Modeling Of Viscoelasticity Dampers Using Fractional Derivative English Language Essay

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Vibration and noise control has become a very important engineering problem, various methods have been widely used in mechanical engineering,electronics, aerospace and some other fields.Among these vibration control technical measures,vibration reduction using damper is an very effective way

The basic principle of a damper is force the system recover the static status via the energy dissipation in the vibration.In a free oscillation system,the energy lost in the motion will decrease the amplitude of the system constantly.On the other hand,in the forced oscillation system,the energy lost in the motion also helps to limit the amplitude of the system.Great amount of materials has the property of damping,including some of the metals.The longitudinal loss factor of common metals are shown as the table 1.

Table 1


Longitudinal Loss Factor


Longitudinal Loss Factor


0.3 to 10




5 to 30



Lead(including antimony)

1 to 4

Copper(single crystal)

2 to 7


1 to 4


0.2 to 1


0.2 to 3



As is shown in the table one,the loss factor of metal is significantly low.This is why the vibration in the automobile or the mechanism made by metal is difficult to remove from the mechanical system.

In the recent 30 years,the development of a new type of damper made by viscoelasticity material is grows rapidly.In the table 2,the longitudinal loss factor of some common viscoelasticity materials are shown.

Table 2


Longitudinal Loss factor

vulcanized rubber


artificial rubber


butylated resin


mucilaginous gum


Compare with the metals,the viscoelasticity materials have a relatively bigger longitudinal loss factor which means the viscoelasticity dampers using the viscoelasticity material such as rubber or resin can increase the damping of a structure to relieve the vibration and decrease the transient via energy dissipation more efficiently.

Viscoelastic dampers have two kinds of structure form[1]: one has non-constrained damping layer, this kind of viscoelastic dampers paste the viscoelastic materials directly the metal surface, when the structure is under vibration, the viscoelastic materials absorb energy through bending and tension .

Pic.1 Non-constrained viscoelasticity damper

Another kind viscoelastic dampers has the constrained damping layer by placing viscoelastic materials between structure surface and metal constraints,when the structure is under vibration,a bending deformation will be produced.Because of the metal constraint limit the translation of viscoelastic material,the viscoelastic material will generate a great shear deformation, which can provide larger damping.

Pic.2 Constrained viscoelasticity damper

Furthermore,since the structure of viscoelasticity dampers is not very complex and the prime cost is low,the dampers are used widely in the the field of automobile,mechanism,earthquake control and so on.

To accurately capture the characteristics of viscoelasticity dampers in time domain and frequency domain,numerous of models are developed in the last a few years.In general,three types of models are proposed:kelvin model,maxwell model,fractional derivative model.Among these three models,both of the first two models employs the combination of linear Hook elastic spring and Newton viscous dashpot to represent the characteristics of viscoelasticity dampers.The standard linear solid model is quite easy to calculate and it is widely used in the condition of small strain.The Kelvin model is commonly employed if the size of the viscoelasticity damper is in big size,on the contrary,the maxwell model will be applied under the situation of small-size dampers.But all the three models fail to obtain the satisfactory accuracy.This is mainly due to the overlook the temperature impact and the viscoelasticity of some viscoelasticity material is not linear,using the linear components such as hook spring and newton dashpot can not describe the nonlinear properties of viscoelasticity dampers.To achieve a pinpoint accuracy,the fraction derivative model is produced using the fractional time-derivative by Bagley[2]and Bagley and Torvik[3].The fractional derivative model is the latest model which has considered the influence of temperature,frequency and input amplitude and can obtain a well representation of the nonlinear characteristic.

2.Viscoelasticity Phenomena

2.1Static viscoelasticity phenomena

2.1.1 Creep[1]

Creep is a phenomena that under a certain temperature and small constant force,the deformation of viscoelasticity material will increase with the time.Depending on the value of force,the creep can be separate into three forms.

(1)The first form is just the elastic deformation due to the force can not lead to segmental motion.

Pic.3 Elastic Creep

(2)The second one is the creep caused by segmental motion,deformation of this type of creep will be recovered after a certain time.

Pic.4 Restorable Creep

In figure.1,three curves describe the relationship between displacement and time under three different forces,each one twice the amplitude of the previous one[4].

Figure.1 Creep under three different constant forces

You can tell from this figure that with the force doubled,the displacement caused by the force doubled too.

(3)The last one is the creep caused by Viscous flow,the deformation of this type of creep is permanent and can not be recovered.

Pic.4 Permanent Creep

2.1.2 Stress relaxation[1]

Stress relaxation is a phenomena that describes how viscoelasticity materials relieve stress under constant strain.Here we set three different displacements and try to find the relationship between the relieve force and time,each displacement is twice the amplitude of the previous one.Result is shown in figure 2

Figure.2 Stress relaxation under three different displacement

2.2 Dynamic viscoelasticity phenomena

2.2.1 Hysteresis loop[1]

Figure.3 Hysteresis loop

Hysteresis loop is a phenomena that the status of strain of the polymer falls behind the one of stress under a alternating stress input.

The internal cause of this phenomena is the friction inside the viscoelasticity material.The phase difference grows with the increase of internal friction of polymer.The aim of the hysteresis loop is determine the parameters in the simulating model and calculate the energy loss per cycle of the viscoelasticity material.The surrounded area of hysteresis loop indicates the energy loss per cycle as the formula as .The mechanical energy in the viscoelasticity is conversed to irreversible heat and lose into the environment.The factor as phase difference indicates the amount of the loss energy.When ,the energy loss per cycle equals zero which means all the energy is stored as elastic energy and the maximum of the energy loss happens at which means all the energy are lost.[5]The loss of energy will lead to a reduction of vibration or noise.

3. Fractional derivative

Fractional derivative was first proposed by the French mathematician Guillaume Francois Antoine[6].Fractional calculus theory establishment has been already 300 years , but early study is focused on the theory, only in recent years the Fractional calculus theory is used in the applications such as automatic control and viscoelasticity modeling.In the development of the theory of fractional calculus,many different definitions arises, such as the definition expanded from the integral calculus :

3.1 Cauchy definition of Fractional calculus

This formulas is expanded from the integral calculus,it can be written as below,

Where the C is the smooth curve surrounded single value of f(t) and the analytical area.

3.2 Grunwald-Letnikov definition of Fractional calculus

Where the is the binomial coefficients,Sjoberg proposed a fractional Kelvin-Voige model whose simulation results meet the measured results pretty good.In this model Sjoberg employ a fractional derivative spring-pot which can be describe by a time discrete equation[7]

3.3 Riemann-Liouville definition of Fractional calculus

In this equation, and the a is the initial of this integral and normally this value equals zero.In this four definitions the Riemann-Liouville definition is the world's most commonly used definition of Fractional calculus.And the fractional derivative formula is shown in pic

To model the viscoelasticity characteristic,the order of fractional derivative is normally between zero to one so that the formula can be rewritten as below,

Although this equation is relatively simple compare to the other ones,it is still can not used directly to simulate the viscoelasticity characteristic,the equation still needs a further simplification.

3.4 Capotu definition of Fractional calculus

Capotu fractional derivative is defined as

In this equation,,m is a integer.Similarly the capotu fractional integral is defined as,

3.5 Simulation of fictional derivative using Filtering algorithm[8]

The mentioned various fractional derivative definition is only applicable when the input signal is a known function.But in actual application ,the input signal is unable to know in advance, so we should employ some other means to simulate the fractional derivative, for example, by constructing filter to process the input signal.To simulate the fractional derivative by this method,many different Filtering algorithm is developed by researchers,one of algorithm is called oustaloup.If the rang of the fitting frequency is to ,the the model of the oustaloup Filtering algorithm can be described as the equation below,


According to the equations above,we can use the Matlab to program the oustaloup Filtering algorithm,then the output signal will be approximate with the fractional derivative of input signal.

Matlab code:

function G=ousta_cp(r,N,w_L,w_H)






%%r is the order of the fractional derivative,2N+1 is the order of the filter,w_L and the %%w_H is the boundaries of selected fitting frequency and the limitation of this model is that the w_L*w_H should be 1 to guarantee the accuracy.


4.1 Maxwell model

This model is a quite old model which is proposed in 1867.The maxwell model employed two components to represent the characteristic of viscoelasticity materials.One is a purely elastic spring which is called hook spring,it exact match the formula where K is the stiffness constants of the spring. The other one is a purely viscous dashpot,it exact match the formula where C is the damping constants of the newton dashpot. This two components are connected consecutively,as shown in the pic.5.

Pic.5 Maxwell model

So simply the relations between force and displacement is given by and ,where s means force and displacement act on the spring and d is for the dashpot.Obviously the equation and exists.Taking the derivative of displacement with respect to time, we obtain

. (1a)

This equation can be employed when a shear stress or a tension is applied to the viscoelasticity materials.

After talking about the time domain,the frequency domain is taking in to consideration.

Apply a harmonic displacement excitation ,where the is the excitation amplitude, is the circular excitation frequency and t is the time.

The complex stiffness of this maxwell model can be easily obtained by using equation(1a),


The energy loss per cycle can be evaluated from the imaginary part of Equation(1b).

[9] (1c)

And the displacement amplitude can also be calculated by the equation(1b),

[9] (1d)

But as what we have known before, in the maxwell model ,the elastic spring does not lose any energy,so the energy loss per cycle is all contributed by the viscous dashpot.And we can indicate from this two equations that the Maxwell model is a frequency dependent system.

4.2 Kelvin-Voigt model

Pic.6 kelvin-Voigt model

Time domain

The Kelvin-voigt was proposed in 1892,a few years after maxwell model had been designed.The model still uses a purely viscous dashpot and a purely elastic spring as its component.But the difference between the two models is that the spring and dashpot in Kelvin model is in parallel while the ones in Maxwell is in series.Due to the transformed layout,the equation between force and displacement is different from the Maxwell.,.From the two equations above,we can obtain (2a)

where K is stiffness of spring and C is the damping of dashpot.

Frequency domain


Applying a harmonic displacement excitation to the Kelvin model.

Since the force on the spring equals to the excitation,the amplitude of the elastic force is produced by the amplitude of excitation,.

where K is the stiffness of spring (2b)

The energy loss of the spring


The results we obtain indicates the elastic force is frequency independent in the Kelvin Model and the energy loss by the spring is zero.


Under the harmonic excitation ,the amplitude of viscous force can be evaluated as below,

,so the (2d)

Energy loss per cycle of the dash pot in Kelvin model can also solved by the given equation below:


4.3 Three-parameter Maxwell Model(Standard Linear Solid model)

Although the maxwell and kelvin model is used widely to describe the viscoelasticity characteristics,the disadvantage of these models is still significant,the Maxwell model failed to describe the characteristic of creep and the Kelvin model failed to describe the characteristic of stress relaxation.To describe both of the two features in one model,the Standard Linear Solid model is proposed by Zener.This is the simplest model to predict both phenomena.In this model,a maxwell model and a elastic spring is assembled in parallel,which can be seen in pic.7 below.

Pic.7 Standard linear solid model

In the time domain,you can easily figure out that for the dashpot.Obviously the equation and .

According to these two equation above,the equation between force and displacement can be modeled as

In the frequency domain since the ,if we apply a harmonic excitation to this system,you will find that the frequency characteristics is very similar to the ones in maxwell and kelvin model.The amplitude of elastic force of spring 1can be described as below,just like the one in kelvin model.

and the energy loss per cycle is zero since it is a spring.Then focus on the maxwell component of this model,the frequency features is the same with the maxwell model.

The energy loss per cycle .

Displacement amplitude,

4.4 Model using a smoothly slipped friction component

Pic.8 Model with a friction component[9]

This model is proposed by Mats Berg in 2002. To take the Payne effect[10] into consideration and make the simulation more accurate,a component of friction force is employed.In this model,three components represent different forces is used to simulate the viscoelasticity characteristics.The elastic force is simulated by the hook spring,the viscous force is described by the maxwell model and the friction force is presented by a smoothly slipped friction-component.This friction component has two parameters to simulate the nonlinear characteristic due to the Payne effect,one is the maximum friction force, ,and the displacement needed to reach half of the

maximum friction force, .The is a factor describes the speed of this system,smaller means a faster friction-force development[11].

Figure.4 Comparison of two type of friction components

Compare the smoothly slipped friction component with the Coloumb type friction component which is made up by friction spring with stiffness and in the elastic spring with stiffness K.By plotting the force versus displacement the smoothly slipped friction component gives a better description of hysteresis loop where the curves are smoother to fit the hysteresis loop more accurately.

Depending on the relationship between and , the friction force against the in the model can be written as below[9],

(Note:The auxiliary quantity ranges from -1 to 1. )

The elastic force of the spring is and the viscous force is

Here we have the force-displacement equation of these three forces,we can obtain a a hysteresis loop if we sum all these three forces. The hysteresis loop is helpful to calculate the stiffness and damping of the damper at a particular excitation.

In the frequency domain,for harmonic displacement excitation , it can be shown that the steady-state force amplitude and the energy loss per cycle are given by[9] ,

Elastic Force:

(1)The energy loss per cycle:

(2)Displacement amplitude:

Viscous Force:

(1)The energy loss per cycle .

(2)Displacement amplitude,

4.5 Model using a Springpot

Many efforts have been made to obtain a accurate description of viscoelasticity materials and reduce the number of parameters at the meantime.In fact,to obtain a high accuracy under the high displacement conditions,simply using a maxwell or kelvin model as the viscous component is not sufficient.The generalized model had been employed before the fractional derivative component to achieve a higher precision.But compare with the fractional derivative component,the generalized maxwell model need 19 parameters to obtain the same accurate description of viscoelastic materials' frequency dependence.

The fractional derivative component is first proposed by Shimizu and Zhang [12] as well as Rossikhin and Shitikova [13].After the proposal,many models based on the theory has been arisen,Bagley and Torvik[3] firstly created a 5-parameter model which is equivalent with the standard linear solid model by using the fractional calculus and then Koeller[14] arose a model called springpot to simulate the fractional derivative component,afterwards a test in frequency domain has be made by Cosson and Michon[15].In 2002,a new model based on the springpot and the smoothly slipped friction-component was employed by Sjöberg, Mattias M. and Kari[11].The model can be described as below,

In this model,the elastic force is still using the hook spring to simulate,the frictional force is described by the same component used in the smoothly slipped friction model and it uses a springpot component to displace the maxwell model to represent the viscous force.

In the time domain,the constitutive equations of this model can be obtained by the follow equations.


And in the frequency domain,it is easy to obtain the frequency characteristic of the elastic force and the friction force as me have mentioned in previous equation.The challenge in this model is to simulate the fractional derivative component in frequency domain.

The definition used to describe the fractional derivatives by Sjöberg is the Grunwald-Letnikov definition

But this shape of formula is not easy enough to simulate the fractional derivative,so another time discrete equation is introduced by Sjöberg to obtain the frequency characteristic of a springpot.

Note:the here is the gamma function.

And for the harmonic excitation ,calculate by using the another widely used definition of fractional derivatives,the Cauchy definition,we can get,

From the equation we get above,the amplitude of the springpot under a harmonic excitation is .

The energy loss per cycle can be evaluated by the function,

4.6 Model using a Parabolic spring

This model was proposed by W. Thaijaroen and A.J.L. Harrison[16] in 2008.As the latest model to simulate properties of viscoelasticity materials,it has a higher accuracy than the other models with only 6 parameters.This model is developed from the springpot model which is mentioned above.The main difference between the two models is that a parabolic model is employed instead of the hook spring which can be seen in the below.

In the time domain,this model also have the relationship between force and displacement as,


In the frequency domain,apply a harmonic excitation to the model and check the amplitude and energy loss per cycle on the nonlinear spring.

So the amplitude

And the energy loss per cycle

The amplitude and energy loss per cycle on friction component and the viscous component is the same with model of springpot.

5. Determination of parameters

5.1 How to read the plot of hysteresis loop[9]

When you drawing a picture of force versus displacement,you will find that the figure is a closed curve which is called hysteresis loop.The hysteresis loop is caused by the damping of the viscoelasticity materials.

Figure.5 Hysteresis loop

By this plot,you can figure out the parameters: and .To determine all these three parameters,one firstly need to make sure the measured hysteresis loop is at the center of the plot,so that the maximum and minimum point( and ) of the hysteresis loop can be written as and .The the tangent stiffness of the loop(the dashed line in the plot) just before these points is approximately equal to the elastic stiffness parameter .

Then you can determine the by the vertical distance between the two dashed line which represent the .The two dashed lines are parallel to each other and the vertical distance of them is approximately twice .

At last,the can also be determined by the equation.The is the max tangent stiffness of the hysteresis loop which can be obtained around the and .

5.2 Determination of other parameters

As the and is determined by the hysteresis loop,the energy loss per cycle of the friction component can be calculated.The total energy loss per cycle can also be obtained by the surrounded area of the hysteresis loop.Determine the viscous energy loss per cycle by the equation of .By changing the value of input frequency and use the equation of

(springpot model) and (maxwell model),the rest parameters , and will be obtained.

5.3 Optimization of parameters[16]

6.Comparison of outcomes by different models

6.1 Frequency domain

Figure.6 [9]

This is the simulation result of the 5-parameter model using a smooth friction component,a maxwell viscous component and a spring in the frequency domain.The graph gives a description when plotting stiffness and damping versus frequency for harmonic excitation.It indicates that a significant error exists in the simulated result of stiffness in the whole range of given frequency.But the simulated result of damping presents a consistency with the measured result.

Figure.7 [11]

This is the simulation result of the 5-parameter model using a smooth friction component,a springpot and a spring in the frequency domain.The graph gives a description when plotting stiffness and damping versus frequency for harmonic excitation.The damping result matches the measurement pretty accurately while the consequence of stiffness shows some discrepancy in the high frequency(>70HZ).

Figure.8 [16]

This is the simulation result of the 6-parameter model using a smooth friction component,a springpot and a parallel spring in the frequency domain.The graph gives a description when plotting stiffness and damping versus amplitude for harmonic excitation.The simulated result obtained a super accuracy in both stiffness and damping consequence except a slightly error in the simulation of PU-HC and PU-LC which have a common ground in the material(polyurethane).

6.2 Amplitude-domain

Figure.9 [9]

This is the simulation result of the 5-parameter model using a smooth friction component,a maxwell viscous component and a spring in the amplitude domain.The graph gives a description when plotting stiffness and damping versus displacement for harmonic excitation.It indicates the both of simulated results of stiffness and damping matches the measured result quite good with a amplitude less than 10mm.

Figure.10 [11]

This is the simulation result of the 5-parameter model using a smooth friction component,a springpot and a spring in the amplitude domain.The graph gives a description when plotting stiffness and damping versus displacement for harmonic excitation.It indicates the simulated result of stiffness matches the measured result pretty well while the damping result shows a significant inconformity in the small amplitude(<0.3mm).


This is the simulation result of the 6-parameter model using a smooth friction component,a springpot and a parallel spring in the amplitude domain.The graph gives a description when plotting stiffness and damping versus displacement for harmonic excitation.It indicates the simulated result of stiffness matches the measured result accurately except a over-estimate of the stiffness under a relatively high amplitude condition.

6.3 Hysteresis loops


You can tell from the pictures above that although the five-parameter model can predict a relatively accurate stiffness and damping properties,it still fails to simulate the imbalance of restoring forces,especially at a high amplitude.On the contrary,the six-parameter model gives a better description of the hysteresis loop.