Modal Analysis and Multi-objective Optimization of Pressurizing Pipeline

WANG Yeping LI Hang

School of Mechanical Engineering,Tongji University,Shanghai 200092,China

Abstract:The pressurizing pipeline of hot press resonates under the excitation load,which poses a serious hidden danger to the safety of the equipment and the operator. In order to increase the natural frequency of the pressurizing pipeline,modal analysis of the pressurizing pipeline is carried out to study the mechanism of pipeline vibration and common vibration reduction measures. A method of increasing the natural frequency of the pressurizing pipeline was analyzed. The influence of pipeline clamp assembly stiffness,pipeline clamp number and pipeline clamp installation position on the mode of the pressurizing pipeline is studied. Sensitivity analysis is carried out to study the influence of the various parameters on the mode of the pressurizing pipeline. Genetic algorithm based on Pareto optimality is introduced for multi-objective optimization of pressurizing pipeline. The optimization results show that the natural frequency of the pressurizing pipeline increases by 2.4% and the displacement response is reduced by 17.7%.

Key words:pressurizing pipeline;modal analysis;sensitivity analysis;Pareto optimality;genetic algorithm;multi-objective optimization

Introduction

Hot press is an important production equipment in the wood-based panel industry,which plays a vital role in the quality and productivity of the wood-based panel[1].The pressing process is the core part of the hot pressing.

In this process,the facing paper,the rubber layer and the plate are tightly combined,the rubber material is infiltrated into the wood cells,and then the plate and the facing paper are glued[2].At the same time,the pressure,flow pulsation and external excitation load of the hydraulic pump cause vibration deformation of the pressurizing pipeline.When the external excitation frequency is close to the natural frequency of the pipeline,resonance occurs[3].Long-term resonance will cause fatigue damage such as looseness and cracks in the pipeline joints,posing a serious threat to the safety of equipment and operators.Therefore,it is necessary to perform modal analysis and multi-objective optimization on the pressurizing pipeline to increase the natural frequency of the pipeline and reduce the vibration deformation of the pipeline.Research on pipeline vibration has developed rapidly in recent years.For example,Jinetal.[4]made a greatcontribution to the establishment of fluid-solid coupling mathematical model of infusion tubes.Lü[5]performed fluid-structure dynamics analysis of aircraft hydraulic pipelines based on ANSYS.Li[6]analyzed the vibration noise and dynamic characteristics of pipeline systems.In this paper,modal analysis and sensitivity analysis of the pipeline are performed,and the influence of various parameters on the modal frequency is studied.The pipeline structure is optimized based on the research results.

1 Mechanism of Pipeline Vibration

1.1 Mechanism of pipeline vibration and common vibration reduction measures

There are two main reasons for the vibration deformation of the pressurizing pipeline.

(1) The excitation load of the pipeline:the pressure pulse of the fluid in the pipeline or the excitation of the frame plate is so large that the pipeline generates strong forced vibration.

(2) Resonance of the pipeline:when the pipeline design and the layout are unreasonable,and the natural frequency of the pipeline system is close to the external excitation frequency of the plunger pump,fluid-structure coupling resonance occurs.

Corresponding to this,common damping measures are:(1) optimizing the pressure control of the hydraulic system to reduce the pressure pulsation in the pipeline;(2) increasing the natural frequency of the pipeline and avoiding the resonance interval of the excitation load.

We mainly improve the natural frequency of the pipeline to reduce the vibration deformation.

1.2 Method of increasing the natural frequency

For a simple pipeline,the natural frequency can be calculated as

(1)

where,Eis the elastic modulus of the pipeline;Iis the cross-section moment of the pipeline;Lis the length of the pipeline;Meqis the equivalent concentrated mass of the system;Mis the concentrated mass of the system;αis the coefficient.

We can see from Eq.(1) that the natural frequency of the pipeline is proportional toE1/2and inversely proportional toL3/2.WhenEbecomes larger,fswill increase.WhenLbecomes larger,fswill decrease.

The natural frequency of the pipeline cannot be directly solved by Eq.(1) because of the complex structure of the pipeline.The finite element simulation software ANSYS can be used to analyze the modal characteristics of complex pipelines and obtain their various modes and natural frequencies[7].

Figure 1 shows the three-dimensional model of the pressurizing pipeline.It mainly includes bent pipes,straight pipes,tee joints,straight pipe joints,end seals and pipe clamps.

We can also see from Eq.(1)that the method of increasing the natural frequency is to increase the stiffness and shorten the length.The length of the pressurizing pipeline is limited by other structural parameters of the hot press and cannot be significantly modified.An effective way to increase the natural frequency is to increase the overall stiffness.The arrangement of the pipe clamps is a major factor affecting the overall stiffness.The main factors are the assembly stiffness of the pipe clamps,the number of pipe clamps,and the installed position of the pipe clamps in particular[8].

Fig.1 Pressurizing pipeline of hot press

2 Influence of Pipeline Structural Parameters on Modal Frequency

2.1 Influence of pipe clamp assembly stiffness on modal frequency

The pipe clamps are welded to the frame plate to transfer the excitation load to the pipeline.During the use,the pipe clamps may loosen due to maintenance,assembly,etc.,and thus it may not provide sufficient fixing and support for the pipeline.According to the working condition of the pipe clamps,the pipe clamps are simplified into spring units,and the tightening degree of the pipe clamps during use are indirectly characterized by the spring stiffness.Since the pipe clamps have little influence on the stiffness of the pipeline along the axial direction,the stiffness value of the pipe clamps along the axial direction of the pipeline can be neglected,and the constraint stiffness along the radial direction of the pipe are defined[9].

Fig.2 Simplified pipe clamps

The first ten modal frequencies of the pressuring piplines was calculated by changing the spring stiffness and using ANSYS.The results are shown in Fig.3.It can be seen from Fig.3 that the pipe clamp stiffness and the natural frequency of the pipeline are linear in a certain range.With the increase of the stiffness,the natural frequency of the pipeline gradually increases and tends to be stable.When the pipe clamp stiffness reaches 107N/m,the modal frequency of the pipeline basically does not change with stiffness.

Fig.3 Influence of pipe clamp assembly stiffness on modal frequency

2.2 Influence of pipe clamp number on modal frequency

The pipeline cannot get enough support to fix with few pipe clamps;for too many pipe clamps,enough mounting positions of the frame plate are required,and the cost increases.Finding the right number of pipe clamps helps to better design.

For this purpose,the modal frequencies of the pipeline under the equal spacing constraint of 2,4,6,8,and 10 pipe clamps are calculated.The results are shown in Fig.4.As the number of pipe clamps increases,the natural frequency of the first ten stages of the pipeline generally increases.The more pipe clamps,the higher the natural frequency is,but when the number of pipe clamps reaches 8,the natural frequency of the pipeline does not change significantly.Therefore,8 pipe clamps can effectively restrain the pipeline.

Fig.4 Influence of pipe clamp number on modal frequency

3 Sensitivity Analysis of Pipe Clamp Parameters

The installed position of the pipe clamps also affects the natural frequency of the pipeline[10].But the influence is very complicated,which is related to the complex shapes of the pipeline.Therefore,it is necessary to conduct sensitivity analysis on the pipeline structural parameters so as to optimize the structure of the pipeline.

According to the analysis in the previous section,the installation of 8 pipe clamps can effectively improve the natural frequency of the pipeline system,but the assembly rigidity of the pipe clamps and the influence of the installed position on the dynamic characteristics of the pipeline are complicated.In this section,sensitivity analysis of the installed position and assembly stiffness of the pipe clamp will be carried out under the condition of installing 8 pipe clamps.

Sensitivity analysis is a method for studying the sensitivity of a system to structural or environmental variables[11].Sensitivity simulation analysis is performed using ANSYS Workbench.The probability design module (PDS) can effectively evaluate the influence of system input parameter changes and uncertainty on the analysis results.

3.1 Response surface methodology(RSM)

RSM is a comprehensive functional simulation technique that effectively fits experimental data to obtain approximate functional expressions[12].RSM is mainly composed of two parts:function fitting and experimental design,which determine the analysis accuracy of RSM.The function fitting method mainly includes the center point precise response surface method,the least square method and the weighted least square method.The experimental design method mainly includes the central combination design method,the full factor design method,the partial factor design method and the D-optimality criterion method.Figure 5 shows the principle of RSM.

Fig.5 Principle of RSM

3.2 Sensitivity analysis simulation process

The pipeline is a nearly symmetrical structure,which is bounded by 8 pipe clamps with 4 pipe clamps on each side.The single-sided layout is shown in Fig.6.The modal analysis and harmonic response analysis of the pipeline were carried out by the equivalent mass method.The detailed structural parameters of the pipeline are shown in Table 1.

Fig.6 Pipe clamp layout of single-sided pipeline

Table 1 Structural parameters of pressurizing pipeline

The pipe clamp spacingsb1,b2andb3,and the pipe clamp assembly stiffnessEare taken as design variables.The first order natural frequency of the pipeline and the pipeline displacement response under sinusoidal excitation are taken as target outputs.The ranges ofb1,b2andb3are 900-1 050 mm,900-1 050 mm and 720-900 mm,respectively,and the range ofEis 9×106-1.1×107N/m.

The Geometry,Static Structural,Model,and Harmonic Response modules are selected in the Workbench.The sensitivity analysis system for the pipeline is built.The structure is shown in Fig.7.The center combination design method is selected to carry out the experimental design.The design points are selected by Monte Carlo random sampling,then the response surface is constructed by the quadratic difference function according to the solution result,and the response curve and local sensitivity analysis are performed.

Fig.7 Structure of sensitivity analysis system

3.3 Sensitivity analysis simulation results

Sensitivity characterizes how much each input parameter affects the system output.In the sensitivity histogram,the absolute value indicates how much the input parameter affects the system output.The larger the absolute value,the higher the impact of the input parameters.

Figure 8 is a sensitivity histogram showing the sensitivity ofb1,b2,b3andEto the natural frequency of the pipeline.It can be seen from Fig.8 that when the assembly stiffnessEis around 107N/m,the assembly stiffness has little effect on the natural frequency of the pipeline,which is consistent with the analysis of the previous section;the installation distance of the pipe clamp has a great influence on the natural frequency of the system.It can be seen from Fig.8 that the pipe clamp spacingb3has the greatest influence on the natural frequency of the pipe.The response curves ofb1,b2,b3and the natural frequency of the pipeline are shown in Figs.9-11.It can be seen from Figs.9-11 that within the variation interval,the natural frequency of the pipeline increases with the increase of the assembly spacing.

Fig.8 Sensitivity histogram of natural frequency

Fig.9 Response curve of pipe clamp spacing b1 and natural frequency

Fig.10 Response curve of pipe clamp spacing b2 and natural frequency

Fig.11 Response curve of pipe clamp spacing b3 and natural frequency

Figure 12 is a sensitivity histogram showing the sensitivity ofb1,b2,b3andEto pipe displacement response.It can be seen from Fig.12 that the pipe clamp spacing and assembly stiffness have a greater impact on the pipe response displacement.The response curves are shown in Figs.13-16.The response displacement decreases with the increase of the pipe clamp assembly stiffnessEin the variation interval,and increases with the increase ofb1,b2andb3.

Fig.12 Sensitivity histogram of response displacement

Fig.13 Response curve of pipe clamp stiffness E and displacement

Fig.14 Response curve of pipe clamp spacing b1 and displacement

Fig.15 Response curve of pipe clamp spacing b2 and displacement

Fig.16 Response curve of pipe clamp spacing b3 and displacement

4 Pipeline Optimization

According to the sensitivity analysis results,aiming at maximizing the first order natural frequency of the pipeline and minimizing the response displacement,parametersb1,b2,b3andEof the pressurizing pipeline are optimized based on the Pareto optimal multi-objective genetic algorithm (GA).

Most of the optimization design in engineering practice needs to meet multiple optimization goals.However,it is difficult to optimize multiple targets at the same time.This problem can be effectively solved by GA based on Pareto optimal,which is finally solved as an algebraic set called Pareto optimal solution set[13].GA is a method based on the theory of biological evolution in nature to find the optimal solution to the problem[14].

4.1 Multi-objective optimization simulation process

Geometry,Static Structural,Modal,Harmonic Response,Response Surface,and Optimization modules in the Workbench are selected to build the optimized design system for the pressurizing pipeline.

The genetic algorithm is selected as the optimization method,the natural frequency of the pressurizing pipeline is maximized and the harmonic excitation displacement response is minimized as the optimization target.The total number of samples is set to 1 000 and the number of iterations is set to 100.The maximum allowable Pareto percentage is 80%.That is,when 80% of the samples are distributed in the Pareto optimization front end,the iterative optimization ends.

4.2 Optimization results

After optimization in Workbench,a set of Pareto optimal solution sets is obtained,and the optimal design points can be selected according to needs.

Figure 17 shows the trade-off table obtained,which can reflect the trade-off relationship between different output parameters and determine the frontier of Pareto optimal solution.In Fig.17,the abscissa is the first order natural frequency objective function,and the ordinate is the harmonic excitation displacement response objective function.

Fig.17 Trade-off diagram of pipeline natural frequency and harmonic excitation response displacement

Each discrete point in the trade-off graph represents the output value of the objective function corresponding to a single sample.According to the optimization goal,the Pareto optimal front appears in the label of Fig.17.

As shown in Table 2,Workbench selects three design points from the Pareto optimal frontier as candidate points.We select the most suitable candidate point as the optimization design point according to the actual situation.Due to the response surface based on Monte Carlo random sampled data point fitting,there may be some error.Therefore,it is necessary to incorporate optimized design points into the simulation analysis to calculate the exact values.According to the optimization goal,candidate A is the best solution for this optimized design.Considering that the pipe clamp distance is usually an integer value,the value of the candidate A is rounded and the simulation is re-simulated.The results are shown in Table 3.

Table 2 Candidate points selected

Table 3 Optimization plan result comparison chart

It can be seen from Table 3 that after the multi-objective optimization,compared with the original scheme,the first order natural frequency of the pipeline increases by 2.4%,and the response displacement is reduced by 17.7%.

5 Conclusions

The vibration mechanism and main damping measures of the pressurizing pipeline of the hot press were introduced.Factors affecting the natural frequency of the pipeline were analyzed.When the assembly stiffness of the pipe clamp is greater than 107N/m,the pipe clamp constraint stiffness has little effect on the pipeline natural frequency.Effective restraint of the pipe requires installation of at least 8 pipe clamps.

Sensitivity analysis was carried out for factors such as pipe clamp assembly stiffnessE,and pipe clamp spacingb1,b2andb3.When the assembly stiffness of the pipe clamp is around 107N/m,the pipe clamp installation stiffness has little effect on the pipeline mode,which verifies the analysis of the previous section.The pipe clamp spacingsb1,b2andb3all have an effect on the pressure pipe mode.Among them,b3has the greatest impact,andb2has the smallest impact.

In order to improve the pipeline natural frequency and reduce the pipeline response displacement,the multi-objective optimization of the pipe clamp assembly stiffnessE,pipe clamp spacingb1,pipe clamp spacingb2and pipe clamp spacingb3is carried out.The natural frequency of the pressurizing pipeline increases by 2.4%,the response displacement is reduced by 17.7%,and good optimization results are obtained.