DDamping control in viscoelastic beam dynamics
Elena Pierro Scuola di Ingegneria, Universit`a degli Studi della Basilicata, 85100 Potenza, Italy
Abstract
Viscoelasticity plays a key role in many practical applications and in different reasearch fields,such as in seals, sliding-rolling contacts and crack propagation. In all these contexts, a properknowledge of the viscoelastic modulus is very important. However, the experimental characteriza-tion of the frequency dependent modulus, carried out through different standard procedures, stillpresents some complexities, then possible alternative approaches are desirable. For example, theexperimental investigation of viscoelastic beam dynamics would be challenging, especially for theintrinsic simplicity of this kind of test. This is why, a deep understanding of damping mechanismsin viscoelastic beams results to be a quite important task to better predict their dynamics. Withthe aim to enlighten damping properties in such structures, an analytical study of the transversalvibrations of a viscoelastic beam is presented in this paper. Some dimensionless parameters aredefined, depending on the material properties and the beam geometry, which enable to shrewdlydesign the beam dynamics. In this way, by properly tuning such disclosed parameters, for examplethe dimensionless beam length or a chosen material, it is possible to enhance or suppress someresonant peaks, one at a time or more simultaneously. This is a remarkable possibility to efficientlycontrol damping in these structures, and the results presented in this paper may help in elucidatingexperimental procedures for the characterization of viscoelastic materials. a r X i v : . [ phy s i c s . a pp - ph ] A p r . INTRODUCTION Nowadays, viscoelastic materials are widely utilized in several engineering applications,such as seals [1] and adhesives/biomimetic adhesives [2–5]. Moreover, they are object ofrecent research investigations, for example: (i) rolling contacts [6–8], (ii) sliding contacts[9–12], (iii) crack propagation [13–15], (iv) viscoelastic dewetting transition [16]. In all theaforementioned research fields, having knowledge of the correct viscoelastic modulus in thefrequency domain is of utmost importance. Very often, viscoelastic materials are also com-bined with fibers or fillers, but also in this case, the mechanical behaviour of the viscoelasticmatrix must be well established, especially as input data for numerical simulations. Usually,viscoelastic modulus is experimentally characterized, and one of the most utilized techniqueis the DMA (Dynamic mechanical analysis) [17], which is quite complex and time con-suming. Alternative approches have been presented in literature, such as the experimentaldynamic evaluation of the viscoelastic beam-like structures [18–20]. Such experiments, how-ever, require a good comprehension from a theoretical point of view of the viscoelastic beamdynamics. Many theoretical studies on the dynamics of non-viscous damped oscillators, forboth SDOF [21, 22], and MDOF [23, 24] systems, have been presented in literature. Alsoviscoelastic continuous systems have been theoretically and experimentally investigated intheir dynamics, such as beams and plates [25–27]. However, most of these studies do notpresent a qualitative analysis of the dynamic characteristics of such systems, in terms ofeigenvalues and their connection with the most representative physical parameters. Onlyin Ref. [28], a deep analysis of a single degree-of-freedom non-viscously damped oscillatorhas been presented. Extending this kind of investigation to continuous systems would beof crucial concern when viscoelastic properties of materials must be properly established.With the aim to shed light on the vibrational behaviour of such systems, in this paper a de-tailed study of the dynamics of a viscoelastic beam is presented. Recall that the viscoelasticmaterials are characterized by the most general stress-strain relation [29] σ ( x, t ) = (cid:90) t −∞ G ( t − τ ) ˙ ε ( x, τ ) d τ (1)where ˙ ε ( t ) is time derivative of the strain, σ ( t ) is the stress, G ( t − τ ) is the time-dependentrelaxation function, which is related, in the Laplace domain, to the viscoelastic modulus E ( s ) through the relation E ( s ) = sG ( s ). Usually, a discrete version of E ( s ) is utilized to2 a) (b) FIG. 1: The real part (a) and the imaginary part (b) of the elastic modulus E ( ω ) of a genericviscoelastic material. characterize linear viscoelastic solids, which can be represented in the Laplace domain as E ( s ) = E + (cid:88) k E k sτ k sτ k (2)where E is the elastic modulus of the material at zero-frequency, τ k and E k are the relaxationtime and the elastic modulus respectively of the generic spring-element in the generalizedlinear viscoelastic model [29]. The general trend of the viscoelastic modulus E ( ω ) is shownin Figure 1. It can be observed that at low frequencies the material is in the ‘rubbery’ re-gion, indeed Re[ E ( ω )] is relatively small and approximately constant (Figure 1-a), and theviscoelastic dissipations related to the imaginary part Im[ E ( ω )] of the viscoelastic modulusbecomes negligible (Figure 1-b). At very high frequencies the material is elastically verystiff (brittle-like). In this ‘glassy’ region Re[ E ( ω )] is again nearly constant but much larger(generally by 3 to 4 orders of magnitude) than in the rubbery region. The intermediate fre-quency range (the so called ‘transition’ region) determines the energy dissipation, and cancompletely deviate the modal behaviour of a viscoelastic solid from the equivalent elasticone. Moreover, the transition region, and hence the functions Re[ E ( ω )] and Im[ E ( ω )], canbe shifted towards higher or smaller frequencies by simply varying temperature, because ofthe viscoelastic modulus E ( ω ) dependence on temperature [29]. Of course, only the knowl-edge of the analytical vibrational response of a viscoelastic structure can provide the rightparametric quantities, useful to accurately enlighten the relationship between the materialproperties and the modal contents. In this direction, the flexural vibrations of a viscoelasticbeam is analytically studied in this paper, and by introducing some non-dimensional pa-3ameters, a qualitative analysis of the eigenvalues is presented. At first, an ideal viscoelasticmaterial is considered, i.e. characterized by one single relaxation time. This kind of study,indeed, is useful for a first understanding of the physical parameters enclosed in the problem.Then, two relaxation times are taken into account, and their influence on the dynamics ofthe beam is deeply evaluated and described. Finally, some considerations are pointed outregarding the vibrational response of the beam in case of real viscoelastic materials. II. THE MODEL
In this section the analytical dynamic response of a viscoelastic beam with rectangularcross section is derived. Let be L , W , and H respectively the length, the width and thethickness of the beam (Figure 2), and let us assume that L (cid:29) W , L (cid:29) H . Assuming alsothat the displacement along the z -axis | u ( x, t ) | (cid:28) L , the Bernoulli theory of transversalvibrations can be applied and therefore it is possible to neglect the influence of shear stressin the beam. It is worth noticing that this hypotesis does not limit the validity of theanalysis, since the attention is paid to the first resonant peaks, which are not affected byshear deformations. Hence, the general equation of motion is [30] J xz (cid:90) t −∞ E ( t − τ ) ∂ u ( x, τ ) ∂x d τ + µ ∂ u ( x, t ) ∂t = f ( x, t ) (3)where µ = ρA , ρ is the bulk density of the material the cantilever is made of, A is thearea of the cross section of the beam, i.e. A = W H , J xz = (1 / W H , and f ( x, t ) is thegeneric forcing term. It must be highlighted that some additional terms could be consideredin Eq.(3), representing different kind of damping contributes [31] (e.g. viscous damping andhysteresis damping). In the present study such terms are neglected, but it is important tounderline that the results obtained in this paper are not affected by this assumption from aqualitative point of view. The forced solution of the above problem Eq.(3) can be found inthe form of a series of the eigenfunction φ n ( x ) of the following problem J xz (cid:90) t −∞ E ( t − τ ) u xxxx ( x, τ ) d τ + µ u tt ( x, t ) = 0 (4)4 u x ( x, t ) = ∂u ( x, t ) /∂x , u t ( x, t ) = ∂u ( x, t ) /∂t ), with the opportune boundary conditions.In this study, the free-free boundary conditions are considered u xx (0 , t ) = 0 (5) u xxx (0 , t ) = 0 u xx ( L, t ) = 0 u xxx ( L, t ) = 0. By Laplace transforming the time-dependence in Eq.(4), and considering equal to zerothe initial conditions, it is easy to show that the eigenfunctions φ ( x, s ) must satisfy thefollowing equation φ xxxx ( x ) − β eq ( s ) φ ( x ) = 0 (6)where it is defined β eq ( s ) = − µ s J xz E ( s ) = − µ s J xz C ( s ) (7)and the compliance of the viscoelastic material C ( s ) = 1 /E ( s ) . The boundary conditionsthen become φ xx (0) = 0 (8) φ xxx (0) = 0 φ xx ( L ) = 0 φ xxx ( L ) = 0The solution of the above Eq.(6) can be written in the form φ ( x, s ) = W cos [ β eq ( s ) x ] + W sin [ β eq ( s ) x ] + W cosh [ β eq ( s ) x ] + W sinh [ β eq ( s ) x ] (9)and by requiring that the determinant of the system matrix obtained from Eqs.(8) is zero,one obtains [1 − cos ( β eq L ) cosh ( β eq L )] = 0 (10)The solutions β n L = c n of the above Eq.(10) are well known [30], and they are the same ofthe perfectly elastic case. In particular, from the following relation − µ s J xz E ( s ) = ( β n ) = (cid:16) c n L (cid:17) (11)5t is possible to calculate the complex conjugate eigenvalues s n corresponding to the n th mode, and the real poles s k related to the material viscoelasticity (a detailed analysis ofthe eigenvalues will be shown in the next section). The values β n allow to determine theeigenfunctions φ n ( x ) φ n ( x ) = cosh ( β n x ) + cos ( β n x ) − cosh ( β n L ) − cos ( β n L )sinh ( β n L ) − sin ( β n L ) [sinh ( β n x ) + sin ( β n x )] (12)which are equal to the eigenfunctions of the elastic case. A simple proof of the previ-ous statement can be shown by considering the initial conditions u ( x,
0) = φ n ( x ) and u t ( x,
0) = 0 of the problem Eq.(4). In this case, indeed, the solution of Eq.(4) is u ( x, t ) = C e Re[ s n ] t φ n ( x ) cos (Im [ s n ] t ). III. BEAM RESPONSE
In this section the solution of Eq.(3) is calculated, by considering (see Ref.[26]) thedecomposition of the system response into the modes φ n ( x ) of the beam u ( x, t ) = + ∞ (cid:88) n =1 φ n ( x ) q n ( t ) (13)For the orthogonality condition one has1 L (cid:90) L φ n ( x ) φ m ( x ) d x = δ nm (14)where δ nm is Kronecker delta function. Moreover, because of Eqs.(6)-(7), the followingrelation holds true1 L (cid:90) L ( φ n ) xxxx ( x ) φ m ( x ) d x = 1 L (cid:90) L φ n ( x ) β n φ m ( x ) d x = δ nm β n (15)Let us project the equation of motion on the function φ m ( x ) of the basis. The projectedsolution u m ( t ) is defined as u m ( t ) = (cid:104) u ( x, t ) φ m ( x ) (cid:105) = 1 L (cid:90) L u ( x, t ) φ m ( x ) d x (16)therefore Eq.(3) becomes µ ¨ q n ( t ) + J xz β n (cid:90) t −∞ E ( t − τ ) q n ( τ ) d τ = f n ( t ) (17)6here f n ( t ) = L (cid:82) L f ( x, t ) φ n ( x ) d x is the projected force term. By taking the LaplaceTransform of Eq.(17), with initial conditions equal to zero, one obtains µs Q n ( s ) + J xz β n E ( s ) Q n ( s ) = F n ( s ) (18)It is possible to rewrite the above equation as Q n ( s ) = H n ( s ) F n ( s ) (19)where the function H n ( s ) = 1[ µs + J xz β n E ( s )] (20)is the Transfer Function of the system, for the n th mode.Eq.(13) can be therefore written in the Laplace domain as U ( x, s ) = + ∞ (cid:88) n =1 φ n ( x ) F n ( x, s ) µs + J xz β n E ( s ) (21)In particular, by considering as external applied force, a Dirac Delta of constant ampli-tude F , in both the time and the spatial domains, the force can be written as f ( x, t ) = F δ ( x − x f ) δ ( t − t ). Therefore in the Laplace domain it becomes F n = (cid:90) L F δ ( x − x f ) φ n ( x ) d x = F φ n ( x f ) (22)and finally the system response is U ( x, s ) = F ∞ (cid:88) n =1 φ n ( x ) φ n ( x f ) µs + J xz β n E ( s ) (23). IV. VISCOELASTIC MODEL - SYSTEM EIGENVALUES
Let us first consider an ideal viscoelastic material with a single relaxation time τ , whoseelastic properties can be represented by the modulus E ( s ) = E + E τ s τ s (24)By substituting the previous complex function in Eq.(11), the characteristic equation foreach n th mode can be obtained τ s + s + ( E + E ) τ r n s + r n E = 0 (25)7 IG. 2: Viscoelastic beam of lenght L , cross section area A = W H . where r n = ( β n ) J xz /µ . Notice that the solutions of the cubic equation Eq.(25) can bei) one real root and two complex conjugate roots, ii) all roots real. This means that oneeigenvalue is always related to an overdamped motion. When the other two eigenvalues arecomplex conjugate, they represent the oscillatory contribute of the n th mode in the beamdynamics. Otherwise, in case of three real roots, the n th mode is not oscillatory.With regards to the transverse motions of a narrow, homogenous beam with a bendingstiffness E J xz and density ρ , the value of the natural frequencies can be calculated using asimple formula which is always valid, regardless of the boundary conditions [32]: ω n = (cid:16) c n L (cid:17) (cid:115) E J xz ρA (26)where coefficient c n depends on the specific boundary conditions. The first natural frequency,in particular, can be written as ω = α δ (27)being δ = c (cid:112) E A/ ( ρJ xz ), and α = R g /L the dimensionless beam length, with R g = (cid:112) J xz /A the radius of gyration. For the rectangular beam cross section under investigation(Figure 2), one has α = H/ (cid:0) √ L (cid:1) and δ = ( c /H ) (cid:112) E /ρ . The non-dimensionaleigenvalue is now defined ¯ s = s/δ (28)and in particular one has, for the n th mode, ω n = E β n J xz /µ = r n E and δ n = c n (cid:112) E A/ ( ρJ xz ). 8y substituting Eq.(28) in Eq.(25), the following non-dimensional characteristic equationis obtained ¯ s + ¯ s θ + (1 + γ ) α ∆ n ¯ s + 1 θ α ∆ n = 0 (29)where ∆ n = δ n /δ , and having defined the dimensional groups θ = δ τ (30) γ = E /E (31). Eq.(29) can be then re-written as ¯ s + (cid:88) j =0 a jj ¯ s j = 0 (32)where a = (1 /θ ) α ∆ n , a = (1 + γ ) α ∆ n , a = 1 /θ .By defining Q = 3 a − a R = 9 a a − a − a
54 (34)the discriminant of Eq.(32) is D = Q + R , and the solutions of Eq.(32) can be thereforewritten as [33] ¯ s = − a −
12 ( S + T ) + i √
32 ( S − T ) (35)¯ s = − a −
12 ( S + T ) − i √
32 ( S − T )¯ s = − a S + T )where S = (cid:112) R + √ D and T = (cid:112) R − √ D . In our case, the discriminant D , indicated as D ( n ), is function of n , i.e. of the number of the n th mode considered D ( n ) = α ∆ n (cid:8) α ∆ n θ (cid:2) − γ (20 + γ ) + 4 α (1 + γ ) ∆ n θ (cid:3)(cid:9) θ (36)This function D ( n ) plays a key role in the understanding the nature of the roots of Eq.(32),as it will be widely discussed in Section III.At last, the beam cross-section acceleration A ( x, s ) in terms of the above defined non-dimensional groups is formulated, considering that A ( x, s ) = s U ( x, s ) (see Eq.(23))9 b)(a) FIG. 3: The region map for the first flexural mode of the beam n = 1, for θ = ¯ θ (a). The shadedarea indicates the parameter ( α, γ ) combinations which determine the suppression of the firstpeak. Similar region maps are shown, for the first three modes n = 1 , , α and γ , some areas at D ( n ) < A ( x, ¯ s ) = F ∞ (cid:88) n =1 ¯ s (1 + θ ¯ s ) φ n ( x ) φ n ( x f ) µθ (cid:16) ¯ s + (cid:80) j =0 a j ¯ s j (cid:17) (37)being φ n ( x ) the eigenfunctions defined in Eq.(12).More realistically, a second relaxation time contribute is now included in the viscoelasticmodulus E ( s ), which therefore becomes E ( s ) = E + E τ s τ s + E τ s τ s (38)Following the same approach previously described, the fourth-order characteristic equationfor each n th mode can be obtained ¯ s + (cid:88) j =0 a j ¯ s j = 0 (39)10here a = α ∆ n θ θ (40) a = (cid:18) θ + 1 θ + 1 θ γ + 1 θ γ (cid:19) α ∆ n a = (cid:18) θ θ + α ∆ n + α ∆ n γ + α ∆ n γ (cid:19) a = (cid:18) θ + 1 θ (cid:19) having defined γ = E /E and θ = τ δ . Moreover, it is possible to define, for the quarticequation Eq.(39), the discriminant D ( n ) [34]-[35] D ( n ) = 256 a − a a a − a a + 144 a a a − a + 144 a a a − a a a − a a a a +(41)+ 18 a a a + 16 a a − a a − a a + 18 a a a a − a a − a a a + a a a which can be utilized to deduce important properties of the roots of Eq.(39).The beam cross-section acceleration A ( x, ¯ s ) is in this case A ( x, ¯ s ) = F ∞ (cid:88) n =1 ¯ s (1 + θ ¯ s ) (1 + θ ¯ s ) φ n ( x ) φ n ( x f ) µθ θ (cid:16) ¯ s + (cid:80) j =0 a j ¯ s j (cid:17) (42) V. RESULTS
In this section the main results of the presented analysis are discussed. The flexuralvibrations of a viscoelastic beam with rectangular cross section and thickness H = 1 [cm],which oscillates in the xz -plane (Figure 2) are studied. The only geometrical parameterwhich is considered varying in calculations, is the beam length L . In particular, the ratio α = R g /L is changed maintaining R g = H/ √
12 constant. The main scope of the paper isnot a quantitative investigation of a specific viscoelastic material, but a qualitative studyof a generic viscoelastic beam behaviour, which can be considered at different lengths L (e.g. in experimental testing campaigns, to cover wide frequency ranges) and at differentworking temperatures (i.e. with varying elastic coefficients E k and relaxation times τ k ). Inthis view, the two material properties considered constant in the numerical calculations are11 a) (b) FIG. 4: The discriminant D (1) for the first mode n = 1, as a function of γ , for θ = ¯ θ (a),and as a function of θ , for γ = ¯ γ (b), for different values of α , i.e α = ˜ α (solid line), α = 1 . α (dashed line), α = 3 ˜ α (dot-dashed line). ρ = 1180 [kg m − ] and E = 2 . ∗ [Pa] of a typical viscoelastic material, i.e. PMMA(polymethyl methacrylate) [36]. Therefore the parameters δ n are constant and, in particular, δ = 3 . ∗ for the first flexural mode of the beam. The other properties E , E , τ and τ are taken varying in the analysis, however ¯ γ = E /E = 87 and ¯ γ = E /E = 126 ofPMMA are considered as reference. Moreover, the relaxation times for the frequency rangeunder study give the reference values ¯ θ = δ τ = 170, ¯ θ = δ τ = 6700. The numericalvalues here considered, are simply representative of a real viscoelastic material but, thanksto the dimensional analysis presented in the paper, can be substituted with the constantsof any other viscoelastic material, thus not modifying the qualitative results.At first, let us consider an ideal viscoelastic material with one relaxation time τ , andelastic coefficients E and E (Eq.24). For each n th mode, the three eigenvalues (see Eq.(35))can be calculated. The two complex conjugate eigenvalues represent the oscillatory counter-part of the beam n th mode. The real eigenvalue gives rise to a pure dissipative contribute.However, when the discriminant D ( n ), defined in Eq.(36), is negative D ( n ) <
0, all rootsof Eq.(29) are real, and the n th mode is not oscillatory. In Figure 3-a, a region map isshown, for the first flexural mode of the beam ( n = 1), with θ = ¯ θ . The shaded area isobtained with the parameter values ( α, γ ) which give the condition D (1) <
0. Analogousmaps, for the first three flexural modes of the beam, are represented by the correspondent12
IG. 5: The acceleration modulus | A (¯ x, ω ) | of the viscoelastic beam with one relaxation time, inthe section x = x f = ¯ x = 0 . L , for θ = ¯ θ , γ = ¯ γ , and for three different values of α , i.e. α = ˜ α (solid line), α = 1 . α (dashed line), α = 3 ˜ α (dot-dashed line). For α = 3 ˜ α , being D (1) < curves in Figure 3-b, which are obtained by finding the two real solutions α = α ( γ ) ofthe equation D ( n ) = 0, for n = 1 , ,
3. By properly combining the parameters ( α, γ ),different peaks can be suppressed simultaneously, since the areas which give the condition D ( n ) < n are overlapped. In particular, this means that, oncethe material is prescribed, i.e. for given values of θ and γ , the dynamics of the beamcan be decisively modified by varying its length L . The sign of the discriminant D (1), forthe first mode, can be directly deduced by means of the curves plotted in Figure 4, where D (1) is shown as a function of γ (Figure 4-a), for θ = ¯ θ , and as a function of θ , for γ = ¯ γ (Figure 4-b), for different values of α , i.e. α = ˜ α (solid line), α = 1 . α (dashedline), α = 3 ˜ α (dot-dashed line), where it has been considered the reference beam lengthequal to ˜ L = 60[cm], and therefore ˜ α = R g / ˜ L = 0 . | A (¯ x, ω ) | (see Eq.37), evaluated at thebeam section x = x f = ¯ x = 0 . L , for θ = ¯ θ and γ = ¯ γ . Three different values of beamlength L are considered, i.e. α = ˜ α (solid line), α = 1 . α (dashed line), α = 3 ˜ α (dot-dashedline), which give a clear first peak for α = ˜ α and α = 1 . α , a suppressed first peak for α = 3 ˜ α , being D (1) < a) (b) FIG. 6: The region map for the first flexural mode of the beam n = 1, with two relaxation times,for θ = ¯ θ , θ = ¯ θ , γ = ¯ γ : the shaded area indicates the parameter ( γ , α ) combinationswhich determine the suppression of the first peak (a). The acceleration modulus | A (¯ x, ω ) | of theviscoelastic beam with two relaxation times is plotted, in the beam section x = x f = ¯ x = 0 . L ,related to points B (20; 0 . α ) and C (20; 0 . α ) . possible variations in the beam dynamics. In Figure 6-a a region map is shown, for the firstflexural mode of the beam ( n = 1), with θ = ¯ θ , θ = ¯ θ , γ = ¯ γ . The shaded area isobtained with the parameter values ( γ , α ) compounds which give the condition D (1) > D ( n ) < A in Figure 6-a is obtained by considering the reference parameters (¯ γ , ˜ α ). Since itis far from the shaded area at D (1) >
0, it is related to a oscillatory first mode for small α variations. In order to better evaluate the influence of the parameter α , points B and C have been considered to calculate the frequency response of the beam. In Figure 6-b, theacceleration modulus | A (¯ x, ω ) | (see Eq.42), calculated at the beam section x = x f = ¯ x =0 . L , is shown for θ = ¯ θ , θ = ¯ θ , γ = ¯ γ , γ = 20 and for α = 0 . α (solid line, point B of Figure 6-a), α = 0 . α (dashed line, point C of Figure 6-a). It is possible to observe thatthe curve obtained with α = 0 . α does not present the first peak, because of the consideredparameters, which give in this case D (1) > a) (b) FIG. 7: The viscoelastic modulus E ( ω ), for one relaxation time, with θ = ¯ θ , γ = ¯ γ (solid curve),and for two relaxation times, with θ = ¯ θ , γ = ¯ γ , θ = ¯ θ , γ = ¯ γ (dashed curve), representedby the function tan δ = Re[ E ( ω )] / Im[ E ( ω )] (a); the acceleration modulus | A (¯ x, ω ) | is shown inthe two cases, for α = ˜ α , x = x f = ¯ x = 0 . L , for one relaxation time (solid curve) and for tworelaxation times (dashed curve) (b). VI. FREQUENCY RESPONSES
Let us observe that, besides the sign of the discriminants D ( n ) and D ( n ), and thedifferent values of α , which establish the possible n th peak suppression, there is no significantdifference in considering one or two relaxation times. In this respect, the spectrum of theviscoelastic modulus E ( ω ) is considered in the two cases, as shown in Figure 7-a, wherethe function tan δ = Re[ E ( ω )] / Im[ E ( ω )] is plotted, for one relaxation time ( θ = ¯ θ , γ =¯ γ , solid curve), and for two relaxation times ( θ = ¯ θ , γ = ¯ γ , θ = ¯ θ , γ = ¯ γ , dashedcurve). It is evident that in the two cases, the transition region, where the function tan δ reaches the maximum, thus determining the prominent energy dissipation, is differentlypositioned in the frequency spectrum. The correspondent acceleration moduli | A (¯ x, ω ) | areshown in Figure 7-b, for α = ˜ α , x = x f = ¯ x = 0 . L , for one relaxation time (solid curve)and for two relaxation times (dashed curve). Notice that, in the case of two relaxationtimes the function tan δ presents higher values, with respect to the one relaxation time case,in the range of frequencies where the first peak lies (0 −
100 [rad s − ]). This is why thefirst peak, when two relaxation times are considered, is more damped. It is important to15nderline that, because of the intrinsic characteristics of viscoelastic materials [29], whichsee the viscoelastic modulus E ( ω ) depending on temperature, the above mentioned dampingeffect can be observed just modifying the surrounding temperature. Indeed, increasingor decreasing the working temperature, the functions Re[ E ( ω )] and Im[ E ( ω )] are shiftedtowards higher or smaller frequencies respectively (as well as and the function tan δ ), andconsequently the material damping is differently spread in frequency.However, once the material is prescribed, i.e. the viscoelastic modulus E ( ω ) is definedwith the related parameters, θ and γ for one relaxation time, θ , θ , γ and γ for two relax-ation times, the dimensionless beam length α plays a crucial role in the possible overlappingof the first natural frequency ω with the transition region. Moreover, in the hypothesisof linearity, such considerations can be extended to all the peaks, since the system can bedecoupled and each vibration mode can be studied independently.In conclusion, through the defined dimensionless parameters, it is possible to completelydisclose the transversal vibrations of a viscoelastic beam. Suppressing certain peaks, byvarying the beam length with α , or by changing the material properties (i.e. θ and γ forone relaxation time, θ , θ , γ and γ for two relaxation times) for example by modifying thesurrounding temperature, is an appealing chance for different applications. In particular, itis important to stress that, although the real viscoelastic materials present more than tworelaxation times, the number of relaxation times to be considered in modelling the beam dy-namics, does not represent a limit of our study. Firstly, because it has been shown that thereis not a considerable difference, from a qualitative point of view, by increasing the numberof time relaxations. Furthermore, it is always possible to divide the frequency spectrumunder analysis in several intervals, thus allowing the decreasing of the predominant timerelaxations number in such intervals. Moreover, by varying the beam length, it is possibleto study a wide frequency range, by focusing the attention only to the first resonant peaks,so that the (Euler-Bernoulli) hypothesis still remains valid.Finally, it must be pointed out that this study can be utilized to properly interpret theviscoelastic beam vibrational spectrum, when a material characterization is carried out.This is an awkward task, indeed, since when a viscoelastic beam with an unknown materialis experimentally studied, the resonant peaks positions are not so straightforward to beidentified, as in the elastic case. This kind of experimental investigation is currently objectof study, with the aim to characterize viscoelastic materials by means of the transversal16ibrations of beams with different lengths. VII. CONCLUSIONS
In this paper an analytical study of the transversal vibrations of a viscoelastic beam hasbeen presented. The analytical solution has been obtained by means of modal superposition.In particular, while the beam eigenfunctions are the same of the perfectly elastic case, theeigenvalues strongly depend on the material viscoelasticity, and they increase in numberwith the relaxation times of the viscoelastic modulus. In order to put in evidence the maincharacteristics of the beam dynamics, two cases have been considered, i.e. a viscoelasticmaterial both with one single relaxation time and with two relaxation times. A dimensionalanalysis has been performed, which has disclosed the fundamental parameters involved in thevibrational behaviour of the beam. Such parameters depend on both the material propertiesand the beam geometry. Some new characteristic maps related to the eigenvalues nature ofthe studied system have been provided, that can be drawn for each natural frequency of thebeam. In comparison to the existing maps presented in literature for a sdof system, thesemaps may help in determining the parameter compounds needed to enhance or suppresscertain frequency peaks, one at a time or more simultaneously, and the same approachcan be exploited for any kind of mdof system. Interestingly, it has been observed that, bymaintaining constant the thickness of the beam cross section, the dimensionless beam lengthcan be utilized as key parameter to properly adjust the resonant peaks, once the materialhas been selected. The presented study, hence, enables to conveniently design a viscoelasticbeam, in order to obtain the most suitable dynamics in the frequency range of interest, thusbecoming a powerful tool for many applications, from system damping control to materialscharacterization.
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