_{-1}^{1} \left[\mathbf{1}\left\|EI(\xi)\right\|u\_0^\*(\xi)\right]^2 + \rho(\xi)\left|u\_1(\xi)\right|^2\right]d\xi. \tag{61}$$

We approximate the action of L*<sup>k</sup>*1*,k*<sup>2</sup> on the finite-dimensional subspace H*<sup>N</sup>* ⊂ H of polynomials of degree at most ð Þ *N* � 1 . Using the CGL grid and the cardinal functions, we substitute for *u*<sup>0</sup> and *u*<sup>1</sup> their truncated expansions:

$$u\_0(\xi) \approx \sum\_{k=1}^{N} \Phi\_k \boldsymbol{\upmu}\_k(\xi), \quad \Phi\_k = u\_0(\xi\_k), \qquad u\_1(\xi) \approx \sum\_{k=1}^{N} \Theta\_k \boldsymbol{\upmu}\_k(\xi), \quad \Theta\_k = u\_1(\xi\_k). \tag{62}$$

Let **Φ** and **Θ** be *N*-dim vectors and **Ψ** be a 2*N*-dim vector defined by

$$\boldsymbol{\Phi} = \begin{bmatrix} \Phi\_1 & \Phi\_2 & \dots & \Phi\_N \end{bmatrix}^T, \qquad \boldsymbol{\Theta} = \begin{bmatrix} \Theta\_1 & \Theta\_2 & \dots & \Theta\_N \end{bmatrix}^T, \qquad \boldsymbol{\Psi} = \begin{bmatrix} \boldsymbol{\Phi} \\ \boldsymbol{\Theta} \end{bmatrix}.\tag{63}$$

Let *L* be the finite-dimensional approximation of the differential operator L*<sup>k</sup>*1*,k*<sup>2</sup> . The discretized operator *L* induced by L*<sup>k</sup>*1*,k*<sup>2</sup> can be given by

$$L = -i \begin{bmatrix} \mathbf{0} & I\_{N \times N} \\ -\mathbf{1} \mathbf{6} \frac{EI}{\rho} D^4 & \mathbf{0} \end{bmatrix},\tag{64}$$

where *IN*�*<sup>N</sup>* is the *N* � *N* identity matrix and *D* is the derivative matrix (58).

### **5.2 Incorporating the boundary conditions**

Discretization of the boundary conditions in the domain description (60) yields

$$\begin{aligned} \Phi\_N &= 0, \quad [D\Phi]\_N = 0, \quad \Theta\_N = 0, \quad [D\Theta]\_N = 0, \\ 4EI\Big[D^2\Phi\big]\_1 + k\_1\Theta\_1 &= 0, \quad 4EI\Big[D^3\Phi\big]\_1 - k\_2[D\Theta]\_1 = 0. \end{aligned} \tag{65}$$

k k **<sup>Ψ</sup>** *<sup>C</sup>* <sup>¼</sup> *<sup>π</sup>=*<sup>4</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.86940*

**6. Conclusions**

who need the elastic modes.

ð Þ *n* 1

highly appreciated by the first author.

Marianna A. Shubov\*† and Laszlo P. Kindrat† University of New Hampshire, Durham, NH, USA

† These authors contributed equally.

provided the original work is properly cited.

\*Address all correspondence to: marianna.shubov@gmail.com

the sequence *k*

**Author details**

**145**

**Acknowledgements**

*<sup>N</sup>* � <sup>1</sup> <sup>∑</sup> *N k*¼1 ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>ξ</sup>*<sup>2</sup> *k*

<sup>16</sup>*EI <sup>ξ</sup><sup>k</sup>* ð Þ *<sup>D</sup>*<sup>2</sup>

In this work we have considered the spectral properties of the Euler-Bernoulli beam model with special feedback-type boundary conditions. The dynamics generator of the model is a non-self-adjoint matrix differential operator acting in a Hilbert space of two-component Cauchy data. This operator has been approximated by a "discrete" operator using Chebyshev polynomial approximation. We have shown that the eigenvalues of the main operator can be approximated by the eigenvalues of its discrete counterpart with high accuracy. This means that the leading asymptotic terms in formulas (20) and (21) can be used by practitioners

Further results deal with existence and formulas of the deadbeat modes. It has been shown that for the case when one control parameter, *k*1, is such that *k*<sup>1</sup> ! 1<sup>þ</sup> and the other one *k*<sup>2</sup> ¼ 0, the number of deadbeat modes approaches infinity. The formula for the rate at which the number of the deadbeat modes tends to infinity has been derived. It has also been established that there exists a sequence *k*

of the values of parameter *k*1, such that the corresponding deadbeat mode has a multiplicity 2, which yields the existence of the associate mode shapes for the operator L*<sup>k</sup>*1*,k*<sup>2</sup> . The formulas for the double deadbeat modes and asymptotics for

Partial support of the National Science Foundation award DMS-1810826 is

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

n o as *<sup>n</sup>* ! <sup>∞</sup> have been derived.

� � � � 2

**Φ** � � *k*

<sup>2</sup> h i*:*

þ *ρ ξ<sup>k</sup>* ð Þj j Θ*<sup>k</sup>*

ð Þ *n* 1 n o<sup>∞</sup>

*n*¼1

q

*Spectral Analysis and Numerical Investigation of a Flexible Structure with…*

Let *rN, zN, lN* ∈ R*<sup>N</sup>* be auxiliary row-vectors

$$\begin{aligned} \mathbf{r}\_N &= [\mathbf{0} & \mathbf{0} & \cdots & \mathbf{0} & \mathbf{1}], \quad \mathbf{z}\_N = [\mathbf{0} & \mathbf{0} & \cdots & \mathbf{0} & \mathbf{0}], \quad l\_N = [\mathbf{1} & \mathbf{0} & \cdots & \mathbf{0} & \mathbf{0}] \end{aligned} \tag{66}$$

and *Dn <sup>j</sup>* designate the *j*th row of the *n*th derivative matrix *Dn*. Using Eqs. (66) we represent Eqs. (65) as the following matrix equation:

$$\mathbb{K}\Psi \equiv \begin{bmatrix} r\_N & z\_N \\ D\_N^1 & z\_N \\ z\_N & r\_N \\ z\_N & D\_N^1 \\ 4EID\_1^2 & k\_1 l\_N \\ 4EID\_1^3 & -k\_2 D\_1^1 \end{bmatrix} \begin{bmatrix} \Phi \\ \Theta \end{bmatrix} = \mathbf{0}.\tag{67}$$

K is called the boundary operator. Let K*<sup>N</sup>* be the kernel of K, i.e., <sup>K</sup>*<sup>N</sup>* <sup>¼</sup> *<sup>v</sup>*<sup>∈</sup> <sup>R</sup><sup>2</sup>*<sup>N</sup>* : <sup>K</sup>*<sup>v</sup>* <sup>¼</sup> <sup>0</sup> � �. We have to identify all eigenvalues of the operator **<sup>L</sup>**, when its domain is restricted to K*N*. It is clear that K*<sup>N</sup>* is isomorphic to R*<sup>k</sup>* with *k* � dimK*<sup>N</sup>* ¼ dimH*<sup>N</sup>* � rankK ¼ 2*N* � 6. Let *B* be the matrix consisting of column vectors that form an orthonormal basis in K*N*. It is clear that *BTB* is the identity matrix on R*<sup>k</sup>* and *BBT* is the identity matrix on K. The following result holds: if *λ* is an eigenvalue of the operator *L*, and the corresponding eigenvector **Ψ** satisfies Eq. (67), then the same *λ* is an eigenvalue of the matrix *BTLB* � �. However, the inverse statement is not necessarily true. Indeed, we observe that *BBT* is the identity in KN , which is not equivalent to the identity in H*N*. Assume now that *λ* is an eigenvalue of *BTL<sup>B</sup>* with corresponding eigenvector *<sup>v</sup>*<sup>∈</sup> <sup>R</sup>*<sup>k</sup>*. If **<sup>Ψ</sup>** <sup>¼</sup> *Bv*, we have

$$BB^T \mathbf{L} \Psi = BB^T \mathbf{L} Bv = \lambda Bv = \lambda \Psi,\tag{68}$$

but *BBT<sup>L</sup>* 6¼ *<sup>L</sup>,* which indicates that fake eigenvalues may exist.

#### **5.3 Filtering of spurious eigenvalues**

In order to decide which eigenvalues of *BTLB* should be discarded, we impose the following condition. Let Λ be the spectrum of *BTLB* and *V* be the set of its eigenfunctions. We construct the set of "trusted" eigenvalues [14, 15], for some *ε<sup>f</sup>* . 0 filtering precision, as

<sup>Λ</sup>*<sup>ε</sup>* <sup>¼</sup> *<sup>λ</sup>*∈<sup>Λ</sup> : k k *<sup>L</sup>Bv<sup>λ</sup>* � *<sup>λ</sup>Bv<sup>λ</sup> <sup>C</sup>* , *<sup>ε</sup><sup>f</sup> ;*for corresponding eigenvector *<sup>v</sup><sup>λ</sup>* <sup>∈</sup>*<sup>V</sup>* � �*,* (69)

where k k� *<sup>C</sup>* is a discrete approximation to the integral norm defined in Eq. (61). (The subscript *C* is short for Chebyshev). Using the CGL quadrature, we obtain the following formula for the norm of a vector **Ψ** defined as in Eq. (63):

*Spectral Analysis and Numerical Investigation of a Flexible Structure with… DOI: http://dx.doi.org/10.5772/intechopen.86940*

$$\|\|\Psi\|\|\_{C} = \frac{\pi/4}{N-1} \sum\_{k=1}^{N} \sqrt{1-\xi\_k^2} \left[ \mathbf{1} \mathbf{\tilde{\xi}} E I(\xi\_k) \left| \left[ D^2 \Phi \right]\_k \right|^2 + \rho(\xi\_k) \left| \Theta\_k \right|^2 \right].$$
