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# Periodic oscillation in suspension bridge model with a periodic damping term

- Lin Wang
^{1}Email author and - Jian Zu
^{2}

**2014**:231

https://doi.org/10.1186/s13661-014-0231-2

© Wang and Zu; licensee Springer. 2014

**Received:**25 September 2014**Accepted:**7 October 2014**Published:**16 October 2014

## Abstract

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.

## Keywords

- periodic damping term
- suspension bridge model
- jumping nonlinearity
- Leray-Schauder degree

## 1 Introduction

$c>0$, $k\ge 0$, $d\ge 0$ are constant, ${U}^{+}=max\{U,0\}$ and ${U}^{-}=max\{-U,0\}$. System (1)-(2) describes the transverse vibrations of a beam hinged at both ends with length *L* and external force $h(t,x)$. The term $d{U}^{+}$ takes into account the fact that the cables’ restoring force exists only in the situation of stretching. Here $k{U}_{t}$ represents the damping term.

*π*-periodic damping term $p(t)$

*π*-periodic external force as same as the assumption in [2]. Looking for a standing-wave solution of (3) and (2), we have

in which $a=c{(\pi /L)}^{4}$ and $b=d+c{(\pi /L)}^{4}$.

In the past years, the jumping nonlinearity has been discussed by many authors [1]–[9]. However, to our knowledge, there is no result about periodic damping term. In the early 1990s, Li [10] 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 [11]. 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 $\gamma ={sup}_{\mathbb{R}}|p(t)|$. To study the existence of periodic solutions of (4), we need the following assumptions:

_{1}): Dolph-type condition:

_{2}): Small periodic damping term condition:

### Theorem 1

*Let* (H_{1}) *and* (H_{2}) *hold*. *Then problem* (4) *has a unique* 2*π*-*periodic solution*.

*π*-periodic functions satisfying

To study the existence of periodic solutions of (5), we make the following assumptions:

_{3}): Dolph-type condition:

_{4}): Small periodic damping term condition:

### Theorem 2

*Let* (H_{3}) *and* (H_{4}) *hold*. *Then problem* (5) *has a unique* 2*π*-*periodic solution*.

### Remark 1

Problem (4) with (H_{1}) and (H_{2}) is a particular case of problem (5) with (H_{3}) and (H_{4}). So we shall only give the proof of Theorem 2.

## 3 Homogeneous equation

The following lemmas will be used in this section.

### Lemma 3

(see [12])

*Let*$x\in {C}^{1}([0,h],\mathbb{R})$, $h>0$,

*with*

*Then*

*and the constant*$\frac{h}{4}$*is optimal*.

### Lemma 4

(see [12])

*Let*$x\in {C}^{1}([a,b],\mathbb{R})$$a,b\in \mathbb{R}$, $a<b$,

*with the boundary value conditions*$x(a)=x(b)=0$.

*Then*

We will prove the following proposition by similar methods to [11].

### Proposition 5

*Suppose that*$p(t)$, $b(t)$, $a(t)$*are*${L}^{2}$-*integrable* 2*π*-*periodic functions satisfying* (6), (H_{3}) *and* (H_{4}), *then* (7) *has only the trivial* 2*π*-*periodic solution*$u(t)\equiv 0$.

### Proof

We assume that (7) has a nonzero 2*π*-periodic solution $u(t)$. A contradiction will be proved in six steps.

which leads to a contradiction.

Without loss of generality, we assume $u(0)=u(2\pi )=0$, ${u}^{\prime}(0)={u}^{\prime}(2\pi )=A>0$.

_{3}), there is a ${t}_{0}\in (0,\pi )$ such that

*t*is decreasing in $[\frac{\pi}{2},\pi )$, we have

_{4}), we have

*θ*is the same as the previous one, and

*t*is decreasing on $[\frac{\pi}{2},\pi )$, we have $0<{t}_{1}<\frac{\pi}{N}$, and

Step 3. We will prove that $u(t)$ has a zero point in $(0,{t}_{1}]$. Assume, on the contrary, $u(t)>0$ for $t\in (0,{t}_{1}]$.

By (12), we know that ${u}^{\prime}({t}_{0})\le 0$.

*i.e.*,

which is a contradiction to $u(t)>0$ on $(0,{t}_{1}]$. Therefore $u(t)$ has at least one zero in $(0,{t}_{1}]$ with ${t}_{1}<\frac{\pi}{N}$.

Step 4. We will prove that $u(t)$ has at least $2N+2$ zeros on $[0,2\pi ]$. Let $u({t}^{1})$ be the first zero point in $(0,{t}_{1}]$ such that $u({t}^{1})=0$, ${u}^{\prime}({t}^{1})=B<0$. We claim that there must exist a zero point in $({t}^{1},2\pi ]$. Otherwise, we consider ${u}^{\u2033}+p(t){u}^{\prime}-a(t){u}^{-}=0$. With a similar argument to Step 3, we have a ${t}_{2}$ such that there must be a zero in $({t}^{1},{t}_{2}]$ and ${t}_{2}-{t}^{1}<\frac{\pi}{N}$. Step by step, we find that $u(t)$ has at least $2N+2$ zeros on $[0,2\pi ]$.

we obtain ${u}^{\prime}({t}^{2N+1})<0$, which contradicts ${u}^{\prime}({t}^{2N+1})={u}^{\prime}({t}^{0})>0$. Therefore $u(t)$ has at least $2N+3$ zeros on $[0,2\pi ]$.

*k*zeros in $({\xi}_{1},{\xi}_{2})$ denoted by ${\tau}_{k}$, $k\in \mathbb{N}$. By Lemma 3,

which implies ${u}^{\prime}(t)=0$ for $t\in [{\xi}_{1},{\xi}_{2}]$. Due to the uniqueness of the solution for the initial value problem, we get $u(t)\equiv 0$ for $t\in [0,2\pi ]$, a contradiction. □

## 4 Non-homogeneous equation

In this section, we will give the complete proof of Theorem 2.

### 4.1 Uniqueness

#### Proof

*π*-periodic solutions of (5). Denote $v={u}_{1}-{u}_{2}$, then

*v*is a solution of the following problem:

Equation (18) satisfies (H_{3}) and (H_{4}). Otherwise, −*v* as a solution satisfies (H_{3}) and (H_{4}). By Proposition 5, we have $v\equiv 0$. □

### 4.2 Boundedness

*i.e.*, $\parallel u\parallel =|u|+|{u}^{\prime}|$. We assert there exists $B>0$ such that every possible periodic solution $u(t)$ of (19) satisfies $\parallel u\parallel \le B$. If not, there exists ${\lambda}_{k}\to {\lambda}_{0}$ and the solution ${u}_{k}(t)$ with $\parallel {u}_{k}\parallel \to \mathrm{\infty}$ ($k\to \mathrm{\infty}$). Let ${y}_{k}=\frac{{u}_{k}}{\parallel {u}_{k}\parallel}$, we have ${y}_{k}^{+}=\frac{{u}_{k}^{+}}{\parallel {u}_{k}\parallel}$ and ${y}_{k}^{-}=\frac{{u}_{k}^{-}}{\parallel {u}_{k}\parallel}$. Obviously, $\parallel {y}_{k}\parallel =1$ ($k=1,2,\dots $). It satisfies the following problem:

which satisfy (H_{3}) and (H_{4}). By Proposition 5, we have $w(t)\equiv 0$ for $t\in [0,2\pi ]$, which contradicts $\parallel w\parallel =1$. Thus, the possible periodic solution is bounded.

### 4.3 Existence

#### Proof

_{3}). Equation (19) can be transformed into the integral equation

*π*-periodic solution of (22), then

So we conclude that ${P}_{1}$ 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 2*π*-periodic solution.

^{−10}. The red line in Figure 1 is the approximate solution of (27) without periodic damping term. Our simulation illustrates that the effect of the small periodic damping term is limited.

### 5.2 Example 2

By Theorem 1, there is a unique 2*π*-periodic solution.

^{−10}.

## 6 Conclusions

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 [11].

## Declarations

### Acknowledgements

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.

## Authors’ Affiliations

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## Copyright

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