Suspension dynamic response design method based on high-order fractional order viscoelastic bushing model

Abstract

The invention relates to the technical field of vehicle dynamics, and discloses a suspension dynamic response design method based on a high-order fractional order viscoelastic bushing model, which comprises the following steps: establishing an automobile initial suspension model and a high-order fractional order rubber bushing dynamic model; obtaining an automobile suspension rubber bushing model, and respectively carrying out static and dynamic loading experiments on the axial direction and the radial direction to generate a static loading curve and a frequency-amplitude-dynamic stiffness curve, so as to carry out parameter identification on the high-order fractional order rubber bushing dynamic model; a high-order fractional order rubber bushing dynamic model for parameter identification is compiled into a user subprogram, then the subprogram is compiled, a dynamic link library is used for linking, and six components are created for calling, so that an original rubber bushing model of an automobile initial suspension model is replaced with a rubber bushing model after parameter identification. And finally generating a complete automobile suspension model. According to the method, the mechanical property precision of the model is improved, and the dynamic response design of the suspension system is also realized.

Description

Technical Field

[0001] The present invention relates to the technical field of vehicle dynamics, and particularly to a suspension dynamic response design method based on a high-order fractional viscoelastic bushing model. Background Art

[0002] With the continuous improvement of people's requirements for automobile performance, the demands for ride comfort and handling stability are also increasing continuously. Rubber bushings have a great impact on riding comfort, so the development of rubber bushings is also ongoing, and simulation analysis is one of the important steps in the R & D process.

[0003] In order to describe the mechanical properties of rubber bushings, a model composed of three elements is usually used to describe the mechanical properties of rubber bushings, namely elastic element, friction element and viscoelastic element. In recent years, the research mainly focuses on these three elements. Among them, the development of the viscoelastic element is a major focus. This element is used to describe the frequency-related characteristics of rubber bushings. A variety of models are used for this element to describe the dynamic characteristics. After multiple iterations, there are still defects in the model accuracy. And when it is used in dynamic simulation software, due to the insufficient accuracy of the rubber bushing model in the software, the mechanical properties of the rubber bushing cannot be fully expressed, resulting in inaccurate vehicle simulation results. Summary of the Invention

[0004] In view of the above deficiencies in the prior art, the present invention provides a suspension dynamic response design method based on a high-order fractional viscoelastic bushing model to solve the problem of insufficient accuracy of the dynamic characteristics of the existing rubber bushing model.

[0005] In order to achieve the above invention object, the technical solution adopted by the present invention is as follows:

[0006] A suspension dynamic response design method based on a high-order fractional viscoelastic bushing model includes the following steps:

[0007] S1. Obtain the characteristic parameters of the suspension and establish an initial vehicle suspension model;

[0008] S2. Establish a high-order fractional rubber bushing dynamic model to be identified, which includes an elastic element, a friction element and a viscoelastic element;

[0009] S3. Obtain the vehicle suspension rubber bushing model, conduct static and dynamic loading experiments on the axial and radial directions respectively, and generate a static loading curve and a frequency-amplitude-dynamic stiffness curve by obtaining experimental data of the stress-displacement curve and the frequency-dynamic stiffness curve;

[0010] S4. Use the static loading curve to identify the parameters of the elastic element and the friction element to obtain static parameters;

[0011] S5. Identify the parameters of the viscoelastic element using the frequency-amplitude-dynamic stiffness curve to obtain the dynamic parameters;

[0012] S6. Substitute the identified static parameters and dynamic parameters into the elastic element, friction element, and viscoelastic element respectively to generate a high-order fractional-order rubber bushing model after parameter identification;

[0013] S7. Program the high-order fractional-order rubber bushing dynamic model after parameter identification, generate a user subroutine and then compile it to obtain the compiled user subroutine;

[0014] S8. Establish a dynamic link library, import the compiled user subroutine into the dynamic link library, create six-component forces, and by calling the user subroutine, replace the original rubber bushing model of the vehicle's initial suspension model with the high-order fractional-order rubber bushing dynamic model after parameter identification to generate a complete vehicle suspension model for realizing the dynamic response analysis of the vehicle suspension system.

[0015] The present invention has the following beneficial effects:

[0016] 1. The suspension dynamic response design method based on the high-order fractional-order viscoelastic bushing model proposed by the present invention uses a polynomial model as the elastic element, a smooth friction model as the friction element, and introduces high-order fractional derivatives into the frequency-dependent model, and proposes a new model as the viscoelastic element, so as to jointly form a high-order fractional-order viscoelastic bushing model with the elastic element and the friction element to better describe the mechanical properties of the rubber bushing;

[0017] 2. Through static and dynamic loading experiments on the axial and radial directions of the vehicle suspension rubber bushing model respectively, obtain the hysteresis curve, that is, the corresponding force-displacement curve and frequency-dynamic stiffness curve experimental data to generate static loading data and dynamic loading data, that is, static loading curves and frequency-amplitude-dynamic stiffness curves, and based on these data, conduct parameter identification. At the same time, corresponding methods are used in the identification process, such as using the least squares method to identify the parameters of the elastic element, using the static loading data and its image to identify the parameters of the friction element, and using the particle swarm optimization method to identify the parameters of the viscoelastic element, improve the accuracy of parameter identification, further improve the accuracy of the identification result, and thus improve the high-order fractional-order rubber bushing model;

[0018] 3. Write the dynamic model of the high-order fractional rubber bushing after parameter identification as a subroutine, link it through a dynamic link library, and use the identified dynamic model of the high-order fractional rubber bushing to replace the rubber bushing in the initial vehicle suspension model, so as to establish a complete vehicle suspension model to achieve the dynamic response design of the suspension system. The generated complete vehicle suspension model can better balance the comfort and handling of the suspension system, while improving the accuracy and durability prediction ability of the bushing model, and promoting the development of vehicle suspensions. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 It is a schematic flow chart of the suspension dynamic response design method based on the high-order fractional viscoelastic bushing model proposed by the present invention;

[0020] Figure 2 It is a schematic diagram of the dynamic model of the high-order fractional rubber bushing in the embodiment. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0021] The following describes the specific embodiments of the present invention to facilitate those skilled in the art of the present technology to understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific embodiments. For those of ordinary skill in the art of the present technology, as long as various changes are within the spirit and scope of the present invention defined and determined by the appended claims, these changes are obvious, and all inventions and creations using the concept of the present invention are within the scope of protection.

[0022] As Figure 1 shown, the suspension dynamic response design method based on the high-order fractional viscoelastic bushing model includes the following steps S1 - S8:

[0023] S1. Obtain the characteristic parameters of the suspension and establish the initial vehicle suspension model.

[0024] Specifically, the characteristic parameters of the suspension in step S1 are the geometric characteristic parameters, mass characteristic parameters, and mechanical characteristic parameters of the vehicle suspension components.

[0025] In this embodiment, according to the geometric characteristic parameters, mass characteristic parameters, and mechanical characteristic parameters of the vehicle suspension components, the vehicle suspension model is initially established, and it can be established by using the ADAMS / Car module of the multi-body dynamics simulation software ADAMS.

[0026] S2. Establish the dynamic model of the high-order fractional rubber bushing to be identified, which includes an elastic unit, a friction unit, and a viscoelastic unit.

[0027] In this embodiment, by introducing the high-order fractional derivative into the frequency-related model, a new model is proposed as the viscoelastic unit. Therefore, the relationship between the force and displacement of the viscoelastic unit is as follows:

[0028]

[0029] Among them, F represents the viscoelastic force, α and β respectively represent different orders of the fractional derivative, represent the Riemann-Liouville fractional derivative operators of different orders respectively, x represents displacement, k1 and k2 respectively represent different elastic moduli in the viscoelastic element, and c1 and c2 respectively represent different viscous coefficients in the viscoelastic element.

[0030] Among them, the expansion formula of the Riemann-Liouville fractional derivative operator is:

[0031]

[0032] Among them, t represents the independent variable of the function, n represents the smallest integer not less than the order α of the fractional derivative, f represents the function to which the operator is applied, and Γ represents the gamma function.

[0033] Meanwhile, when both α and β are 1, it can represent that the viscoelastic element model is a frequency-dependent model, and its mechanical properties include the mechanical properties that the frequency-dependent model can express, that is, the viscoelastic element proposed in the present invention can better describe the viscoelastic characteristics of the high-order fractional-order rubber bushing dynamic model to be identified, and the model structure is as Figure 2 shown, including components such as spring and damping units. Therefore, the viscoelastic element, elastic element, and friction element together constitute the high-order fractional-order rubber bushing dynamic model to be identified.

[0034] In addition, in order to obtain the viscoelastic force and dynamic stiffness, it is also necessary to perform Laplace transform on the relationship between the force and displacement of the viscoelastic element to generate the real part and imaginary part of the viscoelastic force at the x0 amplitude.

[0035] Among them, the real part of the viscoelastic force is:

[0036]

[0037] Among them, F v0Re represents the real part of the viscoelastic force, ω represents frequency, λ1 and λ2 both represent intermediate variables, that is sin represents the sine function, cos represents the cosine function, π represents a constant, and i represents the imaginary unit.

[0038] Among them, the imaginary part of the viscoelastic force is:

[0039]

[0040] Among them, F v0lm represents the imaginary part of the viscoelastic force.

[0041] S3. Obtain the automotive suspension rubber bushing model, and conduct static and dynamic loading experiments on the axial and radial directions respectively. By obtaining the experimental data of the corresponding stress-displacement curve and frequency dynamic stiffness curve, generate the static loading curve and frequency-amplitude-dynamic stiffness curve.

[0042] In this embodiment, a certain type of rubber bushing of the rear suspension subframe of an automobile can be selected as the research object. Through bench tests, dynamic and static load tests are respectively carried out on the axial and radial directions of the rubber bushing of the rear suspension subframe of the automobile to obtain the force-displacement relationship curve of the rubber bushing and the frequency dynamic stiffness relationship curve under various amplitudes. By analyzing and processing these curves, organizing the experimental data, and generating the static loading curve and frequency-amplitude-dynamic stiffness curve, so as to use the data of these two curves in the subsequent steps to identify the parameters of the established high-order fractional-order rubber bushing dynamic model to be identified, thereby generating the identified high-order fractional-order rubber bushing dynamic model.

[0043] S4. Use the static loading curve to identify the parameters of the elastic unit and the friction unit to obtain the static parameters.

[0044] Specifically, in step S4, the force-displacement relationship formula for identifying the parameters of the elastic unit using the static loading curve is:

[0045] F e = a0 + a1x + a2x 2 +......+ a n1 x n1

[0046] Among them, n1 represents the number of polynomial terms, F e represents the elastic force, and a0, a1, a2, a n1 all represent polynomial coefficients.

[0047] Specifically, after identifying the parameters of the elastic unit, the identified static parameters are polynomial coefficients, namely a0, a1, a2, …, a n1 .

[0048] Specifically, in step S4, the force-displacement relationship formula for identifying the parameters of the friction unit using the static loading curve is:

[0049]

[0050] Among them, F f represents the frictional force, (x s , F fs ) respectively represent the state reference point coordinates of the displacement-frictional force change curve of the bushing, x2 represents the displacement when the frictional force increases from 0 to half of the maximum frictional force, and sign(x) represents the change direction of the displacement.

[0051] Specifically, after parameter identification of the friction unit, the identified static parameters are the maximum friction force and the displacement when the friction force increases from 0 to half of the maximum friction force.

[0052] In this embodiment, the static loading curve (static loading data) is used to identify the parameters of the elastic unit and the friction unit respectively. The identified static parameters of the elastic unit are: polynomial coefficients a0, a1, a2, …, a n1 , which can be completed by the least squares fitting tool in the Matlab toolbox for parameter identification; the friction unit can be identified by using a graph. First, the maximum friction force F is calculated by finding the difference between the upper and lower boundary intercepts of the y-axis hysteresis curve. fmax Then, the main stiffness K of the elastic unit is calculated by the slope of the hysteresis curve at the displacement limit and the start of rotation. e and the maximum stiffness K max , and these data are used to find the displacement x2 when the friction force increases from 0 to half of the maximum friction force. The calculation formula of x2 is:

[0053] S5. Use the frequency-amplitude-dynamic stiffness curve to identify the parameters of the viscoelastic unit and obtain the dynamic parameters.

[0054] Specifically, after parameter identification of the viscoelastic unit using the frequency-amplitude-dynamic stiffness curve, the identified dynamic parameters are the elastic modulus, the viscosity coefficient, and the order of the fractional derivative.

[0055] In this embodiment, since there are many parameters to be identified for the viscoelastic unit (the identified parameters include c1, c2, k1, k2, α, β), the particle swarm optimization method can be used for identification to accelerate the identification speed and accuracy. When using the particle swarm optimization method for parameter identification, MATLAB can be used. By adjusting parameters such as weights and accelerations, and then importing the experimental data and the model to be identified for analysis and calculation. In addition, since the particle swarm optimization method is prone to premature convergence and falling into local optima, resulting in poor fitting effects, therefore, it can be proposed to improve the particle swarm optimization method using the genetic algorithm, so as to generate a new particle swarm through mutation for solution to prevent premature convergence and find a better solution. Specifically:

[0056] To improve the optimization speed, a particle is randomly selected during speed update; control the speed update of the particle in three directions, while increasing the optimization speed and avoiding falling into local optima. The speed update equation is:

[0057] v t+1 =wv t +c11r1(p b -x1 t )+c12r2(g b-x1 t ) + c13r3(p s -x1 t )

[0058] where v t+1 is the velocity at time t + 1, v t is the velocity of the particle at time t, w is the inertia weight, c11, c12, and c13 are all learning factors, x1 t is the position of the particle at time t, r1, r2, and r3 are all random numbers in the range (0, 1), and p b is the particle selected from the particle swarm.

[0059] To prevent local optimality, it is necessary to evaluate the iteration results after optimization. If the best fitness remains unchanged after the iteration, the particle swarm optimization method needs to perform crossover and mutation to generate new particles and change the search direction. The formula for the crossover process is:

[0060] x k = x k (1 - σ) + x l , x l = x l (1 - σ) + x k

[0061] where x k is the particle that meets the crossover condition, x l is the randomly selected particle, and σ is a random number in the interval (0, 1).

[0062] Moreover, during mutation, according to the fitness value of the particle, there is a different mutation probability. Particles with a higher fitness value have a higher mutation probability. The specific formula is:

[0063] P m = 0.5 - 0.01(i1 / n2)

[0064] where P m is the mutation probability, i1 is the index value of the particle in the particle swarm, and its value ranges from 1 to n2.

[0065] During mutation, the j - dimension of the i1 particle is mutated, and the formula is:

[0066] x1 i1j = (max(j) + min(j)) / 2 + (max(j) - min(j))(r - 0.5)

[0067] where max(j) is the upper bound of the j - dimension of the particle, min(j) is the lower bound of the j - dimension of the particle, r is a random number in the range (0, 1), and the fitness function of the particle is:

[0068]

[0069] Among them, n3 is the number of working conditions, is the dynamic stiffness calculated under the i'-th working condition, is the dynamic stiffness data measured experimentally under the i'-th working condition; meanwhile, in order to ensure that there are no significant errors in the fitting data during the identification process, it is necessary to set a limiting condition, which is:

[0070] Therefore, the formula for the dynamically calculated dynamic stiffness of the bushing is:

[0071]

[0072] K dyn = F0 / x0

[0073] Among them, K dyn is the dynamic stiffness, x0 is the amplitude, and F e0 is the elastic force, and F f0 is the frictional force.

[0074] S6. Substitute the identified static parameters and dynamic parameters into the elastic element, friction element, and viscoelastic element respectively to generate a high-order fractional-order rubber bushing model after parameter identification.

[0075] S7. Program the high-order fractional-order rubber bushing dynamic model after parameter identification, generate a user subroutine and then compile it to obtain the compiled user subroutine.

[0076] S8. Establish a dynamic link library, import the compiled user subroutine into the dynamic link library, create six-component forces, and by calling the user subroutine, replace the original rubber bushing model of the vehicle's initial suspension model with the high-order fractional-order rubber bushing dynamic model after parameter identification to generate a complete vehicle suspension model for realizing the dynamic response analysis of the vehicle suspension system.

[0077] In this embodiment, the specific operations and principles of steps S7 - S8 are as follows: First, on the Visual Studio platform, the high - order fractional - order rubber bushing dynamic model after parameter identification is programmed using the C++ language and written as a user - subroutine that can be recognized by ADAMS. Then, inside the high - order fractional - order rubber bushing dynamic model after parameter identification, the static and dynamic parameters that need to be identified can be set as parameters to be filled. In this way, even if the parameters to be identified change, there is no need to modify the sub - routine again. Just input the modified parameters when using the user - subroutine. When writing the high - order fractional - order rubber bushing dynamic model after parameter identification as a user - subroutine, parameters such as distance, angle, force, and velocity need to be retrieved according to the user - subroutine template, and the axial model and radial model should be distinguished to ensure the normal operation of the high - order fractional - order rubber bushing dynamic model after parameter identification. Second, use the Visual Studio compiler to compile the completed user - subroutine to generate a dynamic database file, so that the user - subroutine can be recognized by the built - in solver of ADAMS. Finally, by creating a six - component force, retrieving the user - subroutine, and using the high - order fractional - order rubber bushing dynamic model after parameter identification to replace the original rubber bushing model of the initial vehicle suspension model, a complete vehicle suspension model is generated. Among them, the purpose of creating a six - component force is to introduce the user - subroutine, place it at the position of the original rubber bushing, and fill in the Adamsid of the connecting part, the sub - routine address, and the parameters to be input. Therefore, the finally established complete vehicle suspension model can be used for suspension simulation, or after being combined with other subsystems, for vehicle - level simulation to test ride comfort and handling stability.

[0078] In summary, for the suspension dynamic response design method based on the high-order fractional viscoelastic bushing model proposed by the present invention, first, a polynomial model is used as the elastic element, a smooth friction model is used as the friction element, and a high-order fractional derivative is introduced into the frequency-dependent model to propose a new model as the viscoelastic element, so as to jointly form a high-order fractional viscoelastic bushing model with the elastic element and the friction element, in order to better describe the mechanical properties of the rubber bushing; second, through static and dynamic loading experiments on the axial and radial directions of the automotive suspension rubber bushing model respectively, the hysteresis curve is obtained, that is, the corresponding force-displacement curve and frequency dynamic stiffness curve experimental data, so as to generate static loading data and dynamic loading data, that is, the static loading curve and the frequency-amplitude-dynamic stiffness curve, and based on these data, parameter identification is carried out. At the same time, corresponding methods are used in the identification process, such as using the least squares method to identify the parameters of the elastic element, using the static loading data and its image to identify the parameters of the friction element, and using the particle swarm optimization method to identify the parameters of the viscoelastic element, to improve the accuracy of parameter identification and further improve the accuracy of the identification results, so as to improve the high-order fractional rubber bushing model; finally, the high-order fractional rubber bushing dynamic model after parameter identification is written as a subroutine and linked through a dynamic link library, and the rubber bushing in the initial automotive suspension model is replaced with the identified high-order fractional rubber bushing dynamic model, so as to establish a complete automotive suspension model to realize the dynamic response design of the suspension system, and the generated complete automotive suspension model can better balance the comfort and controllability of the suspension system, while improving the accuracy and durability prediction ability of the bushing model, and promoting the development of automotive suspensions.

[0079] Specific embodiments are used in the present invention to elaborate on the principles and implementation manners of the present invention. The description of the above embodiments is only used to help understand the method and its core idea of the present invention; at the same time, for those of ordinary skill in the art, based on the idea of the present invention, there will be changes in the specific implementation manners and application scopes. In summary, the content of this specification should not be construed as a limitation to the present invention.

[0080] Those of ordinary skill in the art will realize that the embodiments described herein are for helping readers understand the principles of the present invention, and it should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those of ordinary skill in the art can make various other specific deformations and combinations without departing from the essence of the present invention based on the technical revelations disclosed in the present invention, and these deformations and combinations are still within the protection scope of the present invention.

Claims

1. A suspension dynamic response design method based on a high-order fractional viscoelastic bushing model, characterized in that It includes the following steps: S1. Obtain the characteristic parameters of the suspension and establish the initial vehicle suspension model; S2. Establish the dynamic model of the high-order fractional rubber bushing to be identified, which includes an elastic element, a friction element, and a viscoelastic element; S3. Obtain the vehicle suspension rubber bushing model, conduct static and dynamic loading experiments on the axial and radial directions respectively, generate the static loading curve and the frequency-amplitude-dynamic stiffness curve by obtaining the experimental data of the stress-displacement curve and the frequency-dynamic stiffness curve; S4. Use the static loading curve to identify the parameters of the elastic element and the friction element and obtain the static parameters; S5. Use the frequency-amplitude-dynamic stiffness curve to identify the parameters of the viscoelastic element and obtain the dynamic parameters; S6. Substitute the identified static parameters and dynamic parameters into the elastic element, the friction element, and the viscoelastic element respectively to generate the high-order fractional rubber bushing model after parameter identification; S7. Program the dynamic model of the high-order fractional rubber bushing after parameter identification, generate a user subroutine and then compile it to obtain the compiled user subroutine; S8. Establish a dynamic link library, import the compiled user subroutine into the dynamic link library, create six-component forces, and by calling the user subroutine, replace the original rubber bushing model of the initial vehicle suspension model with the dynamic model of the high-order fractional rubber bushing after parameter identification to generate a complete vehicle suspension model for realizing the dynamic response analysis of the vehicle suspension system.

2. The suspension dynamic response design method based on the high-order fractional viscoelastic bushing model according to claim 1, characterized in that, In step S1, the characteristic parameters of the suspension are the geometric characteristic parameters, mass characteristic parameters, and mechanical characteristic parameters of the vehicle suspension components.

3. The suspension dynamic response design method based on the high-order fractional viscoelastic bushing model according to claim 2, characterized in that In step S2, the relationship between the force and displacement of the viscoelastic element is: Among them, F represents the viscoelastic force, α and β respectively represent different orders of the fractional derivative, respectively represent the Riemann-Liouville fractional derivative operators of different orders, x represents the displacement, k1 and k2 respectively represent different elastic moduli in the viscoelastic element, and c1 and c2 respectively represent different viscous coefficients in the viscoelastic element; Among them, the expansion formula of the Riemann-Liouville fractional derivative operator is: Among them, t represents the independent variable of the function, n represents the smallest integer not less than the order α of the fractional derivative, f represents the function to which the operator is applied, and Γ represents the gamma function.

4. The suspension dynamic response design method based on the high-order fractional viscoelastic bushing model according to claim 3, characterized in that, Conduct Laplace transform on the relationship between the force and displacement of the viscoelastic element to generate the real part and the imaginary part of the viscoelastic force at the x0 amplitude; Among them, the real part of the viscoelastic force is: where F v0Re represents the real part of the viscoelastic force, ω represents the frequency, and both λ1 and λ2 represent intermediate variables, that is sin represents the sine function, cos represents the cosine function, π represents a constant, and i represents the imaginary unit; Among them, the imaginary part of the viscoelastic force is: Among them, F v0lm represents the imaginary part of the viscoelastic force.

5. The suspension dynamic response design method based on the high-order fractional viscoelastic bushing model according to claim 4, characterized in that In step S4, the relationship between the force and displacement for identifying the parameters of the elastic element using the static loading curve is: F e = a0 + a1x + a2x 2 +......+ a n1 x n1 Among them, n1 represents the number of polynomial terms, and F e represents the elastic force, and a0, a1, a2, a n1 all represent polynomial coefficients.

6. The suspension dynamic response design method based on the high-order fractional viscoelastic bushing model according to claim 5, wherein After parameter identification of the elastic element, the identified static parameters are polynomial coefficients, namely a0, a1, a2, …, a n1 .

7. The suspension dynamic response design method based on the high-order fractional viscoelastic bushing model according to claim 6, characterized in that In step S4, the relationship between the force and displacement for identifying the parameters of the friction element using the static loading curve is: Among them, F f represents the frictional force, (x s , F fs ) respectively represent the state reference point coordinates of the displacement-frictional force change curve of the bushing, F fmax represents the maximum frictional force, x2 represents the displacement when the frictional force increases from 0 to half of the maximum frictional force, and sign(x) represents the change direction of the displacement.

8. The suspension dynamic response design method based on the high-order fractional viscoelastic bushing model according to claim 7, characterized in that, After identifying the parameters of the friction element, the identified static parameters are the maximum friction force and the displacement when the friction force increases from 0 to half of the maximum friction force.

9. The suspension dynamic response design method based on the high-order fractional viscoelastic bushing model according to claim 8, characterized in that, After identifying the parameters of the viscoelastic element using the frequency-amplitude-dynamic stiffness curve, the identified dynamic parameters are the elastic modulus, the viscosity coefficient, and the order of the fractional derivative.

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