# Vibration Suppression of a Single-Cylinder Engine by Means of Multi-objective Evolutionary Optimisation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Single-Cylinder Engine Model

#### 2.1. Kinematic and Kinetic Analyses

_{2}and θ

_{3}are the angular positions of links 2 and 3, respectively, while x is the position vector of the piston. Given that θ

_{2}, ${\dot{\theta}}_{2}$, and ${\ddot{\theta}}_{2}$ are known input variables, we can have the relation:

_{2}, ${\dot{\theta}}_{2}$, and ${\ddot{\theta}}_{2}$, the angular position, velocity, and acceleration of link 3, as well as the position, velocity, and acceleration of piston 4, can be computed using Equations (1)–(4).

_{2}is applied at link 2, so as to meet its prescribed motion. The force analysis can be computed using the following system of equations:

**A**]{

**F**} = {

**RHS**}

**F**= {F

_{12,x}F

_{12,y}F

_{32,x}F

_{32,y}F

_{43,x}F

_{43,y}F

_{14,y}M

_{2}}

^{T}

_{ij}is the constrained force acting at body i by body j, m

_{i}is the mass of body i, I

_{Gi}is the moment of inertia with respect to the axis at the centroid of body i,

**r**

_{i}

_{/j}is the relative position vector of point i with respect to point j, and

**a**

_{C}and

**a**

_{G}

_{2}are the acceleration vector of link 4 (piston) and the centre of gravity i in the x–y coordinates, respectively. The gas pressure P (kPa) in one cycle for some engine has been proposed by Asadi et al. [8] as follows:

_{P}due to gas pressure can be computed by:

_{P}= (P − P

_{atm})A

_{p}

_{atm}is atmosphere pressure (kPa) and A

_{p}is the piston area (m

^{2}).

#### 2.2. Engine Vibration System

**r**

_{0}is the position of the unstretched spring,

**r**is the position vector of the spring under the force

**F**, and δ

**r**is a spring translational vector.

**r**

_{i}is the position vector of spring I,

**r**

_{c}is the position vector of the mass centre, and

**r**

_{ci}is the potion vector of spring i with respect to the centroid.

**θ**is the vector of rotation displacements of the body. The translation and rotation vectors can be defined as:

_{i}is the translation in i-th direction and θ

_{i}is the angular displacement in the i-th axis. The rigid body has six degrees of freedom, as shown in Figure 5. By substituting Equation (12) into Equation (11), we have:

**T**

_{i}is called a transformation matrix for the i-th spring and

**d**is the displacement vector of the body. As a result, elastic potential energy of the i-th spring is:

**K**is the stiffness matrix of the system. The kinetic energy or the work due to inertial forces can be computed as:

**I**is the matrix of moments of inertia. Adding the work done by external forces to the system, a vibration model of a three-dimensional (3D) spring-mass system can be expressed as:

**C**= α

**M**+ β

**K**

## 3. Hybrid RPBIL-DE for Multi-Objective Optimisation

_{c}is a crossover probability, and CR is the probability of selecting an element of an offspring

**c**in binomial crossover.

Algorithm 1. Multi-objective RPBIL-DE [27]. |

Input: N_{G}(number of generation), N_{P} (population size), n_{I}(number of subinterval), N_{T}(number of trays), objective function name (fun), Pareto archive size (N_{A})Output: x^{best}, f^{best}Initialisation: P _{ij}= 1/n_{I} for each tray, where P_{ij} is a probability matrixMain steps : Generate a real-code population X from the probability trays and find f = fun(X): Find a Pareto archive A1: For i = 1 to N _{G}2: Separate the non-dominated solutions into N _{T} groups using a clustering technique, and find the centroid r_{G} of each group3: Update each tray P _{ij} based on r_{G}4: Generate a real-code population X from the probability trays5: For j = 1 to N _{P} recombine X and A using DE operators5.1: Select p from A randomly5.2: Select q and r from X randomly, q ≠ r5.3: Calculate c = p + F(q −r) (DE/best/1/bin)5.4: Set c _{i} into its bound constraints.5.5: If rand <p _{c}, perform crossover5.5.1: For k = 1 to n 5.5.2: If rand <CR, y _{k} = c_{k}5.5.3: Otherwise, y _{j,k} = p_{k}5.5.4: End 6: End 7: New real-code population is Y = { y_{1}, …, y_{j}, …, y_{NP}} and find f = fun(Y)8: Find non-dominated solutions from Y∪A and replace the members in A with these solutions9: If the number of archive members is larger than N , remove some of the members using a clustering technique_{A}10: End |

## 4. Design Problems

_{C}, l

_{P}, R

_{C}

_{1}, R

_{C}

_{2}, R

_{2}, r

_{C}, and ψ. If the values of those parameters are known, the mass centre and moment of inertia of the crank can be calculated. Figure 8 shows the connecting rod where nine design parameters are used to define the shape and dimensions of the rod as l

_{1}, l

_{2}, b

_{1}, b

_{2}, R

_{1}, R

_{2}, t, r

_{p1}, and r

_{p2}. It should be noted that the crank and rod are created for design demonstration in this paper. For practical applications, their shapes may be defined differently. From Figure 7 and Figure 8, l

_{P}= l

_{1}, and R

_{2}= R

_{C}

_{2}, so 14 parameters are assigned as elements of a design vector as

**x**= {R

_{C}

_{1}, R

_{C}

_{2}, r

_{C}, R

_{2}, ψ, l

_{P}, t

_{C}, R

_{1}, r

_{p1}, r

_{p2}, l

_{2}, t, b

_{1}, and b

_{2}}

^{T}.

**x**such that:

**f**= {f

_{1}(

**x**),f

_{2}(

**x**)}

^{T}

_{Crank}≤ σ

_{all}

_{Rod}≤ σ

_{all}

_{Rod}≥ 1

_{p1}≥ r

_{p2}+ 0.005 m

_{2}≥ b

_{1}+ 0.005 m

_{1}≥ b

_{2}+ 0.02 m

_{p1}≥ r

_{p2}+ 0.005 m

_{1}≥ l

_{2}

_{1}≥ r

_{p1}+ 0.002 m

_{C}

_{1}≥ R

_{2}+ 0.002 m

_{C}

_{1}≥ r

_{c}

_{1}+ 0.002 m

_{1}≥ R

_{2}+ 0.002 m

**x**

^{l}≤

**x**≤

**x**

^{u}

_{Crank}is the maximum stress on the crank, σ

_{all}is an allowable stress, and σ

_{Rod}is the maximum stress on the connecting rod. The bound constraints are set as

**x**

^{l}= {0.03, 0.05, 0.015, 0.01, π/6, 0.03, 0.01, 0.03, 0.02, 0.01, 0.02, 0.002, 0.02, and 0.01}

^{T}, and

**x**

^{u}= {0.045, 0.09, 0.04, 0.03, π, 0.05, 0.03, 0.05, 0.03, 0.03, 0.04, 0.005, 0.04, and 0.03}

^{T}. The buckling factor for the rod λ

_{Rod}is defined as the ratio of critical load to applied load. The first three design constraints are set for structural safety, while the other constraints are assigned for manufacturing tolerances and practicality. The objective functions used in this study are set as

**f**= {u

_{rms}+ θ

_{rms}, mass}

^{T}. The root mean squares (RMS) of the vibration translations (u

_{rms}) and rotations (θ

_{rms}) over the period t∈ [0, t

_{max}] can be computed as:

**x**being decoded, the shape and sizing parameters are repaired to meet constraints 4–12, and the inertial properties of the crank and rod can then be computed (the rest of constraints will be handled by using the non-dominated sorting scheme [35]. Then kinematic and dynamic force analyses are carried out as detailed in Section 2. A simple finite element model using a three-dimensional (3D) beam element is applied to determine the maximum stresses on the crank and rod. A buckling factor is also calculated in the cases of the rod. Also, the obtained dynamic forces are used as external excitation for the vibration model of the engine. Having obtained a dynamic response, the objective functions can then be computed.

- OPT1: min {u
_{rms}+ θ_{rms}, mass}, constant crank angular speed 1000 rpm - OPT2: min {u
_{rms}+ θ_{rms}, mass}, constant crank angular speed 1500 rpm - OPT3: min {u
_{rms}+ θ_{rms}, mass}, constant crank angular speed 2000 rpm

_{yt}= 417.1 MPa, and density of 7850 kg/m

^{3}. For each finite element analysis, the maximum compressive force over the period of time [0, t

_{max}] will be used for buckling calculation.

## 5. Pressure Force and Inertia Force Validation

_{max}], as shown in Figure 9, Figure 10 and Figure 11 at 1000 rpm. The maximum gas pressure force exerted on the piston head occurred at the maximum torque, but the maximum tensile force occurred during the maximum revolution speed [8]. Figure 9 shows that the maximum gas pressure force is 22,374 N, which occurs in the combustion process. The inertial force due to the slider–crank mechanism in the x direction is show in Figure 10; meanwhile, the maximum inertia in positive direction is 1141N, while the negative inertia force is 2286 N. Figure 11 shows the total force due to the gas pressure force and inertia force that give the maximum gas pressure force as 20,364 N, while the maximum tensile force is 2867 N. All of the diagrams indicate similar trends to the work by Reference [8], while the magnitude of all of the forces are different, as a result of the differences in the system parameters.

## 6. Design Results

## 7. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 18.**The components of the translational displacements of the six engines in Figure 15.

**Figure 19.**The components of the rotational displacements of the six engines in Figure 15.

**Figure 20.**The components of the translational displacements of the six engines in Figure 16.

**Figure 21.**The components of the rotational displacements of the six engines in Figure 16.

**Figure 22.**The components of the translational displacements of the six engines in Figure 17.

**Figure 23.**The components of the rotational displacements of the six engines in Figure 17.

Parameters | Symbols | Quantities |
---|---|---|

Total engine mass | m | 14.528 kg |

Piston mass | m_{4} | 0.2 kg |

Moment of inertia | I_{xx}, I_{yy}, I_{zz}, I_{xy}, I_{xz}, I_{yz} | 0.0768, 0.0640, 0.0812, 0, 0, 0 kg-m^{2} |

Centre of gravity | R_{G} | [0,0,0]^{T} m |

Crank shaft centre | R_{O/G} | [−0.760, −0.0232, 0.0100]^{T} m |

Mount stiffness | k | 4 × 10^{6} N/m |

Crank length | R | 0.1 m |

Connecting rod length | L | 0.3 m |

Material density | ρ | 7850 kg/m^{3} |

Piston diameter | d | 100 mm |

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**MDPI and ACS Style**

Sleesongsom, S.; Bureerat, S.
Vibration Suppression of a Single-Cylinder Engine by Means of Multi-objective Evolutionary Optimisation. *Sustainability* **2018**, *10*, 2067.
https://doi.org/10.3390/su10062067

**AMA Style**

Sleesongsom S, Bureerat S.
Vibration Suppression of a Single-Cylinder Engine by Means of Multi-objective Evolutionary Optimisation. *Sustainability*. 2018; 10(6):2067.
https://doi.org/10.3390/su10062067

**Chicago/Turabian Style**

Sleesongsom, Suwin, and Sujin Bureerat.
2018. "Vibration Suppression of a Single-Cylinder Engine by Means of Multi-objective Evolutionary Optimisation" *Sustainability* 10, no. 6: 2067.
https://doi.org/10.3390/su10062067