The feasibility design method with multidisciplinary and multiobjective optimization is applied in the research of lightweight design and NVH performances of crankshaft in highpower marine reciprocating compressor. OptLHD is explored to obtain the experimental scheme and perform data sampling. The elliptical basis function neural network (EBFNN) model considering modal frequency, static strength, torsional vibration angular displacement, and lightweight design of crankshaft is built. Deterministic optimization and reliability optimization for lightweight design of crankshaft are operated separately. Multiisland genetic algorithm (MIGA) combined with multidisciplinary cooptimization method is used to carry out the multiobjective optimization of crankshaft structure. Pareto optimal set is obtained. Optimization results demonstrate that the reliability optimization which considers the uncertainties of production process can ensure product stability compared with deterministic optimization. The coupling and decoupling of structure mechanical properties, NVH, and lightweight design are considered during the multiobjective optimization of crankshaft structure. Designers can choose the optimization results according to their demands, which means the production development cycle and the costs can be significantly reduced.
New marine reciprocating compressors must have high power, high pressure ratio, and slight vibration and be environmentally friendly with the development of marine natural gas boosting and gathering [
The highpower reciprocating compressors are designed to run onshore; accordingly, the support of compressors cannot match the crankshaft structure parameters and there are some drawbacks such as loud noise and high vibration intensity caused by gas force, reciprocating inertia force, and centrifugal force when it is used offshore [
The crankshaft structure is a complex engineering system involving structural mechanics, mechanical vibration and noise, and manmachineenvironment engineering. The crankshaft structure design is a complex multidisciplinary and multistage design process relating to high correlation and coupling between all disciplines. The whole process can be described by a complex function. The optimized parameters combination of crankshaft structure is obtained by iterating and optimizing. Consequently, it is a key factor to choose a suitable algorithm to solve this issue. Studies on crankshaft of reciprocating compressors mainly focus on vibration and stress analyses [
Although multiple parameters affect the performance of compressor, the amount of data obtained by compressor experiment is limited. The EBFNN theory is good at solving small sample learning problems [
In this paper, the coupling and interdisciplinary relationships of mechanical properties, NVH, and lightweight are considered and the MDO technology roadmap of the reciprocating compressor was proposed on the basis of virtual proving ground (VPG) technology. Cooptimization based on EBFNN and multiisland genetic algorithm is applied to the multiobjective optimization of crankshaft structure in order to gain the Pareto optimal solution set.
The MDO theory is applied in crankshaft structure design on the basis of VPG. The flow chart of the enforceable technology roadmap is shown in Figure
MDO technology roadmap on the basis of VPG.
The crankshaft MDO issues mainly focus on the effective optimization strategy to achieve concurrent design of multidisciplinary subsystems and obtain the satisfactory solution. The strategy combines the knowledge from different subjects with optimization algorithm and develops an effective method to solve the complex problems [
The MDO optimization framework can be divided into singlestage and multiplestage. The singlestage optimization framework is composed of multidisciplinary feasible method (MDF) and individual discipline feasible method (IDF). The multistage optimization framework consists of concurrent subspace optimization (CSSO), cooptimization (CO), and bilevel integrated (Bliss).
In this paper, the CO is mainly studied. CO is a multistage MDO algorithm on the basis of the optimization algorithm under consistency constraints. It divides the crankshaft MDO issues into one systemlevel optimization and multiple subsystemlevel optimization.
The systemlevel optimization objective of crankshaft can be expressed as
The subsystemlevel optimization objective of crankshaft is listed as
The meaning of the symbols in the formula is shown as follows:
Multiobjective optimization problem (MOP) of crankshaft can be represented as
The meaning of the symbols in the formula is shown as follows:
With the given crankshaft MOP issue, the Pareto optimal solution can be defined as follows: if and only if there exists no feasible solution (
Inevitably, the MDO of the crankshaft is accompanied by the MOP of the crankshaft. MOP of the crankshaft cannot achieve best possible optimization of all objectives simultaneously and arbitrary solution of Pareto set will possibly become the satisfactory solution.
The evaluation methods of MOP can be divided into global optimization algorithms and local optimization algorithms. The global optimization algorithms include genetic algorithm, simulated annealing algorithm, particle swarm optimization, and ant colony algorithm. Due to their high capability of global search, high speeds of convergence, and search results independent on starting point, the global optimization algorithms are capable of solving high dimensional and nonlinear problems. But the computation might be expensive and sometimes unsatisfactory local optimization effect [
Flow chart of MIGA.
The modal analysis is an important part of dynamic analysis in reciprocating compressor machine system, which can help us understand the dynamic characteristic of the system. The natural frequency of crankshaft is usually calculated to avoid resonances during use in the design of NVH. Severe deforming parts of crankshaft are observed to judge the strength of the corresponding structure, which may become noise vibration source or main transfer path and should be modified early.
There are 269612 entity units and 452154 nodes on the finite element model of crankshaft NVH. The characteristic of NVH is studied by crankshaft modal analysis and torsional vibration.
The crankshaft modal is computed by ANSYS under free boundary. Therefore, the firstorder natural frequency is 41.413 Hz. The secondorder natural frequency is 43.54 Hz.
In the process of the torsional vibration analysis, the modal superposition method is used to simplify the finite element of crankshaft. The elastic deformation of the structure is solved approximately by linear combination of suitable modes which can be shown as follows:
The meaning of the symbols in the formula is shown as follows:
An elastic body contains two types of nodes, interface nodes, where forces and boundary conditions interact with the structure during multibody system simulation (MSS), and interior nodes. In MSS the position of the elastic body is computed by superposing its rigid body motion and elastic deformation. In ADAMS, this is performed using “Component Mode Synthesis” technique based on CraigBampton method [
The meaning of the symbols in the formula is shown as follows:
To obtain decoupled set of modes, constrained modes and normal modes are orthogonalized.
The crankshaft system model is shown in Figure
Model of crankshaft system.
The boundary condition of crankshaft strength analysis is shown in Figure
Boundary condition of crankshaft strength analysis.
The deterministic optimization, reliability optimization, and multiobjective optimization are operated independently on the basis of EBFNN and CO. The flow chart of MDO is shown in Figure
Flow chart of crankshaft system MDO.
Systems in crankshaft structure can be divided into mass, NVH, and strength subsystem. The NVH subsystem includes modal analysis and torsional vibration.
The structure of crankshaft system has an important effect on the torsional vibration, strength, natural frequency, and mass of crankshaft. As the crankshaft is constrained by the dimension and assembly of connecting rod, frame, and other parts, the dimension of crank journals, crankpins, and bore spacing cannot be changed in this calculation example. Consequently, transitional fillet (
Design variables of MDO.
OptLHD is adopted to obtain the experimental scheme and perform data sampling. The elliptical basis function neural network (EBFNN) model considering modal frequency, static strength, torsional vibration angular displacement, and lightweight design of crankshaft is built. The experimental scheme is listed in Table
Experimental scheme based on OptLHD.
Sample  Design variables  





 
mm  mm  mm  mm  mm  
1  9.18  16.53  46.84  38.42  75.0 
2  7.82  14.63  40.53  35.26  72.89 
3  7.61  16.74  31.05  40.53  65.53 
4  7.71  17.58  45.79  39.47  70.79 
5  8.87  14.84  42.63  41.58  55.0 
6  8.34  15.68  50.0  33.16  63.42 
7  7.5  15.47  44.74  45.79  60.26 
8  8.97  18.0  34.21  43.68  68.68 
9  9.39  17.37  43.68  34.21  61.32 
10  8.45  16.32  33.16  50.0  57.11 
11  8.66  14.21  48.95  44.74  69.74 
12  9.08  16.11  30.0  36.32  59.24 
13  8.55  16.95  35.26  30.0  71.84 
14  9.5  15.26  37.37  46.84  66.58 
15  8.03  17.79  41.58  37.37  56.05 
16  9.29  14.42  39.47  32.11  67.63 
17  7.92  15.05  36.32  31.05  58.16 
18  8.24  14.0  32.11  42.63  64.47 
19  8.76  17.16  47.89  47.89  62.37 
20  8.13  15.89  38.42  48.95  73.95 
Targets in NVH subsystem include firstorder modal frequency (
The time of training is set to 20 based on the parallel computing. The result is listed from Figures
Result of crankshaft mass.
Result of modal frequency.
Result of the maximum of torsional angular vibration over a period time.
Result of the maximum stress over a period time.
Result of main bearing load over a period time.
Each response is mapped to elliptical basis function surrogate model on the theory of the elliptical basis function neural network. It is shown in Figure
EBFNN model of crankshaft.
The elliptical basis function neural network including
The meaning of the symbols in the formula is shown as follows:
The meaning of the symbols in the formula is shown as follows:
Here,
Having gained output responses
The tansigmoid function is used in this neural network. Hence, ideal output results should be close or equal to 1. The normalization processing of experiment data is carried out, which can be described by
The meaning of the symbols in the formula is shown as follows:
The elliptical basis function surrogate model between design variables and its analysis target can be solved, combining formulas (
Response surface of mass normalization.
Response surface of the maximum stress normalization.
Response surface of torsional angular vibration normalization.
As the elliptical basis function surrogate model between input variables and its analysis targets cannot be described by a specific function, correlation coefficients (
The meaning of the symbols in the formula is shown as follows:
The fitting of the surrogate model can be solved by formula (
The deterministic systemlevel optimization objective can be represented as
The deterministic subsystemlevel optimization objective can be represented as
The meaning of the symbols in the formula is shown as follows:
In the design of crankshaft structure, system uncertainty is caused by various factors, such as structure parameters, system forecast model, sampling technology, judgments criterion. and human factors. Accordingly, reliability optimization is adopted to control and eliminate system uncertainty.
The reliability systemlevel optimization model can be represented as
The reliability subsystemlevel optimization objective can be represented as
The meaning of the symbols in the formula is shown as follows:
The determined optimization and reliability optimization are operated independently on the basis of MIGA. The advanced options of MIGA are listed in Table
Advanced options of MIGA.
Options  Parameter setting 

Subpopulation size  20 
Number of islands  10 
Number of generations  50 
Rate of crossover  0.8 
Rate of mutation  0.0075 
Rate of migration  0.25 
Interval of migration  5 
Relative tournament size  0.5 
Elite  1.0 
The initialization, range. and optimization results of each design variable are listed in Table
Design variables and optimization results.
Variables and response  Initialization  Upper bound  Lower bound  Deterministic optimal results  Reliability optimal results 


8.0  9.5  7.5  7.5  7.76 

16.0  18.0  14.0  17.9  16.9 

40.0  50.0  30.0  49.9  48.1 

40.0  50.0  30.0  49.9  46.3 

65.0  75.0  55.0  71.0  55.2 

846  —  —  838.8  840.1 
The optimization results show that the weight of the crankshaft is reduced 7.2 kg, which accounts for 0.85% of the initial mass. However, the uncertain factors are not considered. The weight of the structure is reduced 5.9 kg through reliability optimization. The reliability optimization can not only achieve the lightweight of the crankshaft, but also ensure the reliability and robustness in engineering quality.
According to formula (
Here,
The range of the design variables and the optimization objectives need to be determined by actual production. MOP can be defined as treeobjective optimization when there are two optimization objectives in formula (
Figures
Pareto set with weight value
Pareto set with weight value
Pareto set with weight value
Pareto set with three optimization objects.
In Figures
In Figure
The multidisciplinary optimization considering the crankshaft modal, torsional angular vibration, maximum stress over a period time and maximum load on the main bearings is operated on the basis of multiisland genetic algorithm, which can effectively improve the comprehensive property of the crankshaft.
The parallel computing in multidisciplinary optimization is operated on the basis of the combination of elliptical basis function neural network theory and cooptimization method, which can enhance the optimization efficiency, so as to reduce product development cycle and costs.
During the design optimization process of the crankshaft structure, the reliability design is combined with the cooptimization method. And the optimization of the crankshaft is operated on the basis of multiisland genetic algorithm, combined with design of experiment. The optimization can not only control the system uncertainty, but also ensure the reliability and robustness of the final optimal results of the crankshaft structure.
The authors declares that there is no conflict of interests regarding the publication of this paper.
This work is supported by Open Fund (OGE20140309) of Key Laboratory of Oil & Gas Equipment, Ministry of Education (Southwest Petroleum University), and (