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Multi-Objective Optimization of Counterweights: A Substitute for the Balance Shaft or Mass Unbalancing in Three-Cylinder Engines.

Introduction

The durability, high performance, and low weight are strict legal requirements for engine design worldwide. In this way, the following functional objectives are considered: high specific power output, low fuel consumption, lightweight construction, minimum friction losses as a result of optimized base engine configuration, greatest possible smooth performance as a result of mass balance systems, low sound radiation, etc. The challenge is to design at the same time the individual derivatives with optimized functions to ensure for each case the top position among the competition [1].

Three-cylinder engines were launched due to the increased demand for fuel economy, friction loss, thermal efficiency, and needing fewer parts (easier to build, to maintain, and to repair) [2]. This has become possible thanks to direct injection turbochargers which can improve fuel economy and emissions without compromising performance feel and power. The focus of the turbocharger design is on quick torque buildup, high efficiency, and excellent fuel economy using latest technologies in the turbo [3, 4]. Nevertheless, inherent noise vibration harshness (NVH) problem caused by naturally unbalancing remained the weakness of the three-cylinder compared with a four-cylinder engine [2].

In three-cylinder engines, unlike four-cylinder engines, primary and secondary inertia forces are zero but resultant moments are applied to engine block. Primary balancing of three-cylinder engines is done by counterweights. In order to eliminate remaining vibrations, the methods of balance shaft and mass unbalancing of flywheel and crankshaft pulley have been used for reducing pitch moment as well as optimized mounts for reducing yaw moment.

Suh et al. [5] balanced a three-cylinder engine with balance shaft. Combustion pressures were measured from the actual tests. The vibration velocities at the engine mounts were evaluated through the real-time vibration analysis. Their results show that the vibration of the three-cylinder engine with the balance shaft is reduced to an acceptable level. Lee et al. [2] argued that by applying a balanced shaft, although the vibration of the components is reduced, the fuel economy will become worse, and the cost and weight will increase. Friedfeldt et al. [4] showed that ambitious NVH targets can be met without a balance shaft, which helped further reduce weight, cost, and friction with the use of an intelligent arrangement of counterweights.

Another important point that should be mentioned is to increase durability and extend the operating speed range of the crankshaft assembly by reducing bearing loads [6]. Together with the small main and connecting rod bearings, the engine weight and powertrain friction have been further reduced and consumption positively influenced. The six counterweights can reduce the inner forces in the camshaft and, with that, the main bearing stress [7]. Articles have been published in the subject of counterweights effects on bearing loads. Stanley and Taraza [8] found a dimensionless formula for calculating the maximum bearing load of an in-line four-cylinder engine. That study focused on the influence of lowering the counterweight mass on bearing load. Yilmaz and Anlas [9] investigated the effects of counterweights configuration on main bearing loads and crankshaft bending stress of an in-line six-cylinder diesel engine. Twelve counterweights with zero offset angle and eight counterweights with 30 degrees offset angle were each considered for 0%, 50%, and 100% balancing rate.

Studies have been done in the optimization of counterweights configuration context. Sharpe et al. [6] developed a numerical model for the crankshaft that considers reciprocating forces as well as gas pressure force. The model was optimized for designing crankshaft counterweights which would minimize main bearing forces. The results indicate reduction in average bearing force as much as 73% while peak force is reduced as much as 41% for a V-8 NASCAR engine. Yu and Zhi-yong [10] concluded that the mean load of a crankshaft's main bearing is an appropriate choice for objective function of a four-cylinder engine. Yannick and Duysinx [11] balanced different configurations of twin-cylinder engines with consideration of gas pressure force. The mass of optimal counterweights was gained by a graphical method of minimizing inertia force. Santos et al. [12] found the optimum mass distribution of the crankshaft by minimizing the weight of components along with satisfying manufacturing and maximum stress constraints for a mono-cylinder engine. Becker [7] reduced the inner forces and the main bearing stress of a three-cylinder engine by optimization of six counterweights. Lee et al. [2] optimized the shape of the crankshaft balance weight, to minimize both the vertical pitching and longitudinal yawing of a three-cylinder engine. Kling and Salif [13] used the design of experiments (DOE) function to find the optimal design for balancing a six-cylinder engine. Different angle and mass variables for individual counterweights are considered and DOE has selected optimum level of them. Niizato et al. [14] optimized the crankshaft reciprocating mass balancing ratio of a three-cylinder engine in order to reduce the vertical vibration at the engine mount points.

As mentioned above, the balance shaft has the undesirable characteristics of cost, weight, friction, noise, and reduction in power, and mass unbalancing is not appropriate dynamically. In order to cope with these methods and their undesirable characteristics, this article recommends counterweights optimization as a balancing process in engine design.

Due to the records, optimization of counterweights configuration for one-, two-, three-, four-, six-, and eight-cylinder engines by minimizing bearing loads, inertia force, vertical pitching, longitudinal yawing, and weight, as well as investigation of counterweights configuration on bearing loads for four-and six-cylinder engines, has already been done. The contribution of this article is to apply an established multi-objective optimization method. So, in this study, optimization of a three-cylinder engine is done considering bearing loads, shaking force, and vibrational moments as objective functions along with counterweights' mass as constraint.

The rest of the article is organized as follows: Next section, contains assumptions, needed quantities of engines, and equations of engine modeling. Then, Counterweights configuration of one industrial three-cylinder engine are optimized by non-dominated sorting genetic algorithm (NSGAII) method to remove mass unbalancing o its flywheel and pulley. After that, a simple formula is introduced with application to quickly find the counterweights configuration of each three-cylinder engine with a new specification, as a substitute for implementing a long-term optimization process. This formula is based on the optimization results of a three-cylinder engine after sensitivity analysis to identify its important parameters. Presented formula is verified with the default three-cylinder engine of Adams/Engine software. The possibility of removing the balance shaft is investigated for the same engine. Also, a comparison of the results of two software, MATLAB and Adams/Engine, and the counterweights configuration of some of the other industrial three-cylinder engines are presented in this section. Finally, the summary and conclusions are expressed.

Modeling of a Three-Cylinder Engine

The first step in optimization is formulating an optimum design problem. In the beginning of this section, assumptions and required parameters of six three-cylinder engines are mentioned. Then, required equations of inertia force, resultant moments, and bearing loads are obtained using a lumped-mass model.

Introduction to Modeling

Lumped-mass dynamic of the basic one-cylinder slider-crank mechanism, one counterweight, and crankshaft of a three-cylinder engine are shown in Figure 1. Generally, cylinders are numbered from the belt pulley (front) of the engine toward the flywheel (rear) of the engine. Crankshaft standard direction of rotation is clockwise, when viewed from the front of the engine [15]. Equations are expressed in global XYZ coordinate. The notations [O.sub.2], A, and B denote main pin, crankpin, and wrist pin, respectively. Main bearings are shown as [B.sub.1], [B.sub.2], [B.sub.3], and [B.sub.4] notations.

In this section, assumptions used for modeling, simulation, and optimization, in addition to required quantities of six three-cylinder engines employed in various sectors, are mentioned.

Assumptions Pressure rolling moment produced by the pressure force is a function of engine load, not engine speed. So, it is impossible to balance the unbalance caused by this moment and engine mounts should be designed for this purpose [8]. Accordingly, this article with the aim of dynamic balancing of engine without external loads does not consider the gas force pressure effect, and it is desirable to balance the inertia moments and forces that cause a net engine unbalance.

The rest of the assumptions are stated in the following. Modeling is done in constant speed. Effects of crank offset, weight, and crankshaft flexibility are all ignored. These items will have negligible effects on the dynamic balancing of the engine. The magnitude of inertia forces and moments increases as the engine speeds. Therefore, the engine is optimized in the maximum usual speed of three-cylinder engines (6000 rpm). The binomial theorem and dynamically equivalent lumped models are used to obtain equations [16]. Two-thirds of the connecting rod mass is assumed to be in pure rotation as part of the crank and a third of it is considered to be in pure translation as part of the piston [8]. The load on each throw is assumed to be supported equally by the associated main bearings [6].

Required Quantities of Three-Cylinder Engines All needed parameters are given in Table A.1 in the Appendix. The information of the first five engines is received from IranKhodro Powertrain Company, Engine Research Center (IPCO). The last engine is the default three-cylinder engine of the Adams/Engine software. Balance methods of each engine are listed in the same table. The optimization method is performed for the first three-cylinder engine, and its optimal configuration is generalized for the next five engines with a simple formula.

Equations

As mentioned in the assumptions, equations for constant speed are achieved. In this section, an equivalent lumped-mass model is proposed to calculate the inertia force, resultant moments, and bearing loads.

Lumped-Mass Dynamic Model [16] The connecting rod in complex motion can be modeled as two lumped point masses, one on the crankpin (in pure rotation) and the other on the wrist pin (in pure translation). These lumped point masses have no dimension and are assumed to be connected with a massless rigid rod. A similar lumped-mass model can be created for a crank. The crank is modeled as two lumped masses, one on the crankpin and the other on the fixed main pin. Lumped mass at the fixed pivot [O.sub.2] is not necessary for any calculations since that pivot is stationary.

These simplifications lead to the lumped parameter model of the slider-crank linkage shown in Figure 1(a). The crankpin-point A-has two masses concentrated on it, the equivalent mass of the crank [m.sub.2a] and the portion of the connecting rod [m.sub.3a]. On the wrist pin-point B-two masses are also concentrated, the piston mass [m.sub.4] and the remaining portion of the connecting rod mass [m.sub.3b]. The model has masses which are either in pure rotation or in pure translation, so it is very easy to dynamically analyze. These masses can be found by the following relations:

[m.sub.A] = [m.sub.2a] + [m.sub.3a] = [m.sub.2] [r.sub.G2]/r + [m.sub.3] [l.sub.b]/l Eq. (1)

[m.sub.B] = [m.sub.3b] + [m.sub.4] = [m.sub.3] [l.sub.a]/l + [m.sub.4] Eq. (2)

where [m.sub.A], [m.sub.B], [m.sub.2], [m.sub.3], [m.sub.4], r, l, [r.sub.G2], [l.sub.a], and [l.sub.b] are rotating mass, reciprocating mass, crank mass, connecting rod mass, piston mass, crank radius, connecting rod length, distance of crank's center of gravity (CG) and main pin, distance of connecting rod's CG and crankpin, and distance of connecting rod's CG and wrist pin, respectively.

Inertia Force, Resultant Moments, and Bearing Loads The crank angle for any constant crankshaft angular velocity can be written as:

[[theta].sub.i] = [omega]t + [[phi].sub.i] Eq. (3)

where [[theta].sub.i], [omega], t, and [[phi].sub.i] are angular position of crank, angular velocity of crankshaft, time, and phase angle of crank, respectively. By employing this equation, forces and moments of each cylinder can be obtained. Position of the piston achieved from the geometry of Figure 1(a) is expressed in the following form [16]:

[mathematical expression not reproducible] Eq. (4)

where [x.sub.i] is the piston's position. To reduce computational time in iterative algorithms including optimization, the nonlinear radical in Equation 4 is usually expanded using the binomial theorem. So, position of the piston is approximately equal to [16]:

[mathematical expression not reproducible] Eq. (5)

For a steady-state analysis, the piston acceleration is given by [16]:

[mathematical expression not reproducible] Eq. (6)

where [[??].sub.i] is the piston's acceleration. The approximately equal symbol in Equation 6 is due to the approximation considered in Equation 5. The inertia force is described by:

[mathematical expression not reproducible] Eq. (7)

where [[??].sub.i], [m.sub.CW,ij], [R.sub.CW,ij] and [[delta].sub.ij] are inertia force, counterweight mass, counterweight radius, and offset angle (angular position) of counterweight, respectively. The moments are separated into three categories: the rolling moment about the Z axis, the pitch moment about the Y axis, and the yaw moment about the X axis of the crankshaft. Rolling moment isn't expressed, because counterweights have no effect on it. Two other moments can be obtained as:

[mathematical expression not reproducible] Eq. (8)

[mathematical expression not reproducible] Eq. (9)

where [M.sub.Pitch], [M.sub.Yaw], [z.sub.i] and [z.sub.CW,ij] are pitch moment, yaw moment, axial position of the crank, and axial position of the counterweight, respectively.

The crankshaft resting on the four main bearings is a statically indeterminate system. To simplify the model, each throw is analyzed separately. The total main bearing loads for the system are obtained by summing the contributions from each individual throw. This simplification turns the problem into a statically determinate system [6]. The first and second bearing loads can be written as shown below:

[mathematical expression not reproducible] Eq. (10)

[mathematical expression not reproducible] Eq. (11)

where [B.sub.i] [z.sub.n], [z.sub.f], and [DELTA][z.sub.mains] are bearing force, axial distance of bearing and its nearest cylinder, axial distance of bearing and its farthest cylinder, and axial main bearing distance, respectively.

Optimization of Counterweights Configuration

In the following lines formulation presented in the previous section, design variables, objective functions, and constraints of optimum design problem are expressed. The counterweights configuration of a three-cylinder engine is optimized by NSGAII method using MATLAB software. By weighting objective functions based on their importance, optimal counterweights are selected. At the end of this section, the feasibility of removing mass unbalancing is investigated.

Introduction to Optimization

Design variables or desired output parameters are offset angles (angular orientation) and the product of the mass at the radius (correction amount) of each counterweight. The offset angle is the angle between the lines of counterweight and crank. Mass and radius are considered as design variables in the form of a product of each other because their multiplication is presented in all equations. Counterweights configuration in industrial three-cylinder engines is considered to be symmetrical around the geometrical center of their crankshaft (almost in the CG). Considering this symmetry, by determining the arrangement of the first three counterweights, configuration of the rest will be specified.

To use optimum design concepts in a dynamic problem, a quasi-static model is used. Therefore, the maximum amounts of forces and moments are considered as objective functions. Because of undesirable roughness related to absolute value functions, using their squares is offered. Due to phase angles, the inertia force of a three-cylinder engine without counterweights is zero. On the other hand, counterweights configuration has no effect on rolling moment. So the inertia force and rolling moment are not considered as objective functions. Due to the symmetry, the maximum loads of the first and fourth bearings, as well as the maximum loads of the second and third bearings, are equal. Therefore, five objective functions that are used are the maximum inertia force of counterweights, the maximum square pitch moment, the maximum square yaw moment, the maximum of the first bearing load, and the maximum of the second bearing load.

By adding counterweights, the unnecessary emphasis on zero inertia force will result in the loss of many good answers. Therefore, a better strategy is to convert the equal constraint to inequality and to set an upper limit for the inertia force of counterweights. In order to avoid divergence of the solution in the stochastic search optimization algorithm, the upper limit of 500 newton is selected for the inertia force. It is worth noting that this selected value, which is much less than bearing loads, prevents excessive inertia force by the inactive constraint. The upper limit of counterweights correction amounts is defined with respect to 100% balance factor. This limit heeds the maximum acceptable counterweights mass (as a symbol of weight and cost) and its radius (as a constraint to avoid collisions). The limits on the offset angles are also defined with respect to the assumption of placing the counterweights in the lower half of the crank.

Multi-objective Optimization

The NSGAII belongs to the class of stochastic search and evolutionary optimization methods. This method is used to discover Pareto-optimal solutions within a multi-objective optimization problem. In the genetic algorithm, the superiority of the solutions is determined by the fitness function, due to the existence of one objective function. But in the multi-objective optimization, responses are not orderable. So, two criterions have been used in NSGAII: quality (proximity to the global Pareto front checked by non-dominated sorting method O ([MN.sup.2])) and discipline (incompressibility or high crowding distance). In the solution domain, a solution is Pareto-optimal if it denies domination from other solutions. The main difficulty in stochastic search algorithms is trapping in the local minimum. To cope with this problem, two parameters including "crossover probability" and "mutation probability" are considered. The former, with the preservation of some initial populations in the reproduction stage, and the latter, by increasing the variety of the response, prevent trapping at the local minimum [17].

Due to comprehensive application, novelty, suitable ranking method, and good convergence, the NSGAII method is used for optimization. It is worth noting that stochastic methods are the proper selection for quasi-static problems. The NSGAII MATLAB code has been written by the authors based on the algorithm described in the reference [17]. In this code, the number of generation, population size, crossover probability, and mutation probability are set to 6000, 300 (50 times the number of design variables), 0.85, and 0.1, respectively. To determine the optimum point of the Pareto front, the lowest weighted sum of the objective functions is selected. These weights are selected according to Table 1. The greatest weight is given to the pitch moment, because its large magnitude is undesirable for passengers. The minimum weights are assigned to the yaw moment and the inertia force, because these vibrations can be eliminated by engine mounts.

Removing Mass Unbalancing by Optimization

The main purpose of this article is optimization of counterweights configuration to remove mass unbalancing of the flywheel and pulley or balance shaft. By implementing the multi-objective optimization process described earlier, the optimal configuration of an industrial three-cylinder engine (engine 1 with the specifications listed in Table A.1) is obtained. Then, the possibility of removing its mass unbalancing is investigated by comparing the optimum and default counterweights.

The counterweights configurations of two similar engines, one with an inappropriate mass unbalancing (the default values used in the industrial engine) and the other with optimized counterweights, are listed in Table 2. The notations [(m*R).sub.CW,f], [(m*R).sub.CW,r], [[delta].sub.CW,f], and [[delta].sub.CW,r] in the table denote correction amounts and angular positions of unbalancing masses of the crankshaft pulley (at the front of the engine) and flywheel (at the rear of the engine), respectively.

The inertia force, vibrational moments, and bearing loads of the similar three-cylinder engines (engine l)-the first with no counterweights, the latter with optimal counterweights, and the last with the default counterweights configuration-are shown in Figure 2.

Table 3 demonstrates the reduction in the maximum and average of the vibrational moments and bearing loads with respect to the no counterweight engine for two similar three-cylinder engines, one with the optimum counterweights and the other with the default counterweights (along with the mass unbalancing of the flywheel and pulley).

According to Figure 2 and Table 3, the reduction in the pitch and yaw moments of the optimum and default counterweights are the same. But the inertia force and bearing loads in the engine with optimum counterweights are much less than the same engine (with default counterweights) used in the industry equipped with dynamically inappropriate mass unbalancing. So, this comparison shows that optimization of counterweights configuration is more suitable than mass unbalancing of the flywheel and pulley.

Referring to Table 1. the default counterweights mass of the industrial three-cylinder engine, considering weight loss criterion, is less than the optimum counterweights mass obtained from this article. However, although the proposed process of this article leads to more counterweights mass, but with a significant reduction in bearing loads, it compensates the weight gain of counterweights using lower journal diameters. As a result, optimization of counterweights configuration is a good alternative to dynamically inappropriate mass unbalancing.

Generalization of the Optimal Configuration

This section introduces a simple formula for determining the appropriate counterweights configuration of any three-cylinder engine. This formula is intended to quickly calculate counterweights configuration without the need for a long process of optimization. The optimization results presented in the previous section are analyzed for extracting this formula by generalizing the gained optimum solution. The optimal configuration should be determined based on the most important parameters. In this regard, sensitivity analysis is performed to achieve these important parameters.

In this section, the resistance of the optimal solution to uncertainty is investigated by sensitivity analysis. After identifying the important parameters, the generalization method is explained for obtaining the mentioned formula. The proposed formula is verified using the three-cylinder engine of Adams/Engine software. The possibility of removing the balance shaft is investigated for the same engine. After that, the results of two software, MATLAB and Adams/Engine, are compared. At last, the counterweights configuration of some of the other industrial three-cylinder engines is presented.

Sensitivity Analysis

Sensitivity analysis investigates how the results change as a function of the range of parameter values. Its application is to test the robustness of an optimal solution in identifying sensitive or important variables. The uncertain parameters in this study are rotating mass, reciprocating mass, crank radius, connecting rod length, axial cylinder distance, and axial distance of cylinder and its counterweights. The partial sensitivity analysis is performed with a percentage change in these parameters from the base up to [+ or -] 15%. By applying this change, for each new value of a parameter, outputs of pitch and yaw moments as well as the first and second bearing loads for the three-cylinder engine 1 with optimum configuration are calculated by keeping the other parameters fixed. Then, the relative variations of the maximum of each output are computed.

Figure 3 depicts the box plots of the relative variation of the maximum pitch and yaw moments as well as the first and second bearing loads, corresponding to the change in each parameter of the three-cylinder engine 1 with optimal counterweights from the base up to [+ or -] 15%. These plots represent the lower, upper, median, and mean values as well as the quartiles. According to Figure 3, the maximum range of the output changes are due to the change in the rotating mass ([m.sub.A]) and crank radius (r). For 15% rotating mass change, the maximum change in pitch and yaw moments as well as the first and second bearing loads are 11%, 37%, 17%, and 17%, respectively. Also, for 15% change in crank radius, the outputs listed change to 14%, 37%, 21%, and 38%, respectively. As a result, the important parameters are the rotating mass and the crank radius which have the greatest effect on the objective functions.

Method of Generalizing Optimal Configuration

This section describes how to generalize the optimum counterweights configuration of the investigated three-cylinder engine 1 to any possible three-cylinder engines. Two important parameters derived from sensitivity analysis (rotating mass and crank radius) have been used in the balancing rate quantity adopted from [9]. Due to the different offset angles of each cylinder, the balancing rate borrowed from [9] is modified as the following form:

[mathematical expression not reproducible] Eq. (12)

where [BF.sub.i,Modified] is the modified balance factor of the crank. In order to maintain the contribution of each counterweight to eliminate vibrations, the balance factor is broken into two distinct parts:

[BF.sub.CW,ij] = [m.sub.CW,ij][R.sub.CW,ij][cos[delta].sub.ij]/[m.sub.A]r Eq. (13)

where [BF.sub.CW,ij] is the counterweight balance factor. Given the symmetry in counterweights configuration of engine 1, balance factors of the first and sixth counterweights, the second and fifth counterweights, as well as the third and fourth counterweights are 73.3%, 47.1%, and 61.2%, respectively. By maintaining the counterweights balance factor equal to the mentioned values and offset angles equal to the values listed in Table 2. the counterweights configuration of each three-cylinder engine is characterized. It must be noted that as the reference engine (engine 1) has a phase of 120 degrees, the offset angles of counterweights for engines with a phase of 240 degrees are opposite sign of them.

Verification of Generalization Method

This section of the article follows the four objectives: validation of generalization method, examination of the possibility of removal of balance shaft, comparison of the results of two software to validate the model used to optimize, and obtaining the counterweights configuration of some of the other three-cylinder engines with simple formula of generalization method.

To verify the method introduced in the previous section, optimization procedure and generalization method are separately performed for the three-cylinder engine of Adams/Engine software. Comparison of their results indicates the reliability of the generalization method. This section compares optimum, generalized, and no counterweights' results with the engine equipped with the best balance shaft to show the feasibility of removing the balance shaft.

Design the Best Balance Shaft For a logical comparison between the balance shaft and optimum counterweights, the best balance shaft for eliminating the primary pitch moment must be designed. First, crankshaft's counterweights with 50% balance factor are considered. Then a primary balance shaft is attached to the cylinder block. Two counterweights are located on the balance shaft, opposite to the first and third cylinders, to eliminate the primary inertia force of them. Using the above method, configuration of the balance shaft counterweights is obtained as follows:

[m.sub.BS,i][R.sub.BS,i] = [0.5m.sub.B]r i = 1,3 Eq. (14)

[[phi].sub.BS,i] = -([pi] + [[phi].sub.i]) i = 1,3 Eq. (15)

where [m.sub.BS,i], [R.sub.BS,i], and [[phi].sub.BS,i] are mass, radius, and phase angle of the balance shaft's counterweight.

Removing Balance Shaft by Generalization Method The optimal counterweights configuration of the Adams/Engine software's three-cylinder engine, the generalized configuration of this engine, and its counterweights configuration when using the best balance shaft are listed in Table 4.

The inertia force, resultant moments, and bearing loads of the three-cylinder engine of Adams/Engine software with no, optimal, and generalized counterweights, and also the same outputs for this engine equipped with the best balance shaft, are shown in Figure 4.

Table 5 demonstrates the reduction in the maximum moments and the maximum bearing loads of the three-cylinder engine of Adams/Engine software with respect to the no counterweight engine for optimal counterweights, generalized configuration, and the same engine equipped with the best balance shaft.

According to Figure 4 and Table 5, two points are analyzed: the reliability of generalization method by comparing the generalized and optimal results, and the possibility of removing balance shaft by comparing the optimum and balance shaft results.

* The results show that the generalized configuration has small excellence in the yaw moment and bearing loads but small weakness in the pitch moment, with respect to optimum configuration. According to Table 4, in terms of weight, the amounts obtained for the generalized counterweights are less than these optimal counterweights values. So, the generalization method is acceptable and can be used to quickly achieve the suitable counterweights configuration of any three-cylinder engine with new specifications.

* The pitch moment reduction of the optimum counterweights and the balance shaft are the same. Although the engine with balance shaft is better in terms of both the yaw moment and a little in bearing loads, the use of optimum counterweights is more suitable because of the undesirable characteristics of a balance shaft, including friction, noise, and reduction in power. It should be noted that the inertia force of the optimum counterweights and the yaw moment of the engine with a balance shaft is smaller than the other, and both of these vibrations can be eliminated by the engine mounts. Therefore, this advantage of the balance shaft can easily be neglected.

Results Comparison with Adams/Engine Software The three-cylinder engine of Adams/Engine software with optimal counterweights is employed to compare results of two software, MATLAB and Adams/Engine. It should be noted that since the engine used by Adams/Engine software was its default three-cylinder engine with slightly modifying, in acceptance that the software was designed properly and we have made the changes carefully, this section is about validating the model. To adapt the models of two software, the effects of gas force, friction, flywheel, and engine block vibrations are removed from Adams/Engine model.

The inertia force, resultant moments, and bearing loads are indicated in Figure 5. Maximum difference of two plots in the inertia force, pitch moment, yaw moment, the first, second, third, and fourth bearing loads are 2.0%, 2.2%, 3.1%, 2.2%, 2.2%, 1.9%, and 2.5%, respectively.

The slight difference observed in Figure 5 between two software models is due to the lumped-mass dynamic model and assumptions used in calculating the bearing loads. The assumption of the lumped-mass model has been considered to simplify modeling and optimization of the problem, while the model is continuous in the Adams/Engine software. Although it was possible to unify the models before the comparison, the validation was made without modifying this item in order to investigate accuracy of this assumption. The negligible difference in the inertia force and moments in the two software shows that it is reasonable to consider the lumped-mass model to reduce the computational time in the optimization problem. In the case of the bearing loads, the difference between the two software is affected by two assumptions: the lumped-mass model and assumptions used in calculating these loads. Therefore, the reason for the slightly larger difference observed compared to the previous outputs is the recent second assumption mentioned. However, the three-cylinder engine models are acceptably coordinated in the two software.

Generalized Configuration of Some Other Engines

In this section, the appropriate counterweights configurations of some industrial engines are computed with the simple generalization formula (Equation 13). not by the long-term optimization method. The purpose of this section is to demonstrate the ability of the introduced simple formula to obtain counterweights configuration of some of the common types of three-cylinder engines, along with a significant reduction in the amounts of vibrational force and moments as well as bearing loads.

The counterweights configuration of four three-cylinder engines (engines 2 to 5 in Table A.1) obtained from generalization method are listed in Table 6.

Table 7 demonstrates the reduction in the maximum moments and the maximum bearing loads of three-cylinder engines with generalized counterweights with respect to the no counterweight engines. The maximum inertia force of all is equal to a small amount of 200 newton.

According to Table 7, although it is possible that the generalization method does not result the best configuration, but quickly gives the suitable arrangement which can reduce forces and moments to an acceptable level. Generalized counterweights configuration as mentioned in Table 6 can remove the balance shaft of engine 3, and also can reduce the vibrations of three other three-cylinder engines. Due to the simple formula of the generalized model in significantly reducing the vibrations of the engines, the formula is expected to be applicable to other three-cylinder engines.

Summary/Conclusions

In this article, the multi-objective optimization of counterweights configuration for dynamic balancing of three-cylinder engines has been studied as an appropriate way to remove their balance shaft or mass unbalancing of the flywheel and pulley. The balance shaft has undesirable characteristics of friction, noise, and reduction in power and mass unbalancing is not proper dynamically. The forces and moments required have been obtained using an equivalent lumped-mass model. The counterweights configuration has been optimized by NSGAII method through minimizing five objective functions as follows: the maximum inertia force of counterweights, the maximum square pitch moment, the maximum square yaw moment, and the maximum of the first and second bearing loads. The accuracy of the model has been confirmed by a maximum difference of 3.1%. The results of the sensitivity analysis showed that the sensitive parameters in the balancing are rotating mass and crank radius. The optimum counterweights configuration of one industrial three-cylinder engine has been generalized based on the concept of balance factor and using the important parameters to other three-cylinder engines. The counterweights configuration introduced in this article with about 90% reduction in the pitch moment has been proposed to reduce the vibration of three-cylinder engines as an appropriate solution for eliminating the balance shaft and mass unbalancing.
Nomenclature

[??] -                  Bearing force "i"
[BF.sub.CW,ij] -        Balance factor of counterweight "j" of crank "i"
[BF.sub.i,Modified] -   Modified balance factor of crank "i"
[??] -                  Inertia force vector
l -                     Connecting rod length
[l.sub.a] -             Distance of connecting rod's CG and crankpin
[l.sub.b] -             Distance of connecting rod's CG and wrist pin
[m.sub.2] -             Crank mass
[m.sub.3] -             Connecting rod mass
[m.sub.4] -             Piston mass
[m.sub.2a] -            Equivalent mass of the crank at crankpin
[m.sub.3a] -            Equivalent mass of the connecting rod at
                        crankpin
[m.sub.3b] -            Equivalent mass of the connecting rod at wrist
                        pin
[m.sub.A] -             Rotating mass
[m.sub.B] -             Reciprocating mass
[m.sub.BS,i] -          Mass of balance shaft's counterweight "i"
[m.sub.CW,ij] -         Mass of counterweight "j" of crank "i"
[M.sub.Pitch] -         Pitch moment
[M.sub.Yaw] -           Yaw moment
r -                     Crank radius
[r.sub.G2] -            Distance of crank's CG and main pin
[R.sub.BS,i] -          Radius of balance shaft's counterweight "i"
[R.sub.CW,ij] -         Radius of counterweight "j" of crank "i"
t -                     Time
[x.sub.i] -             Piston "i" position
[x.sub.i] -             Piston "i" acceleration
[z.sub.f] -             Axial distance of bearing and its farthest
                        cylinder
[z.sub.CW,ij] -         Axial position of counterweight "j" of crank "i"
[z.sub.i] -             Axial position of crank "i"
[z.sub.n] -             Axial distance of bearing and its nearest
                        cylinder
[[delta].sub.ij] -      Offset angle of counterweight "j" of crank "i"
[[DELTA]z.sub.mains] -  Axial main bearing distance
[[theta].sub.i] -       Angular position of crank "i"
[[phi].sub.BS,i] -      Phase angle of balance shaft's counterweight "i"
[[phi].sub.i] -         Phase angle of crank "i"
[omega] -               Angular velocity of crankshaft


Contact Information

Abdolreza Ohadi, Professor

President of Iranian Society of Acoustics and Vibration, Faculty of Mechanical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran.

Acknowledgments

The authors express their thanks to IranKhodro Powertrain Company, Engine Research Center (IPCO), 6th km of Karaj Makhsous Road, Tehran, Iran, for providing support and data during this study and High Performance Computing Research Center, Amirkabir University of Technology, 424 Hafez, Tehran, Iran.

The author and publisher would like to acknowledge that this article is based on an oral-only presentation at ISAV 2016: 6th International Conference on Acoustics and vibration, Tehran, Iran, December 7-8, 2016.
Definitions/Abbreviations

CG -        Center of gravity
CW -        Counterweight
NSGAII -    Second version of non-dominated sorting
            genetic algorithm


References

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Appendix A: Required Parameters of Three-Cylinder Engines

Required parameters for some of the three-cylinder engines are given in Table A.1.

Somaye Mohammadi and Abdolreza Ohadi, Amirkabir University of Technology, Islamic Republic of Iran

Reza Keshavarz, IranKhodro Powertrain Company (IPCO), Islamic Republic of Iran

History

Received: 20 Mar 2018

Revised: 05 Jul 2018

Accepted: 23 Aug 2018

e-Available: 18 Oct 2018

doi:10.4271/03-11-05-0038
TABLE 1 Weights assigned to objective functions.

Objective function            Weight

Inertia force and yaw moment  0.1
Pitch moment                  0.4
The first and second bearing  0.2
loads
Sum of weights                1

TABLE 2 The counterweights configuration of the three-cylinder engine 1.

Parameter                                  Unit  Optimum CWs  Default
                                                              CWs

[(m*R).sub.CW,11] = [(m*R).sub.CW,32]      gxmm  28,257       24,257
[(m*R).sub.CW,12] = [(m*R).sub.CW,31]      gxmm  18,453       14,813
[(m*R).sub.CW,21] = [(m*R).sub.CW,22]      gxmm  25,187        5,982
[(m*R).sub.CW,f]                           gxmm       -          312.7
[(m*R).sub.CW,r]                           g-mm       -        3,584
[[delta].sub.CW,11] = [[delta].sub.CW,32]  deg.      -6.3        -27.6
[[delta].sub.CW,12] = [[delta].sub.CW,31]  deg.      12.0        -18
[[delta].sub.CW,21] = [[delta].sub.CW,22]  deg.      21.4          0
[[delta].sub.CW,f]                         deg.       -          155
[[delta].sub.CW,r]                         deg.       -          -25

TABLE 3 The reduction in the maximum and average of the moments and
loads of the three-cylinder engine 1 with respect to the no
counterweight engine for the optimal and default counterweights
configuration.

                  Reduction in the   Reduction in the
                  maximum            average
                  Optimum   Default  Optimum  Default
Parameter         CWs (%)   CWs (%)  CWs (%)  CWs (%)

Pitch moment      92.0      91.1     92.1     92.1
Yaw moment        64.7      65.3     64.7     65.3
1st bearing load  75.5      46.9     80.4     54.5
2nd bearing load  78.5      52.5     84.1     61.4
3rd bearing load  78.5      52.5     84.1     61.4
4th bearing load  75.5      42.3     80.4     48.1

TABLE 4 The counterweights configuration of the three-cylinder engine
of Adams/Engine software.

                              Optimum   Generalized  With balance
Parameter               Unit  CWs       CWs          shaft

[(m*R).sub.CW,11/32]    gxmm  26,006     25,312      21,359
[(m*R).sub.CW,12/31]    gxmm  24,213     16,530      21,359
[(m*R).sub.CW,21/22]    gxmm  29,922     22,561      21,359
[[delta].sub.CW,11/32]  deg.      11.5        6.3         0
[[delta].sub.CW,12/31]  deg.     -24.6      -12.2         0
[[delta].sub.CW,21/22]  deg.     -19.3      -21.4         0

TABLE 5 The reduction in the maximum moments and loads of the
three-cylinder engine of Adams/Engine software with respect to no
counterweights engine for optimal counterweights, generalized
configuration, and the same engine equipped with the best balance
shaft.

               Optimum CWs  Generalized  With balance
Parameter      (%)          CWs (%)      shaft (%)

Pitch moment   88.6         82.8          89.5
Yaw moment     53.1         65.1         100
1st and 4th    72.0         75.4          75.7
bearing loads
2nd and 3rd    69.6         76.4          75.7
bearing loads

TABLE 6 The generalized optimal counterweights configuration of five
three-cylinder engines.

Parameter               Unit  Engine 2  Engine 3  Engine 4  Engine 5

[(m*R).sub.CW,11/32]    gxmm  29,069    30,814    26,931    28,017
[(m*R).sub.CW,12/31]    gxmm  18,983    20,123    17,587    18,296
[(m*R).sub.CW,21/22]    gxmm  25,910    27,465    24,004    24,973
[[delta].sub.CW,11/32]  deg.      -6.3      -6.3      -6.3      -6.3
[[delta].sub.CW,12/31]  deg.      12.0      12.0      12.0      12.0
[[delta].sub.CW,21/22]  deg.      21.4      21.4      21.4      21.4

TABLE 7 The reduction in the maximum moments and loads of the
three-cylinder engines with generalized optimal counterweights with
respect to the no counterweight engines.

                           Engine  Engine  Engine  Engine
Parameter                  2 (%)   3 (%)   4 (%)   5 (%)

Pitch moment               88.3    91.2    90.6    91.6
Yaw moment                 65.1    65.1    66.3    66.1
1st and 4th bearing loads  76.1    75.4    76.2    76.2
2nd and 3rd bearing loads  78.6    78.5    79.4    79.6

TABLE A.1 Required parameters of three-cylinder engines: five
industrial engines and default engine of the Adams/Engine software.

Parameter            Symbol      Unit  Engine 1        Engine 2

Phase angle          [phi]       deg.   120             120
Crank mass           [m.sub.2]   g     1168.8          1497.6
Connecting rod       [m.sub.3]   g      387.4           413.0
mass
Piston mass          [m.sub.4]   g      206.8           248.0
Crank radius         r           mm      40.9            40.5
Connecting rod       l           mm     137.4           122.5
length
Distance of crank's  [r.sub.G2]  mm      23.8            18.9
CG and main pin
Axial cylinder       -           mm      78              79
distance
Axial distance of    -           mm      19.5            19.5
cylinder and its
counterweight
Balance method                         CWs + mass        CWs
                                       unbalancing of
                                       flywheel and
                                       pulley

Parameter            Engine 3       Engine 4  Engine 5  Adams/Engine

Phase angle           120            120       120       240
Crank mass           1216.1         1239.8    1165.7    1283.9
Connecting rod        327.1          290.1     339.3     402.3
mass
Piston mass           190.3          163.7     170.1     290.4
Crank radius           45.2           41.9      41.9      39.5
Connecting rod        145.9          130.7     141.8     124.0
length
Distance of crank's    26.3           22.9      24.5      18.5
CG and main pin
Axial cylinder         81             78.5      77        76
distance
Axial distance of      20             18.8      18.5      18.5
cylinder and its
counterweight
Balance method       CWs + balance    CWs      CWs      CWs + balance
                     shaft                              shaft
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Author:Mohammadi, Somaye; Ohadi, Abdolreza; Keshavarz, Reza
Publication:SAE International Journal of Engines
Date:Dec 1, 2018
Words:7413
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