\r\n\tBoth diagnosis and clinical manipulation of the patient with vasospasm is a unique and challenging situation. Multi-clinical approach is extremely mandatory. The patient must be treated in a center, which requires a experienced team with both neurological surgeons, interventional radiologists, neurologists and neuroanesthesiologists. Moreover, a well-equiped, isolated neurointensive care is needed for all patients suffering form subarachnoid hemorraghe. \r\n\tIn their daily practice, both neurological surgeons, interventional radiologists, neurologists, neuroanesthesiologists, and even intensive care providers have to deal and challenge of vasospasm. Numerous studies relevant to pathophysiological mechanisms underlying vasospasm had been published, but we still know little about the exact mechanisms causing vasospasm. In the last decades of modern medical era, despite the technological developments concerning the neurological care of the patients with vasospasm, we still have no effective treatment and preventive care of this devastating entity. \r\n\tThe aim of this book project is to provide in detailed knowledge to both physicians and scientists dealing with cerebral vasospasm. This book will attract interest of both students, residents, specialists and academics of neurological sciences.
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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 non-dissipative feedback matrix [1, 2]. Depending on the boundary parameters k1 and k2, 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 initial-boundary 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 Lk1,k2, 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 Lk1,k2.
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 k1 and k2. Having in mind the practical applications of the asymptotic formulas [3, 4, 5], we discuss the case of k1≥0 and k2≥0, such that ∣k1∣+∣k2∣>0 (see Proposition 2). As shown in [2], even though the operator Lk1,k2 is non-dissipative, for the case k1>0 and k2=0 (or k1=0 and k2>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 Lk1,k2, 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 informulas (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 k1 is close to 1 (while k2=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 k1→1+.
We have also shown that there exists a sequence of values of the parameter k1, i.e., k1nn=1∞, such that for each k1=k1n there exist a finite number of deadbeat modes and each corresponds to a double eigenvalue of the dynamics generator Lk1,k2. For each value k1n, the operator Lk1,k2 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 k1n), there exists a mode shape and an associate mode shape. This fact indicates that for some values of k1 and k2, there exists a significant number of associate vectors of Lk1,k2. 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.
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, 11, 12] may be of interest for both analysts and practitioners.
This paper is organized as follows. In Section 1 we formulate the initial-boundary 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 Lk1,k2, which is a non-self-adjoint matrix differential operator depending on two parameters,k1andk2, is the main object of interest. The eigenvalues and the generalized eigenvectors of Lk1,k2 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, 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.
k1=0, k2=0.5, EI=1, ρ=0.1, N=64, εf=10−20
No.
Numerical
Analytic
No.
Numerical
Analytic
1.
−7988.1+237.00i
−7988.1+237.00i
18.
28.860+13.515i
29.453+14.812i
2.
−7020.6+222.19i
−7020.6+222.19i
19.
123.16+29.723i
123.08+29.625i
3.
−6115.5+207.37i
−6115.5+207.37i
20.
279.13+44.431i
279.14+44.437i
4.
−5272.8+192.56i
−5272.8+192.56i
21.
497.61+59.250i
497.61+59.250i
5.
−4492.5+177.75i
−4492.5+177.75i
22.
778.50+74.062i
778.50+74.062i
6.
−3774.7+162.94i
−3774.7+162.94i
23.
1121.8+88.875i
1121.8+88.875i
7.
−3119.3+148.12i
−3119.3+148.12i
24.
1527.6+103.69i
1527.6+103.69i
8.
−2526.3+133.31i
−2526.3+133.31i
25.
1995.7+118.50i
1995.7+118.50i
9.
−1995.7+118.50i
−1995.7+118.50i
26.
2526.3+133.31i
2526.3+133.31i
10.
−1527.6+103.69i
−1527.6+103.69i
27.
3119.3+148.12i
3119.3+148.12i
11.
−1121.8+88.875i
−1121.8+88.875i
28.
3774.7+162.94i
3774.7+162.94i
12.
−778.5+74.062i
−778.5+74.062i
29.
4492.5+177.75i
4492.5+177.75i
13.
−497.61+59.250i
−497.61+59.250i
30.
5272.8+192.56i
5272.8+192.56i
14.
−279.13+44.431i
−279.14+44.437i
31.
6115.5+207.37i
6115.5+207.37i
15.
−123.16+29.723i
−123.08+29.625i
32.
7020.6+222.19i
7020.6+222.19i
16.
−28.860+13.515i
−29.453+14.812i
33.
7988.1+237.00i
7988.1+237.00i
17.
−2.2⋅10−17+4.6007i
Table 1.
Approximations of the eigenvalues for the discrete and “continuous” operators (K>1).
k1=1.3, k2=1.2, EI=10, ρ=0.1, N=64, εf=10−20
No.
Numerical
Analytic
No.
Numerical
Analytic
1.
−25266−229.07i
−25266−229.07i
18.
99.467−19.816i
98.177−14.317i
2.
−22206−214.76i
−22206−214.76i
19.
394.17−28.149i
394.26−28.634i
3.
−19344−200.44i
−19344−200.44i
20.
887.75−42.983i
887.75−42.951i
4.
−16679−186.12i
−16679−186.12i
21.
1578.6−57.267i
1578.6−57.268i
5.
−14212−171.81i
−14212−171.81i
22.
2466.9−71.586i
2466.9−71.585i
6.
−11942−157.49i
−11942−157.49i
23.
3552.5−85.902i
3552.5−85.903i
7.
−9869.1−143.17i
−9869.1−143.17i
24.
4835.6−100.22i
4835.6−100.22i
8.
−7993.9−128.85i
−7993.9−128.85i
25.
6316.0−114.54i
6316.0−114.54i
9.
−6316.0−114.54i
−6316.0−114.54i
26.
7993.9−128.85i
7993.9−128.85i
10.
−4835.6−100.22i
−4835.6−100.22i
27.
9869.1−143.17i
9869.1−143.17i
11.
−3552.5−85.902i
−3552.5−85.903i
28.
11942−157.49i
11942−157.49i
12.
−2466.9−71.586i
−2466.9−71.585i
29.
14212−171.81i
14212−171.81i
13.
−1578.6−57.267i
−1578.6−57.268i
30.
16679−186.12i
16679−186.12i
14.
−887.75−42.983i
−887.75−42.951i
31.
19344−200.44i
19344−200.44i
15.
−394.17−28.149i
−394.26−28.634i
32.
22206−214.76i
22206−214.76i
16.
−99.467−19.816i
−98.177−14.317i
33.
25266−229.07i
25266−229.07i
17.
1.5⋅10−17+7.7256i
Table 2.
Approximations of the eigenvalues for the discrete and “continuous” operators (K<−1).
Figure 1.
Graphical representation of the eigenvalues of the discrete and “continuous” operators (K>1).
Figure 2.
Graphical representation of the eigenvalues of the discrete and “continuous” operators (K<−1).
k2=0, EI=1, ρ=1, N=64, εf=10−30
No.
k1=1+10−4
k1=1+10−7
k1=1+10−10
1.
−222.22+155.56i
−176.06+264.07i
−106.90+372.41i
2.
−133.38+124.45i
−87.723+211.27i
−2.4378⋅10−16+254.44i
3.
−64.540+93.396i
−3.9609⋅10−18+162.37i
−3.0717⋅10−17+123.66i
4.
−9.8081+58.559i
−5.7977⋅10−19+116.23i
−1.3012⋅10−17+4.9349i
5.
−7.7725⋅10−21+5.0488i
−6.3661⋅10−20+44.182i
−8.9232⋅10−18+44.421i
6.
4.9365⋅10−21+38.994i
−6.0066⋅10−20+4.9383i
8.9143⋅10−18+44.406i
7.
7.8007⋅10−21+4.8257i
6.0114⋅10−20+4.9313i
1.3012⋅10−17+4.9347i
8.
9.8081+58.559i
6.6801⋅10−20+44.651i
2.9907⋅10−17+123.09i
9.
64.540+93.396i
2.9381⋅10−18+139.20i
1.1286⋅10−16+234.06i
10.
133.38+124.45i
87.723+211.27i
3.2280⋅10−16+315.13i
11.
222.22+155.56i
176.06+264.07i
106.90+372.41i
Table 3.
Eigenvalues closest to the imaginary axis as k1→1+.
k2=0, EI=1, ρ=1, N=64, εf=10−30
#
k1=1−10−4
k1=1−10−7
k1=1−10−10
1.
−175.34+140.01i
−129.20+237.56i
−60.143+338.80i
2.
−96.394+108.84i
−50.206+186.90i
−8.6602+240.01i
3.
−36.896+78.778i
−8.0431+121.15i
−0.28608+123.37i
4.
−6.0769+42.171i
−0.23455+44.410i
−0.0074192+44.413i
5.
−0.11141+4.9324i
−0.0035253+4.9348i
−0.00011148+4.9348i
6.
0.11141+4.9324i
0.0035253+4.9348i
0.00011148+4.9348i
7.
6.0769+42.171i
0.23455+44.410i
0.0074192+44.413i
8.
36.896+78.778i
8.0431+121.15i
0.28608+123.37i
9.
96.394+108.84i
50.206+186.90i
8.6602+240.01i
10.
175.34+140.01i
129.20+237.56i
60.143+338.80i
Table 4.
Eigenvalues closest to the imaginary axis as k1→1−.
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]
12∫01ϱxAxht2xt−ExIxhxx2xtdx,E1
where hxt is the transverse deflection, Ex is the modulus of elasticity, Ix is the area moment of inertia, ϱx is the linear density, and Ax is the cross-sectional area of the beam.
Assuming that the beam is clamped at the left end x=0 and free at the right end x=1, and applying Hamilton’s variational principle to the action functional defined by (1), we obtain the equation of motion
ϱxAxhttxt+ExIxhxxxtxx=0,0≤x≤1,t>0,E2
and the boundary conditions
h0t=hx0t=0andM1t=Q1t=0,E3
where Mxt and Qxt are the moment and the shear, respectively [10]:
Mxt=ExIxhxxxtandQxt=Mxxt.E4
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
Ut=−Q1tM1tTandYt=ht1thxt1tT,E5
where T stands for transposition. The feedback control law is given by
Ut=KYt,E6
where K is the 2×2 feedback matrix. We select
K=codiag−k2−k1,k1,k2≥0,E7
with k1,k2 being the control parameters. The feedback (6) can be written as
Finally, we arrive to the following initial-boundary value problem: the equation of motion (2), the boundary conditions (3), and the standard initial conditions hx0=h0x,htx0=h1x.
Notice that the choice of a feedback matrix K defines whether the system is dissipative or not. Indeed, let Et be the energy of the system, defined by representation (1). Evaluating Ett on the solutions of Eq. (2) satisfying the left-end conditions from Eqs. (3), we obtain
Thus the system is not dissipative for all nonnegative values of k1 and k2.
2. Operator form of the problem
In what follows, we incorporate the cross-sectional area Ax into the density, write ρx instead of ϱxAx, and also abbreviate EIx≡ExIx. Let H be the Hilbert space of two-component vector functions Ux=u0xu1xT equipped with the following norm:
UH2=12∫01EIxu0″x2+ρxu1x2x.E12
Assuming that EI,ρ∈C201 are positive functions, we obtain that the closure of smooth functions Ux=u0xu1xT satisfying u00=u0′0=0 will produce the energy space H=H0201×L201. Here H0201=u∈H201:u0=u′0=0, 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:
Utxt=iLk1,k2UxtandUx0=u0xu1xT,E13
where 0≥x≥1,t≥0. The dynamics generator Lk1,k2 is given by the following matrix differential expression:
For any k1k2∈R2, the adjoint operator Lk1,k2∗ [13] is given by
Lk1,k2∗=L−k2,−k1,E16
i.e., Lk1,k2∗ is defined by the same differential expression (14) on the domain described in Eq. (15), where k1 and k2 are replaced by −k2 and −k1, respectively. It follows from Eq. (16) that L0,0 is self-adjoint in H and thus L0,0 is the dynamics generator of the clamped-free beam model. For the reader’s convenience, we summarize the properties of Lk1,k2 from [1, 2] needed for the present work.
Proposition 1:
Lk1,k2 is an unbounded operator with compact resolvent, whose spectrum consists of a countable set of normal eigenvalues (i.e., isolated eigenvalues, each of finite algebraic multiplicity [6, 13]).
For each k1k2∈R2,∣k1∣+∣k2∣>0, the operator Lk1,k2 is a rank-two perturbation of the self-adjoint operator L0,0 in the sense that the operators Lk1,k2−1 and L0,0−1 exist and are related by the rule
Lk1,k2−1=L0,0−1+Tk1,k2,E17
where Tk1,k2 is a rank-two operator. A similar decomposition is valid for the adjoint operator, i.e.,
Lk1,k2−1∗=L0,0−1+Tk1,k2∗,Tk1,k2∗=T−k2,−k1.E18
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≠EIρ. Let
K=k1+k2A−k1k2/A,A=EIρ,and∣K∣≠1.E19
The following asymptotic approximations for the eigenvalues λn (as ∣n∣→∞) of the operator Lk1,k2 hold:
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 λn with ∣n∣≥N+1 satisfy the estimate
for the case when 0<∣K∣<1. The following important question holds: From which indexNcan 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, 15, 16], as outlined in Section 5.
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=64 is the number of grid points on 01, and εf 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 k1,k2-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).
Figure 3.
Regions of K on the k1,k2-plane, A=EIρ=1.
3. The deadbeat modes
An eigenvalue λn of the dynamics generator Lk1,k2 is called a deadbeat mode if λn=iβn,βn>0. If the corresponding eigenfunction is Φnx, then the evolution problem (13) has a solution given in the form eiλntΦnx=e−βntΦnx, 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 Lk1,k2Φx=λΦx in the form of an equivalent problem for an operator pencil [17] as
If λn and φnx are an eigenvalue and eigenfunction of the pencil (24), then λn is also an eigenvalue of Lk1,k2 with the eigenfunction Φnx=1iλnφnxφnxT.
To solve problem (24), we first redefine the spectral and control parameters to eliminate ρ and EI from Eq. (24). We define λ˜,k˜1, and k˜2 by λ=EI/ρλ˜ and k˜j=EIρkj, j=1,2. Substituting these relations into Eq. (24) and eliminating the “tilde,” we obtain the following Sturm-Liouville eigenvalue problem:
It can be readily seen that if 0<k1<1, then 2+1+k1cos2x>0 and 1−k1cosh2x>0, which means that Eq. (32) has no solutions. Statement (1) is shown. Statement (2) follows immediately if one considers Eq. (32) for k1=1.
To prove Statement (3), we rewrite Eq. (32) in the form
cosh2x=2k1−1+1+2k1−1cos2x,x>0,k1>1.E33
The left-hand side of Eq. (33) is monotonically increasing, while the right-hand side is sinusoidal, with maximum 1+4k1−1 and minimum −1, and period π. So the graphs of the left- and right-hand side have intersections only on the interval 012cosh−11+4k1−1. There are two intersections for each full period of the 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 k1, the two graphs can have two intersections, one tangential intersection or no intersections on the second half-period. This leads to estimate (31). ■
Figure 4.
Left- and right-hand side of Eq. (33) for different values of k1.
A graphical illustration of the result of Theorem 1 is shown in Figure 5.
Figure 5.
Estimates and actual count of deadbeat modes based on numerical simulations.
4. Structure of the deadbeat mode set
The main result on the existence and distribution of double roots of the function Hxk1 is presented in the statement below.
Theorem 2: For a given k1>1, the multiplicity of each root of Hxk1 does not exceed 2. There exists a sequence k1nn=0,1,2…, such that the function Hxk1 has a double root if and only if k1=k1n for some n. So the original spectral problem with k1=k1n,k2=0 has a double deadbeat mode λn=μn2=2ixn2. The following asymptotic formulas hold
xn=3π4+πn+Pn−1+OPn−2,wherePn=exp3π2+2πn,E34
and
k1n=1+4Pn−1+OPn−2.E35
Proof: If x is a double root of H, then Hxk1=H′xk1=0, i.e., separating the real and imaginary parts, we have
2+k1+1cos2x−k1−1cosh2x=0,E36
k1+1sin2x+k1−1sinh2x=0.E37
Eliminating k1 from system given by (36) and (37), we obtain that the following equation has to be satisfied:
Gx≡1+cos2xsinh2x+1+cosh2xsin2x=0,x>0.E38
Rewriting Eq. (37) in the form k1+1sin2x=−k1−1sinh2x, and taking into account that k1>1, we obtain that if x is the solution of Eq. (38), then sin2x<0.
Now we show that when cos2x<0 and sin2x<0, i.e.,
π2n+1<2x<3π2+2πn,n∈0,1,2…,
Eq. (38) does not have any solutions. Indeed, in the above range of x, we have cos2x+sin2x=2sin2x+π/4<−1 and ∣sin2x−cos2x∣<1. With such estimates we obtain that
Now we consider the case when cos2x>0 and sin2x<0, i.e.,
3π2+2πn<2x<2πn+1,n∈0,1,2….
It is convenient to rewrite system given by (36) and (37) in the form 2x=3π/2+2πn+s, where n∈0,1,2…, 0<s<π/2. If gs≡G3π/4+πn+s, then Eq. (38) generates the following equation for g:
gs≡1+sinssinh3π2+2πn+s−1+cosh3π2+2πn+scoss=0.E40
Let us show that for each n, Eq. (40) has a unique solution. For s=0 we have
g0=sinh3π2+2πn−1−cosh3π2+2πn<0,
and for s=π/2 we have gπ/2=2sinh2πn+1>0. Evaluating g′ we have
g′s=sins+1+2sinscosh3π2+2πn+s>0.E41
Thus gs is a monotonically increasing function, such that g0<0<gπ/2, which means that g has a unique root on 0π/2.
Finally we show that the multiplicity of a multiple root cannot exceed 2. Using a contradiction argument, assume that there exists x0, such that in addition to Eqs. (36) and (37), one has H′′x0k1=0, i.e., the multiplicity of x0 is at least 3. The system H′x0k1=0 and H′′x0k1=0 can be written as
Since k1>1, the second equation of (42) yields cos2x0<0. Also, since x0 is a multiple root, we must have sin2x0<0. Then 2x0 is in the third quadrant, which means that Gx0≠0, as we have seen above. This contradicts our assumption that x0 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:
Representation (44) implies that there exists n0, such that for all n≥n0, we have g2Pn−1>0. Taking into account that g0<0, we obtain that the root sn, n≥n0, of the function gs is located on the interval 02Pn−1. To find the location of this root more precisely [18], we use linear interpolation. Namely, substituting Eq. (43) into the expression for g′s from Eq. (41) yields
g′2Pn−1=Pn2+O1.E45
Replacing gs by the linear function tangential to gs at the point 2Pn−1g2Pn−1, and finding the root of this function, we get
sn=2Pn−1−g2Pn−1g′2Pn−1+OPn−2=2Pn−1+OPn−2.E46
Having this approximation for sn, we immediately get
xn=3π4+πn+sn2=3π4+πn+Pn−1+OPn−2.E47
From the equation Hxnk1n=0, we obtain the formula for k1n as
Corollary 1: Let k1=k1n for some n∈ℕ+∪0, and let xn be the corresponding double root of the function Hxk1n. Then λ0=2ixn2 is an eigenvalue of the operator Lk1n,0′, such that the geometric multiplicity of λ0 is 1 and its algebraic multiplicity is 2. Therefore there exists a unique eigenvector Φ and one associate vector Ψ, such that
Lk1n,0Φ=λ0Φ,Lk1n,0Ψ−λ0Ψ=Φ.E49
Proof: It suffices to show that problem (24) does not have two linearly independent eigenvectors corresponding to λ0 [18, 19]. Using contradiction argument we assume that for some λ0 the boundary-value problem (25) with k2=0 has two linearly independent solutions ψ and χ. Each function satisfies the problem
φ′′′′x=λ02φx,φ0=φ′0=0,φ′′′1=0,φ′′1=iλ0k1φ1.E50
First we observe that ψ1χ1≠0. Indeed, if ψ1=0, then we have
∫01ψ′′σ2dσ=∫01ψ′′′′σψσ¯dσ=λ02∫01ψσ2dσ.E51
Since λ0 is purely imaginary, Eq. (51) is not valid. We define a new function:
gx=ψx−ψ1χ1χx.E52
One can readily check that g satisfies the following boundary-value problem:
φ′′′′x=λ02φx,φ0=φ′0=0,φ1=φ′′1=0,E53
and therefore
∫01g′′σ2dσ=∫01g′′′′σgσ¯σ=λ02∫01gσ2dσ.E54
Eq. (54) is valid if and only if λ02>0; however, for a deadbeat mode, λ02<0. The obtained contradiction means that for each double mode, there is one eigenfunction and one associate function.■
4.1 Deadbeat mode behavior as k1→1
As k1 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 k1 approaches 1 from above or below.
Figure 6.
Eigenvalues with ∣Reλ∣>10 as k1→1+.
Figure 7.
Eigenvalues with ∣Reλ∣>10 as k1→1−.
As follows from Table 3, the real parts of the eigenvalues decrease steadily as k1→1+, to a point where the eigenvalue becomes a deadbeat mode. An increase in the number of deadbeat modes can be seen as k1→1+, 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 k1→1+. (Compare modes no.5 and no.7 for ∣k1−1∣=10−4, modes no.4 and no.7 for ∣k1−1∣=10−6, modes no.5 and no.8 for ∣k1−1∣=10−8, and modes no.4 and no.7 for ∣k1−1∣=10−10). Such behavior indicates convergence of the two simple deadbeat modes to a double mode, which is consistent with Theorem 2.
Analyzing Table 4, one can see that the eigenvalues get closer to the imaginary axis as k1→1−. However the rate at which their real parts approach zero is significantly lower than in the case k1→1+. Even at k1=1−10−10, the eigenvalue closest to the imaginary axis has a real part of about 10−4, which means that it is not a deadbeat mode (see Statement (1) of Theorem 1).
The eigenvalues near the imaginary axis approach the same double deadbeat modes in both cases when k1→1± (see Statement (2) of Theorem 1). In conclusion, one can claim that the eigenvalues are indeed approaching the imaginary axis; however, the rate of this approach is different for k1→1− and k1→1+. In the former case, an eigenvalue’s distance from the imaginary axis decreases very slowly; in the latter case, the eigenvalues quickly “jump” on the imaginary axis and turn into deadbeat modes.
5. Outline of the numerical scheme
To carry out the numerical analysis of the differential operator Lk1,k2, we use the Chebyshev collocation method and cardinal functions [14, 15, 16].
Recall that the Nth Chebyshev polynomial of the first kind is defined by
TNξ=cosNθ,−1≤ξ≤1whereξ=cosθ,θ∈0π.E55
The cardinal functions, ψkξ, and the Chebyshev-Gauss-Lobatto (CGL) grid points ξk are defined as follows:
where coefficients ck are such that c1=cN=2 and ck=1 for 1<k<N. The main property of cardinal functions is ψkξj=δkj (using the Kronecker delta). The family ψkk=1N forms a basis in the space of polynomials of degree N−1, i.e., if f is such polynomial, then f and f′ can be written in the forms
fξ=∑k=1Nfξkψkξandf′ξ=∑k=1Nf′ξkψkξ.E57
If f=fξ1fξ2…fξNT and g=f′ξ1f′ξ2…f′ξNT, then g=Df, where D is the Chebyshev derivative matrix with the elements
We approximate the action of Lk1,k2 on the finite-dimensional subspace HN⊂H of polynomials of degree at most N−1. Using the CGL grid and the cardinal functions, we substitute for u0 and u1 their truncated expansions:
u0ξ≈∑k=1NΦkψkξ,Φk=u0ξk,u1ξ≈∑k=1NΘkψkξ,Θk=u1ξk.E62
Let Φ and Θ be N-dim vectors and Ψ be a 2N-dim vector defined by
Φ=Φ1Φ2…ΦNT,Θ=Θ1Θ2…ΘNT,Ψ=ΦΘ.E63
Let L be the finite-dimensional approximation of the differential operator Lk1,k2. The discretized operator L induced by Lk1,k2 can be given by
L=−i0IN×N−16EIρD40,E64
where IN×N is the N×N identity matrix and D is the derivative matrix (58).
5.2 Incorporating the boundary conditions
Discretization of the boundary conditions in the domain description (60) yields
K is called the boundary operator. Let KN be the kernel of K, i.e., KN=v∈R2N:Kv=0. We have to identify all eigenvalues of the operator L, when its domain is restricted to KN. It is clear that KN is isomorphic to Rk with k≡dimKN=dimHN−rankK=2N−6. Let B be the matrix consisting of column vectors that form an orthonormal basis in KN. It is clear that BTB is the identity matrix on Rk 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 HN. Assume now that λ is an eigenvalue of BTLB with corresponding eigenvector v∈Rk. If Ψ=Bv, we have
BBTLΨ=BBTLBv=λBv=λΨ,E68
but BBTL≠L, which indicates that fake eigenvalues may exist.
5.3 Filtering of spurious eigenvalues
In order to decide which eigenvalues of BTLB should be discarded, we impose the following condition. Let Λ be the spectrum of BTLB and V be the set of its eigenfunctions. We construct the set of “trusted” eigenvalues [14, 15], for some εf>0 filtering precision, as
where ⋅C is a discrete approximation to the integral norm defined in Eq. (61). (The subscript C is short for Chebyshev). Using the CGL quadrature, we obtain the following formula for the norm of a vector Ψ defined as in Eq. (63):
ΨC=π/4N−1∑k=1N1−ξk216EIξkD2Φk2+ρξkΘk2.
6. Conclusions
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, k1, is such that k1→1+ and the other one k2=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 k1nn=1∞ of the values of parameter k1, such that the corresponding deadbeat mode has a multiplicity 2, which yields the existence of the associate mode shapes for the operator Lk1,k2. The formulas for the double deadbeat modes and asymptotics for the sequence k1n as n→∞ have been derived.
Acknowledgments
Partial support of the National Science Foundation award DMS-1810826 is highly appreciated by the first author.
\n',keywords:"matrix differential operator, eigenvalues, Chebyshev polynomials, numerical scheme, boundary control",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/67749.pdf",chapterXML:"https://mts.intechopen.com/source/xml/67749.xml",downloadPdfUrl:"/chapter/pdf-download/67749",previewPdfUrl:"/chapter/pdf-preview/67749",totalDownloads:71,totalViews:0,totalCrossrefCites:0,dateSubmitted:"May 8th 2019",dateReviewed:"May 20th 2019",datePrePublished:"June 20th 2019",datePublished:null,readingETA:"0",abstract:"Analytic and numerical results of the Euler-Bernoulli beam model with a two-parameter family of boundary conditions have been presented. The co-diagonal matrix depending on two control parameters (k1 and k2) 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, k1, is positive and another one, k2, is zero. It has been shown that the number of the deadbeat modes tends to infinity, as k1→1+ and k2=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.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/67749",risUrl:"/chapter/ris/67749",signatures:"Marianna A. Shubov and Laszlo P. Kindrat",book:{id:"7785",title:"Matrix Calculus",subtitle:null,fullTitle:"Matrix Calculus",slug:null,publishedDate:null,bookSignature:"Dr. Kamal Shah",coverURL:"https://cdn.intechopen.com/books/images_new/7785.jpg",licenceType:"CC BY 3.0",editedByType:null,editors:[{id:"231748",title:"Dr.",name:"Kamal",middleName:null,surname:"Shah",slug:"kamal-shah",fullName:"Kamal Shah"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_1_2",title:"1.1 The initial-boundary value problem for the Euler-Bernoulli beam model of a unit length",level:"2"},{id:"sec_3",title:"2. Operator form of the problem",level:"1"},{id:"sec_4",title:"3. The deadbeat modes",level:"1"},{id:"sec_5",title:"4. Structure of the deadbeat mode set",level:"1"},{id:"sec_5_2",title:"4.1 Deadbeat mode behavior as k1→1",level:"2"},{id:"sec_7",title:"5. Outline of the numerical scheme",level:"1"},{id:"sec_7_2",title:"5.1 Discretization of Lk1,k2",level:"2"},{id:"sec_8_2",title:"5.2 Incorporating the boundary conditions",level:"2"},{id:"sec_9_2",title:"5.3 Filtering of spurious eigenvalues",level:"2"},{id:"sec_11",title:"6. Conclusions",level:"1"},{id:"sec_12",title:"Acknowledgments",level:"1"}],chapterReferences:[{id:"B1",body:'Shubov MA, Kindrat LP. Spectral analysis of the Euler–Bernoulli beam model with fully nonconservative feedback matrix. Mathematicsl Methods in the Applied Sciences. 2018;41(12):4691-4713. DOI: 10.1002/mma.4922'},{id:"B2",body:'Shubov MA, Shubov VI. Stability of a flexible structure with destabilizing boundary conditions. Proceedings of the Royal Society A. 2016;472(2191):20160109. DOI: 10.1098/rspa.2016.0109'},{id:"B3",body:'Shubov MA. Exact controllability of nonselfadjoint Euler-Bernoulli beam model via spectral decomposition method. IMA Journal of Mathematical Control and Information. 2008;25(2):185-203. DOI: 10.1093/imamci/dnm018'},{id:"B4",body:'Guiver CH, Opmeer MR. Non-dissipative boundary feedback for Rayleigh and Timoshenko beams. Systems and Control Letters. 2010;59(9):578-586. DOI: 10.1016/j.sysconle.2010.07.002'},{id:"B5",body:'Shubov MA. Exact controllability of coupled Euler-Bernoulli and Timoshenko beam model. IMA Journal of Mathematical Control and Information. 2006;23(3):279-300. DOI: 10.1093/imamci/dnm059'},{id:"B6",body:'Curtain RF, Zwart HJ. An Introduction to Infinite Dimensional Linear Systems Theory. New York, NY: Springer; 1995'},{id:"B7",body:'Dowell EH, editor. A Modern Course on Aeroelasticity. 4th Revised ed. Boston, MA: Klumer Academic Publishers; 2004'},{id:"B8",body:'Balakrishnan AV, Shubov MA. Asymptotic behaviour of aeroelastic modes for aircraft wing model in subsonic air flow. Proceedings of the Royal Society A. 2004;460(2044):1057-1091. DOI: 10.1098/rspa.2003.1217'},{id:"B9",body:'Shubov MA, Rojas-Araneza M. Four-branch vibrational spectrum of double-walled carbon nanotube model. Proceedings of the Royal Society A. 2011;467(2125):99-126. DOI: 10.1098/rspa.2010.0100'},{id:"B10",body:'Benaroya H. Mechanical Vibration: Analysis, Uncertainties, and Control. Upper Saddle River, NJ: Prentice Hall; 1998'},{id:"B11",body:'Han SM, Benaroya H, Wei T. Dynamics of transversely vibrating beams using four engineering theories. Journal of Sound and Vibration. 1999;225(5):935-988. DOI: 10.1006/jsvi.1999.2257'},{id:"B12",body:'Shubov MA. Generation of Gevrey class semigroup by non-selfadjoint Euler-Bernoulli beam model. Mathematicsl Methods in the Applied Sciences. 2006;29(18):2181-2199. DOI: 10.1002/mma.768'},{id:"B13",body:'Gohberg IT, Krein MG. Introduction to the Theory of Nonselfadjoint Operators in Hilbert Space. Translations of Mathematical Monographs No. 18. Providence, RI: AMS; 1996'},{id:"B14",body:'Mason JC, Handscomb DC. Chebyshev Polynomials. Boca Raton, FL: Chapman & Hall/CRC; 2003'},{id:"B15",body:'Trefethen LN, Bau D III. Numerical Linear Algebra. Philadelphia, PA: SIAM; 1997'},{id:"B16",body:'Shubov MA, Wineberg S, Holt R. Numerical investigation of aeroelastic mode distribution for aircraft wing model in subsonic air flow. Mathematical Problems in Engineering. 2010;2010:879519. DOI: 10.1155/2010/879519'},{id:"B17",body:'Marcus AS. Introduction to the Spectral Theory of Polynomial Operator Pencils. Translations of Mathematical Monographs No. 71. Providence, RI: AMS; 1988'},{id:"B18",body:'Fedoryuk MV. Asymptotic Analysis. Berlin, Heidelberg: Springer-Verlag; 1993'},{id:"B19",body:'Naimark MA. Linear Differential Operators. New York, NY: Ungar Pub Co; 1967'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Marianna A. Shubov",address:"marianna.shubov@gmail.com",affiliation:'
University of New Hampshire, Durham, NH, USA
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