**6. Conclusions**

**5.2 Incorporating the boundary conditions**

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

represent Eqs. (65) as the following matrix equation:

K**Ψ** �

4*EI D*<sup>2</sup> **Φ** � �

and *Dn*

*Functional Calculus*

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

**Φ** � �

*<sup>j</sup>* designate the *j*th row of the *n*th derivative matrix *Dn*. Using Eqs. (66) we

**Φ Θ** � �

*BBTL***<sup>Ψ</sup>** <sup>¼</sup> *BBTLBv* <sup>¼</sup> *<sup>λ</sup>Bv* <sup>¼</sup> *<sup>λ</sup>***Ψ***,* (68)

*N*

<sup>1</sup> � *<sup>k</sup>*2½ � *<sup>D</sup>***<sup>Θ</sup>** <sup>1</sup> <sup>¼</sup> <sup>0</sup>*:* (65)

¼ 0*:* (67)

(66)

Φ*<sup>N</sup>* ¼ 0*, D*½ � **Φ** *<sup>N</sup>* ¼ 0*,* Θ*<sup>N</sup>* ¼ 0*, D*½ � **Θ** *<sup>N</sup>* ¼ 0*,*

*rN* ¼ ½ � 0 0 ⋯ 0 1 *, zN* ¼ ½ � 0 0 ⋯ 0 0 *, lN* ¼ ½ � 1 0 ⋯ 0 0

*rN zN*

*<sup>N</sup> zN zN rN zN D*<sup>1</sup>

<sup>1</sup> *k*1*lN*

<sup>1</sup> �*k*2*D*<sup>1</sup> 1

<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

*D*1

4*EID*<sup>2</sup>

4*EID*<sup>3</sup>

K is called the boundary operator. Let K*<sup>N</sup>* be the kernel of K, i.e.,

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

following formula for the norm of a vector **Ψ** defined as in Eq. (63):

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

<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

**5.3 Filtering of spurious eigenvalues**

*ε<sup>f</sup>* . 0 filtering precision, as

**144**

<sup>1</sup> <sup>þ</sup> *<sup>k</sup>*1Θ<sup>1</sup> <sup>¼</sup> <sup>0</sup>*,* <sup>4</sup>*EI D*<sup>3</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 who need the elastic modes.

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* ð Þ *n* 1 n o<sup>∞</sup> *n*¼1 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 the sequence *k* ð Þ *n* 1 n o as *<sup>n</sup>* ! <sup>∞</sup> have been derived.

#### **Acknowledgements**

Partial support of the National Science Foundation award DMS-1810826 is highly appreciated by the first author.

## **Author details**

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

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

† These authors contributed equally.

© 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, provided the original work is properly cited.
