The standard linear solid model, on the other hand, considers a maxwell element which has a spring and dashpot in sequence in parallel with a. Modeling, control, and applications by yang fang a dissertation submitted to. The linear elastic solid the linear spring the simplest way to create a model of a material is to suppose that it consists of nothing but a linear spring of stiffness e. Based on the fractional lossy inductance model and the fractional standard linear solid model, this work presented a fractional equivalent electrical circuit feec for modeling the linear behaviors of movingcoil loudspeakers. Although the maxwell model did not give us a realistic result for a viscoelastic solid it is more representative of a viscoelastic fluid it did allow us to illustrate some important aspects of a viscoelastic material including the stress relaxation function gt and the creep function jt. The standard linear solid model slsm is a typical and useful model for analyzing. The tf model object can represent siso or mimo transfer functions in continuous time or. Transfer function representation of linear physical systems. Alternative form of standard linear solid model for.
Linear viscoelasticity numerical tours of continuum. In the linear case the transfer function fs of a control system is usually expressed as the ratio of two polynomials in the laplace operator s with real coef. This constitutive equation has been put in what is known as standard form stress on. The relations between transfer functions and other system descriptions of dynamics is also discussed. Seismic modelling, time domain, standard linear solid, constant q. Fractional modeling on movingcoil loudspeaker linear system.
These models are often proved insufficient, however. For the kelvin or standard linear solid model, there are two different but equivalent configurations. The simplest model that predicts creep and stressrelaxation is the standard linear solid sls with a parallel combination of a maxwell arrangement spring and dashpot in series with a spring fung, 1981. The simulations show that the nonlinear version of the maxwell sls model can result in a time dependent. The standard linear solid sls, also known as the zener model, is a method of modeling the behavior of a viscoelastic material using a linear combination of springs and dashpots to represent elastic and viscous components, respectively. Formal transfer function rational transfer function iir infinite impulse response model broad class of inputoutput linear models differentiation operator formal transfer function rational function of s for a causal system m. Show that the constitutive equations for a standard linear solid can be expressed by show that the constitutive equation for a standard linear solid model can be expressed by. Fractional modeling and analysis of coupled mr damping. The standard linear model consider next the standard linear model, which consists of a spring in series with a kelvin unit, fig. Some basics of rheology university of nevada, reno. A variety of mathematical models were suggested to describe the stressstrain response of. Viscoelastic effects are often present in loudspeaker suspensions, this can be seen in the displacement transfer function which often shows a frequency dependent value below the resonance frequency.
Classroom methods for demonstrating, modelling and. Response and recovery times of elastic and viscoelastic. Frequencies above 1012 hz correspond to molecular vibration and represent an upper limit for stress waves in real solids. Strain in both elements of the model is the same and the total stress is the sum of the two contributions. A better model is the standard linear solid which is shown below.
It must be stressed in the class that the solid linear standard is a rather simple model and that almost no material could be described according to that model. The generalized standardlinearsolid model, also called the zener model. The strain as a function of time during the creep test using a unit step stress function. In this paper nonlinear versions of the standard linear solid model sls are investigated. The dash pot then takes up the stress, transferring the load to the second spring as it slowly opens over time. The first is that the transfer function representation of an equivalent timedomain nonlinear system consists of a sequence of transfer functions instead of only one function in the linear case. A novel nonlinear viscoelastic solid model sciencedirect. Massachusetts institute of technology department of. Strain response for a stressstep for a elastic b elastoplastic, c viscoelastic and d viscoplastic material behavior another way of representing the measurement data of the tensile experiment is by plotting the stress against the strain, resulting in the stressstrain curve. Approximating constantq seismic propagation in the time. Linear elastic body, hooke body hooke model, hooke element, elastic spring, represents behavior of a perfectly elastic lossless solid material.
Creep, relaxation and viscosity properties for basic. In this respect, the response functions to an excitation expressed by the heaviside step function. Mathematical modeling of linear viscoelastic impact. According to the linear theory, the viscoelastic body can be considered as a linear system with the stress or strain as the excitation function input and the strain or stress as the response function output. Stress waves from 20 hz to 20 khz are perceived as sound above 20 khz is the ultrasonic range.
Jaroslav vond rejc born on the 27th of june 1983 in opocno january 2009 czech technical university in prague. Transfer functions of discretetime nonlinear control systems. Mathematically the transfer function is a function of complex variables. Constitutive models of linear viscoelasticity using laplace transform author. Bme 315 biomechanics material properties of tendon. Voigt model maxwell mdel essentially assumes a uniform distribution of stress. The spring represents the elastic component of the models response. Constitutive models of linear viscoelasticity using. The standard linear solid model sls also know as the kelvin model is a method of modeling the behavior of a viscoelastic material using a linear combination of springs and dashpots to represent elastic and viscous components, respectively. Standard linear solid model, but other models could have been used as well. Deriving the equations modeling these systems is beyond the scope of this course.
Here, we revisit the generalized standard linear solid model and seek to overcome the confusion concerning the expression for the relaxation function or modulus. Fractional modeling and analysis of coupled mr damping system. However we will look at the time dependent behavior of the tendon and. Each element carries the same load while strains are added. Cartilage has no known function other than maintaining mechanical. Often, the simpler maxwell model and the kelvinvoigt model are used. In the kelvinvoigt model, membrane stress is computed as the sum of elastic and viscous stress resultants. Estimate transfer function models with prior knowledge of model structure and constraints this example shows how to estimate a transfer function model when the structure of the expected model is known and apply constraints to the numerator and denominator coefficients. Viscoelastic response is often used as a probe in polymer science, since it is. The dashpot model describes purely viscous fluid figure 1b. Rational approximations of transfer functions of some. In particular, simple analytical approximations are derived for the impact duration, t c, and for the coe cient of restitution, e.
Is carried out and the axial force f is plotted as a function of. Bme 315 biomechanics material properties of tendon the three most common models combining these elements are the maxwell, voight, and kelvin standard linear solid model fig. The constitutive equation for the linear elastic solid is then simply e 1 the creep and recovery behaviour of the material to a suddenly applied load o is. Linear viscoelasticity an overview sciencedirect topics. Transfer functions the roots of the polynomial as are called poles of the system and the roots of bs are called the zeros of the system. Although either can be used, for ease of calculation the 3 element generalized maxwell model gm is used for sr and epsilon dot, while the sls kelvin form of the model is often used in the case of creep. Inaseries connection such as themaxwell model, thestresson each element is the same and equal to the imposed stress, while thetotal strain is the sumof the strainineachelement. The maxwell model does not exhibit creep since it models strain as linear function of time. After some manual rearrangement, the radial displacement. When the material is viscoelastic the strain shows time delayed response on a stress step, which indicates a. Inaseries connection such as themaxwell model, thestresson each element is the same and equal to the imposed stress, while thetotal strain is the sumof the. Equation 7 was used to curve fit the stress response i. Upon loading one expects the lefthand spring to stretch immediately.
In the sls configuration a maxwell element is connected in parallel with a second spring this setup is also known as the zener model. We assumed that the intestine was a standard linear solid and described the change of opening angles of intestine as function of time using this model. Introduction the transfer function plays an important role in the linear theory. Mechanical models, such as the maxwell model, the voigt model, and the kelvin model also called the standard linear solid 25,26 are often used to describe the viscoelastic behavior of materials. Instead, one could, for example, use the standard linear solid sls model, which is a combination of the maxwell and kelvinvoigt models. However, the geophysical literature shows two different expressions for the relaxation function or modulus in the generalized sls model. For instance, consider a continuoustime siso dynamic system represented by the transfer function syss nsds, where s jw and ns and ds are called the numerator and denominator polynomials, respectively. This will introduce the notion of the distribution of characteristic times and all the formalisms associated with the relaxationretardation spectra in the. The constitutive equation for the linear elastic solid is then simply e 1 the creep and recovery behaviour of. The transfer function of this linear system thus will be rational function, note that, as and bs are given above as polynomial of system. Timedependent viscoelastic properties along rat small. The spring model describes an ideally elastic body dt d g s h. In order to get analytical approximations, we consider the standard solid model as a perturbation of the kelvinvoigt section 5 or the maxwell model section 6. Moreover, it benefits from less computational costs in computer time and memory.
Often more similar maxwell model and the kelvinvoigt model are used. An alternative configuration, also commonly referred to as the sls, consists of a linear spring connected in series with a kelvin element, which consists of a. Transfer functions are a frequencydomain representation of linear timeinvariant systems. Pdf the generalized standardlinearsolid model and the. The formulation can also be quite easily extended to a generalized maxwell model. The linear elastic solid has been the main material model analysed in this book thus far. The creep compliance function and the relaxation modulus have.
The transfer function appoach for general nonlinear systems has two principle differences compared to the traditional linear techniques. Some basics of rheology standard linear solid will not relax to zero stress kelvin element maxwell element hook spring newtonian. The standard linear solid sls model, also known as the zener model, is constructed by connecting a linear spring in parallel with a maxwell element, which consists of a linear spring and a linear dashpot in series, as shown in fig. The fractional order derivatives have been used in modeling the lossy inductance as well as the creep effect for movingcoil loudspeakers. If a small stress is applied for a sufficiently long time, then the irreversible strains become large. Solution of transfer function problems as stated previously, the transfer function is a common and extremely powerful method of representing a system mathematically. The dash pot then takes up the stress, transferring the load to the second spring as it. Non linear modal analysis methods for engineering structures by hugo ramon elizalde siller. Creep and relaxation functions of the basic constitutive models 2.
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