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Journal ArticleDOI

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

01 Jul 2014-Journal of Computational and Nonlinear Dynamics (American Society of Mechanical Engineers)-Vol. 9, Iss: 3, pp 031013
TL;DR: In this paper, the analysis of revolute joint clearance is formulated in terms of a Hertzian-based contact force model and a polynomial function Kriging meta-model is established instead of the actual simulation model.
Abstract: Over the past two decades, extensive work has been conducted on the dynamic effect of joint clearances in multibody mechanical systems. In contrast, little work has been devoted to optimizing the performance of these systems. In this study, the analysis of revolute joint clearance is formulated in terms of a Hertzian-based contact force model. For illustration, the classical slider-crank mechanism with a revolute clearance joint at the piston pin is presented and a simulation model is developed using the analysis/design software MSC.ADAMS. The clearance is modeled as a pin-in-a-hole surface-to-surface dry contact, with an appropriate contact force model between the joint and bearing surfaces. Different simulations are performed to demonstrate the influence of the joint clearance size and the input crank speed on the dynamic behavior of the system with the joint clearance. In the modeling and simulation of the experimental setup and in the followed parametric study with a slightly revised system, both the Hertzian normal contact force model and a Coulomb-type friction force model were utilized. The kinetic coefficient of friction was chosen as constant throughout the study. An innovative design-of-experiment (DOE)-based method for optimizing the performance of a mechanical system with the revolute joint clearance for different ranges of design parameters is then proposed. Based on the simulation model results from sample points, which are selected by a Latin hypercube sampling (LHS) method, a polynomial function Kriging meta-model is established instead of the actual simulation model. The reason for the development and use of the meta-model is to bypass computationally intensive simulations of a computer model for different design parameter values in place of a more efficient and cost-effective mathematical model. Finally, numerical results obtained from two application examples with different design parameters, including the joint clearance size, crank speed, and contact stiffness, are presented for the further analysis of the dynamics of the revolute clearance joint in a mechanical system. This allows for predicting the influence of design parameter changes, in order to minimize contact forces, accelerations, and power requirements due to the existence of joint clearance.

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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Zhenhua Zhang
Department of Mechanical Engineering,
Wichita State University,
Wichita, KS 67260
e-mail: zxzhang1@wichita.edu
Liang Xu
Department of Mechanical Engineering,
Wichita State University,
Wichita, KS 67260
e-mail: lxxu3@wichita.edu
Paulo Flores
Departamento de Engenharia Mec
^
anica,
Universidade do Minho,
Campus de Azur
em,
4800-058 Guimar
~
aes, Portugal
e-mail: pflores@dem.uminho.pt
Hamid M. Lankarani
Department of Mechanical Engineering,
Wichita State University,
Wichita, KS 67260
e-mail: hamid.lankarani@wichita.edu
A Kriging Model for Dynamics
of Mechanical Systems With
Revolute Joint Clearances
Over the past two decades, extensive work has been conducted on the dynamic effect of
joint clearances in multibody mechanical systems. In contrast, little work has been
devoted to optimizing the performance of these systems. In this study, the analysis of rev-
olute joint clearance is formulated in terms of a Hertzian-based contact force model. For
illustration, the classical slider-crank mechanism with a revolute clearance joint at the
piston pin is presented and a simulation model is developed using the analysis/design
software
MSC.ADAMS. The clearance is modeled as a pin-in-a-hole surface-to-surface dry
contact, with an appropriate contact force model between the joint and bearing surfaces.
Different simulations are performed to demonstrate the influence of the joint clearance
size and the input crank speed on the dynamic behavior of the system with the joint clear-
ance. In the modeling and simulation of the experimental setup and in the followed para-
metric study with a slightly revised system, both the Hertzian normal contact force model
and a Coulomb-type friction force model were utilized. The kinetic coefficient of friction
was chosen as constant throughout the study. An innovative design-of-experiment
(DOE)-based method for optimizing the performance of a mechanical system with the
revolute joint clearance for different ranges of design parameters is then proposed. Based
on the simulation model results from sample points, which are selected by a Latin hyper-
cube sampling (LHS) method, a polynomial function Kriging meta-model is established
instead of the actual simulation model. The reason for the development and use of the
meta-model is to bypass computationally intensive simulations of a computer model for
different design parameter values in place of a more efficient and cost-effective mathe-
matical model. Finally, numerical results obtained from two application examples with
different design parameters, including the joint clearance size, crank speed, and contact
stiffness, are presented for the further analysis of the dynamics of the revolute clearance
joint in a mechanical system. This allows for predicting the influence of design parameter
changes, in order to minimize contact forces, accelerations, and power requirements due
to the existence of joint clearance. [DOI: 10.1115/1.4026233]
Keywords: Revolute joint clearance, contact forces, multibody dynamics, Kriging meta-
model, genetic algorithms
1 Introduction
In the past decade, many researchers have examined the opti-
mal dynamical solution of different mechanical systems and
mechanisms [13]. Additionally, different optimization methods
have been implemented to obtain optimal solutions. In the study
by Laribi et al. [2], a solution for the path generation problem in
mechanisms was presented using the generic algorithm-fuzzy
logic method. Selcuk et al. [3] proposed a neural-genetic method
to investigate the effects of joints with clearance on its path gener-
ation and kinematic transmission quality. In order to reduce the
computational complexity, the neural network has been used as a
surrogate model in this study. The Genetic algorithm, as a global
optimization method, has been widely used in many research
fields, but its associated computational cost dramatically
increases, especially for expensive model functions.
As a result of manufacturing tolerances, material deformations,
and wear after a certain working period, clearances between me-
chanical components of mechanical systems occur in most kine-
matic joints. Excessive clearance values result in large contact
forces at the joints, especially during high-speed mechanical oper-
ations. The presence of clearances leads to a decrease in the sys-
tem reliability and durability of the system’s components and
machines [4,5]. Over the past decades, advances, mainly due to
the development of intercross applications between computer-
aided analysis of mechanical systems and optimization methodol-
ogies, have been achieved. These results could be utilized for the
application of different mathematical programming techniques to
the parametrical and topological syntheses and analyses of me-
chanical systems [6]. The optimization of mechanical system
modeling with clearances can be used to bypass the computation-
ally intensive simulation of the computer dynamic model. It also
helps in the analysis, design, and control of the dynamic perform-
ance of a complex mechanical system and in quantifying the influ-
ence of clearance parameters.
During the past two decades, many studies on the influence of
the joint clearance in planar and spatial multibody mechanical
systems have been conducted. Dubowsky and Freudenstein devel-
oped the impact ring model, which is a simple model to demon-
strate the effects of joint clearance in planar mechanisms [7].
Springs and dashpots were arranged in their model to predict the
dynamics response of the mechanical system. Dubowsky and
Moening quantified the interaction between the clearance joints
and the mechanical system elasticity using a Scotch–Yoke simula-
tion model [8]. Large impact forces developed at the clearance
joints caused a failure in the Scotch–Yoke model. Furubashi and
Morita presented a four-bar mechanism with multiple clearance
revolute joints [9]. They analyzed and compared the results for
Contributed by the Design Engineering Division of ASME for publication in the
J
OURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August
1, 2013; final manuscript received December 10, 2013; published online February
13, 2014. Assoc. Editor: Ahmet S. Yigit.
Journal of Computational and Nonlinear Dynamics JULY 2014, Vol. 9 / 031013-1
Copyright
V
C
2014 by ASME
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different numbers and various combinations of clearance joints
and demonstrated the effect of clearances on the performance of
the four-bar mechanism system.
Lankarani and Nikravesh extended the Hertz contact law to
include a hysteresis damping function and represent the dissipated
energy during impact [10]. A nonlinear continuous force acted on
the model and the local indentation and relative penetration veloc-
ity was related to the contact force. Flores and his coworkers
developed a precise model for the dynamic analysis of a mechani-
cal system with dry and lubricated revolute joints [1113]. The
influences of the selected parameters on the dynamic response of
mechanical systems with multiple clearance joints, including the
clearance size, input crank speed, and number of joints modeled
as clearance joints, were quantified in this study.
Mahrus designed a set of experimental investigations to show
the performance of the journal bearing and the effect of the load
diagram on hydrodynamic lubrication [14]. Different loads were
applied to the test journal-bearing joint and both steady and vary-
ing unidirectional and full two-component dynamic loading were
considered in the study. Wilson and Fawcett modeled a slider-
crank mechanism with a clearance in the sliding bearing to mea-
sure the transverse motion of the slider [15]. They tested a number
of parameters such as the geometry, speed, and mass distribution
of the mechanical system, which influence the transverse motion
and they derived the equation of motion with these parameters
based on the results. Haines derived the equations of motion for a
multibody mechanical system that describes the contributions at a
revolute clearance joint with no lubrication present [16]. The
study also included an experimental investigation on the dynamic
response of revolute clearance joints. Under static loading, the
deflection associated with contact elasticity in the dry journal-
bearing joint was found to be much greater and linear than pre-
dicted [17]. Bengisu et al. developed a four-bar mechanism based
on zero-clearance analysis to compare the theoretical results with
the experimental results [18]. A model with multiple joints was
used in clearances to study contact energy loss in the mechanical
system.
Feng et al. developed a method for optimizing the mass distri-
bution of planar linkage with clearance joints to control the
change of inertia forces [19]. Tsai and Lai investigated the kine-
matic sensitivity of the transmission performance of linkages with
joint clearances [20]. In their study, loop-closure equations were
used in the position analysis of a four-bar mechanism in which all
joints have clearances. Yildirim et al. predicted the transmission
angle of a slider-crank mechanism with an eccentric connector
based on neural networks [21]. The neural network structure was
a feed-forward network and the best approximation was obtained
with five types of algorithms. 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 net-
works and genetic algorithms (GAs), respectively [3,21].
A computer-aided analysis of multibody mechanical systems is
utilized in this study as a simulation model. The goal of this study
is to use the Kriging mathematical model as a design-of-experi-
ments optimization tool, in order to demonstrate the influence of
the design variables on the dynamic performance of mechanical
systems with revolute clearance joints. The reason the Kriging
model was used in this research is that the computer simulation
for a given set of design parameters is usually quite computation-
ally intensive and each simulation for a given set of design varia-
bles could take extensive computation time. Because there are
wide ranges of values in the design variables such as the clearance
sizes, ratios of length, material properties, contact stiffnesses, and
speeds of operation, studying the effects of each of these variables
would take enormous computational time and effort.
In the present study, the mathematical formulation of the revo-
lute clearance joint is fully described and the relationship between
the design parameters and contact forces in joints is examined.
First, the classical slider-crank mechanical system is modeled and
simulated in
MSC.ADAMS and the performance of the system with
different sets of parameters is examined. Next, the theoretical ba-
sis of the methods are stated, illustrating the framework for the
DOE methods of the Latin hypercube sampling, Kriging meta-
model, and genetic algorithm. Next, two simple examples are pre-
sented using these previous methods to further expand the analysis
of the dynamic behavior of the mechanism with the revolute clear-
ance joint at different ranges of the design parameters.
2 Modeling Revolute Joints With Clearance
A revolute joint with clearance, as shown in Fig. 1, can be
described as a movable journal assembled inside a bearing, with
the journal’s and bearing’s radii of R
J
and R
B
, respectively. 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 journal can move inside the bearing and this will add
some degrees of freedom to the system. The difference between
the radius of the bearing and the radius of the journal is the radial
clearance size c. The penetration between the bearing and journal
appears when they are in contact.
The indentation depth due to the contact impact between the
journal and the bearing can be defined as
d ¼ e c (1)
where e is the magnitude of the eccentricity and c is the radial
clearance. The eccentricity is evaluated as
e ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
DX
2
þ DY
2
p
(2)
where DX and DY are the horizontal and vertical displacements,
respectively, measured from the state when the centers of the
Fig. 1 Revolute joint with clearance (clearance exaggerated for clarity)
031013-2 / Vol. 9, JULY 2014 Transactions of the ASME
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journal and the bearing coincide. In turn, the radial clearance is
defined as
c ¼ R
B
R
J
(3)
Two situations can occur at the joint. In the first case, when the
journal does not make contact with the bearing and the penetration
depth is a negative value, the journal has a free-flight motion
inside the bearing and, thus, there is no contact-impact force
developed at the joint. In the second case, when the journal con-
tacts with the bearing wall, a contact force between the journal
and the bearing is developed in the direction of the centers of the
bearing and the journal [4] and the indentation depth value will be
greater than zero.
The contact-impact force F
N
, in relation to the penetration in-
dentation, can be modeled by the Hertz law as [10]
F
N
¼ Kd
n
(4)
where K is the stiffness coefficient and d is the indentation depth
given by Eq. (1). The exponent n is usually set for analysis in the
range of 1.5–2.5 for most metal-to-metal contact. The stiffness
coefficient K depends on the material properties and the contact-
ing surface and is defined as
K ¼
4
3ðr
B
þ r
J
Þ
R
B
R
J
R
B
R
J

1=2
(5)
The material parameters r
B
and r
J
are defined as
r
k
¼
1
2
k
E
k
k ¼ B; JðÞ (6)
where the variables
k
and E
k
are Poisson’s ratio and Young’s
modulus, respectively, for the journal and the bearing.
The Hertz law given by Eq. (4) does not include any energy dis-
sipation. Lankarani and Nikravesh [10] extended the Hertz model
to include a hysteresis damping function as follows:
F
N
¼ Kd
n
1 þ
3ð1 c
2
e
Þ
4
_
d
_
d
ðÞ
"#
(7)
where the stiffness coefficient K can be obtained from Eqs. (5)
and (6)], c
e
is the restitution coefficient,
_
d is the relative penetra-
tion velocity, and
_
d
ðÞ
is the initial impact velocity.
3 A Multibody System With Joint Clearance
In this section, 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. In this case, the multibody model has only
one clearance joint. There are four joints: two ideal revolute joints
between the ground and the crank and the crank and the connect-
ing rod; one ideal translational joint between the slider and
ground; and one nonideal revolute joint clearance between the
connecting rod and slider. The geometric and inertia properties of
each body in this system are shown in Table 1 [17]. The moment
of inertia is taken with respect to the center of gravity of the body.
A model of the slider-crank mechanism is constructed in
MSC.ADAMS, as shown in Fig. 3. In the model, all bodies are consid-
ered to be rigid. The initial crank angle and velocity of the journal
Fig. 2 Slider-crank mechanism with clearance joint
Table 1 Geometric and inertial properties of mechanism
Body number Length (m) Mass (kg) Moment of inertia (kg m
2
)
2 0.05 0.30 0.00010
3 0.12 0.21 0.00025
4 0.06 0.14 0.00010
Fig. 3 (a) Model of the slider-crank mechanism developed in MSC. ADAMS, and (b) exaggerated joint clearance at the piston
pin
Journal of Computational and Nonlinear Dynamics JULY 2014, Vol. 9 / 031013-3
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center are set to zero and the journal and bearing centers are coin-
cident. The dynamic parameters used in the simulation are listed
in Table 2.
Comparing the results from this
MSC.ADAMS computer simula-
tion model (Fig. 4) with the experimental results of the slider-
crank mechanism obtained by Flores et al. [22] (see Fig. 5) indi-
cates a similar pattern between the computer model in this study
and the experimental results. Hence, the slider-crank mechanism
model from this study can simulate the dynamic behavior of the
system with reasonable accuracy. This simulation model will be
utilized here. The values for the radial clearance size and the input
crank speed are used for investigating the impact of the revolution
clearance joint in the slider-crank mechanical system.
As shown in Figs. 68, the results of the slider acceleration,
contact force at the clearance, and crank reaction force demon-
strate different dynamic behaviors of the slider-crank mechanism
with different values of the radial clearance; namely, 0.05 mm,
0.1 mm, 0.2 mm, and 0.5 mm. The crank rotates with a 2000 rpm
constant angular speed. The results indicate that when the clear-
ance size is increased, the curves become noisier and the dynamic
behavior tends to be nonperiodic. As the clearance size decreases,
the dynamic behavior tends to be closer to ideal. Those plots typi-
cally reach the highest value when the crank angle is at multiples
of 180 deg rotation, which is when the slider is in the critical posi-
tion. These observations can also be confirmed in the plots of the
joint reaction forces shown in Figs. 7 and 8. As can be seen, when
the clearance size is increased, the contact force and required
crank input power are significantly increased.
Figures 911 show the influence of the crank angular speed.
The values chosen for the crank speed are 200 rpm, 1000 rpm,
2000 rpm, and 5000 rpm. In this set, the clearance size is set to
0.1 mm. The different behaviors of the slider acceleration and the
joint reaction force are displayed in these plots. The decrease in
crank speed results in the curves having more noise and higher
peak values for the slider acceleration.
4 Kriging Model-Based Optimization
This section presents a procedure for constructing an objective
function using the Kriging model. In the first part, the Latin
Table 2 Parameters used in dynamic simulation of slider-crank
mechanism with clearance joint
Nominal-bearing radius 10.0 mm
Journal-bearing width 40.0 mm
Restitution coefficient 0.9
Friction coefficient 0.0
Young’s modulus 207 GPa
Poisson’s ratio 0.3
Baumgarte a, b 5
Total simulation time 0.24 s
Total steps 50,000
Fig. 4 Dynamic response of the slider-crank from MCS.ADAMS modeling with a crank speed of 200 rpm: (a) slider position for a
clearance of 0.25 mm, and (b) slider acceleration for a clearance of 0.25 mm
Fig. 5 Dynamic response of the experimental slider-crank from Ref. [22] with a crank speed of 200 rpm: (a) slider position for a
clearance of 0.25 mm, and (b) slider acceleration for a clearance of 0.25 mm
031013-4 / Vol. 9, JULY 2014 Transactions of the ASME
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hypercube is introduced for acquiring the initial design points.
After locating several initial design points, their performance
data can be obtained by the computer simulation experiment
(
MSC.ADAMS). Each computer simulation for a given set of deign
parameters requires up to 50,000 numerical integration steps for
one complete crank rotation, which is quite inefficient and compu-
tationally intensive; therefore, the Kriging model is developed and
utilized instead in this research to optimize the process. The
objective here is to develop a prediction model to estimate the
value of the objective function for any given design point in the
design space using the Kriging model instead of the computer
simulation experiment. For this purpose, the Kriging model can
be constructed based on the initial design points and their per-
formances. The implementation of the method is shown in Fig.
12. The Kriging model is constructed by using the results from
sample points coming from individual computer simulations. A
genetic algorithm is used to obtain optimal results on the design
parameter.
The neural network and Kriging model are two potential techni-
ques, among others, and both can capture the unknown nonlinear-
ity in the system performance. Based on a study by Yuan and Bai
[1], in which they compared the neural network and Kriging
model, the Kriging model can usually produce meta-model optima
that are superior in precision. Additionally, for a given sample
size, the Kriging model tends to provide a better overall fit than
the neural networks.
In this research, the objective is to utilize a nonparameter Krig-
ing model to predict the dynamic response for any given design
point within the design space. In order to construct the surrogate
Kriging model, several points inside the design space must first be
utilized and their corresponding responses must be obtained at
these points first, using the computer simulations (here, ADAMS).
The constructed Kriging model can then be used as a surrogate
model instead of the computer simulation model, in order to pre-
dict the response at any other design point within the design space.
In addition to the use of the Kriging model for the prediction of
Fig. 6 Slider acceleration for different clearance sizes: (a) 0.05 mm, (b) 0.1 mm, (c) 0.2 mm, and (d) 0.5 mm
Fig. 7 Contact force for different clearance sizes: (a) 0.05 mm, (b) 0.1 mm, (c) 0.2 mm, and (d) 0.5 mm
Journal of Computational and Nonlinear Dynamics JULY 2014, Vol. 9 / 031013-5
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Citations
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TL;DR: A comprehensive survey of the literature of the most relevant analytical, numerical, and experimental approaches for the kinematic and dynamic analyses of multibody mechanical systems with clearance joints is presented in this review.

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02 Sep 2008
TL;DR: This chapter discusses the design and exploration of a Surrogate-based kriging model, and some of the techniques used in that process, as well as some new approaches to designing models based on the data presented.
Abstract: Preface. About the Authors. Foreword. Prologue. Part I: Fundamentals. 1. Sampling Plans. 1.1 The 'Curse of Dimensionality' and How to Avoid It. 1.2 Physical versus Computational Experiments. 1.3 Designing Preliminary Experiments (Screening). 1.3.1 Estimating the Distribution of Elementary Effects. 1.4 Designing a Sampling Plan. 1.4.1 Stratification. 1.4.2 Latin Squares and Random Latin Hypercubes. 1.4.3 Space-filling Latin Hypercubes. 1.4.4 Space-filling Subsets. 1.5 A Note on Harmonic Responses. 1.6 Some Pointers for Further Reading. References. 2. Constructing a Surrogate. 2.1 The Modelling Process. 2.1.1 Stage One: Preparing the Data and Choosing a Modelling Approach. 2.1.2 Stage Two: Parameter Estimation and Training. 2.1.3 Stage Three: Model Testing. 2.2 Polynomial Models. 2.2.1 Example One: Aerofoil Drag. 2.2.2 Example Two: a Multimodal Testcase. 2.2.3 What About the k -variable Case? 2.3 Radial Basis Function Models. 2.3.1 Fitting Noise-Free Data. 2.3.2 Radial Basis Function Models of Noisy Data. 2.4 Kriging. 2.4.1 Building the Kriging Model. 2.4.2 Kriging Prediction. 2.5 Support Vector Regression. 2.5.1 The Support Vector Predictor. 2.5.2 The Kernel Trick. 2.5.3 Finding the Support Vectors. 2.5.4 Finding . 2.5.5 Choosing C and epsilon. 2.5.6 Computing epsilon : v -SVR 71. 2.6 The Big(ger) Picture. References. 3. Exploring and Exploiting a Surrogate. 3.1 Searching the Surrogate. 3.2 Infill Criteria. 3.2.1 Prediction Based Exploitation. 3.2.2 Error Based Exploration. 3.2.3 Balanced Exploitation and Exploration. 3.2.4 Conditional Likelihood Approaches. 3.2.5 Other Methods. 3.3 Managing a Surrogate Based Optimization Process. 3.3.1 Which Surrogate for What Use? 3.3.2 How Many Sample Plan and Infill Points? 3.3.3 Convergence Criteria. 3.3.4 Search of the Vibration Isolator Geometry Feasibility Using Kriging Goal Seeking. References. Part II: Advanced Concepts. 4. Visualization. 4.1 Matrices of Contour Plots. 4.2 Nested Dimensions. Reference. 5. Constraints. 5.1 Satisfaction of Constraints by Construction. 5.2 Penalty Functions. 5.3 Example Constrained Problem. 5.3.1 Using a Kriging Model of the Constraint Function. 5.3.2 Using a Kriging Model of the Objective Function. 5.4 Expected Improvement Based Approaches. 5.4.1 Expected Improvement With Simple Penalty Function. 5.4.2 Constrained Expected Improvement. 5.5 Missing Data. 5.5.1 Imputing Data for Infeasible Designs. 5.6 Design of a Helical Compression Spring Using Constrained Expected Improvement. 5.7 Summary. References. 6. Infill Criteria With Noisy Data. 6.1 Regressing Kriging. 6.2 Searching the Regression Model. 6.2.1 Re-Interpolation. 6.2.2 Re-Interpolation With Conditional Likelihood Approaches. 6.3 A Note on Matrix Ill-Conditioning. 6.4 Summary. References. 7. Exploiting Gradient Information. 7.1 Obtaining Gradients. 7.1.1 Finite Differencing. 7.1.2 Complex Step Approximation. 7.1.3 Adjoint Methods and Algorithmic Differentiation. 7.2 Gradient-enhanced Modelling. 7.3 Hessian-enhanced Modelling. 7.4 Summary. References. 8. Multi-fidelity Analysis. 8.1 Co-Kriging. 8.2 One-variable Demonstration. 8.3 Choosing X c and X e . 8.4 Summary. References. 9. Multiple Design Objectives. 9.1 Pareto Optimization. 9.2 Multi-objective Expected Improvement. 9.3 Design of the Nowacki Cantilever Beam Using Multi-objective, Constrained Expected Improvement. 9.4 Design of a Helical Compression Spring Using Multi-objective, Constrained Expected Improvement. 9.5 Summary. References. Appendix: Example Problems. A.1 One-Variable Test Function. A.2 Branin Test Function. A.3 Aerofoil Design. A.4 The Nowacki Beam. A.5 Multi-objective, Constrained Optimal Design of a Helical Compression Spring. A.6 Novel Passive Vibration Isolator Feasibility. References. Index.

2,335 citations


"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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Journal ArticleDOI
TL;DR: In this paper, kriging is equated with spatial optimal linear prediction, where the unknown random-process mean is estimated with the best linear unbiased estimator, and early appearances of (spatial) prediction techniques are assessed in terms of how close they came to Kriging.
Abstract: In this article, kriging is equated with spatial optimal linear prediction, where the unknown random-process mean is estimated with the best linear unbiased estimator. This allows early appearances of (spatial) prediction techniques to be assessed in terms of how close they came to kriging.

1,526 citations

Journal ArticleDOI
TL;DR: In this article, a continuous contact force model for the impact analysis of a two-particle collision is presented, where a hysteresis damping function is incorporated in the model which represents the dissipated energy in impact.
Abstract: A continuous contact force model for the impact analysis of a two-particle collision is presented. The model uses the general trend of the Hertz contact law. A hysteresis damping function is incorporated in the model which represents the dissipated energy in impact. The parameters in the model are determined, and the validity of the model is established. The model is then generalized to the impact analysis between two bodies of a multibody system. A continuous analysis is performed using the equations of motion of either the multibody system or an equivalent two-particle model of the colliding bodies. For the latter, the concept of effective mass is presented in order to compensate for the effects of joint forces in the system. For illustration, the impact situation between a slider-crank mechanism and another sliding block is considered.

807 citations


"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:...

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

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

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  • ...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)
Q1. What have the authors contributed in "A kriging model for dynamics of mechanical systems with revolute joint clearances" ?

In this study, the analysis of revolute joint clearance is formulated in terms of a Hertzian-based contact force model. For illustration, the classical slider-crank mechanism with a revolute clearance joint at the piston pin is presented and a simulation model is developed using the analysis/design software MSC. In the modeling and simulation of the experimental setup and in the followed parametric study with a slightly revised system, both the Hertzian normal contact force model and a Coulomb-type friction force model were utilized. The kinetic coefficient of friction was chosen as constant throughout the study. Finally, numerical results obtained from two application examples with different design parameters, including the joint clearance size, crank speed, and contact stiffness, are presented for the further analysis of the dynamics of the revolute clearance joint in a mechanical system. 

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.