Approximate method for temperature-dependent characteristics of structures with viscoelastic dampers

Transcription

1 Arch Appl Mech : ORIGINAL Roan Lewandowsi Maciej Przychodzi Approxiate ethod for teperature-dependent characteristics of structures with viscoelastic dapers Received: 18 October 2017 / Accepted: 1 May 2018 / Published online: 14 May 2018 The Authors 2018 Abstract A ethod for analyzing the influence of teperature on the dynaic characteristics of structures with viscoelastic dapers is proposed in this paper. Dapers which are described by so-called fractional rheological odels are considered. The teperature frequency superposition principle is used to describe the influence of teperature on the dynaic characteristics. The concept of continuous dependence of daping on an artificially introduced paraeter is adopted for solving, in an approxiate way, the nonlinear eigenvalue proble fro which dynaic characteristics are deterined. The correctness and effectiveness of the ethod was verified by eans of two exaples. Keywords Viscoelastic daper Nonlinear eigenvalue proble Teperature influence Fractional rheological odel 1 Introduction The proble of reduction of structural vibration due to environental factors such as strong wind or earthquaes has been studied for any years. A nuber of ethods which have been developed for this purpose are divided into three categories: ethods of active, sei-active and passive control. These ethods are reviewed in [1] and other papers. One of the ost widely nown, the passive ethod, is based on the use of aorphous polyers that exhibit viscoelastic properties [2] in dapers. The use of viscoelastic dapers is characterized by high cost-effectiveness. But the ajor disadvantage of viscoelastic dapers is that the daping properties of the polyers depend on teperature and vibration frequency. Studies on the effects of teperature on the perforance of viscoelastic dapers installed in building structures are reported in [3,4], presenting the results of experients on the seisic behavior of a viscoelastically daped steel frae under various abient teperatures. The experiental results were copared with analytical calculations of equivalent daping ratios and seisic response of the structure. The daping ratios were calculated using the odal strain energy ethod. The influence of teperature on the dapers properties was taen into account by using epirical forulae based on regression analysis. In papers [5,6], a one-diensional viscoelastic finite eleent with a odified fractional odel coupled to an epirical teperature odel was presented. In paper [7], viscoelastic dapers were described using the fractional derivative Kelvin odel. The effects of teperature on viscoelastic aterials were accounted for by adopting the teperature frequency superposition principle. The results of calculations were validated by coparison with the experiental results obtained for the three-story daped R. Lewandowsi M. Przychodzi B Poznan University of Technology, ul. Piotrowo 5, Poznan, Poland E-ail: aciej.przychodzi@put.poznan.pl R. Lewandowsi E-ail: roan.lewandowsi@put.poznan.pl

2 1696 R. Lewandowsi, M. Przychodzi steel frae odel. The analysis was done in the tie and the frequency doain, but the diensionless daping ratios and natural frequencies were not deterined. It should be noted that changes in the teperature of the viscoelastic aterial in the daper are the result of external influences on the one hand and of the so-called self-heating phenoenon on the other. This phenoenon, consisting in the rise in teperature due to energy dissipation in the viscoelastic aterial, has been the subject of nuerous studies. A nuber of coupled theral-echanical odels have been developed, whicharedescribedin[8 12]. It is well nown that teperature changes have a significant ipact on the paraeters of the daper odel adopted [3 6]. The ost coonly used odels of viscoelastic dapers are classic rheological odels described by thepronyseries, fractional derivativeodels orpower lawrepresentations [10,13,14]. In recent ties, fractional derivative odels have often been used because of the sall nuber of paraeters needed to be identified. Model paraeters are identified using the values of the so-called coplex odulus, easured in experiental tests for exaple, in dynaic echanical analysis tests [14,15]. Fro the experiental results, the so-called aster curves of the real part storage odulus and iaginary part loss odulus of the coplex odulus are created. The values of the coplex odulus for other teperatures are obtained by shifting the aster curves [2,14 17]. Shifting the aster curves changes the odel paraeters, and this requires their re-identification. This paper presents an easy ethod for deterination of the fractional derivative odel paraeters of the viscoelastic daper for different teperatures without their re-identification. The ain purpose of the paper is to develop a ethod for deterining the basic dynaic characteristics, i.e., natural frequencies and diensionless daping factors, for structures with viscoelastic dapers depending on teperature changes. The calculation of these characteristics requires the solution of a nonlinear eigenvalue proble. This type of eigenvalue proble can be solved using the atheatically advanced continuation ethod [18]. In [19], the ethod of approxiating an eigenvalue proble solution was presented. It was extended in [20] to the case the viscoelastic properties of the attenuators are described by fractional derivatives. In order to deterine the dynaic characteristics of a syste with viscoelastic dapers for different teperatures, it is necessary to solve the eigenvalue proble repeatedly due to changes in the paraeters of the daper odels. However, on the basis of the ethod described in [20], a ethod of solving the nonlinear eigenvalue proble will be developed that taes into account the effect of teperature changes on the paraeters of the daper odels in a direct way. Details of the proposed ethod will be presented in this paper. The reaining part of the paper is organized as follows. The fractional daper odel is described in Sect. 2. The influence of teperature on the viscoelastic daper paraeters is described in Sect. 3. The nonlinear eigenproble is forulated in Sect. 4. The analytical solution of the nonlinear eigenproble for proportionally daped systes is presented in Sect. 5. Section 6 describes the approxiate ethod of solving the nonlineareigenproble. Section 7 presents an approxiation of the dynaic characteristics of a structure according to the teperature changes in a direct way. An alternative ethod for approxiating the diensionless daping ratios of a structure is described in Sect. 8. In Sect. 9, nuerical exaples are presented. The final rears are provided in Sect Fractional daper odel Adopting a suitable viscoelastic daper odel is a difficult proble: on the one hand, it should accurately describe the dynaic behavior of the daper; on the other, it is iportant to iniize the nuber of odel paraeters. The fulfillent of these two conditions can be facilitated by the use of fractional derivatives in the atheatical daper odel. The siplest fractional daper odels are the fractional Maxwell odel and the fractional Kelvin odel. Each of the consists of the elastic eleent described by the stiffness paraeter, the Scott Blair eleent described by the viscosity paraeter c, and the paraeter α, which expresses the order of the fractional derivative in the daper otion equation. These odels are relatively siple fro a atheatical point of view but in soe instances they reproduce poorly the behavior of real viscoelastic aterials. More coplex odels are a cobination of the Maxwell and Kelvin odels. In this paper, the odel shown in Fig. 1 is taen into account. The considered odel consists of the siple Maxwell and the siple Kelvin ones connected in parallel. Later in this paper, these two siple parts of the odel will be called the Maxwell and Kelvin eleents, respectively. The total daper response force is the su of the forces of the Kelvin and Maxwell eleents, i.e.: u t = u 0 t + u 1 t, 1

3 Approxiate ethod for teperature-dependent characteristics u q i i c 0,α 1 c 1,α j q j u Fig. 1 Generalized fractional odel sybol u 0 t denotes the response force of the Kelvin eleent and u 1 t is the response force of the Maxwell eleent, respectively. The response force of the Kelvin eleent is described by the following forula: u 0 t = 0 q t + c 0 Dt α q t. 2 Dt α denotes the fractional derivative of the α order with respect to tie t. Here, the Rieann Liouville definition of the fractional derivative is used. The sybol q t is the relative daper displaceent: q t = q j t q i t, 3 q i t and q j t are the displaceents of ith and jth nodes of the daper odel see Fig. 1. The response force of the Maxwell eleent is governed by the equation: ν 1 u 1 t + Dt α u 1 t = 1 Dt α q t, 4 ν 1 = 1 /c 1. After using the Laplace transforation with zero initial conditions to Eqs. 2 and4, the transfor of total force of the fractional daper odel response can be written as follows: ū s = 0 + s α c 0 + 1s α ν 1 + s α q s, 5 ū s and q s are the Laplace transfors of u t and q t, respectively, and s = iλ is the Laplace variable. The sybol λ denotes the frequency of vibration. The constitutive equation of daper 5 can also be written using the coplex odulus K λ: ū λ = K λ q λ = [ K λ + ik λ ] q λ. 6 A coparison of Eqs. 5and6 enables the storage odulus K λ and loss odulus K λ to be expressed using the daper paraeters: K λ = 0 + c 0 λ α cos απ 2 + 1λ α λ α + ν 1 cos απ 2 ν ν 1λ α cos απ, λ2α K λ = c 0 λ α sin απ ν 1 λ α sin απ 2 ν ν 1λ α cos απ 2 + λ2α. 8 3 Influence of teperature on viscoelastic daper paraeters The coplex odulus depends not only on the frequency but also on the teperature of a viscoelastic aterial. For ost viscoelastic aterials, the dependence of the coplex odulus on teperature can be described by the so-called teperature frequency correspondence principle [21]. If the storage odulus or loss odulus aster curve, which is its plot in the frequency doain, is nown for a certain teperature T 0, called the reference teperature, then, according to this principle, it is possible to define the plot of this odule for any other teperature T. This is done by shifting the graph at the reference teperature. According to the teperature frequency correspondence principle, the following relationship for the coplex odulus can be written: K λ, T = K α T λ 0, T 0, 9 α T denotes the so-called shift factor and λ 0 is the reference frequency.

4 1698 R. Lewandowsi, M. Przychodzi K λ K T 0 T Fig. 2 Frequency teperature superposition principle illustration λ 0 λ λ In general, the horizontal and vertical shift factors ust be used for the description of viscoelastic aterials [2]. However, there are not enough experiental data concerning the vertical shift factor of the viscoelastic aterial used in dapers copare [8,9,11,13,22 24]. However, if the viscoelastic aterial can be classified as a therorheologically siple aterial, then only the horizontal shift factor is required [23]. The value of the horizontal shift factor α T is ost often deterined using the epirical Willia Landel Ferry forula: log α T = C 1 T C 2 + T, 10 C 1 and C 2 are the epirical constants and T = T T 0. Moreover, it is assued that the fractional free volue increases linearly with respect to tie. An additional assuption is that, as the free volue of aterial increases, its viscosity rapidly decreases copare [14].The teperature frequency correspondenceprinciple for therorheologically siple viscoelastic aterials is illustrated in Fig. 2 and the following relationship can be written: λ 0 = α T λ. 11 The storage and loss oduli are described for the reference teperature T 0 by the forulae: K λ 0, T 0 = 0 + c 0 λ α 0 cos απ 2 + 1λ α λ α 0 + ν 1 cos απ 2 0 ν ν 1λ α 0 cos απ 2 +, 12 λ2α 0 K λ 0, T 0 = c 0 λ α 0 sin απ ν 1 λ α 0 sin απ 2 ν ν 1λ α 0 cos απ λ2α 0 For teperature T, the daper odel paraeters are different, and Eqs. 12and13 tae the for: K λ, T = 0 + c 0 λ α cos απ λ α λ α + ν 1 cos απ 2 ν ν 1λ α cos απ, λ2α K λ, T = c 0 λ α sin απ ν 1 λ α sin απ 2 ν ν 1λ α cos απ λ2α However, the oduli K λ, T and K λ, T for any teperature T can be derived fro Eqs. 12 and 13 taing into account the teperature frequency superposition principle. Replacing λ 0 by α T λ gives the relationships: K λ 0 = α T λ, T 0 = 0 + c 0 α T λ α cos απ α T λ α α T λ α + ν 1 cos απ 2 ν ν 1 α T λ α cos απ 2 + α, 16 2α T λ K λ 0 = α T λ, T 0 = c 0 α T λ α sin απ ν 1 α T λ α sin απ 2 ν ν 1 α T λ α cos απ 2 + α. 17 2α T λ The coparison of Eq. 14 with Eqs. 16and15 with Eq. 17 leads to the following results: 0 = 0, 1 = 1, 18 c 0 =ˆα T c 0, c 1 =ˆα T c 1, ˆα T = α α T.

5 Approxiate ethod for teperature-dependent characteristics Nonlinear eigenproble The equation of otion of structure with viscoelastic dapers could be written in the following for [18,20]: M q t + C q t + Kq t = f t + p t, 19 M, C and K are the ass, structural daping and stiffness atrices of structure, respectively. The sybol q t denotes the vector of displaceents in a tie doain. Moreover, p t is the vector of external load and f t is the vector of interaction forces between the dapers and the structure in short, the daping forces. The vector f t can be written in the for of the su of vectors of forces induced by the individual dapers: f t = f r t, 20 is the total nuber of dapers. As a result of the Laplace transforation of Eq. 19 for the zero initial conditions, the following expression is obtained: s 2 M + sc + K q s = f s + p s. 21 Taing into account Eq. 5, the vector of daping forces can be written as: f s = f r s = 0r + s α c 0r + 1r s α ν 1r + s α L r q s, 22 L r is the location atrix of the rth daper. The equation of otion of structure with viscoelastic dapers in the frequency doain taes, finally, the following for: s 2 M + sc + K + G s q s = p s, 23 K = K + G s = s α 0r L r, 24 c 0r + 1r ν 1r + s α L r, 25 The basic dynaic characteristics, such as natural frequencies and diensionless daping factors, are calculated after solving the appropriately defined eigenproble. The nonlinear eigenproble of the structure with viscoelastic dapers is obtained fro Eq. 23 by subtracting the zero external load vector, i.e.: D s q s = 0, 26 D s = s 2 M + sc + K + G s. 27 Taing into account Eq. 18, it is possible to introduce the influence of teperature on the daper paraeters into the equation of otion: D s, ˆα T q s = 0, 28 D s, ˆα T = s 2 M + sc + K + ˆα T c 0r s α + 1r ˆα T s α ν 1r +ˆα T s α L r. 29

6 1700 R. Lewandowsi, M. Przychodzi In avast ajority of cases, an analytical solutionto the eigenproble 26 is ipossible. However, nuerical ethods have been developed see, for exaple [18]. In this paper, the ethod of which the ain assuptions were proposed by Lazaro [19] is presented and used. The basic dynaic characteristics of structures are their natural frequencies ω and diensionless daping factors γ, which are calculated fro the forulae: ω = μ 2 + η2,γ = μ ω, 30 μ and η are the real and iaginary parts of the th eigenvalue s being the solution to the eigenproble Solution to eigenproble of proportionally daped syste In general, the nonlinear eigenproble 28 has to be solved nuerically. But in the special case of a structure in which the effect of dapers is proportional to the stiffness atrix, a sei-analytical solution can be given. The sei-analytical solution eans that the characteristic equation is obtained but is nonlinear and has to be solved nuerically. If the syste is proportionally daped, the eigenproble 26 can be written in the following for: [ s 2 M + K c 0 s α + 1s α ] ν 1 + s α κk q = 0, 31 κ denotes the proportionality coefficient. The coefficient κ fulfils the following relationship: L r = κk, 32 After transforation, Eq. 31 can be written in the for of the linear eigenvalue proble: K + s 2 M q = 0, 33 s 2 = s κ 0 + κc 0 s α + κ 1s α ν 1 +s α α T. 34 Taing into account that s = iω, the solution of the eigenproble 31 coes to the solution of the equation: s 2 + ω κ 0 + κc 0 s α + κ 1s α ν 1 + s α = Above, the sybol ω denotes the natural frequency of the syste without daping. The nonlinear characteristic Eq. 35 has to be solved nuerically, for exaple, using the Newton ethod. In Eq. 35, the influence of teperature change on the daper odel paraeters can directly be taen into account by considering Eq. 18. Then, Eq. 35 taes the following for: s 2 + ω κ 0 + κc 0 ˆα T s α + κ 1 ˆα T s α ν 1 +ˆα T s α = 0. 36

7 Approxiate ethod for teperature-dependent characteristics Approxiate ethod of solving nonlinear eigenproble In the ethod proposed by Lazaro [19], the artificial paraeter p is introduced to Eq. 26: D s, p = s 2 M + spc + K + s α D s, p q p = 0, 37 pc 0r + p 1r ν 1r + ps α L r. 38 The paraeter p is treated as a variable, and its value is in the range fro 0 to 1. For p = 0, Eq. 37 describes the linear eigenproble. For p = 1, the solution of nonlinear eigenproble is obtained. It should be noted that s is a function of p. The first derivative of Eq. 37 with respect to p is as follows: D s, p s ds D s, p + dp p q p + D s, p d q p dp = the sybol D s, p/ s denotes the derivative of D s, p with respect to p occurring as an explicit variable. After preultiplication by q T p, Eq.38 taes the for: D s, p q T ds D s, p p + q p + q T p D s, p s dp p The atrix D s, p is syetrical. Thus, Eq. 37 can be written as: and fro Eq. 40, the following expression is obtained: d q p dp = q T p D s, p = 0, 41 ds dp = qt Ds, p p p q p q T p Ds,p s q p. 42 It is assued that the eigenvector q is scaled in the following anner: q T D s, p p q p = 2a, 43 s a is a scalar. After taing into account the condition 43 and deriving the derivative of a atrix D with respect to p, Eq. 42 finally taes the for: ds dp = 1 sd1 q, p + s α D 2 q, s, p, 44 2a D 1 q, p = q T p C q p, 45 D 2 q, s, p = q T 1r ν 1r p c 0r + ν 1r + ps α 2 L r q p. 46 The next step of the presented ethod is to expand the D 1 q, p and D 2 q, s, p functions in the Taylor series in vicinity of p = 0: D 1 q, p = D 1 a, 0 + p dd 1 dp, 47 p=0 D 2 q, s, p = D 2 a, iω,0 + p dd 2 dp. 48 p=0

8 1702 R. Lewandowsi, M. Przychodzi This operation leads to the following first-order ordinary differential equation: ds dp = 1 [ s d1 + pd 2 + s α d 3 + pd 4 ], 49 2a d 1 = a T Ca, 50 d 2 = 2a T C d q dp, 51 p=0 d 3 = a T d 4 = 2a T c 0r + c 1r L r a, 52 d q c 0r + c 1r L r dp 2a T 1r iω 2 L r a. 53 p=0 Equation 49 is the nonlinear first-order differential equation which can be solved nuerically. In this paper, the Euler ethod [25] isused. In order to calculate the agnitude d q, i.e., the sensitivity of the eigenvector q to changes in the paraeter p, Eq.39 and thefirstderivativeofeq. 43 with respect to p are used. After soe siple atheatical dp p=0 operations, the following atrix sensitivity equation is obtained: [ ] d q A11 A { } 12 dp p=0 A 21 A 22 ds = b1, 54 b 2 dp p=0 A 11 = K ω 2 M, 55 A 12 = 2iωMa, 56 A 21 = 2iωa T M, 57 A 22 = a T Ma, 58 [ ] b 1 = iωc + iω α c 0r + c 1r L r a, 59 b 2 = 1 2 at [C + ν 2 1r ] α iω α 1 c 0r + c 1r L r a Analysis of teperature influence on dynaic characteristics of structures with viscoelastic dapers Deterining the dynaic characteristics of structures with viscoelastic dapers according to teperature changes withintheir specific range requiresultiple solvingof the nonlineareigenproble 26 because recalculation of the paraeters of dapers for each teperature is required. This is quite troublesoe because, as entioned earlier, this tas can only be solved by iterative ethods. However, the use of the Lazaro assuptions [19] enables a direct deterination of the course of the natural frequency and the diensionless daping factor functions within a certain teperature range. Naely, the shift factor ˆα T in Eq. 29 can be treated as a hootopycoefficient, as does the coefficient p in Eq. 38. After perforing the corresponding atheatical operations described in Sect. 6, the following equation is obtained: ds d ˆα T = sα 2a D q, s, ˆα T, 61

9 Approxiate ethod for teperature-dependent characteristics 1703 D q, s, ˆα T = q T 1r ν 1r ˆα T c 0r + ν1r +ˆα T s α L 2 r q ˆα T. 62 The next step is to expand the function D q, s, ˆα T in the Taylor series for ˆαT = 1: D q, s, ˆα T = D q0, s 0, 1 + ˆα T 1 dd d ˆα T ˆαT =1, 63 q 0 and s 0 are the eigenvector and the eigenvalue of the syste, respectively, for the reference teperature T 0 ˆαT = 1. Finally, Eq. 61 taes the following for: ds = sα [ d1 + ˆα T 1 ] d 2, 64 d ˆα T 2a d 1 = q T 0 d 2 = 2 q T 0 c 0r + 2 q T 0 c 0r + 1r ν 1r ν1r +ˆα T s0 α 2 L r q 0, 65 1r ν 1r d q 0 ν1r +ˆα T s0 α 2 L r d ˆα T 1r ν 1r ˆα T αs0 α 1 ν1r +ˆα T s0 α 3 L r q 0 ds 0 d ˆα T + ˆαT =1 2 q 0 T ˆαT =1 1r ν 1r αs α 0 ν1r +ˆα T s α 0 3 L r q 0, 66 The sensitivities of the eigenvector and eigenvalue due to changes in the paraeter ˆα T are calculated fro the following atrix sensitivity equation: A 11 = s 2 0 M + s 0C + K + A 12 = A 21 = 2s 0 M + C + A 22 = 1 2 qt 0 [2M+ b 1 = b 2 = 1 2 qt 0 [ ] d q A11 A 12 d ˆα T ˆαT =1 A 21 A 22 ds = d ˆα T ˆαT =1 αs α 2 0 { b1 b 2 }, 67 1r s0 α ν 1r + s0 α L r, 68 1r ν 1r αs α 2 0 1r ν 1r αs α 1 0 ν1r + s α 0 2 L r q 0, 69 ] α 1 ν 1r + s0 α α + 1 ν1r + s0 α 3 L r q 0, 70 1r ν 1r ν1r + s0 α 2 L r q 0, 71 1r ν 1r αs0 α 1 ν1r s0 α ν1r + s0 α 3 L r q 0, 72

10 1704 R. Lewandowsi, M. Przychodzi 8 Modal strain energy ethod The odal strain energy ethod is used to approxiate the diensionless daping ratios of a structure [3,4]. The daping ratios are calculated fro the following forula: γ = at D Ia 2a TD, 73 Ra a is the th odal shape vector of the structure without daping. The sybols D I and D R denote the iaginary and real parts of the dynaic stiffness atrix D s described by Eq. 27. In order to deterine the atrices D I and D R, the following expression has to be substituted into Eqs. 25and27: s = iω, 74 ω is the th natural frequency of the structure without daping. After separating the real and iaginary parts of the dynaic stiffness atrix D s, the atrices D I and D R can be described as follows: [ ] D I = ω C + ω α c 0r sin απ 2 + 1r ν 1r ω α sin απ 2 ν 2 1r + 2ν 1r ω α cos απ 2 + L r, 75 [ ω2α D R = K + ω α c 0r cos απ 2 + 1r ω α ν 1r sin απ 2 + ] ωα ν 2 1r + 2ν 1r ω α cos απ 2 + L r. 76 ω2α The influence of the teperature changes on the paraeter of the viscoelastic daper odel can be taen into account using Eq. 18.Then,Eqs.75and76 tae the for: [ ] D I = ω C + ˆα T ω α c 0r sin απ 2 + 1r ν 1r ˆα T ω α sin απ 2 ν1r 2 + 2ν 1r ˆα T ω α cos απ 2 + ˆα T ω α L 2 r, 77 [ D R = K + ˆα T ω α c 0r cos απ 2 + 1r ˆα T ω α ν 1r sin απ 2 +ˆα ] T ω α ν1r 2 + 2ν 1r ˆα T ω α cos απ 2 + ˆα T ω α L 2 r Nuerical exaples 9.1 Exaple 1: the proportionally daped syste In the first nuerical exaple, the results obtained by the approxiate ethod described in Sect. 6 are copared with the exact solution for proportional daping systes described in Sect. 5. The odel of a four-story shear frae, i.e., the syste with rigid beas and elastic coluns, was used for the analysis. The syste is shown in the scheatic in Fig. 3a. The ass of the structure is luped on the floor levels. The value of each luped ass is = 10,000 g. The bending rigidity of the syste is described by the story rigidity coefficients,, which are also the sae for each story = 1600 N/. Structural daping was oitted as irrelevant due to the nature of the study. In order to ipleent the daping proportionality condition, the dapers of the sae paraeters were located on all stories. The following paraeters of the daper odel were adopted: 0 = 0.0N/,c 0 = 0.0 Ns/, 1 = 10,000 N/, c 1 = 60 Ns α /, α = 0.7. The nonlinear equation 35 was solved using the Newton ethod. In the calculation by the approxiate ethod, it was assued that the increent in the hootopy paraeter, p, was p = The nonlinear differential equation 49 was solved using the iplicit Euler ethod. The results of the calculations are presented in Tables 1 and 2. For natural frequencies, thedifference in theresults is a axiu of 0.05%. In the case of diensionless daping ratios, the axiu difference is 0.17%. It should be noted that the syste with high daping is considered. The exaple presented above shows that, although the ethod proposed by Lazaro [19] is an approxiate one, its accuracy for systes with proportional daping, including systes with high daping, is very good at least in engineering applications.

11 Approxiate ethod for teperature-dependent characteristics 1705 b q 10 q 9 q 8 q 7 q 6 q 5 a q 4 q 4 q 3 q 3 q 2 q 2 q 1 q 1 Fig. 3 Scheatics of analyzed odels: a proportionally daped syste, b non-proportionally daped syste Table 1 Coparison of natural frequencies Mode nuber Approxiate ethod rad/s Exact ethod rad/s Difference % Table 2 Coparison of diensionless daping factors Mode nuber Approxiate ethod % Exact ethod % Difference % Using the sae odel with proportional daping, the efficiency of the ethod of approxiation of the dynaic characteristics of the syste depending on the teperature changes, described in Sect. 7, was further analyzed. The teperature changes translate into changes in the coefficient ˆα T. The range of ˆα T is fro 0.4 to 2.0 in calculations, which corresponds to the range of α T between 0.3 and 2.7. The increent of ˆα T was For exaple, for the aterial described in [10], the given range of α T corresponds to the teperature range of C at the reference teperature T 0 = 0.2 C. The results obtained fro the nuerical solution of Eq. 64 were copared with those calculated fro Eq. 36. These coparisons are illustrated in Figs. 4 and 5. For proportionally daped systes, convergence of the results of the approxiate ethod with those obtained fro the exact solution can be considered quite good: the axiu difference in results for the natural frequencies is 0.63%, and that for the diensionless daping factors is 3.81%.

12 1706 R. Lewandowsi, M. Przychodzi In addition, the plots of diensionless daping ratios calculated using the odal strain energy ethod are presented in Fig. 5.It can be seen thattheconvergenceoftheresults obtainedwith this ethodwith theexact solution is worse than that for the ethod proposed in Sect. 7 for the proportionally daped syste. 9.2 Exaple 2: the non-proportionally daped syste The effectiveness of the approxiation ethod of calculating the dynaic characteristics of the nonproportionally daped systes according to teperature changes was verified by perforing nuerical calculations for the shear frae odel of the building structure with ten degrees of freedo and six dapers. The syste is shown in the scheatic in Fig. 3b. The values of the luped asses are the sae: = 230,000 g. The bending rigidity of each story is also the sae. The rigidity coefficient is = 130,000 N/. The diensionless factor of structural daping of the first and second vibration odes is γ 1,2 = The dapers are located on the three lowest and the three highest stories of the structure. The odel paraeters of each daper are: 0 = 2000 N/, c 0 = 0.0Ns/, 1 = 200,000 N/, c 1 = 5000 Ns α /, α = 0.7. It was assued that changes in the teperature of the viscoelastic aterial affect the changes shift factor, α T. Calculations were perfored twice for the sae α T interval. In the first series of calculations, the daper odel paraeters were changed for the subsequent ˆα T values according to Eq. 49, and then the proble was solved by the ethod described in Sect. 6. In the second series of calculations, the ethod described in Sect. 7 was used, the subsequent ˆα T values were substituted for Eq. 64. As in Exaple 1, the paraeter ˆα T was between 0.4 and 2.0, which translates into α T of The liits of the ˆα T range were deterined in such a way as to obtain a satisfactory consistency of the results. The increent of ˆα T for the first calculation series was assued to be For exaple, for the aterial described in [10], the given range of α T corresponds to the teperature range of C at the reference teperature T 0 = 0.2 C. For the second series, the value was used to increase the accuracy of the calculation. It should be ephasized that despite the decrease in the increent value of ˆα T, the speed of coputer-aided calculations of the second series was uch higher. For the first three natural frequencies, the axiu differences between the results were 0.45, 1.31 and 2.54%, respectively. For the diensionless daping factors of the first three vibration odes, the effects were a bit worse. Naely, the axiu differences in the results were 7.69, 7.87 and 9.36%, respectively. The natural frequency functions versus α T are shown in Fig. 6. The graphs of the diensionless daping ratios are given in Fig. 7. The values obtained fro the odal strain energy ethod are also shown in this figure. It can be seen that the results obtained using all the presented ethods are very siilar. Coparing the results for proportionally and non-proportionally daped fraes, it is obvious that results for the proportionally daped frae are noticeably ore accurate. 10 Concluding rears Viscoelastic dapers are often used nowadays to reduce the excessive vibrations of structures. The odal analysis in the frequency doain needs an eigensolution to the nonlinear eigenvalue proble resulting fro the equation of otion for a daped, free vibration proble. Although exact solutions to the above-entioned eigenvalue proble can be obtained nuerically, approxiated expressions for frequency of vibrations and non-diensional daping ratios are very valuable in reducing the coputational effort. It is ore iportant in the case of systes with viscoelastic dapers because their paraeters depend on teperature. In this paper, the ethod of analysis of the influence of teperature on the dynaic characteristics of structures with viscoelastic daper is presented. The ethod enables the calculation of the sequences of natural frequencies and diensionless daping ratios of a structure in a certain teperature range without the need of ultiply solving the nonlinear eigenvalue proble. The obtained dynaic characteristics values are approxiate. The degree of consistency of the results with those obtained by directly solving the eigenvalue proble for each set of daper s paraeters depends on the considered range of teperature. Significant increase in the efficiency of the proposed approxiation ethod in relation to the conventional eigenvalue proble solution is due to the fact that instead of ultiple coplex atrix operations, the ordinary differential equation 64 is solved only once. In addition, the effect of teperature changes is directly taen into account when solving this equation. In the conventional approach, the eigenvalue proble would have to be solved any ties for each teperature. The effectiveness of the presented ethod was verified nuerically for two exaples of structures with four and ten degrees of freedo, respectively. Several classic and fractional daper odels were used to

13 Approxiate ethod for teperature-dependent characteristics 1707 Fig. 4 Plots of natural frequencies for Exaple 1: case 1 the values calculated using the presented ethod taing into account the teperature changes directly, case 2 the values calculated using the exact ethod

14 1708 R. Lewandowsi, M. Przychodzi Fig. 5 Plots of diensionless daping factors for Exaple 1: case 1 the values calculated using the presented ethod taing into account the teperature changes directly, case 2 the values calculated using the exact ethod, case 3 the values calculated using the odal strain energy ethod

15 Approxiate ethod for teperature-dependent characteristics 1709 Fig. 6 Plots of three first natural frequencies for Exaple 2: case 1 the values calculated using the presented ethod taing into account the teperature changes directly, case 2 the values calculated for the case when the paraeters of the daper odel were changed each tie before solving the eigenproble

16 1710 R. Lewandowsi, M. Przychodzi Fig. 7 Plots of diensionless daping factors of three first vibration odes for Exaple 2: case 1 the values calculated using the presented ethod taing into account the teperature changes directly, case 2 the values calculated for the case when the paraeters of the daper odel were changed each tie before solving the eigenproble, case 3 the values calculated using the odal strain energy ethod

17 Approxiate ethod for teperature-dependent characteristics 1711 describe the properties of viscoelastic dapers ounted on structures. The results of coputation show that the suggested ethod can be successfully applied to proportionally and non-proportionally daped structures. The relative error of approxiately calculated eigenvalues depends on the daping level and on the range of considered changes of teperature. The results obtained using the proposed ethod were also copared with those obtained by the odal strain ethod, and the proposed ethod was found to lead to saller relative errors. Acnowledgeents The study has been supported by the National Science Centre, Poland, as part of Project No. DEC/2013/ 09/B/ST8/01733, carried out in the years This support is gratefully acnowledged. Open Access This article is distributed under the ters of the Creative Coons Attribution 4.0 International License creativecoons.org/licenses/by/4.0/, which perits unrestricted use, distribution, and reproduction in any ediu, provided you give appropriate credit to the original authors and the source, provide a lin to the Creative Coons license, and indicate if changes were ade. References 1. Soong, T.T., Spencer Jr., B.F.: Suppleental energy dissipation: state-of-the art and state-of-the practice. Eng. Struct. 24, Cho, K.S.: Viscoelasticity of Polyers. Theory and Nuerical Algoriths. Springer, Berlin Chang, K.C., Soong, T.T., Oh, S.-T., Lai, M.L.: Effect of abient teperature on viscoelastically daped structure. J. Struct. Eng. 1187, Chang, K.C., Soong, T.T., Oh, S.-T., Lai, M.L.: Seisic behavior of steel frae with added viscoelastic dapers. J. Struct. Eng , Tsai, C.S.: Teperature effect of viscoelastic dapers during earthquaes. J. Struct. Eng. 1202, Tsai, C.S., Lee, H.H.: Applications of viscoelastic dapers to high-rise buildings. J. Struct. Eng. 1194, Chang, K.C., Tsai, M.H., Chang, Y.H., Lai, M.L.: Teperature rise effect of viscoelastically daped structures under strong earthquae ground otions. J. Mech. 143, de Cazenove, J., Rade, D.A., de Lia, A.M.G., Araujo, C.A.: A nuerical and experiental investigation on self-heating effects in viscoelastic dapers. Mech. Syst. Signal Process. 27, Guo, J.W., Daniel, Y., Montgoery, M., Christopoulos, C.: Theral-echanical odel for predicting the wind and seisic response of viscoelastic dapers. J. Eng. Mech , Kasai, K., Sato, D.: A constitutive rule for viscoelastic aterial considering heat conduction and heat transfer. In: The Second International Conference on Urban Earthquae Engineering, pp de Lia, A.M.G., Rade, D.A., Lacerda, H.B., Araújo, C.A.: An investigation of the self-heating phenoenon in viscoelastic aterials subjected to cyclic loadings accounting for prestress. Mech. Syst. Signal Process , Ovalle Rodas, C., Zaïri, F., Naït-Abdelaziz, M.: A finite strain thero-viscoelastic constitutive odel to describe the self-heating in elastoeric aterials during low-cycle fatigue. J. Mech. Phys. Solids 64, Par, S.W.: Analytical odeling of viscoelastic dapers for structural and vibration control. Int. J. Solids Struct , Pir, R., Rouleau, L., Deset, W., Pluyers, B.: Validating the odeling of sandwich structures with constrained layer daping using fractional derivative odels. J. Braz. Soc. Mech. Sci. Eng. 387, Garcia-Barruetabena, J., Cortes, F., Abete, J.M., Fernandez, P., Laela, M.J., Fernandez-Canteli, A.: Experiental characterization and odelization of the relaxation and coplex oduli of a flexible adhesive. Mater. Des. 32, Dealy, J., Plaze, D.: Tie-teperature superposition a user s guide. Rheol. Bull. 782, Rouleau, L., Deu, J.-F., Legay, A., Le Lay, F.: Application of Kraers Kronig relations to tie teperature superposition for viscoelastic aterials. Mech. Mater. 65, Pawla, Z., Lewandowsi, R.: The continuation ethod for the eigenvalue proble of structures with viscoelastic dapers. Coput. Struct. 125, Lazaro, M.: Eigensolutions of non-proportionally daped systes based on continuous. J. Sound Vib. 363, Lewandowsi, R.: Approxiate ethod for deterination of dynaic characteristics of structures with viscoelastic dapers. Vib. Phys. Syst. 27, Moreira, R.A.S., Corte-Real, J.D., Dias Rodrigues, J.: A generalized frequency teperature viscoelastic odel. Shoc Vib. 17, Blac, C.J., Maris, N.: Viscous heating of fluid dapers under sall and large aplitude otions: experiental studies and paraetric odeling. J. Eng. Mech. 1335, Henry, T.C., Bais, C.E., Sith, E.C.: Viscoelastic characterization and self-heating behavior of lainated fiber coposite driveshafts. Mater. Des. 66, Maris, N.: Viscous heating of fluid dapers, I: sall-aplitude otions. J. Eng. Mech , Butcher, J.C.: Nuerical Methods for Ordinary Differential Equations. Wiley, London 2008 Publisher s Note Springer Nature reains neutral with regard to jurisdictional clais in published aps and institutional affiliations.