Periodic oscillation in suspension bridge model with a periodic damping term
© Wang and Zu; licensee Springer. 2014
Received: 25 September 2014
Accepted: 7 October 2014
Published: 16 October 2014
We study periodic solutions of the suspension bridge model proposed by Lazer and McKenna with a periodic damping term. Under the Dolph-type condition and a small periodic damping term condition, the existence and the uniqueness of a periodic solution have been proved by a constructive method. Two numerical examples are presented to illustrate the effect of the periodic damping term.
MSC: 34B15, 34C15, 34C25.
, , are constant, and . System (1)-(2) describes the transverse vibrations of a beam hinged at both ends with length L and external force . The term takes into account the fact that the cables’ restoring force exists only in the situation of stretching. Here represents the damping term.
in which and .
In the past years, the jumping nonlinearity has been discussed by many authors –. However, to our knowledge, there is no result about periodic damping term. In the early 1990s, Li  obtained an ingenious method to discuss the existence and uniqueness of nonlinear two-point boundary value problems with variable coefficient. Recently, the second author of this paper extended this method to the periodic situation . In this paper we refine this method to solve problem (4) and take some numerical simulations to illustrate the effect of periodic damping term.
The rest of this paper is organized as follows. In Section 2, we briefly state the main results. In Section 3, we study the properties of the homogeneous equation by a constructive method. In Section 4, we prove our main results by Leray-Schauder degree theory. In Section 5, we present some numerical experiments. In Section 6, we give the conclusion.
2 Main results
We denote by N a positive integer and . To study the existence of periodic solutions of (4), we need the following assumptions:
Let (H1) and (H2) hold. Then problem (4) has a unique 2π-periodic solution.
To study the existence of periodic solutions of (5), we make the following assumptions:
Let (H3) and (H4) hold. Then problem (5) has a unique 2π-periodic solution.
3 Homogeneous equation
The following lemmas will be used in this section.
and the constantis optimal.
We will prove the following proposition by similar methods to .
We assume that (7) has a nonzero 2π-periodic solution . A contradiction will be proved in six steps.
which leads to a contradiction.
Without loss of generality, we assume , .
Step 3. We will prove that has a zero point in . Assume, on the contrary, for .
By (12), we know that .
which is a contradiction to on . Therefore has at least one zero in with .
Step 4. We will prove that has at least zeros on . Let be the first zero point in such that , . We claim that there must exist a zero point in . Otherwise, we consider . With a similar argument to Step 3, we have a such that there must be a zero in and . Step by step, we find that has at least zeros on .
we obtain , which contradicts . Therefore has at least zeros on .
which implies for . Due to the uniqueness of the solution for the initial value problem, we get for , a contradiction. □
4 Non-homogeneous equation
In this section, we will give the complete proof of Theorem 2.
Equation (18) satisfies (H3) and (H4). Otherwise, −v as a solution satisfies (H3) and (H4). By Proposition 5, we have . □
which satisfy (H3) and (H4). By Proposition 5, we have for , which contradicts . Thus, the possible periodic solution is bounded.
So we conclude that has at least one fixed point in Ω, that is, (5) has a unique solution. □
5 Numerical experiment
5.1 Example 1
By Theorem 2, there is a unique 2π-periodic solution.
5.2 Example 2
By Theorem 1, there is a unique 2π-periodic solution.
Periodic solutions of the suspension bridge model with a periodic damping term have been studied. After transforming this system into an equivalent ordinary differential equation, we get the existence and the uniqueness of a periodic solution by the Dolph-type condition and a small periodic damping term condition. Our constructive method is very adaptable to this kind of non-smooth problem. Two numerical examples have been presented to simulate our main results. By the numerical experiment, we know that the effect of the small periodic damping term is limited. Furthermore, we compare the approximate solution of our system to the suspension bridge model without the cables’ restoring force, the latter one is a particular case of .
The authors express deep gratitude to the anonymous referees for their useful suggestions and comments, especially for pointing out the flaws in Step 3 of Section 3. The authors also thank Professor Yong Li for his useful discussion.
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