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Constructing rods with given natural frequencies
Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
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Constructing rods with given natural frequencies Antonino Morassi n Università degli Studi di Udine, Dipartimento di Ingegneria Civile e Architettura, via Cotonificio 114, 33100 Udine, Italy
a r t i c l e i n f o
abstract
Article history: Received 20 December 2012 Received in revised form 4 April 2013 Accepted 21 April 2013
In this paper we present a new method for constructing axially vibrating rods having prescribed values of the first N natural frequencies, under a given set of boundary conditions. The analysis is based on the determination of the so-called quasi-isospectral rods, that is rods which have the same spectrum as a given rod, with the exception of a single eigenvalue which is free to move in a prescribed interval. The reconstruction procedure needs the specification of an initial rod whose eigenvalues must be close to the assigned eigenvalues. The rods and their normal modes can be constructed explicitly by means of closed-form expressions. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Vibrating rods Eigenvalues Structural identification Inverse problems
1. Introduction The construction of a system having prescribed values of natural frequencies, usually the lower frequencies, is an important issue in many engineering contexts. A key problem in Structural Identification, for example, consists in determining either the stiffness or the inertia coefficients so that the first analytical eigenvalues coincide with the corresponding values determined in experimental tests [1–3]. Other applications concern the construction of vibrating systems such that possible occurrence of resonance phenomena with excitation frequencies is avoided. The assignation of certain chosen poles of a vibrating system while other poles (usually at higher frequencies) remain unchanged is an important issue also in eigenvalue assignment problems in vibration by passive modifications or by active control procedures [4,5]. The reconstruction of a vibrating system with a given set of natural frequencies via structural identification methods is usually based on optimization techniques of variational type, [6–10]. A function which measures the error between the assigned and analytical frequency values is minimized via gradient-type methods and, therefore, the coefficients of a starting configuration of the system are iteratively updated, under some a priori assumptions on the coefficients to be identified (symmetry assumptions, lower and upper bounds, etc.). The numerical problem is highly non-linear and the identification algorithms can be very time consuming, see, for example, [11,12]. In this paper we show how to construct longitudinally vibrating rods having prescribed values of the first N natural frequencies, under a given set of boundary conditions. The rods and their normal modes can be constructed explicitly by means of closed-form expressions. The reconstruction procedure needs the specification of an initial rod whose eigenvalues must be close to the assigned eigenvalues. The analysis is based on the determination of the so-called quasi-isospectral rods, that is rods which have the same spectrum as a given rod, with the exception of a single eigenvalue which is free to move in a prescribed interval. The reconstruction is obviously not unique, since classical results show that two full spectra
n
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0888-3270/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ymssp.2013.04.010
Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
2
A. Morassi / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
corresponding to two specific sets of boundary conditions are required, in general, to determine the shape of a rod uniquely [13]. With a view to practical applications, this intrinsic non-uniqueness could be useful in engineering design in order to formulate and solve structural optimization problems, such as, for example, the determination of rods with maximal global stiffness or the determination of rods having minimal mass. The main ideas and mathematical tools of the reconstruction method are presented, for the sake of simplicity, for rods under cantilever end conditions. The analysis of this case allows for a clear presentation of the key aspects of the method. Extensions to rods under other end conditions and generalizations are discussed in the last section of the paper.
2. Main ideas of the reconstruction procedure The free (undamped, infinitesimal) longitudinal vibrations u(x) of frequency ω, of a thin straight rod of unit length under Supported-Elastically Restrained (S-ER) end conditions are governed by the Sturm–Liouville equation 8 b b > > < ðAðxÞu′ðxÞÞ′ þ λAðxÞuðxÞ ¼ 0; uð0Þ ¼ 0; > > : Að1Þu′ð1Þ b b uð1Þ ¼ 0; þK
x∈ð0; 1Þ;
ðaÞ
ðS ¼ SupportedÞ
ðbÞ
ðER ¼ Elastically RestrainedÞ
ðcÞ
ð2:1Þ
b where λ ¼ γω2 =E and ðÞ′≡dðÞ=dx. Here E is Young's modulus, γ is the volume mass density, both assumed constant; AðxÞ is the cross-sectional area at section x. Condition (2.1c) means that the end x¼ 1 is attached to a fixed support by means of an b, K b ≥0. The case of K b ¼ 0 corresponds to a cantilever rod. elastic spring having stiffness K b We shall assume throughout that AðxÞ is a strictly positive, twice continuously differentiable function of x in ½0; 1. It is b well known that for such AðxÞ, and end conditions (2.1b)–(2.1c), there is an infinite sequence fλm g∞ m ¼ 1 of eigenvalues (or spectrum), with 0 o λ1 o λ2 o⋯; limm-∞ λm ¼ ∞, for which (2.1a)–(2.1c) has a non-trivial solution u(x). Let n≥1 be given. The key step of our method is the explicit construction of a new rod quasi-isospectral to the given rod, b that is a rod with cross-sectional profile A(x) having the same eigenvalues as the given rod AðxÞ under (S-ER) end conditions, with the exception of the nth eigenvalue. In fact, by keeping fixed all the eigenvalues λm with m≠n and moving the nth eigenvalue λn to the desired value, say λ~ n , and using repeatedly the procedure, after N steps we will produce a rod with the N first N given eigenvalues fλ~ m gm ¼ 1 , and the construction is finished. We will see in the following that the reconstruction procedure works only if the rod to be determined is not far (in a way that can be rigorously specified) from the initial guess rod. Moreover, even if the initial rod is fixed once and for all, it is evident that the construction is not unique, since the flow from the initial rod depends also on the particular order chosen to move every individual eigenvalue to the corresponding target value. b The main steps of our construction of rods A(x) quasi-isospectral to a given rod AðxÞ, under (S-ER) end conditions, are the bðxÞ by a standard Sturm–Liouville following. First, Eq. (2.1a) is reduced to canonical form with a Schrödinger potential q transformation (see Section 3). Second, a Darboux Lemma (Section 4) is used to construct explicit families of Schrödinger bðxÞ (Section 5). Third, the Darboux Lemma is applied once more in potentials q(x) quasi-isospectral to the initial potential q iterate form to determine rods A(x) corresponding to the quasi-isospectral potentials q(x) and, ultimately, to find rods A(x) b quasi-isospectral to the initial rod AðxÞ (Section 6). The second step of the above procedure is fully described in a series of papers by Trubowitz and colleagues for Sturm– Liouville operators in canonical form and square-summable potentials, under either Dirichlet end-conditions (corresponding to the (S–S) case for rods, [14]) or Robin end conditions (corresponding to the (ER–ER) case for rods, [15]). The case corresponding to (S-ER) end conditions is only briefly sketched in [16] (Lemma 1) and, therefore, our original contribution lies also on the explicit determination of the quasi-isospectral potentials for this set of boundary conditions. For the sake of completeness, we recall that the problem of constructing quasi-isospectral rods under Dirichlet or Neumann (e.g., u′ð0Þ ¼ u′ð1Þ ¼ 0) boundary conditions has been solved by Coleman and McLaughlin [17] by using a method different from ours, which extends the results obtained by Trubowitz and co-workers to Eq. (2.1a) with less regular coefficients (e.g., pffiffiffiffi pffiffiffiffi b Þ′= A b ∈L2 ð0; 1Þ). ð A Finally, it is appropriate to compare the results of this paper and those obtained in the analysis of the so-called isospectral rods, that is rods having exactly the same full spectrum of a given rod. In 1995, Ram and Elhay [18] have shown that for a given non-uniform rod there is a dual rod which has the same infinite spectrum. Later on, Gladwell and Morassi [19] showed how to construct numerable families of isospectral rods, see also the paper by Coleman and McLaughlin [17] for more general treatment applied to Dirichlet or Neumann end conditions. Here, instead, we show how to construct a rod with given values of a finite number of natural frequencies. As it was stated above, the crucial step is the ability to construct quasi-isospectral rods, that is rods having all the eigenvalues in common with the exception of a single eigenvalue. Although the Darboux Lemma is the main mathematical tool of both the present analysis and that developed in [19], the results found here are clearly different and, in some respects, are more general than those obtained in the mentioned paper. In fact, the procedure for the construction of isospectral rods proposed in [19] cannot be used to construct rods having prescribed values of a finite set of natural frequencies. Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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3. Reduction to normal form In Eq. (2.1a) put b ¼a b2 ðxÞ; AðxÞ
bðxÞuðxÞ: yðxÞ ¼ a
ð3:1Þ
Then, Eq. (2.1a) reduces to the Sturm–Liouville canonical form bðxÞyðxÞ ¼ λyðxÞ; −y″ðxÞ þ q
ð3:2Þ
x∈ð0; 1Þ;
where bðxÞ ¼ q
b″ðxÞ a bðxÞ a
ð3:3Þ
is a continuous function in ½0; 1. b or a b, there is a unique q b, but for a given q b there are many a b. In particular, if a b0 is one a b Note that for a given A b, then all the coefficients a b satisfying (3.3) are given by corresponding to a given q ! Z x ds b¼a b 0 c1 þ c2 ; ð3:4Þ a b20 ðsÞ 0 a b is one-sign – say, positive – in ½0; 1. where the constants c1, c2 must be chosen such that a
4. The Darboux Lemma We now introduce the key mathematical tool of our analysis: the so-called Darboux Lemma. Lemma 1 (Darboux Lemma [20]). Let μ be a real number, and suppose g≡gðxÞ is a non-trivial solution of the Sturm–Liouville equation bg ¼ μg −g″ þ q
ð4:1Þ
b≡q bðxÞ. Let λ be a real number. If f is a non-trivial solution of with potential q bf ¼ λf −f ″ þ q
ð4:2Þ
and λ≠μ, then y¼
1 1 ½g; f ≡ ðgf ′−g′f Þ g g
ð4:3Þ
is a non-trivial solution of the Sturm–Liouville equation −y″ þ q ̌y ¼ λy;
ð4:4Þ
where b−2 q̌ ¼ q
2
d
2
dx
lnðgðxÞÞ:
ð4:5Þ
Moreover, the general solution of the equation −y″ þ q ̌y ¼ μy
ð4:6Þ
is y¼
Z x 1 b1 þ b2 g 2 ðsÞ ds ; g 0
where b1 and b2 are arbitrary constants. In particular, y ¼
ð4:7Þ 1 g
is a solution of (4.6).
This lemma, which can be verified by direct calculation (see [19]), allows to associate to every equation of the form (4.1), that one knows how to integrate for all values of μ, another equation of the same form that one also knows how to integrate for all the values of the parameter μ. In particular, the addition law of the logarithm makes iteration of Lemma 1 simple, as it will be shown in the sequel. It should be noted that if g vanishes in ½0; 1, then Eq. (4.4) is understood to hold between the roots of g. We will see that these singular situations disappear by applying the Darboux Lemma twice, as we shall describe in the following section. Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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5. Quasi-isospectral potentials under (S-ER) end conditions b ¼ AðxÞ b Suppose that the left end of the original rod A is supported, while the right end is elastically restrained by means b , 0≤K b o ∞. The eigenvalue problem written in Sturm–Liouville canonical form is of an elastic spring with stiffness K 8 by ¼ λy; x∈ð0; 1Þ; ðaÞ −y″ þ q > > > < yð0Þ ¼ 0; ðbÞ ð5:1Þ yð1Þ > > b b; yð1Þ þ K > ¼ 0; ðcÞ : ½a bð1Þ a b¼a b″=a b, or equivalently with q 8 by ¼ λy; x∈ð0; 1Þ; −y″ þ q > < yð0Þ ¼ 0; > : b y′ð1Þ þ Hyð1Þ ¼ 0;
ðaÞ ðbÞ
ð5:2Þ
ðcÞ
where b b b ¼ − a ′ð1Þ þ K : H 2 bð1Þ a a b ð1Þ
ð5:3Þ
b of Eqs. (5.2a)–(5.2c) are denoted by fðλm ; zm Þg∞ , with 0 o λ1 o λ2 o ⋯; limm-∞ λm ¼ ∞. b; ∞; Hg The eigenvalues of the problem fq m¼1 Let n, n≥1, be a given integer. We want to construct families of Sturm–Liouville problems fqðxÞ; ∞; Hg which share the b with the exception of the nth eigenvalue which is free to move in a prescribed bðxÞ; ∞; Hg, same eigenvalues of the problem fq interval. The strategy of the proof is to use appropriately the Darboux Lemma in an iterated form. It is useful to introduce some notations. Let t∈R be such that λn−1 o λn þ t o λnþ1 ;
ð5:4Þ
b; λn þ tÞ, η2 ¼ η2 ðx; q b; λn þ tÞ the solutions to the initial value problems with λ0 ¼ 0. Denote by η1 ¼ η1 ðx; q 8 b > < −η″1 þ q η1 ¼ ðλn þ tÞη1 ; x∈ð0; 1Þ; ðaÞ ðbÞ η1 ð0Þ ¼ −1; > : η′ ð0Þ ¼ 0; ðcÞ 1
ð5:5Þ
and 8 b > < −η″2 þ q η2 ¼ ðλn þ tÞη2 ; η2 ð0Þ ¼ 0; > : η′ ð0Þ ¼ −1:
x∈ð0; 1Þ;
ðaÞ ðbÞ
ð5:6Þ
ðcÞ
2
Moreover, we introduce the function b; λn Þ; η02 ¼ η2 ðx; q
ð5:7Þ
which is proportional to the eigenfunction associated to λn . Let b; λn þ tÞ wn;t ¼ wn;t ðx; q
ð5:8Þ
be a solution of bwn;t ¼ ðλn þ tÞwn;t ; −w″n;t þ q
x∈ð0; 1Þ;
ð5:9Þ
that will be uniquely determined later on. Finally, let us define the function ωn;t ¼ ½wn;t ; η02 :
ð5:10Þ
The Darboux Lemma applied to the equations satisfied by wn;t and η02 shows that the function h¼
1 ωn;t ½wn;t ; η02 ¼ 0 η02 η2
ð5:11Þ
is a non-trivial solution to −h″ þ q ̌h ¼ ðλn þ tÞh;
ð5:12Þ
where b−2 q̌ ¼ q
d
2 2
dx
ðlnðη02 ÞÞ:
ð5:13Þ
Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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Now, let m be an integer number, m≥1, and m≠n. Let h be as in (5.11) and consider a solution g¼
1 ½zm ; η02 η02
ð5:14Þ
to equation −g″ þ q ̌g ¼ λm g:
ð5:15Þ
By Darboux Lemma applied to Eqs. (5.12) and (5.15) we have that km;t ¼
1 ½g; h h
ð5:16Þ
is a non-trivial solution to −k″m;t þ qkm;t ¼ λm km;t ;
ð5:17Þ
x∈ð0; 1Þ;
with b−2 q¼q
2
d
2
dx
ðlnðωn;t ÞÞ:
ð5:18Þ
If m ¼n, then it is easy to verify that the function kn;t ¼
zn ωn;t
ð5:19Þ
is a non-trivial solution to −k″n;t þ qkn;t ¼ ðλn þ tÞkn;t ;
ð5:20Þ
x∈ð0; 1Þ:
The above analysis is correct provided that ωn;t is a regular and strictly positive function in ½0; 1, for every t such that λn−1 o λn þ t o λnþ1 . It is exactly at this point that the choice of wn;t is important. The function wn;t can be written as wn;t ¼ η1 þ dðtÞη2 ;
ð5:21Þ
where the function d ¼ dðtÞ can be determined so that ωn;t ðx ¼ 1Þ ¼ 1;
for every t such that λn−1 oλn þ t o λnþ1 ;
ð5:22Þ
see Appendix. Regarding the function ωn;t , in addition to (5.22), we note that ωn;t ðx ¼ 0Þ ¼ 1 and ωn;t ðt ¼ 0Þ ¼ 1. Therefore, by b∈L2 ð0; 1Þ, the function ωn;t ¼ ωn;t ðx; tÞ, n≥1, is a adapting the proof of a lemma in [14] (p. 109), one can show that, for every q continuous and strictly positive function for ðx; tÞ∈½0; 1 ðλn−1 −λn ; λnþ1 −λn Þ. Moreover, ωn;t is a C2-function of the variable x in ½0; 1. Then, the potential q defined in (5.18) is a continuous function in ½0; 1. We now evaluate the functions km;t , for m≠n and t such that λn−1 o λn þ t oλnþ1 . By (5.16), recalling the expressions (5.11) and (5.5c) of h and g, respectively, we easily get ½η02 ; zm d ðlnðωn;t ÞÞ: η02 dx R x Since ½η02 ; zm ðxÞ ¼ ðλn −λm Þ 0 η02 ðsÞzm ðsÞ ds, the function km;t has a factor ðλn −λm Þ≠0. Therefore, observing that km;t ¼ ðλn −λm Þzm −
ð5:23Þ
d ðωn;t Þ ¼ twn;t zn ; dx
ð5:24Þ
we have km;t ¼ zm −t
wn;t ωn;t
Z
x 0
η02 ðsÞzm ðsÞ ds:
ð5:25Þ ∞
Collecting the above results and by (5.19) and (5.20), for a given n, n≥1, the functions fkm;t gm ¼ 1 are solutions to −k″m;t þ qkm;t ¼ ðλm þ tδmn Þkm;t ;
x∈ð0; 1Þ;
ð5:26Þ
for m≥1, where t satisfies the upper and lower bounds (5.4) and q is defined in (5.18). The end conditions for km;t at x ¼0 and x ¼1 are km;t ð0Þ ¼ 0;
ð5:27Þ
b þ twn;t ð1Þη0 ð1ÞÞkm;t ð1Þ ¼ 0; k′m;t ð1Þ þ ðH 2
ð5:28Þ
m≥1. Since the function km;t has exactly ðm−1Þ simple zeros in ð0; 1Þ (see the Deformation Lemma in the Appendix), the function km;t is the mth eigenfunction of the problem (5.26)–(5.28), m≥1. b (see Eqs. (5.2a)–(5.2c)) and fqðxÞ; ∞; H ¼ H bþ bðxÞ; ∞; Hg Therefore, to this point, we have two Sturm–Liouville problems fq 0 b b twn;t ð1Þη2 ð1Þg (see Eqs. (5.26)–(5.28)). Let n, n≥1, be a given integer and let m≠n. For each eigenpair ðλm ; zm Þ of fq ðxÞ; ∞; Hg we b and q are quasi-isospectral. have an eigenpair ðλm ; km;t Þ of fqðxÞ; ∞; Hg, so that the potentials q Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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6. Quasi-isospectral rods under (S-ER) end conditions b ¼a b g the initial rod under (S-ER) end conditions. In order to construct (S-ER) rods fAðxÞ ¼ b2 ðxÞ; ∞; K Denote by fAðxÞ a ðxÞ; ∞; Kg quasi-isospectral to the initial rod, we must now find a one-sign function a ¼ aðxÞ in ½0; 1 corresponding to the quasi-isospectral potential q given in (5.18) (e.g., qðxÞ ¼ a″ðxÞ=aðxÞ in ð0; 1Þ) and a positive boundary constant K such that an elastically restrained-type condition is satisfied at x ¼1. b, the intermediate a ̌ and the final a are solutions of To find a we use again the Darboux Lemma, noting that the original a the equations 2
b″ þ q ba b ¼ 0; −a
ð6:1Þ
−a ̌″ þ q ̌a ̌ ¼ 0;
ð6:2Þ
−a″ þ qa ¼ 0;
ð6:3Þ
respectively. The Darboux Lemma applied to Eq. (6.1) and to the equation satisfied by that a non-trivial a ̌ is given by ǎ ¼
η02 ,
e.g.
1 0 b: ½η ; a η02 2
−ðη02 Þ″
þ
bη02 q
¼ λn η02 ,
n≥1, shows
ð6:4Þ
Next, the Darboux Lemma applied to Eq. (6.2) and to (5.12), shows that a non-trivial a is a¼
1 ½h; a ̌; h
ð6:5Þ
where h is defined in (5.11). Note that this a ̌ and a are just one of each of the families of a ̌'s and a's corresponding to q ̌ and q, respectively; all others may be found by using the expression in Eq. (3.4). We can find a just as we found km;t . Substituting (6.4) in (6.5) and elaborating we have, a part a factor λn , b−t a¼a
wn;t 0 b; ½η ; a λn ωn;t 2
n≥1;
λn−1 oλn þ t oλnþ1 :
ð6:6Þ
We now have a one-parameter family of functions aðxÞ≡at ðxÞ defined for 0≤x≤1 and t such that λn−1 o λn þ t oλnþ1 , n≥1, with λ0 ¼ 0. These functions correspond to a “physical” rod if each member of the family is of one-sign, say positive, in ½0; 1. To show this, we evaluate a(x) at x ¼0, x ¼1. By (6.6) we obtain að0Þ λn þ t ¼ ; bð0Þ λn a
ð6:7Þ
að1Þ ðη ð1Þ þ dðtÞη2 ð1ÞÞ b 0 K η2 ð1Þ≡μ1 : ¼ 1−t 1 bð1Þ a b2 ð1Þ λn a
ð6:8Þ
It is evident from (6.8) that if jtj is small enough (and, of course, λn−1 o λn þ t oλnþ1 , with λ0 ¼ 0)), then μ1 4 0. Next, we can use the Deformation Lemma stated in the first part of the Appendix to show that a(x) given by (6.6) is strictly positive in ½0; 1 whenever jtj o δ1 , for a sufficiently small number δ1 . A concrete determination of δ1 is presented in the Appendix. To complete the determination of the quasi-isospectral rods, we need to examine the end conditions for the new a(x) and show that these are of the form (5.1a)–(5.1c). The eigenfunctions of the new rod are um, where aum ¼ km;t . The left end, at x ¼0, is supported, namely um ð0Þ ¼ 0. A length, but straightforward calculation shows that the end condition at x ¼1 is given by b km;t ð1Þ b n;t ð1Þ þ w′n;t ð1ÞÞ ≡− km;t ð1Þ K; m≥1; b þ t a ð1Þ ½a b; η02 ð1ÞðHw μ1 K ð6:9Þ ½a; km;t ð1Þ ¼ − λn að1Þ að1Þ that is Að1Þu′m ð1Þ þ Kum ð1Þ ¼ 0, where A ¼ a2 , for every t such that λn−1 oλn þ t o λnþ1 . Therefore, the end condition (6.9) corresponds to an elastically restrained end for the rod a if K is positive. This is certainly true if jtj is small enough, say jtj oδ2 (see Appendix). b¼a b g with end conditions (5.1b)–(5.1c) and b2 ; ∞; K Collecting the above results, we can conclude that, given a rod fA spectrum fλm g∞ , for a given integer n, n≥1, there exists a positive number δ, δ ¼ minfδ1 ; δ2 g, such that for jtjo δ we can m¼1 b ∞; K b g, with the exception construct a new rod fA ¼ a2 ; ∞; Kg, which has exactly the same eigenvalues of the original rod fA; of the nth eigenvalue. Therefore, the two rods are quasi-isospectral. b ¼ 0), then ½a b is a cantilever (e.g., K b; η02 ð1Þ ¼ 0 and, by (6.9), the end condition of the new We note that if the original rod a rod a at x¼1 is of cantilever-type ½a; km;t ð1Þ ¼ 0:
ð6:10Þ
b¼a b2 ; ∞; 0g, fA ¼ a2 ; ∞; 0g have exactly the same eigenvalues, with Therefore, for a given integer n, n≥1, the two cantilevers fA the exception of the nth eigenvalue λn , which must satisfy condition (5.4) only. Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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7. Constructing rods with a given finite set of eigenvalues Consider a cantilever rod with given cross sectional profile A0 ðxÞ ¼ a20 ðxÞ and eigenvalues fλm ða0 Þg∞ m ¼ 1 , 0 o λ1 ða0 Þ o λ2 ða0 Þ o⋯. We now ask whether it is possible to construct from this cantilever rod a new cantilever rod having prescribed N values of the first N eigenvalues fλ~ m g , with m¼1
0 o λ~ 1 o λ~ 2 o ⋯ o λ~ N :
ð7:1Þ
By the analysis of Section 6 we know how to construct from the cantilever rod A0 ðxÞ a new cantilever rod, say A1 ðxÞ ¼ a21 ðxÞ, so that the eigenvalues fλm ða0 Þg, m≥2, are kept fixed while λ1 ða1 Þ is moving to the desired value λ~ 1 . More precisely, by (6.6), the function a1 ¼ a1 ðxÞ given by a1 ¼ a0 −t
w1;t ½η0 ðn ¼ 1; a0 Þ; a0 λ1 ða0 Þω1;t 2
ð7:2Þ
corresponds to a one-parameter family of rods such that λm ða1 Þ ¼ λm ða0 Þ þ δm1 t;
ð7:3Þ
m≥1, for t such that 0 o λ1 ða0 Þ þ t o λ2 ða0 Þ:
ð7:4Þ
λ~ 1 oλ2 ða0 Þ;
ð7:5Þ
If
then we can determine the parameter t, say t ¼ t 1 , such that λ1 ða1 Þ ¼ λ~ 1 , i.e. t 1 ¼ λ~ 1 −λ1 ða0 Þ:
ð7:6Þ
The rod A1 ¼ a21 has eigenvalues fλ~ 1 ; λ2 ða0 Þ; λ3 ða0 Þ; …g, with 0 o λ~ 1 oλ2 ða0 Þ oλ3 ða0 Þ o ⋯, and can be used as starting point for the next step of the procedure. By repeating the same arguments, it is possible to modify a1 so as to keep λm ða1 Þ fixed for m≠2 and to move λ2 ða0 Þ to the desired value λ~ 2 , i.e. a2 ¼ a1 −t 2
w2;t 2 ½η0 ðn ¼ 2; a1 Þ; a1 ; λ2 ða0 Þω2;t2 2
ð7:7Þ
where t 2 ¼ λ~ 2 −λ2 ða0 Þ:
ð7:8Þ
The eigenvalues of the new rod a2 ðxÞ are fλ~ 1 ; λ~ 2 ; λ3 ða0 Þ; λ4 ða0 Þ; …g. Using repeatedly this procedure, after N steps we produce a rod, with cross-sectional profile area AN ðxÞ ¼ a2N ðxÞ, such that λm ðaN Þ ¼ λ~ m ;
for 1≤m≤N;
ð7:9Þ
and the construction is finished. We note that the choice of the initial rod a0 ðxÞ is restricted by the conditions λ~ 1 oλ2 ða0 Þ; λ~ 2 o λ3 ða0 Þ; …; λ~ N−1 oλN ða0 Þ; λ~ N oλNþ1 ða0 Þ;
ð7:10Þ
which allow to determine the numbers t 1 ; t 2 ; …; t N by the expressions analogous to Eq. (7.6). The above construction is not unique, since the flow from the initial cantilever rod a0 to a cantilever rod with prescribed first N eigenvalues depends on the particular order chosen to move every individual eigenvalue to the target value. Similarly, the compatibility conditions on the initial rod a0 may change depending on the sequence of eigenvalue shifts. Finally, we observe that the above construction of quasi-isospectral rods can be adapted to rods under (S-ER) end conditions. In this case, the construction of quasi-isospectral rods works when the shift parameter t (of the generic eigenvalue) belongs to a sufficiently small neighborhood of zero, as it was shown in Section 6. 8. Examples In this section we present some applications of the above results and we determine examples of rods AðxÞ ¼ a2 ðxÞ under b cantilever end conditions which are quasi-isospectral to the uniform rod with γ ¼ 1, E¼1 and AðxÞ≡1 in ½0; 1. The families of rods shown in Figs. 1a–c, 2a–c, 2a–c, 3a–c have been derived using Eq. (6.6) for a ¼ aðxÞ, with n ¼ 1; 2; 5 and for some discrete values of t satisfying (5.4). It can be seen that when the parameter t is close to the lower or upper limit in (5.4), or n is taken to be large, then the profile departs significantly from the uniform rod. For these situations the vibration of the rod could be not adequately described by the classical mechanical model of Eq. (2.1a)–(21.c) and other vibrational models would have to be considered. Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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8
4
3
2
1
0 0.0
0.2
0.4
0.6
0.8
1.0
x t = 0.0
t = -0.2 λ 1
t = -0.4 λ 1
t = -0.9 λ 1
20
15
10
5
0 0.0
0.2
0.4
0.6
0.8
1.0
x t = 0.0
t = 0.1 ( λ 2 − λ 1 )
t = 0.2 ( λ 2 − λ 1 )
t = 0.4 ( λ 2 − λ 1 )
500
375
250
125
0 0.0
0.2
0.4
t = 0.0 t = 0.7 ( λ 2 − λ 1 )
0.6
x
0.8
1.0
t = 0.5 ( λ 2 − λ 1 ) t = 0.9 ( λ 2 − λ 1 )
b Fig. 1. (a–c) Examples of quasi-isospectral cantilever rods A ¼ AðxÞ with γ ¼ 1, E¼ 1 and AðxÞ≡1 in ½0; 1, for n¼ 1 and for some discrete values of t.
9. Extensions The analysis developed in the previous sections can be extended to rods under different sets of boundary conditions. We recall that the end conditions are assumed to be Elastically Restrained (ER) when ðERÞ
Að0Þu′ð0Þ−kuð0Þ ¼ 0 ¼ Að1Þu′ð1Þ þ Kuð1Þ;
ð9:1Þ
where k, K≥0 and k þ K 4 0. Conditions (9.1) mean that the ends x ¼0, x ¼1 are attached to fixed supports by means of elastic springs having stiffness k, K, respectively. The cases of Supported (S), Free (F) and Supported-Elastically Restrained (SER) end conditions are obtained as limit cases of (9.1), precisely ðSÞ
k¼∞¼K
uð0Þ ¼ 0 ¼ uð1Þ;
ð9:2Þ
ðFÞ
k¼0¼K
u′ð0Þ ¼ 0 ¼ u′ð1Þ:
ð9:3Þ
Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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9
4
3
2
1
0 0.0
0.2
0.4
t = 0.0
x
0.6
0.8
1.0
t = 0.2 ( λ 1 − λ 2 )
t = 0.4 ( λ 1 − λ 2 )
t = 0.9 ( λ 1 − λ 2 )
4
3
2
1
0 0.0
0.2
0.4
t = 0.0
x
0.6
0.8
1.0
t = 0.1 ( λ 3 − λ 2 )
t = 0.2 ( λ 3 − λ 2 )
t = 0.4 ( λ 3 − λ 2 )
40
30
20
10
0 0.0
0.2
0.4
t = 0.0
0.6
x
t = 0.7 ( λ 3 − λ 2 )
0.8
t = 0.5 ( λ 3 − λ 2 ) t = 0.9 ( λ 3 − λ 2 )
b Fig. 2. (a–c) Examples of quasi-isospectral cantilever rods A ¼ AðxÞ with γ ¼ 1, E¼ 1 and AðxÞ≡1 in ½0; 1, for n¼ 2 and for some discrete values of t.
ðS−ERÞ
k ¼ ∞;
0≤K o∞
uð0Þ ¼ 0;
Að1Þu′ð1Þ þ Kuð1Þ ¼ 0;
ð9:4Þ
In this paper, we have considered in detail the problem of constructing rods with given natural frequencies for the (S-ER) case. It has been shown that the crucial point of the procedure lies in the ability to construct quasi-isospectral systems, that is rods which share with a given rod all the natural frequencies, with the exception of a single eigenvalue. Therefore, for the sake of brevity, in this section we focus on the main analogies and differences of the treatment for (ER), (S) and (F) end conditions with respect to the (S-ER) case. More details can be found in [21]. 1. The analysis of the (S) case can be developed following the same lines of the (S-ER) end conditions. The main difference, when the stiffness of the spring acting at the right side of the (S-ER) rod is strictly positive, is that, in the (S) case, the parameter t appearing in the definition of the cross-sectional profile needs to satisfy a condition analogous to (5.4) only. That is, the further restriction jtj oδ, where δ is a sufficiently small number (see the second part of Section 6 and Appendix), is not necessary for the (S) end conditions case. Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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10
4
3
2
1
0 0.0
0.2
0.4
0.6
x
t= 0.0
0.8
1.0
t= 0.2 ( λ 4 − λ 5 )
t= 0.4 ( λ 4 − λ 5 )
t =0 .9 ( λ 4 − λ 5 )
4
3
2
1
0 0.0
0.2
0.4
0.6
x
t= 0.0
0.8
1.0
t= 0.1 ( λ 6 − λ 5 ) t =0 .4 ( λ 6 − λ 5 )
t= 0.2 ( λ 6 − λ 5 )
6 5 4 3 2 1 0 0.0
0.2
0.4
t= 0.0 t= 0.7 ( λ 6 − λ 5 )
0.6
x
0.8
1.0
t= 0.5 ( λ 6 −
λ 5)
t =0 .9 ( λ 6 −
λ 5)
b Fig. 3. (a–c) Examples of quasi-isospectral cantilever rods A ¼ AðxÞ with γ ¼ 1, E ¼1 and AðxÞ≡1 in ½0; 1, for n¼ 5 and for some discrete values of t.
2. The determination of quasi-isospectral rods under (F) end conditions can be developed on the basis of the results of the (S) case. In fact, under suitable regularity assumptions on the cross-sectional profile, it can be shown that the eigenvalue problem for a (F) rod can be transformed in an “equivalent” eigenvalue problem for a (S) rod. 3. The case of rods under (ER) end-conditions is different from the previous ones. There are two main results for this set of boundary conditions. First, it can be shown that there exists a way to explicitly construct a rod under (S) end conditions such that all the eigenvalues of the initial (ER) rod coincide with those of the new (S) rod, with the exception of the smallest eigenvalue. Therefore, on adapting the arguments for (S) rods recalled in (1), it is possible to construct new (S) rods which share the same higher spectrum (e.g., all the eigenvalues with the exception of the smallest one) of the initial (ER) rod, with the exception of a single eigenvalue. The above characterization of quasi-isospectral (ER) rods shows that there is no control over the smallest eigenvalue. The second result to be mentioned is the possibility to construct families of (ER) rods quasi-isospectral to a given (ER) rod such that they preserve the higher spectrum and change only the Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
A. Morassi / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
11
smallest eigenvalue in a prescribed manner. As in the (S-ER) case, the quasi-isospectral families depend on a real parameter t and, in order to have physical rods (e.g., positive cross-sectional area and positive elastic stiffness at the ends), t must belong to a sufficiently small neighborhood of the origin.
10. Concluding remarks We have shown that we may construct one-parameter t-families of rods with prescribed values of a finite set of natural frequencies under a given set of boundary conditions. Details of the method have been presented for rods under supportedelastically restrained end conditions, but the results can be extended to other end conditions. The analysis is based on a double application of the Darboux Lemma after reducing the equation of the free vibrations to normal form by a Liouville transformation. The rods with given natural frequencies can be determined explicitly by means of closed-form expressions. The reconstruction procedure needs the specification of an initial rod whose eigenvalues must be close to the assigned ones. In connection with this aspect, we point out that a theoretical aspect worth of investigation stands on the possibility of characterizing the full set of rods that could be chosen as starting point in our procedure. The analysis of the analogous problem for a special class of beams under bending vibration is presented in [22].
Acknowledgments The collaboration of Michele Dilena in preparing the figures and numerical simulations is gratefully acknowledged. Appendix A In this Appendix we recall a technical result which has been used in Section 5 and in Section 6 to count the number of zeros of the functions km;t ðxÞ (defined by (5.25)) and a(x) (defined by (6.6)), respectively. Moreover, we provide the details of the determination of the function d(t), defined by (5.21) and (5.22), and of the determination of the parameter δ ¼ minfδ1 ; δ2 g introduced at the end of Section 6. Lemma 2 (Deformation Lemma). Let ft, t 1 ≤t≤t 2 , be a family of real valued functions of x in ½0; 1 which is jointly continuously differentiable in t and in x. Suppose that for every t, ft has a finite number of zeros in ½0; 1, all of which are simple, and has boundary values with signs that are independent of t. Then, the number of zeros of ft in ½0; 1 is independent of t, for all t satisfying t 1 ≤t≤t 2 . This lemma has been introduced in [19], see also [14] for a slightly different version in which it is supposed that ft has boundary values that are independent of t. Here, we apply the lemma to count the number of zeros in ½0; 1 of the function km;t defined by (5.25). We start by noticing that, as solution to the differential equation (5.26), the function km;t can have only a finite number of roots in ½0; 1, and that these roots must be simple. Moreover, km;t can be normalized by assuming km;t ð1Þ ¼ 1, for every integer number m≥1 and for every real number t satisfying the upper and lower bounds (5.4). Therefore, the Deformation Lemma implies that km;t has exactly ðm−1Þ zeros in ð0; 1Þ. We conclude that km;t is the mth eigenfunction of the problem (5.26)–(5.28) associated to the potential q. Concerning the function d(t), by imposing condition (5.22) and observing that Z 1 ½η1 ; η02 ðx ¼ 1Þ ¼ 1 þ t η1 η02 ds; ðA:1Þ 0
Z ½η2 ; η02 ðx ¼ 1Þ ¼ t
1 0
η2 η02 ds;
ðA:2Þ
we find R1 dðtÞ ¼ − R01 0
η1 η02 ds η2 η02 ds
;
ðA:3Þ
and d(t) is well-defined provided that the denominator in (A.3) does not vanish for all t, with λn−1 o λn þ t o λnþ1 (λ0 ¼ 0). R1 If the denominator does vanish at some t , t ≠0, and λn−1 o λn þ t o λnþ1 , then 0 η2 ðs; t Þη02 ðsÞ ds ¼ 0. This means that η2 ðx; t Þ belongs to the subspace of functions orthogonal to the nth eigenfunction zn(x), that is η2 ðx; t Þ ¼ ∑∞ k ¼ 1;k≠n αk zk ðxÞ. Substituting this expression in Eq. (5.6a) and elaborating we have ∞
∑
k ¼ 1;k≠n
αk ðλk −ðλn þ tÞÞzk ðxÞ ¼ 0;
x∈ð0; 1Þ;
ðA:4Þ
and then αk ¼ 0 for every k, k≥1 and k≠n. Thus η2 ðx; t Þ≡0, a contradiction. We conclude that d(t) is always well-determined for all t's such that λn−1 o λn þ t o λnþ1 . Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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The determination of δ is based on estimates of the functions η1 , η2 , η02 , and their first x-derivative, and of d(t), see, for example, [14] (Theorem 3, Chapter 1). We first show how to determine δ1 such that the constant μ1 in (6.8) be strictly positive. Hereinafter, we assume that there exists ϵ 4 0 such that 0 oϵ oλ0 . For a given integer number n, n≥1, and for q∈L2 ð0; 1Þ we have pffiffiffiffiffiffiffiffiffiffiffiffiffi MðxÞ∥q∥ η1 ≤1 þ pMðxÞ ffiffiffiffiffiffiffiffiffiffiffiffiffi ; jη′1 j≤ λn þ t 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðA:5Þ λn þ t λn þ t 1 MðxÞ η2 ≤ pffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; λn þ t λn þ t 0 1 η ≤ pffiffiffiffiffi 1 þ MðxÞ ffiffiffiffi ffi p ; 2 λn λn
ðA:6Þ
MðxÞ∥q∥ pffiffiffiffiffi ; λn
ðA:7Þ
jðη02 Þ′j≤1 þ
where
pffiffiffi MðxÞ ¼ expð∥q∥ xÞ;
MðxÞ∥q∥ jη′2 j≤1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; λn þ t
Z ∥q∥ ¼
1
!1=2 2
jqðxÞj dx
0
:
By (6.8) and in order to control d(t) from above, we need to estimate from below the denominator function
ðA:8Þ R1 0
η2 η02 dx in Eq. (A.3). The
v ¼ η2 −η02
ðA:9Þ
is the unique solution of the initial value problem 8 b > < −v″ þ q v ¼ λn v þ tη2 ; x∈ð0; 1Þ; ðaÞ vð0Þ ¼ 0; > : v′ð0Þ ¼ 0:
ðbÞ
ðA:10Þ
ðcÞ
The solution of (A.10a)–(A.10c) is given by Z x ðη01 ðxÞη02 ðsÞ−η01 ðsÞη02 ðxÞÞη2 ðs; tÞ ds; vðx; tÞ ¼ t
ðA:11Þ
0
bÞ ¼ η1 ðx; q b; λn Þ. By estimates (A.5)–(A.7) and recalling that λn−1 o λn þ t oλnþ1 , we obtain the following uniform where η01 ðx; q estimate in ½0; 1: 0 12 2jtj Mð1Þ Mð1Þ C vðx; tÞ≤ pffiffiffiffiffiffiffiffiffipffiffiffiffiffi 1 þ pffiffiffiffiffiffiffiffiffi B @1 þ pffiffiffiffiffi A ≡C 1 jtj: λn−1 λn−1 λn λn By (A.12) and by Cauchy–Schwartz inequality we have Z 1 η2 η02 dx≥∥η02 ∥ð∥η02 ∥−C 1 jtjÞ
ðA:12Þ
ðA:13Þ
0
and then, by (A.3), we find dðtÞ≤
∥η1 ∥ : ∥η02 ∥−C 1 jtj
ðA:14Þ
If, for example, jtjo ∥η02 ∥=2C 1 , then we obtain a uniform estimate of d(t) Mð1Þffi 2 1 þ pffiffiffiffiffiffi λn−1 dðtÞ≤ ≡C 2 ; ∥η02 ∥ ! ∥ζ 0 ∥ ∥ζ 0 ∥ for t∈ − 2 ; 2 and λn−1 o λn þ t o λnþ1 : 2C 1 2C 1
ðA:15Þ
By the above estimates, the coefficient μ1 is strictly positive if the parameter t satisfies the conditions (A.15) and jt j o
1þ
Mð1Þ p ffiffiffiffiffiffiffi λn−1
3=2 bð1ÞÞ2 λ ða n : 2 ffi pffiffiffiffi 1 þ Mð1Þ 1 þ pCffiffiffiffiffiffi λ λ n
ðA:16Þ
n−1
This completes the determination of δ1 . Similar calculations, not reported here for brevity, show that there exists a neighborhood of the origin ð−δ2 ; δ2 Þ, δ2 4 0, such that for t∈ð−δ2 ; δ2 Þ the stiffness K in (6.9) is positive. To conclude it is enough to take δ ¼ minðδ1 ; δ2 Þ. This completes the determination of δ. Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
H.G. Natke, Identification of Vibrating Structures, Springer, New York, 1982. D. Capecchi, F. Vestroni, Identification of finite element models in structural dynamics, Eng. Struct. 15 (1993) 21–30. M.I. Friswell, J.E. Mottershead, Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, Dordrecht, 1995. J.E. Mottershead, Y.M. Ram, Inverse eigenvalue problems in vibration absorption: passive modification and active control, Mech. Syst. Signal Process. 20 (2006) 5–44. M.G. Tehrani, J.E. Mottershead, A.T. Shenton, Y.M. Ram, Robust pole placement in structures by the method of receptances, Mech. Syst. Signal Process. 25 (2011) 112–122. D. Capecchi, F. Vestroni, Monitoring of structural systems by using frequency data, Earthquake Eng. Struct. Dyn. 28 (2000) 447–461. A. Morassi, Damage detection and generalized Fourier coefficients, J. Sound Vib. 302 (2007) 229–259. F. Vestroni, Structural identification, parametric models and Idefem code, in: Dynamics Methods for Damage Detection in Structures, CISM Courses and Lectures, vol. 499, Springer, Wien-New York, 2008, pp. 95–109. M. Dilena, A. Morassi, Structural health monitoring of rods based on natural frequency and antiresonant frequency measurements, Struct. Health Monitoring: Int. J. 8 (2009) 149–173. M. Dilena, A. Morassi, Reconstruction method for damage detection in beams based on natural frequency and antiresonant frequency measurements, J. Eng. Mech. ASCE 136 (2010) 329–344. D. W Marquardt, An algorithm for least-squares estimation of nonlinear parameters, SIAM J. Appl. Math. 11 (1963) 431–441. P. Eykhoff, System Identification, Parameter and State Estimation, John Wiley & Sons, New York, 1987. G. Borg, Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Math. 78 (1946) 1–96. J. Pöschel, E. Trubowitz, Inverse Spectral Theory, Academic Press, London, UK, 1987. E.L. Isaacson, H.P. Mc Kean, E. Trubowitz, The inverse Sturm–Liouville problem II, Commun. Pure Appl. Math. XXXVII (1984) 1–11. B.E.J. Dahlberg, E. Trubowitz, The inverse Sturm–Liouville problem III, Commun. Pure Appl. Math. XXXVII (1984) 255–267. C.F. Coleman, J.R. McLaughlin, Solution of the inverse spectral problem for an impedance with integrable derivative – part II, Commun. Pure Appl. Math. XLVI (1993) 185–212. Y.M. Ram, S. Elhay, Dualities in vibrating rods and beams: continuous and discrete models, J. Sound Vib. 184 (5) (1995) 759–766. G.M.L. Gladwell, A. Morassi, On isospectral rods, horns and strings, Inverse Problems 11 (1995) 533–554. G. Darboux, Sur la répresentation sphérique des surfaces, C.R. Acad. Sci. Paris 94 (1882) 1343–1345. A. Morassi, 2012. Quasi-isospectral Vibrating Systems. Technical Report, University of Udine. A. Morassi, in press. Exact construction of beams with a finite number of given natural frequencies, J. Vib. Control, submitted for publication.
Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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Constructing rods with given natural frequencies Antonino Morassi n Università degli Studi di Udine, Dipartimento di Ingegneria Civile e Architettura, via Cotonificio 114, 33100 Udine, Italy
a r t i c l e i n f o
abstract
Article history: Received 20 December 2012 Received in revised form 4 April 2013 Accepted 21 April 2013
In this paper we present a new method for constructing axially vibrating rods having prescribed values of the first N natural frequencies, under a given set of boundary conditions. The analysis is based on the determination of the so-called quasi-isospectral rods, that is rods which have the same spectrum as a given rod, with the exception of a single eigenvalue which is free to move in a prescribed interval. The reconstruction procedure needs the specification of an initial rod whose eigenvalues must be close to the assigned eigenvalues. The rods and their normal modes can be constructed explicitly by means of closed-form expressions. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Vibrating rods Eigenvalues Structural identification Inverse problems
1. Introduction The construction of a system having prescribed values of natural frequencies, usually the lower frequencies, is an important issue in many engineering contexts. A key problem in Structural Identification, for example, consists in determining either the stiffness or the inertia coefficients so that the first analytical eigenvalues coincide with the corresponding values determined in experimental tests [1–3]. Other applications concern the construction of vibrating systems such that possible occurrence of resonance phenomena with excitation frequencies is avoided. The assignation of certain chosen poles of a vibrating system while other poles (usually at higher frequencies) remain unchanged is an important issue also in eigenvalue assignment problems in vibration by passive modifications or by active control procedures [4,5]. The reconstruction of a vibrating system with a given set of natural frequencies via structural identification methods is usually based on optimization techniques of variational type, [6–10]. A function which measures the error between the assigned and analytical frequency values is minimized via gradient-type methods and, therefore, the coefficients of a starting configuration of the system are iteratively updated, under some a priori assumptions on the coefficients to be identified (symmetry assumptions, lower and upper bounds, etc.). The numerical problem is highly non-linear and the identification algorithms can be very time consuming, see, for example, [11,12]. In this paper we show how to construct longitudinally vibrating rods having prescribed values of the first N natural frequencies, under a given set of boundary conditions. The rods and their normal modes can be constructed explicitly by means of closed-form expressions. The reconstruction procedure needs the specification of an initial rod whose eigenvalues must be close to the assigned eigenvalues. The analysis is based on the determination of the so-called quasi-isospectral rods, that is rods which have the same spectrum as a given rod, with the exception of a single eigenvalue which is free to move in a prescribed interval. The reconstruction is obviously not unique, since classical results show that two full spectra
n
Tel.: +39 0432 558739; fax: +39 0432 558700. E-mail address: [email protected]
0888-3270/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ymssp.2013.04.010
Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
2
A. Morassi / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
corresponding to two specific sets of boundary conditions are required, in general, to determine the shape of a rod uniquely [13]. With a view to practical applications, this intrinsic non-uniqueness could be useful in engineering design in order to formulate and solve structural optimization problems, such as, for example, the determination of rods with maximal global stiffness or the determination of rods having minimal mass. The main ideas and mathematical tools of the reconstruction method are presented, for the sake of simplicity, for rods under cantilever end conditions. The analysis of this case allows for a clear presentation of the key aspects of the method. Extensions to rods under other end conditions and generalizations are discussed in the last section of the paper.
2. Main ideas of the reconstruction procedure The free (undamped, infinitesimal) longitudinal vibrations u(x) of frequency ω, of a thin straight rod of unit length under Supported-Elastically Restrained (S-ER) end conditions are governed by the Sturm–Liouville equation 8 b b > > < ðAðxÞu′ðxÞÞ′ þ λAðxÞuðxÞ ¼ 0; uð0Þ ¼ 0; > > : Að1Þu′ð1Þ b b uð1Þ ¼ 0; þK
x∈ð0; 1Þ;
ðaÞ
ðS ¼ SupportedÞ
ðbÞ
ðER ¼ Elastically RestrainedÞ
ðcÞ
ð2:1Þ
b where λ ¼ γω2 =E and ðÞ′≡dðÞ=dx. Here E is Young's modulus, γ is the volume mass density, both assumed constant; AðxÞ is the cross-sectional area at section x. Condition (2.1c) means that the end x¼ 1 is attached to a fixed support by means of an b, K b ≥0. The case of K b ¼ 0 corresponds to a cantilever rod. elastic spring having stiffness K b We shall assume throughout that AðxÞ is a strictly positive, twice continuously differentiable function of x in ½0; 1. It is b well known that for such AðxÞ, and end conditions (2.1b)–(2.1c), there is an infinite sequence fλm g∞ m ¼ 1 of eigenvalues (or spectrum), with 0 o λ1 o λ2 o⋯; limm-∞ λm ¼ ∞, for which (2.1a)–(2.1c) has a non-trivial solution u(x). Let n≥1 be given. The key step of our method is the explicit construction of a new rod quasi-isospectral to the given rod, b that is a rod with cross-sectional profile A(x) having the same eigenvalues as the given rod AðxÞ under (S-ER) end conditions, with the exception of the nth eigenvalue. In fact, by keeping fixed all the eigenvalues λm with m≠n and moving the nth eigenvalue λn to the desired value, say λ~ n , and using repeatedly the procedure, after N steps we will produce a rod with the N first N given eigenvalues fλ~ m gm ¼ 1 , and the construction is finished. We will see in the following that the reconstruction procedure works only if the rod to be determined is not far (in a way that can be rigorously specified) from the initial guess rod. Moreover, even if the initial rod is fixed once and for all, it is evident that the construction is not unique, since the flow from the initial rod depends also on the particular order chosen to move every individual eigenvalue to the corresponding target value. b The main steps of our construction of rods A(x) quasi-isospectral to a given rod AðxÞ, under (S-ER) end conditions, are the bðxÞ by a standard Sturm–Liouville following. First, Eq. (2.1a) is reduced to canonical form with a Schrödinger potential q transformation (see Section 3). Second, a Darboux Lemma (Section 4) is used to construct explicit families of Schrödinger bðxÞ (Section 5). Third, the Darboux Lemma is applied once more in potentials q(x) quasi-isospectral to the initial potential q iterate form to determine rods A(x) corresponding to the quasi-isospectral potentials q(x) and, ultimately, to find rods A(x) b quasi-isospectral to the initial rod AðxÞ (Section 6). The second step of the above procedure is fully described in a series of papers by Trubowitz and colleagues for Sturm– Liouville operators in canonical form and square-summable potentials, under either Dirichlet end-conditions (corresponding to the (S–S) case for rods, [14]) or Robin end conditions (corresponding to the (ER–ER) case for rods, [15]). The case corresponding to (S-ER) end conditions is only briefly sketched in [16] (Lemma 1) and, therefore, our original contribution lies also on the explicit determination of the quasi-isospectral potentials for this set of boundary conditions. For the sake of completeness, we recall that the problem of constructing quasi-isospectral rods under Dirichlet or Neumann (e.g., u′ð0Þ ¼ u′ð1Þ ¼ 0) boundary conditions has been solved by Coleman and McLaughlin [17] by using a method different from ours, which extends the results obtained by Trubowitz and co-workers to Eq. (2.1a) with less regular coefficients (e.g., pffiffiffiffi pffiffiffiffi b Þ′= A b ∈L2 ð0; 1Þ). ð A Finally, it is appropriate to compare the results of this paper and those obtained in the analysis of the so-called isospectral rods, that is rods having exactly the same full spectrum of a given rod. In 1995, Ram and Elhay [18] have shown that for a given non-uniform rod there is a dual rod which has the same infinite spectrum. Later on, Gladwell and Morassi [19] showed how to construct numerable families of isospectral rods, see also the paper by Coleman and McLaughlin [17] for more general treatment applied to Dirichlet or Neumann end conditions. Here, instead, we show how to construct a rod with given values of a finite number of natural frequencies. As it was stated above, the crucial step is the ability to construct quasi-isospectral rods, that is rods having all the eigenvalues in common with the exception of a single eigenvalue. Although the Darboux Lemma is the main mathematical tool of both the present analysis and that developed in [19], the results found here are clearly different and, in some respects, are more general than those obtained in the mentioned paper. In fact, the procedure for the construction of isospectral rods proposed in [19] cannot be used to construct rods having prescribed values of a finite set of natural frequencies. Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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3
3. Reduction to normal form In Eq. (2.1a) put b ¼a b2 ðxÞ; AðxÞ
bðxÞuðxÞ: yðxÞ ¼ a
ð3:1Þ
Then, Eq. (2.1a) reduces to the Sturm–Liouville canonical form bðxÞyðxÞ ¼ λyðxÞ; −y″ðxÞ þ q
ð3:2Þ
x∈ð0; 1Þ;
where bðxÞ ¼ q
b″ðxÞ a bðxÞ a
ð3:3Þ
is a continuous function in ½0; 1. b or a b, there is a unique q b, but for a given q b there are many a b. In particular, if a b0 is one a b Note that for a given A b, then all the coefficients a b satisfying (3.3) are given by corresponding to a given q ! Z x ds b¼a b 0 c1 þ c2 ; ð3:4Þ a b20 ðsÞ 0 a b is one-sign – say, positive – in ½0; 1. where the constants c1, c2 must be chosen such that a
4. The Darboux Lemma We now introduce the key mathematical tool of our analysis: the so-called Darboux Lemma. Lemma 1 (Darboux Lemma [20]). Let μ be a real number, and suppose g≡gðxÞ is a non-trivial solution of the Sturm–Liouville equation bg ¼ μg −g″ þ q
ð4:1Þ
b≡q bðxÞ. Let λ be a real number. If f is a non-trivial solution of with potential q bf ¼ λf −f ″ þ q
ð4:2Þ
and λ≠μ, then y¼
1 1 ½g; f ≡ ðgf ′−g′f Þ g g
ð4:3Þ
is a non-trivial solution of the Sturm–Liouville equation −y″ þ q ̌y ¼ λy;
ð4:4Þ
where b−2 q̌ ¼ q
2
d
2
dx
lnðgðxÞÞ:
ð4:5Þ
Moreover, the general solution of the equation −y″ þ q ̌y ¼ μy
ð4:6Þ
is y¼
Z x 1 b1 þ b2 g 2 ðsÞ ds ; g 0
where b1 and b2 are arbitrary constants. In particular, y ¼
ð4:7Þ 1 g
is a solution of (4.6).
This lemma, which can be verified by direct calculation (see [19]), allows to associate to every equation of the form (4.1), that one knows how to integrate for all values of μ, another equation of the same form that one also knows how to integrate for all the values of the parameter μ. In particular, the addition law of the logarithm makes iteration of Lemma 1 simple, as it will be shown in the sequel. It should be noted that if g vanishes in ½0; 1, then Eq. (4.4) is understood to hold between the roots of g. We will see that these singular situations disappear by applying the Darboux Lemma twice, as we shall describe in the following section. Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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4
5. Quasi-isospectral potentials under (S-ER) end conditions b ¼ AðxÞ b Suppose that the left end of the original rod A is supported, while the right end is elastically restrained by means b , 0≤K b o ∞. The eigenvalue problem written in Sturm–Liouville canonical form is of an elastic spring with stiffness K 8 by ¼ λy; x∈ð0; 1Þ; ðaÞ −y″ þ q > > > < yð0Þ ¼ 0; ðbÞ ð5:1Þ yð1Þ > > b b; yð1Þ þ K > ¼ 0; ðcÞ : ½a bð1Þ a b¼a b″=a b, or equivalently with q 8 by ¼ λy; x∈ð0; 1Þ; −y″ þ q > < yð0Þ ¼ 0; > : b y′ð1Þ þ Hyð1Þ ¼ 0;
ðaÞ ðbÞ
ð5:2Þ
ðcÞ
where b b b ¼ − a ′ð1Þ þ K : H 2 bð1Þ a a b ð1Þ
ð5:3Þ
b of Eqs. (5.2a)–(5.2c) are denoted by fðλm ; zm Þg∞ , with 0 o λ1 o λ2 o ⋯; limm-∞ λm ¼ ∞. b; ∞; Hg The eigenvalues of the problem fq m¼1 Let n, n≥1, be a given integer. We want to construct families of Sturm–Liouville problems fqðxÞ; ∞; Hg which share the b with the exception of the nth eigenvalue which is free to move in a prescribed bðxÞ; ∞; Hg, same eigenvalues of the problem fq interval. The strategy of the proof is to use appropriately the Darboux Lemma in an iterated form. It is useful to introduce some notations. Let t∈R be such that λn−1 o λn þ t o λnþ1 ;
ð5:4Þ
b; λn þ tÞ, η2 ¼ η2 ðx; q b; λn þ tÞ the solutions to the initial value problems with λ0 ¼ 0. Denote by η1 ¼ η1 ðx; q 8 b > < −η″1 þ q η1 ¼ ðλn þ tÞη1 ; x∈ð0; 1Þ; ðaÞ ðbÞ η1 ð0Þ ¼ −1; > : η′ ð0Þ ¼ 0; ðcÞ 1
ð5:5Þ
and 8 b > < −η″2 þ q η2 ¼ ðλn þ tÞη2 ; η2 ð0Þ ¼ 0; > : η′ ð0Þ ¼ −1:
x∈ð0; 1Þ;
ðaÞ ðbÞ
ð5:6Þ
ðcÞ
2
Moreover, we introduce the function b; λn Þ; η02 ¼ η2 ðx; q
ð5:7Þ
which is proportional to the eigenfunction associated to λn . Let b; λn þ tÞ wn;t ¼ wn;t ðx; q
ð5:8Þ
be a solution of bwn;t ¼ ðλn þ tÞwn;t ; −w″n;t þ q
x∈ð0; 1Þ;
ð5:9Þ
that will be uniquely determined later on. Finally, let us define the function ωn;t ¼ ½wn;t ; η02 :
ð5:10Þ
The Darboux Lemma applied to the equations satisfied by wn;t and η02 shows that the function h¼
1 ωn;t ½wn;t ; η02 ¼ 0 η02 η2
ð5:11Þ
is a non-trivial solution to −h″ þ q ̌h ¼ ðλn þ tÞh;
ð5:12Þ
where b−2 q̌ ¼ q
d
2 2
dx
ðlnðη02 ÞÞ:
ð5:13Þ
Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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5
Now, let m be an integer number, m≥1, and m≠n. Let h be as in (5.11) and consider a solution g¼
1 ½zm ; η02 η02
ð5:14Þ
to equation −g″ þ q ̌g ¼ λm g:
ð5:15Þ
By Darboux Lemma applied to Eqs. (5.12) and (5.15) we have that km;t ¼
1 ½g; h h
ð5:16Þ
is a non-trivial solution to −k″m;t þ qkm;t ¼ λm km;t ;
ð5:17Þ
x∈ð0; 1Þ;
with b−2 q¼q
2
d
2
dx
ðlnðωn;t ÞÞ:
ð5:18Þ
If m ¼n, then it is easy to verify that the function kn;t ¼
zn ωn;t
ð5:19Þ
is a non-trivial solution to −k″n;t þ qkn;t ¼ ðλn þ tÞkn;t ;
ð5:20Þ
x∈ð0; 1Þ:
The above analysis is correct provided that ωn;t is a regular and strictly positive function in ½0; 1, for every t such that λn−1 o λn þ t o λnþ1 . It is exactly at this point that the choice of wn;t is important. The function wn;t can be written as wn;t ¼ η1 þ dðtÞη2 ;
ð5:21Þ
where the function d ¼ dðtÞ can be determined so that ωn;t ðx ¼ 1Þ ¼ 1;
for every t such that λn−1 oλn þ t o λnþ1 ;
ð5:22Þ
see Appendix. Regarding the function ωn;t , in addition to (5.22), we note that ωn;t ðx ¼ 0Þ ¼ 1 and ωn;t ðt ¼ 0Þ ¼ 1. Therefore, by b∈L2 ð0; 1Þ, the function ωn;t ¼ ωn;t ðx; tÞ, n≥1, is a adapting the proof of a lemma in [14] (p. 109), one can show that, for every q continuous and strictly positive function for ðx; tÞ∈½0; 1 ðλn−1 −λn ; λnþ1 −λn Þ. Moreover, ωn;t is a C2-function of the variable x in ½0; 1. Then, the potential q defined in (5.18) is a continuous function in ½0; 1. We now evaluate the functions km;t , for m≠n and t such that λn−1 o λn þ t oλnþ1 . By (5.16), recalling the expressions (5.11) and (5.5c) of h and g, respectively, we easily get ½η02 ; zm d ðlnðωn;t ÞÞ: η02 dx R x Since ½η02 ; zm ðxÞ ¼ ðλn −λm Þ 0 η02 ðsÞzm ðsÞ ds, the function km;t has a factor ðλn −λm Þ≠0. Therefore, observing that km;t ¼ ðλn −λm Þzm −
ð5:23Þ
d ðωn;t Þ ¼ twn;t zn ; dx
ð5:24Þ
we have km;t ¼ zm −t
wn;t ωn;t
Z
x 0
η02 ðsÞzm ðsÞ ds:
ð5:25Þ ∞
Collecting the above results and by (5.19) and (5.20), for a given n, n≥1, the functions fkm;t gm ¼ 1 are solutions to −k″m;t þ qkm;t ¼ ðλm þ tδmn Þkm;t ;
x∈ð0; 1Þ;
ð5:26Þ
for m≥1, where t satisfies the upper and lower bounds (5.4) and q is defined in (5.18). The end conditions for km;t at x ¼0 and x ¼1 are km;t ð0Þ ¼ 0;
ð5:27Þ
b þ twn;t ð1Þη0 ð1ÞÞkm;t ð1Þ ¼ 0; k′m;t ð1Þ þ ðH 2
ð5:28Þ
m≥1. Since the function km;t has exactly ðm−1Þ simple zeros in ð0; 1Þ (see the Deformation Lemma in the Appendix), the function km;t is the mth eigenfunction of the problem (5.26)–(5.28), m≥1. b (see Eqs. (5.2a)–(5.2c)) and fqðxÞ; ∞; H ¼ H bþ bðxÞ; ∞; Hg Therefore, to this point, we have two Sturm–Liouville problems fq 0 b b twn;t ð1Þη2 ð1Þg (see Eqs. (5.26)–(5.28)). Let n, n≥1, be a given integer and let m≠n. For each eigenpair ðλm ; zm Þ of fq ðxÞ; ∞; Hg we b and q are quasi-isospectral. have an eigenpair ðλm ; km;t Þ of fqðxÞ; ∞; Hg, so that the potentials q Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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6
6. Quasi-isospectral rods under (S-ER) end conditions b ¼a b g the initial rod under (S-ER) end conditions. In order to construct (S-ER) rods fAðxÞ ¼ b2 ðxÞ; ∞; K Denote by fAðxÞ a ðxÞ; ∞; Kg quasi-isospectral to the initial rod, we must now find a one-sign function a ¼ aðxÞ in ½0; 1 corresponding to the quasi-isospectral potential q given in (5.18) (e.g., qðxÞ ¼ a″ðxÞ=aðxÞ in ð0; 1Þ) and a positive boundary constant K such that an elastically restrained-type condition is satisfied at x ¼1. b, the intermediate a ̌ and the final a are solutions of To find a we use again the Darboux Lemma, noting that the original a the equations 2
b″ þ q ba b ¼ 0; −a
ð6:1Þ
−a ̌″ þ q ̌a ̌ ¼ 0;
ð6:2Þ
−a″ þ qa ¼ 0;
ð6:3Þ
respectively. The Darboux Lemma applied to Eq. (6.1) and to the equation satisfied by that a non-trivial a ̌ is given by ǎ ¼
η02 ,
e.g.
1 0 b: ½η ; a η02 2
−ðη02 Þ″
þ
bη02 q
¼ λn η02 ,
n≥1, shows
ð6:4Þ
Next, the Darboux Lemma applied to Eq. (6.2) and to (5.12), shows that a non-trivial a is a¼
1 ½h; a ̌; h
ð6:5Þ
where h is defined in (5.11). Note that this a ̌ and a are just one of each of the families of a ̌'s and a's corresponding to q ̌ and q, respectively; all others may be found by using the expression in Eq. (3.4). We can find a just as we found km;t . Substituting (6.4) in (6.5) and elaborating we have, a part a factor λn , b−t a¼a
wn;t 0 b; ½η ; a λn ωn;t 2
n≥1;
λn−1 oλn þ t oλnþ1 :
ð6:6Þ
We now have a one-parameter family of functions aðxÞ≡at ðxÞ defined for 0≤x≤1 and t such that λn−1 o λn þ t oλnþ1 , n≥1, with λ0 ¼ 0. These functions correspond to a “physical” rod if each member of the family is of one-sign, say positive, in ½0; 1. To show this, we evaluate a(x) at x ¼0, x ¼1. By (6.6) we obtain að0Þ λn þ t ¼ ; bð0Þ λn a
ð6:7Þ
að1Þ ðη ð1Þ þ dðtÞη2 ð1ÞÞ b 0 K η2 ð1Þ≡μ1 : ¼ 1−t 1 bð1Þ a b2 ð1Þ λn a
ð6:8Þ
It is evident from (6.8) that if jtj is small enough (and, of course, λn−1 o λn þ t oλnþ1 , with λ0 ¼ 0)), then μ1 4 0. Next, we can use the Deformation Lemma stated in the first part of the Appendix to show that a(x) given by (6.6) is strictly positive in ½0; 1 whenever jtj o δ1 , for a sufficiently small number δ1 . A concrete determination of δ1 is presented in the Appendix. To complete the determination of the quasi-isospectral rods, we need to examine the end conditions for the new a(x) and show that these are of the form (5.1a)–(5.1c). The eigenfunctions of the new rod are um, where aum ¼ km;t . The left end, at x ¼0, is supported, namely um ð0Þ ¼ 0. A length, but straightforward calculation shows that the end condition at x ¼1 is given by b km;t ð1Þ b n;t ð1Þ þ w′n;t ð1ÞÞ ≡− km;t ð1Þ K; m≥1; b þ t a ð1Þ ½a b; η02 ð1ÞðHw μ1 K ð6:9Þ ½a; km;t ð1Þ ¼ − λn að1Þ að1Þ that is Að1Þu′m ð1Þ þ Kum ð1Þ ¼ 0, where A ¼ a2 , for every t such that λn−1 oλn þ t o λnþ1 . Therefore, the end condition (6.9) corresponds to an elastically restrained end for the rod a if K is positive. This is certainly true if jtj is small enough, say jtj oδ2 (see Appendix). b¼a b g with end conditions (5.1b)–(5.1c) and b2 ; ∞; K Collecting the above results, we can conclude that, given a rod fA spectrum fλm g∞ , for a given integer n, n≥1, there exists a positive number δ, δ ¼ minfδ1 ; δ2 g, such that for jtjo δ we can m¼1 b ∞; K b g, with the exception construct a new rod fA ¼ a2 ; ∞; Kg, which has exactly the same eigenvalues of the original rod fA; of the nth eigenvalue. Therefore, the two rods are quasi-isospectral. b ¼ 0), then ½a b is a cantilever (e.g., K b; η02 ð1Þ ¼ 0 and, by (6.9), the end condition of the new We note that if the original rod a rod a at x¼1 is of cantilever-type ½a; km;t ð1Þ ¼ 0:
ð6:10Þ
b¼a b2 ; ∞; 0g, fA ¼ a2 ; ∞; 0g have exactly the same eigenvalues, with Therefore, for a given integer n, n≥1, the two cantilevers fA the exception of the nth eigenvalue λn , which must satisfy condition (5.4) only. Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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7
7. Constructing rods with a given finite set of eigenvalues Consider a cantilever rod with given cross sectional profile A0 ðxÞ ¼ a20 ðxÞ and eigenvalues fλm ða0 Þg∞ m ¼ 1 , 0 o λ1 ða0 Þ o λ2 ða0 Þ o⋯. We now ask whether it is possible to construct from this cantilever rod a new cantilever rod having prescribed N values of the first N eigenvalues fλ~ m g , with m¼1
0 o λ~ 1 o λ~ 2 o ⋯ o λ~ N :
ð7:1Þ
By the analysis of Section 6 we know how to construct from the cantilever rod A0 ðxÞ a new cantilever rod, say A1 ðxÞ ¼ a21 ðxÞ, so that the eigenvalues fλm ða0 Þg, m≥2, are kept fixed while λ1 ða1 Þ is moving to the desired value λ~ 1 . More precisely, by (6.6), the function a1 ¼ a1 ðxÞ given by a1 ¼ a0 −t
w1;t ½η0 ðn ¼ 1; a0 Þ; a0 λ1 ða0 Þω1;t 2
ð7:2Þ
corresponds to a one-parameter family of rods such that λm ða1 Þ ¼ λm ða0 Þ þ δm1 t;
ð7:3Þ
m≥1, for t such that 0 o λ1 ða0 Þ þ t o λ2 ða0 Þ:
ð7:4Þ
λ~ 1 oλ2 ða0 Þ;
ð7:5Þ
If
then we can determine the parameter t, say t ¼ t 1 , such that λ1 ða1 Þ ¼ λ~ 1 , i.e. t 1 ¼ λ~ 1 −λ1 ða0 Þ:
ð7:6Þ
The rod A1 ¼ a21 has eigenvalues fλ~ 1 ; λ2 ða0 Þ; λ3 ða0 Þ; …g, with 0 o λ~ 1 oλ2 ða0 Þ oλ3 ða0 Þ o ⋯, and can be used as starting point for the next step of the procedure. By repeating the same arguments, it is possible to modify a1 so as to keep λm ða1 Þ fixed for m≠2 and to move λ2 ða0 Þ to the desired value λ~ 2 , i.e. a2 ¼ a1 −t 2
w2;t 2 ½η0 ðn ¼ 2; a1 Þ; a1 ; λ2 ða0 Þω2;t2 2
ð7:7Þ
where t 2 ¼ λ~ 2 −λ2 ða0 Þ:
ð7:8Þ
The eigenvalues of the new rod a2 ðxÞ are fλ~ 1 ; λ~ 2 ; λ3 ða0 Þ; λ4 ða0 Þ; …g. Using repeatedly this procedure, after N steps we produce a rod, with cross-sectional profile area AN ðxÞ ¼ a2N ðxÞ, such that λm ðaN Þ ¼ λ~ m ;
for 1≤m≤N;
ð7:9Þ
and the construction is finished. We note that the choice of the initial rod a0 ðxÞ is restricted by the conditions λ~ 1 oλ2 ða0 Þ; λ~ 2 o λ3 ða0 Þ; …; λ~ N−1 oλN ða0 Þ; λ~ N oλNþ1 ða0 Þ;
ð7:10Þ
which allow to determine the numbers t 1 ; t 2 ; …; t N by the expressions analogous to Eq. (7.6). The above construction is not unique, since the flow from the initial cantilever rod a0 to a cantilever rod with prescribed first N eigenvalues depends on the particular order chosen to move every individual eigenvalue to the target value. Similarly, the compatibility conditions on the initial rod a0 may change depending on the sequence of eigenvalue shifts. Finally, we observe that the above construction of quasi-isospectral rods can be adapted to rods under (S-ER) end conditions. In this case, the construction of quasi-isospectral rods works when the shift parameter t (of the generic eigenvalue) belongs to a sufficiently small neighborhood of zero, as it was shown in Section 6. 8. Examples In this section we present some applications of the above results and we determine examples of rods AðxÞ ¼ a2 ðxÞ under b cantilever end conditions which are quasi-isospectral to the uniform rod with γ ¼ 1, E¼1 and AðxÞ≡1 in ½0; 1. The families of rods shown in Figs. 1a–c, 2a–c, 2a–c, 3a–c have been derived using Eq. (6.6) for a ¼ aðxÞ, with n ¼ 1; 2; 5 and for some discrete values of t satisfying (5.4). It can be seen that when the parameter t is close to the lower or upper limit in (5.4), or n is taken to be large, then the profile departs significantly from the uniform rod. For these situations the vibration of the rod could be not adequately described by the classical mechanical model of Eq. (2.1a)–(21.c) and other vibrational models would have to be considered. Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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8
4
3
2
1
0 0.0
0.2
0.4
0.6
0.8
1.0
x t = 0.0
t = -0.2 λ 1
t = -0.4 λ 1
t = -0.9 λ 1
20
15
10
5
0 0.0
0.2
0.4
0.6
0.8
1.0
x t = 0.0
t = 0.1 ( λ 2 − λ 1 )
t = 0.2 ( λ 2 − λ 1 )
t = 0.4 ( λ 2 − λ 1 )
500
375
250
125
0 0.0
0.2
0.4
t = 0.0 t = 0.7 ( λ 2 − λ 1 )
0.6
x
0.8
1.0
t = 0.5 ( λ 2 − λ 1 ) t = 0.9 ( λ 2 − λ 1 )
b Fig. 1. (a–c) Examples of quasi-isospectral cantilever rods A ¼ AðxÞ with γ ¼ 1, E¼ 1 and AðxÞ≡1 in ½0; 1, for n¼ 1 and for some discrete values of t.
9. Extensions The analysis developed in the previous sections can be extended to rods under different sets of boundary conditions. We recall that the end conditions are assumed to be Elastically Restrained (ER) when ðERÞ
Að0Þu′ð0Þ−kuð0Þ ¼ 0 ¼ Að1Þu′ð1Þ þ Kuð1Þ;
ð9:1Þ
where k, K≥0 and k þ K 4 0. Conditions (9.1) mean that the ends x ¼0, x ¼1 are attached to fixed supports by means of elastic springs having stiffness k, K, respectively. The cases of Supported (S), Free (F) and Supported-Elastically Restrained (SER) end conditions are obtained as limit cases of (9.1), precisely ðSÞ
k¼∞¼K
uð0Þ ¼ 0 ¼ uð1Þ;
ð9:2Þ
ðFÞ
k¼0¼K
u′ð0Þ ¼ 0 ¼ u′ð1Þ:
ð9:3Þ
Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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9
4
3
2
1
0 0.0
0.2
0.4
t = 0.0
x
0.6
0.8
1.0
t = 0.2 ( λ 1 − λ 2 )
t = 0.4 ( λ 1 − λ 2 )
t = 0.9 ( λ 1 − λ 2 )
4
3
2
1
0 0.0
0.2
0.4
t = 0.0
x
0.6
0.8
1.0
t = 0.1 ( λ 3 − λ 2 )
t = 0.2 ( λ 3 − λ 2 )
t = 0.4 ( λ 3 − λ 2 )
40
30
20
10
0 0.0
0.2
0.4
t = 0.0
0.6
x
t = 0.7 ( λ 3 − λ 2 )
0.8
t = 0.5 ( λ 3 − λ 2 ) t = 0.9 ( λ 3 − λ 2 )
b Fig. 2. (a–c) Examples of quasi-isospectral cantilever rods A ¼ AðxÞ with γ ¼ 1, E¼ 1 and AðxÞ≡1 in ½0; 1, for n¼ 2 and for some discrete values of t.
ðS−ERÞ
k ¼ ∞;
0≤K o∞
uð0Þ ¼ 0;
Að1Þu′ð1Þ þ Kuð1Þ ¼ 0;
ð9:4Þ
In this paper, we have considered in detail the problem of constructing rods with given natural frequencies for the (S-ER) case. It has been shown that the crucial point of the procedure lies in the ability to construct quasi-isospectral systems, that is rods which share with a given rod all the natural frequencies, with the exception of a single eigenvalue. Therefore, for the sake of brevity, in this section we focus on the main analogies and differences of the treatment for (ER), (S) and (F) end conditions with respect to the (S-ER) case. More details can be found in [21]. 1. The analysis of the (S) case can be developed following the same lines of the (S-ER) end conditions. The main difference, when the stiffness of the spring acting at the right side of the (S-ER) rod is strictly positive, is that, in the (S) case, the parameter t appearing in the definition of the cross-sectional profile needs to satisfy a condition analogous to (5.4) only. That is, the further restriction jtj oδ, where δ is a sufficiently small number (see the second part of Section 6 and Appendix), is not necessary for the (S) end conditions case. Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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10
4
3
2
1
0 0.0
0.2
0.4
0.6
x
t= 0.0
0.8
1.0
t= 0.2 ( λ 4 − λ 5 )
t= 0.4 ( λ 4 − λ 5 )
t =0 .9 ( λ 4 − λ 5 )
4
3
2
1
0 0.0
0.2
0.4
0.6
x
t= 0.0
0.8
1.0
t= 0.1 ( λ 6 − λ 5 ) t =0 .4 ( λ 6 − λ 5 )
t= 0.2 ( λ 6 − λ 5 )
6 5 4 3 2 1 0 0.0
0.2
0.4
t= 0.0 t= 0.7 ( λ 6 − λ 5 )
0.6
x
0.8
1.0
t= 0.5 ( λ 6 −
λ 5)
t =0 .9 ( λ 6 −
λ 5)
b Fig. 3. (a–c) Examples of quasi-isospectral cantilever rods A ¼ AðxÞ with γ ¼ 1, E ¼1 and AðxÞ≡1 in ½0; 1, for n¼ 5 and for some discrete values of t.
2. The determination of quasi-isospectral rods under (F) end conditions can be developed on the basis of the results of the (S) case. In fact, under suitable regularity assumptions on the cross-sectional profile, it can be shown that the eigenvalue problem for a (F) rod can be transformed in an “equivalent” eigenvalue problem for a (S) rod. 3. The case of rods under (ER) end-conditions is different from the previous ones. There are two main results for this set of boundary conditions. First, it can be shown that there exists a way to explicitly construct a rod under (S) end conditions such that all the eigenvalues of the initial (ER) rod coincide with those of the new (S) rod, with the exception of the smallest eigenvalue. Therefore, on adapting the arguments for (S) rods recalled in (1), it is possible to construct new (S) rods which share the same higher spectrum (e.g., all the eigenvalues with the exception of the smallest one) of the initial (ER) rod, with the exception of a single eigenvalue. The above characterization of quasi-isospectral (ER) rods shows that there is no control over the smallest eigenvalue. The second result to be mentioned is the possibility to construct families of (ER) rods quasi-isospectral to a given (ER) rod such that they preserve the higher spectrum and change only the Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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11
smallest eigenvalue in a prescribed manner. As in the (S-ER) case, the quasi-isospectral families depend on a real parameter t and, in order to have physical rods (e.g., positive cross-sectional area and positive elastic stiffness at the ends), t must belong to a sufficiently small neighborhood of the origin.
10. Concluding remarks We have shown that we may construct one-parameter t-families of rods with prescribed values of a finite set of natural frequencies under a given set of boundary conditions. Details of the method have been presented for rods under supportedelastically restrained end conditions, but the results can be extended to other end conditions. The analysis is based on a double application of the Darboux Lemma after reducing the equation of the free vibrations to normal form by a Liouville transformation. The rods with given natural frequencies can be determined explicitly by means of closed-form expressions. The reconstruction procedure needs the specification of an initial rod whose eigenvalues must be close to the assigned ones. In connection with this aspect, we point out that a theoretical aspect worth of investigation stands on the possibility of characterizing the full set of rods that could be chosen as starting point in our procedure. The analysis of the analogous problem for a special class of beams under bending vibration is presented in [22].
Acknowledgments The collaboration of Michele Dilena in preparing the figures and numerical simulations is gratefully acknowledged. Appendix A In this Appendix we recall a technical result which has been used in Section 5 and in Section 6 to count the number of zeros of the functions km;t ðxÞ (defined by (5.25)) and a(x) (defined by (6.6)), respectively. Moreover, we provide the details of the determination of the function d(t), defined by (5.21) and (5.22), and of the determination of the parameter δ ¼ minfδ1 ; δ2 g introduced at the end of Section 6. Lemma 2 (Deformation Lemma). Let ft, t 1 ≤t≤t 2 , be a family of real valued functions of x in ½0; 1 which is jointly continuously differentiable in t and in x. Suppose that for every t, ft has a finite number of zeros in ½0; 1, all of which are simple, and has boundary values with signs that are independent of t. Then, the number of zeros of ft in ½0; 1 is independent of t, for all t satisfying t 1 ≤t≤t 2 . This lemma has been introduced in [19], see also [14] for a slightly different version in which it is supposed that ft has boundary values that are independent of t. Here, we apply the lemma to count the number of zeros in ½0; 1 of the function km;t defined by (5.25). We start by noticing that, as solution to the differential equation (5.26), the function km;t can have only a finite number of roots in ½0; 1, and that these roots must be simple. Moreover, km;t can be normalized by assuming km;t ð1Þ ¼ 1, for every integer number m≥1 and for every real number t satisfying the upper and lower bounds (5.4). Therefore, the Deformation Lemma implies that km;t has exactly ðm−1Þ zeros in ð0; 1Þ. We conclude that km;t is the mth eigenfunction of the problem (5.26)–(5.28) associated to the potential q. Concerning the function d(t), by imposing condition (5.22) and observing that Z 1 ½η1 ; η02 ðx ¼ 1Þ ¼ 1 þ t η1 η02 ds; ðA:1Þ 0
Z ½η2 ; η02 ðx ¼ 1Þ ¼ t
1 0
η2 η02 ds;
ðA:2Þ
we find R1 dðtÞ ¼ − R01 0
η1 η02 ds η2 η02 ds
;
ðA:3Þ
and d(t) is well-defined provided that the denominator in (A.3) does not vanish for all t, with λn−1 o λn þ t o λnþ1 (λ0 ¼ 0). R1 If the denominator does vanish at some t , t ≠0, and λn−1 o λn þ t o λnþ1 , then 0 η2 ðs; t Þη02 ðsÞ ds ¼ 0. This means that η2 ðx; t Þ belongs to the subspace of functions orthogonal to the nth eigenfunction zn(x), that is η2 ðx; t Þ ¼ ∑∞ k ¼ 1;k≠n αk zk ðxÞ. Substituting this expression in Eq. (5.6a) and elaborating we have ∞
∑
k ¼ 1;k≠n
αk ðλk −ðλn þ tÞÞzk ðxÞ ¼ 0;
x∈ð0; 1Þ;
ðA:4Þ
and then αk ¼ 0 for every k, k≥1 and k≠n. Thus η2 ðx; t Þ≡0, a contradiction. We conclude that d(t) is always well-determined for all t's such that λn−1 o λn þ t o λnþ1 . Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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The determination of δ is based on estimates of the functions η1 , η2 , η02 , and their first x-derivative, and of d(t), see, for example, [14] (Theorem 3, Chapter 1). We first show how to determine δ1 such that the constant μ1 in (6.8) be strictly positive. Hereinafter, we assume that there exists ϵ 4 0 such that 0 oϵ oλ0 . For a given integer number n, n≥1, and for q∈L2 ð0; 1Þ we have pffiffiffiffiffiffiffiffiffiffiffiffiffi MðxÞ∥q∥ η1 ≤1 þ pMðxÞ ffiffiffiffiffiffiffiffiffiffiffiffiffi ; jη′1 j≤ λn þ t 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðA:5Þ λn þ t λn þ t 1 MðxÞ η2 ≤ pffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; λn þ t λn þ t 0 1 η ≤ pffiffiffiffiffi 1 þ MðxÞ ffiffiffiffi ffi p ; 2 λn λn
ðA:6Þ
MðxÞ∥q∥ pffiffiffiffiffi ; λn
ðA:7Þ
jðη02 Þ′j≤1 þ
where
pffiffiffi MðxÞ ¼ expð∥q∥ xÞ;
MðxÞ∥q∥ jη′2 j≤1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; λn þ t
Z ∥q∥ ¼
1
!1=2 2
jqðxÞj dx
0
:
By (6.8) and in order to control d(t) from above, we need to estimate from below the denominator function
ðA:8Þ R1 0
η2 η02 dx in Eq. (A.3). The
v ¼ η2 −η02
ðA:9Þ
is the unique solution of the initial value problem 8 b > < −v″ þ q v ¼ λn v þ tη2 ; x∈ð0; 1Þ; ðaÞ vð0Þ ¼ 0; > : v′ð0Þ ¼ 0:
ðbÞ
ðA:10Þ
ðcÞ
The solution of (A.10a)–(A.10c) is given by Z x ðη01 ðxÞη02 ðsÞ−η01 ðsÞη02 ðxÞÞη2 ðs; tÞ ds; vðx; tÞ ¼ t
ðA:11Þ
0
bÞ ¼ η1 ðx; q b; λn Þ. By estimates (A.5)–(A.7) and recalling that λn−1 o λn þ t oλnþ1 , we obtain the following uniform where η01 ðx; q estimate in ½0; 1: 0 12 2jtj Mð1Þ Mð1Þ C vðx; tÞ≤ pffiffiffiffiffiffiffiffiffipffiffiffiffiffi 1 þ pffiffiffiffiffiffiffiffiffi B @1 þ pffiffiffiffiffi A ≡C 1 jtj: λn−1 λn−1 λn λn By (A.12) and by Cauchy–Schwartz inequality we have Z 1 η2 η02 dx≥∥η02 ∥ð∥η02 ∥−C 1 jtjÞ
ðA:12Þ
ðA:13Þ
0
and then, by (A.3), we find dðtÞ≤
∥η1 ∥ : ∥η02 ∥−C 1 jtj
ðA:14Þ
If, for example, jtjo ∥η02 ∥=2C 1 , then we obtain a uniform estimate of d(t) Mð1Þffi 2 1 þ pffiffiffiffiffiffi λn−1 dðtÞ≤ ≡C 2 ; ∥η02 ∥ ! ∥ζ 0 ∥ ∥ζ 0 ∥ for t∈ − 2 ; 2 and λn−1 o λn þ t o λnþ1 : 2C 1 2C 1
ðA:15Þ
By the above estimates, the coefficient μ1 is strictly positive if the parameter t satisfies the conditions (A.15) and jt j o
1þ
Mð1Þ p ffiffiffiffiffiffiffi λn−1
3=2 bð1ÞÞ2 λ ða n : 2 ffi pffiffiffiffi 1 þ Mð1Þ 1 þ pCffiffiffiffiffiffi λ λ n
ðA:16Þ
n−1
This completes the determination of δ1 . Similar calculations, not reported here for brevity, show that there exists a neighborhood of the origin ð−δ2 ; δ2 Þ, δ2 4 0, such that for t∈ð−δ2 ; δ2 Þ the stiffness K in (6.9) is positive. To conclude it is enough to take δ ¼ minðδ1 ; δ2 Þ. This completes the determination of δ. Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i
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Please cite this article as: A. Morassi, , Constructing rods with given natural frequencies, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.04.010i