Asymptotic Formulas for the Eigenvalues of the Timoshenko Beam

Asymptotic Formulas for the Eigenvalues of the Timoshenko Beam Bruce Geist

1

Daimler Chrysler Corporation, 800 Chrysler Drive Auburn Hills, MI 48326; [email protected]

and Joyce R. McLaughlin

1

Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, NY 12180; [email protected]

Asymptotic formulas are derived for the eigenvalues of a free-ended Timoshenko beam which has variable mass density and constant beam parameters otherwise. These asymptotic formulas show how the eigenvalues (and hence how the natural frequencies) of such a beam depend on the material and geometric parameters which appear as coefficients in the Timoshenko differential equations.

Key Words: eigenvalue asymptotics, Timoshenko beam models

Subject classifications:. 34L (ordinary differential operators), 73D (wave propagation in and vibration of solids) 1. INTRODUCTION Suppose a structural beam is driven by a laterally oscillating sinusoidal force. As the frequency of this applied force is varied, the response varies. Experimental frequencies for which the response is maximized are called 1 Acknowledgment: The work of both authors was completed at Rensselaer Polytechnic Institute, and was partially supported by funding from the Office of Naval Research, grant number N00014-96-1-0349. The work of the first author was also partially supported by the Department of Education fellowship grant number 6-28069. The work of the second author was also partially supported by the National Science Foundation, grant number DMS-9802309.

1

2

GEIST AND MCLAUGHLIN

natural frequencies of the beam. Our goal is to address the question: if a beam’s natural frequencies are known, what can be inferred about its bending stiffnesses or its mass density? To answer this question we need to know asymptotic formulas for these frequencies. Here we establish such formulas for beams with variable mass density but otherwise constant beam parameters. We make this assumption as a first step toward solving the problem with both variable density and variable stiffness. We also make this assumption because it is consistent with some applications of interest to us. An example of an application consistent with our assumption is an aircraft wing with struts which have been added so that there is an appreciable change in the density and a minimal change in stiffness. One widely used mathematical model for describing the transverse vibration of beams was developed by Stephen Timoshenko in the 1920s. This model is chosen because it is a more accurate model than the Euler-Bernouli beam model and because systems of Timoshenko beam models are used to model aircraft wings. The mathematical equations that arise are two coupled partial differential equations, (EIψx )x + kAG(wx − ψ) − ρIψtt = 0,

(kAG(wx − ψ))x − ρAwtt = P (x, t). The dependent variable w = w(x, t) represents the lateral displacement at time t of a cross section located x units from one end of the beam. ψ = ψ(x, t) is the cross sectional rotation due to bending. E is Young’s modulus, i.e., the modulus of elasticity in tension and compression, and G is the modulus of elasticity in shear. The non-uniform distribution of shear stress over a cross section depends on cross sectional shape. The coefficient k is introduced to account for this geometry dependent distribution of shearing stress. I and A represent cross sectional inertia and area, ρ is the mass density of the beam per unit length, and P (x, t) is an applied force. If we suppose the beam is anchored so that the so called “free-free” boundary conditions hold (i.e., shearing forces and moments are assumed to be zero at each end of the beam), then w and ψ must satisfy the following four boundary conditions,

wx − ψ|x=0,L = 0,

ψx |x=0,L = 0.

(1)

EIGENVALUES OF THE TIMOSHENKO BEAM

3

After making a standard separation of variables argument, one finds that the Timoshenko differential equations for w and ψ lead to a coupled system of two second order ordinary differential equations for y(x) and ψ(x),

(EIΨx )x + kAG(yx − Ψ) + p2 IρΨ = 0,

(2)

(kAG(yx − Ψ))x + p2 Aρy = 0.

(3)

Here, p2 is an eigenvalue parameter. The conditions on w and ψ in (1) imply y and Ψ must satisfy the same free-free boundary conditions. We must have

yx − Ψ|x=0,L = 0,

Ψx |x=0,L = 0.

(4)

This boundary value problem for y and Ψ is self-adjoint, which implies that the values of p2 for which nontrivial solutions to this problem exist; the eigenvalues for this model, are real. Furthermore, it is not difficult to show that the collection of all eigenvalues for this problem forms a discrete, countable, unbounded set of real non-negative numbers. Moreover, it can be shown that if σ is a natural frequency for a beam, then p2 = (2πσ)2 is one of the beam’s eigenvalues. Therefore, it is possible to determine eigenvalues from natural frequency data obtained in an experiment like the one indicated in the opening paragraph. Suppose from vibration experiments we have determined a set of natural frequencies for a beam with unknown elastic moduli and mass density, and have constructed a sequence of eigenvalues from this data. What information can the eigenvalues provide about these unknown material parameters? To address this question, we must determine how eigenvalues depend on E, I, kG, A and ρ. This determination is not easy, since the dependence of eigenvalues on these coefficients is highly nonlinear. Another difficulty arises because the Timoshenko boundary value problem involves two second order differential equations. When the coefficients in these differential equations are non-constant, the system of two second order equations cannot be transformed into a single fourth order equation. Therefore, to make progress in the case where coefficients are non-constant, the boundary-value problem must be handled as a system of equations. For a simpler, Sturm-Liouville type boundary value problem,

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GEIST AND MCLAUGHLIN

y

(z) + (λ − q(z))y(z) = 0, 0 ≤ z ≤ 1,

(5)

y
(0) − hy(0) = y
(1) + Hy(1) = 0, it is known that for square integrable q(z), nontrivial solutions y(z) for this problem exist if and only if λ = µn , where 

1

µn = n2 π 2 + Cq −

q(z)cos(2nπz)dz + αn ,

(6)

0

 Cq = 2h + 2H +

1

q(z)dz, 0

∞ 

αn2 < ∞,

n=1

(See Hald-McLaughlin [12, pp. 313-314] as well as Isaacson-Trubowitz [14], Borg [2], and Fulton-Pruess [6, 7]; for other Sturm-Liouville equations with, effectively, less smooth coefficients, see Coleman-McLaughlin [5].) The importance of equation (6) is that it shows how the eigenvalues for the Sturm-Liouville problem are related to the coefficient q(z) appearing in (5). Algorithms for reconstructing q from spectral data rely strongly on asymptotic formulas like the one given in (6). (For example, see Hald [11] and Rundell-Sacks [18].) Returning now to the Timoshenko beam equations, we ask the question: what information is contained in the eigenvalues for the Timoshenko beam? Given a sequence of eigenvalues, can we infer knowledge about the beam parameters which give rise to these eigenvalues? Asymptotic formulas for the Sturm-Liouville eigenvalues are critical to determining q from spectral data. We expect that analogous formulas for the Timoshenko eigenvalues will play a key role in recovering beam parameters like E, kG, or ρ from such data. In this paper, our objective is to determine asymptotic formulas for the eigenvalues of the Timoshenko beam when free-free boundary conditions are enforced and when ρ is allowed to vary. We suppose that E, kG, A and I are constants and assume ρ is a positive function of x on [0, L] such that 0 = ρx (0) = ρx (L) and ρxx is in L∞ (0, L). Under these assumptions, we derive asymptotic formulas for the eigenvalues of the free-free Timoshenko beam. In the next three sections, approximations are derived (accurate to within an error that is O(1/p)) for the square roots of eigenvalues of free-free beams

EIGENVALUES OF THE TIMOSHENKO BEAM

5

with variable density. An important step in deriving these preliminary approximations of the eigenvalues is the use of a transformation (see section 1) which changes the Timoshenko differential equations (2) and (3) into a new pair of differential equations, where in the new equations, the coefficient which contains the eigenvalue parameter p2 no longer depends on the independent variable. The key feature of the transformed system is that the largest terms in the new differential equations, and hence the most important terms, are multiplied by coefficients which do not depend on the new independent variable. As the eigenvalue parameter grows, solutions to the transformed differential equations approach the solutions of a certain set of constant coefficient differential equations. It is therefore possible to derive an approximate solution (accurate to within an O(1/p) error) to an initial value problem in which initial conditions are chosen so that the left transformed boundary conditions are enforced. In section 2, this initial value problem is presented and its approximate solution is derived. By applying the two remaining transformed right boundary conditions to the approximate solution of the initial value problem, a frequency equation is determined. In section 3, estimates of square-roots of eigenvalues are made from this frequency equation. These estimates appear in Theorem 4.3. Our approach to deriving the final asymptotic formulas (section 4) is L built from the following idea. Suppose ρ0 ≡ ( L1 0 ρ1/2 (x)dx)2 , and let ρˆ(x; t) ≡ ρ0 + t˜ ρ(x), where ρ˜ = ρ − ρ0 and t is an auxiliary parameter which we allow to vary from 0 to 1. In the Timoshenko differential equations, let ρ(x) be replaced by ρˆ(x; t). Define p´2 to be an eigenvalue for a free-free beam with mass density ρˆ and constant material and geometric parameters otherwise. When t = 0, ρˆ = ρ0 and p´2 is an eigenvalue for a beam where E, I, kG, A, and ρˆ = ρ0 are all independent of x. As t increases to 1, ρˆ goes to ρ(x), and p´2 changes continuously in t to an eigenvalue for a beam with  ˆ 1 = L ρˆ1/2 (x; t)dx, and define µ2 ≡ L ˆ 2 p´2 . We variable density ρ(x). Let L 1 0 show that there is a function G such that d(µ2 ) ˆ = G(Yˆ , Φ), dt

(7)

ˆ is a transformed eigenfunction pair corresponding to the where (Yˆ , Φ) eigenvalue p´2 of a free-free beam with material parameters E, I, kG, A, and ρˆ. Integrating (7) formally with respect to t from 0 to 1, we find that p´2 |t=1 − p´2 |t=0 =

1 L1

 0

1

ˆ G(Yˆ , Φ)dt.

(8)

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GEIST AND MCLAUGHLIN

The term p´2 |t=0 is an eigenvalue for a beam where E, I, kG, A and ρ = ρ0 are independent of x; i.e. p´2 |t=0 represents an eigenvalue for a uniform beam. Asymptotic formulas for the eigenvalues of free-free and clamped-clamped uniform beams are derived in Geist [8] and published in [10]. From these uniform beam formulas, an asymptotic approximation for the term p´2 |t=0 can be made. The final asymptotic eigenvalue formulas for beams with variable density ρ(x) are obtained by replacing the term 1 1 ˆ ˆ L1 0 G(Y , Φ)dt with an asymptotic approximation derived below, and by replacing p´2 |t=0 with the appropriate uniform beam eigenvalue formulas given in [10]. The function G depends only on E, I, kG, A, ρ and the transformed ˆ and Yˆ . Approximations to the square roots of eigenvalues eigenfunctions Φ ˆ and Yˆ and hence G to given in Theorem 4.3 allows us to determine Φ within an error that is O(1/p). Then (8) is used to sharpen our estimates of the eigenvalues for the Timoshenko beam. From (8) we compute the final asymptotic formulas, which are given in Theorem 5.2. Note that the advantage of this method over say a variational method is that we can determine more than the first term in the eigenvalue expansion and prove a bound in the remainder no matter how large the difference is between ρ and ρ0 . 2. THE TRANSFORMED PROBLEM To begin, the free-free Timoshenko boundary value problem is proved equivalent to a certain transformed boundary value problem derived below. This equivalency holds when ρ depends on x; all other beam parameters are assumed constant. A key feature of the transformed problem is that the coefficient containing the eigenvalue parameter no longer depends on the independent variable. To derive this equivalent problem, a lemma is proved which applies to single second order equations. We will use this lemma to prove Theorem 2.1, in which the Timoshenko system of differential equations is transformed to a new pair of equations. Lemma 2.1. Suppose ρ(x) is positive for all x ∈ [0, L] and ρxx (x) ∈ x L L2 [0, L]. Let L1 = 0 ρ1/2 (x1 )dx1 and z(x) = L11 0 ρ1/2 (x2 )dx2 . Let β, α, I, and p2 be constants. Then v(x) = ρ−1/4 V (z(x)) satisfies the equation

αv

+ (p2 τ ρ − β)v = f

(9)

EIGENVALUES OF THE TIMOSHENKO BEAM

7

if and only if V (z) satisfies  Vzz +

 (ρ−1/4 )xx L21 τ L21 p2 βL21 f L21 + . − V = α αρ ρ3/4 αρ3/4

(10)

proof: Suppose v = A(x)V (z(x)), where A(x) and z(x) are as yet unspecified smooth functions of x. Then αv

= αA(z
)2 Vzz +

(αA2 z
)
Vz + αA

V, A

and provided α(z
)2 A = 0 for x ∈ [0, L], it follows that αv

+ (p2 τ ρ − β)v = f (x) if and only if   f (A2 z
)
A

p2 τ ρ β Vzz + 2
2 Vz + V = + − . A (z ) A(z
)2 α(z
)2 α(z
)2 α(z
)2 A x Now let A = ρ−1/4 (x) and z(x) = L11 0 ρ1/2 (x2 )dx2 . Then (A2 z
)
≡ 0, A

/A(z
)2 = (ρ−1/4 )xx L21 /ρ3/4 , p2 τ ρ/(α(z
)2 ) = τ L21 p2 /α, and β/(α(z
)2 ) = βL21 /αρ, so equation (9) becomes equation (10). ✷ In the next theorem, new differential equations are determined that are related to the Timoshenko equations by the transformation indicated in the previous lemma. Let L1 and z(x) be defined as they are in Lemma 2.1, and let µ2E = p2 L21 /E, µ2kG = p2 L21 /kG, and γ = kAG/EI. Since ρ(x) is positive for all x ∈ [0, L], z(x) is an invertible function. Let x(z) denote the inverse of z(x), and let ρ3 = L21 [ρ−1/4 (x(z))]xx /[ρ3/4 (x(z))] and ρ4 = ρx (x(z))L21 /[4ρ2 (x(z))]. Theorem 2.1. Let E, kG, I, A, and p2 all be constants, let ρ(x) be

positive for all x ∈ [0, L], and let ρxx (x) ∈ L2 (0, L). Then y(x) = ρ−1/4 Y (z(x)) and Ψ(x) = ρ−1/4 Φ(z(x)) satisfy the equations EIΨxx + (p2 Iρ − kAG)Ψ = −kAGyx + F1 ,

(11)

kAGyxx + p2 Aρy = kAGΨx + F2

(12)

and

if and only if Φ and Y satisfy

8

GEIST AND MCLAUGHLIN

Φzz +

p2 L21 γL1 F1 L21 L2 γ , Φ − 1 Φ + ρ3 Φ − γρ4 Y + 1/2 Yz = E ρ ρ EIρ3/4

(13)

p2 L21 L1 F2 L21 . Y + ρ3 Y + ρ4 Φ − 1/2 Φz = kG ρ kAGρ3/4

(14)

and

Yzz +

proof: This Theorem is an immediate consequence of Lemma 2.1. ✷ Theorem 2.1 shows how to transform the original Timoshenko differential equations into new equations so that in the new equations, coefficients involving p2 are constant with respect to the new independent variable z. Equations (11) - (14) include generic “right hand side” terms so that Theorem 2.1 is general enough that it applies to differential equations that arise in the next section. In the case where F1 = F2 ≡ 0, equations (11) and (12) are the homogeneous Timoshenko differential equations (2) and (3). Theorem 2.2. Suppose ρ(x) is a positive function of x such that ρxx ∈ L2 (0, L) and ρx (0) = ρx (L) = 0. Let E, kG, A, I and p2 be positive constants. Then nontrivial Y and Φ are solutions to the boundary value problem

Φzz +

L2 γ p2 L21 γL1 Φ − 1 Φ + ρ3 Φ − γρ4 Y + 1/2 Yz = 0, E ρ ρ

(15)

p2 L21 L1 Y + ρ3 Y + ρ4 Φ − 1/2 Φz = 0, kG ρ

(16)

Yzz + 

 L1 Yz − 1/2 Φ = 0, and Φz |z=0,1 = 0, ρ z=0,1

(17)

if and only if y(x) = ρ−1/4 Y (z(x)) and Ψ(x) = ρ−1/4 Φ(z(x)) are nontrivial solutions to the free-free Timoshenko boundary value problem (2) - (4). The value p2 is an eigenvalue for the free-free Timoshenko boundary value problem if and only if there exist nontrivial functions Y and Φ which satisfy

EIGENVALUES OF THE TIMOSHENKO BEAM

9

the differential equations (15) and (16) and the boundary conditions given in (17). proof: This theorem is an easy consequence of Theorem 2.1 and the fact that when ρx |x=0,L = 0, (yx − ψ)|x=0,L = 0 if and only if (Yz − L1 Φ/ρ1/2 )|z=0,1 = 0 and ψx |x=0,L = 0 if and only if Ψz |z=0,1 = 0. ✷ 3. A FREQUENCY EQUATION Consider the following initial value problem. Let α = L1 /ρ1/2 (0). Suppose that Y and Φ satisfy differential equations (15) and (16) and that for real c and d, they also satisfy Y (0) = d, Φ(0) = c/α, Yz (0) = c, and Φz (0) = 0. These initial conditions ensure that the boundary conditions on Y and Φ at z = 0 given in (17) are satisfied no matter how c and d are chosen. They are the least restrictive initial conditions on Y (z) and Φ(z) which enforce the transformed boundary conditions at the left boundary point z = 0. Furthermore, if nontrivial c and d can be chosen so that the boundary conditions at z = 1 are satisfied, then by definition L21 p2 is an eigenvalue for the transformed boundary value problem, and by Theorem 2.2, p2 is an eigenvalue for the free-free Timoshenko boundary value problem. In the next two lemmas, integral equations for Y and Φ are derived which are equivalent to the initial value problem discussed above. These integral equations are used to determine approximate solutions to the above initial value problem. The approximate solutions to the initial value problem make possible estimates of the values of p for which nontrivial solutions exist to the transformed boundary value problem. Lemma 3.1. Let µ, c and d be fixed constants, and let q(z) and f (z) be integrable. Then w(z) is the solution to the initial value problem

w

+ µ2 w = q(z)w + f (z), w
(0) = c, and w(0) = d,

(18)

if and only if w satisfies the integral equation  z sin(µz) sin[µ(z − t)] w(z) = c + d cos µz + [q(t)w(t) + f (t)]dt. (19) µ µ 0

proof: The elementary proof is based on the well known technique of variation of parameters, and is omitted. ✷

10

GEIST AND MCLAUGHLIN

Theorem 3.1. Suppose ρ(x) is a positive function of x such that ρxx ∈ L2 (0, L) and ρx (0) = ρx (L) = 0. Let E, kG, A, I and p2 be positive constants, let α = L1 /ρ1/2 (0) and let c and d be arbitrary but fixed constants. Then Φ(z) and Y (z) satisfy the differential equations (15) and (16) and the initial conditions

Y (0) = d, Φ(0) = c/α, Yz (0) = c, Φz (0) = 0

(20)

if and only if Φ and Y also satisfy the integral equations c Φ = cos µE z + α  2    z L1 γ sin[µE (z − t)] γL1 − ρ3 Φ + γρ4 Y − 1/2 Yt dt µE ρ ρ 0

(21)

and c sin(µkG z) + µkG    z sin[µkG (z − t)] L1 d cos(µkG z) + −ρ3 Y − ρ4 Φ + 1/2 Φt dt (22) µkG ρ 0

Y =

proof: First observe that since ρxx ∈ L2 (0, L), Theorem 2.1 shows that Y and Φ satisfy (15) and (16) if and only if y(x) = ρ−1/4 Y (z(x)) and Ψ(x) = ρ−1/4 Φ(z(x)) satisfy the homogeneous Timoshenko differential equations. All coefficients in the Timoshenko equations (2) and (3) are continuously differentiable. Since yx and Ψx must be continuous, it follows that Yz and Ψz are also continuous. Next, apply Lemma 3.1 to the transformed differential equation (15). Observe that L21 γ/ρ − ρ3 ∈ L2 (0, 1) and that (from the discussion of the previous paragraph) (γρ4 Y − [γL1 /ρ1/2 ]Yz ) ∈ L2 (0, 1). Lemma 3.1 shows that the differential equation (15) and initial conditions (20) are satisfied if and only if integral equation (20) holds. Similarly, since −ρ3 and −ρ4 Φ + [L1 /ρ1/2 ]Φz are in L2 (0, 1), Lemma 3.1 shows that (16) and (20) hold if and only if (21) holds. ✷ We will use the integral equations of Theorem 3.1 to determine approximate solutions to the initial value problem given in (15), (16), and (20). If c and d are allowed to vary over , the solution to this initial value problem generates every solution to differential equations (15) and (16) that satisfies

EIGENVALUES OF THE TIMOSHENKO BEAM

11

the left boundary conditions at z = 0. Consider the boundary terms Yz (1) −

L1 Φ(1) ρ1/2 (1)

µkG

and

αΦz (1) , µE

(23)

where Y and Φ are solutions to the initial value problem (15), (16), and (20). Setting the above expressions equal to zero leads to a homogeneous linear system for the arbitrary constants c and d. Values of the eigenvalue parameter p which make possible nontrivial choices for c and d correspond to the square roots of eigenvalues for the free-free Timoshenko beam. The homogeneous linear system in c and d has nontrivial solutions if and only if the determinant of the corresponding coefficient matrix is zero. This determinant will define a frequency function. The objective in this section is to determine this frequency function from estimates of the coefficients of c and d in the expressions given in (23). The following technical fact is used many times in the estimates that follow. Lemma 3.2. Suppose that fz ∈ L∞ (0, 1), and that δ is a real constant. Then for z ∈ [0, 1],  z f ∞ + fz ∞ sin[µ(z − t) + δ]f (t)dt < . µ 0

proof: 

z

sin[µ(z − t) + δ]f (t)dt = 0

z  z cos[µ(z − t) + δ] cos[µ(z − t) + δ] f (t) − ft (t)dt µ µ 0 0

This implies the result. ✷ Lemma 3.3. √ Suppose h and δ are real constants and that h is not equal to 0 or L1 / kG. Let dg/dz and ρxx (x(z)) ∈ L∞ (0, 1), and let ρ(x) > 0 when x ∈ [0, L]. Then  z sin[hµ(z − t) + δ] g(t)Yt dt < O(1/µ) Y ∞ + O(1/µ) Φ ∞ µ 0 + O(1/µ)|c| + O(1/µ)|d|.

12

GEIST AND MCLAUGHLIN

proof: Yz = c cos(µkG z) − dµkG sin(µkG z) +  0

z



 L1 cos[µkG (z − t)] −ρ3 (t)Y (t) − ρ4 (t)Φ(t) + 1/2 Φt (t) dt ρ (t) 

z

⇒ 0



z

0

sin[hµ(z − t) + δ] g(t)Yt dt = µ

sin[hµ(z − t) + δ] g(t){c cos(µkG t) − dµkG sin(µkG t) µ





t

cos[µkG (t − s)][−ρ3 (s)Y (s) − ρ4 (s)Φ(s)]ds} dt

+ 0

+

sin[hµ(z − t) + δ] g(t) µ



t

cos[µkG (t − s)] 0

 L1 (s)ds dt. Φ s ρ1/2 (s)

Since by hypothesis hµ = µkG , a double angle formula, the assumption that dg/dz ∈ L∞ (0, 1), and Lemma 3.2 can be used to show that the absolute value of the first integral on the right of the above equation is bounded above by O(1/µ) Y ∞ + O(1/µ) Φ ∞ + O(1/µ)|c| + O(1/µ)|d|. To demonstrate a similar result for the second integral on the right of the above equation, first note that 

t

cos[µkG (t − s)] 0

L1 Φs (s)ds = ρ1/2 (s)

L1 Φ(t) L1 Φ(0) − 1/2 cos(µkG t) ρ1/2 (t) ρ (0) −

 t µkG sin[µkG (t − s)] 0

L1 + cos[µkG (t − s) ρ1/2 (s)



L1 ρ1/2 (s)

  Φds. s

Therefore, if we show that 

0

z

sin[hµ(z − t) + δ] g(t) µ

 0

t



 L1 1 , µkG sin[µkG (t − s)] 1/2 Φ(s)ds dt < O µ ρ (s)

13

EIGENVALUES OF THE TIMOSHENKO BEAM

then the Lemma follows. After changing the order of integration, the integral above may be rewritten as 



z

s=0

z

t=s

sin[hµ(z − t) + δ] L1 g(t)µkG sin[µkG (t − s)] 1/2 Φ(s)dtds. µ ρ (s)

Again, using a double angle formula and the fact that g
(z) ∈ L∞ (0, 1), we apply Lemma 3.2 to find that the above integral is bounded in absolute value by a function of the form O(1/µ) Φ ∞ . ✷ Lemma 3.4. √ Suppose h and δ are real constants and that h is not equal

to 0 or L1 / E. Let dg/dz and ρxx (x(z)) ∈ L∞ (0, 1), and let ρ(x) > 0 when x ∈ [0, L]. Then 

z

0

sin[hµ(z − t) + δ] g(t)Φt dt < O(1/µ) Y ∞ + O(1/µ) Φ ∞ µ + O(1/µ)|c| + O(1/µ)|d|.

proof: The proof of Lemma 3.4 is similar to the proof of Lemma 3.3, and is therefore omitted. ✷ In the next Theorem, we show that the infinity norms of Y and Φ remain finite as p approaches infinity. Lemma 3.5. Suppose E = kG, that ρxx ∈ L∞ (0, L), and that ρ(x) > 0 for all x ∈ [0, 1]. Then

Φ ∞ ≤ O(1)|c| + O(1/p)|d| and Y ∞ ≤ O(1/p)|c| + O(1)|d|. proof: The integral equations for Φ and Y together with Lemma 3.3 and Lemma 3.4 imply that Φ ∞ ≤ O(1)|c| + O(1/p)|d| + O(1/p) Φ ∞ + O(1/p) Y ∞

(24)

Y ∞ ≤ O(1/p)|c| + O(1)|d| + O(1/p) Φ ∞ + O(1/p) Y ∞ .

(25)

and

14

GEIST AND MCLAUGHLIN

Inequality (24) implies that (1 − O(1/p)) Φ ∞ ≤ O(1)|c| + O(1/p)|d| + O(1/p) Y ∞ . For p large enough, this inequality implies that Φ ∞ ≤ O(1)|c| + O(1/p)|d| + O(1/p) Y ∞ .

(26)

Similarly, from inequality (25), we find that for p large enough Y ∞ ≤ O(1/p)|c| + O(1)|d| + O(1/p) Φ ∞ .

(27)

The Theorem follows from inequalities (26) and (27). ✷ In the next Theorem, we calculate estimates of the coefficients of c and d in the functions and (Yz − ΦL1 /ρ1/2 (z))/µkG and αΦz /µE . Lemma 3.6. Suppose E = kG, that ρxx ∈ L∞ (0, L) and ρ(x) > 0 for all x ∈ [0, L], and that α = L1 /ρ1/2 (0). Then

   Y (z) − L1 Φ(z)  ρ1/2 (0) z) − cos(µ z) cos(µ  z  kG E 1/2 1/2 ρ (z) ρ (z)   − c + d sin(µ z) kG   µkG µkG  

∞

and

   Φz (z)  c   ≤ O(1/p)|c| + O(1/p)|d|. + z) sin(µ E   µE α ∞

proof: Integral equations for Yz /µkG and Φz /µE and can be determined from the integral equations for Y and Φ. The proof follows from the integral equations for Y and Φ and Lemmas 3.3, 3.4, and 3.5. ✷ Lemma 3.6 facilitates the derivation of a frequency equation for the freefree Timoshenko beam with variable density ρ(x). Theorem 3.2. Let E, kG, A and I be positive constants such that E = kG. Let ρ(x) > 0 for all x ∈ [0, L], let ρx (0) = ρx (L) = 0, and suppose ρxx (x) ∈ L∞ (0, L). Then p2 is an eigenvalue for the free-free Timoshenko beam if and only if

F (p) ≡ sin µkG sin µE + sin(µE )δ1,2 + sin(µkG )δ2,1 + δ = 0,

(28)

where the functions δ1,2 (p), and δ2,1 (p) are O(1/p) and δ(p) is O(1/p2 ).

EIGENVALUES OF THE TIMOSHENKO BEAM

15

proof: We seek to determine the values of p2 for which there exist nontrivial functions Φ and Y that solve the transformed differential equations and all transformed boundary conditions, including those at z = 1. To determine all such solutions, we seek nontrivial solutions to the initial value problem in (15), (16), and (20) which also solve the transformed boundary conditions at z = 1. Theorem 2.2 shows that the values of p2 which admit nontrivial solutions when boundary conditions at z = 1 are imposed are the eigenvalues of the free-free Timoshenko beam. Lemma 3.6 implies that for any choice of c and d, solutions Y and Φ to the initial value problem (15), (16), and (20) satisfy Yz (1) − [L1 /ρ1/2 (1)] Φ(1) = δ1,1 c + [sin(µkG ) − δ1,2 ]d µkG

(29)

αΦz = [− sin µE − δ2,1 ]c − δ2,2 d µE

(30)

and

where the δi,j are all O(1/p) functions. Equations (29) and (30) imply that the right boundary conditions, i.e., the conditions in (20) when z = 1, may be satisfied if and only if     δ1,1 c 0 − sin µkG − δ1,2 = . d 0 − sin µE − δ2,1 −δ2,2 This linear system can be non-trivially solved if and only if sin µkG sin µE + sin(µE )δ1,2 + sin(µkG )δ2,1 + O(1/p2 ) = 0. ✷

4. ROOTS OF THE FREQUENCY FUNCTION In this section, four results are presented. Theorem 4.1 shows that all roots of the frequency equation F must occur near roots of sin(µE ) · sin(µkG ). Theorem 4.2 shows that near each root of sin(µE )·sin(µkG ) which is isolated from neighboring roots, there must exist at least one root of F . In Lemma 4.1, an approximation for ∂F/∂p is calculated. Theorems 4.1, 4.2 and Lemma 4.1 facilitate the proof of Theorem 4.3, in which it is shown that exactly one root of F occurs near isolated roots of sin(µE ) · sin(µkG ). Theorem 4.1. Let

F (p) ≡ sin µkG sin µE + sin(µE )δ1,2 + sin(µkG )δ2,1 + δ

16

GEIST AND MCLAUGHLIN

√ √ where δ1,2 and δ2,1 are O(1/p) and δ is O(1/p2 ). Let c¯ = max{ E, kg}, and suppose p is a root of the frequency function F (p). Then there exists a root of the function sin(µkG ) sin(µE ), say pˆ, such that p − pˆ = /p , where    2 | + |δ | (|δ | + |δ |) |δ c¯ 1,2 2,1 1,2 2,1 |/p | < + + |δ| . sin−1 L1 2 4 (Thus, |/p | is O(1/p).) proof: If p is a root of the frequency equation F , then



 



pL1 pL1 pL1 pL1 √ √ √ √ sin = − sin δ1,2 − sin sin δ2,1 − δ.(31) E E kG kG √ √ If either sin(pL1 / kG) or√sin(pL1 / E) are √ zero, then the result follows. Suppose neither sin(pL1 / kG) nor sin(pL1 / E) is zero. Assume for now that sin(pL /√E) 1 √ (32) ≤ 1, sin(pL1 / kG) √ and divide both sides of equation (31) above by sin(pL1 / kG). It follows that √ √ sin(pL1 / E) δ √ √ | sin(pL1 / E)| = δ1,2 + δ2,1 + sin(pL1 / kG) sin(pL1 / kG) ≤ |δ1,2 | + |δ2,1 | +

|δ| √ | sin(pL1 / E)|

√ √ ⇒ sin2 (pL1 / E) < (|δ1,2 | + |δ2,1 |) | sin(pL1 / E)| + |δ|. √ Let s = | sin(pL1 / E)| , b = |δ1,2 | + |δ2,1 |, and c = |δ|. Then s > 0 and s2 − bs − c < 0. This implies that √ b + b2 + 4c 0 1, the same argument given above holds in this case provided E and kG are interchanged. In √ either case, it follows that there exists a root of sin(µ ) sin(µ ), say p ˆ = EM π/L1 kG E √ or kGM π/L1 , such that p − pˆ = /p and    (|δ1,2 | + |δ2,1 |)2 |δ1,2 | + |δ2,1 | c¯ −1 |/p | < + + |δ| . ✷ sin L1 2 4 Theorem 4.2. Suppose pˆ and p˜ are two adjacent roots of the function

sin(µkG ) · sin(µE ) and that pˆ is a simple root √ of √ not√closer to any other √ root this function than it is to p˜. Let c¯ = max{ E, kG}, c = min{ E, kG} and let p¯ be the largest root of sin(µkG ) sin(µE ) which satisfies p¯ < min{ˆ p, p˜}. Let   c¯ π |δ| c¯ −1 ∆p = sin supp>p¯|δ1,2 (p)| + supp>p¯|δ2,1 (p)| + supp>p¯ , c2 sin(¯ p−1 ) L1 and suppose that |ˆ p − p˜| > 2∆p + Lc¯1 p1¯ Then there is at least one root of the characteristic equation (28) in the interval [ˆ p − ∆p, pˆ + ∆p]. Furthermore, this interval must contain an odd number of roots of the characteristic equation. Remark: The quantity ∆p defined above is O(1/¯ p), since all of the δ s appearing in its definition are O(1/¯ p). Thus, the hypothesis above requires that pˆ and p˜ be separated by a distance which is O(1/¯ p).

18

GEIST AND MCLAUGHLIN

proof sketch: The proof follows from the basic observation that sin(µkG )· sin(µE ) is strictly monotonic near its isolated roots, and that F = sin(µkG )· sin(µE ) + O(1/p). For pˆ large enough, F (p) must change sign near where sin(µkG ) sin(µE ) changes sign, i.e. in the interval [ˆ p − ∆p, pˆ + ∆p]. See Geist [8, pages 164-168] for a detailed presentation of this proof. ✷ An estimate of ∂F/∂p derived in the next lemma is an important step toward proving that for each isolated root of sin(µE ) · sin(µkG ), there is exactly one zero of F nearby. Lemma 4.1. Let F be the frequency function defined in (28). Then

∂F L1 L1 cos(µkG ) · sin(µE ) + r(p), (34) = √ cos(µE ) · sin(µkG ) + √ ∂p E kG where r(p) = O(1/p). proof sketch: If δ, δ1,2 and δ2,1 were all known exactly, it might be possible to calculate ∂F/∂p by direct differentiation. Unfortunately, only order estimates for the δ s are known; δ, δ1,2 and δ2,1 are not known explicitly. However, ∂F/∂p can be estimated using the formula      L L coef. Yz (1)− ρ1/21(1) Φ(1) coef. Yz (1)− ρ1/21(1) Φ(1) ,   µkG µkG ∂F of d  of c  − = det   ∂p  coef. ∂  αΦz (1)  coef. ∂  αΦz (1)   , µE µE of c ∂p of d ∂p 

  L coef. ∂ Yz (1)− ρ1/21(1) Φ(1) coef. ,  µkG of d  of c ∂p + det   coef.  αΦz (1)  , µE of c

 ∂ ∂p

Yz (1)−

L1 Φ(1) ρ1/2 (1)



  , (35)   coef. αΦz (1)  µE of d µkG

where in the expression above each entry in each matrix is a coefficient of either c or d. Let “·” denote differentiation with respect to p. Integral ˙ and Φ˙ z can be determined from the integral equaequations for Y˙ z , Φ, tions for Y and Φ. (It follows from Theorem 2.2 that Y and Φ, which satisfy the initial value problem (15), (16), and (20) may be written as Y (z) = [ρ(x(z))]1/4 y(x(z)) and Φ(z) = [ρ(x(z))]1/4 Ψ(x(z)), where y(x) and Ψ(x) satisfy the original Timoshenko differential equations (2) and (3) and the initial conditions y(0) = ρ−1/4 (0)d, yx (0) = [ρ1/4 (0)/L1 ]c, Ψ(0) = [ρ1/4 (0)/L1 ]c, and Ψx (0) = 0. For each x ∈ [0, L], y, yx , Ψ and Ψx are analytic functions of p. See [4, page 37]. This implies that for each

19

EIGENVALUES OF THE TIMOSHENKO BEAM

z ∈ [0, 1], Yz , Φz and Φ are analytic functions of p, and so differentiation with respect to the eigenvalue parameter p is allowed.) Estimates for Y˙ z , ˙ and Φ˙ z can be obtained by first showing that the infinity norms of these Φ, functions are bounded. Once this is demonstrated, one can generate esti˙ and Φ˙ z accurate to within an error that is O(1/p) in the mates for Y˙ z , Φ, same way that estimates for Y , Yz , and Φz were produced in section 3. Then, estimates for ∂F/∂p can be calculated using equation (35). For the details of this calculation, see Geist [8, pages 173 -189]. ✷ We focus now on proving that for each isolated root of sin(µE ) · sin(µkG ), there is exactly one zero of F nearby. In particular, we demonstrate that |∂F/∂p| > 0 near roots of sin(µE ) · sin(µkG ) that are not too close to one another. Theorem 4.3. Suppose pˆ and p˜ are two adjacent roots of the function sin(µkG )·sin(µE ). Suppose pˆ is a simple root of sin(µkG )·sin(µE√ ) not√closer to any other root of this function than it is to p ˜ . Let c ¯ = max{ E, kG}, √ √ c = min{ E, kG} and let p¯ be the largest root of sin(µkG ) · sin(µE ) which satisfies p¯ < min{ˆ p, p˜}. Let −1



∆p = sin

c¯ π |δ| supp>p¯|δ1,2 (p)| + supp>p¯|δ2,1 (p)| + supp>p¯ c2 sin(¯ p−1 ) 

Γ=

sup p>p−∆p ¯

c¯ sin−1 L1



|δ1,2 | + |δ2,1 | + 2





c¯ , L1

 (|δ1,2 | + |δ2,1 |)2 + |δ| (36) , 4

and r(p) =

∂F L1 L1 cos(µkG ) · sin(µE ). − √ cos(µE ) · sin(µkG ) − √ ∂p E kG

Let pˆ be large enough so that Γ < cπ/[4L1 ]. If for p ∈ [ˆ p − Γ, pˆ + Γ] |ˆ p − p˜| > 

c¯ 1 max 2∆p + , L1 p¯

c¯ π L1 2

  √

|r(p)| tan L1 Γ + EkG π +Γ , c L21 2 cos (ΓL1 /c) (37)

then there is exactly one root of the characteristic function F in the [ˆ p− Γ, pˆ + Γ]. Remark: r is O(1/p) and Γ and ∆p are O(1/¯ p).

20

GEIST AND MCLAUGHLIN

proof: Theorem 4.1 demonstrates that all roots of F occur an O(1/p) distance from the zeros of sin(µkG ) sin(µE ). In particular, it follows from Theorem 4.1 that any root of F nearer to pˆ than it is to any other root of sin(µkG ) sin(µE ) must occur in the narrow interval [ˆ p − Γ, pˆ + Γ]. Theorem 4.2 guarantees that in this interval, there is at least one root of F . We prove now that |∂F/∂p| > 0 throughout [ˆ p − Γ, pˆ + Γ] when the hypotheses of this theorem hold. Lemma 4.1 shows that ∂F L1 L1 = √ cos(µE ) · sin(µkG ) + √ cos(µkG ) · sin(µE ) + r(p), ∂p E kG where r(p) is an O(1/p) function. Let p ∈ [ˆ p − Γ, pˆ + Γ]. Without loss of generality, suppose pˆ is a zero of sin(µE ). Hypothesis (37) implies that 

2L21 |r(p)| L1 L1  . √ [|ˆ p − p˜| − Γ] > √ tan √ Γ + 1 E π EkG kG cos √LE Γ

(38)

Let pkG be a zero of sin(µkG ) at least as√near to p as is any other root of sin(µE ). This implies that |(p − pkG )L1 / kG| ≤ π/2, and hence that

 L1 L1 L 2 L L1 √ | sin(µkG )| = √ sin √ (p − pkG ) ≥ √ 1 √ 1 |p − pkG | E E E π kG kG ≥ ≥

2L2 √ 1 |ˆ p − pkG + p − pˆ| π EkG

2L2 √ 1 [|ˆ p − pkG | − |p − pˆ|] π EkG

2L2 √ 1 [|ˆ p − p˜| − Γ]. (39) π EkG √ p ∈ [ˆ p − Γ, pˆ + Γ], pˆ a zero of tan(µE ), and ΓL1 / E < π/4 imply that ≥



L1 |r(p)| L1  = √ tan √ Γ + 1 E kG cos √LE Γ 

L |r(p)| L  ≥ √ 1 tan √ 1 (ˆ p ± Γ) + L 1 E kG cos √E (ˆ p ± Γ)

EIGENVALUES OF THE TIMOSHENKO BEAM

21

L |r(p)| r(p) L1 √ 1 | tan µE | + ≥ √ | tan µE || cos µkG | + . (40) | cos µ | | cos µE | kG kG E Therefore, from (38), (39) and (40) it follows that L L |r(p)| √ 1 | sin µkG | > √ 1 | tan µE cos µkG | + | cos µE | E kG L1 L1 ⇒ √ sin µkG cos µE > √ sin µE cos µkG + |r(p)| E kG ∂F ⇒ (p) > 0, ∂p as desired. Thus, |∂F /∂p| > 0 for all p ∈ [ˆ p − Γ, pˆ + Γ], which in turn implies that there is at most one root of F in this interval. Since it has been established that there is at least one root in this interval, the Theorem follows when pˆ is a zero of sin(µE ). The proof for the case where pˆ is a zero of sin(µkG ) is identical to the proof above, except that the role of E and kG are interchanged. ✷ 5. THE ASYMPTOTIC FORMULAS We carry out the following steps in order to sharpen our estimates of the eigenvalues. First, we define a new density function ρˆ in terms of ρ. Let  1 L 1/2 ρ0 = [ L 0 ρ (x)dx]2 and ρ˜ = ρ(x) − ρ0 ; then define ρˆ = ρ0 + t˜ ρ, where t is an auxiliary parameter which is allowed to vary from 0 to 1. Note that ˆ and Yˆ be when t = 0, ρˆ is the constant ρ0 , and when t = 1, ρˆ ≡ ρ(x). Let Φ solutions to the boundary value problem given in (15-17), except z, ρ, ρ3 , ρ4 ˆ 1 , respectively, where zˆ, ρˆ3 , ρˆ4 and and L1 are replaced by zˆ, ρˆ, ρˆ3 , ρˆ4 , and L ˆ the constant L1 are defined as are z, ρ3 , ρ4 and L1 except that ρˆ replaces ˆ and ρ in their definitions. The resulting boundary value problem for Φ Yˆ corresponds to the transformed boundary value problem (see Theorem 2.2) one obtains from the original Timoshenko differential equations and boundary conditions when ρ(x) is taken to be ρˆ(x; t). ˆ It will prove useful to view the transformed differential equations for Φ ˆ and Y as a single vector equation. Define

ˆ (ˆ U z) =

ˆ z) Φ(ˆ Yˆ (ˆ z)

 .

22

GEIST AND MCLAUGHLIN

Let

B1 =

1 E

,

1 kG

0

S=





0

0 γ −1 0

Q=

ˆ2γ −L 1 ρˆ

+ ρˆ3 −γ ρˆ4 ρˆ4 ρˆ3

 (41)

 ˆ 21 p2 . , and µ2 = L

(42)

Then the boundary value problem (15-17), where ρ is now replaced by ρˆ, may be written in vector notation as

 ˆ ˆzˆzˆ + µ2 B1 U ˆzˆ = 0 ˆ + QU ˆ + L1 S U U 0 ρˆ1/2   ˆ1 L ˆ zˆ|zˆ=0,1 = 0, ˆ Yˆz − 1/2 Φ = 0, and Φ ρˆ

(43) (44)

zˆ=0,1

ˆ 2 p2 , is chosen so Suppose the transformed eigenvalue parameter, µ2 = L 1 ˆ (ˆ that a nontrivial solution U z ) exists to the above boundary value problem. Then by Theorem 2.2, if u ˆ is defined as

u ˆ(x) ≡

ˆ Ψ(x) yˆ(x)



−1/4

= ρˆ

ˆ (ˆ U z (x)) = ρˆ−1/4 (x)

then u ˆ must satisfy

∂ ∂ 2 ˆ − kAG) ∂x EI ∂x + (p I ρ ∂ kAG ∂x

∂ kAG ∂x ∂ ∂ 2 ρ ∂x kAG ∂x + p Aˆ

ˆ x=0,L = 0, [ˆ yx − Ψ]| ˆ 2 p2 L 1

2



ˆ x=0,L = 0. Ψ|

ˆ z (x)) Φ(ˆ Yˆ (ˆ z (x))

 ,

 u ˆ(x) = 80

(45)

(46)

Thus, if = µ is an eigenvalue for the transformed boundary value ˆ (ˆ problem (43-44) and U z ) is the corresponding eigenfunction, then p2 is an ˆ (ˆ z (x)) is the corresponding eigenfunction for eigenvalue and u ˆ(x) = ρˆ−1/4 U the free-free Timoshenko beam with density ρˆ. Our method for sharpening the estimates of the eigenvalues of the Timoshenko beam will rely on determining the derivative of µ2 with respect to the auxiliary parameter t, introduced in the definition of ρˆ. The approach may be summarized as follows. Let µ2 be an eigenvalue for the transˆ (ˆ formed boundary value problem (43-44) and let U z ) be a corresponding ˆ (ˆ eigenfunction. The differential equation and boundary conditions for U z)

EIGENVALUES OF THE TIMOSHENKO BEAM

23

are differentiated with respect to ρˆ in the direction of ρ˜. This differentiation of (43-44) will lead to a new, non-homogeneous boundary value problem. The right hand side of this new boundary value problem will contain dρˆµ2 [˜ ρ], which we will show is equal to dµ2 /dt. Since the new boundary value problem comes from differentiating the differential equations and boundary conditions (43-44), a boundary value problem with ˆ (ˆ a known solution U z) ≡ / 0, it follows that a nontrivial solution to the new, non-homogeneous problem exists, and must be equal to dρˆµ2 [˜ ρ]. Using Theorem 2.1, the new non-homogeneous boundary value problem for dρˆµ2 [˜ ρ] may be written as a non-homogeneous Timoshenko boundary value problem which must also have a nontrivial solution. The fact that the non-homogeneous Timoshenko boundary value problem has a nontrivial solution implies that the right hand side of the non-homogeneous problem must satisfy a certain orthogonality requirement (see Lemma 5.1 below). This orthogonality condition allows us to determine dµ2 /dt = dρˆµ2 [˜ ρ] in ˆ terms of the transformed eigenfunction U (ˆ z ); i.e., from the orthogonality condition it follows that d(µ2 ) ˆ (ˆ = G(U z (x))). dt Integrating both sides of the above equation gives  µ |t=1 − µ |t=0 = 2

2

1

ˆ (ˆ G(U z (x)))dt.

(47)

0

Sharp estimates of µ2 |t=0 can be obtained by applying formulas from GeistMcLaughlin [10]. In particular, when t = 0, ρˆ is ρ0 , a constant. Therefore µ2 |t=0 may be determined from the formulas for the eigenvalues of the uniform Timoshenko beam. From results in the previous five sections, ˆ (ˆ G(U z (x))) can be determined to within an error that converges to zero as µ (and p) get large. Thus, equation (47) provides a means for obtaining sharp estimates for µ2 when t = 1. From these estimates of µ2 , we will obtain asymptotic formulas for the free-free Timoshenko eigenvalues when mass density is ρ(x). Our next goal then is to calculate an expression for dµ2 /dt. Suppose ρxx ∈ L∞ [0, L]. For these calculations, we first assume that for all such ρ, i) ii) iii)

dρˆµ2 [˜ ρ] exists, that ˆzˆ)[˜ ˆzˆzˆ)[˜ ˆ dρˆU [˜ ρ], dρˆ(U ρ] all exist, and that ρ], dρˆ(U ˆ [˜ ˆ [˜ ˆzˆzˆ)[˜ ˆzˆ)[˜ ρ])zˆzˆ = dρˆ(U ρ])zˆ = dρˆ(U (dρˆU ρ] and (dρˆU ρ].

24

GEIST AND MCLAUGHLIN

Later, we will give conditions which guarantee that assumptions i), ii) and iii) are valid. As indicated in the discussion above, we formally differentiate the differential equations and boundary conditions in (43-44) with respect to ρˆ in the direction of ρ˜. We use assumptions i), ii), and iii) from above to ˆ [˜ obtain the differential equation and boundary conditions that dρˆU ρ] must satisfy. We find that ˆ1 L ˆ [˜ ˆ [˜ ˆ [˜ ˆ [˜ (dρˆU ρ])zˆzˆ + µ2 B1 (dρˆU ρ]) + Q(dρˆU ρ]) + 1/2 S(dρˆU ρ])zˆ = ρˆ ˆ − dρˆ(L ˆ 1 /ˆ ˆzˆ, ˆ − (dρˆQ[˜ −(dρˆµ2 [˜ ρ])B1 U ρ])U ρ1/2 )[˜ ρ]S U and that ˆ ρ]) − (dρˆ(L ˆ 1 /ˆ ˆ 1 /ˆ ˆ zˆ=0,1 = 0, [(dρˆYˆ [˜ ρ])zˆ − (L ρ1/2 )(dρˆΦ[˜ ρ1/2 )[˜ ρ])Φ]| ˆ ρ])|zˆ=0,1 = 0. (dρˆΦ[˜ When dρˆµ2 [˜ ρ] exists, it follows that µ2 (ˆ ρ + /˜ ρ) − µ2 (ˆ ρ) →0 / µ2 (ρ0 + (t + /)˜ ρ) − µ2 (ρ0 + t˜ ρ) = lim →0 / dµ2 = . dt

dρˆµ2 [˜ ρ] = lim

ˆ 1 /ˆ Similarly, from the definitions of Q and L ρ1/2 , it follows that dρˆQ[˜ ρ] = ∂ 1/2 1/2 ˆ 1 /ˆ ˆ 1 /ˆ ∂Q/∂t and dρˆ(L ρ )[˜ ρ] = ∂t (L ρ ). Define

B2 =

1/EI 0 0 1/kAG

 (48)

and ρˆ3/4 −1 F =− B ˆ2 2 L



   

 ˆ1 d 2 ∂ ∂ L ˆ ˆ ˆ S Uzˆ . (49) µ B1 U + Q U+ dt ∂t ∂t ρˆ1/2

Now define

ˆ [˜ V¯ (ˆ z ) ≡ dρˆU ρ] =

ˆ ρ] dρˆΦ[˜ dρˆYˆ [˜ ρ]



≡

¯ z) Φ(ˆ Y¯ (ˆ z)

 .

25

EIGENVALUES OF THE TIMOSHENKO BEAM

Then V¯ satisfies ˆ1 ˆ2 L L 1 V¯zˆzˆ + µ2 B1 V¯ + QV¯ + 1/2 S V¯zˆ = 3/4 B2 F, ρˆ ρˆ 



ˆ1 L ¯− ∂ Y¯z¯ − 1/2 Φ ∂t ρˆ

ˆ1 L ρˆ1/2



= 0,

(50)

zˆ=0,1

and ¯ z¯|zˆ=0,1 = 0. Φ

(51)

Let v¯ be defined as

v¯(x) ≡

¯ Ψ(x) y¯(x)



≡ ρˆ−1/4 (x)V¯ (ˆ z (x)).

From Theorem 2.1, we know that v¯ must satisfy the vector differential equation

+ (p2 I ρˆ − kAG) ∂ ∂x kAG

∂ ∂ ∂x EI ∂x

∂ kAG ∂x ∂ ∂ 2 ρ ∂x kAG ∂x + p Aˆ

 v¯(x) = F.

(52)

The boundary conditions (50) and (51) may also be rewritten in terms ¯ and y¯. By multiplying (50) through by ρˆ1/4 /L ˆ 1 and noting that of Ψ ρˆx (0) = ρˆx (L) = 0, we find that 

ˆ1 L ¯− ∂ Y¯zˆ − 1/2 Φ ∂t ρˆ

 ⇔



ρˆ1/4 ∂ ¯ y¯x (x) − Ψ(x) − ˆ 1 ∂t L

ˆ1 L ρˆ1/2



  ˆ Φ

ˆ1 L ρˆ1/2



=0 zˆ=0,1

 ˆ z (x)) Φ(ˆ

=0

(53)

x=0,L

¯ zˆ|zˆ=0,1 = 0 Similarly, from (51) we have Φ ¯ x |x=0,L = 0. ⇔Ψ

(54)

In the next lemma, we show that if the boundary value problem for v¯(x) is to have a solution when p2 is an eigenvalue for the boundary value problem (45)-(46), then the right hand side of the differential equation (52) must satisfy a certain orthogonality condition.

26

GEIST AND MCLAUGHLIN

Lemma 5.1. Let E, I, kG and A be positive constants,

and let ρˆ be a

¯ Ψ(x) y¯(x)

positive function such that ρˆxx ∈ L∞ (0, L). Suppose v¯ =

satisfies

the non-homogeneous differential equations ¯ xx + kAG(¯ ¯ + p2 I ρˆΨ ¯ = F1 , EI Ψ yx − Ψ)

(55)

¯ x + p2 Aˆ kAG(¯ yx − Ψ) ρy¯ = F2 ,

(56)

and



 ˆ Ψ(x) ˆ (ˆ = ρˆ−1/4 U z (x)) yˆ(x) solves the homogeneous boundary value problem given in (45)

Timoshenko  F1 and (46). Then F = must satisfy F2

and boundary conditions (53)-(54). Suppose u ˆ(x) =

−1/4

(F(x), u ˆ(x)) = (F, ρˆ

kAG ∂ ˆ (ˆ U z (x))) = ˆ 1 ∂t L



ˆ1 L ρˆ1/2



1 ˆ ˆ ΦY

.

zˆ=0

(In the last equation, (·, ·) denotes the standard inner product in L2 (0, L).) proof:  (F(x), u ˆ(x)) =

L

¯ xx ψˆ + kAG(¯ ¯ ψˆ + p2 I ρˆΨ ¯ ψˆ (EI Ψ yx − Ψ)

0

¯ x )ˆ + kAG(¯ yx − Ψ) y + p2 Aˆ ρy¯yˆ)dx 

L

=

¯ ψˆxx + kAG(¯ ¯ ψˆ − kAG(¯ ¯ yx (EI Ψ yx − Ψ) yx − Ψ)ˆ

0

¯ ψˆ + p2 Aˆ ¯ y |L +p2 I ρˆΨ ρy¯yˆ)dx + kAG(¯ yx − Ψ)ˆ z=0  =

L

ˆ yx − Ψ) ¯ ψˆxx − kAG(ˆ ¯ + p2 I ρˆΨ ¯ ψˆ + p2 Aˆ [EI Ψ yx − ψ)(¯ ρy¯yˆ]dx

0

¯ y |L +kAG(¯ yx − Ψ)ˆ x=0  = 0

L

ˆ + p2 I ρˆψ] ˆΨ ¯ {[EI ψˆxx + kAG(ˆ yx − ψ)

27

EIGENVALUES OF THE TIMOSHENKO BEAM

ˆ x + p2 Aˆ ¯ y |L +[kAG(ˆ yx − ψ) ρyˆ]¯ y }dx + kAG(¯ yx − Ψ)ˆ x=0 ρˆ1/4 ∂ = kAG ˆ1 ∂t L



ˆ1 L ρˆ1/2

kAG ∂ ⇒ (F(x), u ˆ(x)) = ˆ 1 ∂t L



1 −1/4 ˆ ˆ Φˆ ρ Y

.

zˆ=0



ˆ1 L ρˆ1/2



1 ˆ Yˆ Φ

✷

.

zˆ=0

Returning now to (52)-(54), assumptions i) - iii) above imply that ˆ [˜ v¯ = ρˆ−1/4 V¯ (ˆ z (x)) = ρˆ−1/4 (dρˆU ρ](ˆ z )) ≡ /0 is a nontrivial solution to this boundary value problem. Lemma 5.1 implies that 1   ˆ1 kAG ∂ L −1/4 ˆ ˆ ˆ ⇒ (F(x), ρˆ U (x)) = Φ Y . (57) ˆ 1 ∂t ρˆ1/2 L zˆ=0

Equation (57) holds if and only if 

ρˆ3/4 −1 − B ˆ2 2 L



   

  ˆ1 ∂ d 2 ∂ L −1/4 ˆ+ ˆ ˆzˆ , ρˆ ˆ+ U SU µ B1 U Q U dt ∂t ∂t ρˆ1/2

kAG ∂ = ˆ 1 ∂t L  ⇔

d 2 dt µ

−

=

ρ ˆ1/2 ˆ L 1



 

∂ ˆ ,U ˆ − B2−1 ( ∂t Q)U

ˆ1 L ρˆ1/2

ρ ˆ1/2 ˆ L 1







1 ˆ Yˆ Φ

zˆ=0

ˆ ∂ L 1 ∂t ρ ˆ1/2

ρ ˆ1/2 ˆ L 1





ˆzˆ ,U ˆ − kAG ∂ B2−1 S U ∂t

ˆ ,U ˆ B2−1 B1 U





ˆ L 1 ρ ˆ1/2



The inner products above involve an integration with respect to x. Since ρˆ1/2 z , all of the above inner products may be rewritten as integraˆ 1 dx = dˆ L tions with respect to zˆ. Thus,

d 2 dt µ

−

=

1 0

1

ˆ T B −1 ( ∂ Q)U ˆ dˆ U z− ∂t 2

0

 ∂ ∂t

1 0

ˆ L 1 ρ ˆ1/2



∂ ˆ T B −1 S U ˆzˆ dˆ U z − kAG ∂t 2



ˆ L 1 ρ ˆ1/2



1

ˆ Yˆ Φ

1

ˆ Yˆ Φ

z ˆ=0

ˆ T B −1 B1 U ˆ dˆ U z 2

Next we show that the third term in the numerator above, the boundary term, may be combined with the second term so that all terms in the

z ˆ=0

.

28

GEIST AND MCLAUGHLIN

numerator are integrals. From the definitions of B2 and S, we find that      1  1 ˆ1 ˆ1 L L ∂ ∂ T −1 ˆ ˆ −Φ ˆ zˆYˆ )dˆ U B2 SUzˆdˆ (YˆzˆΦ z z = kAG ρˆ1/2 ρˆ1/2 0 ∂t 0 ∂t 

1

∂ ∂t

= 2kAG 0

 = 2kAG 0



1

ˆ1 L ρˆ1/2

∂ ∂t





 0



ˆ1 L ρˆ1/2 

1

+ kAG 0



1

∂ ∂t

⇒− 0

 −2kAG 0

1

1

ˆ z − kAG YˆzˆΦdˆ

ˆ z −kAGYˆ Φ ˆ ∂ YˆzˆΦdˆ ∂t

2 ˆ Yˆ ∂ Φ ∂t∂ zˆ





ˆ Yˆ )zˆ ∂ (Φ ∂t

ˆ1 L ρˆ1/2



ˆ1 L ρˆ1/2

ˆ1 L ρˆ1/2

 dˆ z

 1 0

 dˆ z

 1 

ˆ  ˆ 1 ∂ L1 L ˆ T B −1 S U ˆzˆdˆ ˆ U z − kAGYˆ Φ = 2 ∂t ρˆ1/2 ρˆ1/2 0

∂ ∂t



ˆ1 L ρˆ1/2



 ˆ z − kAG YˆzˆΦdˆ 0

1

2 ˆ Yˆ ∂ Φ ∂t∂ zˆ



ˆ1 L ρˆ1/2

 dˆ z.

Therefore,

−

1 0

ˆ T B −1 ( ∂ Q)U ˆ dˆ U z −2kAG ∂t 2

1  01

d 2 µ = dt   ∂ ∂t

ˆ L 1 ρ ˆ1/2

ˆ z −kAG Yˆzˆ Φdˆ

ˆ T B −1 B1 U ˆ dˆ U z 2 0

1 0

ˆ Yˆ Φ

∂2 ∂t∂ z ˆ



ˆ L 1 ρ ˆ1/2

 dˆ z

(58) .

The formula above for dµ2 /dt holds so long as assumptions i) - iii) hold. In the following theorem, conditions are given which guarantee the validity of these assumptions, and hence show when dµ2 /dt may be calculated using the last formula given above. Theorem 5.1. Suppose ρ(x) is a positive function such that ρx (0) =

L ρx (L) = 0 and ρxx (x) ∈ L∞ (0, L). Let ρ0 = [(1/L) 0 ρ1/2 (x)dx]2 , ρ˜ ≡  L 1/2 ˆ1 = ρ(x) − ρ0 , and ρˆ(x, t) = ρ0 + t˜ ρ(x). Let L ρˆ (x; t)dx, and let 0 2 2 2 ˆ µ (t) = L1 p´ be an eigenvalue for the transformed boundary value problem ˆ is an eigenfunction of (43(43-44). Then dµ2 /dt satisfies (7), where U 2 44) corresponding to the eigenvalue µ , and Q, B1 and B2 are the 2 × 2 matrices defined in (41) and in (48).

EIGENVALUES OF THE TIMOSHENKO BEAM

29

proof: The proof follows from the discussion preceding this theorem, provided we show assumptions i) - iii) are valid. We observe that u ˆ(x) = −1/4 ˆ ρˆ (x)U (ˆ z (x)), where u ˆ(x) satisfies (45) and (46). The differential equation for the vector u ˆ(x) may be written as a system of four first order linear scalar equations in which the parameter t appears linearly. This implies that for each x, u ˆ, u ˆx , and u ˆxx must be analytic in t. See Coddington and ˆ (ˆ ˆzˆ(ˆ Levinson [4, page 37]. This in turn implies that for each zˆ, U z ), U z) , ˆ and Uzˆzˆ(ˆ z ) must also be analytic in t when t ∈ [0, 1]. When zˆ = 0 and t ˆ (0) cannot simultaneis any value between 0 and 1, both components of U ously be zero. (If they were both zero, boundary conditions at zˆ = 0 would ˆ (ˆ imply U z ) must be identically zero.) From the scalar differential equaˆ (ˆ tions for the components of U z ), we conclude that µ2 must be analytic in t. We have shown already that dρˆµ2 [˜ ρ] = dµ2 /dt; hence, assumption i), 2 that dρˆµ [˜ ρ] exits, is valid. ∂ ˆ ∂ ˆ ˆ [˜ ˆzˆ[˜ ˆzˆzˆ[˜ Next, we observe that dρˆU ρ] = ∂t ρ] = ρ] = ∂t U , dρˆU Uzˆ, and dρˆU ∂ ˆ ˆ ˆ ˆ ρ], and dρˆUzˆzˆ[˜ ρ] all exist, ρ], dρˆUzˆ[˜ ∂t Uzˆzˆ. From this we conclude that dρˆU [˜ ˆ, U ˆzˆ, and U ˆzˆzˆ are all analytic in t. This proves assumption ii). since U To demonstrate that assumption iii) holds under the stated hypotheses, ˆ satisfy the integral equations (20) and observe that the components of U (21) when ρ is taken to be ρˆ. When ρxx (x) is continuous, it follows from ˆzˆzˆt and U ˆtˆzzˆ are continuous functions of zˆ these integral equations that U ˆzˆzˆt = U ˆtˆzzˆ. Furthermore, and t. This implies that when ρ(x) ∈ C2 [0, L], U ˆzˆzˆt and U ˆtˆzzˆ are continuous in ρ from the integral equations, it follows that U with respect to the standard Sobelev norm of order 2. They are continuous maps which take ρ(x) ∈ H2 (0, L) to an element of L2 (0, 1). Since C2 [0, L] is ˆzˆzˆt = U ˆtˆzzˆ when ρ(x) ∈ H2 (0, L). Since dense in H2 (0, L), it follows that U H2 (0, L) contains the set of all ρ such that ρxx ∈ L∞ (0, L), assumption iii) must hold when ρxx ∈ L∞ (0, L), as desired. ✷ In the next three lemmas, our goal is to determine the constants c and d which appear in the integral equations (20) and (21). Theorem 4.1 shows 2 that if p√ is a large enough √ eigenvalue, then p must lie near a root of ˆ 1 p/ kG) · sin(L ˆ 1 p/ E). We will show that provided p is not near sin(L √ √ ˆ 1 p/ kG) · sin(L ˆ 1 p/ E), then p must be a more than one root of sin(L simple eigenvalue, and the vector ( c, d )T must be a multiple of either ( 1, O(1/p)√)T or ( O(1/p), 1 )T√, depending upon whether p is near a zero ˆ 1 p/ kG) or of sin(L ˆ 1 p/ E). This result will be used to calculate of sin(L 2 an estimate of dµ /dt. In the next Lemma, we show that if pˆ is large enough and satisfies either (59) or (60) below, then the square root of the nearest eigenvalue to pˆ is well separated from square roots of other eigenvalues.

30

GEIST AND MCLAUGHLIN

Lemma√5.2. Let e ∈ √(0, 1) and suppose pˆ is a root of the function ˆ 1 p/ kG) · sin(L ˆ 1 p/ E) such that either sin(L √ √ ˆ 1 pˆ/ kG) = 0 and | sin(L ˆ 1 pˆ/ E)| > e sin(L (59)

or √ √ ˆ 1 pˆ/ E) = 0 and | sin(L ˆ 1 pˆ/ kG)| > e. sin(L

(60)

Then there exists an M > 0 such that when pˆ > M , exactly one eigenvalue 2 of the free-free Timoshenko beam  with√ density√ρˆ, say p´ , is such that its e c e c square root p´ lies in the interval pˆ − 2Lˆ , pˆ + 2Lˆ . Furthermore, |´ p − pˆ| < 1

1

O(1/´ p). proof: It is not difficult to show √ that when that √ when (59) or (60) hold, ˆ 1 p/ kG) · sin(L ˆ 1 p/ E) as near to pˆ as is any and when p˜ is a root of sin(L √ ˆ other root of this function, then |ˆ p − p˜| > e c/L 1 , where c = min E, kG. Let Γ be defined as it is in the hypothesis of Theorem 4.3. Let M be chosen large enough so that pˆ > M implies that hypothesis (37) of Theorem 4.3 √ ˆ 1 ]. Theorem 4.1 shows that if there is is satisfied and that |Γ| < e c/[2L  

a root p´ of the frequency equation such that p´ ∈ pˆ −

√ √ e c e c ˆ + 2Lˆ ˆ1 , p 2L 1

, then p´ must be contained in the narrower interval (ˆ p − Γ, pˆ + Γ). On the other hand, Theorem 4.3 shows that there is exactly one zero of the frequency function in the interval (ˆ p − Γ, pˆ + Γ). Together, Theorem 4.1 and Theorem 4.3 imply that there is exactly one root of the frequency function in the  √ √  e c e c interval pˆ − 2Lˆ , pˆ + 2Lˆ . ✷ 1 1 √ √ ˆ 1 p/ kG·sin(L ˆ 1 p/ E) Lemma 5.3. Let e ∈ (0, 1), and suppose pˆ is a root of sin(L such that either (59) or (60) holds. For pˆ large enough, let p´2 be the unique eigenvalue of the free-free beam with √density ρˆ whose square root p´ is con  √ e c e c tained in the interval pˆ − 2Lˆ , pˆ + 2Lˆ . If (59) holds, then for the eigen1

1

value p´2 , the vector ( c, d )T (where c and d are the constants appearing in (20) and (21)) must satisfy



 c O(1/´ p) = constant . d 1 If (60) holds,



c d



= constant

1 O(1/´ p)

 .

31

EIGENVALUES OF THE TIMOSHENKO BEAM

In either case, whether (59) holds or (60) holds, p´2 must be a simple eigenvalue. proof: p´2 is an eigenvalue if and only if nontrivial solutions exist to a linear system of the form

0 0



 =

ˆ

1 − sin( √p´LkG ) − O(1/´ p)

O(1/´ p) ˆ

´L1 − sin( p√ ) − O(1/´ p) E

−O(1/´ p)



c d

 (61)

(see Theorem 2.15). The conclusion is immediate. ✷ Our next goal is to use formula (58) in Theorem 5.1 to calculate an 2 estimate for dµ2 /dt. We make √ an estimate dµ √ /dt in the case where p´ ˆ 1 p/ kG) · sin(L ˆ 1 p/ E) which for an arbitrary lies near a root pˆ of sin(L but fixed e ∈ (0, 1) satisfies either (59) or (60). A p´ which meets this criterion gives rise to an eigenvalue which we will refer to as being “wellseparated” from its neighboring eigenvalues. The estimate of dµ2 /dt is used to calculate asymptotic formulas for these eigenvalues. In the next lemma, we calculate estimates for one of the terms appearing in the numerator of the right hand side of (58). ˆ and µ2 = L ˆ 2 p´2 be an eigenfunction and correLet U 1 sponding eigenvalue for the transformed boundary value problem (43-44). Then p´2 is an eigenvalue for the free-free Timoshenko beam with density ρˆ. Let e√be a fixed but arbitrary number in (0, 1), and let pˆ be a root of √ ˆ 1 p/ kG) · sin(L ˆ 1 p/ E) as near to p´ as is any other root of this funcsin(L tion. If pˆ satisfies either (59) or (60), then Lemma 5.4.

 −2kAG 0

  c 2 α

−d

2A

I



kAG E kG − 1

1

∂ ∂t

 0

1





ˆ1 L ρˆ1/2

ˆ1 L ρˆ1/2

 ˆ = Yˆs Φds 

∂ ∂t



ˆ1 L ρˆ1/2

 ds + O(1/´ p).

proof: Theorem 2.2 shows that p´2 is an eigenvalue for the free-free Timoshenko beam when L21 p´2 is an eigenvalue for the transformed problem. ˆ ˆ 1 p´ √1 p´ and µ Let µ ˆE = L ˆkG = √LkG . Then Theorem 3.1 shows that E Yˆzˆ = c cos(ˆ µkG zˆ) − dˆ µkG sin(ˆ µkG zˆ)+

32

GEIST AND MCLAUGHLIN



zˆ

0



 ˆ1 L ˆ ˆ s (s) ds cos[ˆ µkG (ˆ z − s)] −ˆ ρ3 (s)Yˆ (s) − ρˆ4 (s)Φ(s) + 1/2 Φ ρˆ (s)

and µE zˆ) ˆ = c cos(ˆ Φ + α



zˆ

0

sin[ˆ µE (ˆ z − s)] µ ˆE



 ˆ2γ ˆ1 L γ L 1 ˆ ˆ · −ˆ ρ3 (s)Φ(s) + Φ(s) + γ ρˆ4 (s)Yˆ (s) − 1/2 Yˆ (s) ds. ρˆ(s) ρˆ (s)   ˆ1 L ∂ Let f (ˆ z ) ≡ 2kAG ∂t . Then ρˆ1/2 

1

ˆ z )dˆ f (ˆ z )Yˆzˆ(ˆ z )Φ(ˆ z=

0





1

f 0

µ ˆkG −d µ ˆE  0

1



 c2 cd µE zˆ) − µ µkG zˆ) cos(ˆ µE zˆ) dˆ z cos(ˆ µkG zˆ) cos(ˆ ˆkG sin(ˆ α α 

1

zˆ

f (ˆ z ) sin(ˆ µkG zˆ) 0

0

c µE zˆ) f (ˆ z ) cos(ˆ α

 0

zˆ



ˆ2 ˆ ˆ + L1 γ Φ ˆ + γ ρˆ4 Yˆ − L1 γ Yˆs sin[ˆ µE (ˆ z −s)] −ˆ ρ3 Φ ρˆ ρˆ1/2 

ˆ ˆ + L1 Φ ˆs cos[ˆ µkG (ˆ z −s)] −ˆ ρ3 Yˆ − ρˆ4 Φ ρˆ1/2



 1 ds dˆ z +O . p´

By Lemma 5.3, when pˆ satisfies (59) or (60), then cd ˆkG and αc must both αµ be O(1) or smaller. Since µ ˆE = µ ˆkG by assumption (otherwise neither (59) nor (60) could hold), the first integral on the right of the above equation must be O(1/´ p). Let  µ ˆkG 1 I1 = −d f (ˆ z ) sin(ˆ µkG zˆ) µ ˆE 0  · 0

zˆ



ˆ2 ˆ ˆ + L1 γ Φ ˆ + γ ρˆ4 Yˆ − γ L1 Yˆs sin[ˆ µE (ˆ z − s)] −ˆ ρ3 Φ ρˆ ρˆ1/2



By switching the order of integration, we find that I1 = −d

µ ˆkG µ ˆE



1



1

f (ˆ z ) sin(ˆ µkG zˆ) sin[ˆ µE (ˆ z − s)] 0

s



 ds dˆ z.

ds dˆ z

33

EIGENVALUES OF THE TIMOSHENKO BEAM



 ˆ2γ ˆ1 L γ L 1 ˆ ˆ · −ˆ ρ3 (s)Φ(s) + z ds Φ(s) + γ ρˆ4 (s)Yˆ (s) − 1/2 Yˆs (s) dˆ ρˆ(s) ρˆ (s) 

µ ˆkG = −d µ ˆE  · s

1

0

1



ˆ2 ˆ ˆ + L1 γ Φ ˆ + γ ρˆ4 Yˆ − γ L1 Yˆs −ˆ ρ3 Φ 1/2 ρˆ ρˆ



 f (ˆ z) ˆE )ˆ z+µ ˆE s] − cos[(ˆ µkG + µ ˆE )ˆ z−µ ˆE s]}dˆ z ds. {cos[(ˆ µkG − µ 2

Now f

∈ L∞ (0, 1) because ρˆxx ∈ L∞ (0, L). Therefore we may integrate the square bracketed term above by parts with respect to zˆ, and apply Lemmas 3.2, 3.3, 3.4, and 3.5 to show that    ˆ2γ ˆ1 L γL µ ˆkG 1 1 ˆ ˆ ˆ ˆ I1 = −d −ˆ ρ3 Φ + Φ + γ ρˆ4 Y − 1/2 Ys µ ˆE 0 ρˆ ρˆ ·

  f 1 1 sin(ˆ µkG s)ds + O(1/´ − p) 2 µ ˆkG + µ ˆE µ ˆkG − µ ˆE

µ ˆkG = −d µ ˆE



1

0

ˆ1 f γL 2ˆ µE sin(ˆ µkG s)Yˆs ds + O(1/´ p). ˆ2kG − µ ˆ2E ρˆ1/2 2 µ

From the integral equation for Yˆzˆ, it follows that Yˆzˆ = −dˆ µkG sin(ˆ µkG zˆ) + O(1). Substituting this expression for Yˆzˆ into the last expression for I1 , we find that  ˆ1 ˆkG 1 γ L ˆE µ ˆkG µ 2µ I1 = d f 2 sin2 (ˆ µkG s)ds + O(1/´ p) µ ˆE 0 ρˆ1/2 µ ˆkG − µ ˆ2E A = d kAG I



2

1 E kG − 1



1

0

ˆ1 ∂ L ρˆ1/2 ∂t



ˆ1 L ρˆ1/2

 ds + O(1/´ p).

Using a similar argument as was used to derive the estimate of I1 , it is possible to show that    1  zˆ ˆ1 L c ˆ+ ˆ s ds dˆ I2 = Φ z µE zˆ) f (ˆ z ) cos(ˆ cos[ˆ µkG (ˆ z −s)] −ˆ ρ3 Yˆ − ρˆ4 Φ α ρˆ1/2 0 0

=−

 c 2 α

 kAG

1 E kG − 1

 0

1

ˆ1 ∂ L ρˆ1/2 ∂t



ˆ1 L ρˆ1/2

 ds + O(1/´ p).

34

GEIST AND MCLAUGHLIN

Since



1

0

∂ 2kAG ∂t



ˆ1 L ρˆ1/2

 ˆ = I1 + I2 + O(1/´ Yˆs Φds p),

the lemma follows. ✷ In the next theorem, we make estimates for transformed eigenfunctions ˆ (ˆ U z ) associated with well separated eigenvalues for the untransformed Timˆ to calculate oshenko beam with density ρˆ. We will use these estimates for U 2 an estimate for dµ /dt. Lemma 5.5. Suppose p´ is an eigenvalue for the free-free Timoshenko L beam with density ρˆ = ρ0 + t˜ ρ(x), where ρ0 = ([1/L] 0 ρ1/2 (x)dx)2 and ρ˜(x) = ρ(x)−ρ0 . Suppose ρ(x) is a positive function on [0, L], that ρxx (x) ∈ L∞ (0, L), and that ρx (0) = ρx (L) = 0. Assume also√that E, I, kG,√and ˆ 1 p/ kG) · sin(L ˆ 1 p/ E) A are positive constants. Let pˆ be a root of sin(L as near to p´ as is any other root of this function. Let e be an arbitrary but fixed 0 and 1. If (59) holds, then for some integer n, pˆ = √ constant between ˆ 1 , and µ2 = L ˆ 2 p´2 , U ˆ (ˆ z ) + O(1/n), O(1/n) )T is nπ kG/L z ) = ( cos(nπˆ 1 an eigenvalue-eigenfunction pair for √ (43-44). Similarly, if pˆ satisfies (60), ˆ 1 and an eigenvalue-eigenfunction then for some integer n, pˆ = nπ E/L 2 2 2 ˆ ˆ z ) + O(1/n) )T . pair for (43-44) is µ = L1 p´ , U (ˆ z ) = ( O(1/n), cos(nπˆ

proof: If µ ˆE =

ˆ L √1 p´ E

ˆ 1 p´ L √ , kG

and µ ˆkG =

then Theorem 3.1 shows that for a ˆ 1 /ˆ nontrivial choice of the constants c and d and for α ˆ=L ρ1/2 (0), µE zˆ) ˆ = c cos(ˆ Φ α ˆ  0

zˆ

sin[ˆ µE (ˆ z − s)] µ ˆE



ˆ2 ˆ ˆ + L1 γ + γ ρˆ4 Yˆ − γ L1 Yˆs −ˆ ρ3 Φ 1/2 ρˆ ρˆ

 ds

(62)

and c sin(ˆ µkG zˆ) Yˆ = + d cos(ˆ µkG zˆ) µ ˆkG  0

zˆ

sin[ˆ µkG (ˆ z − s)] µ ˆkG



ˆ ˆ + L1 Φ ˆs −ˆ ρ3 Yˆ − ρˆ4 Φ ρˆ1/2

 ds.

(63)

For fixed c and d, Lemmas 3.3, 3.4 and 3.5 show that all terms in (62) and (63) under the integral sign are O(1/´ p). If (59) holds,√then by Lemma 5.3 ˆ 1 / kG+O(1/n). This and Theorem 4.1 we find that p´ = pˆ+O(1/ˆ p) = nπ L

35

EIGENVALUES OF THE TIMOSHENKO BEAM

ˆ (ˆ z ) + O(1/n) )T . Similarly, when (60) implies that U z ) = ( O(1/n), cos(nπˆ ˆ (ˆ z ) + O(1/n), O(1/n) )T . ✷ holds, U z ) = ( cos(nπˆ The next Lemma gives an estimate of dµ2 /dt. Lemma 5.6. Suppose e is an arbitrary but fixed real number between 0 and 1. Suppose p´2 is an eigenvalue for the free-free Timoshenko √ beam ˆ 1 p/ kG) · with density ρˆ(x; t) and that pˆ is a root of the function sin(L √ √ ˆ 1 p/ E) nearest p´. Let µ2 = L ˆ 2 p´2 . If (59) holds, then pˆ = nπ kG for sin(L 1 ˆ1 L some integer n, and

dµ2 = −kG dt 2kAG

− E I kG − 1



1

{(ˆ ρ3 )t [1 + cos(2nπz)]}dz 0



1

0

ˆ1 ∂ L ρˆ1/2 ∂t



ˆ1 L ρˆ1/2

On the other hand, if (60) holds, then pˆ = dµ2 =E dt



1



ˆ2γ L 1 − ρˆ3 ρˆ

0

2kAG

+ E I kG − 1



1

0

 dz + O(1/n).

√ nπ E ˆ L1

for some integer n, and



ˆ1 ∂ L ρˆ1/2 ∂t

(64)

 − cos(2nπz)(ˆ ρ3 )t

dz

t



ˆ1 L ρˆ1/2

 dz + O(1/n).

(65)

proof: The proof follows from Theorem 5.1 and Lemmas 5.4 and 5.5. ✷ Finally, for eigenvalues which are well separated from neighboring eigenvalues, we prove the following asymptotic formulas. Theorem 5.2. Let A, I, E and kG be positive constants. Suppose that ρ(x) > 0 for all x ∈ [0, L], that ρx (0) = ρx (L) = 0, and that ρxx ∈ L∞ [0, L]. L Let ρ0 = ([1/L] 0 ρ1/2 (x)dx)2 , and define

 E(p) = sin

1/2

pLρ0 √ E



 sin

1/2

pLρ0 √ kG

 .

Let e be a fixed but arbitrary real number between 0 and 1, and let pn i ≡

√ ni π E 1/2

Lρ0

, i = 1, , 2, . . . ,

36

GEIST AND MCLAUGHLIN

be a sequence of roots of E(p) such that   1/2 pni Lρ0 √ sin > e. kG Suppose each pni is large enough that exactly one simple eigenvalue pˇ of the Timoshenko beam lies in the interval   √ √ c c pn i − e, pni + e , 1/2 1/2 2Lρ0 2Lρ0 where c = min{E, kG}. Then pˇ2 =

n2i π 2 E E − 2 2 L ρ0 L ρ0



1

ρ3 (x) cos(2ni πz)dz + CE + O(1/ni ),

(66)

0

where x(z) is the inverse of the function z(x) = [1/L1 ]

x 0

ρ1/2 (s)ds and

  1

 E ρ0 A kG CE = − 1 dz I ρ0 E − kG 0 ρ(x) E − 2 L ρ0

 0

1

A kG ρ3 (x)dz + I ρ0

Let pmi ≡

√ mi π kG 1/2

Lρ0



kG + E 1− 2(E − kG)

 .

(67)

, i = 1, , 2, . . . ,

be a sequence of roots of E(p) such that   1/2 pmi Lρ0 √ sin > e. E Suppose each pmi is large enough that there is exactly one simple eigenvalue pˇ2 of the Timoshenko beam satisfying   √ √ c c pˇ ∈ pmi − e, pmi + e . 1/2 1/2 2Lρ0 2Lρ0 In this case, pˇ2 =

m2i π 2 kG kG − 2 L2 ρ0 L ρ0



1

ρ3 (x) cos(2mi πz)dz + CkG + O(1/mi ), (68) 0

37

EIGENVALUES OF THE TIMOSHENKO BEAM

where CkG

  1

 kG ρ0 A kG =− − 1 dz I ρ0 E − kG 0 ρ(x)

kG − 2 L ρ0

 0

1

A kG ρ3 (x)dz + I ρ0



kG + E 1+ 2(E − kG)

 .

(69)

ˆ 2 p´2 where p´2 is an eigenvalue for proof: To prove formula (66) let µ2 = L 1 a beam with density ρˆ(x; t). Define pˇ2 = p´2 |t=1 and p¯2 = p´2 |t=0 . Because ˆ 2 |t=1 = L ˆ 2 |t=0 = L2 ρ0 = L2 , as t changes from 0 to 1, µ2 /[L2 ρ0 ] will L 1 1 1 change continuously in t from p¯2 , an eigenvalue for a uniform beam with constant density ρ0 , to ρˇ2 , and eigenvalue for a beam with density ρ(x). Furthermore, Lemma 5.6 shows that dµ2 /dt satisfies (65). This implies that  

 1 1 1 L2 ρ0 (ˇ L21 γ − ρ3 − cos(2nπz)ρ3 dz p2 − p¯2 ) = E − ρ ρ0 0 L2 kAG

+ E I kG − 1

 0

1



 ρ0 − 1 dz + O(1/ni ) ρ

(70)

where L2 ρ0 pˇ2 = µ2 |t=1 is an eigenvalue for the transformed problem with density ρ(x) and L2 ρ0 p¯2 = µ2 |t=0 is an eigenvalue for the transformed problem with constant density ρ0 . Dividing through by L2 ρ0 = L21 , we find that   1  1

E 1 1 2 2 dz − 2 pˇ − p¯ = E γ ρ3 dz − ρ ρ0 L ρ0 0 0 kAG

+ E I kG − 1

 0

1



1 1 − ρ ρ0



E dz − 2 L ρ0





1

cos(2ni πz)ρ3 dz + O 0

 1 (71) ni

From Geist-McLaughlin [10], it follows that

  n2i π 2 E 1 kG + E A kG 1 2 p¯ = 2 . + 1− +O L ρ0 2 kG − E I ρ0 ni Formula (66) follows from the last equation and (71). A similar argument proves formulas (68) and (69). For eigenvalues that are close to the pmi s, dµ2 /dt satisfies (64) (instead of (65)), and the formula for p¯2 becomes (see [10])

  m2 π 2 kG 1 kG + E A kG 1 p¯2 = i 2 .✷ + 1+ +O L ρ0 2 kG − E I ρ0 mi

38

GEIST AND MCLAUGHLIN

REFERENCES 1. George Bachman and Lawerence Narici. Functional Analysis. Academic Press, New York, 1966. 2. G. Borg. Eine umkehrung der Sturm-Liouvilleschen eigenwertaufgabe. Acta Math., 78:1–96, 1946. 3. William E. Boyce and Richard C. Diprima. Elementary Differential Equations and Boundary Value Problems. John Wiley & Sons, Inc., New York, 5th edition, 1992. 4. E. A. Coddington and N. Levinson. Theory of Ordinary Differential Equations. McGraw-Hill Book Company, New York, 1955. 5. C. F. Coleman and J. R. McLaughlin. Solution of the inverse spectral problem for an impedance with integrable derivative, part I, II. Comm. Pure and Appl. Math., XLVI:145–212, 1993. 6. C. T. Fulton and S. A. Pruess. Eigenvalue and eigenfunction asymptotics for regular Sturm-Liouville problems. J. Math. Anal. Appl., 188:227–340, 1994. 7. C. T. Fulton and S. A. Pruess. Erratum. J. Math. Anal. Appl., 189:313–314, 1995. 8. Bruce Geist. The asymptotic expansion of the eigenvalues of the Timoshenko beam. Ph.D. dissertation, Rensselaer Polytechnic Institue, Troy, N.Y., 1994. 9. Bruce Geist and J. R. McLaughlin. Double eigenvalues for the uniform Timoshenko beam. Applied Mathematics Letters, 10(3):129–134, 1997. 10. Bruce Geist and J. R. McLaughlin. Eigenvalue formulas for the uniform Timoshenko beam: The free-free problem. Electronic Research Announcement AMS, 4, 1998. 11. O. H. Hald. The inverse Sturm-Liouville problem with symmetric potentials. Acta Math., 141:263–291, 1978. 12. O. H. Hald and J. R. McLaughlin. Solutions of inverse nodal problems. Inverse Problems, 5:307–347, 1989. 13. T. C. Huang. The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions. Journal of Applied Mechanics, 28:579–584, 1961. 14. E. L. Isaacson and E. Trubowitz. The inverse Sturm-Liouville problem, I. Comm. Pure and Appl. Math., XXXVI:767–783, 1983. 15. Edwin T. Kruszewski. Effect of transverse shear and rotary inertia on the natural frequency of a uniform beam. In National Advisory Committee for Aeronautics, Technical Note no. 1909, July 1949. 16. Ben Noble and James W. Daniel. Applied Linear Algebra. Prentice-Hall, Inc., Englewood Cliffs, N.J., 2nd edition, 1977. 17. Jurgen Poschel and Eugene Trubowitz. Inverse Spectral Theory. Academic Press, New York, 1987. 18. W. Rundell and P. Sacks. Reconstruction techniques for classical inverse SturmLiouville problems. Math. Comp., 58(197):161–183, 1992. 19. Charles R. Steele. Application of the WKB method in solid mechanics. Mechanics Today, 3:243–295, 1976. 20. William T. Thomson. Vibration Theory and Applications. Prentice-Hall, Englewood Cliffs, N.J., 1965.

EIGENVALUES OF THE TIMOSHENKO BEAM

39

21. Stephen Timoshenko. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philisophical Magazine, 41:744–746, 1921. 22. Stephen Timoshenko. On the transverse vibrations of bars of uniform cross-section. Philisophical Magazine, 43:125–131, 1922. 23. Stephen Timoshenko. Strength of Materials: Elementary Theory and Problems. D. Van Nostrand Company, Inc., New York, 2nd edition, 1940. 24. R. W. Trail-Nash and A. R. Collar. The effects of shear flexibility and rotatory inertia on the bending vibrations of beams. Quarterly Journal of Mechanics and Applied Mathematics, VI(2):186–222, 1953.