# A Kriging Model for Dynamics of Mechanical Systems With Revolute Joint Clearances

## Summary (2 min read)

### 1 Introduction

- In the past decade, many researchers have examined the optimal dynamical solution of different mechanical systems and mechanisms [1–3].
- Over the past decades, advances, mainly due to the development of intercross applications between computeraided analysis of mechanical systems and optimization methodologies, have been achieved.
- Erkaya and Uzmay studied the effects of joint clearances on the performance of a mechanism in terms of path generation and transmission angle, using neural networks and genetic algorithms (GAs), respectively [3,21].

### 2 Modeling Revolute Joints With Clearance

- In reality, there is a clearance between the journal and the bearing in mechanical systems in order to cause a relative motion between the two.
- The penetration between the bearing and journal appears when they are in contact.
- The Hertz law given by Eq. (4) does not include any energy dissipation.

### 3 A Multibody System With Joint Clearance

- A computer model for the classic slider-crank mechanism with one revolute clearance joint is considered in order to analyze the dynamic behavior of the mechanical system.
- Figure 2 shows the configuration of the slider-crank mechanism, which comprises four bodies that represent the crank, connecting rod, slider, and ground.
- The multibody model has only one clearance joint.
- The geometric and inertia properties of each body in this system are shown in Table 1 [17].
- The initial crank angle and velocity of the journal.

### 4 Kriging Model-Based Optimization

- This section presents a procedure for constructing an objective function using the Kriging model.
- Hypercube is introduced for acquiring the initial design points.
- For this purpose, the Kriging model can be constructed based on the initial design points and their performances.
- In addition, these two parameters can be determined by maximizing the likelihood function and, therefore, the correlation matrix R can be calculated.
- Then the initial population is changed depending on the fitness function and, using a crossover strategy and mutation, a new generation of population is created.

### 5 DOE- and Kriging-Based Study of Slider-Crank Mechanism

- The dynamic behavior of the slider-crank mechanical system with a revolute clearance joint will be studied further, using two simple examples that use the DOE method introduced in the previous sections.
- Both the computer simulation model and Kriging meta-model are used to examine the highest absolute value of the slider acceleration of the slider-crank mechanism.
- The Kriging model is established as a prediction model to estimate the objective function for any given design point and it is Table 5 General experiment objects in example 2 Design variables 1.

### 6 Conclusion

- The influence of the dynamic behavior of a multibody mechanical system with a revolute clearance joint was investigated in this study.
- For the studied mechanism, the predictions were shown to be within 5% of the actual values from dynamic simulations, for which close to an hour of computational time is to be spent for each simulation.
- The global results obtained from this study indicate that the dynamic behavior of the mechanical system with clearance is quite sensitive to the crank speed and clearance size.
- The contact force significantly increases with the increase in the clearance size and, as the clearance size decreases, the dynamic behavior tends to be close to the ideal situation.
- By utilizing the Kriging meta-model, the computer simulation time can be significantly reduced, while the response of the system can be studied and optimized for a range of input design variables.

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### Cites methods from "A Kriging Model for Dynamics of Mec..."

...[458] proposed a design-of-experiment (DOE)-based method for optimizing the performance a slider-crank mechanism with a revolute joint clearance between the connecting rod and slider....

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##### References

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### "A Kriging Model for Dynamics of Mec..." refers methods in this paper

...In order to explain the sampling method and the Kriging model, a mathematical model, which is shown as the Branin function, is considered in term of two variables x1 and x2 as [27]...

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### "A Kriging Model for Dynamics of Mec..." refers background in this paper

...Lankarani and Nikravesh [10] extended the Hertz model to include a hysteresis damping function as follows:...

[...]

...The contact-impact force FN, in relation to the penetration indentation, can be modeled by the Hertz law as [10]...

[...]

...Lankarani and Nikravesh extended the Hertz contact law to include a hysteresis damping function and represent the dissipated energy during impact [10]....

[...]

...Lankarani and Nikravesh [10] extended the Hertz model to include a hysteresis damping function as follows: FN ¼ Kdn 1þ 3ð1 c2eÞ 4 _d _dð Þ " # (7) where the stiffness coefficient K can be obtained from Eqs....

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##### Frequently Asked Questions (2)

###### Q2. What are the future works mentioned in the paper "A kriging model for dynamics of mechanical systems with revolute joint clearances" ?

A computer-aided analysis of the mechanical system and the framework of the DOE modeling were presented to study the effect of the joint clearance size, input crank speed, and material/ contact stiffness coefficient on the dynamic response of a multibody system with one clearance joint. The method presented in this paper can be utilized for optimizing the performance of mechanical systems with joint clearances. By utilizing the Kriging meta-model, the computer simulation time can be significantly reduced, while the response of the system can be studied and optimized for a range of input design variables.