**Abstract**

Analytic and numerical results of the Euler-Bernoulli beam model with a twoparameter family of boundary conditions have been presented. The co-diagonal matrix depending on two control parameters (*k*<sup>1</sup> and *k*2) relates a two-dimensional input vector (the shear and the moment at the right end) and the observation vector (the time derivatives of displacement and the slope at the right end). The following results are contained in the paper. First, high accuracy numerical approximations for the eigenvalues of the discretized differential operator (the dynamics generator of the model) have been obtained. Second, the formula for the number of the deadbeat modes has been derived for the case when one control parameter, *k*1, is positive and another one, *k*2, is zero. It has been shown that the number of the deadbeat modes tends to infinity, as *k*<sup>1</sup> ! 1<sup>þ</sup> and *k*<sup>2</sup> ¼ 0. Third, the existence of *double* deadbeat modes and the asymptotic formula for such modes have been proven. Fourth, numerical results corroborating all analytic findings have been produced by using Chebyshev polynomial approximations for the continuous problem.

**Keywords:** matrix differential operator, eigenvalues, Chebyshev polynomials, numerical scheme, boundary control

## **1. Introduction**

The present paper is concerned with the spectral analysis and numerical investigation of the eigenvalues of the Euler-Bernoulli beam model. The beam is clamped at the left end and subject to linear feedback-type conditions with a nondissipative feedback matrix [1, 2]. Depending on the boundary parameters *k*<sup>1</sup> and *k*2, the model can be either conservative, dissipative, or completely non-dissipative. We focus on the non-dissipative case, i.e., when the energy of a vibrating system is not a decreasing (or nonincreasing) function of time. In our approach, the initialboundary value problem describing the beam dynamics is reduced to the first order in time evolution equation in the state Hilbert space H. The evolution of the system is completely determined by the dynamics generator L*<sup>k</sup>*1*,k*<sup>2</sup> , *which is an unbounded non-self-adjoint matrix differential operator* (see Eqs. (2), (3), and (8)).

The eigenmodes and the mode shapes of the flexible structure are defined as the eigenvalues (up to a multiple *i*) and the generalized eigenvectors of L*<sup>k</sup>*1*,k*<sup>2</sup> .

Based on the results of [1, 2], the dynamics generator has a purely discrete spectrum, whose location on the complex plane is determined by the controls *k*<sup>1</sup> and *k*2. Having in mind the practical applications of the asymptotic formulas [3–5], we discuss the case of *k*<sup>1</sup> ≥ 0 and *k*<sup>2</sup> ≥0, such that ∣*k*1∣ þ ∣*k*2∣ . 0 (see Proposition 2). As shown in [2], even though the operator L*<sup>k</sup>*1*,k*<sup>2</sup> is non-dissipative, for the case *k*<sup>1</sup> . 0 and *k*<sup>2</sup> ¼ 0 (or *k*<sup>1</sup> ¼ 0 and *k*<sup>2</sup> . 0), the entire set of eigenvalues is located in the closed upper half of the complex plane C, which means that all eigenmodes are stable or neutrally stable. (We recall that to obtain an elastic mode from an eigenvalues of L*<sup>k</sup>*1*,k*<sup>2</sup> , one should multiply the eigenvalue by a factor *i*).

In the paper we address the question of accuracy of the asymptotic formulas for the eigenvalues. *Namely, under what conditions the leading asymptotic terms in formulas (20) and (21) can be used for practical estimation of the actual frequencies of the flexible beam?* Numerical simulations show that the accuracy of the asymptotic formulas is really high; the leading asymptotic terms can be used by practitioners almost immediately, i.e., almost from the first vibrational mode. The second question is concerned with the role of the *deadbeat modes*. A deadbeat mode is a purely negative elastic mode that generates a solution of the evolution equation exponentially decaying in time. The deadbeat modes are important in engineering applications. As we prove in the paper, when the boundary parameter *k*<sup>1</sup> is close to 1 (while *k*<sup>2</sup> ¼ 0), the number of the deadbeat modes is so large that the corresponding mode shapes become important for the description of the beam dynamics. More precisely, the number of deadbeat modes tends to infinity as *k*<sup>1</sup> ! 1þ.

We have also shown that there exists a sequence of values of the parameter *k*1, i.e., *k* ð Þ *n* 1 n o<sup>∞</sup> *n*¼1 , such that for each *k*<sup>1</sup> ¼ *k* ð Þ *n* <sup>1</sup> there exist a finite number of deadbeat modes and each corresponds to a *double eigenvalue* of the dynamics generator L*<sup>k</sup>*1*,k*<sup>2</sup> . For each value *k* ð Þ *n* <sup>1</sup> , the operator L*<sup>k</sup>*1*,k*<sup>2</sup> has a two-dimensional root subspace spanned by an eigenvector and an associate vector. This result means that for a double deadbeat mode (corresponding to *k* ð Þ *n* <sup>1</sup> ), there exists a mode shape and an associate mode shape. This fact indicates that for some values of *k*<sup>1</sup> and *k*2, there exists a significant number of associate vectors of L*<sup>k</sup>*1*,k*<sup>2</sup> . Therefore, if one can prove that the set of the generalized eigenvectors (eigenvectors and associate vectors together) forms an unconditional basis for the state space, then construction of the bi-orthogonal basis [6] would be a more complicated problem than for the case when no associate vectors exist.

we study the deadbeat modes and derive the estimates for the number of the deadbeat modes from below and above for different values of the boundary parameters (see **Figure 5**). Section 4 is concerned with the asymptotic approximation for the set of double deadbeat modes (see **Tables 3** and **4** and **Figures 6** and **7**). In Section 5, we outline the numerical scheme used for the spectral analysis of the

*Approximations of the eigenvalues for the discrete and "continuous" operators (*K . 1*).*

**No. Numerical Analytic No. Numerical Analytic** 1. �7988*:*1 þ 237*:*00*i* �7988*:*1 þ 237*:*00*i* 18. 28*:*860 þ 13*:*515*i* 29*:*453 þ 14*:*812*i* 2. �7020*:*6 þ 222*:*19*i* �7020*:*6 þ 222*:*19*i* 19. 123*:*16 þ 29*:*723*i* 123*:*08 þ 29*:*625*i* 3. �6115*:*5 þ 207*:*37*i* �6115*:*5 þ 207*:*37*i* 20. 279*:*13 þ 44*:*431*i* 279*:*14 þ 44*:*437*i* 4. �5272*:*8 þ 192*:*56*i* �5272*:*8 þ 192*:*56*i* 21. 497*:*61 þ 59*:*250*i* 497*:*61 þ 59*:*250*i* 5. �4492*:*5 þ 177*:*75*i* �4492*:*5 þ 177*:*75*i* 22. 778*:*50 þ 74*:*062*i* 778*:*50 þ 74*:*062*i* 6. �3774*:*7 þ 162*:*94*i* �3774*:*7 þ 162*:*94*i* 23. 1121*:*8 þ 88*:*875*i* 1121*:*8 þ 88*:*875*i* 7. �3119*:*3 þ 148*:*12*i* �3119*:*3 þ 148*:*12*i* 24. 1527*:*6 þ 103*:*69*i* 1527*:*6 þ 103*:*69*i* 8. �2526*:*3 þ 133*:*31*i* �2526*:*3 þ 133*:*31*i* 25. 1995*:*7 þ 118*:*50*i* 1995*:*7 þ 118*:*50*i* 9. �1995*:*7 þ 118*:*50*i* �1995*:*7 þ 118*:*50*i* 26. 2526*:*3 þ 133*:*31*i* 2526*:*3 þ 133*:*31*i* 10. �1527*:*6 þ 103*:*69*i* �1527*:*6 þ 103*:*69*i* 27. 3119*:*3 þ 148*:*12*i* 3119*:*3 þ 148*:*12*i* 11. �1121*:*8 þ 88*:*875*i* �1121*:*8 þ 88*:*875*i* 28. 3774*:*7 þ 162*:*94*i* 3774*:*7 þ 162*:*94*i* 12. �778*:*5 þ 74*:*062*i* �778*:*5 þ 74*:*062*i* 29. 4492*:*5 þ 177*:*75*i* 4492*:*5 þ 177*:*75*i* 13. �497*:*61 þ 59*:*250*i* �497*:*61 þ 59*:*250*i* 30. 5272*:*8 þ 192*:*56*i* 5272*:*8 þ 192*:*56*i* 14. �279*:*13 þ 44*:*431*i* �279*:*14 þ 44*:*437*i* 31. 6115*:*5 þ 207*:*37*i* 6115*:*5 þ 207*:*37*i* 15. �123*:*16 þ 29*:*723*i* �123*:*08 þ 29*:*625*i* 32. 7020*:*6 þ 222*:*19*i* 7020*:*6 þ 222*:*19*i* 16. �28*:*860 þ 13*:*515*i* �29*:*453 þ 14*:*812*i* 33. 7988*:*1 þ 237*:*00*i* 7988*:*1 þ 237*:*00*i*

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

**1.1 The initial-boundary value problem for the Euler-Bernoulli beam model**

*<sup>t</sup>*ð Þ� *<sup>x</sup>; <sup>t</sup> E x*ð Þ*I x*ð Þ*h*<sup>2</sup>

where *h x*ð Þ *; t* is the transverse deflection, *E x*ð Þ is the modulus of elasticity, *I x*ð Þ is the area moment of inertia, *ϱ*ð Þ *x* is the linear density, and *A x*ð Þ is the cross-sectional

Assuming that the beam is clamped at the left end ð Þ *x* ¼ 0 and free at the right end ð Þ *x* ¼ 1 , and applying Hamilton's variational principle to the action functional

*ϱ*ð Þ *x A x*ð Þ*htt*ð Þþ *x; t* ð Þ *E x*ð Þ*I x*ð Þ*hxx*ð Þ *x; t xx* ¼ 0*,* 0≤ *x*≤ 1*, t* . 0*,* (2)

*xx*ð Þ *<sup>x</sup>; <sup>t</sup>* � �*dx,* (1)

finite-dimensional approximation of the dynamics generator.

*<sup>k</sup>***<sup>1</sup>** <sup>¼</sup> **0,** *<sup>k</sup>***<sup>2</sup>** <sup>¼</sup> **<sup>0</sup>***:***5,** *EI* <sup>¼</sup> **1,** *<sup>ρ</sup>* <sup>¼</sup> **<sup>0</sup>***:***1,** *<sup>N</sup>* <sup>¼</sup> **64,** *<sup>ε</sup><sup>f</sup>* <sup>¼</sup> **<sup>10</sup>**�**<sup>20</sup>**

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

The Lagrangian of the system is defined by [10, 11]

*<sup>ϱ</sup>*ð Þ *<sup>x</sup> A x*ð Þ*h*<sup>2</sup>

**of a unit length**

17. �2*:*<sup>2</sup> � <sup>10</sup>�<sup>17</sup> <sup>þ</sup> <sup>4</sup>*:*6007*<sup>i</sup>*

**Table 1.**

area of the beam.

**129**

defined by (1), we obtain the equation of motion

Finally, we mention that the feedback control of beams is a well-studied area [6], with multiple applications to the control of robotic manipulators, long and slender aircraft wings, propeller blades, large space structure [7, 8], and the dynamics of carbon nanotubes [9]. The analysis of a classical beam model with nonstandard feedback control law that originated in engineering literature [4, 10–12] may be of interest for both analysts and practitioners.

This paper is organized as follows. In Section 1 we formulate the initialboundary value problem for the Euler-Bernoulli beam model. In Section 2, we reformulate the problem as an evolution equation in the Hilbert space of Cauchy data (the energy space). The dynamics generator L*<sup>k</sup>*1*,k*<sup>2</sup> , which is *a non-self-adjoint matrix differential operator depending on two parameters, k*<sup>1</sup> *and k*2*, is the main object of interest*. The eigenvalues and the generalized eigenvectors of L*<sup>k</sup>*1*,k*<sup>2</sup> correspond to the modes and the mode shapes of the beam. We also give numerical approximations and graphical representations of the eigenvalues of a discrete approximation of the main operator (see **Tables 1** and **2** and **Figures 1** and **2**). In Section 3,


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

#### **Table 1.**

The eigenmodes and the mode shapes of the flexible structure are defined as the eigenvalues (up to a multiple *i*) and the generalized eigenvectors of L*<sup>k</sup>*1*,k*<sup>2</sup> . Based on the results of [1, 2], the dynamics generator has a purely discrete spectrum, whose location on the complex plane is determined by the controls *k*<sup>1</sup> and *k*2. Having in mind the practical applications of the asymptotic formulas [3–5], we discuss the case of *k*<sup>1</sup> ≥ 0 and *k*<sup>2</sup> ≥0, such that ∣*k*1∣ þ ∣*k*2∣ . 0 (see Proposition 2). As shown in [2], even though the operator L*<sup>k</sup>*1*,k*<sup>2</sup> is non-dissipative, for the case *k*<sup>1</sup> . 0 and *k*<sup>2</sup> ¼ 0 (or *k*<sup>1</sup> ¼ 0 and *k*<sup>2</sup> . 0), the entire set of eigenvalues is located in the closed upper half of the complex plane C, which means that all eigenmodes are stable or neutrally stable. (We recall that to obtain an elastic mode from an eigenvalues of L*<sup>k</sup>*1*,k*<sup>2</sup> , one should multiply the eigenvalue by a factor *i*).

In the paper we address the question of accuracy of the asymptotic formulas for the eigenvalues. *Namely, under what conditions the leading asymptotic terms in formulas (20) and (21) can be used for practical estimation of the actual frequencies of the flexible beam?* Numerical simulations show that the accuracy of the asymptotic formulas is really high; the leading asymptotic terms can be used by practitioners almost immediately, i.e., almost from the first vibrational mode. The second question is concerned with the role of the *deadbeat modes*. A deadbeat mode is a purely negative elastic mode that generates a solution of the evolution equation exponentially decaying in time. The deadbeat modes are important in engineering applications. As we prove in the paper, when the boundary parameter *k*<sup>1</sup> is close to 1 (while *k*<sup>2</sup> ¼ 0), the number of the deadbeat modes is so large that the corresponding mode shapes become important for the description of the beam dynamics. More

We have also shown that there exists a sequence of values of the parameter *k*1,

<sup>1</sup> there exist a finite number of deadbeat

<sup>1</sup> ), there exists a mode shape and an associate

ð Þ *n*

modes and each corresponds to a *double eigenvalue* of the dynamics generator L*<sup>k</sup>*1*,k*<sup>2</sup> .

by an eigenvector and an associate vector. This result means that for a double

mode shape. This fact indicates that for some values of *k*<sup>1</sup> and *k*2, there exists a significant number of associate vectors of L*<sup>k</sup>*1*,k*<sup>2</sup> . Therefore, if one can prove that the set of the generalized eigenvectors (eigenvectors and associate vectors together) forms an unconditional basis for the state space, then construction of the bi-orthogonal basis [6] would be a more complicated problem than for the case

Finally, we mention that the feedback control of beams is a well-studied area [6], with multiple applications to the control of robotic manipulators, long and slender aircraft wings, propeller blades, large space structure [7, 8], and the dynamics of carbon nanotubes [9]. The analysis of a classical beam model with nonstandard feedback control law that originated in engineering literature

This paper is organized as follows. In Section 1 we formulate the initialboundary value problem for the Euler-Bernoulli beam model. In Section 2, we reformulate the problem as an evolution equation in the Hilbert space of Cauchy data (the energy space). The dynamics generator L*<sup>k</sup>*1*,k*<sup>2</sup> , which is *a non-self-adjoint matrix differential operator depending on two parameters, k*<sup>1</sup> *and k*2*, is the main object of interest*. The eigenvalues and the generalized eigenvectors of L*<sup>k</sup>*1*,k*<sup>2</sup> correspond to the modes and the mode shapes of the beam. We also give numerical approximations and graphical representations of the eigenvalues of a discrete approximation of the main operator (see **Tables 1** and **2** and **Figures 1** and **2**). In Section 3,

ð Þ *n*

[4, 10–12] may be of interest for both analysts and practitioners.

<sup>1</sup> , the operator L*<sup>k</sup>*1*,k*<sup>2</sup> has a two-dimensional root subspace spanned

precisely, the number of deadbeat modes tends to infinity as *k*<sup>1</sup> ! 1þ.

, such that for each *k*<sup>1</sup> ¼ *k*

i.e., *k* ð Þ *n* 1 n o<sup>∞</sup>

**128**

*n*¼1

ð Þ *n*

deadbeat mode (corresponding to *k*

when no associate vectors exist.

For each value *k*

*Functional Calculus*

*Approximations of the eigenvalues for the discrete and "continuous" operators (*K . 1*).*

we study the deadbeat modes and derive the estimates for the number of the deadbeat modes from below and above for different values of the boundary parameters (see **Figure 5**). Section 4 is concerned with the asymptotic approximation for the set of double deadbeat modes (see **Tables 3** and **4** and **Figures 6** and **7**). In Section 5, we outline the numerical scheme used for the spectral analysis of the finite-dimensional approximation of the dynamics generator.

#### **1.1 The initial-boundary value problem for the Euler-Bernoulli beam model of a unit length**

The Lagrangian of the system is defined by [10, 11]

$$\frac{1}{2} \int\_{0}^{1} \left[ \varrho(\varkappa) A(\varkappa) h\_{t}^{2}(\varkappa, t) - E(\varkappa) I(\varkappa) h\_{\infty}^{2}(\varkappa, t) \right] d\varkappa,\tag{1}$$

where *h x*ð Þ *; t* is the transverse deflection, *E x*ð Þ is the modulus of elasticity, *I x*ð Þ is the area moment of inertia, *ϱ*ð Þ *x* is the linear density, and *A x*ð Þ is the cross-sectional area of the beam.

Assuming that the beam is clamped at the left end ð Þ *x* ¼ 0 and free at the right end ð Þ *x* ¼ 1 , and applying Hamilton's variational principle to the action functional defined by (1), we obtain the equation of motion

$$\left(\varrho(\mathbf{x})A(\mathbf{x})h\_{tt}(\mathbf{x},t) + \left(E(\mathbf{x})I(\mathbf{x})h\_{\mathbf{x}\mathbf{x}}(\mathbf{x},t)\right)\_{\mathbf{x}\mathbf{x}} = \mathbf{0}, \qquad \mathbf{0} \le \mathbf{x} \le \mathbf{1}, \quad t \ge \mathbf{0}, \tag{2}$$


#### **Table 2.**

*Approximations of the eigenvalues for the discrete and "continuous" operators (*K , � 1*).*

*Graphical representation of the eigenvalues of the discrete and "continuous" operators (*K . 1*).*

and the boundary conditions

$$h(\mathbf{0}, t) = h\_x(\mathbf{0}, t) = \mathbf{0} \qquad \text{and} \qquad M(\mathbf{1}, t) = Q(\mathbf{1}, t) = \mathbf{0},\tag{3}$$

Now we replace the free right-end conditions from Eq. (3) with the following

*U t*ðÞ¼ �½ � *<sup>Q</sup>*ð Þ <sup>1</sup>*; <sup>t</sup> ; <sup>M</sup>*ð Þ <sup>1</sup>*; <sup>t</sup> <sup>T</sup>* and *Y t*ðÞ¼ ½ � *ht*ð Þ <sup>1</sup>*; <sup>t</sup> ; hxt*ð Þ <sup>1</sup>*; <sup>t</sup> T,* (5)

*U t*ðÞ¼ *KY t*ð Þ*,* (6)

*K* ¼ codiagð Þ �*k*2*;* �*k*<sup>1</sup> *, k*1*, k*<sup>2</sup> ≥0*,* (7)

boundary feedback control law [2, 4]. Define the input and the output as

*Graphical representation of the eigenvalues of the discrete and "continuous" operators (*K , � 1*).*

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

**No.** *<sup>k</sup>***<sup>1</sup>** <sup>¼</sup> **<sup>1</sup>** <sup>þ</sup> **<sup>10</sup>**�**<sup>4</sup>** *<sup>k</sup>***<sup>1</sup>** <sup>¼</sup> **<sup>1</sup>** <sup>þ</sup> **<sup>10</sup>**�**<sup>7</sup>** *<sup>k</sup>***<sup>1</sup>** <sup>¼</sup> **<sup>1</sup>** <sup>þ</sup> **<sup>10</sup>**�**<sup>10</sup>** 1. �222*:*22 þ 155*:*56*i* �176*:*06 þ 264*:*07*i* �106*:*90 þ 372*:*41*i* 2. �133*:*<sup>38</sup> <sup>þ</sup> <sup>124</sup>*:*45*<sup>i</sup>* �87*:*<sup>723</sup> <sup>þ</sup> <sup>211</sup>*:*27*<sup>i</sup>* �2*:*<sup>4378</sup> � <sup>10</sup>�<sup>16</sup> <sup>þ</sup> <sup>254</sup>*:*44*<sup>i</sup>* 3. �64*:*<sup>540</sup> <sup>þ</sup> <sup>93</sup>*:*396*<sup>i</sup>* �3*:*<sup>9609</sup> � <sup>10</sup>�<sup>18</sup> <sup>þ</sup> <sup>162</sup>*:*37*<sup>i</sup>* �3*:*<sup>0717</sup> � <sup>10</sup>�<sup>17</sup> <sup>þ</sup> <sup>123</sup>*:*66*<sup>i</sup>* 4. �9*:*<sup>8081</sup> <sup>þ</sup> <sup>58</sup>*:*559*<sup>i</sup>* �5*:*<sup>7977</sup> � <sup>10</sup>�<sup>19</sup> <sup>þ</sup> <sup>116</sup>*:*23*<sup>i</sup>* �1*:*<sup>3012</sup> � <sup>10</sup>�<sup>17</sup> <sup>þ</sup> <sup>4</sup>*:*9349*<sup>i</sup>* 5. �7*:*<sup>7725</sup> � <sup>10</sup>�<sup>21</sup> <sup>þ</sup> <sup>5</sup>*:*0488*<sup>i</sup>* �6*:*<sup>3661</sup> � <sup>10</sup>�<sup>20</sup> <sup>þ</sup> <sup>44</sup>*:*182*<sup>i</sup>* �8*:*<sup>9232</sup> � <sup>10</sup>�<sup>18</sup> <sup>þ</sup> <sup>44</sup>*:*421*<sup>i</sup>* 6. <sup>4</sup>*:*<sup>9365</sup> � <sup>10</sup>�<sup>21</sup> <sup>þ</sup> <sup>38</sup>*:*994*<sup>i</sup>* �6*:*<sup>0066</sup> � <sup>10</sup>�<sup>20</sup> <sup>þ</sup> <sup>4</sup>*:*9383*<sup>i</sup>* <sup>8</sup>*:*<sup>9143</sup> � <sup>10</sup>�<sup>18</sup> <sup>þ</sup> <sup>44</sup>*:*406*<sup>i</sup>* 7. <sup>7</sup>*:*<sup>8007</sup> � <sup>10</sup>�<sup>21</sup> <sup>þ</sup> <sup>4</sup>*:*8257*<sup>i</sup>* <sup>6</sup>*:*<sup>0114</sup> � <sup>10</sup>�<sup>20</sup> <sup>þ</sup> <sup>4</sup>*:*9313*<sup>i</sup>* <sup>1</sup>*:*<sup>3012</sup> � <sup>10</sup>�<sup>17</sup> <sup>þ</sup> <sup>4</sup>*:*9347*<sup>i</sup>* 8. <sup>9</sup>*:*<sup>8081</sup> <sup>þ</sup> <sup>58</sup>*:*559*<sup>i</sup>* <sup>6</sup>*:*<sup>6801</sup> � <sup>10</sup>�<sup>20</sup> <sup>þ</sup> <sup>44</sup>*:*651*<sup>i</sup>* <sup>2</sup>*:*<sup>9907</sup> � <sup>10</sup>�<sup>17</sup> <sup>þ</sup> <sup>123</sup>*:*09*<sup>i</sup>* 9. <sup>64</sup>*:*<sup>540</sup> <sup>þ</sup> <sup>93</sup>*:*396*<sup>i</sup>* <sup>2</sup>*:*<sup>9381</sup> � <sup>10</sup>�<sup>18</sup> <sup>þ</sup> <sup>139</sup>*:*20*<sup>i</sup>* <sup>1</sup>*:*<sup>1286</sup> � <sup>10</sup>�<sup>16</sup> <sup>þ</sup> <sup>234</sup>*:*06*<sup>i</sup>* 10. <sup>133</sup>*:*<sup>38</sup> <sup>þ</sup> <sup>124</sup>*:*45*<sup>i</sup>* <sup>87</sup>*:*<sup>723</sup> <sup>þ</sup> <sup>211</sup>*:*27*<sup>i</sup>* <sup>3</sup>*:*<sup>2280</sup> � <sup>10</sup>�<sup>16</sup> <sup>þ</sup> <sup>315</sup>*:*13*<sup>i</sup>* 11. 222*:*22 þ 155*:*56*i* 176*:*06 þ 264*:*07*i* 106*:*90 þ 372*:*41*i*

where *T* stands for transposition. The feedback control law is given by

where *K* is the 2 � 2 feedback matrix. We select

**Figure 2.**

**Table 3.**

**131**

*<sup>k</sup>***<sup>2</sup>** <sup>¼</sup> **0,** *EI* <sup>¼</sup> **1,** *<sup>ρ</sup>* <sup>¼</sup> **1,** *<sup>N</sup>* <sup>¼</sup> **64,** *<sup>ε</sup><sup>f</sup>* <sup>¼</sup> **<sup>10</sup>**�**<sup>30</sup>**

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

*Eigenvalues closest to the imaginary axis as k*<sup>1</sup> ! 1þ*.*

where *M x*ð Þ *; t* and *Q x*ð Þ *; t* are the moment and the shear, respectively [10]:

$$M(\mathbf{x},t) = E(\mathbf{x})I(\mathbf{x})h\_{\mathbf{x}\mathbf{x}}(\mathbf{x},t) \qquad \text{and} \qquad Q(\mathbf{x},t) = M\_{\mathbf{x}}(\mathbf{x},t). \tag{4}$$

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

**Figure 2.** *Graphical representation of the eigenvalues of the discrete and "continuous" operators (*K , � 1*).*


**Table 3.** *Eigenvalues closest to the imaginary axis as k*<sup>1</sup> ! 1þ*.*

Now we replace the free right-end conditions from Eq. (3) with the following boundary feedback control law [2, 4]. Define the input and the output as

$$U(t) = \begin{bmatrix} -Q(\mathbf{1}, t), & M(\mathbf{1}, t) \end{bmatrix}^T \qquad \text{and} \qquad Y(t) = \begin{bmatrix} h\_l(\mathbf{1}, t), & h\_{xt}(\mathbf{1}, t) \end{bmatrix}^T,\tag{5}$$

where *T* stands for transposition. The feedback control law is given by

$$U(t) = KY(t),\tag{6}$$

where *K* is the 2 � 2 feedback matrix. We select

$$K = \text{codim}(-k\_2, -k\_1), \qquad k\_1, k\_2 \ge 0,\tag{7}$$

and the boundary conditions

17. <sup>1</sup>*:*<sup>5</sup> � <sup>10</sup>�<sup>17</sup> <sup>þ</sup> <sup>7</sup>*:*7256*<sup>i</sup>*

*<sup>k</sup>***<sup>1</sup>** <sup>¼</sup> **<sup>1</sup>***:***3,** *<sup>k</sup>***<sup>2</sup>** <sup>¼</sup> **<sup>1</sup>***:***2,** *EI* <sup>¼</sup> **10,** *<sup>ρ</sup>* <sup>¼</sup> **<sup>0</sup>***:***1,** *<sup>N</sup>* <sup>¼</sup> **64,** *<sup>ε</sup><sup>f</sup>* <sup>¼</sup> **<sup>10</sup>**�**<sup>20</sup>**

*Functional Calculus*

**No. Numerical Analytic No. Numerical Analytic** 1. �25266 � 229*:*07*i* �25266 � 229*:*07*i* 18. 99*:*467 � 19*:*816*i* 98*:*177 � 14*:*317*i* 2. �22206 � 214*:*76*i* �22206 � 214*:*76*i* 19. 394*:*17 � 28*:*149*i* 394*:*26 � 28*:*634*i* 3. �19344 � 200*:*44*i* �19344 � 200*:*44*i* 20. 887*:*75 � 42*:*983*i* 887*:*75 � 42*:*951*i* 4. �16679 � 186*:*12*i* �16679 � 186*:*12*i* 21. 1578*:*6 � 57*:*267*i* 1578*:*6 � 57*:*268*i* 5. �14212 � 171*:*81*i* �14212 � 171*:*81*i* 22. 2466*:*9 � 71*:*586*i* 2466*:*9 � 71*:*585*i* 6. �11942 � 157*:*49*i* �11942 � 157*:*49*i* 23. 3552*:*5 � 85*:*902*i* 3552*:*5 � 85*:*903*i* 7. �9869*:*1 � 143*:*17*i* �9869*:*1 � 143*:*17*i* 24. 4835*:*6 � 100*:*22*i* 4835*:*6 � 100*:*22*i* 8. �7993*:*9 � 128*:*85*i* �7993*:*9 � 128*:*85*i* 25. 6316*:*0 � 114*:*54*i* 6316*:*0 � 114*:*54*i* 9. �6316*:*0 � 114*:*54*i* �6316*:*0 � 114*:*54*i* 26. 7993*:*9 � 128*:*85*i* 7993*:*9 � 128*:*85*i* 10. �4835*:*6 � 100*:*22*i* �4835*:*6 � 100*:*22*i* 27. 9869*:*1 � 143*:*17*i* 9869*:*1 � 143*:*17*i* 11. �3552*:*5 � 85*:*902*i* �3552*:*5 � 85*:*903*i* 28. 11942 � 157*:*49*i* 11942 � 157*:*49*i* 12. �2466*:*9 � 71*:*586*i* �2466*:*9 � 71*:*585*i* 29. 14212 � 171*:*81*i* 14212 � 171*:*81*i* 13. �1578*:*6 � 57*:*267*i* �1578*:*6 � 57*:*268*i* 30. 16679 � 186*:*12*i* 16679 � 186*:*12*i* 14. �887*:*75 � 42*:*983*i* �887*:*75 � 42*:*951*i* 31. 19344 � 200*:*44*i* 19344 � 200*:*44*i* 15. �394*:*17 � 28*:*149*i* �394*:*26 � 28*:*634*i* 32. 22206 � 214*:*76*i* 22206 � 214*:*76*i* 16. �99*:*467 � 19*:*816*i* �98*:*177 � 14*:*317*i* 33. 25266 � 229*:*07*i* 25266 � 229*:*07*i*

**Figure 1.**

**130**

**Table 2.**

*h*ð Þ¼ 0*; t hx*ð Þ¼ 0*; t* 0 and *M*ð Þ¼ 1*; t Q*ð Þ¼ 1*; t* 0*,* (3)

*M x*ð Þ¼ *; t E x*ð Þ*I x*ð Þ*hxx*ð Þ *x; t* and *Q x*ð Þ¼ *; t Mx*ð Þ *x; t :* (4)

where *M x*ð Þ *; t* and *Q x*ð Þ *; t* are the moment and the shear, respectively [10]:

*Graphical representation of the eigenvalues of the discrete and "continuous" operators (*K . 1*).*

*Approximations of the eigenvalues for the discrete and "continuous" operators (*K , � 1*).*


Hilbert space of two-component vector functions *U x*ð Þ¼ ½ � *u*0ð Þ *x ; u*1ð Þ *x*

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

*EI x*ð Þj j *<sup>u</sup>*″ <sup>0</sup>ð Þ *x*

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

*Ut*ð Þ¼ *x; t i*ð Þ L*<sup>k</sup>*1*,k*2*U* ð Þ *x; t* and *U x*ð Þ¼ *;* 0 ½ � *u*0ð Þ *x ; u*1ð Þ *x*

� <sup>1</sup> *ρ*ð Þ *x*

<sup>0</sup>ð Þ *x* <sup>0</sup> � �

, the adjoint operator L<sup>∗</sup>

described in Eq. (15), where *k*<sup>1</sup> and *k*<sup>2</sup> are replaced by ð Þ �*k*<sup>2</sup> and ð Þ �*k*<sup>1</sup> ,

nience, we summarize the properties of L*<sup>k</sup>*1*,k*<sup>2</sup> from [1, 2] needed for the

<sup>0</sup>*,*<sup>0</sup> exist and are related by the rule

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

L�<sup>1</sup>

each of finite algebraic multiplicity [6, 13]).

L∗

2 4

<sup>2</sup> <sup>þ</sup> *<sup>ρ</sup>*ð Þ *<sup>x</sup>* j j *<sup>u</sup>*1ð Þ *<sup>x</sup>* <sup>2</sup> h i*x:* (12)

*<sup>T</sup>* satisfying *<sup>u</sup>*0ð Þ¼ <sup>0</sup> *<sup>u</sup>*<sup>0</sup>

ð Þ <sup>0</sup>*;* <sup>1</sup> . Here *<sup>H</sup>*<sup>2</sup>

0 1

3

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

ð Þg 1 *:* (15)

*∂x*<sup>2</sup> � � <sup>0</sup>

*<sup>k</sup>*1*,k*<sup>2</sup> [13] is given by

*<sup>k</sup>*1*,k*<sup>2</sup> ¼ L�*k*2*,*�*k*<sup>1</sup> *,* (16)

ð Þ¼ 0 0g, and the equality of function spaces is understood in the sense of

Problem (2) with conditions (3) can be represented as the time evolution problem:

where 0 ≥*x*≥1*, t*≥0. The dynamics generator L*<sup>k</sup>*1*,k*<sup>2</sup> is given by the following

*∂*2

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

*<sup>x</sup>*¼<sup>1</sup> <sup>¼</sup> *<sup>k</sup>*2*u*<sup>1</sup>

*<sup>k</sup>*1*,k*<sup>2</sup> is defined by the same differential expression (14) on the domain

respectively. It follows from Eq. (16) that L0*,*<sup>0</sup> is self-adjoint in H and thus L0*,*<sup>0</sup> is the dynamics generator of the clamped-free beam model. For the reader's conve-

1. L*<sup>k</sup>*1*,k*<sup>2</sup> is an unbounded operator with compact resolvent, whose spectrum consists of a countable set of normal eigenvalues (i.e., isolated eigenvalues,

perturbation of the self-adjoint operator L0*,*<sup>0</sup> in the sense that the operators

where T *<sup>k</sup>*1*,k*<sup>2</sup> is a rank-two operator. A similar decomposition is valid for the

*,* ∣*k*1∣ þ ∣*k*2∣ . 0, the operator L*<sup>k</sup>*1*,k*<sup>2</sup> is a rank-two

<sup>0</sup>*,*<sup>0</sup> þ T *<sup>k</sup>*1*,k*<sup>2</sup> *,* (17)

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

0

ð Þ 0*;* 1 are positive functions, we obtain that the closure

with the following norm:

*u*ð Þ¼ 0 *u*<sup>0</sup>

*EI*ð Þ<sup>1</sup> *<sup>u</sup>*″

i.e., L<sup>∗</sup>

present work.

L�<sup>1</sup>

**133**

**Proposition 1:**

2. For each ð Þ *<sup>k</sup>*1*; <sup>k</sup>*<sup>2</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup>

*<sup>k</sup>*1*,k*<sup>2</sup> and <sup>L</sup>�<sup>1</sup>

adjoint operator, i.e.,

Assuming that *EI, ρ* ∈C<sup>2</sup>

a Hilbert-space isomorphism.

matrix differential expression:

defined on the domain

D L*<sup>k</sup>*1*,k*<sup>2</sup> ð Þ¼f*U* ¼ ð Þ *u*0*; u*<sup>1</sup>

For any ð Þ *<sup>k</sup>*1*; <sup>k</sup>*<sup>2</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup>

produce the energy space <sup>H</sup> <sup>¼</sup> *<sup>H</sup>*<sup>2</sup>

k k *<sup>U</sup>* <sup>2</sup> <sup>H</sup> <sup>¼</sup> <sup>1</sup> 2 Z <sup>1</sup> 0

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

of smooth functions *U x*ð Þ¼ ½ � *u*0ð Þ *x ; u*1ð Þ *x*

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

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

*<sup>T</sup>* equipped

<sup>0</sup>ð Þ¼ 0 0 will

ð Þ <sup>0</sup>*;* <sup>1</sup> : �

*T,* (13)

<sup>1</sup>ð Þ¼ 0 0;

5*,* (14)

<sup>0</sup>ð Þ¼ <sup>0</sup>*;* <sup>1</sup> *<sup>u</sup>*<sup>∈</sup> *<sup>H</sup>*<sup>2</sup>

#### **Table 4.**

*Eigenvalues closest to the imaginary axis as k*<sup>1</sup> ! 1�*.*

with *k*1*, k*<sup>2</sup> being the control parameters. The feedback (6) can be written as

$$E(\mathbf{1})I(\mathbf{1})h\_{\text{xx}}(\mathbf{1},t) = -k\_1 h\_t(\mathbf{1},t) \quad \text{and} \quad (E(\mathbf{x})I(\mathbf{x})h\_{\text{xx}}(\mathbf{x},t))\_{\mathbf{x}}|\_{\mathbf{x}=1} = k\_2 h\_{\text{xt}}(\mathbf{1},t). \tag{8}$$

Finally, we arrive to the following initial-boundary value problem: the equation of motion (2), the boundary conditions (3), and the standard initial conditions *h x*ð Þ¼ *;* 0 *h*0ð Þ *x , ht*ð Þ¼ *x;* 0 *h*1ð Þ *x :*

Notice that the choice of a feedback matrix *K* defines whether the system is dissipative or not. Indeed, let Eð Þ*t* be the energy of the system, defined by representation (1). Evaluating E*t*ð Þ*t* on the solutions of Eq. (2) satisfying the left-end conditions from Eqs. (3), we obtain

$$\mathcal{L}\_t(t) = \int\_0^1 \left[ \varrho(\mathbf{x}) A(\mathbf{x}) h\_t(\mathbf{x}, t) h\_{tt}(\mathbf{x}, t) + E(\mathbf{x}) I(\mathbf{x}) h\_{\mathbf{x}\mathbf{x}}(\mathbf{x}, t) h\_{\mathbf{x}\mathbf{x}t}(\mathbf{x}, t) \right] \mathbf{x} \tag{9}$$

$$= -(E(\mathbf{x}) I(\mathbf{x}) h\_{\mathbf{x}\mathbf{x}}(\mathbf{x}t))\_{\mathbf{x}} h\_t(\mathbf{x}t)|\_{\mathbf{x}=1} + E(\mathbf{1}) I(\mathbf{1}) h\_{\mathbf{x}\mathbf{x}}(\mathbf{1}, t) h\_{\mathbf{x}t}(\mathbf{1}, t).$$

Taking into account Eqs. (4) and (6), we represent the right-hand side of Eq. (9) as the dot product in R<sup>2</sup> :

$$\mathcal{E}\_t(t) = -Q(\mathbf{1}, t)h\_t(\mathbf{1}, t) + M(\mathbf{1}, t)h\_{\text{xt}}(\mathbf{1}, t) = U(t) \cdot Y(t) = KY(t) \cdot Y(t). \tag{10}$$

With the choice of *K* as in Eq. (7), we have

$$\mathcal{E}\_t(t) = \begin{bmatrix} \mathbf{0} & -k\_2 \\ -k\_1 & \mathbf{0} \end{bmatrix} \begin{bmatrix} 2h\_t(\mathbf{1}, t) \\ h\_{\text{xt}}(\mathbf{1}, t) \end{bmatrix} \cdot \begin{bmatrix} 2h\_t(\mathbf{1}, t) \\ h\_{\text{xt}}(\mathbf{1}, t) \end{bmatrix} = -(k\_1 + k\_2)h\_t(\mathbf{1}, t)h\_{\text{xt}}(\mathbf{1}, t). \tag{11}$$

Thus the system is not dissipative for all nonnegative values of *k*<sup>1</sup> and *k*2.

#### **2. Operator form of the problem**

In what follows, we incorporate the cross-sectional area *A x*ð Þ into the density, write *ρ*ð Þ *x* instead of *ϱ*ð Þ *x A x*ð Þ, and also abbreviate *EI x*ð Þ� *E x*ð Þ*I x*ð Þ. Let H be the *Spectral Analysis and Numerical Investigation of a Flexible Structure with… DOI: http://dx.doi.org/10.5772/intechopen.86940*

Hilbert space of two-component vector functions *U x*ð Þ¼ ½ � *u*0ð Þ *x ; u*1ð Þ *x <sup>T</sup>* equipped with the following norm:

$$\|\|U\|\|\_{\mathcal{H}}^2 = \frac{1}{2} \int\_0^1 \left[ EI(\mathbf{x}) \left| u\_0''(\mathbf{x}) \right|^2 + \rho(\mathbf{x}) \left| u\_1(\mathbf{x}) \right|^2 \right] \mathbf{x}.\tag{12}$$

Assuming that *EI, ρ* ∈C<sup>2</sup> ð Þ 0*;* 1 are positive functions, we obtain that the closure of smooth functions *U x*ð Þ¼ ½ � *u*0ð Þ *x ; u*1ð Þ *x <sup>T</sup>* satisfying *<sup>u</sup>*0ð Þ¼ <sup>0</sup> *<sup>u</sup>*<sup>0</sup> <sup>0</sup>ð Þ¼ 0 0 will produce the energy space <sup>H</sup> <sup>¼</sup> *<sup>H</sup>*<sup>2</sup> <sup>0</sup>ð Þ� <sup>0</sup>*;* <sup>1</sup> *<sup>L</sup>*<sup>2</sup> ð Þ <sup>0</sup>*;* <sup>1</sup> . Here *<sup>H</sup>*<sup>2</sup> <sup>0</sup>ð Þ¼ <sup>0</sup>*;* <sup>1</sup> *<sup>u</sup>*<sup>∈</sup> *<sup>H</sup>*<sup>2</sup> ð Þ <sup>0</sup>*;* <sup>1</sup> : � *u*ð Þ¼ 0 *u*<sup>0</sup> ð Þ¼ 0 0g, and the equality of function spaces is understood in the sense of a Hilbert-space isomorphism.

Problem (2) with conditions (3) can be represented as the time evolution problem:

$$U\_t(\mathbf{x}, t) = i(\mathcal{L}\_{k\_1, k\_2} U)(\mathbf{x}, t) \qquad \text{and} \qquad U(\mathbf{x}, \mathbf{0}) = [u\_0(\mathbf{x}), u\_1(\mathbf{x})]^T,\tag{13}$$

where 0 ≥*x*≥1*, t*≥0. The dynamics generator L*<sup>k</sup>*1*,k*<sup>2</sup> is given by the following matrix differential expression:

$$\mathcal{L}\_{k\_1,k\_2} = -i \left[ -\frac{1}{\rho(\mathbf{x})} \frac{\partial^2}{\partial \mathbf{x}^2} \left( EI(\mathbf{x}) \frac{\partial^2}{\partial \mathbf{x}^2} \right) \quad \mathbf{0} \right], \tag{14}$$

defined on the domain

with *k*1*, k*<sup>2</sup> being the control parameters. The feedback (6) can be written as

**#** *<sup>k</sup>***<sup>1</sup>** <sup>¼</sup> **<sup>1</sup>** � **<sup>10</sup>**�**<sup>4</sup>** *<sup>k</sup>***<sup>1</sup>** <sup>¼</sup> **<sup>1</sup>** � **<sup>10</sup>**�**<sup>7</sup>** *<sup>k</sup>***<sup>1</sup>** <sup>¼</sup> **<sup>1</sup>** � **<sup>10</sup>**�**<sup>10</sup>** 1. �175*:*34 þ 140*:*01*i* �129*:*20 þ 237*:*56*i* �60*:*143 þ 338*:*80*i* 2. �96*:*394 þ 108*:*84*i* �50*:*206 þ 186*:*90*i* �8*:*6602 þ 240*:*01*i* 3. �36*:*896 þ 78*:*778*i* �8*:*0431 þ 121*:*15*i* �0*:*28608 þ 123*:*37*i* 4. �6*:*0769 þ 42*:*171*i* �0*:*23455 þ 44*:*410*i* �0*:*0074192 þ 44*:*413*i* 5. �0*:*11141 þ 4*:*9324*i* �0*:*0035253 þ 4*:*9348*i* �0*:*00011148 þ 4*:*9348*i* 6. 0*:*11141 þ 4*:*9324*i* 0*:*0035253 þ 4*:*9348*i* 0*:*00011148 þ 4*:*9348*i* 7. 6*:*0769 þ 42*:*171*i* 0*:*23455 þ 44*:*410*i* 0*:*0074192 þ 44*:*413*i* 8. 36*:*896 þ 78*:*778*i* 8*:*0431 þ 121*:*15*i* 0*:*28608 þ 123*:*37*i* 9. 96*:*394 þ 108*:*84*i* 50*:*206 þ 186*:*90*i* 8*:*6602 þ 240*:*01*i* 10. 175*:*34 þ 140*:*01*i* 129*:*20 þ 237*:*56*i* 60*:*143 þ 338*:*80*i*

Finally, we arrive to the following initial-boundary value problem: the equation

of motion (2), the boundary conditions (3), and the standard initial conditions

dissipative or not. Indeed, let Eð Þ*t* be the energy of the system, defined by

Notice that the choice of a feedback matrix *K* defines whether the system is

representation (1). Evaluating E*t*ð Þ*t* on the solutions of Eq. (2) satisfying the left-end

½ � *ϱ*ð Þ *x A x*ð Þ*ht*ð Þ *x; t htt*ð Þþ *x; t E x*ð Þ*I x*ð Þ*hxx*ð Þ *x; t hxxt*ð Þ *x; t x*

Taking into account Eqs. (4) and (6), we represent the right-hand side of Eq. (9)

E*t*ðÞ¼� *t Q*ð Þ 1*; t ht*ð Þþ 1*; t M*ð Þ 1*; t hxt*ð Þ¼ 1*; t U t*ðÞ� *Y t*ðÞ¼ *KY t*ðÞ� *Y t*ð Þ*:* (10)

� <sup>2</sup>*ht*ð Þ <sup>1</sup>*; <sup>t</sup> hxt*ð Þ 1*; t* � �

Thus the system is not dissipative for all nonnegative values of *k*<sup>1</sup> and *k*2.

In what follows, we incorporate the cross-sectional area *A x*ð Þ into the density, write *ρ*ð Þ *x* instead of *ϱ*ð Þ *x A x*ð Þ, and also abbreviate *EI x*ð Þ� *E x*ð Þ*I x*ð Þ. Let H be the

*<sup>x</sup>*¼<sup>1</sup> <sup>þ</sup> *<sup>E</sup>*ð Þ<sup>1</sup> *<sup>I</sup>*ð Þ<sup>1</sup> *hxx*ð Þ <sup>1</sup>*; <sup>t</sup> hxt*ð Þ <sup>1</sup>*; <sup>t</sup> :*

¼ �ð Þ *k*<sup>1</sup> þ *k*<sup>2</sup> *ht*ð Þ 1*; t hxt*ð Þ 1*; t :* (11)

*<sup>x</sup>*¼<sup>1</sup> <sup>¼</sup> *<sup>k</sup>*2*hxt*ð Þ <sup>1</sup>*; <sup>t</sup> :* (8)

(9)

*E*ð Þ1 *I*ð Þ1 *hxx*ð Þ¼� 1*; t k*1*ht*ð Þ 1*; t* and ð Þ *E x*ð Þ*I x*ð Þ*hxx*ð Þ *x; t <sup>x</sup>*j

¼ �ð*E x*ð Þ*I x*ð Þ*hxx*ð*xt*ÞÞ*<sup>x</sup> ht*ð*xt*Þ|

:

With the choice of *K* as in Eq. (7), we have

*hxt*ð Þ 1*; t* � �

*h x*ð Þ¼ *;* 0 *h*0ð Þ *x , ht*ð Þ¼ *x;* 0 *h*1ð Þ *x :*

*Eigenvalues closest to the imaginary axis as k*<sup>1</sup> ! 1�*.*

*<sup>k</sup>***<sup>2</sup>** <sup>¼</sup> **0,** *EI* <sup>¼</sup> **1,** *<sup>ρ</sup>* <sup>¼</sup> **1,** *<sup>N</sup>* <sup>¼</sup> **64,** *<sup>ε</sup><sup>f</sup>* <sup>¼</sup> **<sup>10</sup>**�**<sup>30</sup>**

*Functional Calculus*

conditions from Eqs. (3), we obtain

Z <sup>1</sup> 0

0 �*k*<sup>2</sup> �*k*<sup>1</sup> 0

**2. Operator form of the problem**

� � 2*ht*ð Þ 1*; t*

E*t*ðÞ¼ *t*

**Table 4.**

as the dot product in R<sup>2</sup>

E*t*ðÞ¼ *t*

**132**

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

$$\left. E\mathbf{1}(\mathbf{1})u\_0''(\mathbf{1}) = -k\_1 u\_1(\mathbf{1}), \ (E\mathbf{1}(\mathbf{x})u\_0''(\mathbf{x}))' \right|\_{\mathbf{x}=\mathbf{1}} = k\_2 u\_1'(\mathbf{1}) \}. \tag{15}$$

For any ð Þ *<sup>k</sup>*1*; <sup>k</sup>*<sup>2</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup> , the adjoint operator L<sup>∗</sup> *<sup>k</sup>*1*,k*<sup>2</sup> [13] is given by

$$
\mathcal{L}\_{k\_1,k\_2}^\* = \mathcal{L}\_{-k\_2,-k\_1} \tag{16}
$$

i.e., L<sup>∗</sup> *<sup>k</sup>*1*,k*<sup>2</sup> is defined by the same differential expression (14) on the domain described in Eq. (15), where *k*<sup>1</sup> and *k*<sup>2</sup> are replaced by ð Þ �*k*<sup>2</sup> and ð Þ �*k*<sup>1</sup> , respectively. It follows from Eq. (16) that L0*,*<sup>0</sup> is self-adjoint in H and thus L0*,*<sup>0</sup> is the dynamics generator of the clamped-free beam model. For the reader's convenience, we summarize the properties of L*<sup>k</sup>*1*,k*<sup>2</sup> from [1, 2] needed for the present work.

#### **Proposition 1:**


$$
\mathcal{L}\_{k\_1,k\_2}^{-1} = \mathcal{L}\_{0,0}^{-1} + \mathcal{T}\_{k\_1,k\_2} \tag{17}
$$

where T *<sup>k</sup>*1*,k*<sup>2</sup> is a rank-two operator. A similar decomposition is valid for the adjoint operator, i.e.,

$$\left(\mathcal{L}\_{k\_1,k\_2}^{-1}\right)^\* = \mathcal{L}\_{0,0}^{-1} + \mathcal{T}\_{k\_1,k\_2}^\* \qquad \mathcal{T}\_{k\_1,k\_2}^\* = \mathcal{T}\_{-k\_2,-k\_1}.\tag{18}$$

From now on, we assume that the structural parameters are constant. In the case of variable parameters, the spectral asymptotics will have the same leading terms and remainder terms depending on parameter smoothness.

**Proposition 2:** Assume that k1*,* k2 . 0 and k1k2 6¼ EIρ. Let

$$\mathcal{K} = \frac{k\_1 + k\_2}{A - k\_1 k\_2 / A}, \qquad A = \sqrt{E I \rho}, \qquad \text{and} \qquad |\mathcal{K}| \neq 1. \tag{19}$$

In **Figures 1** and **2**, we represent the graphical distribution of the eigenvalues corresponding to the discretized operator ("numerical" eigenvalues) and the leading asymptotic terms from Eqs. (20) and (21) ("analytic" eigenvalues). In **Tables 1** and **2**, the numerical values of the corresponding graphical points on **Figures 1** and **2** are listed. We have used the following notations: *N* ¼ 64 is the number of grid points on 0½ � *;* 1 , and *ε<sup>f</sup>* is the filtering parameter as described in Eq. (69). It can be easily seen from the graphs and tables that the two sets of data coincide almost immediately, i.e., the leading asymptotic terms in the approximations are very close

*EI<sup>ρ</sup>* <sup>p</sup> <sup>¼</sup> <sup>1</sup>*.*

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

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

**Figure 3** shows the sub-domains of the *k*1*, k*2-plane, which correspond to different intervals for the values of K defined by Eq. (19). On the sub-domain with dark gray color K such that ∣K∣ . 1, i.e., to evaluate the asymptotic approximation for the eigenvalues, one needs formula (20), while on the complementary sub-domain, one

An eigenvalue *λ<sup>n</sup>* of the dynamics generator L*<sup>k</sup>*1*,k*<sup>2</sup> is called a deadbeat mode if *λ<sup>n</sup>* ¼ *iβn, β<sup>n</sup>* . 0. If the corresponding eigenfunction is Φ*n*ð Þ *x* , then the evolution

As shown in paper [2], for the case when one of the control parameters is zero and the other one is positive, the entire set of the eigenvalues is located in the closed upper half plane. This result is not obvious since the operator is not dissipative; in fact, it requires a fairly nontrivial proof. However, due to this fact, we assume that any deadbeat mode can be given in the form *iβ*, with *β* . 0. To deal with the deadbeat modes analytically, we rewrite the spectral equation ð Þ L*<sup>k</sup>*1*,k*2Φ ð Þ¼ *x λ*Φð Þ *x*

*ρφ*ð Þ *x , φ*ð Þ¼ 0 *φ*<sup>0</sup>

*EIφ*00ð Þ¼� 1 *iλk*1*φ*ð Þ1 *, EIφ*000ð Þ¼ 1 *iλk*2*φ*<sup>0</sup>

in the form of an equivalent problem for an operator pencil [17] as

<sup>Φ</sup>*n*ð Þ¼ *<sup>x</sup>* <sup>e</sup>�*βnt*

ð Þ¼ 0 0*,*

Φ*n*ð Þ *x* , which tends

ð Þ<sup>1</sup> *:* (24)

to the numerically approximated eigenvalues.

*Regions of* <sup>K</sup> *on the <sup>k</sup>*1*, k*2*-plane, <sup>A</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffi

problem (13) has a solution given in the form e*<sup>i</sup>λnt*

*EIφ*0000ð Þ¼ *<sup>x</sup> <sup>λ</sup>*<sup>2</sup>

needs formula (21).

**135**

**Figure 3.**

**3. The deadbeat modes**

to zero without any oscillation.

The following asymptotic approximations for the eigenvalues *λ<sup>n</sup>* (as ∣*n*∣ ! ∞) of the operator L*<sup>k</sup>*1*,k*<sup>2</sup> hold:

1. If 1 , ∣K∣ , ∞, then for ∣*n*∣ ! ∞ one has

$$\lambda\_{\pi} = \text{sign}(\mathcal{K}n) \sqrt{\frac{EI}{\rho}} \Big[ \left( \pi n \right)^{2} - \frac{1}{4} \ln^{2} \left( \frac{\mathcal{K} + 1}{\mathcal{K} - 1} \right) + i \pi n \ln \left( \frac{\mathcal{K} + 1}{\mathcal{K} - 1} \right) \Big] + O \left( n \mathbf{e}^{-\pi |n|} \right). \tag{20}$$

2. If 0 , K , 1, then for *n* ! ∞ one has

$$\lambda\_n = \text{sign}(\mathcal{K}n) \sqrt{\frac{EI}{\rho}} \left[ \left( \frac{2n+1}{2} \pi \right)^2 - \frac{1}{4} \text{Im}^2 \left( \frac{\mathcal{K}+1}{\mathcal{K}-1} \right) + i\pi \left( \frac{2n+1}{2} \right) \text{Im} \left( \frac{\mathcal{K}+1}{\mathcal{K}-1} \right) \right] + O \left( n \text{e}^{-\pi|n|} \right). \tag{21}$$

First of all, we address the question of accuracy of the asymptotic formulas (20) and (21). By its nature, formula (20) (as well as formula (21)) means that for any small *ε* . 0, one can find a positive integer *N*, such that all eigenvalues *λ<sup>n</sup>* with ∣*n*∣ ≥ *N* þ 1 satisfy the estimate

$$\left| \lambda\_n - \text{sign}(\mathcal{K}n) \sqrt{\frac{EI}{\rho}} \left[ \left( \pi n \right)^2 - \frac{1}{4} \ln^2 \left( \frac{\mathcal{K} + 1}{\mathcal{K} - 1} \right) + i \pi n \ln \left( \frac{\mathcal{K} + 1}{\mathcal{K} - 1} \right) \right] \right| \le \varepsilon \tag{22}$$

for the case when 1 , ∣K∣ , ∞ and

$$\left| \lambda\_{\text{f}} - \text{sign}(\mathcal{K}n) \sqrt{\frac{EI}{\rho}} \left[ \left( \frac{2n+1}{2} \pi \right)^{2} - \frac{1}{4} \ln^{2} \left( \frac{\mathcal{K}+1}{\mathcal{K}-1} \right) + i\pi \left( \frac{2n+1}{2} \right) \ln \left( \frac{\mathcal{K}+1}{\mathcal{K}-1} \right) \right] \right| \le \epsilon \tag{23}$$

for the case when 0 , ∣K∣ , 1. The following important question holds: *From which index N can the eigenvalues be approximated by the leading asymptotic terms with acceptable accuracy?* In other words, can one claim that the asymptotic formulas (20) and (21) are valuable to practitioners, or are they just important mathematical results of the spectral analysis?

The results of numerical simulations (see **Tables 1** and **2** and **Figures 1** and **2**) show that the asymptotic formulas are indeed quite accurate. That is, if one places on the complex plane the numerically produced sets of the eigenvalues, then the theoretically predicted distribution of eigenvalues can be seen almost immediately. To obtain these results, we used the numerical procedure based on Chebyshev polynomial approximations [14–16], as outlined in Section 5.

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

**Figure 3.** *Regions of* <sup>K</sup> *on the <sup>k</sup>*1*, k*2*-plane, <sup>A</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffi *EI<sup>ρ</sup>* <sup>p</sup> <sup>¼</sup> <sup>1</sup>*.*

L�<sup>1</sup> *k*1*,k*<sup>2</sup> � �<sup>∗</sup>

<sup>K</sup> <sup>¼</sup> *<sup>k</sup>*<sup>1</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup>

1. If 1 , ∣K∣ , ∞, then for ∣*n*∣ ! ∞ one has

ð Þ *πn*

2. If 0 , K , 1, then for *n* ! ∞ one has

2*n*þ1 <sup>2</sup> *<sup>π</sup>* � �<sup>2</sup> � <sup>1</sup>

> ffiffiffiffiffi *EI ρ*

ð Þ *πn*

2*n* þ 1 <sup>2</sup> *<sup>π</sup>* � �<sup>2</sup>

polynomial approximations [14–16], as outlined in Section 5.

� � � � � �

� 1 4

� " # � � �

for the case when 0 , ∣K∣ , 1. The following important question holds: *From which index N can the eigenvalues be approximated by the leading asymptotic terms with acceptable accuracy?* In other words, can one claim that the asymptotic formulas (20) and (21) are valuable to practitioners, or are they just important mathematical

The results of numerical simulations (see **Tables 1** and **2** and **Figures 1** and **2**) show that the asymptotic formulas are indeed quite accurate. That is, if one places on the complex plane the numerically produced sets of the eigenvalues, then the theoretically predicted distribution of eigenvalues can be seen almost immediately. To obtain these results, we used the numerical procedure based on Chebyshev

s

for the case when 1 , ∣K∣ , ∞ and

ffiffiffiffiffi *EI ρ*

s

results of the spectral analysis?

4

ffiffiffiffiffi *EI ρ*

s

ffiffiffiffiffi *EI ρ* r

∣*n*∣ ≥ *N* þ 1 satisfy the estimate

*λ<sup>n</sup>* � signð Þ K*n*

�

the operator L*<sup>k</sup>*1*,k*<sup>2</sup> hold:

*λ<sup>n</sup>* ¼ signð Þ K*n*

*Functional Calculus*

*λ<sup>n</sup>* ¼ signð Þ K*n*

�

� �

�

� �

**134**

*λ<sup>n</sup>* � signð Þ K*n*

<sup>¼</sup> <sup>L</sup>�<sup>1</sup>

and remainder terms depending on parameter smoothness. **Proposition 2:** Assume that k1*,* k2 . 0 and k1k2 6¼ EIρ. Let

*<sup>A</sup>* � *<sup>k</sup>*1*k*2*=<sup>A</sup> , A* <sup>¼</sup> ffiffiffiffiffiffiffiffi

ln <sup>2</sup> <sup>K</sup> <sup>þ</sup> <sup>1</sup> K � 1 � �

ln <sup>2</sup> <sup>K</sup> <sup>þ</sup> <sup>1</sup> K � 1 � �

� � � �

First of all, we address the question of accuracy of the asymptotic formulas (20) and (21). By its nature, formula (20) (as well as formula (21)) means that for any small *ε* . 0, one can find a positive integer *N*, such that all eigenvalues *λ<sup>n</sup>* with

> ln <sup>2</sup> <sup>K</sup> <sup>þ</sup> <sup>1</sup> K � 1 � �

> > ln <sup>2</sup> <sup>K</sup> <sup>þ</sup> <sup>1</sup> K � 1 � �

<sup>0</sup>*,*<sup>0</sup> <sup>þ</sup> <sup>T</sup> <sup>∗</sup>

*k*1*,k*<sup>2</sup>

From now on, we assume that the structural parameters are constant. In the case of variable parameters, the spectral asymptotics will have the same leading terms

The following asymptotic approximations for the eigenvalues *λ<sup>n</sup>* (as ∣*n*∣ ! ∞) of

<sup>þ</sup> *<sup>i</sup>πn*ln <sup>K</sup> <sup>þ</sup> <sup>1</sup>

<sup>þ</sup> *<sup>i</sup><sup>π</sup>* <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> 2 � �

K � 1

<sup>þ</sup> *<sup>i</sup>πn*ln <sup>K</sup> <sup>þ</sup> <sup>1</sup>

<sup>þ</sup> *<sup>i</sup><sup>π</sup>* <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> 2 � �

K � 1

� � � � �

ln <sup>K</sup> <sup>þ</sup> <sup>1</sup> K � 1

*,* T <sup>∗</sup>

*<sup>k</sup>*1*,k*<sup>2</sup> ¼ T �*k*2*,*�*k*<sup>1</sup> *:* (18)

<sup>þ</sup> *O n*e�*π*∣*n*<sup>∣</sup> � �*:* (20)

<sup>þ</sup> *O n*e�*π*∣*n*<sup>∣</sup> � �*:*

≤ *ε* (22)

� � � � � ≤ *ε*

(23)

(21)

*EI<sup>ρ</sup>* <sup>p</sup> *,* and <sup>∣</sup>K<sup>∣</sup> 6¼ <sup>1</sup>*:* (19)

ln <sup>K</sup> <sup>þ</sup> <sup>1</sup> K � 1 � ��

In **Figures 1** and **2**, we represent the graphical distribution of the eigenvalues corresponding to the discretized operator ("numerical" eigenvalues) and the leading asymptotic terms from Eqs. (20) and (21) ("analytic" eigenvalues). In **Tables 1** and **2**, the numerical values of the corresponding graphical points on **Figures 1** and **2** are listed. We have used the following notations: *N* ¼ 64 is the number of grid points on 0½ � *;* 1 , and *ε<sup>f</sup>* is the filtering parameter as described in Eq. (69). It can be easily seen from the graphs and tables that the two sets of data coincide almost immediately, i.e., the leading asymptotic terms in the approximations are very close to the numerically approximated eigenvalues.

**Figure 3** shows the sub-domains of the *k*1*, k*2-plane, which correspond to different intervals for the values of K defined by Eq. (19). On the sub-domain with dark gray color K such that ∣K∣ . 1, i.e., to evaluate the asymptotic approximation for the eigenvalues, one needs formula (20), while on the complementary sub-domain, one needs formula (21).

#### **3. The deadbeat modes**

An eigenvalue *λ<sup>n</sup>* of the dynamics generator L*<sup>k</sup>*1*,k*<sup>2</sup> is called a deadbeat mode if *λ<sup>n</sup>* ¼ *iβn, β<sup>n</sup>* . 0. If the corresponding eigenfunction is Φ*n*ð Þ *x* , then the evolution problem (13) has a solution given in the form e*<sup>i</sup>λnt* <sup>Φ</sup>*n*ð Þ¼ *<sup>x</sup>* <sup>e</sup>�*βnt* Φ*n*ð Þ *x* , which tends to zero without any oscillation.

As shown in paper [2], for the case when one of the control parameters is zero and the other one is positive, the entire set of the eigenvalues is located in the closed upper half plane. This result is not obvious since the operator is not dissipative; in fact, it requires a fairly nontrivial proof. However, due to this fact, we assume that any deadbeat mode can be given in the form *iβ*, with *β* . 0. To deal with the deadbeat modes analytically, we rewrite the spectral equation ð Þ L*<sup>k</sup>*1*,k*2Φ ð Þ¼ *x λ*Φð Þ *x* in the form of an equivalent problem for an operator pencil [17] as

$$\begin{aligned} \text{EI}\varrho^{\prime\prime\prime}(\mathbf{x}) &= \lambda^2 \rho \,\varrho(\mathbf{x}), & \rho(\mathbf{0}) = \rho^\prime(\mathbf{0}) = \mathbf{0}, \\ \text{EI}\varrho^{\prime\prime}(\mathbf{1}) &= -i\lambda k\_1 \rho(\mathbf{1}), & \text{EI}\varrho^{\prime\prime\prime}(\mathbf{1}) = i\lambda k\_2 \rho^\prime(\mathbf{1}). \end{aligned} \tag{24}$$

If *λ<sup>n</sup>* and *φn*ð Þ *x* are an eigenvalue and eigenfunction of the pencil (24), then *λ<sup>n</sup>* is also an eigenvalue of <sup>L</sup>*<sup>k</sup>*1*,k*<sup>2</sup> with the eigenfunction <sup>Φ</sup>*n*ð Þ¼ *<sup>x</sup>* <sup>1</sup> *<sup>i</sup>λ<sup>n</sup> φn*ð Þ *x ; φn*ð Þ *x* h i*<sup>T</sup>* .

*Proof:* Let *<sup>μ</sup>* <sup>¼</sup> ffiffi

*λ*

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

cosh 2*<sup>x</sup>* <sup>¼</sup> <sup>2</sup>

0*;* <sup>1</sup>

**Figure 4.**

**Figure 5.**

**137**

<sup>2</sup> cosh �<sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>4</sup>

we reduce Eq. (28) to the following form:

*k*<sup>1</sup> � 1

*Left- and right-hand side of Eq. (33) for different values of k*1*.*

*Estimates and actual count of deadbeat modes based on numerical simulations.*

side is sinusoidal, with maximum 1 <sup>þ</sup> <sup>4</sup>

*k*1�1

<sup>p</sup> <sup>¼</sup> *<sup>x</sup>*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>i</sup> , x* . 0. Taking into account the relations

2 þ ð Þ 1 þ *k*<sup>1</sup> cos 2*x* þ ð Þ 1 � *k*<sup>1</sup> cosh 2*x* ¼ 0*, x* . 0*:* (32)

� � cos 2*x, x* . <sup>0</sup>*, k*<sup>1</sup> . <sup>1</sup>*:* (33)

� � and minimum ð Þ �<sup>1</sup> , and period *<sup>π</sup>*. So

2cosh *μ* cos *μ* ¼ cosh 1ð Þ þ *i μ* þ cosh 1ð Þ � *i μ,* 2*i*sinh *μ* sin *μ* ¼ cosh 1ð Þ þ *i μ* � cosh 1ð Þ � *i μ,*

It can be readily seen that if 0 , *k*<sup>1</sup> , 1, then 2 þ ð Þ 1 þ *k*<sup>1</sup> cos 2*x* . 0 and ð Þ 1 � *k*<sup>1</sup> cosh 2*x* . 0, which means that Eq. (32) has no solutions. Statement (1) is shown. Statement (2) follows immediately if one considers Eq. (32) for *k*<sup>1</sup> ¼ 1.

> 2 *k*<sup>1</sup> � 1

The left-hand side of Eq. (33) is monotonically increasing, while the right-hand

*k*1�1

the graphs of the left- and right-hand side have intersections only on the interval

h i � � . There are two intersections for each full period of the

To prove Statement (3), we rewrite Eq. (32) in the form

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

þ 1 þ

To solve problem (24), we first redefine the spectral and control parameters to eliminate *ρ* and *EI* from Eq. (24). We define ~*λ,* ~ *<sup>k</sup>*1, and <sup>~</sup> *<sup>k</sup>*<sup>2</sup> by *<sup>λ</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi *EI=ρ* p ~*λ* and ~ *kj* <sup>¼</sup> ffiffiffiffiffiffiffiffi *EI<sup>ρ</sup>* <sup>p</sup> *kj*, *<sup>j</sup>* <sup>¼</sup> <sup>1</sup>*,* 2. Substituting these relations into Eq. (24) and eliminating the "tilde," we obtain the following Sturm-Liouville eigenvalue problem:

$$\begin{aligned} \, \, \rho \prime \prime \prime (\mathbf{x}) = \lambda^2 \rho(\mathbf{x}), \quad \, \rho(\mathbf{0}) = \rho'(\mathbf{0}) = \mathbf{0}, \quad \, \rho \prime \prime(\mathbf{1}) = -i\lambda k\_1 \rho(\mathbf{1}), \quad \, \rho \prime \prime(\mathbf{1}) = i\lambda k\_2 \rho \prime(\mathbf{1}). \end{aligned} \tag{25}$$

The solution of Eq. (25) satisfying the left-end boundary conditions *φ*ð Þ¼ 0 *φ*<sup>0</sup> ð Þ¼ 0 0 can be written in the form

$$\boldsymbol{\rho}(\boldsymbol{\lambda}, \mathbf{x}) = \mathcal{A}(\boldsymbol{\lambda}) \left[ \cosh \left( \sqrt{\boldsymbol{\lambda}} \mathbf{x} \right) - \cos \left( \sqrt{\boldsymbol{\lambda}} \mathbf{x} \right) \right] + \mathcal{B}(\boldsymbol{\lambda}) \left[ \sinh \left( \sqrt{\boldsymbol{\lambda}} \mathbf{x} \right) - \sin \left( \sqrt{\boldsymbol{\lambda}} \mathbf{x} \right) \right]. \tag{26}$$

Substituting formula (26) into the right-end boundary conditions of Eq. (25), one gets a system for the coefficients Að Þ*λ* and Bð Þ*λ* :

$$\begin{aligned} \mathcal{A}(\lambda) \left[ (\mathbf{1} + ik\_1) \cosh \sqrt{\lambda} - (\mathbf{1} - ik\_1) \cos \sqrt{\lambda} \right] + \\ \mathcal{B}(\lambda) \left[ (\mathbf{1} + ik\_1) \sinh \sqrt{\lambda} - (\mathbf{1} - ik\_1) \sin \sqrt{\lambda} \right] &= \mathbf{0}, \\ \mathcal{A}(\lambda) \left[ (\mathbf{1} - ik\_2) \sinh \sqrt{\lambda} - (\mathbf{1} + ik\_2) \sin \sqrt{\lambda} \right] + \\ \mathcal{B}(\lambda) \left[ (\mathbf{1} - ik\_2) \cosh \sqrt{\lambda} + (\mathbf{1} + ik\_2) \cos \sqrt{\lambda} \right] &= \mathbf{0}. \end{aligned} \tag{27}$$

Let Δð Þ*λ* be the determinant of the matrix of coefficients for Að Þ*λ* and Bð Þ*λ* in Eqs. (27). System (27) has nontrivial solutions if and only if Δð Þ¼ *λ* 0, i.e.,

$$(1 + k\_1 k\_2) + (1 - k\_1 k\_2) \cosh \sqrt{\lambda} \cos \sqrt{\lambda} + i(k\_1 + k\_2) \sinh \sqrt{\lambda} \sin \sqrt{\lambda} = 0. \tag{28}$$

**Theorem 1:** The following results hold in the case when *k*<sup>1</sup> . 0 and *k*<sup>2</sup> ¼ 0. Similar results hold in the case when *k*<sup>1</sup> ¼ 0 and *k*<sup>2</sup> . 0.


$$
\lambda\_n = \mu\_n^2, \qquad \mu\_n = \mathfrak{x}\_n(\mathfrak{1} + i), \qquad \mathfrak{x}\_n = \frac{\pi}{2}(2n + \mathfrak{1}), \qquad n = 0, 1, 2, \ldots \tag{29}
$$

3. For any *k*<sup>1</sup> . 1, there exist a finite number N ð Þ *k*<sup>1</sup> of deadbeat modes. Each mode has the form *<sup>λ</sup>* <sup>¼</sup> *<sup>μ</sup>*<sup>2</sup>*, <sup>μ</sup>* <sup>¼</sup> *<sup>x</sup>*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>i</sup>* , where *<sup>x</sup>* is a root of the function

$$H(\mathbf{x}; k\_1) \equiv \mathbf{2} + (\mathbf{1} + k\_1) \cos 2\mathbf{x} + (\mathbf{1} - k\_1) \cosh 2\mathbf{x}, \quad \mathbf{x} \ge \mathbf{0}. \tag{30}$$

Let *X k*ð Þ¼ <sup>1</sup> 1 <sup>2</sup>*<sup>π</sup>* cosh �<sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>4</sup> *k*1�1 � �, and then <sup>N</sup> ð Þ *<sup>k</sup>*<sup>1</sup> satisfies the estimate

$$\mathbb{1}[X(k\_1)] + \mathbb{1} \le \mathcal{N}(k\_1) \le \mathbb{2}[X(k\_1)] + \mathbb{3}.\tag{31}$$

Hence N ð Þ! *k*<sup>1</sup> ∞ as *k*<sup>1</sup> ! 1þ. (By ½ � *X* we denote the greatest integer less than or equal to *X*).

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

*Proof:* Let *<sup>μ</sup>* <sup>¼</sup> ffiffi *λ* <sup>p</sup> <sup>¼</sup> *<sup>x</sup>*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>i</sup> , x* . 0. Taking into account the relations

$$\begin{aligned} 2\cosh\mu\cos\mu &= \cosh\left(\mathbf{1} + i\right)\mu + \cosh\left(\mathbf{1} - i\right)\mu, \\ 2i\sinh\mu\sin\mu &= \cosh\left(\mathbf{1} + i\right)\mu - \cosh\left(\mathbf{1} - i\right)\mu, \end{aligned}$$

we reduce Eq. (28) to the following form:

If *λ<sup>n</sup>* and *φn*ð Þ *x* are an eigenvalue and eigenfunction of the pencil (24), then *λ<sup>n</sup>* is

To solve problem (24), we first redefine the spectral and control parameters to

*EI<sup>ρ</sup>* <sup>p</sup> *kj*, *<sup>j</sup>* <sup>¼</sup> <sup>1</sup>*,* 2. Substituting these relations into Eq. (24) and eliminating the

*<sup>k</sup>*1, and <sup>~</sup>

ð Þ¼ 0 0*, φ*00ð Þ¼� 1 *iλk*1*φ*ð Þ1 *, φ*000ð Þ¼ 1 *iλk*2*φ*<sup>0</sup>

<sup>þ</sup> <sup>B</sup>ð Þ*<sup>λ</sup>* sinh ffiffi

*<sup>i</sup>λ<sup>n</sup> φn*ð Þ *x ; φn*ð Þ *x* h i*<sup>T</sup>*

*EI=ρ* p ~*λ* and

� sin ffiffi

*λ* <sup>p</sup> *<sup>x</sup>*

*<sup>k</sup>*<sup>2</sup> by *<sup>λ</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi

*λ* <sup>p</sup> *<sup>x</sup>* � �

*λ* <sup>p</sup> sin ffiffi *λ*

ð Þ 2*n* þ 1 *, n* ¼ 0*;* 1*;* 2*,* …*:* (29)

h i � �

.

ð Þ1 *:* (25)

*:* (26)

(27)

<sup>p</sup> <sup>¼</sup> <sup>0</sup>*:* (28)

also an eigenvalue of <sup>L</sup>*<sup>k</sup>*1*,k*<sup>2</sup> with the eigenfunction <sup>Φ</sup>*n*ð Þ¼ *<sup>x</sup>* <sup>1</sup>

"tilde," we obtain the following Sturm-Liouville eigenvalue problem:

The solution of Eq. (25) satisfying the left-end boundary conditions

� cos ffiffi

*λ* <sup>p</sup> *<sup>x</sup>*

Substituting formula (26) into the right-end boundary conditions of Eq. (25),

*<sup>λ</sup>* � � <sup>p</sup> <sup>þ</sup>

*λ*

*<sup>λ</sup>* � � <sup>p</sup> <sup>þ</sup>

*λ*

Let Δð Þ*λ* be the determinant of the matrix of coefficients for Að Þ*λ* and Bð Þ*λ* in

**Theorem 1:** The following results hold in the case when *k*<sup>1</sup> . 0 and *k*<sup>2</sup> ¼ 0.

2. For *k*<sup>1</sup> ¼ 1, there exist infinitely many deadbeat modes given explicitly by

3. For any *k*<sup>1</sup> . 1, there exist a finite number N ð Þ *k*<sup>1</sup> of deadbeat modes. Each mode has the form *<sup>λ</sup>* <sup>¼</sup> *<sup>μ</sup>*<sup>2</sup>*, <sup>μ</sup>* <sup>¼</sup> *<sup>x</sup>*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>i</sup>* , where *<sup>x</sup>* is a root of the function

2

Hence N ð Þ! *k*<sup>1</sup> ∞ as *k*<sup>1</sup> ! 1þ. (By ½ � *X* we denote the greatest integer less than or

*H x*ð Þ� ; *k*<sup>1</sup> 2 þ ð Þ 1 þ *k*<sup>1</sup> cos 2*x* þ ð Þ 1 � *k*<sup>1</sup> cosh 2*x, x* . 0*:* (30)

, and then N ð Þ *k*<sup>1</sup> satisfies the estimate

2½ �þ *X k*ð Þ<sup>1</sup> 1≤ N ð Þ *k*<sup>1</sup> ≤ 2½ �þ *X k*ð Þ<sup>1</sup> 3*:* (31)

<sup>p</sup> � ð Þ <sup>1</sup> � *ik*<sup>1</sup> cos ffiffi

<sup>p</sup> � ð Þ <sup>1</sup> <sup>þ</sup> *ik*<sup>2</sup> sin ffiffi

<sup>p</sup> � ð Þ <sup>1</sup> � *ik*<sup>1</sup> sin ffiffi *<sup>λ</sup>* � � <sup>p</sup> <sup>¼</sup> <sup>0</sup>*,*

<sup>p</sup> <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> *ik*<sup>2</sup> cos ffiffi *<sup>λ</sup>* � � <sup>p</sup> <sup>¼</sup> <sup>0</sup>*:*

<sup>p</sup> <sup>þ</sup> *i k*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> sinh ffiffi

*λ*

*λ*

Eqs. (27). System (27) has nontrivial solutions if and only if Δð Þ¼ *λ* 0, i.e.,

*λ* <sup>p</sup> cos ffiffi *λ*

eliminate *ρ* and *EI* from Eq. (24). We define ~*λ,* ~

*φ*ð Þ *x , φ*ð Þ¼ 0 *φ*<sup>0</sup>

*φ λ*ð Þ¼ *; <sup>x</sup>* <sup>A</sup>ð Þ*<sup>λ</sup>* cosh ffiffi

ð Þþ <sup>1</sup> <sup>þ</sup> *<sup>k</sup>*1*k*<sup>2</sup> ð Þ <sup>1</sup> � *<sup>k</sup>*1*k*<sup>2</sup> cosh ffiffi

*<sup>λ</sup><sup>n</sup>* <sup>¼</sup> *<sup>μ</sup>*<sup>2</sup>

Let *X k*ð Þ¼ <sup>1</sup>

equal to *X*).

**136**

1

<sup>2</sup>*<sup>π</sup>* cosh �<sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>4</sup>

*k*1�1 � �

ð Þ¼ 0 0 can be written in the form

*λ* <sup>p</sup> *<sup>x</sup>* � �

one gets a system for the coefficients Að Þ*λ* and Bð Þ*λ* :

<sup>A</sup>ð Þ*<sup>λ</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *ik*<sup>1</sup> cosh ffiffi

<sup>A</sup>ð Þ*<sup>λ</sup>* ð Þ <sup>1</sup> � *ik*<sup>2</sup> sinh ffiffi

Similar results hold in the case when *k*<sup>1</sup> ¼ 0 and *k*<sup>2</sup> . 0.

1. For 0 , *k*<sup>1</sup> , 1, the deadbeat modes do not exist.

*n, <sup>μ</sup><sup>n</sup>* <sup>¼</sup> *xn*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>i</sup> , xn* <sup>¼</sup> *<sup>π</sup>*

<sup>B</sup>ð Þ*<sup>λ</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *ik*<sup>1</sup> sinh ffiffi

<sup>B</sup>ð Þ*<sup>λ</sup>* ð Þ <sup>1</sup> � *ik*<sup>2</sup> cosh ffiffi

h i � �

~ *kj* <sup>¼</sup> ffiffiffiffiffiffiffiffi

*<sup>φ</sup>*0000ð Þ¼ *<sup>x</sup> <sup>λ</sup>*<sup>2</sup>

*Functional Calculus*

*φ*ð Þ¼ 0 *φ*<sup>0</sup>

$$2 + (1 + k\_1) \cos 2\pi + (1 - k\_1) \cosh 2\pi = 0, \quad \pi \ge 0. \tag{32}$$

It can be readily seen that if 0 , *k*<sup>1</sup> , 1, then 2 þ ð Þ 1 þ *k*<sup>1</sup> cos 2*x* . 0 and ð Þ 1 � *k*<sup>1</sup> cosh 2*x* . 0, which means that Eq. (32) has no solutions. Statement (1) is shown. Statement (2) follows immediately if one considers Eq. (32) for *k*<sup>1</sup> ¼ 1.

To prove Statement (3), we rewrite Eq. (32) in the form

$$
\cosh 2x = \frac{2}{k\_1 - 1} + \left( 1 + \frac{2}{k\_1 - 1} \right) \cos 2x, \quad x \ge 0, \quad k\_1 \ge 1. \tag{33}
$$

The left-hand side of Eq. (33) is monotonically increasing, while the right-hand side is sinusoidal, with maximum 1 <sup>þ</sup> <sup>4</sup> *k*1�1 � � and minimum ð Þ �<sup>1</sup> , and period *<sup>π</sup>*. So the graphs of the left- and right-hand side have intersections only on the interval 0*;* <sup>1</sup> <sup>2</sup> cosh �<sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>4</sup> *k*1�1 h i � � . There are two intersections for each full period of the

**Figure 4.** *Left- and right-hand side of Eq. (33) for different values of k*1*.*

**Figure 5.** *Estimates and actual count of deadbeat modes based on numerical simulations.*

right-hand side that fits into the above interval (**Figure 4**). As it can be seen in **Figure 4**, one should add at least one more intersection for the first half-period after the full periods. Depending on the value of *k*1, the two graphs can have two intersections, one tangential intersection or no intersections on the second halfperiod. This leads to estimate (31). ■

A graphical illustration of the result of Theorem 1 is shown in **Figure 5**.

### **4. Structure of the deadbeat mode set**

The main result on the existence and distribution of double roots of the function *H x*ð Þ ; *k*<sup>1</sup> is presented in the statement below.

**Theorem 2:** For a given *k*<sup>1</sup> . 1, the multiplicity of each root of *H x*ð Þ ; *k*<sup>1</sup> does not exceed 2. There exists a sequence *k* ð Þ *n* <sup>1</sup> ; *n* ¼ 0*;* 1*;* 2*;* … n o, such that the function *H x*ð Þ ; *k*<sup>1</sup> has a double root if and only if *k*<sup>1</sup> ¼ *k* ð Þ *n* <sup>1</sup> for some *n*. So the original spectral problem with *k*<sup>1</sup> ¼ *k* ð Þ *n* <sup>1</sup> *, k*<sup>2</sup> <sup>¼</sup> 0 has a double deadbeat mode *<sup>λ</sup><sup>n</sup>* <sup>¼</sup> *<sup>μ</sup>*<sup>2</sup> *<sup>n</sup>* <sup>¼</sup> <sup>2</sup>*ix*<sup>2</sup> *<sup>n</sup>*. The following asymptotic formulas hold

$$\mu\_n = \frac{3\pi}{4} + \pi n + P\_n^{-1} + \mathcal{O}(P\_n^{-2}), \qquad \text{where} \qquad P\_n = \exp\left\{\frac{3\pi}{2} + 2\pi n\right\}, \tag{34}$$

and

$$k\_1^{(n)} = \mathbf{1} + 4P\_n^{-1} + O(P\_n^{-2}).\tag{35}$$

Now we consider the case when cos 2*x* . 0 and sin 2*x* , 0, i.e.,

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

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*π<sup>n</sup>* <sup>þ</sup> *<sup>s</sup>* 

> <sup>2</sup> <sup>þ</sup> <sup>2</sup>*π<sup>n</sup>*

ðÞ¼ *<sup>s</sup>* sin *<sup>s</sup>* <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> 2 sin *<sup>s</sup>* cosh <sup>3</sup>*<sup>π</sup>*

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*π<sup>n</sup>* , <sup>2</sup>*<sup>x</sup>* , <sup>2</sup>*π*ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *, n*<sup>∈</sup> f g <sup>0</sup>*;* <sup>1</sup>*;* <sup>2</sup>*;* … *:*

2*x* ¼ 3*π=*2 þ 2*πn* þ *s*, where *n*∈f g 0*;* 1*;* 2*;* … , 0 , *s* , *π=*2. If *g s*ðÞ� *G*ð Þ 3*π=*4 þ *πn* þ *s* ,

Let us show that for each *n*, Eq. (40) has a unique solution. For *s* ¼ 0 we have

and for *s* ¼ *π=*2 we have *g*ð Þ¼ *π=*2 2sinh 2ð Þ *π*ð Þ *n* þ 1 . 0. Evaluating *g*<sup>0</sup> we have

Thus *g s*ð Þ is a monotonically increasing function, such that *g*ð Þ 0 , 0 , *g*ð Þ *π=*2 ,

contradiction argument, assume that there exists *x*0, such that in addition to Eqs. (36) and (37), one has *H*00ð Þ¼ *x*0; *k*<sup>1</sup> 0, i.e., the multiplicity of *x*<sup>0</sup> is at least 3.

ð Þ¼ *x*0; *k*<sup>1</sup> 0 and *H*00ð Þ¼ *x*0; *k*<sup>1</sup> 0 can be written as

ð Þ *k*<sup>1</sup> þ 1 sin 2*x*<sup>0</sup> þ ð Þ *k*<sup>1</sup> � 1 sinh 2*x*<sup>0</sup> ¼ 0*,*

Since *k*<sup>1</sup> . 1, the second equation of (42) yields cos 2*x*<sup>0</sup> , 0. Also, since *x*<sup>0</sup> is a multiple root, we must have sin 2*x*<sup>0</sup> , 0. Then 2*x*<sup>0</sup> is in the third quadrant, which means that *G x*ð Þ<sup>0</sup> 6¼ 0, as we have seen above. This contradicts our assumption that

To derive asymptotic distribution of the roots of Eq. (40), we check that with *Pn*

*n*

<sup>¼</sup> *Pn* <sup>þ</sup> <sup>2</sup> <sup>þ</sup> *<sup>P</sup>*�<sup>1</sup>

<sup>¼</sup> *Pn* <sup>þ</sup> <sup>2</sup> <sup>þ</sup> <sup>3</sup>*P*�<sup>1</sup>

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*π<sup>n</sup>* <sup>þ</sup> <sup>2</sup>*P*�<sup>1</sup>

*n*

*n* <sup>¼</sup> <sup>2</sup>*P*�<sup>1</sup>

<sup>¼</sup> <sup>1</sup> � <sup>2</sup>*P*�<sup>2</sup>

*<sup>n</sup>* <sup>þ</sup> *O P*�<sup>2</sup> *n ,*

> *<sup>n</sup>* <sup>þ</sup> *O P*�<sup>2</sup> *n :*

> > �

*<sup>n</sup>* and using Eq. (43), we get

*,* cos 2*P*�<sup>1</sup>

Finally we show that the multiplicity of a multiple root cannot exceed 2. Using a

� <sup>1</sup> <sup>þ</sup> cosh <sup>3</sup>*<sup>π</sup>*

� <sup>1</sup> � cosh <sup>3</sup>*<sup>π</sup>*

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*π<sup>n</sup>* <sup>þ</sup> *<sup>s</sup>* cos*<sup>s</sup>* <sup>¼</sup> <sup>0</sup>*:*

. <sup>0</sup>*:* (41)

*<sup>n</sup>* <sup>þ</sup> *O P*�<sup>4</sup> *n ,*

> *<sup>n</sup>* <sup>þ</sup> *O P*�<sup>2</sup> *n :*

(43)

(44)

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*π<sup>n</sup>* , <sup>0</sup>*,*

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*π<sup>n</sup>* <sup>þ</sup> *<sup>s</sup>*

ð Þ *<sup>k</sup>*<sup>1</sup> <sup>þ</sup> <sup>1</sup> cos 2*x*<sup>0</sup> <sup>þ</sup> ð Þ *<sup>k</sup>*<sup>1</sup> � <sup>1</sup> cosh 2*x*<sup>0</sup> <sup>¼</sup> <sup>0</sup>*:* (42)

(40)

It is convenient to rewrite system given by (36) and (37) in the form

3*π*

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

*g s*ðÞ� ð Þ <sup>1</sup> <sup>þ</sup> sin *<sup>s</sup>* sinh <sup>3</sup>*<sup>π</sup>*

*g*0

*x*<sup>0</sup> is a root of Eqs. (36) and (37).

2sinh <sup>3</sup>*<sup>π</sup>*

2cosh <sup>3</sup>*<sup>π</sup>*

<sup>¼</sup> <sup>1</sup> <sup>þ</sup> sin 2*P*�<sup>1</sup>

<sup>1</sup> <sup>þ</sup> cosh <sup>3</sup>*<sup>π</sup>*

sin 2*P*�<sup>1</sup> *n* <sup>¼</sup> <sup>2</sup>*P*�<sup>1</sup>

*g* 2*P*�<sup>1</sup> *n*

**139**

The system *H*<sup>0</sup>

then Eq. (38) generates the following equation for *g*:

*<sup>g</sup>*ð Þ¼ <sup>0</sup> sinh <sup>3</sup>*<sup>π</sup>*

which means that *g* has a unique root on 0½ � *; π=*2 .

from Eq. (34), the following approximations are valid:

*<sup>n</sup>* <sup>þ</sup> *O P*�<sup>3</sup> *n*

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*π<sup>n</sup>* <sup>þ</sup> <sup>2</sup>*P*�<sup>1</sup>

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*π<sup>n</sup>* <sup>þ</sup> <sup>2</sup>*P*�<sup>1</sup>

*n* sinh <sup>3</sup>*<sup>π</sup>*

Evaluating *g s*ð Þ from Eq. (40) for *<sup>s</sup>* <sup>¼</sup> <sup>2</sup>*P*�<sup>1</sup>

*n*

*n*

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*π<sup>n</sup>* <sup>þ</sup> <sup>2</sup>*P*�<sup>1</sup>

cos 2*P*�<sup>1</sup>

*n*

*Proof:* If *x* is a double root of *H*, then *H x*ð Þ¼ ; *k*<sup>1</sup> *H*<sup>0</sup> ð Þ¼ *x*; *k*<sup>1</sup> 0, i.e., separating the real and imaginary parts, we have

$$2 + (k\_1 + 1)\cos 2\chi - (k\_1 - 1)\cosh 2\chi = 0,\tag{36}$$

$$(k\_1 + 1)\sin 2\pi + (k\_1 - 1)\sinh 2\pi = 0.\tag{37}$$

Eliminating *k*<sup>1</sup> from system given by (36) and (37), we obtain that the following equation has to be satisfied:

$$G(\varkappa) \equiv (\mathbb{1} + \cos 2\varkappa) \sinh 2\varkappa + (\mathbb{1} + \cosh 2\varkappa) \sin 2\varkappa = 0, \qquad \varkappa \ge 0. \tag{38}$$

Rewriting Eq. (37) in the form ð Þ *k*<sup>1</sup> þ 1 sin 2*x* ¼ �ð Þ *k*<sup>1</sup> � 1 sinh 2*x*, and taking into account that *k*<sup>1</sup> . 1, we obtain that if *x* is the solution of Eq. (38), then sin 2*x* , 0.

Now we show that when cos 2*x* , 0 and sin 2*x* , 0, i.e.,

$$
\pi(2n+1) < 2\pi < \frac{3\pi}{2} + 2\pi n, \qquad n \in \{0, 1, 2, \ldots\},
$$

Eq. (38) does not have any solutions. Indeed, in the above range of *x*, we have cos 2*<sup>x</sup>* <sup>þ</sup> sin 2*<sup>x</sup>* <sup>¼</sup> ffiffi 2 <sup>p</sup> sin 2ð Þ *<sup>x</sup>* <sup>þ</sup> *<sup>π</sup>=*<sup>4</sup> , � 1 and <sup>∣</sup>sin 2*<sup>x</sup>* � cos 2*x*<sup>∣</sup> , 1. With such estimates we obtain that

$$G(\mathbf{x}) = \sin 2\mathbf{x} + \sinh 2\mathbf{x} + \frac{1}{2} \mathbf{e}^{2\mathbf{x}} (\cos 2\mathbf{x} + \sin 2\mathbf{x}) + \frac{1}{2} \mathbf{e}^{-2\mathbf{x}} (\sin 2\mathbf{x} - \cos 2\mathbf{x}) < \sin 2\mathbf{x} < 0,\tag{39}$$

which mean that Eq. (38) cannot be satisfied.

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

Now we consider the case when cos 2*x* . 0 and sin 2*x* , 0, i.e.,

$$
\frac{3\pi}{2} + 2\pi n \le 2\pi \le 2\pi (n+1), \qquad n \in \{0, 1, 2, \ldots\}.
$$

It is convenient to rewrite system given by (36) and (37) in the form 2*x* ¼ 3*π=*2 þ 2*πn* þ *s*, where *n*∈f g 0*;* 1*;* 2*;* … , 0 , *s* , *π=*2. If *g s*ðÞ� *G*ð Þ 3*π=*4 þ *πn* þ *s* , then Eq. (38) generates the following equation for *g*:

$$\log(s) \equiv (1 + \sin s) \sinh\left(\frac{3\pi}{2} + 2\pi n + s\right) - \left[1 + \cosh\left(\frac{3\pi}{2} + 2\pi n + s\right)\right] \cos s = 0. \tag{40}$$

Let us show that for each *n*, Eq. (40) has a unique solution. For *s* ¼ 0 we have

$$g(\mathbf{0}) = \sinh\left(\frac{3\pi}{2} + 2\pi n\right) - 1 - \cosh\left(\frac{3\pi}{2} + 2\pi n\right) < \mathbf{0},$$

and for *s* ¼ *π=*2 we have *g*ð Þ¼ *π=*2 2sinh 2ð Þ *π*ð Þ *n* þ 1 . 0. Evaluating *g*<sup>0</sup> we have

$$g'(s) = \sin s + (1 + 2\sin s)\cosh\left(\frac{3\pi}{2} + 2\pi n + s\right) \ge 0. \tag{41}$$

Thus *g s*ð Þ is a monotonically increasing function, such that *g*ð Þ 0 , 0 , *g*ð Þ *π=*2 , which means that *g* has a unique root on 0½ � *; π=*2 .

Finally we show that the multiplicity of a multiple root cannot exceed 2. Using a contradiction argument, assume that there exists *x*0, such that in addition to Eqs. (36) and (37), one has *H*00ð Þ¼ *x*0; *k*<sup>1</sup> 0, i.e., the multiplicity of *x*<sup>0</sup> is at least 3. The system *H*<sup>0</sup> ð Þ¼ *x*0; *k*<sup>1</sup> 0 and *H*00ð Þ¼ *x*0; *k*<sup>1</sup> 0 can be written as

$$\begin{aligned} \left((k\_1+1)\sin 2\mathbf{x}\_0 + (k\_1-1)\sinh 2\mathbf{x}\_0\right) &= \mathbf{0}, \\ \left((k\_1+1)\cos 2\mathbf{x}\_0 + (k\_1-1)\cosh 2\mathbf{x}\_0\right) &= \mathbf{0}. \end{aligned} \tag{42}$$

Since *k*<sup>1</sup> . 1, the second equation of (42) yields cos 2*x*<sup>0</sup> , 0. Also, since *x*<sup>0</sup> is a multiple root, we must have sin 2*x*<sup>0</sup> , 0. Then 2*x*<sup>0</sup> is in the third quadrant, which means that *G x*ð Þ<sup>0</sup> 6¼ 0, as we have seen above. This contradicts our assumption that *x*<sup>0</sup> is a root of Eqs. (36) and (37).

To derive asymptotic distribution of the roots of Eq. (40), we check that with *Pn* from Eq. (34), the following approximations are valid:

$$\begin{aligned} \sin\left(2P\_{\pi}^{-1}\right) &= 2P\_{\pi}^{-1} + O\left(P\_{\pi}^{-3}\right), & \cos\left(2P\_{\pi}^{-1}\right) &= 1 - 2P\_{\pi}^{-2} + O\left(P\_{\pi}^{-4}\right), \\ 2\sinh\left(\frac{3\pi}{2} + 2\pi n + 2P\_{\pi}^{-1}\right) &= P\_{n} + 2 + P\_{\pi}^{-1} + O\left(P\_{\pi}^{-2}\right), \\ 2\cosh\left(\frac{3\pi}{2} + 2\pi n + 2P\_{\pi}^{-1}\right) &= P\_{n} + 2 + 3P\_{\pi}^{-1} + O\left(P\_{\pi}^{-2}\right). \end{aligned} \tag{43}$$

Evaluating *g s*ð Þ from Eq. (40) for *<sup>s</sup>* <sup>¼</sup> <sup>2</sup>*P*�<sup>1</sup> *<sup>n</sup>* and using Eq. (43), we get

$$\begin{split} \lg\left(2P\_{n}^{-1}\right) &= \left[1 + \sin\left(2P\_{n}^{-1}\right)\right] \sinh\left(\frac{3\pi}{2} + 2\pi n + 2P\_{n}^{-1}\right) - \\ &\left[1 + \cosh\left(\frac{3\pi}{2} + 2\pi n + 2P\_{n}^{-1}\right)\right] \cos\left(2P\_{n}^{-1}\right) = 2P\_{n}^{-1} + O\left(P\_{n}^{-2}\right). \end{split} \tag{44}$$

right-hand side that fits into the above interval (**Figure 4**). As it can be seen in **Figure 4**, one should add at least one more intersection for the first half-period after

The main result on the existence and distribution of double roots of the function

**Theorem 2:** For a given *k*<sup>1</sup> . 1, the multiplicity of each root of *H x*ð Þ ; *k*<sup>1</sup> does not

<sup>1</sup> ; *n* ¼ 0*;* 1*;* 2*;* … n o

<sup>1</sup> *, k*<sup>2</sup> <sup>¼</sup> 0 has a double deadbeat mode *<sup>λ</sup><sup>n</sup>* <sup>¼</sup> *<sup>μ</sup>*<sup>2</sup>

� �*,* where *Pn* <sup>¼</sup> exp

Eliminating *k*<sup>1</sup> from system given by (36) and (37), we obtain that the following

*G x*ð Þ� ð Þ 1 þ cos 2*x* sinh 2*x* þ ð Þ 1 þ cosh 2*x* sin 2*x* ¼ 0*, x* . 0*:* (38)

Rewriting Eq. (37) in the form ð Þ *k*<sup>1</sup> þ 1 sin 2*x* ¼ �ð Þ *k*<sup>1</sup> � 1 sinh 2*x*, and taking into account that *k*<sup>1</sup> . 1, we obtain that if *x* is the solution of Eq. (38), then sin 2*x* , 0.

Eq. (38) does not have any solutions. Indeed, in the above range of *x*, we have

ð Þ *n*

*<sup>n</sup>* <sup>þ</sup> *O P*�<sup>2</sup> *n*

2 þ ð Þ *k*<sup>1</sup> þ 1 cos 2*x* � ð Þ *k*<sup>1</sup> � 1 cosh 2*x* ¼ 0*,* (36) ð Þ *k*<sup>1</sup> þ 1 sin 2*x* þ ð Þ *k*<sup>1</sup> � 1 sinh 2*x* ¼ 0*:* (37)

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*πn, n* <sup>∈</sup>f g <sup>0</sup>*;* <sup>1</sup>*;* <sup>2</sup>*;* … *,*

<sup>p</sup> sin 2ð Þ *<sup>x</sup>* <sup>þ</sup> *<sup>π</sup>=*<sup>4</sup> , � 1 and <sup>∣</sup>sin 2*<sup>x</sup>* � cos 2*x*<sup>∣</sup> , 1. With such

2

, such that the function

*<sup>n</sup>* <sup>¼</sup> <sup>2</sup>*ix*<sup>2</sup>

*<sup>n</sup>*. The

*,* (34)

<sup>1</sup> for some *n*. So the original spectral

� �*:* (35)

ð Þ¼ *x*; *k*<sup>1</sup> 0, i.e., separating the

<sup>e</sup>�2*x*ð Þ sin 2*<sup>x</sup>* � cos 2*<sup>x</sup>* , sin 2*<sup>x</sup>* , <sup>0</sup>*,*

(39)

3*π* <sup>2</sup> <sup>þ</sup> <sup>2</sup>*π<sup>n</sup>* � �

ð Þ *n*

<sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>4</sup>*P*�<sup>1</sup>

the full periods. Depending on the value of *k*1, the two graphs can have two intersections, one tangential intersection or no intersections on the second halfperiod. This leads to estimate (31). ■ A graphical illustration of the result of Theorem 1 is shown in **Figure 5**.

**4. Structure of the deadbeat mode set**

*H x*ð Þ ; *k*<sup>1</sup> is presented in the statement below.

*H x*ð Þ ; *k*<sup>1</sup> has a double root if and only if *k*<sup>1</sup> ¼ *k*

*<sup>n</sup>* <sup>þ</sup> *O P*�<sup>2</sup> *n*

> *k* ð Þ *n*

*Proof:* If *x* is a double root of *H*, then *H x*ð Þ¼ ; *k*<sup>1</sup> *H*<sup>0</sup>

Now we show that when cos 2*x* , 0 and sin 2*x* , 0, i.e.,

3*π*

<sup>e</sup>2*x*<sup>ð</sup> cos 2*<sup>x</sup>* <sup>þ</sup> sin 2*x*Þ þ <sup>1</sup>

*π*ð Þ 2*n* þ 1 , 2*x* ,

1 2

which mean that Eq. (38) cannot be satisfied.

2

ð Þ *n*

exceed 2. There exists a sequence *k*

following asymptotic formulas hold

<sup>4</sup> <sup>þ</sup> *<sup>π</sup><sup>n</sup>* <sup>þ</sup> *<sup>P</sup>*�<sup>1</sup>

real and imaginary parts, we have

equation has to be satisfied:

cos 2*<sup>x</sup>* <sup>þ</sup> sin 2*<sup>x</sup>* <sup>¼</sup> ffiffi

estimates we obtain that

*G x*ð Þ¼ sin 2*x* þ sinh 2*x* þ

**138**

problem with *k*<sup>1</sup> ¼ *k*

*Functional Calculus*

*xn* <sup>¼</sup> <sup>3</sup>*<sup>π</sup>*

and

Representation (44) implies that there exists *n*0, such that for all *n*≥ *n*0, we have *g* 2*P*�<sup>1</sup> *n* � � . 0. Taking into account that *<sup>g</sup>*ð Þ <sup>0</sup> , 0, we obtain that the root *sn*, *<sup>n</sup>* <sup>≥</sup>*n*0, of the function *g s*ð Þ is located on the interval 0*;* <sup>2</sup>*P*�<sup>1</sup> *n* � �. To find the location of this root more precisely [18], we use linear interpolation. Namely, substituting Eq. (43) into the expression for *g*<sup>0</sup> ð Þ*s* from Eq. (41) yields

$$\lg'(2P\_n^{-1}) = \frac{P\_n}{2} + O(1). \tag{45}$$

Since *λ*<sup>0</sup> is purely imaginary, Eq. (51) is not valid. We define a new function:

One can readily check that *g* satisfies the following boundary-value problem:

*ψ*ð Þ1

*<sup>g</sup>*0000ð Þ *<sup>σ</sup> <sup>g</sup>*ð Þ *<sup>σ</sup> <sup>σ</sup>* <sup>¼</sup> *<sup>λ</sup>*<sup>2</sup>

obtained contradiction means that for each double mode, there is one eigenfunction and one associate function. ■

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

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,

<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

Analyzing **Table 4**, one can see that the eigenvalues get closer to the imaginary

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

axis as *k*<sup>1</sup> ! 1�. However the rate at which their real parts approach zero is

modes to a double mode, which is consistent with Theorem 2.

<sup>0</sup> . 0; however, for a deadbeat mode, *λ*<sup>2</sup>

*<sup>χ</sup>*ð Þ<sup>1</sup> *<sup>χ</sup>*ð Þ *<sup>x</sup> :* (52)

ð Þ¼ 0 0*, φ*ð Þ¼ 1 *φ*00ð Þ¼ 1 0*,* (53)

j j *<sup>g</sup>*ð Þ *<sup>σ</sup>* <sup>2</sup>

d*σ:* (54)

<sup>0</sup> , 0. The

*g x*ð Þ¼ *ψ*ð Þ� *x*

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

Z <sup>1</sup> 0

<sup>0</sup>*φ*ð Þ *x , φ*ð Þ¼ 0 *φ*<sup>0</sup>

*<sup>φ</sup>*0000ð Þ¼ *<sup>x</sup> <sup>λ</sup>*<sup>2</sup>

and therefore

above or below.

**Figure 6.**

**141**

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

Z <sup>1</sup> 0 *g*00 ð Þ *<sup>σ</sup>* � � � � 2 d*σ* ¼

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

Eq. (54) is valid if and only if *λ*<sup>2</sup>

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

Replacing *g s*ð Þ by the linear function tangential to *g s*ð Þ at the point 2*P*�<sup>1</sup> *<sup>n</sup> ; g* 2*P*�<sup>1</sup> *n* � � � � , and finding the root of this function, we get

$$\kappa\_n = 2P\_n^{-1} - \frac{g\left(2P\_n^{-1}\right)}{g'\left(2P\_n^{-1}\right)} + O\left(P\_n^{-2}\right) = 2P\_n^{-1} + O\left(P\_n^{-2}\right). \tag{46}$$

Having this approximation for *sn*, we immediately get

$$
\infty\_n = \frac{3\pi}{4} + \pi n + \frac{s\_n}{2} = \frac{3\pi}{4} + \pi n + P\_n^{-1} + O\left(P\_n^{-2}\right). \tag{47}
$$

From the equation *H xn*; *k* ð Þ *n* 1 � � <sup>¼</sup> 0, we obtain the formula for *<sup>k</sup>* ð Þ *n* <sup>1</sup> as

$$k\_1^{(n)} = \frac{\sinh\left(3\pi/2 + 2\pi n + s\_n\right) + \cos s\_n}{\sinh\left(3\pi/2 + 2\pi n + s\_n\right) - \cos s\_n}.\tag{48}$$

Substituting formulas (43) and (46) into formula (48), we obtain representation (35). ■

**Corollary 1:** Let *k*<sup>1</sup> ¼ *k* ð Þ *n* <sup>1</sup> for some *n*∈ℕþ∪f g0 , and let *xn* be the corresponding double root of the function *H x*; *k* ð Þ *n* 1 � �. Then *<sup>λ</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*ix*<sup>2</sup> *<sup>n</sup>* is an eigenvalue of the operator L*<sup>k</sup>* ð Þ *n* <sup>1</sup> *,*00, such that the geometric multiplicity of *<sup>λ</sup>*<sup>0</sup> is 1 and its algebraic multiplicity is 2. Therefore there exists a unique eigenvector Φ and one associate vector Ψ, such that

$$
\mathcal{L}\_{k\_1^{(n)},0} \Phi = \lambda\_0 \Phi, \qquad \mathcal{L}\_{k\_1^{(n)},0} \Psi - \lambda\_0 \Psi = \Phi. \tag{49}
$$

*Proof:* It suffices to show that problem (24) does not have two linearly independent eigenvectors corresponding to *λ*<sup>0</sup> [18, 19]. Using contradiction argument we assume that for some *λ*<sup>0</sup> the boundary-value problem (25) with *k*<sup>2</sup> ¼ 0 has two linearly independent solutions *ψ* and *χ*. Each function satisfies the problem

$$\begin{aligned} \, \rho \prime \prime \prime (\mathbf{x}) = \lambda\_0^2 \rho(\mathbf{x}), \qquad \rho(\mathbf{0}) = \rho'(\mathbf{0}) = \mathbf{0}, \qquad \rho \prime \prime (\mathbf{1}) = \mathbf{0}, \qquad \rho \prime \prime (\mathbf{1}) = i \lambda\_0 k\_1 \rho(\mathbf{1}). \end{aligned} \tag{50}$$

First we observe that *ψ*ð Þ1 *χ*ð Þ1 6¼ 0. Indeed, if *ψ*ð Þ¼ 1 0, then we have

$$\int\_0^1 |\psi''(\sigma)|^2 d\sigma = \int\_0^1 \psi^{\prime\prime\prime}(\sigma) \overline{\psi(\sigma)} d\sigma = \lambda\_0^2 \int\_0^1 |\psi(\sigma)|^2 d\sigma. \tag{51}$$

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

Since *λ*<sup>0</sup> is purely imaginary, Eq. (51) is not valid. We define a new function:

$$\mathbf{g}(\mathbf{x}) = \boldsymbol{\Psi}(\mathbf{x}) - \frac{\boldsymbol{\Psi}(\mathbf{1})}{\boldsymbol{\chi}(\mathbf{1})} \boldsymbol{\chi}(\mathbf{x}).\tag{52}$$

One can readily check that *g* satisfies the following boundary-value problem:

$$
\rho^{\prime\prime\prime\prime}(\mathbf{x}) = \lambda\_0^2 \rho(\mathbf{x}), \qquad \rho(\mathbf{0}) = \rho^\prime(\mathbf{0}) = \mathbf{0}, \qquad \rho(\mathbf{1}) = \rho^{\prime\prime}(\mathbf{1}) = \mathbf{0},\tag{53}
$$

and therefore

Representation (44) implies that there exists *n*0, such that for all *n*≥ *n*0, we have

*n*

� �. To find the location of this root

<sup>2</sup> <sup>þ</sup> *<sup>O</sup>*ð Þ<sup>1</sup> *:* (45)

*<sup>n</sup>* <sup>þ</sup> *O P*�<sup>2</sup> *n*

*<sup>n</sup>* <sup>þ</sup> *O P*�<sup>2</sup> *n*

<sup>1</sup> for some *n*∈ℕþ∪f g0 , and let *xn* be the corresponding

� �*:* (46)

� �*:* (47)

*:* (48)

(50)

*dσ:* (51)

ð Þ *n* <sup>1</sup> as

*<sup>n</sup>* is an eigenvalue of the

<sup>1</sup> *,*0<sup>Ψ</sup> � *<sup>λ</sup>*0<sup>Ψ</sup> <sup>¼</sup> <sup>Φ</sup>*:* (49)

ð Þ¼ 0 0*, φ*000ð Þ¼ 1 0*, φ*00ð Þ¼ 1 *iλ*0*k*1*φ*ð Þ1 *:*

j j *ψ σ*ð Þ <sup>2</sup>

� � . 0. Taking into account that *<sup>g</sup>*ð Þ <sup>0</sup> , 0, we obtain that the root *sn*, *<sup>n</sup>* <sup>≥</sup>*n*0, of

more precisely [18], we use linear interpolation. Namely, substituting Eq. (43) into

*n* � � <sup>¼</sup> <sup>2</sup>*P*�<sup>1</sup>

<sup>4</sup> <sup>þ</sup> *<sup>π</sup><sup>n</sup>* <sup>þ</sup> *<sup>P</sup>*�<sup>1</sup>

<sup>1</sup> <sup>¼</sup> sinh 3<sup>ð</sup> *<sup>π</sup>=*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*π<sup>n</sup>* <sup>þ</sup> *sn*Þ þ cos*sn* sinh 3ð *π=*2 þ 2*πn* þ *sn*Þ � cos*sn*

representation (35). ■

multiplicity is 2. Therefore there exists a unique eigenvector Φ and one associate

. Then *<sup>λ</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*ix*<sup>2</sup>

<sup>1</sup> *,*00, such that the geometric multiplicity of *<sup>λ</sup>*<sup>0</sup> is 1 and its algebraic

ð Þ *n*

*<sup>ψ</sup>*0000ð Þ *<sup>σ</sup> ψ σ*ð Þ*d<sup>σ</sup>* <sup>¼</sup> *<sup>λ</sup>*<sup>2</sup>

¼ 0, we obtain the formula for *k*

the function *g s*ð Þ is located on the interval 0*;* <sup>2</sup>*P*�<sup>1</sup>

*sn* <sup>¼</sup> <sup>2</sup>*P*�<sup>1</sup>

*xn* <sup>¼</sup> <sup>3</sup>*<sup>π</sup>*

*k* ð Þ *n*

L*k* ð Þ *n*

<sup>0</sup>*φ*ð Þ *x , φ*ð Þ¼ 0 *φ*<sup>0</sup>

*<sup>ψ</sup>*<sup>00</sup> j j ð Þ *<sup>σ</sup>* <sup>2</sup>

*dσ* ¼

Z <sup>1</sup> 0

Z <sup>1</sup> 0

ð Þ *n*

From the equation *H xn*; *k*

**Corollary 1:** Let *k*<sup>1</sup> ¼ *k*

ð Þ *n*

vector Ψ, such that

*<sup>φ</sup>*0000ð Þ¼ *<sup>x</sup> <sup>λ</sup>*<sup>2</sup>

**140**

operator L*<sup>k</sup>*

double root of the function *H x*; *k*

ð Þ*s* from Eq. (41) yields

� � � � , and finding the root of this function, we get

*<sup>n</sup>* � *<sup>g</sup>* <sup>2</sup>*P*�<sup>1</sup> *n* � � *g*<sup>0</sup> 2*P*�<sup>1</sup> *n* � � <sup>þ</sup> *O P*�<sup>2</sup>

Having this approximation for *sn*, we immediately get

<sup>4</sup> <sup>þ</sup> *<sup>π</sup><sup>n</sup>* <sup>þ</sup>

ð Þ *n* 1 � �

*g*<sup>0</sup> 2*P*�<sup>1</sup> *n* � � <sup>¼</sup> *Pn*

Replacing *g s*ð Þ by the linear function tangential to *g s*ð Þ at the point

*sn* <sup>2</sup> <sup>¼</sup> <sup>3</sup>*<sup>π</sup>*

Substituting formulas (43) and (46) into formula (48), we obtain

ð Þ *n* 1 � �

<sup>1</sup> *,*0<sup>Φ</sup> <sup>¼</sup> *<sup>λ</sup>*0Φ*,* <sup>L</sup>*<sup>k</sup>*

*Proof:* It suffices to show that problem (24) does not have two linearly

First we observe that *ψ*ð Þ1 *χ*ð Þ1 6¼ 0. Indeed, if *ψ*ð Þ¼ 1 0, then we have

independent eigenvectors corresponding to *λ*<sup>0</sup> [18, 19]. Using contradiction argument we assume that for some *λ*<sup>0</sup> the boundary-value problem (25) with *k*<sup>2</sup> ¼ 0 has two linearly independent solutions *ψ* and *χ*. Each function satisfies the problem

*g* 2*P*�<sup>1</sup> *n*

2*P*�<sup>1</sup>

*<sup>n</sup> ; g* 2*P*�<sup>1</sup> *n*

the expression for *g*<sup>0</sup>

*Functional Calculus*

$$\int\_0^1 \left| \mathbf{g}'(\sigma) \right|^2 \mathbf{d}\sigma = \int\_0^1 \mathbf{g}'^{\prime\prime\prime}(\sigma) \overline{\mathbf{g}(\sigma)} \sigma = \lambda\_0^2 \int\_0^1 \left| \mathbf{g}(\sigma) \right|^2 \mathbf{d}\sigma. \tag{54}$$

Eq. (54) is valid if and only if *λ*<sup>2</sup> <sup>0</sup> . 0; however, for a deadbeat mode, *λ*<sup>2</sup> <sup>0</sup> , 0. The obtained contradiction means that for each double mode, there is one eigenfunction and one associate function. ■
