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A methodology is proposed for designing a mathematical model for shock absorbers; the proposal is guided by characteristic diagrams of the shock absorbers. These characteristic diagrams (ForceDisplacement, VelocityAcceleration) are easily constructed from experimental data generated by standard tests. By analyzing the diagrams at different frequencies of interest, they can be classified into one of seven patterns, to guide the design of a model. Finally, the identification of the mathematical model can be obtained using conventional algorithms. This methodology has generated highly nonlinear models for 2 degrees of freedom magnetorheological dampers with high precision (2–10% errors).
A dynamic mathematical model for an automotive shock absorber must accurately simulate its behavior and accommodate nonlinearities (e.g., friction, hysteresis, and inertia) over a frequency range with a maximum value lower than 30 Hz in the automotive field. The characteristics of the
Acronym definitions.

Degree of freedom 


Electrohydraulic 

Force displacement 

Force velocity 

Magnetorheological 

Passive 

Semiactive 
Comparison of the different shock absorber technologies.
Characteristic 




Hysteresis  Low  Low  Low 
Principle  Constant flow  Change of viscosity  Area of variable flow 
Excitation  —  Electric current  Electric current 
Excitation range  —  0–2.5 A @ 12 V  0–5 A 
Power  —  30 W  >60 W 
Speed of response  —  15–40 m s  10–60 m s 
Other applications  Safety  Clutches, brakes, prosthesis  Flow control 
Advantages  Maintenance cost  Actuation system  Proportional response 
Disadvantage  Performance  Cost  Maintenance 
Technological maturity  High  High  High 
Service life  80,000 km  32,000 km  40,000 km 
Relative cost [%]  100  ∼5,000  ∼7,000 
Some models have been developed with parameters that have no physical meaning, such as 1)
Variables definition.
Variable  Description  Units 

Ω  Frequency  rads/s 
Α  Amplitude  mm 
M  Excitation (exogenous variable)  — 

High and low stiffness  N/m 

Semiactive stiffness  N/m 
In the preyield zone for 

Z  Displacement  m 

Speed  m/s 

Acceleration  m^{2}/s 

High and low damping slope  Ns/m 

Semiactive damping in the preyield zone for 
Ns/m 

Body mass of the shock absorber  kg 

Virtual mass of the shock absorber when 
kg 

Nonlinear force of kwok model 
N 

Coefficient in the preyield zone related to damping  s/m 

Coefficient in the preyield zone related to hysteresis  1/m 

Nonlinear force of the guo model 
N 

Nonlinear force of the Çesmeci model 
N 

Damping force  N 

Damping force, 
N 

Straight lines in the characteristic diagrams  — 

Points in the characteristic diagrams  — 

Stiffness in 
N/m 

Damping in 
Ns/m 

Damping coefficient in preyield zone  Ns/m 

Damping coefficient in postyield zone  Ns/m 

Constant damping force  N 
τ  Response time constant of 
S 

Bandwidth for low, medium and high frequencies  Hz 

Speed threshold for changing from preyield to postyield zone for 
m/s 

Speed threshold for changing from preyield to postyield zone for 
m/s 

Slope of the semiactive force due to the excitation applied, 
N/(Excitation units) 
θ  Auxiliary variable  

Coefficient with respect to 
— 

Coefficient with respect to 
— 

Sigmoidal damping force on 
N 

Sigmoidal damping force with hysteresis due to 
N 

Sigmoidal damping force with hysteresis due to 
N 

Damping force in the preyield zone dependent on 
N 

Damping force in the preyield zone dependent on 
N 

Sigmoidal damping force magnitude  N 

Damping force magnitude for 
N 

Sigmoidal damping force magnitude for 
N 

Damping coefficient sigmoidal for 
s/m 

Damping coefficient sigmoidal with hysteresis for 
1/m 

Damping coefficient in the preyield zone due to 
s/m 

Damping coefficient in the preyield zone with hysteresis due to 
1/m 

Damping coefficient in the preyield zone due to 
s/m 

Damping coefficient in the preyield zone with hysteresis due to 
1/m 

Damping coefficient sigmoidal due to 
s/m 

Damping coefficient in the preyield zone due to 
s/m 
Because the
The results are satisfactory in terms of the
Models comparison.
Author/Year 




Goal  Prototyping and simulation  Simulation  Simulation 
Experiments  Standard/Variable  Standard  Standard 
Parameters 

Depend on the model 

Bandwidth  0–15 Hz  0–5 Hz  0–15 Hz 
Nonlinearity  Friction, hysteresis  Semiactive yield  Hysteresis 
Technology 



Model  Algebraic with 
Dependent of excitation  Nonlinear dynamics 
Advantages  Parametric  —  Computation 
Disadvantages  Multiple experiments, complex  No meaning on characteristic diagrams 
A generic model design method based on characteristic diagrams to obtain a model that can be identified and simulated with a generic tools is proposed,
General block diagram of the
This paper deals particularly with the suitability of this method to understand and model a damper using its characteristic diagrams when it has one damping control input. The specimen to be used in this work has two control inputs,
The total force of a semipassive shock absorber can be expressed with two terms,
When
The characteristic diagrams show the kinematic performance when the excitation is zero
The exogenous variable affects the
Characteristic diagrams.
There are three types of points in the characteristic diagrams. Yield point is the point at which the slope of the line decreases. In the
Characteristic points in the
Operation  Compression  Extension 



Lines 


ω 

Yield  H  D 


Restoring  F  B 


Return  A  E 


The analysis of the characteristic diagrams will be performed in three frequency ranges relevant to the automotive field: low frequency (ω
In
Lines 











A 






B 






C 






D 






E 






F 






G 






H 






Variable with more  
influence In the  Ω 



Characteristic diagrams 
The
Lines 











A 






B 






C 






D 






E 






F 






G 






H 






Variable with more  
influence In the  ω 



Characteristic diagrams 
In the
Lines 











A 






B 






c 






d 






e 






f 






g 






h 






Variable with more  
influence In the  Ω  
Characteristic diagrams 
A model is presented for each frequency range; for
For frequencies
The three equations are similar to that presented by
Type0. In diagrams
Characteristic diagrams at
The
Types of
Characteristic diagrams at
Characteristic diagrams at
Characteristic diagrams at
Characteristic diagrams at
Characteristic diagrams at
The proposed generic model of the shock absorber includes two terms.
Basic methodology
This methodology is divided in four steps,
Step 1: Pattern classification.
The first step of the methodology is to classify the pattern of the characteristic diagram that was generated experimentally from the shock absorber. This classification allows the definition of the specific model equation from a set of options. The classification uses the type patterns defined and built for the
Step 2a: Modeling Rules for the
A set of rules defines the model for the
Step 2b: Modeling Rules for the
Rules for modeling the
Model  Type of diagram  Function  

ωB  ωM  ωA  
Simple  0 or 1  1  1 

0 or 1  1  ^{a}  
Inertial simple  0 or 1  1  2 

0 or 1  2  2  
0 or 1  2  6  
0 or 1  1  6  
Stiff simple  0 or 1  1  3 

0 or 1  3  3  
Complete  0 or 1  2  5 

0 or 1  3  5 
^{a}Indicates a simple model fits for low and medium frequencies domains, i. e., the precision at
Similarly, a set of rules defines the model for the
Rules for modeling the
Model  Type of diagram  Function  

ωB  ωM  ωA  Option 1  Option 2  
Simple  0 or 4  4  4 

— 
Complete  0 or 4  4, 6 or 5  6 or 5 


Symmetry of the damping force in the characteristic diagrams. If the shock absorber is symmetric, damping force equals in shape and magnitude in tension and compression zones (positive and negative forces), the method proposes the following formulation:
If the shock absorber is asymmetric, the method proposes to consider a generic model as in
Step 3: Model identification
The identification process of the model uses the trustregion reflective optimization algorithm,
Step 4: Model validation
To validate the results, the ErrortoSignal Ratio (
Block diagram describing the generation of characteristic diagrams from experimental data. For details on the Model identification and model validation (white) block output see the equations in
A
Architecture of the
The system works without the volume compensator and presents a precise internal pressure control. The architecture includes no protruding elements, a thermal compensation system, and cavitation prevention. For full details on the design and explanation of the functioning of this specimen, see
Specification  Value 

Maximum force [N]  2000 
Maximum velocity [mm/s]  150 
Stroke [mm]  50 
Maximum input current [A]  2 
Maximum body diameter [mm]  50 
Maximum pressure [bar]  40 
This device shows cavitation phenomena when no pressure is applied. On the other hand, when pressure is applied, it shows a similar behavior expected from a
The Design of Experiments (
Experiment  

Characteristic  Variable  Units  1  2  3  4  5  6  7  8  9 
Amplitude 

[mm]  5  
Frequency 

[Hz]  1  2  
Current 

[A]  0  1  2  
Pressure 

[Bar]  0  20  40  0  20  40  0  20  40 
Replicates  3  3  3  3  3  3  3  3  3  
Experimental points  54 
We would like to add that the sinusoidal test pattern and constant current permits the identification of precise models for the motion dynamics. These data patterns allow describing the nonlinearities of the damper force with a persistent frequency at different manipulations,
This subsection shows, step by step, the method application for the modeling of the described specimen in
Step 1. Generation of the characteristic diagrams and its pattern classification.
The first step consists of the plotting of the characteristic diagrams
When the internal pressure changes to 20 and 40 bar respectively, the cavitation and the dynamics in the vicinity of zero phenomena decrease considerably. Regarding to the presence of pressure, it supplies a damping force increment in a quasilinear ratio,
In the
The total damping force,
The transient behavior of the damping forces in the specimen for a 5 mm experiment, 20 bar input pressure and the three values of electric current:
The
Characteristic diagrams according to the proposed method. The
The characteristic diagrams of the pressure vs. the force when the electric current is constant,
Characteristic diagrams of the semiactive force generated by the constant increments of the pressure according to the proposed method.
Steps 2a and 2b: Modeling Rules for the
According to
Step 3: Model identification
Classification of characteristic diagrams and proposed models. It only takes into account one frequency, since the analyzed test is a 1 Hz signal.




Component  Rule  Model  Rule  Model 
1 
3  Stiff simple  5  Complete 
2 
5  Complete  5  Complete 
The model identification process used the experimental set with 20 bar, since when the pressure is present, the complexity of the dynamics diminishes to that of a typical MR damper. All the amplitudes and electric current values were included. All the possible models were identified,
Model estimation performances using
Model 



Passive  Semiactive  
Simple  Simple  0.0213 
Simple  Complete  0.0188 
Stiffness simple  Simple  0.0244 
Stiffness simple  Complete  0.0160 
Inertial simple  Simple  0.0212 
Inertial simple  Complete  0.0186 
Complete  Simple  0.0184 
Complete  Complete  0.0160 
The identified parameters for the CompleteComplete
Step 4: Model validation
Identified model parameters for the CompleteComplete approach.
Parameter number  Parameter name  Function on the model  Velocity 
Velocity 
Unit 

C1 

Offset  −0.1941  −45.1065  N 
C2 

Damping  −3.7911  −0.2774  Ns/mm 
C3 

Stiffness  89.2199  −1.5460  N/mm 
C4 

Mass  0.7599  0.0957  kg 
C5 

Stiffness gain  2,303.0421  −672.6543  N 
C6 

Stiffness  0.0023  −0.1279  s/mm 
C7 

Stiffness  −0.0368  −0.2162  1/mm 
C8 

Inertia gain  −622.9134  −48.9551  N 
C9 

Inertia  −0.2141  730.8866  s/mm 
C10 

Inertia  −2.8172  −495.4853  Unitless 
C11 

Semiactive gain  448.6180  435.0991  N/A 
C12 

Sigmoidal shape  0.0496  0.0645  s/(mmA) 
The characteristic diagrams,
Characteristic diagrams, transient response and estimated vs. experimental forces plots for the CompleteComplete
The proposed method of modeling using the generation and classification of
The characterization of the dynamics and the effect of the control inputs based on the subtraction of the damping forces when the input under study remains constant, allows for better understanding the diagrams, in this case, it has also been used for the pressure,
A methodology for the modeling of
The present method needs to be extended to include the modeling of the damping force generated for the variation of internal pressure as control input, so a set of new models will be added for such classification.
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
JLS, OS, RMM and RRM conceived and designed the analysis. JLS and JTM collected the non 2DoF shock absorber data. AS collected the data for the 2DOF MR damper. JLS and JTM performed the analysis. JLS, AS and JTM contributed data and analysis tools. JLS wrote the paper.
This work was supported by a CONACYT Grant 20122014, “Programa de Estimulo a la Innovacion” in category “BILATERAL” with grant number 142183, and the 2011 ”Programa de Estimulo a la Innovacion” in category ”PROINNOVA” with grant number 132758.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The authors want to acknowledge the support from Luc Dugard from GipsaLab, Grenoble Institute of Technology, and Ricardo Prado, Professor in Tecnologico de Monterrey. Also, JLS wants to thanks to the support from GIPSALab in Grenoble, France.