Vibration Suppression of an Axially Moving String with Transverse Wind Loadings by a Nonlinear Energy Sink

Abstract

Nonlinear targeted energy transfer (TET) is applied to suppress the excessive vibration of an axially moving string with transverse wind loads. The coupling dynamic equations used are modeled by a nonlinear energy sink (NES) attached to the string to absorb vibrational energy. By a two-term Galerkin procedure, the equations are discretized, and the effects of vibration suppression by numerical methods are demonstrated. Results show that the NES can effectively suppress the vibration of the axially moving string with transverse wind loadings, thereby protecting the string from excessive movement.

1. Introduction

The structures of axial speed-dependent behaviors have been analyzed in numerous studies [1–6]. Various engineering systems can be simplified into an axially moving string model such as aerial gondola cableways and conveyor belts. Specifically, the lateral vibrations of an axially moving string have been extensively examined. Both the aerial gondola cableway and the conveyor belt operate in rural environments and suffer from transverse wind loads. Excessive wind loads can destroy the string running security, causing catastrophic failure. The dynamic response of the axially moving string has been widely investigated [7]; however, studies on the vibration suppression of an axially moving string with transverse wind loads are rarely reported.

Traditional linear vibration absorbers have been used in various engineering fields. However, vibration suppression strongly responds only at the natural frequency of the vibration absorber. The nonlinear energy sink (NES) functions as an effective vibration absorber for a nonlinear system. The NES has recently been reported to engage in resonance over a very broad frequency range, has a small additional mass, and can perform targeted energy transfer (TET). Nonlinear TET has been used in numerous engineering structures for vibration suppression, such as drill-string [8], beam [9], rod [10], and plate [11]. Specifically, Lee et al. [12] attached the NES to the fixed wing of the plane for vibration suppression of the limit cycle, which resulted in increased damping. Nucera et al. [13] applied the NES to a multistory frame structure to absorb the vibration. Savadkoohi et al. [14] examined the four-story frame structure by using two parallel NES.

The present study focuses on the vibration suppression of an axially moving string with certain and steady transverse wind loads by using NES. The coupling dynamic differential equations of an axially moving string and the NES with transverse wind loads are established. In addition, the governing equations are approximately discretized by the two-term Galerkin procedure. The effects of vibration suppression are finally demonstrated by numerical simulation.

2. Equation of Motion

Figure 1 shows the system under study, consisting of a simply supported axially moving string with an essentially nonlinear damped attachment. The attachment called NES is expected to irreversibly absorb the vibrational energy of the string.

Figure 1 

An axially moving string with a nonlinear energy sink (NES).

The length of the axially moving string is represented by 𝐿. Let 𝑈(𝑋,𝑇) and 𝑈(𝑋,𝑇) be the displacements of the string and the NES relative to the horizontal 𝑋-axis, respectively. The governing equation of motion can be derived by Newton’s second law: 𝜕𝜌𝐴²𝑈(𝑋,𝑇)𝜕𝑇²𝜕+2𝑉²𝑈(𝑋,𝑇)𝜕𝑋𝜕𝑇+𝑉²𝜕²𝑈(𝑋,𝑇)𝜕𝑋²𝜕−𝑃²𝑈(𝑋,𝑇)𝜕𝑋²𝑉()+𝜂𝜕𝑈𝑋,𝑇+()𝜕𝑋𝜕𝑈𝑋,𝑇(𝑡)𝛿()(),𝜕𝑇=𝑅𝑋−𝑑+𝐹𝑋,𝑇 (1) where 𝜂 is the viscosity coefficient of the string material, 𝜌 is the linear density, 𝐴 is the cross-sectional area, 𝑃 is the initial tension, and 𝑉 is the axial speed. 𝑅(𝑡) is the interaction force between the string and the NES.

In (1), 𝐹(𝑋,𝑇) is expressed as follows [15]: 𝐹(𝑋,𝑇)=𝐹₁𝜕𝑈(𝑋,𝑇)𝜕𝑇+𝐹₃𝜕𝑈(𝑋,𝑇)𝜕𝑇³,𝐹₁=0.5𝑎𝜌₀ℎ𝑣₀,𝐹₃=0.5𝑏𝜌₀ℎ𝑣₀⁻¹. (2)

The NES equation of motion is given by 𝑚_NES𝜕²𝑈𝜕𝑇²+𝑅(𝑡)=0. (3)

The interaction force 𝑅(𝑡) can be written as 𝑅(𝑡)=𝐾𝑈(𝑇)−𝑍(𝑇)³𝜕+𝐷𝑈(𝑇)−(𝑇)𝜕𝑇𝜕𝑍.𝜕𝑇 (4)

The attachment point displacement and velocity are expressed as follows [16]: 𝑍(𝑇)()(𝑇)=𝑈𝑑,𝑇𝜕𝑍=()𝜕𝑇𝜕𝑈𝑑,𝑇()𝜕𝑇+𝑉𝜕𝑈𝑑,𝑇,𝜕𝑋 (5) where 𝐾 is the nonlinear (cubic) spring stiffness, 𝐷 is the NES dissipation, and 𝑑 is the adding position to the NES on the string.

The nondimensional quantities are given as follows: 𝑋𝑥=𝐿𝑈,𝑢=𝐿𝑉,𝑡=𝜔𝑇,𝑣=𝑣𝜔𝐿,𝜆=⁰𝜔𝐿,𝛼₀=𝜌₀𝐿ℎ,𝜌𝐴𝑘=𝐾𝐿⁴𝑃,𝜎=𝐷𝐿𝑚𝜌𝐴𝑃,𝜀=^NES,𝜂𝜌𝐴𝐿₀=𝜂𝜌𝐴𝜔,𝜔²=𝑃𝜌𝐴𝐿². (6)

Substituting (6) into (1) to (5) yields the following dimensionless form: 𝜕²𝑢(𝑥,𝑡)𝜕𝑡²𝜕+2𝑣²𝑢(𝑥,𝑡)+𝑣𝜕𝑥𝜕𝑡²𝜕−1²𝑢(𝑥,𝑡)𝜕𝑥²+𝜂₀𝑣𝜕𝑢+𝜕𝑥𝜕𝑢=𝑘[𝑢()−𝜕𝑡𝑑,𝑡𝑢(𝑡)]³𝜕+𝜎𝑢(𝑡)−𝜕𝑡𝜕𝑢()𝜕𝑡𝑑,𝑡−𝑣𝜕𝑢()()𝜕𝑥𝑑,𝑡×𝛿𝑥−𝑑+0.5𝛼₀()𝑎𝜆𝜕𝑢𝑥,𝑡+𝑏𝜕𝑡𝜆𝜕𝑢(𝑥,𝑡)𝜕𝑡³,𝜀𝜕²𝑢𝜕𝑡²+𝑘[𝑢(𝑡)−𝑢(𝑑,𝑡)]³𝜕+𝜎𝑢(𝑡)−()𝜕𝑡𝜕𝑢𝑑,𝑡()𝜕𝑡−𝛾𝜕𝑢𝑑,𝑡𝜕𝑥=0. (7)

3. The Galerkin Method

By using a tractable finite-dimensional dynamical system, governing equations (7) can be approximated by the standard Galerkin-type projections as follows: 𝑢(𝑥,𝑡)=^𝑁_𝑟=1𝜙_𝑟(𝑥)𝑞_𝑟(𝑡), (8) where 𝜙_𝑟(𝑥) denotes the eigenfunctions for the free undamped vibrations of a string satisfying the same boundary conditions and 𝑞_𝑟(𝑡) represents the generalized coordinates of the discretized system.

Substituting (8) into (7) yields ^𝑁_𝑟=1𝜙_𝑟(𝑥)𝑞̈_𝑟(𝑡)+2𝛾𝜙^′_𝑟(𝑥)𝑞̇_𝑟(𝑡)+𝛾²−1𝜙^′′_𝑟(𝑥)𝑞_𝑟(𝑡)+𝜂₀𝑣𝜙^′_𝑟(𝑥)𝑞_𝑟(𝑡)+𝜂₀𝜙_𝑟(𝑥)𝑞̇_𝑟(𝑡=𝑘_𝑁𝑟=1)𝜙_𝑟(𝑑)𝑞_𝑟(𝑡)−𝑢(𝑡)³+𝜎^𝑁_𝑟=1𝜙_𝑟(𝑑)𝑞̇_𝑟(𝑡)+𝛾^𝑁_𝑟=1𝜙^′_𝑟(𝑑)𝑞_𝑟(𝑡)−𝑢̇(𝑡)×𝛿(𝑥−𝑑)+0.5𝛼₀𝑎𝜆^𝑁_𝑟=1𝜙_𝑟(𝑥)𝑞̇_𝑟(𝑡)+0.5𝛼₀𝑏𝜆^𝑁_𝑟=1𝜙_𝑟(𝑥)𝑞̇_𝑟(𝑡)³,𝜀𝑢̈(𝑡)+𝑘𝑢(𝑡)−^𝑁_𝑟=1𝜙_𝑟(𝑑)𝑞_𝑟(𝑡)³+𝜎𝑢̇(𝑡)−^𝑁_𝑟=1𝜙_𝑟(𝑑)𝑞̇_𝑟(𝑡)−𝛾^𝑁_𝑟=1𝜙^′_𝑟(𝑑)𝑞_𝑟(𝑡)=0. (9)(10) In (10), for example, 𝜙_𝑟=√2sin𝜆_𝑟𝑥 and 𝜆_𝑟=𝑟𝜋 if a string is supported by pinned ends. The √2 factor ensures orthonormality. Multiplying both (9) and (10) by 𝜙_𝑠(𝑥) and integrating over the domain [0,1] yields the following equations: 𝛿_𝑠𝑟𝑞̈_𝑟(𝑡)+2𝑣𝑏_𝑠𝑟𝑞̇_𝑟(𝑡)+𝑐_𝑠𝑟𝑣²−1𝑞_𝑟(𝑡)+𝜂₀𝑣𝑏_𝑠𝑟𝑞_𝑟(𝑡)+𝜂₀𝛿_𝑠𝑟𝑞̇_𝑟(𝑡)=𝑘^𝑁_𝑟=1𝜙_𝑟(𝑑)𝑞_𝑟(𝑡)−𝑢(𝑡)³+𝜎^𝑁_𝑟=1𝜙_𝑟(𝑑)𝑞̇_𝑟(𝑡)+𝛾^𝑁_𝑟=1𝜙^′_𝑟(𝑑)𝑞_𝑟(𝑡)−𝑢̇(𝑡)×𝜙_𝑟(𝑑)+0.5𝛼₀𝑎𝜆𝛿_𝑠𝑟𝑞̇_𝑟(𝑡)+0.5𝛼₀𝑏𝜆𝑒_𝑠1𝑞̇³₁(𝑡)+0.5𝛼₀𝑏𝜆𝑒_𝑠2𝑞̇³₂(𝑡)+1.5𝛼₀𝑏𝜆𝑓_𝑠1𝑞̇²₁(𝑡)𝑞̇₂(𝑡)+1.5𝛼₀𝑏𝜆𝑓_𝑠2𝑞̇₁(𝑡)𝑞̇²₂(𝑡),𝜀𝑢̈(𝑡)+𝑘𝑢(𝑡)−^𝑁_𝑟=1𝜙_𝑟(𝑑)𝑞_𝑟(𝑡)³+𝜎𝑢̇(𝑡)−^𝑁_𝑟=1𝜙_𝑟(𝑑)𝑞̇_𝑟(𝑡)−𝛾^𝑁_𝑟=1𝜙^′_𝑟(𝑑)𝑞_𝑟(𝑡)=0, (11) where 𝛿_𝑠𝑟=¹₀𝜙_𝑠(𝑥)𝜙_𝑟(𝑥)𝑑𝑥,𝑏_𝑠𝑟=¹₀𝜙_𝑠(𝑥)𝜙^′_𝑟(𝑥)𝑐𝑑𝑥,_𝑠𝑟=¹₀𝜙_𝑠(𝑥)𝜙^′′_𝑟(𝑥)𝑑𝑥,𝑒_𝑠𝑟=¹₀𝜙_𝑠(𝑥)𝜙³_𝑟(𝑥)𝑓𝑑𝑥,_𝑠2=¹₀𝜙_𝑠(𝑥)𝜙₁(𝑥)𝜙²₂(𝑥)𝑓𝑑𝑥,_𝑠1=¹₀𝜙_𝑠(𝑥)𝜙²₁(𝑥)𝜙₂(𝑥)𝑑𝑥, (12) where 𝛿_𝑠𝑟 is the Kronecker delta and 𝜆_𝑟 is the 𝑟th eigenvalue for the free undamped vibrations of a string with the same boundary conditions.

Equation (10) can be written as follows:𝑀𝑞̈_𝑟(𝑡)+𝐶𝑞̇_𝑟(𝑡)+𝐾𝑞_𝑟(𝑡)×𝑘^𝑁_𝑟=1𝜙_𝑟(𝑑)𝑞_𝑟(𝑡)−𝑢(𝑡)³+𝜎^𝑁_𝑟=1𝜙_𝑟(𝑑)𝑞̇_𝑟(𝑡)+𝛾^𝑁_𝑟=1𝜙^′_𝑟(𝑑)𝑞_𝑟(𝑡)−𝑢̇(𝑡)×𝜙_𝑟(𝑑)𝜀̈=0,𝑢(𝑡)+𝑘𝑢(𝑡)−^𝑁_𝑟=1𝜙_𝑟(𝑑)𝑞_𝑟(𝑡)³+𝜎𝑢̇(𝑡)−^𝑁_𝑟=1𝜙_𝑟(𝑑)𝑞̇_𝑟(𝑡)−𝛾^𝑁_𝑟=1𝜙^′_𝑟(𝑑)𝑞_𝑟(𝑡)=0, (13a)(13b) where𝑀=𝛿_𝑠𝑟,𝐶₁=2𝑣𝑏_𝑠𝑟+𝜂₀𝛿_𝑠𝑟+0.5𝛼₀𝑎𝜆𝛿_𝑠𝑟,𝐶₂=0.5𝛼₀𝑏𝜆𝑒_𝑠1,𝐶₃=0.5𝛼₀𝑏𝜆𝑒_𝑠2,𝐶₄=0.5𝛼₀𝑏𝜆𝑓_𝑠1,𝐶₅=0.5𝛼₀𝑏𝜆𝑓_𝑠2,𝐾=𝑣²−1𝐶_𝑠𝑟+𝜂₀𝑣𝑏_𝑠𝑟=𝜔²_𝑟, (14) where 𝑀, 𝐶, and 𝐾 are the mass, damping, and stiffness matrices, respectively. In addition, 𝜔_𝑟 is the rth natural frequency of the axially moving string.

4. Effectiveness of the NES

The effectiveness of the NES coupled to an axially moving string at varying axial speeds is demonstrated. Equations (13a) and (13b) are a high-dimensional nonlinear dynamical system; thus, numerical methods should be used to truncate the expansion (13a), (13b) to a finite number of modes. In the Galerkin procedure for gyroscopic systems, at least two modes of the displacement amplitude can yield a good approximation [17–19]. Therefore, by choosing 𝑁=2, the equations can achieve good numerical convergence. To initiate oscillations, an initial distributed velocity is imposed as follows: 𝑞̇₁(0)𝑞=𝑋,_𝑟(0)=𝑞̇₂(0)=⋯=𝑞̇_𝑟=𝑢(0)=𝑢̇(0)=0. (15) Figure 2 shows the transient responses of the axially moving string and the NES for 𝜎 = 0.65, 𝑘=80000, 𝑑=0.4, 𝛼 = 0.001, 𝜀 = 0.06, 𝑋=0.15, 𝜂₀=0.005, 𝑎=0.2, 𝑏=−0.2, 𝑝₀=1.293, and 𝐴=2.7745×10⁻⁴. As shown in the figure, the effectiveness of NES for the axial speed 𝑣 varying from 0 to 0.4 is examined. Solid and dashed lines denote the responses of the beam and the NES, respectively. Figures 2(a)–2(c) show that the amplitude of the NES is markedly higher than that of the string, indicating the occurrence of energy transfer from the string to the NES. Both the string and the NES perform decaying vibration attributed to damping dissipation. During this process, the vibrational energy is irreversibly transferred and eventually damped by the NES. The NES can effectively absorb the vibrational energy and prevent the string from excessive vibrations at varying axial speeds. Energy absorption is achieved over a wide range of axial speeds.

(a) 𝑣=0

(b) 𝑣=0.2

(c) 𝑣=0.4

Figure 2 

Response of the axially moving string and NES for different axial speeds (solid line: axially moving string’s response, 𝑢; dashed line: NES’s response, 𝑢).

To further demonstrate the effectiveness of the NES, the responses of the string with the NES and without the NES are presented in Figures 3(a)–3(c) for 𝜎=0.65, 𝑘=80000, 𝑑=0.4, 𝛼=0.001, 𝜀=0.06, 𝑋=0.15, 𝜂₀=0.005, 𝑎=0.2, 𝑏=−0.2, 𝑝₀=1.293, and 𝐴=2.7745×10⁻⁴. The string attached with the NES exhibits a drastic reduction in transient response, as indicated by solid lines. By contrast, the string without the NES slowly decays as time increases, as indicated by dashed lines. Therefore, the NES can robustly absorb vibrational energy over a broad range of axial speeds.

(a) 𝑣=0

(b) 𝑣=0.2

(c) 𝑣=0.4

Figure 3 

Comparison of the transient response of axially moving string with and without NES under varying speed (solid line: string coupled with NES; dotted line: string without NES).

In Figure 4, the effect of adding position to the NES is examined. Figures 4(a)–4(c) for 𝜎=0.65, 𝑘=80000, 𝑣=0.2, 𝛼=0.001, 𝜀=0.06, 𝑋=0.15, 𝜂₀ = 0.005, 𝑎=0.2, 𝑏=−0.2, 𝑝₀ = 1.293, and 𝐴=2.7745×10⁻⁴ illustrate the different effects of varying the position of the NES on absorbed vibration. When the position 𝑑 is 0.4, the maximum effect of vibration suppression is achieved.

(a) 𝑑=0.35

(b) 𝑑=0.4

(c) 𝑑=0.45

Figure 4 

Comparison of the transient response of axially moving string with and without NES under varying attaching location (solid line: string coupled with NES; dotted line: string without NES).

5. Conclusions

The vibration of an axially moving string with transverse wind loads is effectively suppressed using the NES. The results of the simulation experiments indicate that at various flow speeds, the NES can irreversibly transfer and dissipate vibrational energy from the axially moving string. By considering the adding internal degrees of freedom to the NES, vibration suppression is most clearly demonstrated at 𝑑=0.4. Therefore, the vibration of an axially moving string can be suppressed based on the additional internal degrees of freedom to NES and the speed of the axially moving string.

Acknowledgment

The authors acknowledge the funding support of the Natural Science Foundation of Liaoning Province (201102170).