Abstract
The purpose of this paper is to characterize the modal parameters and transverse vibrations of a monochain ropeway system for hilly orchards. The moving chain is modeled with uniform distribution and concentrated inertial loads. In order to study the dynamical behavior of the moving chain, Hamilton’s principle is applied to obtain the homogenous differential equation of transverse vibration. With the boundary conditions subjected to the polygonal action caused by chain-support engagement, the coupling effect of concentrated load, variable tension, and time-dependent speed on transverse vibration is investigated. The contribution of residue of singularity to total vibrations in phase space is numerically analyzed by using the Laplace transform method. The influence of the boundary condition considering the polygonal action is investigated in terms of excitation frequency and amplitude coupled with transport speed. The transverse vibrations are calculated numerically and measured experimentally. The numerical results are in agreement with the experimental data, which suggest that the amplitude and frequency of vibration are proportional to the value of propagation speed. The analytical solution to the moving chain problem provides a feasible reference for its stability analysis and wind-induced vibration control.
1. Introduction
Ropeways are widely used for transportation of goods as well as public transport in difficult terrain conditions in mountain regions. Considering the adverse working conditions and personal safety, special attention should be paid to ropeways during the design and initial testing phase as well as routine maintenance during its operation. The steel wire has been applied in the passenger ropeway system for recent years. When it comes to the mining cableway or heavy lift crane, transmission chain has an advantage of good stability with small amplitude of vibration, easier installation of the cargo hooks, convenient maintenance, and large breaking force compared to the steel wire. However, the coupling of multibody dynamics and external incentives may result in instability of the whole ropeway system. It is necessary to reduce the effects.
The neoteric application of chain ropeway system was proved to be labor-saving and high efficiency on moving harvested fruits and production materials including fertilizers and pesticides in mountainous orchards. High transport speed coupled with concentrated loads and low tension in chain may lead to a loss of stability and cause accidents. Hence, the dynamics and boundary conditions of the transverse and longitudinal oscillations should be investigated within the admissible range of speed, span, initial tension, and load to avoid losing stability. It is critically important to solve the governing equations that would permit an accurate description of the vibration in practical applications.
The dynamics of ropeways has been a topic of interest because of its importance in many engineering applications. Extensive literature reviews on transverse vibrations and control of axially moving strings were provided by Rega [1–3] and Chen [4]. Nonideal boundary conditions of the transverse oscillations in travelling beam were considered by Öz et al. [5] using perturbation method. As truncating the approximation to a single mode of vibration may reduce accuracy on long time-scales, van Horssen [6] improved the solution by using the method of Laplace transforms to obtain infinitely many modes of vibration. Van Horssen and Ponomareva [7] extended their study by constructing the solution of an initial boundary value problem for a linear, nonhomogeneous axially moving string equation. And they also investigated the transversal vibrations of an axially moving continuum with a time varying velocity [8]. Pakdemirli and Boyaci [9] concentrated on the dynamics of a transportation system carrying moving distributed and concentrated inertial loads. They also found that the exact solution to the frequency problem can be obtained using Laplace transforms. However, it is difficult to integrate the time varying boundary conditions in the derivative equations. Nonlinear vibrations of an axially moving multisupported string have been investigated by Yurddaş et al. [10] with method of multiple scales for the stability and bifurcations. They continued this study with the boundary conditions of nonideal support between the opposite sides allowing small displacements [11].
Chains are driven by wrapping them on sprockets. When the chain link engages a sprocket tooth, there is an impact caused by polygonal action. This impact depends on the pitch of chain link, the sprocket speed, the effective stiffness of the chain link against the sprocket tooth, and the effective mass of the part of the chain involved in the impact. The engagement forms a regular polygon and this motion is called polygonal action. The polygonal action and its resulting vibrations are the source of most of the noise, which have caught attentions of experts for many years. Earlier analysis that focused on the natural frequency of a travelling chain in the application of power transmission was proposed by Mahalingam [12], where the amplitude of forced vibration excited by polygonal action was obtained by considering the chain as uniform string, and further explanation on the polygonal action was carried out [13]. Besides that, the periodic fluctuation of power transmitting chains related to polygonal action was investigated by Ariaratnam and Asokanthan [14]. Choi and Johnson formulated the polygonal action under different chain speeds and the case interacted with tensioner [15, 16]. In order to have a useful model of the vibration behavior of the chain, the description of polygonal action must be considered into the model [17, 18].
The interaction between the boundary conditions with polygonal action and static stability of the chain ropeway system has not been formulated. The problem is not easy to be solved due to the complicate dynamics response to the heterogeneous chain link, concentrated loads varying in location, large sag to span ratio, and boundary conditions imposed by the interaction between links and supporting rollers. An accumulation of various waves exists in vibration of the axially moving chain. The key factors affecting the stability of the moving chain are the tension, speed, load, boundary conditions, and initial sag.
The emphasis of this study is on modeling the interaction with the consideration of concentrated loads, large initial sag, and boundary conditions. By applying the Laplace transform method, the contribution of singularities to total vibrations in phase space is numerically analyzed to figure out the influences of key parameters on transverse vibrations.
2. Modeling and Solving
2.1. Equations of Motion
The chain ropeway system can be modeled as an axially moving string with uniform distribution and concentrated inertial loads. The behavior of transverse vibration is predicted by a two-dimensional model in vertical plane without considering the spatial nonlinear oscillation. The transport speed and chain tension are assumed to be constants. The chain ropeway system is supported at each span end with a pair of rollers. The schematic drawing of the monochain ropeway is shown in Figure 1. The supporting roller is specifically designed for the chain to get through the hooks. Each time a chain link engages the supporting roller, the polygonal action occurs. When the position in which the chain and the supporting roller engage fluctuates, the chain vibrates along with this fluctuation. Supporting roller with smaller wrap angle for chain links is likely to have increased vibration and noise because of polygonal action.
(a)
(b)
(c)
The homogenous differential equation that represents transverse vibrations of a chain ropeway within the span can be written as follows [19]:where is the transverse displacement of the chain; is the axial transport velocity of the chain; is the spatial coordinate; is the time coordinate; is the distance between the two supporting ends; is the initial tension; is the mass of the chain per unit length; is the Dirac delta; is the coordinate of the th load; and is the initial position of the th load.
Due to the time-variant parameters, the dynamic equation has variable coefficients for the second-order nonhomogeneous differential equation. In order to simply construct the solution by using the Laplace transform method, the mass of the chain is assumed be uniform distribution within each span which can be written as
The dimensionless quantities for the parameters and functions are defined as
When the external impacts act on the supporting rollers, the chain links will vibrate with harmonic function due to the polygonal action between the supporting rollers and chain links. The boundary conditions have the formwhere is the dimensionless amplitude of the impact load; is the angular frequency; is the pitch of the chain link; and are the phase angles for the two end supports respectively, assumed to be the same here.
With respect to the concentrated loads and gravity of moving chain, the initial configuration is given by the following parabolic function:
2.2. Solving of Equations
The method of Laplace transforms will be used to construct the exact solution for this problem. Define as the Laplace transform of
Performing elementary transformations, we obtain the following expression:where
The solution to the second-order nonhomogeneous differential equation is determined by a general solution of homogeneous differential equation and a particular solution of the nonhomogeneous differential equation.
The solution to (7) has the form where and .
The boundary conditions in phase space for the deflection function can be written in the following form:
Using (9) and (10), the expressions for and are found aswhere and .
Substituting and into (9), we can get where
For the case of the boundary conditions without any perturbation, we can obtain the solution in phase space with the form of by setting in (4).
Furthermore, and are the contributions corresponding to boundary conditions of the left and right end, respectively, in the phase space.
Applying the inverse Laplace transform of by the definitionwhere , and is the residue of at .
The residue can be obtained as follows:
The singularities of are given by
And we can also note that the function and both have singularity of order two at , and has singularity of order one at .
The singularities of and are given by (18) and
By applying the inverse Laplace transform of , the is evaluated with integration by the residues of singularities of , , and . Then, the solutions of transverse vibration equations for the chain ropeway system can be obtained.
3. Numerical Analysis
In order to analyze how do parameters of the moving chain system affect the vibration frequency and amplitude of displacement, numerical procedures are performed.
The practical monochain ropeway systems have been applied in Longmen County in Guangdong province and Anyuan County in Jiangxi province. In this study, the simulation parameters of moving chain system originate from the experimental set-up built in South China Agricultural University. The physical parameters are given as follows: , , , , , = 5–15 kg, , = 200–1350 N, , and = 0.3–1.2 m/s. In fact, the variation of , , and is ultimately the modification of propagation speed .
3.1. Without Polygonal Action
In the case of boundary conditions without polygonal action, vibrations of the moving chain are dominated by the residues of singularity for , , and and singularities for . The residues of singularity for and diverge if they are calculated individually. is a monotonically increasing function and its sign is always positive, while is a monotonically decreasing function with sign always negative. The total residues have been proven to converge to a constant if and are evaluated with integration, which corresponds to the sag to be monitored. Hence, their effects on the characteristics of transverse vibration can be neglected.
In addition, the residues of singularity of do not contribute to the vibration in the time coordinate, because it has no time dependence. Then, without considering the polygonal action, the determinant term corresponding to the vibration with the boundary conditions is the total residue of singularities of . The diagram of transverse vibrations at the midpoint for various residues of singularity at different orders of , for the transport speed , is depicted in Figure 2. Sufficiently accurate results can be obtained for the lower order .
In the case of and , the transverse vibrations at the midpoint for various propagation speed are shown in Figure 3. The larger the value of chosen, the higher the frequency. This can be obtained by increasing initial tension and decreasing gravity of cargo with consideration of the derivation of .
3.2. With Polygonal Action
To obtain an insight into the influence of the boundary conditions with polygonal action, the dynamic behavior of the system is analyzed at different propagation speeds. The and represent the features of boundary conditions and at the end supports in the phase space. The and are the superposition of the residues of singularities and . The singularities are influenced by the transport speed , initial tension , and average density .
For , transverse vibrations at the midpoint imposed by boundary conditions with polygonal action at the end supports for different propagation speed are given in Figures 4 and 5, which are the numerical results obtained for and , respectively. In terms of amplitude and phase, there are differences between and for the discrepant expression in (14) and (15). Figures 4 and 5 depict that the amplitude of displacement increases continuously and monotonically as propagation speed increases.
Imposed by the boundary conditions with polygonal action, the total contributions of both end supports are shown in Figure 6, which also represent the numerical results of and . The amplitudes of displacement in Figure 6(a) are larger than those in Figures 4 and 5 due to the superposition of the responses for . Figure 6(b) shows the total vibrations are subjected to the boundary conditions with different pitches of chain link under the condition and . It is easy to distinguish the dynamic responses corresponding to various link pitches and line densities. The bigger chain pitch can make system more stable because it can achieve lower angular frequency and smaller value of with boundary conditions.
(a) Effects of the propagation speed
(b) Effects of the type of chain link
The transverse vibrations of the chain ropeway system with integration of all the inverse transform at different values of the parameters and are shown in Figure 7, which are the numerical results obtained for . In the case of in Figure 7(a), we can see that the vibratory displacement of the travelling chain fluctuates, which may lead to the acoustical resonance and load detachment from hook. As shown in Figure 7(b), for the condition of propagation speed , small vibrations are found with respect to different transport speed . Because the transport speed is far away from propagation speed , the impact of transport speed contributes a little among the numerical process.
(a) Displacements at different propagation speed with
(b) Displacements at different transport speed with
Comparisons of the vibratory displacement with or without the polygonal action are illustrated in Figure 8, which also compare the trajectories of the midpoint for and . The difference between both situations depicted in Figure 8(a) can be considered as fluctuations based on the vibration model without the polygonal action. As a result, the level of noise and abrasion resulting from the chain-support engagement increase as the vibration frequency increases. Figure 8(b) shows that the vibratory displacement will increase with increasing propagation speed. As depicted in Figure 8(c), the amplitudes at are twice as vibration model without polygonal action shown in Figure 3. To effectively reduce this effect of propagation speed, there are some ways to prevent instability by decreasing tension, increasing load as well as increasing chain pitch.
(a) Displacements at with
(b) Displacements at with
(c) Displacements at with
4. Experimental Validation
For the purpose of validation of the numerical results, the full-scale experiments were carried out with two concentrated inertial loads of 5 kg each hung on adjacent hooks. The variation of and can be obtained by choosing different tensile forces and frequencies of the frequency converter, respectively. The visualization and measurement of the transverse vibrations of chain ropeway were done by high-speed cameras. The video frames are extracted for the purpose of obtaining vibration data of chain links by image processing. The total vibrations at the midpoint of the span are showed in Figures 9 and 10 for different and . The results in Figures 9(a) and 9(b) demonstrate the same features of the vibratory displacement as in Figure 7(a) regarding the propagation speed as a reference variable.
(a) Displacements at
(b) Displacements at
(a) Displacements at
(b) Displacements at
The slight effect of the transport speed on transverse vibration can be observed in Figure 10, which verifies the numerical results in Figure 7(b).
Admittedly, the simulation results are not very close to experimental data. The reason is the neglected uncertainties consisting of stiffness, variations of tension, and cross coupling effect within spans. In general, the numerical results have the same tendencies with those of experimental data that the wave propagation speed and transport speed have positive impact on the amplitude of transverse vibration.
5. Conclusion
The dynamics of axially moving chain ropeway systems for hilly orchards are modeled with the polygonal action, which result from the chain-support engagement. Supporting roller with smaller wrap angle for chain links increases the vibration and noise because of polygonal action. By applying the Laplace transform, the solutions of the transverses vibration problem are investigated in a decoupling form with different terms considering the residues of various singularities. Numerical approaches and full-scale experiments were carried out in order to characterize the transverse vibration of chain ropeway system having support boundary conditions with polygonal action.
The results of numerical analysis indicate that the interaction between wave propagation speed and transport speed significantly affect the frequency and amplitude of transverse vibration. The amplitude and frequency are proportional to the value of propagation speed. There is a positive correlation between the propagation speed and the initial tension, along with a negative correlation between the propagation speed and the concentrated load. Compared to the propagation speed, the effect of transport speed on variation is relatively smaller. A big chain pitch has a stabilizing influence on the moving chain system. Based on the image processing and pattern recognition, the experimental data are analyzed. The numerical results show good agreement with experimental observations and indicate the same general tendencies, which provide additional insight into the dynamic behavior of these systems.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The study is supported by the National Science Foundation of China (51205139) and the project of high level talents in Colleges in Guangdong Province (Guangdong finance education [2011] no. 431).