CN111967121B - Viscoelastic dielectric elastomer uncertainty quasi-static and nonlinear dynamics analysis method based on interval method - Google Patents

Viscoelastic dielectric elastomer uncertainty quasi-static and nonlinear dynamics analysis method based on interval method Download PDF

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CN111967121B
CN111967121B CN202010542320.5A CN202010542320A CN111967121B CN 111967121 B CN111967121 B CN 111967121B CN 202010542320 A CN202010542320 A CN 202010542320A CN 111967121 B CN111967121 B CN 111967121B
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李云龙
刘晨浩
王磊
王晓军
邱志平
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Beihang University
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Abstract

The invention discloses a viscoelastic dielectric elastomer uncertainty quasi-static and nonlinear dynamics analysis method based on an interval method. In the method, a viscoelastic dynamic model of the dielectric elastomer is provided, and in consideration of uncertainty of material parameters, external load and voltage, creep analysis, relaxation analysis and dynamic response analysis are carried out on the dielectric elastomer with the uncertain parameters of intervals by introducing an interval perturbation method and a first-order Taylor expansion method. The effectiveness of the interval method provided by the invention is verified by a Monte Carlo simulation method, and the uncertainty prediction method can be used in the design of an active control system using a dielectric elastomer as a driver or a sensor in the future.

Description

Viscoelastic dielectric elastomer uncertainty quasi-static and nonlinear dynamics analysis method based on interval method
Technical Field
The invention relates to the field of quasi-static and nonlinear dynamics analysis of dielectric elastomers, in particular to a dielectric elastomer quasi-static and nonlinear dynamics analysis method based on an interval method when uncertainty of parameters such as external force, materials, voltage and the like is considered, and provides a feasible design method for an active control system using a dielectric elastomer as a driver or a sensor.
Background
Dielectric Elastomer (DE), a soft active material, has been widely used in many engineering fields such as artificial muscle drivers, acoustic drivers, sensors, speakers, active vibration control, etc. due to its characteristics of large deformation, light weight, high flexibility, etc. After voltage is applied to compatible electrodes at two ends of the dielectric elastomer, charges on the upper and lower layers of electrodes interact with each other, so that the elastomer expands in a plane and reduces the thickness, electric energy is converted into mechanical energy, and the dielectric elastomer can be used for designing or manufacturing an intelligent driver based on the principle; on the contrary, when a certain external force is applied, the dielectric elastomer deforms, so that the electrodes on the two sides change in charge, a capacitance signal is output and monitored by the tester, after the external force is cancelled, the capacitance changes again, the operation is repeated, the relation between the tensile strain and the capacitance is expressed, and the sensor can be used as a skin sensor of the artificial muscle of the robot. With the rapid development of dielectric elastomer applications, accurate characterization of its mechanical properties is increasingly important.
Over the last several decades, researchers have studied the dynamics of dielectric elastomers in both theoretical and experimental respects, respectively. Test results show that viscoelasticity can significantly influence the dynamic response of the dielectric elastomer, and in order to obtain better control, researchers consider the influence of viscoelasticity and obtain a constitutive model of the dielectric elastomer. For quasi-static cases, many models have been proposed for the creep and relaxation behavior of dielectric elastomers based on non-equilibrium thermodynamic theory, complex-mode theory, standard linear solid-state models, and combinations of different theories. For the dynamic case, the viscoelastic effect will generate a damping force to dissipate the input energy, some researchers consider the damping force as an external force and study the dynamic response with different damping factors, and then get a more general dynamic analysis.
While the above research models are based on deterministic theory, many practical factors may introduce uncertainties that have also been shown to have significant impact on alignment static and dynamic properties. Some have studied the uncertainty of dielectric elastomers, such as jinshiling and huangshilong, for free response of spherical dielectric elastomers disturbed by electronic or mechanical fluctuations. Typically, small errors in the input parameters will result in significant deviations in the kinetic analysis, so to ensure accurate analysis of the upper and lower bounds of the structural response, a large amount of data needs to be collected to obtain the probability density function, but for exploratory materials such as dielectric elastomers, not enough data is collected in practical applications, and it may be better to deal with the uncertainty of the parameters using non-probabilistic methods such as interval methods or convex models.
To date, the dynamic response of viscoelastic dielectric elastomers has not been analyzed in view of uncertainty in material parameters. To fill this gap, in the present invention, a theoretical model is built to analyze the effect of material uncertainty on the quasi-static and nonlinear dynamic responses of viscoelastic dielectric elastomers. The method provided by the invention provides a new thought for designing the dielectric elastomer driver with uncertain parameters, and provides a theoretical basis for further reliability evaluation, model verification and active control of the dielectric elastomer.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the method overcomes the defects of the prior art, provides a viscoelastic dielectric elastomer uncertainty quasi-static and nonlinear dynamics analysis method based on an interval method, fully considers parameter uncertainties (such as uncertainties of material parameters, external force loads, geometric parameters and the like) commonly existing in engineering practical problems, provides a feasible design method for an active control system using a dielectric elastomer as a driver or a sensor, and obtains a design result which is more consistent with a real situation and stronger in engineering applicability.
The technical scheme adopted by the invention for solving the technical problems is as follows: a viscoelastic dielectric elastomer uncertainty quasi-static and nonlinear dynamics analysis method based on an interval method is used in the design of an active control system using a dielectric elastomer as a driver or a sensor, and comprises the following implementation steps:
the first step is as follows: obtaining a dynamics control equation of a planar rectangular dielectric elastomer film based on a virtual work principle;
the second step is that: simplifying problems and equations by neglecting inertial force terms under the quasi-static condition;
the third step: solving an interval boundary of the creep response of the dielectric elastomer based on interval analysis and a Taylor expansion method;
the fourth step: solving an interval boundary of dielectric elastomer relaxation response based on interval analysis and a Taylor expansion method;
the fifth step: and solving the interval boundary of the dynamic response of the dielectric elastomer based on the interval analysis and the Taylor expansion method.
Because of the uncertainty in the parameters, the responses (displacement, elongation ratio, stress) of the dielectric elastomer film are also uncertain, and the interval boundaries are the upper and lower bounds of these responses; the invention only provides a response interval solving and predicting method aiming at parameter uncertainty of the dielectric elastomer at present, and can be used in the design of an active control system taking the dielectric elastomer as a driver or a sensor.
Further, the first step of obtaining a kinetic control equation based on the virtual work principle specifically includes:
a planar rectangular dielectric elastomer film having dimensions of 2L in the X, Y and Z directions in an initial state, i.e., an undeformed state 1 ,2L 2 And 2H; respectively P in X and Y directions 1 And P 2 Under the external force of (2), the dimensions of the dielectric elastomer film in the X, Y and Z directions become 2l 1 ,2l 2 And 2h; the ratio of elongation in the X and Y directions can be expressed as λ 1 ＝l 1 /L 1 And λ 2 ＝l 2 /L 2 (ii) a Since elastomers are generally assumed to be incompressible, λ 1 λ 2 λ 3 =1, wherein the elongation ratio in the Z direction is λ 3 ＝h/H＝1/λ 1 λ 2 . By using a Zener model and an ideal dielectric elastomer model, a free energy function considering the viscoelasticity of the dielectric elastomer is obtained as follows:
Figure BDA0002539421240000031
wherein W stretch Represents the viscoelastic free energy of the elastomer, W electric Denotes the dielectric elastic free energy of the elastomer, D is the electrical displacement, ε is the dielectric constant of the elastomer, μ 1 And mu 2 Is the shear modulus of two springs, J lim1 And J lim2 Is a constant, ξ, related to the extension limit of the two springs 1 And xi 2 Is a deformation of the damper and is,
Figure BDA0002539421240000032
and
Figure BDA0002539421240000033
is and λ 1 And λ 2 The elongation ratio of the relevant spring is given by the following relation:
Figure BDA0002539421240000034
wherein λ is 1 ,λ 2 And xi 1 ,ξ 2 The relationship between them is that eta is viscosity,
Figure BDA0002539421240000035
assuming that the charge on the dielectric elastomer changes from an equilibrium state by an amount δ Q, the work performed by the voltage Φ is Φ δ Q; when the length of the elastomer in both the X and-X directions is from equilibrium,. Delta.l 1 When the external force is changed, the work of the external force in the X and-X directions is 2P 1 δl 1 (ii) a The inertial force works as follows, also in the Y and-Y directions:
Figure BDA0002539421240000036
where u is the displacement field of the elastomer,
Figure BDA0002539421240000037
indicating acceleration, V 0 Is the volume of the elastomer, phi is the external voltage, p 0 For density, according to the thermodynamic theory, for any equilibrium state, i.e. the fully relaxed state, the free energy of the material changes due to the damper not having time to deformThe work of external load, inertia force and voltage is divided; namely:
Figure BDA0002539421240000038
wherein the relationship between δ λ and δ l is:
Figure BDA0002539421240000041
according to the variation principle, the following are obtained:
Figure BDA0002539421240000042
since the damper of the model is assumed not to have enough time to deform, δ ξ can be made to be 1 ＝0,δξ 2 =0; the relationship between the electrical displacement and the amount of charge on the electrode is:
Figure BDA0002539421240000043
based on the above formula, we obtain:
δQ＝4l 1 l 2 δD+4l 2 δl 1 D+4l 1 δl 2 D＝4l 1 l 2 δD+4l 2 L 1 δλ 1 D+4l 1 L 2 δλ 2 D (9)
substituting formula (7) and formula (9) for formula (5) yields:
Figure BDA0002539421240000044
from the free energy function in equation (1), we obtain:
Figure BDA0002539421240000045
then, formula (11) is substituted for formula (10), and the independence between the three parameters is used:
Figure BDA0002539421240000051
by substituting the third equation in equation (12) into the other two equations, thereby eliminating the potential shift D, the final control equation is obtained:
Figure BDA0002539421240000052
furthermore, the simplification problem and the equation of the neglected inertia force term under the second quasi-static condition specifically comprise;
for quasi-static problems including creep and relaxation, the inertial terms are ignored, i.e. omitted
Figure BDA0002539421240000053
And
Figure BDA0002539421240000054
the problem thus becomes solving the following equation, which is the governing equation for viscoelastic dielectric elastomers in the quasi-static case:
Figure BDA0002539421240000055
wherein,
Figure BDA0002539421240000061
reduce the problem to λ 1 ＝λ 2 ＝λ，ξ 1 ＝ξ 2 ＝ξ，P 1 ＝P 2 ＝P，L 1 ＝L 2 = L and g 1 ＝g 2 Case of = g, the kinetic equation is then converted to the following equation:
g(P,Φ,λ,ξ)＝0 (16)
wherein, the specific expression of the equation g (P, Φ, λ, ξ) =0 is:
Figure BDA0002539421240000062
further, the third step of solving the interval boundary of the creep response based on the interval analysis and the taylor expansion method specifically includes;
for creep response in quasi-static problems, let force P 0 At t 0 Is applied to the dielectric elastomer at a time and remains constant over time; when t is>t 0 Then, the elongation ratio shown by the following formula is obtained:
Figure BDA0002539421240000063
all parameters of the equation, including external load, material parameter and geometry parameter, cause a certain amount of uncertainty; defining an uncertain parameter vector as b = [ b ] 1 ,b 2 ,...,b m ] T And their interval is:
Figure BDA0002539421240000071
wherein, b I In order to be an uncertainty interval,bandb i in order to obtain the lower limit of the interval,
Figure BDA0002539421240000072
and
Figure BDA0002539421240000073
is the upper limit of the interval, b c And
Figure BDA0002539421240000074
for nominal values of the interval, Δ b and Δ b i M is an integer and is the interval width.
Since the uncertainty parameter varies within a given interval to find the limit Γ of the response, i.e. the elongation ratio, as shown in the following equation:
Γ＝{λ:g(b)＝0,b∈b I } (20)
find an interval with exact bounds of the envelope, i.e.:
Figure BDA0002539421240000075
wherein,λand
Figure BDA0002539421240000076
comprises the following steps:
Figure BDA0002539421240000077
to determine the upper and lower limits of the elongation ratio, a first order taylor expansion is used here:
Figure BDA0002539421240000078
wherein,
Figure BDA0002539421240000079
then, according to the natural expansion principle of interval calculation, the interval boundary of the creep response is obtained as shown in the following formula:
Figure BDA00025394212400000710
wherein,
Figure BDA00025394212400000711
thus, with interval uncertainty parametersInterval limit of electric elastic body creep response
Figure BDA00025394212400000712
It is determined that:
Figure BDA0002539421240000081
further, the fourth step of solving the interval boundary of the relaxation response based on the interval analysis and the taylor expansion method specifically includes;
for relaxation behavior in quasi-static problems, let the elongation ratio λ 0 At t 0 Is constantly applied to the dielectric elastomer and remains constant; when t is>t 0 Then, the stress shown by the following formula is obtained:
Figure BDA0002539421240000082
using the same method as in the third step, the interval for obtaining the relaxation response is bounded by
Figure BDA0002539421240000083
Comprises the following steps:
Figure BDA0002539421240000084
further, the fifth step of solving the interval boundary of the dynamic response based on the interval analysis and the taylor expansion method specifically comprises the following steps of;
for the nonlinear dynamical response problem of the uncertain dielectric elastomers, the external excitation is a function of uncertain parameters, so the dynamical equation is described by the following nonlinear parametric excitation problem:
Figure BDA0002539421240000085
wherein,
Figure BDA0002539421240000086
and x is acceleration, velocity and displacement, respectively, b is the uncertainty parameter vector of the aforementioned dielectric elastomer, f (t, d) is the external excitation, d is the uncertainty parameter vector of the excitation load; the exact boundaries of the time domain response are:
Figure BDA0002539421240000087
as with the quasi-static creep behavior analysis in the third step, find an interval vector Λ containing the true kinetic response rather than the exact boundary:
Figure BDA0002539421240000088
wherein,
Figure BDA0002539421240000091
then, using a first order taylor expansion method and according to the natural expansion theory of interval arithmetic, the interval boundary of the kinetic response is described as:
Figure BDA0002539421240000092
wherein,
Figure BDA0002539421240000093
thus, the interval boundaries of the kinetic response are obtained
Figure BDA0002539421240000094
Comprises the following steps:
Figure BDA0002539421240000095
advantageous effects
Compared with the prior art, the method has the advantages that a new idea of viscoelastic dielectric elastomer quasi-static and nonlinear dynamics analysis in consideration of uncertainty of parameters such as materials, external force and the like is provided, and the limitations of the traditional dielectric elastomer quasi-static and nonlinear dynamics analysis theory and method based on certainty and a probability uncertainty model are made up and perfected. The dielectric elastomer quasi-static and nonlinear dynamics analysis model is established, on one hand, the influence of material viscoelasticity on the characteristics of the dielectric elastomer quasi-static and nonlinear dynamics analysis model is considered, on the other hand, the influence of non-probability uncertainty of parameters such as materials, external force, geometry and the like on the characteristics of the dielectric elastomer quasi-static and nonlinear dynamics analysis model is considered, a new idea is provided for the design of a dielectric elastomer driver with uncertain parameters, and a theoretical basis is provided for further reliability evaluation, model verification and active control of the dielectric elastomer.
Drawings
FIG. 1 (a) is a schematic view of a planar dielectric elastomer in an initial state;
FIG. 1 (b) is a schematic view of the current state of a planar dielectric elastomer;
FIG. 2 is a viscoelastic model of a dielectric elastomer;
FIG. 3 is a section boundary stretched by the creep process;
FIG. 4 is a comparison of results obtained by the creep process interval method and the Monte Carlo simulation method;
FIG. 5 is a span boundary of relaxation process stress;
FIG. 6 is a comparison of results obtained by the relaxation process interval method and the Monte Carlo simulation method;
fig. 7 (a) is the damping effect of the viscosity η =100 dielectric elastomer free vibration;
fig. 7 (b) is the damping effect of viscosity η =500 of the free vibration of the dielectric elastomer;
fig. 7 (c) is the damping effect of the viscosity η =1000 of the free vibration of the dielectric elastomer;
FIG. 8 is a time domain dynamic response of a dielectric elastomer film under 1Hz harmonic voltage excitation;
FIG. 9 is a frequency domain dynamic response of a dielectric elastomer film;
FIG. 10 is a flow chart of the quasi-static and non-linear kinetic analysis of the viscoelastic dielectric elastomer of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be described clearly and completely with reference to the accompanying drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, rather than all embodiments, and all other embodiments obtained by a person skilled in the art without creative efforts based on the embodiments in the present invention belong to the protection scope of the present invention.
The invention provides a viscoelastic dielectric elastomer uncertainty quasi-static and nonlinear dynamics analysis method based on an interval method, as shown in figure 10, comprising the following steps:
(1) And obtaining a general dynamics control equation based on the virtual work principle. Consider a rectangular planar dielectric elastomer film as shown in fig. 1, which is widely used in practical applications. As shown in FIG. 1 (a), the dimensions of the dielectric elastomer film in the X, Y and Z directions in the initial state (undeformed state) are 2L, respectively 1 ,2L 2 And 2H. The film expands in the X-Y plane while decreasing in thickness in the Z direction. The external forces in the X and Y directions are respectively P 1 And P 2 . As shown in FIG. 1 (b), in the present state, the dimensions of the dielectric elastomer film in the X, Y and Z directions become 2l 1 ,2l 2 And 2h. The ratio of elongation in the X and Y directions can be expressed as λ 1 ＝l 1 /L 1 And λ 2 ＝l 2 /L 2 . Since elastomers are generally assumed to be incompressible, λ 1 λ 2 λ 3 =1, wherein the elongation ratio in the Z direction is λ 3 ＝h/H＝1/λ 1 λ 2 . In order to take into account the influence of the viscoelasticity of the dielectric elastomer, a Zener model as shown in fig. 2 is used.
Figure BDA0002539421240000101
Wherein mu 1 And mu 2 Is the shear modulus of two springs, J lim1 And J lim2 Is a constant, ξ, related to the stretch limits of the two springs 1 And xi 2 Is a deformation of the damper, λ 1 e And
Figure BDA0002539421240000118
is and λ 1 And λ 2 The elongation ratio of the associated spring.
λ 1 ＝λ 1 e ξ 1 ，
Figure BDA0002539421240000119
Due to the adoption of the ideal dielectric elastomer model just proposed by the lock mark, the free energy function considering the viscoelasticity is:
Figure BDA0002539421240000111
wherein W stretch Represents the viscoelastic free energy of the elastomer, W electric Represents the dielectric elastic free energy of the elastomer, where D is the true electrical displacement and ε is the dielectric constant of the elastomer. Based on the virtual work principle, the invention obtains the following kinetic control equation:
Figure BDA0002539421240000112
where phi is the external voltage and where phi is the external voltage,
Figure BDA0002539421240000113
indicating acceleration, V 0 Is the volume of the elastomer, H is the initial thickness of the membrane, ρ 0 Is the density. When the viscosity is η, the relationship between λ and ξ is:
Figure BDA0002539421240000114
Figure BDA0002539421240000115
(2) The problem and equation are simplified by neglecting the inertial force term in the quasi-static condition. For quasi-static problems including creep and relaxation, the inertial term can be ignored, i.e., omitted
Figure BDA0002539421240000116
And
Figure BDA0002539421240000117
the problem thus becomes solved for the following equation, which is the governing equation for the viscoelastic dielectric elastomer in the quasi-static case:
g 1 (P 1 ,Φ,λ 1 ,λ 2 ,ξ 1 ,ξ 2 )＝0
g 2 (P 1 ,Φ,λ 1 ,λ 2 ,ξ 1 ,ξ 2 )＝0
wherein,
Figure BDA0002539421240000121
without loss of generality, the present invention reduces the problem to λ 1 ＝λ 2 ＝λ，ξ 1 ＝ξ 2 ＝ξ，P 1 ＝P 2 ＝P，L 1 ＝L 2 = L and g 1 ＝g 2 Case of = g, the invention may then convert the kinetic equation to the following equation:
g(P,Φ,λ,ξ)＝0
wherein,
Figure BDA0002539421240000122
Figure BDA0002539421240000123
(3) And solving the interval boundary of the creep response based on the interval analysis and the Taylor expansion method. In order to study the creep behaviour in quasi-static problems, the invention specifies a constant external force P 0 At t 0 Is applied to the dielectric elastomer film at a time and is held constant over time. When t is>t 0 In the present invention, the elongation ratio shown by the following formula can be obtained:
Figure BDA0002539421240000124
all parameters of the equation, including external forces, material parameters, and geometric parameters, may cause a certain amount of uncertainty. The invention defines the uncertain parameter vector as b = [ b ] 1 ,b 2 ,...,b m ] T And their interval is:
Figure BDA0002539421240000131
Figure BDA0002539421240000132
Figure BDA0002539421240000133
since the uncertain parameters vary within a given interval, the invention can find the limit of the response, i.e. the elongation ratio, as shown in the following equation:
Γ＝{λ:g(b)＝0,b∈b I }
the exact limits of the set described by the above equation are usually complex, and in the present invention, it is desirable to find an interval that can envelop the exact limits, namely:
Figure BDA0002539421240000134
wherein,
λ＝min{λ:g(b)＝0,b∈b I }
Figure BDA0002539421240000135
to determine the upper and lower limits of the elongation ratio, a first order taylor expansion may be employed here:
Figure BDA0002539421240000136
wherein,
Figure BDA0002539421240000137
then, according to the natural development principle of interval calculation, the interval boundary at which the creep response can be obtained is as follows:
Figure BDA0002539421240000138
wherein,
Figure BDA0002539421240000139
thus, the interval limits of the creep response of the dielectric elastomer with interval uncertainty parameters are determined as shown in the following equation:
Figure BDA00025394212400001310
Figure BDA00025394212400001311
(4) And solving the interval boundary of the relaxation response based on the interval analysis and the Taylor expansion method. In order to investigate the relaxation behavior in the quasi-static problem, the invention specifies a constant elongation ratio λ 0 At t 0 The time is applied to the dielectric elastomer and remains constant. When t is>t 0 In this case, the present invention can obtain a stress represented by the following formula:
Figure BDA0002539421240000141
using the same method as in the third step, the interval limit for obtaining the relaxation response of the present invention is shown by the following formula:
Figure BDA0002539421240000142
Figure BDA0002539421240000143
(5) And solving the interval boundary of the dynamic response based on the interval analysis and the Taylor expansion method. For the nonlinear dynamics of an indeterminate dielectric elastomer, its general dynamics equations can be described by the following nonlinear parametric excitation problem:
Figure BDA0002539421240000144
wherein,
Figure BDA0002539421240000145
and x is acceleration, velocity and displacement, respectively, c (t), k (t) and g (t) are different functions of time t, and f (t) is an external stimulus. From the third step, it can be seen that the external excitation, etc. are all functions of uncertain parameters, so the nonlinear parameter excitation equation can be rewritten as follows:
Figure BDA0002539421240000146
where b is the uncertain parameter vector of the dielectric elastomer described earlier and d is the uncertain parameter vector of the excitation load. The exact boundaries of the time domain response are:
Figure BDA0002539421240000147
as with the quasi-static creep behavior analysis in the third step, the present invention seeks to find an interval vector Λ that contains the true dynamic response rather than the exact boundaries:
Figure BDA0002539421240000148
wherein,
Figure BDA00025394212400001410
Figure BDA0002539421240000149
adopting a first-order Taylor expansion method to obtain:
Figure BDA0002539421240000151
wherein,
Figure BDA0002539421240000152
Figure BDA0002539421240000153
then, according to the natural expansion theory of interval arithmetic, the interval boundary of the kinetic response can be described as:
Figure BDA0002539421240000154
wherein,
Figure BDA0002539421240000155
Figure BDA0002539421240000156
thus, the present invention yields interval boundaries for the kinetic response shown by the following formula:
Figure BDA0002539421240000157
Figure BDA0002539421240000158
example (b):
in order to more fully understand the characteristics of the method and the applicability of the method to engineering practice, the invention aims at the creep and relaxation situations in the quasi-static problem to verify the proposed uncertainty quasi-static analysis method based on the interval method. Then, in order to verify the provided uncertainty nonlinear dynamics analysis method based on the interval method, the invention aims at the in-plane deformation dielectric elastomer to make two numerical simulation examples, one is a quasi-static problem and the other is a dynamics problem.
In an embodiment of the quasi-static problem, the material parameter is μ 1 ＝18000Pa，μ 2 ＝42000Pa，J lim1 ＝110， J lim2 ＝55，ρ 0 ＝960kg/m 3 ，ε＝3.9825×10 -11 F/m; the dimension parameter is L 1 ＝L 2 = L =1m, h =3mm. The present invention assumes 2% fluctuation in all material parameters, external loads and geometric parameters. For creep conditionShape, external force is P 1 ＝P 2 = P =0.9kN and voltage Φ =20kV; for the relaxation case, the initial conditions are d λ (0)/dt =0 and ξ (0) =1.
FIG. 3 shows the elongation ratio and its boundary between regions during creep of an in-plane deformed dielectric elastomer. Initially, there is not enough time for the damper to deform and all external forces are borne by the two springs in FIG. 2. Over time, the damper deforms and the load on the spring in series with the damper decreases until the stresses of the damper and the series spring are completely subtracted, all the external load being taken up by another spring in parallel with the Maxwell cell. Fig. 4 shows the results of a monte carlo simulation with 10000 samples, where it can be seen that the interval boundary obtained by the proposed interval method can envelop all sample results. Fig. 5 shows the interval boundaries and nominal values during relaxation, and a comparison between the interval method and the monte carlo simulation method is given in fig. 6. Generally, if the number of samples is large enough, the monte carlo simulation method can get an accurate boundary at a given parameter interval, but this method is very time consuming. In contrast, the results obtained by the method proposed by the present invention also give quite accurate results, albeit slightly more conservative than the exact results of the MCS, but much less time consuming. In addition, if the taylor expansion in the third step retains the second order terms and even higher order terms, more accurate results can be obtained.
In the embodiment of the dynamical problem, the present invention uses the same parameters as the quasi-static problem. For dielectric elastomers with in-plane deformation, in order to study the damping effect of different viscosities, the invention solves the free vibration problem with different viscosity values η =100, η =500, η = 1000. The boundaries of the interval of the dynamic response are shown in fig. 7 (a), 7 (b) and 7 (c), from which it can be seen that the influence of damping on the dynamic response is more significant as the viscosity coefficient increases. Fig. 8 shows the interval boundaries of the dynamic response of the dielectric elastomer at 10kV dc voltage and at 5kV ac amplitude and 1Hz frequency, and it can be seen that there is both a transient response and a steady state response in the graph, where the transient solution disappears over time. Taking the steady state solution as an example, the present invention plots the amplitude of the steady state solution to a wide range actuation frequency map as shown in fig. 9, and then, considering the uncertainty of the parameters, the interval boundary of the frequency domain response as shown in fig. 9 can be obtained.
The invention has not been described in detail and is part of the common general knowledge of a person skilled in the art.
The above are only the specific steps of the present invention, and the protection scope of the present invention is not limited in any way; the method can be expanded and applied to the field of quasi-static and dynamic analysis of the viscoelastic dielectric elastomer with parameter uncertainty, and all technical schemes formed by adopting equivalent transformation or equivalent replacement fall within the protection scope of the invention.

Claims (3)

1. A viscoelastic dielectric elastomer uncertainty quasi-static and nonlinear dynamics analysis method based on an interval method is used in the design of an active control system using a dielectric elastomer as a driver or a sensor, and is characterized by comprising the following implementation steps:
the first step is as follows: obtaining a dynamics control equation of a planar rectangular dielectric elastomer film based on a virtual work principle;
the second step is that: simplifying problems and equations by neglecting inertial force terms under the quasi-static condition;
the third step: solving an interval boundary of the creep response of the dielectric elastomer based on interval analysis and a Taylor expansion method;
the fourth step: solving an interval boundary of dielectric elastomer relaxation response based on interval analysis and a Taylor expansion method;
the fifth step: solving an interval boundary of the dynamic response of the dielectric elastomer based on interval analysis and a Taylor expansion method;
the third step of solving the interval boundary of the creep response based on the interval analysis and the Taylor expansion method specifically comprises the following steps of;
for creep response in quasi-static problems, let force P 0 At t 0 Is applied to the dielectric elastomer at a time and remains constant over time; when t is>t 0 Then, the elongation ratio shown by the following formula is obtained:
Figure FDA0004035609270000011
all parameters of the equation, including external load, material parameter and geometry parameter, cause a certain amount of uncertainty; defining an uncertain parameter vector as b = [ b ] 1 ,b 2 ,...,b m ] T And their interval is:
Figure FDA0004035609270000012
wherein, b I As an uncertainty interval, b and b i In order to obtain the lower limit of the interval,
Figure FDA0004035609270000013
and
Figure FDA0004035609270000014
is the upper limit of the interval, b c And
Figure FDA0004035609270000015
for nominal values of the interval, Δ b and Δ b i Is interval width, m is an integer;
since the uncertainty parameter varies within a given interval to find the limit Γ of the response, i.e. the elongation ratio, as shown in the following equation:
Γ＝{λ:g(b)＝0,b∈b I } (20)
find an interval with exact bounds of the envelope, i.e.:
Figure FDA0004035609270000016
wherein λ and
Figure FDA0004035609270000021
comprises the following steps:
Figure FDA0004035609270000022
to determine the upper and lower limits of the elongation ratio, a first order taylor expansion is used here:
Figure FDA0004035609270000023
wherein,
Figure FDA0004035609270000024
then, according to the natural expansion principle of interval operation, the interval boundary of the creep response is obtained as shown in the following formula:
Figure FDA0004035609270000025
wherein,
Figure FDA0004035609270000026
thus, the interval limits of the creep response of the dielectric elastomer with interval uncertainty parameters
Figure FDA0004035609270000027
It is determined that:
Figure FDA0004035609270000028
the fourth step of solving the interval boundary of the relaxation response based on the interval analysis and the Taylor expansion method specifically comprises the following steps of;
for loosening in quasi-static problemsRelaxation behavior, let the elongation ratio lambda 0 At t 0 Is constantly applied to the dielectric elastomer and remains constant; when t is>t 0 Then, the stress shown by the following formula is obtained:
Figure FDA0004035609270000029
using the same method as in the third step, the interval for obtaining the relaxation response is bounded by
Figure FDA00040356092700000210
Comprises the following steps:
Figure FDA0004035609270000031
the fifth step of solving the interval boundary of the dynamic response based on the interval analysis and the Taylor expansion method specifically comprises the following steps of;
for the nonlinear dynamical response problem of the uncertain dielectric elastomers, the external excitation is a function of uncertain parameters, so the dynamical equation is described by the following nonlinear parametric excitation problem:
Figure FDA0004035609270000032
wherein,
Figure FDA0004035609270000033
and x is acceleration, velocity and displacement, respectively, b is the uncertainty parameter vector of the aforementioned dielectric elastomer, f (t, d) is the external excitation, d is the uncertainty parameter vector of the excitation load; the exact boundaries of the time domain response are:
Figure FDA0004035609270000034
as with the quasi-static creep behavior analysis in the third step, find an interval vector Λ containing the true kinetic response rather than the exact boundary:
Figure FDA0004035609270000035
wherein,
Figure FDA0004035609270000036
then, using a first order taylor expansion method and according to the natural expansion theory of interval arithmetic, the interval boundary of the kinetic response is described as:
Figure FDA0004035609270000037
wherein,
Figure FDA0004035609270000038
thus, the interval boundaries of the kinetic response are obtained
Figure FDA0004035609270000039
Comprises the following steps:
Figure FDA0004035609270000041
2. the viscoelastic dielectric elastomer uncertainty quasi-static and nonlinear dynamics analysis method based on the interval method as claimed in claim 1, characterized in that:
the first step of obtaining a dynamic control equation based on the virtual work principle specifically comprises the following steps:
planar rectangular dielectric elastomer filmIn the initial state, i.e., the undeformed state, the dimensions of the dielectric elastomer film in the X, Y and Z directions are 2L, respectively 1 ,2L 2 And 2H; respectively P in X and Y directions 1 And P 2 Under the external force of (2), the dimension of the dielectric elastomer film in the X, Y and Z directions becomes 2l 1 ,2l 2 And 2h; the ratio of elongation in the X and Y directions is expressed as λ 1 ＝l 1 /L 1 And λ 2 ＝l 2 /L 2 (ii) a Since the elastomer is incompressible, λ 1 λ 2 λ 3 =1, wherein the elongation ratio in the Z direction is λ 3 ＝h/H＝1/λ 1 λ 2 Adopting a Zener model and an ideal dielectric elastomer model to obtain a free energy function considering the viscoelasticity of the dielectric elastomer as follows:
Figure FDA0004035609270000042
wherein W stretch Represents the viscoelastic free energy of the elastomer, W electric Denotes the dielectric elastic free energy of the elastomer, D is the electric displacement, ε is the dielectric constant of the elastomer, μ 1 And mu 2 Is the shear modulus of two springs, J lim1 And J lim2 Is a constant, ξ, related to the extension limit of the two springs 1 And xi 2 Is a deformation of the damper and is,
Figure FDA0004035609270000043
and
Figure FDA0004035609270000044
is and λ 1 And λ 2 The elongation ratio of the relevant spring is as follows:
Figure FDA0004035609270000045
wherein λ 1 ,λ 2 And xi 1 ,ξ 2 The relationship between isWherein eta is the viscosity of the mixture,
Figure FDA0004035609270000051
assuming that the charge on the dielectric elastomer changes from an equilibrium state by an amount δ Q, the work performed by the voltage Φ is Φ δ Q; when the length of the elastomer in both the X and-X directions is from equilibrium,. Delta.l 1 When the external force is changed, the work of the external force in the X and-X directions is 2P 1 δl 1 (ii) a The inertial force works as follows, also in the Y and-Y directions:
Figure FDA0004035609270000052
where u is the displacement field of the elastomer,
Figure FDA0004035609270000053
indicating acceleration, V 0 Is the volume of the elastomer, phi is the external voltage, p 0 For density, according to the thermodynamic theory, for any equilibrium state, i.e., a fully relaxed state, since the damper is not ready to deform, the strain of the free energy of the material is equal to the work done by the external load, the inertial force and the voltage; namely:
Figure FDA0004035609270000054
wherein the relationship between δ λ and δ l is:
Figure FDA0004035609270000055
according to the variation principle, the following are obtained:
Figure FDA0004035609270000056
since the damper of the assumed model does not have enough time to deform, δ ξ can be made to be 1 ＝0,δξ 2 =0; the relationship between the electrical displacement D and the amount of charge Q on the electrode is:
Figure FDA0004035609270000057
based on the above formula, we obtain:
δQ＝4l 1 l 2 δD+4l 2 δl 1 D+4l 1 δl 2 D＝4l 1 l 2 δD+4l 2 L 1 δλ 1 D+4l 1 L 2 δλ 2 D (9)
substituting formula (7) and formula (9) for formula (5) yields:
Figure FDA0004035609270000061
from the free energy function in equation (1), we obtain:
Figure FDA0004035609270000062
then, formula (11) is substituted for formula (10), and the independence between the three parameters is used:
Figure FDA0004035609270000063
by substituting the third equation in equation (12) into the other two equations, thereby eliminating the electrical displacement D, the final governing equation is obtained:
Figure FDA0004035609270000071
3. the viscoelastic dielectric elastomer uncertainty quasi-static and nonlinear dynamics analysis method based on the interval method as claimed in claim 2, characterized in that:
the simplification problem and equation of neglecting the inertia force item under the quasi-static condition of the second step specifically comprise;
for quasi-static problems including creep and relaxation, the inertial terms are ignored, i.e., omitted
Figure FDA0004035609270000072
And
Figure FDA0004035609270000073
the problem thus becomes solved for the following equation, which is the governing equation for the viscoelastic dielectric elastomer in the quasi-static case:
Figure FDA0004035609270000074
wherein,
Figure FDA0004035609270000075
reduce the problem to λ 1 ＝λ 2 ＝λ，ξ 1 ＝ξ 2 ＝ξ，P 1 ＝P 2 ＝P，L 1 ＝L 2 = L and g 1 ＝g 2 Case of = g, then the kinetic equation is converted to the following equation:
g(P,Φ,λ,ξ)＝0 (16)
wherein, the specific expression of the equation g (P, Φ, λ, ξ) =0 is:
Figure FDA0004035609270000081
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