MATHEMATICAL ANALYSIS OF DYNAMIC MODELS OF ...
SIAM J. APPL. MATH. Vol. 58, No. 3, pp. 853–874, June 1998
c 1998 Society for Industrial and Applied Mathematics ° 007
MATHEMATICAL ANALYSIS OF DYNAMIC MODELS OF SUSPENSION BRIDGES∗ N. U. AHMED† AND H. HARBI† Abstract. In this paper we present some mathematical analysis of dynamic (PDE) models of suspension bridges as proposed by Lazer and McKenna. Our results are illustrated by numerical simulation with physical interpretation. Key words. suspension bridges, dynamic models, semigroups, stability, simulations AMS subject classifications. 93D05, 58D25,35Q72 PII. S0036139996308698
1. Introduction. In [1] Lazer and McKenna studied the problem of nonlinear oscillation in a suspension bridge. They presented a (one-dimensional) mathematical model for the bridge that takes account of the fact that the coupling provided by the stays (ties) connecting the suspension (main) cable to the deck of the road bed is fundamentally nonlinear. The federal works agency report [2] authored by Amann, Von Karman, and Woodruff on the collapse of the Tacoma Narrows bridge in the state of Washington on November 7, 1940, as a result of wind action, created a widespread demand from both the general public and the engineering profession that steps be taken to inaugurate a comprehensive investigation of the problem of dynamic oscillation in suspension bridges, with a view to understanding the causes of such destructive oscillations and developing design techniques to prevent their recurrence in future. It appears that serious study of the mathematical theory of suspension bridges was initiated by Bleich et al. [3], Wiles [4], Selberg [5], and later by Abdel-Ghaffer [6], and more recently by Lazer and McKenna [1]. In [6] Abdel-Ghaffar gave a new methodology of free vibration analysis of suspension bridges with horizontal decks, utilizing a continuum approach to include the coupling between the vertical and torsional vibration and the effects of cross-sectional distortion. Variational principles were used to obtain the coupled equations of motion in their most general and nonlinear form. In [7] Kawada and Hirai gave a new approach to suspension bridge rehabilitation. In [8] McKenna and Walter investigated nonlinear oscillation in a suspension bridge, where the model was suggested in [9]. In [10] McKenna and Walter considered the existence of traveling wave solutions to a nonlinear beam equation which was proposed as a model for a suspension bridge in two previous articles [8], [9]. Oscillation-induced fatigue of the structural members is a major factor limiting the useful life of a suspension bridge. Some relevant studies regarding oscillatory solutions were carried out in [11, 12, 13]. Instability and vibration problems are also encountered in space structures such as communication satellites, space shuttles, and space stations, which are equipped with large antennas mounted on long flexible masts (beams) [14, 15, 16, 17]. In this paper, we use the model proposed by Lazer and McKenna as described ∗ Received
by the editors September 3, 1996; accepted for publication (in revised form) December 17, 1996; published electronically March 24, 1998. http://www.siam.org/journals/siap/58-3/30869.html † Department of Electrical Engineering and Mathematics, University of Ottawa, 161 Louis Pasteur St., P.O. Box 450, Station A, Ottawa, ON K1N 6N5, Canada ([email protected], harbi@trix. genie.uottawa.ca). 853
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in [1] and study the PDE version of it. We present some mathematical analysis, numerical results, and discussions thereof. We also present an abstract model which allows us to determine the (regularity) properties of solutions of the proposed models. 2. Some relevant function spaces. Let Σ ⊂ Rn be an open bounded set with smooth boundary ∂Σ and let L2 (Σ) denote the space of equivalence classes of Lebesgue square integrable functions with the standard norm topology. Let H m (Σ) ≡ H m , m ∈ N, denote the standard Sobolev space with the usual norm topology and H0m (Σ) ≡ H0m ⊂ H m denote the completion in the topology of H m of C ∞ functions on Σ with compact support. From classical results on Sobolev spaces it is well known that the elements of H0m are those of H m which, along with their conormal derivatives up to order m − 1, vanish on the boundary ∂Σ. 3. Dynamic models of suspension bridges. Suspension bridge model. A simplified model of a suspension bridge is given by a coupled system of PDEs of the form: (3.1)
mb ztt + αD4 z − F0 (y − z) = mb g + f1 , mc ytt − βD2 y + F0 (y − z) = mc g + f2 ,
x ∈ (0, ℓ) ≡ Σ, t ≥ 0, x ∈ (0, ℓ), t ≥ 0,
where the first equation describes the vibration of the road bed in the vertical plain and the second equation describes that of the main cable from which the road bed is suspended by tie cables (stays). Dk denotes the spatial derivative of order k. Here mb , mc are the masses per unit length of the road bed and the cable, respectively; α, β are the flexural rigidity of the structure and coefficient of tensile strength of the cable, respectively. The function F0 represents the restraining force experienced by both the road bed and the suspension cable as transmitted through the tie lines (stays), thereby producing the coupling between the two. The functions f1 and f2 represent external as well as nonconservative forces, which are generally time-dependent. Let zs , ys represent the static displacements (equilibrium positions) which are the solutions of the system of equations: (3.2)
αD4 z − F0 (y − z) = mb g, −βD2 y + F0 (y − z) = mc g,
x ∈ (0, ℓ), x ∈ (0, ℓ).
Subtracting (3.2) from (3.1), we obtain the following system of equations: (3.3)
y − z˜) = f1 , mb z˜tt + αD4 z˜ − F (˜ y − z˜) = f2 , mc y˜tt − βD2 y˜ + F (˜
x ∈ Σ ≡ (0, ℓ), t ≥ 0, x ∈ (0, ℓ), t ≥ 0,
where z˜ ≡ z − zs , y˜ ≡ y − ys , and the function F is given by F (ζ) ≡ F0 (ζ + zs − ys ) − F0 (zs − ys ). Note that F (0) = 0. Throughout the rest of the paper we assume that the displacements, again denoted by z, y instead of z˜, y˜, are measured relative to the static positions. We use (3.3) as the general model. 3.1. Linear model. A linear model is obtained by supporting the road bed with ties (stays) connected to two symmetrically placed main (suspension) cables, one above and one below the road bed. In the absence of external forces, the dynamics of a linear suspension bridge around the equilibrium position can be described by a system of coupled PDEs as given below: (3.4)
mb ztt + αD4 z − k(y − z) = 0, mc ytt − βD2 y + k(y − z) = 0,
x ∈ (0, ℓ), x ∈ (0, ℓ),
t ≥ 0, t ≥ 0.
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The quantity k denotes the stiffness coefficient of the stays (ties) connecting the road bed to the suspension cable. Assuming that the beam is clamped at both ends the boundary conditions are given by z(t, 0) = z(t, ℓ) = 0, y(t, 0) = y(t, ℓ) = 0.
(3.5)
Dz(t, 0) = Dz(t, ℓ) = 0,
In case the beam is hinged at both ends the boundary conditions are given by D2 z(t, 0) = D2 z(t, ℓ) = 0,
z(t, 0) = z(t, ℓ) = 0, y(t, 0) = y(t, ℓ) = 0.
(3.6)
Other combinations, such as hinged on one side and clamped on the other, are also used. The initial conditions are given by z(0, x) = z1 (x), y(0, x) = y1 (x),
(3.7)
zt (0, x) = z2 (x), yt (0, x) = y2 (x),
x ∈ (0, ℓ), x ∈ (0, ℓ),
where z1 , z2 , y1 , y2 are suitable real-valued functions defined on Σ = (0, ℓ). Using the Fadeo–Galerkin method one can establish the existence and uniqueness of solutions of the system (3.4)–(3.7). Given that z1 ∈ H02 , z2 ∈ L2 (Σ), y1 ∈ H01 , and y2 ∈ L2 (Σ) the system (3.4)–(3.7) has unique solutions {z, y} ∈ L∞ (I, H02 × H01 ) and {zt , yt } ∈ L∞ (I, L2 (Σ) × L2 (Σ)). See the abstract model and analysis thereof in section 4. This system is conservative. Indeed, the total energy is given by Z ℓ ª (3.8) E(t) ≡ (1/2) mb |zt |2 + mc |yt |2 + α|D2 z|2 + β|Dy|2 + k|(y − z)|2 dx. 0
Differentiating E with respect to time and using the boundary conditions (3.5) or (3.6), we obtain Z ℓ ¡ ¡ 4 2 mb ztt + αD z − k(y − z) zt + mc ytt + βD y + k(y − z) yt dx. (d/dt)E = 0
Since {z, y} is the solution of the system (3.4)–(3.7), it follows from (3.4) and the above expression that (d/dt)E = 0 and hence (3.9a)
E(t) = E(0) for all t ≥ 0.
This shows that the system (3.4)–(3.7) is conservative and hence stable in the Lyapunov sense. However, the system is not asymptotically stable with respect to the rest state, though this is what is desirable for engineering structures. This is illustrated in Fig. 1. 3.2. Nonlinear model. If one of the sets of tie cables (stays) above or below the road bed is removed, the linear system of section 3.1 turns into a nonlinear one and can be described as follows: (3.10)
mb ztt + αD4 z − kΨ(y − z) = 0, mc ytt − βD2 y + kΨ(y − z) = 0,
x ∈ Σ ≡ (0, ℓ), t ≥ 0, x ∈ (0, ℓ), t ≥ 0,
where (3.11)
Ψ(ξ) =
ξ if ξ > 0, 0, otherwise.
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Fig. 1. General definition diagram of suspension bridge.
This is subject to the same set of boundary and initial conditions (3.5), (3.6), and (3.7), respectively. Again the total system energy is given by (3.12) E(t) ≡ (1/2)
Z
0
ℓ
ª mb |zt |2 + mc |yt |2 + α|D2 z|2 + β|Dy|2 + k(Ψ(y − z))2 dx.
With the help of Heaviside function it is not difficult to verify that (d/dt)E(t) =
Z ℓ 0
¡
¡ 2 mb ztt + αD z − kΨ(y − z) zt + mc ytt + βD y + kΨ(y − z) yt dx. 4
Thus it follows from equation (3.10) and the above expression that (3.9b)
˙ E(t) =0
and hence again the nonlinear system is also conservative.
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3.3. General nonlinear model. In general the function F of model (3.3) can be taken as any function with its graph lying in the first and third quadrants of the plane R2 . However, from a physical point of view, it makes sense only if F is a nondecreasing function of its argument. In any case let us consider the corresponding homogeneous system: (3.13)
mb ztt + αD4 z − F (y − z) = 0, mc ytt − βD2 y + F (y − z) = 0,
x ∈ Σ ≡ (0, ℓ), t ≥ 0, x ∈ (0, ℓ), t ≥ 0.
This is subject to the same set of boundary and initial conditions as in (3.5), (3.6), and (3.7), respectively. Define (3.14)
G(ζ) ≡
Z
ζ
F (ξ)dξ.
0
The total system energy is given by Z ℓ ª (3.15) E(t) ≡ (1/2) mb |zt |2 + mc |yt |2 + α|D2 z|2 + β|Dy|2 + 2G(y − z) dx. 0
Again, it is easy to verify that ˙ E(t) = 0,
(3.9c)
and hence the nonlinear system (3.13) is also conservative. In view of the above results we observe that in the absence of external forces a suspension bridge, linear or nonlinear, is conservative. 3.4. Aerodynamic damping. In all the models given above aerodynamic damping has been neglected. Considering the model (3.3) and including aerodynamic damping we can write (3.16)
mb ztt + αD4 z − F (y − z) + f1 (zt ) = 0, mc ytt − βD2 y + F (y − z) + f2 (yt ) = 0,
x ∈ Σ ≡ (0, ℓ), t ≥ 0, x ∈ (0, ℓ), t ≥ 0.
This is subject to the same set of boundary and initial conditions as in (3.5) and (3.7), respectively. Using the energy function (3.15) and carrying out the differentiation one can verify that Z ℓ ª (3.17) f1 (zt )zt + f2 (yt )yt dx. (d/dt)E(t) = − 0
It follows from this expression that if fi (ξ)ξ ≥ 0 then E˙ ≤ 0. Here we need a result similar to that of LaSalle invariance to conclude that the system is asymptotically stable given that fi (ζ)ζ > 0 for ζ 6= 0. We give here an elementary proof of this. For this purpose we can choose the energy functional for the Lyapunov function. We introduce W ≡ H02 × L2 (Σ) × H01 × L2 (Σ) as the energy space and define V (z1 , z2 , y1 , y2 ) ≡ (1/2) (3.18)
Z
0
ℓ
ª mb |z2 |2 +mc |y2 |2 +α|D2 z1 |2 +β|Dy1 |2 +2G(y1 −z1 ) dx
as the Lyapunov function on W.
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Theorem 3.1. Consider the system (3.16) and suppose the following assumptions hold: (A1) fi (0) = 0, fi (ζ)ζ > 0 for ζ 6= 0, (A2) F (ξ)ξ ≥ 0 for ξ ∈ R. Then the system is asymptotically stable with respect to the rest state (zero state). ˙ ≡ Proof. By virtue of (3.17) and the assumption (A1), it is clear that E(t) (d/dt)V (z, zt , y, yt ) < 0 whenever zt 6= 0 or yt 6= 0. Hence E(t), t ≥ 0, is a nonnegative monotone nonincreasing function of t and has a limit, say, limt→∞ E(t) = r0 . If r0 6= 0, then along any solution trajectory of (3.16) starting from any point on the boundary of the energy ellipsoid, Er0 ≡ {(z1 , z2 , y1 , y2 ) ∈ W : V ≤ r0 }, E˙ < 0, and the system continues to dissipate energy and move towards the origin. We show that E˙ can vanish on an interval, say, J ≡ [t0 , t0 + τ ], and the energy decay stall, only if the rest state has been reached. We prove this by establishing a contradiction. Suppose that E˙ vanishes on the interval J, without having reached the rest state. Then, by virtue of assumption (A1), it follows from (3.17) that {zt ≡ 0, yt ≡ 0} on J, and hence {z, y} must satisfy the equation a2 D4 z − F (y − z) = 0, −b2 D2 y + F (y − z) = 0,
(3.19)
x ∈ Σ = (0, ℓ), x ∈ Σ = (0, ℓ),
and either of the boundary conditions (3.5) or (3.6). Let {z 0 , y 0 } be a nontrivial solution of (3.19). Scalar-multiplying the first equation by z = z 0 and the second by y = y 0 , integrating by parts using the boundary conditions, and then adding the resulting expressions we arrive at (3.20)
a2 k D2 z 0 k2 +b2 k Dy 0 k2 +(F (y 0 − z 0 ), y 0 − z 0 )L2 (Σ) = 0.
Since by assumption (A2), F (ξ)ξ ≥ 0, ξ ∈ R, we have Z ℓ F (y 0 (ξ) − z 0 (ξ))(y 0 (ξ) − z 0 (ξ))dξ ≥ 0. (F (y 0 − z 0 ), y 0 − z 0 )L2 (Σ) = 0
Hence (3.21)
a2 k D2 z 0 k2 +b2 k Dy 0 k2 ≤ 0.
This implies that D2 z 0 ≡ 0,
Dy 0 ≡ 0 on Σ,
and hence it follows from the boundary conditions that z 0 (ξ) ≡ 0, y 0 (ξ) ≡ 0 on Σ = (0.ℓ). This contradiction proves the result. In the presence of both viscous and structural damping f1 is a function of both zt and D4 zt . Assuming linearity f1 may be given by f1 (zt , D4 zt ) = γ11 zt + γ12 D4 zt . For the suspension cable the structural damping is negligible. Assuming linear viscous damping, f2 is given by f2 (yt ) = γ21 yt . Clearly, from physical consideration, γ11 , γ12 , γ21 ≥ 0. Substituting in equation (3.16), ˙ ≤ 0 and the system is asymptotically stable with it follows from (3.17) that E(t) respect to the origin.
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4. Abstract model. In this section we consider the system (3.3) with either of the boundary conditions (3.5) or (3.6) and reformulate (3.3) as an ODE on an appropriate Hilbert space. The abstract formulation has many advantages, as we shall see below. First we write equation (3.3) in its normalized form as follows: (4.1)
ztt + a2 D4 z = F1 (t, x, y, z), ytt − b2 D2 y = F2 (t, x, y, z),
x ∈ D, x ∈ D,
t ≥ 0, t ≥ 0,
where (4.2)
b2 ≡ (β/mc ), a2 ≡ (α/mb ), F1 (t, x, y, z) ≡ (1/mb )(F (y − z) + f1 ), F2 (t, x, y, z) ≡ (1/mc )(−F (y − z) + f2 ).
We introduce the Hilbert space H and the vector space V as follows: H ≡ L2 (Σ) × L2 (Σ), V ≡ H02 × H01 ,
(4.3)
with the first space being equipped with the standard inner product. The second space is equipped with the scalar product and norms as follows: (4.4)
hφ, ψiV ≡ (D2 φ1 , D2 ψ1 )L2 (Σ) + (Dφ2 , Dψ2 )L2 (Σ) , 1/2 2 2 2 . k φ kV ≡ k D φ1 kL2 (Σ) + k Dφ2 kL2 (Σ)
By virtue of the Poincar´e inequality the norms k v kH m ≡
X
k Dα v kL2 (Σ)
1/2
X
k Dα v kL2 (Σ)
1/2
|α|≤m
k v kH0m ≡
|α|=m
and
are equivalent, and hence the space V endowed with the scalar product and norm as defined by (4.4) is also a Hilbert space. Note that the embedding V ֒→ H is continuous and dense, and moreover it is actually compact. Letting V ∗ denote the topological dual of V and identifying H with its dual we have V ֒→ H ֒→ V ∗ with injections being compact. Note that V ∗ = H −2 ×H −1 , where H −s , s > 0, denote the Sobolev spaces with negative exponents which are actually distributions. Now we introduce the bilinear form (4.5)
c(φ, ψ) ≡ a2 (D2 φ1 , D2 ψ1 )L2 (D) + b2 (Dφ2 , Dψ2 )L2 (D) .
It is clear that the map (4.6)
c : V × V −→ R
is a continuous bilinear form. Indeed, it is easy to verify that for d ≡ 2(a2 + b2 ) (4.7)
c(φ, ψ) ≤ d k φ kV k ψ kV
for all φ, ψ ∈ V.
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Thus there exists a linear operator A ∈ L(V, V ∗ ) such that (4.8)
c(φ, ψ) = hAφ, ψiV ∗ ,V for all φ, ψ ∈ V.
The operator A is precisely the realization of the formal differential operator 2 4 a D φ1 (4.9) , A(D)φ ≡ −b2 D2 φ2 with the boundary conditions (3.5), (3.6). Similarly, for d0 ≡ min{a2 , b2 } we have c(φ, φ) ≥ d0 k φ k2V
for all φ ∈ V.
Thus A is coercive. Define the operator F1 (t, ., φ1 (t, .), φ2 (t, .)) (4.10) . F (t, φ) ≡ F2 (t, ., φ1 (t, .), φ2 (t, .)) Now, defining the state variable for each t ≥ 0, as φ1 (t) z(t, .) φ(t) ≡ = , φ2 (t) y(t, .) which are functions defined on Σ, we can reformulate the system (4.1) as an abstract second-order differential equation on the Hilbert space H given by (d2 /dt2 )φ + Aφ = F (t, φ), t ≥ 0, ˙ φ(0) ≡ φ01 , φ(0) = φ02 ,
(4.11)
where the initial state {φ01 , φ02 } is given by equation (3.7). With this preparation we can now prove a result on the existence, uniqueness, and regularity of solutions of the Cauchy problem (4.11). For any Banach space X and 1 ≤ p ≤ ∞, let Lloc p (X) ≡ loc Lp (R0 , X) denote the class of X-valued functions on R0 ≡ [0, ∞) having X-norms locally pth-power integrable. Definition 4.1. The Cauchy problem (4.11) is said to have a weak solution if there exists a φ satisfying loc loc ∗ ¨ ˙ (i) φ ∈ Lloc 2 (V ) φ ∈ L2 (H), φ ∈ L2 (V ); ∞ (ii) for all η ∈ C0 (I) and any v ∈ V Z Z Z ˙ (4.12) v)η(t)dt ˙ + hφ(t), A∗ viη(t)dt = (F (t, φ(t)), v)H η(t)dt; − (φ(t), I
I
I
˙ = φ02 . (iii) φ(0) = φ01 , φ(0) First we prove an a priori bound. Let I ≡ [0, T ], T < ∞. We introduce the following assumptions for F. (F1) F : I × H −→ H is Lebesgue measurable in t on I and Borel measurable in the second variable on H. (F2) There exists a constant K ∈ R such that k F (t, ζ) k2H ≤ K 2 (1+ k ζ k2H ),
ζ ∈ H.
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MATHEMATICAL ANALYSIS OF SUSPENSION BRIDGES
Lemma 4.2. Suppose the hypotheses (F1) and (F2) hold and let φ01 ∈ V and φ02 ∈ H. Then any solution φ of (4.11), if one exists, satisfies the following conditions. (i) There exist constants M, ω > 0 such that (4.13)
˙ k2 + k φ(t) k2 ≤ M eωt , t ≥ 0. k φ(t) H V
(ii) φ ∈ L∞ (I, V ), φ˙ ∈ L∞ (I, H), and φ¨ ∈ L∞ (I, V ∗ ). Proof. Let φ be any solution of the Cauchy problem (4.11). Using the V ∗ − V pairing and scalar-multiplying the equation with φ˙ on either side, we obtain (4.14)
˙ v∗ ,V + c(φ, φ) ˙ = hF (t, φ), φi ˙ V ∗ ,V = (F (t, φ), φ) ˙ H, ¨ φi hφ,
where c is the bilinear form as given by (4.5). It is not difficult to show that 2 ˙ ˙ ¨ hφ(t), φ(t)i = (1/2)d/dt k φ(t) kH in the sense of distributions. Clearly, the bilinear form c is symmetric and hence ˙ 2c(φ(t), φ(t)) = (d/dt)c(φ(t), φ(t)). Using these facts in (4.14) we obtain 2 ˙ ˙ H, (1/2)d/dt k φ(t) kH +c(φ(t), φ(t)) = (F (t, φ(t)), φ)
t ≥ 0.
Integrating this, recalling the norm topology of V , and using the Schwarz inequality and the embedding constant, say δ > 0, for the injection V ֒→ H, we arrive at Z t 2 2 2 2 ˙ ˙ k φ(s) k2H ds k φ(t) kH +d0 k φ(t) kV ≤ k φ02 kH +d k φ01 kV + 0 Z t (4.15) (1 + δ 2 k φ(s) k2V )ds, t ≥ 0. + K2 0
Defining d˜0 = min{1, d0 }, d˜ = max{1, d}, δ˜2 = max{1, δ 2 }, ˜ 2 )/d˜0 , ˜ d˜0 ), d2 ≡ (1 + (K δ) d1 ≡ (d/ one can rewrite equation (4.15) as ¡ ¡ ˙ k2 + k φ(t) k2 ≤ d1 1+ k φ02 k2 + k φ01 k2 1+ k φ(t) H V H V Z t (4.16) ¡ ˙ k2H + k φ(s) k2V )ds, 1+ k φ(s) + d2 0
Defining ˙ k2 + k φ(t) k2 , η(t) ≡ 1+ k φ(t) H V it follows from (4.16) that η(t) ≤ d1 η(0) + d2
Z
0
t
η(s)ds,
t ≥ 0.
t ≥ 0.
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N. U. AHMED AND H. HARBI
Hence, by the Gronwall inequality, we have η(t) ≤ d1 η(0)ed2 t ≡ M eωt ,
t ≥ 0,
which implies (4.13). Hence (i) follows. The first two inclusions of (ii) follow from (i). The last inclusion follows from (i) and the inequality ¨ k2 ∗ ≤ c0 + c1 k φ(t) k2 k φ(t) V V
for all t ≥ 0,
where the constants c0 = c0 (δ, K), c1 = c1 (d, δ, K) are dependent on the parameters as indicated. This completes the proof. The following result is believed to have independent interest in other applications. Lemma 4.3. Let X, Y be two arbitrary Banach spaces with the embedding X ֒→ Y being compact. Let I = [0, T ] be a finite interval, p ∈ (1, ∞), and B ⊂ L∞ (I, X) be a bounded set for which there exists a constant ˜b > 0, such that Z k f˙ kpY dt ≤ ˜b for all f ∈ B. I
Then B is a relatively compact subset of Lp (I, Y ). Proof. Clearly, the set B, considered as a subset of Lp (I, Y ), is bounded. Since B is a bounded subset of L∞ (I, X) for almost all t ∈ I, the set B(t) ≡ {f (t), f ∈ B} ⊂ X is bounded, and since the embedding X ֒→ Y is compact, B(t) is a relatively compact subset of Y. For each f ∈ B, f˙ ∈ Lp (I, Y ) and we have f (t + h) = f (t) +
Z
t+h
f˙(s)ds for almost all t, t + h ∈ I.
t
For h ∈ R define I(h) ≡ I ∩ [0, T − h]. Then we have °Z °p Z ° t+h ° R ° p ˙(s)ds° dt = k f (t + h) − f (t) k f ° ° dt Y I(h) ° I(h) ° t Y Z (4.17) kf˙(t)kp dt ≤ |h|p−1 T ≤ |h|p−1 T ˜b
I
for all f ∈ B. Therefore, lim
h→0
Z
I(h)
k f (t + h) − f (t) kpY dt = 0
uniformly with respect to f ∈ B. Thus we have shown that the set B satisfies the following conditions: (i) B is a bounded subset of Lp (I, Y ), (ii) the t sections B(t) of B are relatively compact subsets of Y , and R (iii) limh→0 I(h) k f (t + h) − f (t) kpY dt = 0 uniformly with respect to f ∈ B. Thus B is a relatively compact subset of Lp (I, Y ). This completes the proof. Now we are prepared to prove the existence of a solution of equation (4.11) in the sense of Definition 4.1.
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Theorem 4.4. Suppose the assumptions (F1) and (F2) hold and the map x ∋ H −→ F (t, x) ∈ H is continuous. Then, for each φ01 ∈ V, φ02 ∈ H, the Cauchy problem (4.11) has at least one weak solution. Proof. Since the injection V ֒→ H is compact there exists a complete system of functions {φi , i ∈ N } which is orthogonal in V and V ∗ and orthonormal in H. We use the Galerkin approach and define ξin ≡ (φ01 , φi ), ξ˙in ≡ (φ02 , φi ),
(4.18)
i = 1, 2, 3, . . . , n ∈ N, i = 1, 2, 3, . . . , n ∈ N,
expand φ01 and φ02 with respect to the system {φi , i ∈ N }, and note that X s φn01 ≡ ξin φi −→ φ01 in V, 1≤i≤n
(4.19)
φn02
≡
X
s ξ˙in φi −→ φ02 in H.
1≤i≤n
By projection to the n-dimensional subspace spanned by {φi , i = 1, 2, . . . , n} we obtain the system of ODEs (4.20)
hd2 /dt2 φn (t), φj i + hAφn , φj i = (F (t, φn (t)), φj ),
1 ≤ j ≤ n,
with initial conditions given by the left-hand members of (4.19). Since F is measurable in t and continuous in the second variable and satisfies the growth condition (F2), this system of equations has at least one solution φn satisfying φn ∈ AC(I, V ), φ˙ n ∈ AC(I, V ) ⊂ AC(I, H), with φn (0) = φn01 and φ˙ n (0) = φn02 . By virtue of the a priori bounds (see Lemma 4.2) the sequence {φn , φ˙ n } is contained in a bounded subset of L∞ (I, V ) × L∞ (I, H) and hence there exists a subsequence of this sequence (relabeled as above) and a φ ∈ L∞ (I, V ) with φ˙ ∈ L∞ (I, H) such that w∗
φn −→ φ in L∞ (I, V ), w∗ φ˙ n −→ φ˙ in L∞ (I, H).
(4.21)
Multiplying equation (4.20) by η ∈ C0∞ (I) and integrating by parts over I we have Z Z Z (4.22) − (φ˙ n (t), φj )η(t)dt ˙ + hφn (t), A∗ φj iη(t)dt = (F (t, φn (t)), φj )η(t)dt. I
I
I
Letting n → ∞ in the first two terms of (4.22) it follows from (4.21) that Z Z ˙ φj )η(t)dt, ˙ ˙ = − (φ(t), limn→∞ − (φ˙ n (t), φj )η(t)dt Z I Z I (4.23) limn→∞ hφn (t), A∗ φj iη(t)dt = hφ(t), A∗ φj iη(t)dt. I
I
It remains to verify that Z Z (4.24) (F (t, φn (t)), φj )η(t)dt = (F (t, φ(t)), φj )η(t)dt. lim n→∞
I
I
Here we use the embedding result (Lemma 4.3). Take X = V, Y = H, p = 2, and B = {φn , n = 1, 2, . . .}. By virtue of Lemma 4.2, B satisfies the conditions of Lemma
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N. U. AHMED AND H. HARBI
4.3 and hence the family B = {φn } is relatively compact in L2 (I, H). Thus, through a subsequence if necessary, φn converges to φ strongly in L2 (I, H) and hence there exists a subsequence that converges almost everywhere to φ. Using this fact and the assumptions on F, (4.24) follows from the Lebesgue-dominated convergence theorem. Since the set {φj } is complete, we conclude that Z Z Z ∗ ˙ (4.25) ˙ + hφ(t), A viη(t)dt = (F (t, φ(t)), v)H η(t)dt − (φ(t), v)η(t)dt I
I
I
for any v ∈ V. Thus (ii) of Definition 4.1 is satisfied. Now integrating the first term of (4.25) by parts once, we have Z ¨ hφ(t), vi + hAφ(t), vi − (F (t, φ(t)), v)H η(t)dt = 0 (4.26) I
for all η ∈
C0∞ .
Hence the equality ¨ + Aφ(t) = F (t, φ(t)), φ(t)
(4.27)
t ∈ I,
holds in the sense of V ∗ -valued distributions. Thus (i) of Definition 4.1 follows from Lemma 4.2. To prove that φ as constructed above also satisfies the given initial conditions, multiply (4.20) by η ∈ C 2 (I) satisfying η(T ) = 0, η(T ˙ ) = 0, and scalarmultiply (4.27) by φj η(t) and integrate by parts. Subtracting one from the other and letting n → ∞ one arrives at n n ˙ (4.28) ˙ = 0. lim (φ(0) − φ02 , φj )η(0) + (φ01 − φ(0), φj )η(0) n→∞
Since η ∈ C 2 (satisfying the end conditions) is arbitrary and {φj } is complete in the triple {V, H, V ∗ }, the conclusion follows from (4.19) and (4.28). This completes the proof of Theorem 4.4. An additional regularity property of the solution is given in the following corollary. Corollary 4.5. The solutions of the Cauchy problem (4.11) satisfy the following regularity properties: φ ∈ C(I, V ), φ˙ ∈ C(I, H). Proof. The proof is given by parabolic regularization using the result (ii) of Lemma 4.2, and Lemma 8.1 of [18, p. 275]. Theorem 4.6. If, in addition to the assumptions of Theorem 4.4, the operator F also satisfies the Lipschitz condition k F (t, ζ) − F (t, ξ) k2H ≤ K 2 (k ζ − ξ k2H ),
ζ, ξ ∈ H,
then the system (4.11) has a unique weak solution and the solution is continuously dependent on the initial data. In view of the above regularity result we can extend the result of Theorem 4.4 to cover the system model (4.29)
˙ (d2 /dt2 )φ + Aφ = F (t, φ, φ), t ≥ 0, ˙ φ(0) = φ02 . φ(0) ≡ φ01 ,
MATHEMATICAL ANALYSIS OF SUSPENSION BRIDGES
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φ Defining E ≡ V × H and ψ = φ˙ we can rewrite equation (4.29) as a first-order evolution equation on the Hilbert space E as given below: (4.30)
ψ˙ + Aψ = F˜ (t, ψ), ψ(0) = ψ0 ≡
φ(0) ˙ φ(0)
t ≥ 0, ,
where the operators A and F˜ are given by 0 0 − Id F˜ (t, ψ) ≡ , , A≡ ˙ F (t, φ, φ) A0 with Id being the identity operator in H. Theorem 4.7. Suppose F˜ satisfy the following hypotheses. (G1) k F˜ (t, ζ) kE ≤ K(1+ k ζ kE ). (G2) For every r > 0 there exists a constant Kr ≥ 0 such that k F˜ (t, ζ) − F˜ (t, ξ) kE ≤ Kr (k ζ − ξ kE )
for all ζ, ξ ∈ Br ,
where Br ≡ {η ∈ E :k η kE ≤ r}. Then for each ψ0 ∈ E the Cauchy problem (4.30) has a unique mild solution ψ ∈ C(I, E). Proof (outline). We follow the procedure as in [19, Theorem 5.3.3]. In view of Corollary 4.5, the operator −A generates a C0 -semigroup, S(t), t ≥ 0, on E and hence, by the variation of constants formula, one can rewrite the evolution equation (4.30) as an integral equation Z t (4.31) S(t − s)F˜ (s, ψ(s))ds, t ∈ I. ψ(t) = S(t)ψ0 + 0
Using the assumptions (G1) and (G2) and the Banach fixed point theorem one can easily establish the existence and uniqueness of a solution ψ ∈ C(Iτ1 , E) of the integral equation (4.31) over the interval, say Iτ1 ≡ [0, τ1 ], for sufficiently small but positive τ1 . The hypothesis (G1) implies finiteness of ψ(τ1 ). Starting with this as the initial state one solves the integral equation Z t S(t − s)F˜ (s, ψ(s))ds, t ∈ Iτ1 ,τ2 ≡ [τ1 , τ2 ], (4.32) ψ(t) = S(t − τ1 )ψ(τ1 ) + τ1
where again τ2 is chosen suitably so that the integral operator is a contraction. Since I is compact, by repeating this and piecing together these uniquely defined local solutions on [0, τ1 ], [τ1 , τ2 ], [τ2 , τ3 ], . . . , one obtains a unique solution for the original interval I. This is the mild solution of equation (4.30). This completes the proof. Remark. Before we conclude this section we wish to briefly indicate that the abstract model (4.30) can be conveniently used to construct meaningful stochastic evolution equations on the Hilbert space E to include random wind forces (load). This is given by (4.33)
dψ = −Aψdt + F˜ (t, ψ)dt + G(t, ψ)dW, ψ(0) = ψ0 .
t ≥ 0,
Here W ≡ {W (t), t ≥ 0} is an H-valued Wiener process and G is an operator-valued function G : I × E 7→ L(H, E). More specific representation of this operator will be given in a forthcoming paper, where we also show that the abstract system (4.33) also absorbs the more elaborate model given in [6].
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Fig. 2. Linear system (total energy, displacement, and velocity).
5. Numerical results and discussions. In this section we present some simulation results and discussions thereof. For numerical solution we use a FORTRAN program using the finite difference scheme on a SUN SPARC station 5. For numerical stability the ratio (△t/ △ x) was chosen as 2 × 10−3 . Figure 1 shows the typical geometry of a suspension bridge, which is self-explanatory. (i) Linear case. According to equation (3.9a) a linear suspension bridge in the absence of damping is conservative. This is illustrated by the total energy plot as shown in Fig. 2a. Figures 2b and 2c show the corresponding displacements and their rates at the midpoint of the span. Note that the frequency of oscillation of the structure is roughly twice that of the cable.
MATHEMATICAL ANALYSIS OF SUSPENSION BRIDGES
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Fig. 3. Nonlinear system (total energy, displacement, and velocity).
(ii) Nonlinear case. The nonlinear system is also subject to the same initial data as in the linear case. We have seen in section 3.1, equation (3.9b), that the nonlinear bridge is also conservative in the absence of damping. This is illustrated by the numerical simulation result shown in Fig. 3a. Figures 3b and 3c show the displacements and their rates again at the midpoint of the span. Note the distortion caused by nonlinearity. The initial energy level in this case is lower because of cutoff due to nonlinearity (see (3.10), (3.11)). This is because the Lebesgue measure of the set Σ0 ≡ {x ∈ (0, ℓ) : y(0, x) ≥ z(0, x)} is less than ℓ. (iii) Linear/nonlinear. To start with, here we assume that the system is linear and at time ts = 4 the stays (tie cables) below the road bed break down (due pos-
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Fig. 4. Linear/nonlinear system (total energy and displacement).
sibly to seismic or other activities). After the breakdown, the system is nonlinear. Again, Fig. 4a shows the conservation of energy with some wiggles due possibly to computational error caused by abrupt change. This constancy of energy level even after switching to a nonlinear system is due to the fact that at the precise moment of switching, the set Σs ≡ {x ∈ (0, ℓ) : y(ts , x) − z(ts , x) ≥ 0} has (full) Lebesgue measure λ(Σs ) = ℓ, equal to the length of the span. This means that at time ts −, just before the breakdown, the stays were active and the system behaved like a linear system, and hence, being conservative, the energy stayed at the prebreakdown level. As a consequence, since the nonlinear system is also conserva-
MATHEMATICAL ANALYSIS OF SUSPENSION BRIDGES
869
Fig. 5. Linear/nonlinear system (total energy and displacement).
tive, the energy level remained constant thereafter. Just at the time of breakdown, the energy from the stay cables as transferred to the upper cables, resulting in a sudden increase of the amplitude of oscillation of these cables and a decrease of frequency, as shown in Fig. 4b. The same experiment was performed with different initial conditions (see Fig. 5). We observe a break in the energy level at ts . In this case λ(Σs ) < ℓ, meaning that the tie cables were loose on some sections of the span at the instant of breakdown. Consequently, in this case, comparatively, there was less energy transfer to the upper cables. Thus there is a drop in the total energy level, and after the breakdown, the nonlinear system continued its motion at a lower energy level. Even though, comparatively, the energy transfer was less in this case, the upper
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N. U. AHMED AND H. HARBI
Fig. 6. Damped system (total energy and displacement).
cables continued to oscillate at reduced amplitude compared to the previous case, as expected. (iv) Aerodynamic damping. We carried out a numerical experiment with system (3.16) for two different cases: (1) f1 (ζ) = γ ζ, γ > 0, and f2 = 0. (2) f1 (ζ) = γ ζ, γ > 0, and f2 (ζ) = δ ζ, δ > 0. The results are shown in Figs. 6 and 7. The system is asymptotically stable with respect to the rest state, as expected (see Theorem 3.1). (v) Moving load. System (3.3) with (i) F (ζ) = k ζ (linear case) and (ii) F (ζ) = kΨ(ζ) (nonlinear case), f2 = 0, and f1 6= 0, were subjected to a moving load (for
MATHEMATICAL ANALYSIS OF SUSPENSION BRIDGES
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Fig. 7. Damped system (total energy and displacement).
example, a moving truck) represented by f1 (t, x) ≡ −m0 g Ce(t) (x), e(t) ≡ (vt, vt + L) ∩ (0, ℓ), where m0 is the mass of the vehicle per unit length, L is its length, v is its velocity, and Ce stands for the characteristic function of the set e. The velocity v was chosen so that the vehicle passes the bridge in four units of time. Figure 8a shows the variation of energy level with time, and Fig. 8b shows the corresponding response. Note that the system is left with some residual energy after the passage of the vehicle (Fig. 8a). This is due to the absence of friction or damping. For the nonlinear system, similar results are shown in Fig. 9.
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N. U. AHMED AND H. HARBI
Fig. 8. Moving vehicle (total energy and displacement).
6. Concluding remarks. In this paper we have presented rigorous mathematical analysis of a simplified (PDE) model for suspension bridges as proposed by Lazer and McKenna. We have studied the dynamic behavior of the system under different situations and presented associated simulation results and physical interpretations. We have also presented an analysis of an abstract model and indicated its usefulness in determining the regularity of solutions and in constructing stochastic models. Currently, we are planning to use a more elaborate model such as the one suggested in [6] and its stochastic version and carry out a similar analysis, including simulation.
MATHEMATICAL ANALYSIS OF SUSPENSION BRIDGES
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Fig. 9. Moving vehicle (total energy and displacement).
REFERENCES [1] A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridge: Some new connections with nonlinear analysis, SIAM Rev., 32 (1990), pp. 537–578. [2] O. H. Amann, T. Von Karman, and G. B. Woodruff, The Failure of the Tacoma Narrows Bridge, Federal Works Agency, Washington, D.C., 1941. [3] F. Bleich, C. B. McCullough, R. Rosecrans, and G. S. Vincent, The Mathematical Theory of Suspension Bridges, Bureau of Public Roads, U.S. Dept. of Commerce, Washington, D.C., 1950. [4] E. G. Wiles, Report of Aerodynamic Studies on Proposed San Pedro-Terminal Island Suspension Bridge, California, Bridge Research Branch, Division of Physical Research, Bureau of Public Roads, U.S. Dept. of Commerce, Washington, D.C., 1960.
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[5] A. Selberg, Oscillation and aerodynamic stability of suspension bridges, Acta Polytech. Scand., 13 (1961), pp. 308–377. [6] A. M. Abdel-Ghaffer, Suspension bridge vibration: Continuum formulation, J. Engrg. Mech., 108 (1982), pp. 1215–1232. [7] T. Kawada and A. Hirai, Additional Mass Method-A New Approach to Suspension Bridge Rehabilitation, Official Proceedings, 2nd Annual International Bridge Conference, Engineers Society of Western Pennsylvania, Pittsburgh, PA, 1985. [8] P. J. McKenna and W. Walter, Nonlinear oscillation in a suspension bridge, Arch. Rational Mech. Anal., 98 (1987), pp. 167–177. [9] A. C. Lazer and P. J. McKenna, Large scale oscillation behavior in loaded asymmetric systems, Ann. Inst. H. Poincar´ e, Anal. Non Lin´ eaire, 4 (1987), pp. 244–274. [10] P. J. McKenna and W. Walter, Traveling waves in a suspension bridge, SIAM J. Appl. Math., 50 (1990), pp. 703–715. [11] J. Glover, A. C. Lazer, and P. J. McKenna, Existence and stability of large scale nonlinear oscillations in suspension bridges, J. Appl. Math. Phys. (ZAMP), 40 (1989), pp. 172–200. [12] Y. S. Choi, K. C. Jen, and P. J. McKenna, The structure of the solution set for periodic oscillation in a suspension bridge model, IMA J. Appl. Math., 47 (1991), pp. 283–306. [13] D. Jacover and P. J. McKenna, Nonlinear torsional flexings in a periodically forced suspended beam, J. Comp. Appl. Math., 52 (1994), pp. 241–265. [14] S. K. Biswas and N. U. Ahmed, Optimal control of large space structures governed by a coupled system of ordinary and partial differential equations, J. Math. Control, Signals, and Systems, 2 (1989), pp. 1–18. [15] N. U. Ahmed and S. S. Lim, Modeling and control of flexible space stations (slew maneuvers), in Proc. 3rd Annual Conference on Aerospace Computational Control, D. E. Bernard and G. K. Mann, eds., NASA Pub. 89-45, Vol. 2, NASAJPL, Pasadena, CA, pp. 900–914. [16] N. U. Ahmed and S. K. Biswas, Mathematical modeling and control of large space structures with multiple appendages, J. Math. Comput. Modeling, 10 (1988), pp. 891–900. [17] P. Li and N. U. Ahmed, On exponential stability of infinite dimensional systems with bounded or unbounded perturbations, Appl. Anal., 30 (1988), pp. 175–187. [18] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York, Heidelberg, Berlin, 1972. [19] N. U. Ahmed, Semigroup Theory with Applications to Systems and Control, Pitman Res. Notes Math. Ser. 246, Longman Scientific and Technical, Harlow, UK, 1991.
c 1998 Society for Industrial and Applied Mathematics ° 007
MATHEMATICAL ANALYSIS OF DYNAMIC MODELS OF SUSPENSION BRIDGES∗ N. U. AHMED† AND H. HARBI† Abstract. In this paper we present some mathematical analysis of dynamic (PDE) models of suspension bridges as proposed by Lazer and McKenna. Our results are illustrated by numerical simulation with physical interpretation. Key words. suspension bridges, dynamic models, semigroups, stability, simulations AMS subject classifications. 93D05, 58D25,35Q72 PII. S0036139996308698
1. Introduction. In [1] Lazer and McKenna studied the problem of nonlinear oscillation in a suspension bridge. They presented a (one-dimensional) mathematical model for the bridge that takes account of the fact that the coupling provided by the stays (ties) connecting the suspension (main) cable to the deck of the road bed is fundamentally nonlinear. The federal works agency report [2] authored by Amann, Von Karman, and Woodruff on the collapse of the Tacoma Narrows bridge in the state of Washington on November 7, 1940, as a result of wind action, created a widespread demand from both the general public and the engineering profession that steps be taken to inaugurate a comprehensive investigation of the problem of dynamic oscillation in suspension bridges, with a view to understanding the causes of such destructive oscillations and developing design techniques to prevent their recurrence in future. It appears that serious study of the mathematical theory of suspension bridges was initiated by Bleich et al. [3], Wiles [4], Selberg [5], and later by Abdel-Ghaffer [6], and more recently by Lazer and McKenna [1]. In [6] Abdel-Ghaffar gave a new methodology of free vibration analysis of suspension bridges with horizontal decks, utilizing a continuum approach to include the coupling between the vertical and torsional vibration and the effects of cross-sectional distortion. Variational principles were used to obtain the coupled equations of motion in their most general and nonlinear form. In [7] Kawada and Hirai gave a new approach to suspension bridge rehabilitation. In [8] McKenna and Walter investigated nonlinear oscillation in a suspension bridge, where the model was suggested in [9]. In [10] McKenna and Walter considered the existence of traveling wave solutions to a nonlinear beam equation which was proposed as a model for a suspension bridge in two previous articles [8], [9]. Oscillation-induced fatigue of the structural members is a major factor limiting the useful life of a suspension bridge. Some relevant studies regarding oscillatory solutions were carried out in [11, 12, 13]. Instability and vibration problems are also encountered in space structures such as communication satellites, space shuttles, and space stations, which are equipped with large antennas mounted on long flexible masts (beams) [14, 15, 16, 17]. In this paper, we use the model proposed by Lazer and McKenna as described ∗ Received
by the editors September 3, 1996; accepted for publication (in revised form) December 17, 1996; published electronically March 24, 1998. http://www.siam.org/journals/siap/58-3/30869.html † Department of Electrical Engineering and Mathematics, University of Ottawa, 161 Louis Pasteur St., P.O. Box 450, Station A, Ottawa, ON K1N 6N5, Canada ([email protected], harbi@trix. genie.uottawa.ca). 853
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in [1] and study the PDE version of it. We present some mathematical analysis, numerical results, and discussions thereof. We also present an abstract model which allows us to determine the (regularity) properties of solutions of the proposed models. 2. Some relevant function spaces. Let Σ ⊂ Rn be an open bounded set with smooth boundary ∂Σ and let L2 (Σ) denote the space of equivalence classes of Lebesgue square integrable functions with the standard norm topology. Let H m (Σ) ≡ H m , m ∈ N, denote the standard Sobolev space with the usual norm topology and H0m (Σ) ≡ H0m ⊂ H m denote the completion in the topology of H m of C ∞ functions on Σ with compact support. From classical results on Sobolev spaces it is well known that the elements of H0m are those of H m which, along with their conormal derivatives up to order m − 1, vanish on the boundary ∂Σ. 3. Dynamic models of suspension bridges. Suspension bridge model. A simplified model of a suspension bridge is given by a coupled system of PDEs of the form: (3.1)
mb ztt + αD4 z − F0 (y − z) = mb g + f1 , mc ytt − βD2 y + F0 (y − z) = mc g + f2 ,
x ∈ (0, ℓ) ≡ Σ, t ≥ 0, x ∈ (0, ℓ), t ≥ 0,
where the first equation describes the vibration of the road bed in the vertical plain and the second equation describes that of the main cable from which the road bed is suspended by tie cables (stays). Dk denotes the spatial derivative of order k. Here mb , mc are the masses per unit length of the road bed and the cable, respectively; α, β are the flexural rigidity of the structure and coefficient of tensile strength of the cable, respectively. The function F0 represents the restraining force experienced by both the road bed and the suspension cable as transmitted through the tie lines (stays), thereby producing the coupling between the two. The functions f1 and f2 represent external as well as nonconservative forces, which are generally time-dependent. Let zs , ys represent the static displacements (equilibrium positions) which are the solutions of the system of equations: (3.2)
αD4 z − F0 (y − z) = mb g, −βD2 y + F0 (y − z) = mc g,
x ∈ (0, ℓ), x ∈ (0, ℓ).
Subtracting (3.2) from (3.1), we obtain the following system of equations: (3.3)
y − z˜) = f1 , mb z˜tt + αD4 z˜ − F (˜ y − z˜) = f2 , mc y˜tt − βD2 y˜ + F (˜
x ∈ Σ ≡ (0, ℓ), t ≥ 0, x ∈ (0, ℓ), t ≥ 0,
where z˜ ≡ z − zs , y˜ ≡ y − ys , and the function F is given by F (ζ) ≡ F0 (ζ + zs − ys ) − F0 (zs − ys ). Note that F (0) = 0. Throughout the rest of the paper we assume that the displacements, again denoted by z, y instead of z˜, y˜, are measured relative to the static positions. We use (3.3) as the general model. 3.1. Linear model. A linear model is obtained by supporting the road bed with ties (stays) connected to two symmetrically placed main (suspension) cables, one above and one below the road bed. In the absence of external forces, the dynamics of a linear suspension bridge around the equilibrium position can be described by a system of coupled PDEs as given below: (3.4)
mb ztt + αD4 z − k(y − z) = 0, mc ytt − βD2 y + k(y − z) = 0,
x ∈ (0, ℓ), x ∈ (0, ℓ),
t ≥ 0, t ≥ 0.
MATHEMATICAL ANALYSIS OF SUSPENSION BRIDGES
855
The quantity k denotes the stiffness coefficient of the stays (ties) connecting the road bed to the suspension cable. Assuming that the beam is clamped at both ends the boundary conditions are given by z(t, 0) = z(t, ℓ) = 0, y(t, 0) = y(t, ℓ) = 0.
(3.5)
Dz(t, 0) = Dz(t, ℓ) = 0,
In case the beam is hinged at both ends the boundary conditions are given by D2 z(t, 0) = D2 z(t, ℓ) = 0,
z(t, 0) = z(t, ℓ) = 0, y(t, 0) = y(t, ℓ) = 0.
(3.6)
Other combinations, such as hinged on one side and clamped on the other, are also used. The initial conditions are given by z(0, x) = z1 (x), y(0, x) = y1 (x),
(3.7)
zt (0, x) = z2 (x), yt (0, x) = y2 (x),
x ∈ (0, ℓ), x ∈ (0, ℓ),
where z1 , z2 , y1 , y2 are suitable real-valued functions defined on Σ = (0, ℓ). Using the Fadeo–Galerkin method one can establish the existence and uniqueness of solutions of the system (3.4)–(3.7). Given that z1 ∈ H02 , z2 ∈ L2 (Σ), y1 ∈ H01 , and y2 ∈ L2 (Σ) the system (3.4)–(3.7) has unique solutions {z, y} ∈ L∞ (I, H02 × H01 ) and {zt , yt } ∈ L∞ (I, L2 (Σ) × L2 (Σ)). See the abstract model and analysis thereof in section 4. This system is conservative. Indeed, the total energy is given by Z ℓ ª (3.8) E(t) ≡ (1/2) mb |zt |2 + mc |yt |2 + α|D2 z|2 + β|Dy|2 + k|(y − z)|2 dx. 0
Differentiating E with respect to time and using the boundary conditions (3.5) or (3.6), we obtain Z ℓ ¡ ¡ 4 2 mb ztt + αD z − k(y − z) zt + mc ytt + βD y + k(y − z) yt dx. (d/dt)E = 0
Since {z, y} is the solution of the system (3.4)–(3.7), it follows from (3.4) and the above expression that (d/dt)E = 0 and hence (3.9a)
E(t) = E(0) for all t ≥ 0.
This shows that the system (3.4)–(3.7) is conservative and hence stable in the Lyapunov sense. However, the system is not asymptotically stable with respect to the rest state, though this is what is desirable for engineering structures. This is illustrated in Fig. 1. 3.2. Nonlinear model. If one of the sets of tie cables (stays) above or below the road bed is removed, the linear system of section 3.1 turns into a nonlinear one and can be described as follows: (3.10)
mb ztt + αD4 z − kΨ(y − z) = 0, mc ytt − βD2 y + kΨ(y − z) = 0,
x ∈ Σ ≡ (0, ℓ), t ≥ 0, x ∈ (0, ℓ), t ≥ 0,
where (3.11)
Ψ(ξ) =
ξ if ξ > 0, 0, otherwise.
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N. U. AHMED AND H. HARBI
Fig. 1. General definition diagram of suspension bridge.
This is subject to the same set of boundary and initial conditions (3.5), (3.6), and (3.7), respectively. Again the total system energy is given by (3.12) E(t) ≡ (1/2)
Z
0
ℓ
ª mb |zt |2 + mc |yt |2 + α|D2 z|2 + β|Dy|2 + k(Ψ(y − z))2 dx.
With the help of Heaviside function it is not difficult to verify that (d/dt)E(t) =
Z ℓ 0
¡
¡ 2 mb ztt + αD z − kΨ(y − z) zt + mc ytt + βD y + kΨ(y − z) yt dx. 4
Thus it follows from equation (3.10) and the above expression that (3.9b)
˙ E(t) =0
and hence again the nonlinear system is also conservative.
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3.3. General nonlinear model. In general the function F of model (3.3) can be taken as any function with its graph lying in the first and third quadrants of the plane R2 . However, from a physical point of view, it makes sense only if F is a nondecreasing function of its argument. In any case let us consider the corresponding homogeneous system: (3.13)
mb ztt + αD4 z − F (y − z) = 0, mc ytt − βD2 y + F (y − z) = 0,
x ∈ Σ ≡ (0, ℓ), t ≥ 0, x ∈ (0, ℓ), t ≥ 0.
This is subject to the same set of boundary and initial conditions as in (3.5), (3.6), and (3.7), respectively. Define (3.14)
G(ζ) ≡
Z
ζ
F (ξ)dξ.
0
The total system energy is given by Z ℓ ª (3.15) E(t) ≡ (1/2) mb |zt |2 + mc |yt |2 + α|D2 z|2 + β|Dy|2 + 2G(y − z) dx. 0
Again, it is easy to verify that ˙ E(t) = 0,
(3.9c)
and hence the nonlinear system (3.13) is also conservative. In view of the above results we observe that in the absence of external forces a suspension bridge, linear or nonlinear, is conservative. 3.4. Aerodynamic damping. In all the models given above aerodynamic damping has been neglected. Considering the model (3.3) and including aerodynamic damping we can write (3.16)
mb ztt + αD4 z − F (y − z) + f1 (zt ) = 0, mc ytt − βD2 y + F (y − z) + f2 (yt ) = 0,
x ∈ Σ ≡ (0, ℓ), t ≥ 0, x ∈ (0, ℓ), t ≥ 0.
This is subject to the same set of boundary and initial conditions as in (3.5) and (3.7), respectively. Using the energy function (3.15) and carrying out the differentiation one can verify that Z ℓ ª (3.17) f1 (zt )zt + f2 (yt )yt dx. (d/dt)E(t) = − 0
It follows from this expression that if fi (ξ)ξ ≥ 0 then E˙ ≤ 0. Here we need a result similar to that of LaSalle invariance to conclude that the system is asymptotically stable given that fi (ζ)ζ > 0 for ζ 6= 0. We give here an elementary proof of this. For this purpose we can choose the energy functional for the Lyapunov function. We introduce W ≡ H02 × L2 (Σ) × H01 × L2 (Σ) as the energy space and define V (z1 , z2 , y1 , y2 ) ≡ (1/2) (3.18)
Z
0
ℓ
ª mb |z2 |2 +mc |y2 |2 +α|D2 z1 |2 +β|Dy1 |2 +2G(y1 −z1 ) dx
as the Lyapunov function on W.
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Theorem 3.1. Consider the system (3.16) and suppose the following assumptions hold: (A1) fi (0) = 0, fi (ζ)ζ > 0 for ζ 6= 0, (A2) F (ξ)ξ ≥ 0 for ξ ∈ R. Then the system is asymptotically stable with respect to the rest state (zero state). ˙ ≡ Proof. By virtue of (3.17) and the assumption (A1), it is clear that E(t) (d/dt)V (z, zt , y, yt ) < 0 whenever zt 6= 0 or yt 6= 0. Hence E(t), t ≥ 0, is a nonnegative monotone nonincreasing function of t and has a limit, say, limt→∞ E(t) = r0 . If r0 6= 0, then along any solution trajectory of (3.16) starting from any point on the boundary of the energy ellipsoid, Er0 ≡ {(z1 , z2 , y1 , y2 ) ∈ W : V ≤ r0 }, E˙ < 0, and the system continues to dissipate energy and move towards the origin. We show that E˙ can vanish on an interval, say, J ≡ [t0 , t0 + τ ], and the energy decay stall, only if the rest state has been reached. We prove this by establishing a contradiction. Suppose that E˙ vanishes on the interval J, without having reached the rest state. Then, by virtue of assumption (A1), it follows from (3.17) that {zt ≡ 0, yt ≡ 0} on J, and hence {z, y} must satisfy the equation a2 D4 z − F (y − z) = 0, −b2 D2 y + F (y − z) = 0,
(3.19)
x ∈ Σ = (0, ℓ), x ∈ Σ = (0, ℓ),
and either of the boundary conditions (3.5) or (3.6). Let {z 0 , y 0 } be a nontrivial solution of (3.19). Scalar-multiplying the first equation by z = z 0 and the second by y = y 0 , integrating by parts using the boundary conditions, and then adding the resulting expressions we arrive at (3.20)
a2 k D2 z 0 k2 +b2 k Dy 0 k2 +(F (y 0 − z 0 ), y 0 − z 0 )L2 (Σ) = 0.
Since by assumption (A2), F (ξ)ξ ≥ 0, ξ ∈ R, we have Z ℓ F (y 0 (ξ) − z 0 (ξ))(y 0 (ξ) − z 0 (ξ))dξ ≥ 0. (F (y 0 − z 0 ), y 0 − z 0 )L2 (Σ) = 0
Hence (3.21)
a2 k D2 z 0 k2 +b2 k Dy 0 k2 ≤ 0.
This implies that D2 z 0 ≡ 0,
Dy 0 ≡ 0 on Σ,
and hence it follows from the boundary conditions that z 0 (ξ) ≡ 0, y 0 (ξ) ≡ 0 on Σ = (0.ℓ). This contradiction proves the result. In the presence of both viscous and structural damping f1 is a function of both zt and D4 zt . Assuming linearity f1 may be given by f1 (zt , D4 zt ) = γ11 zt + γ12 D4 zt . For the suspension cable the structural damping is negligible. Assuming linear viscous damping, f2 is given by f2 (yt ) = γ21 yt . Clearly, from physical consideration, γ11 , γ12 , γ21 ≥ 0. Substituting in equation (3.16), ˙ ≤ 0 and the system is asymptotically stable with it follows from (3.17) that E(t) respect to the origin.
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4. Abstract model. In this section we consider the system (3.3) with either of the boundary conditions (3.5) or (3.6) and reformulate (3.3) as an ODE on an appropriate Hilbert space. The abstract formulation has many advantages, as we shall see below. First we write equation (3.3) in its normalized form as follows: (4.1)
ztt + a2 D4 z = F1 (t, x, y, z), ytt − b2 D2 y = F2 (t, x, y, z),
x ∈ D, x ∈ D,
t ≥ 0, t ≥ 0,
where (4.2)
b2 ≡ (β/mc ), a2 ≡ (α/mb ), F1 (t, x, y, z) ≡ (1/mb )(F (y − z) + f1 ), F2 (t, x, y, z) ≡ (1/mc )(−F (y − z) + f2 ).
We introduce the Hilbert space H and the vector space V as follows: H ≡ L2 (Σ) × L2 (Σ), V ≡ H02 × H01 ,
(4.3)
with the first space being equipped with the standard inner product. The second space is equipped with the scalar product and norms as follows: (4.4)
hφ, ψiV ≡ (D2 φ1 , D2 ψ1 )L2 (Σ) + (Dφ2 , Dψ2 )L2 (Σ) , 1/2 2 2 2 . k φ kV ≡ k D φ1 kL2 (Σ) + k Dφ2 kL2 (Σ)
By virtue of the Poincar´e inequality the norms k v kH m ≡
X
k Dα v kL2 (Σ)
1/2
X
k Dα v kL2 (Σ)
1/2
|α|≤m
k v kH0m ≡
|α|=m
and
are equivalent, and hence the space V endowed with the scalar product and norm as defined by (4.4) is also a Hilbert space. Note that the embedding V ֒→ H is continuous and dense, and moreover it is actually compact. Letting V ∗ denote the topological dual of V and identifying H with its dual we have V ֒→ H ֒→ V ∗ with injections being compact. Note that V ∗ = H −2 ×H −1 , where H −s , s > 0, denote the Sobolev spaces with negative exponents which are actually distributions. Now we introduce the bilinear form (4.5)
c(φ, ψ) ≡ a2 (D2 φ1 , D2 ψ1 )L2 (D) + b2 (Dφ2 , Dψ2 )L2 (D) .
It is clear that the map (4.6)
c : V × V −→ R
is a continuous bilinear form. Indeed, it is easy to verify that for d ≡ 2(a2 + b2 ) (4.7)
c(φ, ψ) ≤ d k φ kV k ψ kV
for all φ, ψ ∈ V.
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Thus there exists a linear operator A ∈ L(V, V ∗ ) such that (4.8)
c(φ, ψ) = hAφ, ψiV ∗ ,V for all φ, ψ ∈ V.
The operator A is precisely the realization of the formal differential operator 2 4 a D φ1 (4.9) , A(D)φ ≡ −b2 D2 φ2 with the boundary conditions (3.5), (3.6). Similarly, for d0 ≡ min{a2 , b2 } we have c(φ, φ) ≥ d0 k φ k2V
for all φ ∈ V.
Thus A is coercive. Define the operator F1 (t, ., φ1 (t, .), φ2 (t, .)) (4.10) . F (t, φ) ≡ F2 (t, ., φ1 (t, .), φ2 (t, .)) Now, defining the state variable for each t ≥ 0, as φ1 (t) z(t, .) φ(t) ≡ = , φ2 (t) y(t, .) which are functions defined on Σ, we can reformulate the system (4.1) as an abstract second-order differential equation on the Hilbert space H given by (d2 /dt2 )φ + Aφ = F (t, φ), t ≥ 0, ˙ φ(0) ≡ φ01 , φ(0) = φ02 ,
(4.11)
where the initial state {φ01 , φ02 } is given by equation (3.7). With this preparation we can now prove a result on the existence, uniqueness, and regularity of solutions of the Cauchy problem (4.11). For any Banach space X and 1 ≤ p ≤ ∞, let Lloc p (X) ≡ loc Lp (R0 , X) denote the class of X-valued functions on R0 ≡ [0, ∞) having X-norms locally pth-power integrable. Definition 4.1. The Cauchy problem (4.11) is said to have a weak solution if there exists a φ satisfying loc loc ∗ ¨ ˙ (i) φ ∈ Lloc 2 (V ) φ ∈ L2 (H), φ ∈ L2 (V ); ∞ (ii) for all η ∈ C0 (I) and any v ∈ V Z Z Z ˙ (4.12) v)η(t)dt ˙ + hφ(t), A∗ viη(t)dt = (F (t, φ(t)), v)H η(t)dt; − (φ(t), I
I
I
˙ = φ02 . (iii) φ(0) = φ01 , φ(0) First we prove an a priori bound. Let I ≡ [0, T ], T < ∞. We introduce the following assumptions for F. (F1) F : I × H −→ H is Lebesgue measurable in t on I and Borel measurable in the second variable on H. (F2) There exists a constant K ∈ R such that k F (t, ζ) k2H ≤ K 2 (1+ k ζ k2H ),
ζ ∈ H.
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Lemma 4.2. Suppose the hypotheses (F1) and (F2) hold and let φ01 ∈ V and φ02 ∈ H. Then any solution φ of (4.11), if one exists, satisfies the following conditions. (i) There exist constants M, ω > 0 such that (4.13)
˙ k2 + k φ(t) k2 ≤ M eωt , t ≥ 0. k φ(t) H V
(ii) φ ∈ L∞ (I, V ), φ˙ ∈ L∞ (I, H), and φ¨ ∈ L∞ (I, V ∗ ). Proof. Let φ be any solution of the Cauchy problem (4.11). Using the V ∗ − V pairing and scalar-multiplying the equation with φ˙ on either side, we obtain (4.14)
˙ v∗ ,V + c(φ, φ) ˙ = hF (t, φ), φi ˙ V ∗ ,V = (F (t, φ), φ) ˙ H, ¨ φi hφ,
where c is the bilinear form as given by (4.5). It is not difficult to show that 2 ˙ ˙ ¨ hφ(t), φ(t)i = (1/2)d/dt k φ(t) kH in the sense of distributions. Clearly, the bilinear form c is symmetric and hence ˙ 2c(φ(t), φ(t)) = (d/dt)c(φ(t), φ(t)). Using these facts in (4.14) we obtain 2 ˙ ˙ H, (1/2)d/dt k φ(t) kH +c(φ(t), φ(t)) = (F (t, φ(t)), φ)
t ≥ 0.
Integrating this, recalling the norm topology of V , and using the Schwarz inequality and the embedding constant, say δ > 0, for the injection V ֒→ H, we arrive at Z t 2 2 2 2 ˙ ˙ k φ(s) k2H ds k φ(t) kH +d0 k φ(t) kV ≤ k φ02 kH +d k φ01 kV + 0 Z t (4.15) (1 + δ 2 k φ(s) k2V )ds, t ≥ 0. + K2 0
Defining d˜0 = min{1, d0 }, d˜ = max{1, d}, δ˜2 = max{1, δ 2 }, ˜ 2 )/d˜0 , ˜ d˜0 ), d2 ≡ (1 + (K δ) d1 ≡ (d/ one can rewrite equation (4.15) as ¡ ¡ ˙ k2 + k φ(t) k2 ≤ d1 1+ k φ02 k2 + k φ01 k2 1+ k φ(t) H V H V Z t (4.16) ¡ ˙ k2H + k φ(s) k2V )ds, 1+ k φ(s) + d2 0
Defining ˙ k2 + k φ(t) k2 , η(t) ≡ 1+ k φ(t) H V it follows from (4.16) that η(t) ≤ d1 η(0) + d2
Z
0
t
η(s)ds,
t ≥ 0.
t ≥ 0.
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Hence, by the Gronwall inequality, we have η(t) ≤ d1 η(0)ed2 t ≡ M eωt ,
t ≥ 0,
which implies (4.13). Hence (i) follows. The first two inclusions of (ii) follow from (i). The last inclusion follows from (i) and the inequality ¨ k2 ∗ ≤ c0 + c1 k φ(t) k2 k φ(t) V V
for all t ≥ 0,
where the constants c0 = c0 (δ, K), c1 = c1 (d, δ, K) are dependent on the parameters as indicated. This completes the proof. The following result is believed to have independent interest in other applications. Lemma 4.3. Let X, Y be two arbitrary Banach spaces with the embedding X ֒→ Y being compact. Let I = [0, T ] be a finite interval, p ∈ (1, ∞), and B ⊂ L∞ (I, X) be a bounded set for which there exists a constant ˜b > 0, such that Z k f˙ kpY dt ≤ ˜b for all f ∈ B. I
Then B is a relatively compact subset of Lp (I, Y ). Proof. Clearly, the set B, considered as a subset of Lp (I, Y ), is bounded. Since B is a bounded subset of L∞ (I, X) for almost all t ∈ I, the set B(t) ≡ {f (t), f ∈ B} ⊂ X is bounded, and since the embedding X ֒→ Y is compact, B(t) is a relatively compact subset of Y. For each f ∈ B, f˙ ∈ Lp (I, Y ) and we have f (t + h) = f (t) +
Z
t+h
f˙(s)ds for almost all t, t + h ∈ I.
t
For h ∈ R define I(h) ≡ I ∩ [0, T − h]. Then we have °Z °p Z ° t+h ° R ° p ˙(s)ds° dt = k f (t + h) − f (t) k f ° ° dt Y I(h) ° I(h) ° t Y Z (4.17) kf˙(t)kp dt ≤ |h|p−1 T ≤ |h|p−1 T ˜b
I
for all f ∈ B. Therefore, lim
h→0
Z
I(h)
k f (t + h) − f (t) kpY dt = 0
uniformly with respect to f ∈ B. Thus we have shown that the set B satisfies the following conditions: (i) B is a bounded subset of Lp (I, Y ), (ii) the t sections B(t) of B are relatively compact subsets of Y , and R (iii) limh→0 I(h) k f (t + h) − f (t) kpY dt = 0 uniformly with respect to f ∈ B. Thus B is a relatively compact subset of Lp (I, Y ). This completes the proof. Now we are prepared to prove the existence of a solution of equation (4.11) in the sense of Definition 4.1.
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Theorem 4.4. Suppose the assumptions (F1) and (F2) hold and the map x ∋ H −→ F (t, x) ∈ H is continuous. Then, for each φ01 ∈ V, φ02 ∈ H, the Cauchy problem (4.11) has at least one weak solution. Proof. Since the injection V ֒→ H is compact there exists a complete system of functions {φi , i ∈ N } which is orthogonal in V and V ∗ and orthonormal in H. We use the Galerkin approach and define ξin ≡ (φ01 , φi ), ξ˙in ≡ (φ02 , φi ),
(4.18)
i = 1, 2, 3, . . . , n ∈ N, i = 1, 2, 3, . . . , n ∈ N,
expand φ01 and φ02 with respect to the system {φi , i ∈ N }, and note that X s φn01 ≡ ξin φi −→ φ01 in V, 1≤i≤n
(4.19)
φn02
≡
X
s ξ˙in φi −→ φ02 in H.
1≤i≤n
By projection to the n-dimensional subspace spanned by {φi , i = 1, 2, . . . , n} we obtain the system of ODEs (4.20)
hd2 /dt2 φn (t), φj i + hAφn , φj i = (F (t, φn (t)), φj ),
1 ≤ j ≤ n,
with initial conditions given by the left-hand members of (4.19). Since F is measurable in t and continuous in the second variable and satisfies the growth condition (F2), this system of equations has at least one solution φn satisfying φn ∈ AC(I, V ), φ˙ n ∈ AC(I, V ) ⊂ AC(I, H), with φn (0) = φn01 and φ˙ n (0) = φn02 . By virtue of the a priori bounds (see Lemma 4.2) the sequence {φn , φ˙ n } is contained in a bounded subset of L∞ (I, V ) × L∞ (I, H) and hence there exists a subsequence of this sequence (relabeled as above) and a φ ∈ L∞ (I, V ) with φ˙ ∈ L∞ (I, H) such that w∗
φn −→ φ in L∞ (I, V ), w∗ φ˙ n −→ φ˙ in L∞ (I, H).
(4.21)
Multiplying equation (4.20) by η ∈ C0∞ (I) and integrating by parts over I we have Z Z Z (4.22) − (φ˙ n (t), φj )η(t)dt ˙ + hφn (t), A∗ φj iη(t)dt = (F (t, φn (t)), φj )η(t)dt. I
I
I
Letting n → ∞ in the first two terms of (4.22) it follows from (4.21) that Z Z ˙ φj )η(t)dt, ˙ ˙ = − (φ(t), limn→∞ − (φ˙ n (t), φj )η(t)dt Z I Z I (4.23) limn→∞ hφn (t), A∗ φj iη(t)dt = hφ(t), A∗ φj iη(t)dt. I
I
It remains to verify that Z Z (4.24) (F (t, φn (t)), φj )η(t)dt = (F (t, φ(t)), φj )η(t)dt. lim n→∞
I
I
Here we use the embedding result (Lemma 4.3). Take X = V, Y = H, p = 2, and B = {φn , n = 1, 2, . . .}. By virtue of Lemma 4.2, B satisfies the conditions of Lemma
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N. U. AHMED AND H. HARBI
4.3 and hence the family B = {φn } is relatively compact in L2 (I, H). Thus, through a subsequence if necessary, φn converges to φ strongly in L2 (I, H) and hence there exists a subsequence that converges almost everywhere to φ. Using this fact and the assumptions on F, (4.24) follows from the Lebesgue-dominated convergence theorem. Since the set {φj } is complete, we conclude that Z Z Z ∗ ˙ (4.25) ˙ + hφ(t), A viη(t)dt = (F (t, φ(t)), v)H η(t)dt − (φ(t), v)η(t)dt I
I
I
for any v ∈ V. Thus (ii) of Definition 4.1 is satisfied. Now integrating the first term of (4.25) by parts once, we have Z ¨ hφ(t), vi + hAφ(t), vi − (F (t, φ(t)), v)H η(t)dt = 0 (4.26) I
for all η ∈
C0∞ .
Hence the equality ¨ + Aφ(t) = F (t, φ(t)), φ(t)
(4.27)
t ∈ I,
holds in the sense of V ∗ -valued distributions. Thus (i) of Definition 4.1 follows from Lemma 4.2. To prove that φ as constructed above also satisfies the given initial conditions, multiply (4.20) by η ∈ C 2 (I) satisfying η(T ) = 0, η(T ˙ ) = 0, and scalarmultiply (4.27) by φj η(t) and integrate by parts. Subtracting one from the other and letting n → ∞ one arrives at n n ˙ (4.28) ˙ = 0. lim (φ(0) − φ02 , φj )η(0) + (φ01 − φ(0), φj )η(0) n→∞
Since η ∈ C 2 (satisfying the end conditions) is arbitrary and {φj } is complete in the triple {V, H, V ∗ }, the conclusion follows from (4.19) and (4.28). This completes the proof of Theorem 4.4. An additional regularity property of the solution is given in the following corollary. Corollary 4.5. The solutions of the Cauchy problem (4.11) satisfy the following regularity properties: φ ∈ C(I, V ), φ˙ ∈ C(I, H). Proof. The proof is given by parabolic regularization using the result (ii) of Lemma 4.2, and Lemma 8.1 of [18, p. 275]. Theorem 4.6. If, in addition to the assumptions of Theorem 4.4, the operator F also satisfies the Lipschitz condition k F (t, ζ) − F (t, ξ) k2H ≤ K 2 (k ζ − ξ k2H ),
ζ, ξ ∈ H,
then the system (4.11) has a unique weak solution and the solution is continuously dependent on the initial data. In view of the above regularity result we can extend the result of Theorem 4.4 to cover the system model (4.29)
˙ (d2 /dt2 )φ + Aφ = F (t, φ, φ), t ≥ 0, ˙ φ(0) = φ02 . φ(0) ≡ φ01 ,
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865
φ Defining E ≡ V × H and ψ = φ˙ we can rewrite equation (4.29) as a first-order evolution equation on the Hilbert space E as given below: (4.30)
ψ˙ + Aψ = F˜ (t, ψ), ψ(0) = ψ0 ≡
φ(0) ˙ φ(0)
t ≥ 0, ,
where the operators A and F˜ are given by 0 0 − Id F˜ (t, ψ) ≡ , , A≡ ˙ F (t, φ, φ) A0 with Id being the identity operator in H. Theorem 4.7. Suppose F˜ satisfy the following hypotheses. (G1) k F˜ (t, ζ) kE ≤ K(1+ k ζ kE ). (G2) For every r > 0 there exists a constant Kr ≥ 0 such that k F˜ (t, ζ) − F˜ (t, ξ) kE ≤ Kr (k ζ − ξ kE )
for all ζ, ξ ∈ Br ,
where Br ≡ {η ∈ E :k η kE ≤ r}. Then for each ψ0 ∈ E the Cauchy problem (4.30) has a unique mild solution ψ ∈ C(I, E). Proof (outline). We follow the procedure as in [19, Theorem 5.3.3]. In view of Corollary 4.5, the operator −A generates a C0 -semigroup, S(t), t ≥ 0, on E and hence, by the variation of constants formula, one can rewrite the evolution equation (4.30) as an integral equation Z t (4.31) S(t − s)F˜ (s, ψ(s))ds, t ∈ I. ψ(t) = S(t)ψ0 + 0
Using the assumptions (G1) and (G2) and the Banach fixed point theorem one can easily establish the existence and uniqueness of a solution ψ ∈ C(Iτ1 , E) of the integral equation (4.31) over the interval, say Iτ1 ≡ [0, τ1 ], for sufficiently small but positive τ1 . The hypothesis (G1) implies finiteness of ψ(τ1 ). Starting with this as the initial state one solves the integral equation Z t S(t − s)F˜ (s, ψ(s))ds, t ∈ Iτ1 ,τ2 ≡ [τ1 , τ2 ], (4.32) ψ(t) = S(t − τ1 )ψ(τ1 ) + τ1
where again τ2 is chosen suitably so that the integral operator is a contraction. Since I is compact, by repeating this and piecing together these uniquely defined local solutions on [0, τ1 ], [τ1 , τ2 ], [τ2 , τ3 ], . . . , one obtains a unique solution for the original interval I. This is the mild solution of equation (4.30). This completes the proof. Remark. Before we conclude this section we wish to briefly indicate that the abstract model (4.30) can be conveniently used to construct meaningful stochastic evolution equations on the Hilbert space E to include random wind forces (load). This is given by (4.33)
dψ = −Aψdt + F˜ (t, ψ)dt + G(t, ψ)dW, ψ(0) = ψ0 .
t ≥ 0,
Here W ≡ {W (t), t ≥ 0} is an H-valued Wiener process and G is an operator-valued function G : I × E 7→ L(H, E). More specific representation of this operator will be given in a forthcoming paper, where we also show that the abstract system (4.33) also absorbs the more elaborate model given in [6].
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Fig. 2. Linear system (total energy, displacement, and velocity).
5. Numerical results and discussions. In this section we present some simulation results and discussions thereof. For numerical solution we use a FORTRAN program using the finite difference scheme on a SUN SPARC station 5. For numerical stability the ratio (△t/ △ x) was chosen as 2 × 10−3 . Figure 1 shows the typical geometry of a suspension bridge, which is self-explanatory. (i) Linear case. According to equation (3.9a) a linear suspension bridge in the absence of damping is conservative. This is illustrated by the total energy plot as shown in Fig. 2a. Figures 2b and 2c show the corresponding displacements and their rates at the midpoint of the span. Note that the frequency of oscillation of the structure is roughly twice that of the cable.
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Fig. 3. Nonlinear system (total energy, displacement, and velocity).
(ii) Nonlinear case. The nonlinear system is also subject to the same initial data as in the linear case. We have seen in section 3.1, equation (3.9b), that the nonlinear bridge is also conservative in the absence of damping. This is illustrated by the numerical simulation result shown in Fig. 3a. Figures 3b and 3c show the displacements and their rates again at the midpoint of the span. Note the distortion caused by nonlinearity. The initial energy level in this case is lower because of cutoff due to nonlinearity (see (3.10), (3.11)). This is because the Lebesgue measure of the set Σ0 ≡ {x ∈ (0, ℓ) : y(0, x) ≥ z(0, x)} is less than ℓ. (iii) Linear/nonlinear. To start with, here we assume that the system is linear and at time ts = 4 the stays (tie cables) below the road bed break down (due pos-
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Fig. 4. Linear/nonlinear system (total energy and displacement).
sibly to seismic or other activities). After the breakdown, the system is nonlinear. Again, Fig. 4a shows the conservation of energy with some wiggles due possibly to computational error caused by abrupt change. This constancy of energy level even after switching to a nonlinear system is due to the fact that at the precise moment of switching, the set Σs ≡ {x ∈ (0, ℓ) : y(ts , x) − z(ts , x) ≥ 0} has (full) Lebesgue measure λ(Σs ) = ℓ, equal to the length of the span. This means that at time ts −, just before the breakdown, the stays were active and the system behaved like a linear system, and hence, being conservative, the energy stayed at the prebreakdown level. As a consequence, since the nonlinear system is also conserva-
MATHEMATICAL ANALYSIS OF SUSPENSION BRIDGES
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Fig. 5. Linear/nonlinear system (total energy and displacement).
tive, the energy level remained constant thereafter. Just at the time of breakdown, the energy from the stay cables as transferred to the upper cables, resulting in a sudden increase of the amplitude of oscillation of these cables and a decrease of frequency, as shown in Fig. 4b. The same experiment was performed with different initial conditions (see Fig. 5). We observe a break in the energy level at ts . In this case λ(Σs ) < ℓ, meaning that the tie cables were loose on some sections of the span at the instant of breakdown. Consequently, in this case, comparatively, there was less energy transfer to the upper cables. Thus there is a drop in the total energy level, and after the breakdown, the nonlinear system continued its motion at a lower energy level. Even though, comparatively, the energy transfer was less in this case, the upper
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Fig. 6. Damped system (total energy and displacement).
cables continued to oscillate at reduced amplitude compared to the previous case, as expected. (iv) Aerodynamic damping. We carried out a numerical experiment with system (3.16) for two different cases: (1) f1 (ζ) = γ ζ, γ > 0, and f2 = 0. (2) f1 (ζ) = γ ζ, γ > 0, and f2 (ζ) = δ ζ, δ > 0. The results are shown in Figs. 6 and 7. The system is asymptotically stable with respect to the rest state, as expected (see Theorem 3.1). (v) Moving load. System (3.3) with (i) F (ζ) = k ζ (linear case) and (ii) F (ζ) = kΨ(ζ) (nonlinear case), f2 = 0, and f1 6= 0, were subjected to a moving load (for
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Fig. 7. Damped system (total energy and displacement).
example, a moving truck) represented by f1 (t, x) ≡ −m0 g Ce(t) (x), e(t) ≡ (vt, vt + L) ∩ (0, ℓ), where m0 is the mass of the vehicle per unit length, L is its length, v is its velocity, and Ce stands for the characteristic function of the set e. The velocity v was chosen so that the vehicle passes the bridge in four units of time. Figure 8a shows the variation of energy level with time, and Fig. 8b shows the corresponding response. Note that the system is left with some residual energy after the passage of the vehicle (Fig. 8a). This is due to the absence of friction or damping. For the nonlinear system, similar results are shown in Fig. 9.
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Fig. 8. Moving vehicle (total energy and displacement).
6. Concluding remarks. In this paper we have presented rigorous mathematical analysis of a simplified (PDE) model for suspension bridges as proposed by Lazer and McKenna. We have studied the dynamic behavior of the system under different situations and presented associated simulation results and physical interpretations. We have also presented an analysis of an abstract model and indicated its usefulness in determining the regularity of solutions and in constructing stochastic models. Currently, we are planning to use a more elaborate model such as the one suggested in [6] and its stochastic version and carry out a similar analysis, including simulation.
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Fig. 9. Moving vehicle (total energy and displacement).
REFERENCES [1] A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridge: Some new connections with nonlinear analysis, SIAM Rev., 32 (1990), pp. 537–578. [2] O. H. Amann, T. Von Karman, and G. B. Woodruff, The Failure of the Tacoma Narrows Bridge, Federal Works Agency, Washington, D.C., 1941. [3] F. Bleich, C. B. McCullough, R. Rosecrans, and G. S. Vincent, The Mathematical Theory of Suspension Bridges, Bureau of Public Roads, U.S. Dept. of Commerce, Washington, D.C., 1950. [4] E. G. Wiles, Report of Aerodynamic Studies on Proposed San Pedro-Terminal Island Suspension Bridge, California, Bridge Research Branch, Division of Physical Research, Bureau of Public Roads, U.S. Dept. of Commerce, Washington, D.C., 1960.
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[5] A. Selberg, Oscillation and aerodynamic stability of suspension bridges, Acta Polytech. Scand., 13 (1961), pp. 308–377. [6] A. M. Abdel-Ghaffer, Suspension bridge vibration: Continuum formulation, J. Engrg. Mech., 108 (1982), pp. 1215–1232. [7] T. Kawada and A. Hirai, Additional Mass Method-A New Approach to Suspension Bridge Rehabilitation, Official Proceedings, 2nd Annual International Bridge Conference, Engineers Society of Western Pennsylvania, Pittsburgh, PA, 1985. [8] P. J. McKenna and W. Walter, Nonlinear oscillation in a suspension bridge, Arch. Rational Mech. Anal., 98 (1987), pp. 167–177. [9] A. C. Lazer and P. J. McKenna, Large scale oscillation behavior in loaded asymmetric systems, Ann. Inst. H. Poincar´ e, Anal. Non Lin´ eaire, 4 (1987), pp. 244–274. [10] P. J. McKenna and W. Walter, Traveling waves in a suspension bridge, SIAM J. Appl. Math., 50 (1990), pp. 703–715. [11] J. Glover, A. C. Lazer, and P. J. McKenna, Existence and stability of large scale nonlinear oscillations in suspension bridges, J. Appl. Math. Phys. (ZAMP), 40 (1989), pp. 172–200. [12] Y. S. Choi, K. C. Jen, and P. J. McKenna, The structure of the solution set for periodic oscillation in a suspension bridge model, IMA J. Appl. Math., 47 (1991), pp. 283–306. [13] D. Jacover and P. J. McKenna, Nonlinear torsional flexings in a periodically forced suspended beam, J. Comp. Appl. Math., 52 (1994), pp. 241–265. [14] S. K. Biswas and N. U. Ahmed, Optimal control of large space structures governed by a coupled system of ordinary and partial differential equations, J. Math. Control, Signals, and Systems, 2 (1989), pp. 1–18. [15] N. U. Ahmed and S. S. Lim, Modeling and control of flexible space stations (slew maneuvers), in Proc. 3rd Annual Conference on Aerospace Computational Control, D. E. Bernard and G. K. Mann, eds., NASA Pub. 89-45, Vol. 2, NASAJPL, Pasadena, CA, pp. 900–914. [16] N. U. Ahmed and S. K. Biswas, Mathematical modeling and control of large space structures with multiple appendages, J. Math. Comput. Modeling, 10 (1988), pp. 891–900. [17] P. Li and N. U. Ahmed, On exponential stability of infinite dimensional systems with bounded or unbounded perturbations, Appl. Anal., 30 (1988), pp. 175–187. [18] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York, Heidelberg, Berlin, 1972. [19] N. U. Ahmed, Semigroup Theory with Applications to Systems and Control, Pitman Res. Notes Math. Ser. 246, Longman Scientific and Technical, Harlow, UK, 1991.
