Quasi-static response of linear viscoelastic cantilever beams subject to a concentrated harmonic end load

International Journal of Non-Linear Mechanics 54 (2013) 43–54

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Quasi-static response of linear viscoelastic cantilever beams subject to a concentrated harmonic end load M.A. Vaz n, A.J. Ariza Ocean Engineering Department, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

art ic l e i nf o

a b s t r a c t

Article history: Received 25 July 2011 Received in revised form 4 December 2012 Accepted 19 March 2013 Available online 3 April 2013

This study evaluates the response of a uniform cantilever beam with a symmetric cross-section fixed at one end, and submitted to a lateral concentrated sinusoidal load at the free extremity. The beam material is assumed to be homogeneous, isotropic and linear viscoelastic. Due to the nature of the loading and the beam slenderness, large displacements are developed but the strains are considered small. Consequently, the mathematical formulation only involves geometrical non-linearity. It is also assumed that the beam is inextensible (neutral axis length is constant) and that inertial forces are negligible, i.e., dynamic effects are insignificant and the system can thus be modeled quasi-statically. The beam is therefore subject to oscillations caused by the sinusoidal time-dependent load, leading to a transient response until the material stabilizes and the system exhibits a periodic response, which can be conveniently described in the frequency domain. The time domain solution of this problem is elaborated by considering the quasi-static response for each time interval. The mathematical equations are presented in dimensional and dimensionless forms, and for the latter case, a numerical solution is generated and several case studies are presented. The problem is governed by a set of non-linear ordinary differential equations encompassing functions of space and time that relate the curvature, rotation angle, bending moment and geometrical coordinates. In this study, an elegant solution is deduced using perturbation theory, yielding a precise steady-state solution in the frequency domain with considerable computational economy. The solutions for both time and frequency domain methods are developed and compared using a case study for a series of dimensionless parameters that influence the response of the system. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Elastica Cantilever beam Linear viscoelasticity Perturbation method

1. Introduction The structural response of beams has long been the subject of intense research because they are a fundamental component of engineering systems. The first studies examined the response of beams exposed to small displacements (as compared to their lengths) and moderate loadings because of the ease in obtaining analytical solutions from the linear ordinary differential equations that govern such configurations. With both mathematical and computational advances combined with the necessity to describe more complex situations, numerical solutions have been developed for beams composed of non-linear materials and submitted to different boundary conditions and significant loads yielding finite displacements and large strains. In these situations, the differential equations that govern the problem exhibit strong nonlinearity due to the response of the material or due to changes in the geometric configuration. Barten [2,3] and Bisshopp and Drucker [6] presented analytical solutions using elliptical integrals for the response of a cantilever beam composed of a linear elastic

n

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material submitted to large displacements. Analytical solutions obtained using elliptical and numerical integrals were considered in the post-buckling study of a beam-column by Lee [13] and Sepahi et al. [24]. The response of beams composed of non-linear elastic materials and submitted to large displacements was studied by Lewis and Monasa [16], Lee [14], Eren [11] and Brojan et al. [7], who used a Ludwick constitutive model for different loadings. Brojan et al. [7] also developed a formula to relate moment and curvature for beams with non-uniform cross-sections, and employed it in the study of a cantilever beam submitted to large displacements. In general, beams may be submitted to different levels and types of loading. Lee [14], Eren [11] and Beléndez et al. [5] numerically modeled the response of an inextensible cantilever beam, with a uniform cross-section, that was subjected to large displacements and uniformly distributed loading with a vertical load at the free extremity. Dado and Al-Sadder [10] and Wang et al. [27] considered a cantilever beam composed of linear elastic material submitted to large displacements. They performed a series approximation of the rotational angles and utilized different mathematical models to obtain their coefficients. Series approximation allows for a semi-analytical formulation, as presented by Carrillo [8,9]. A beam may present different equilibrium configurations that depend on the form in which the load is imposed.

44

M.A. Vaz, A.J. Ariza / International Journal of Non-Linear Mechanics 54 (2013) 43–54

Batista and Kosel [4], Shvartsman [25], Nallathambi et al. [19] and Mutyalarao et al. [18] studied the multiple configurations exhibited by a cantilever beam composed of linear elastic material with a uniform cross-section submitted to different loading levels and forms of load application. A more in-depth study, which considered the tridimensional spatial configuration of an elastic cantilever beam with a uniform cross-section, was performed by Pai and Palazotto [20]. Light and resistant polymer components have been considered as a structural material to replace metals in weight critical parts in the aerospace, automotive and marine industry. For the study of beams constructed of viscoelastic materials (elastomers, thermoplastics and thermorigids), it is important that the material mechanical properties be known, in addition to the external factors that may affect its properties. Ward and Pinnock [28] presented a general study on the mechanical properties of polymers and a detailed study of the principal models that best represent their mechanical behaviors and also investigated how they may vary with time and temperature. When considering a differential element of a linear viscoelastic material with a nonlinear geometry, while disregarding inertial forces, it is possible to develop a quasi-static equilibrium equation whose solution changes with time, allowing for bifurcations in the new equilibrium conditions. Pina et al. [22] focused on solving quasi-static problems principally for non-linear materials or geometrically non-linear problems with responses in the time domain. For the analytical development of problems involving linear viscoelastic materials Laplace transformation may be employed. Adey and Brebbia [1] presented a solution for static loading conditions with significant reduction in the computational time. Renganathan et al. [23] employed a similar methodology, which satisfactorily described the time domain system behavior. Both papers demonstrated that Laplace transformation is reliable and valid for geometrically linear analysis. The mechanical properties of polymers may be linear or nonlinear viscoelastic, and the differences in their responses are mainly due to the loading magnitude, analysis time and temperature changes. Winemam and Kolberg [29,30] and Wineman and Min [31] compared the time domain behaviors of linear and non-linear viscoelastic beams with different loading histories, and showed the response differences. The problems were developed by assuming quasi-static equilibrium at a constant temperature. The mathematical equations of a cantilever beam composed of a linear viscoelastic material and submitted to large displacements were developed in the works of Lee [15] and Vaz and Caire [26], who studied the behavior of such a system submitted to different loading forces. The dynamic response of beams made of viscoelastic materials and subject to small displacements has been studied analytically and numerically by several authors. Pérez-Gavilán and Aliabadi [21], for instance, employed the symmetric Galerkin boundary element method for two-dimensional viscoelastic problems in the frequency domain and compared the solution for three cases with the collocation method. Kargarnovin et al. [12] analyzed an infinity beam on a non-linear viscoelastic foundation submitted to harmonic moving loads. They compared solutions for Bernoulli–Euler and Timoshenko beams' formulations and employed a perturbation technique for a closed form solution. In this study, a cantilever beam composed of a linear viscoelastic material with a uniform cross-section is submitted to a vertical load at the free extremity, and quasi-static solutions are found. Non-linear ordinary differential equations are developed using the fourth-order Runge–Kutta method. The boundary value problem is resolved by using a shooting method. Assuming an initial estimate of curvature at the support (missing boundary condition), the problem is then solved and the curvature at the free extremity of the beam is found and compared to the actual

boundary condition (zero curvature) until the difference between the two solutions is less than an acceptable value, which in this paper is taken as 10−10. Time and frequency domain solutions are developed and solutions are compared for a case study. The following paragraph was added to the text: even though there are many simplifying assumptions and the problem looks rather specific as it is typified for a particular class of problem, the methodology developed here may be extended to other systems, i.e., boundary conditions, load types, thick and extensible beams and even non-linear viscoelastic materials.

2. The mathematical formulation The beam studied herein is composed of a linear viscoelastic material, fixed at one end and submitted to a time (t)- and frequency (ω)-dependent harmonic lateral force P at the free extremity (Fig. 1). For the mathematical development, the dynamic effects are disregarded because the original inertial forces are considered insignificant. This is typically valid when the excitation frequencies are much lower than the system natural frequencies. The mechanical property of the viscoelastic, homogeneous and isotropic material is represented by a standard linear solid model. Therefore, the equations in the time domain are developed as a quasi-static problem for each time instant, taking into consideration large transverse displacements (large rotations) and small strains. The angle ϕ represents the rotation for each beam crosssection, which depends upon the loading level. The length (L) of the beam remains constant because the axial stiffness is assumed to be high. The formulation developed here may be promptly extended to other loading and boundary conditions as well as to different material constitutive relations. The stress relaxation and creep functions for the standard linear solid model, represented by Prony series with one exponential term, are respectively: −t

GðtÞ ¼ G∞ þ ðG0 −G∞ Þeτr −t

JðtÞ ¼ J ∞ þ ðJ 0 −J ∞ Þeτc

ð1Þ ð2Þ

where the material constants G0 , G∞ , τr , J 0 , J ∞ , and τc , which can be obtained from relaxation and creep tests, are related by: G0 J 0 ¼ 1, G∞ J ∞ ¼ 1 and G∞ τc ¼ G0 τr . The force imposed on the beam extremity consists of a constant (static) load P0 applied instantaneously at time t¼0, superimposed with a sinusoidal load with amplitude εP0 and frequency ω, which causes oscillations in the beam. The magnitude of the sinusoidal load is considered as a percentage (ε) of the static load P0: Pð t; ωÞ ¼ P 0 þ εP 0 sinðω tÞ

ð3Þ

The equations of motion for the beam are derived by considering two mathematical models, one in the time domain and

Fig. 1. Configuration of a cantilever beam submitted to a concentrated loading at the free extremity.

M.A. Vaz, A.J. Ariza / International Journal of Non-Linear Mechanics 54 (2013) 43–54

another in the frequency domain. The response of the beam in the time domain includes two regions (transient and permanent) with distinct behaviors. The beam response in the permanent region is the focus of this study. Viscoelastic materials generally require a long time to completely relax before exhibiting a permanent pattern. Therefore, this type of analysis involves a considerable computational cost. Using a classic method based on perturbation theory, solutions for the beam equations in the frequency domain are developed and it will be shown that they adequately describe the permanent response of the system. 2.1. Time domain solution To obtain the equations of motion for the beam, the following steps are taken. First, a fixed coordinate system at the beam support is employed (Fig. 1). Second, the differential element is evaluated (Fig. 2), yielding the geometric compatibility equations. Third, a quasi-static equilibrium equation relating force, bending moment and angle (Fig. 3) is invoked. Fourth, the constitutive relation of linear viscoelastic material is substituted in the equilibrium equations. Finally, equations satisfying the boundary conditions are numerically developed for each time step. 2.1.1. Geometric relationships The geometric relationships are obtained as shown in Fig. 2. The first two equations are obtained via trigonometric relationships, and the third equation is obtained by the definition of curvature for a plane curve: ∂xðs; t; ωÞ ¼ cos½ϕðs; t; ωÞ ∂s

ð4Þ

45

∂yðs; t; ωÞ ¼ sin½ϕðs; t; ωÞ ∂s

ð5Þ

∂ϕðs; t; ωÞ ¼ κðs; t; ωÞ ∂s

ð6Þ

where s is the longitudinal arc length measured from the built-in end, x and y describe the configuration of the beam in space and time, and κ represents the curvature in each section of the beam. This system of non-linear differential equations presents the independent “space, time and frequency” (s; t; ω) variables. To develop the differential equations (Eqs. (4)–(6) for the dependent variables (x; y; ϕ; κ), one more differential equation is necessary. This equation is obtained from the quasi-static equilibrium condition. 2.1.2. Equilibrium of forces and moments The mathematical relation between the bending moment and imposed load can be found from the quasi-static equilibrium relation shown in Fig. 3, considering large displacements. ∂Mðs; t; ωÞ ¼ −Pðt; ωÞ cos½ϕðs; t; ωÞ ∂s

ð7Þ

2.1.3. Bending moment versus curvature The mathematical relation between the bending moment and curvature for a linear viscoelastic material (small strains) is given by the following equations: ∂Mðs; t; ωÞ ∂κðs; t; ωÞ ¼ I  dGðtÞ ∂s ∂s

ð8Þ

∂κðs; t; ωÞ ∂Mðs; t; ωÞ ¼  d JðtÞ ∂s ∂s

ð9Þ

I

where the constant I represents the cross-section moment of inertia for a uniform beam. These equations are similar to the stress–strain relations presented by Wineman and Rajagopal [32]. The substitution of Eq. (7) in (9) produces the curvature–load relation:  ∂κðs; t; ωÞ 1
¼− Pðt; ωÞ cos½ϕðs; t; ωÞ  d JðtÞ ∂s I An expansion of the previous equation for numerical development gives: ∂κðs; t; ωÞ 1 ¼ − P ðt; ωÞ cos½ϕðs; t; ωÞJ 0 ∂s I Z 1 t ∂Jðt−υÞ dυ − Pðυ; ωÞcos½ϕðs; υ; ωÞ I 0 ∂ðt−υÞ

ð10Þ

The integro-differential equation (Eq. (10)) must be developed for each time, together with Eq. (6). It is necessary to specify two boundary conditions and to define the time steps of the analyses.

Fig. 2. Beam differential element.

()

2.1.4. Boundary conditions The boundary conditions are obtained from the geometrical configuration. In Fig. 1, the origin of the coordinate system coincides with the beam built-in end, and there is no curvature (moment) at the free extremity. Therefore: xð0; t; ωÞ ¼ yð0; t; ωÞ ¼ ϕð0; t; ωÞ ¼ κðL; t; ωÞ ¼ 0 ( ) ( )

( ) Fig. 3. Beam section for analysis.

ð11Þ

2.1.5. Dimensionless solution Eqs. (4)–(6) and (10) and the boundary conditions (11) govern the behavior of the beam for each time step, but to include different case studies, it is convenient to make the problem dimensionless by introducing the following variables: s¼

s ; L

x¼

x ; L

y¼

y ; L

κ ¼ κ L;

t¼

t ; τr

ω ¼ ωτr ;

υ¼

υ ; τr

46

M.A. Vaz, A.J. Ariza / International Journal of Non-Linear Mechanics 54 (2013) 43–54

α¼

G∞ ; G0

P0 ¼

P 0 L2 þ I G0

Substitution into the previous equations combined with some algebraic manipulations and term rearrangements gives:   ∂xðs; t; ωÞ ¼ cos ϕðs; t; ωÞ ∂s

ð12Þ

  ∂yðs; t; ωÞ ¼ sin ϕðs; t; ωÞ ∂s

ð13Þ

∂ϕðs; t; ωÞ ¼ κðs; t; ωÞ ∂s

ð14Þ

  ∂κðs; t; ωÞ ¼ −Pðt; ωÞ cos ϕðs; t; ωÞ ∂s Z þðα−1Þe−tα

0t

Pðυ; ωÞcos½ϕðs; υ; ωÞeυα dυ

ð15Þ

The boundary conditions are then rewritten as: xð0; t; ωÞ ¼ yð0; t; ωÞ ¼ ϕð0; t; ωÞ ¼ κð1; t; ωÞ ¼ 0

ð16Þ

Eqs. (14) and (15) are developed simultaneously, and using the boundary conditions in (16), it is possible to obtain the angle of rotation and curvature. Then, by substituting the calculated angles into Eqs. (12) and (13) and using the initial position conditions (16), the beam configuration in space is obtained for each time point. Eq. (15) is an integro-differential equation that is developed for each time step. The time necessary to reach a periodic response is large, and thus, to obtain the results of this system of equations, significant computational time is required. To solve this system of non-linear ordinary differential equations, a fourth-order Runge–Kutta algorithm was implemented for developing the equations in space and time using the shooting method, which transforms a boundary value problem into an initial value problem. Mathcad©: Mathcad computational program [17] was used for this procedure. 2.2. Frequency domain solution The time domain response presents a considerably long transient regime before periodic oscillations are reached. Therefore, processing time is heavily penalized. The proposed model develops the time domain governing equations for t-∞, hence a steady-state solution is obtained. The coefficient ε (Eq. (3)), which regulates the amplitude of the variable load, is the parameter in the mathematical model that causes perturbations in the system. For this perturbation, the differential Eqs. (4)–(7) are employed. The angle of rotation, curvature and bending moment are respectively estimated by using a second-order perturbation approximation: ( ϕ0 ðsÞ þ ε½ϕ1 ðsÞsinðωtÞ þ ϕ2 ðsÞcosðωtÞ ϕðs; t; ωÞ ¼ ð17Þ þε2 ½ϕ3 ðsÞ þ ϕ4 ðsÞsinð2ωtÞ þ ϕ5 ðsÞcosð2ωtÞ ( κðs; t; ωÞ ¼

κ0 ðsÞ þ ε½κ 1 ðsÞsinðωtÞ þ κ 2 ðsÞcosðωtÞ þε2 ½κ3 ðsÞ þ κ 4 ðsÞsinð2ωtÞ þ κ 5 ðsÞcosð2ωtÞ

( Mðs; t; ωÞ ¼

M 0 ðsÞ þ ε½M 1 ðsÞsinðωtÞ þ M 2 ðsÞcosðωtÞ þε ½M 3 ðsÞ þ M 4 ðsÞsinð2ωtÞ þ M 5 ðsÞcosð2ωtÞ 2

ð18Þ

M 0 ¼ I G∞ κ 0 " 0 " # M1 G ð ωÞ ¼ I 00 M2 G ðωÞ M 3 ¼ IG∞ κ3 " 0 " # M4 G ð2ωÞ ¼ I 00 M5 G ð2ωÞ

−G00 ðωÞ G 0 ð ωÞ

κ1

#

κ2

−G00 ð2ωÞ

#"

G0 ð2ωÞ

κ4

# ð21Þ

κ5

The angle of rotation represented by Eq. (17) is redefined, for a better interpretation, as: ϕðs; t; ωÞ ¼ ϕ 0 ðsÞ þ Δϕ ðs; t; ωÞ

ð22Þ

Approximating the sine and cosine functions using the Taylor series: sinðφÞ≈φ cosðφÞ≈1−

φ2 2!

The substitution of Eqs. (3) and (22) in (7) and the expansion of the cosine by a Taylor series approximation result in: ( ) cos½ϕ0 ðs; t; ωÞ−sin½ϕ0 ðs; t; ωÞΔϕðs; t; ωÞ ∂Mðs; t; ωÞ   ¼ −P 0 ½1 þ ε sinðωtÞ 2 1 −2cos ϕ0 ðs; t; ωÞ Δϕðs; t; ωÞ ∂s

ð23Þ Substituting Eqs. (17) in (23) and arranging the terms by the degree of perturbation, the derivative of the bending moment coefficients is obtained, which can then be substituted, together with the curvature derivative (18), into Eq. (21), resulting in the following system of differential equations: dκ0 P0 þ cosðϕ0 Þ ¼ 0 ds I G∞ 2

dκ 1 4 ds dκ 2 ds

ð24Þ

3

" # "  # ϕ 5− P 0 sinðϕ0 ÞΩðωÞ 1 ¼ − P 0 ΩðωÞ cos ϕ0 ϕ2 I I 0

ð25Þ

 dκ3 P0 P0  2 − sinðϕ0 Þϕ3 ¼ ðϕ1 þ ϕ2 2 Þcosðϕ0 Þ þ 2ϕ1 sinðϕ0 Þ ds I G∞ 4I G∞ ð26Þ 2 4

dκ 4 ds dκ 5 ds

3

" # ϕ 5− P 0 sinðϕ0 ÞΩð2ωÞ 4 ϕ5 I " # 2ϕ1 ϕ2 cosðϕ0 Þ þ 2ϕ2 sinðϕ0 Þ P0 Ωð2ωÞ ¼ ðϕ2 2 −ϕ1 2 Þcosðϕ0 Þ−2ϕ1 sinðϕ0 Þ 4I

where ΩðωÞ ¼

ð20Þ

#"

The coefficients of the bending moment and curvature are time independent, which is the first advantage of this model. The coefficients are related by G0 ðωÞ and G00 ðωÞ, defined as the storage modulus and loss modulus, respectively, which are expressed as: " # " # α þ ðωτr Þ2 ð1−αÞωτr 00 G0 ðωÞ ¼ G0 ¼ G G 0 1 þ ðωτr Þ2 1 þ ðωτr Þ2

ð19Þ

The x and y coordinates are likewise described and will be presented later. The relaxation modulus, described by Eq. (1), may be rewritten as a function of two parameters, facilitating the acquisition of the relation between the bending moment and the curvature: GðtÞ ¼ G∞ þ ΔGðtÞ

where ΔGðtÞ ¼ ðG0 −G∞ Þe−t=τr . By substituting Eqs. (17)–(19) in (8), performing algebraic manipulations and using limt-∞ ΔGðtÞ ¼ 0, the following is obtained:

1 G0 ðωÞ2 þ G00 ðωÞ2

"

G0 ðωÞ

G00 ðωÞ

−G00 ðωÞ

G0 ðωÞ

ð27Þ

#

To develop this system of equations, it is necessary to understand the relation between the coefficients of the angle and curvature, which are obtained by substituting Eqs. (17) and (18) into (6): ∂ϕi ¼ κi ∂s

∀

i ¼ ½0−5

ð28Þ

M.A. Vaz, A.J. Ariza / International Journal of Non-Linear Mechanics 54 (2013) 43–54

Given the system of Eqs. (24)–(28), two conditions are necessary for its development. Note that Eqs. (25)–(28) are linear, they are solved only once and the response for any percentage variation of load can be easily calculated. 2.2.1. Boundary conditions The boundary conditions are the same as those presented in the time domain. It is sufficient to consider the coefficients of the angle at the support and of the curvature at the free extremity as equal to zero: ϕi ð0Þ ¼ κ i ðLÞ ¼ 0

∀

i ¼ ½0−5

ð29Þ

After developing Eqs. (24)–(28) subject to boundary conditions (29), the coefficients of the curvature and angle of rotation are obtained, which are then substituted into Eqs. (17)–(19), to obtain the oscillatory behavior of the angle, curvature and bending moment. A dimensionless analysis is performed because it can better characterize the system behavior. 2.2.2. Dimensionless solution To make the equations dimensionless, the criteria developed for the time domain are used, and the storage modulus and the loss modulus are employed in their dimensionless form: 0

G ðωÞ ¼

αþω 1 þ ω2 2

00

G ð ωÞ ¼

∂κ 0 P 0 þ cosðϕ0 Þ ¼ 0 ∂s α 2 3 " # dκ 1  ϕ cosðϕ0 Þ 4 ds 5−P 0 sinðϕ0 ÞΩðωÞ 1 ¼ −P 0 ΩðωÞ dκ 2 ϕ2 0

ð30Þ

( yðs; ωtÞ ¼

y0 ðsÞ þ ε½y1 ðsÞsinðω tÞ þ y2 ðsÞcosðω tÞ þε2 ½y3 ðsÞ þ y4 ðsÞsinð2ω tÞ þ y5 ðsÞcosð2ω tÞ

ð36Þ

Eqs. (35), (36) present the configuration of the deformed beam for each section. These equations are then substituted, together with Eq. (17), into Eqs. (12), (13). To approximate the sine and cosine functions a Taylor series expansion is used, and after algebraic manipulation and rearrangement, the coefficients for the x and y coordinates are obtained as follows: 2 3 dx0 2 3 −cosðϕ0 Þ 6 ds 7 6 dx1 7 6 7 6 ds 7 ϕ1 sinðϕ0 Þ 6 7 6 7 6 7 6 dx2 7 6 7 sinðϕ Þ ϕ 2 0 6 ds 7 6 7 6 7 ¼ −6 2 2 ð37Þ 1 7 6 dx3 7 sinðϕ Þ þ ðϕ þ ϕ Þcosðϕ Þ ϕ 0 2 0 7 6 3 4 1 6 ds 7 6 7 6 7 6 7 ϕ4 sinðϕ0 Þ þ 12ϕ1 ϕ2 cosðϕ0 Þ 6 dx4 7 4 5 6 ds 7 2 2 1 4 5 ϕ5 sinðϕ0 Þ þ 4ðϕ2 −ϕ1 Þcosðϕ0 Þ dx5 ds

 dk3 P 0 P0  2 − sinðϕ0 Þϕ3 ¼ ðϕ1 þ ϕ2 2 Þcosðϕ0 Þ þ 2ϕ1 sinðϕ0 Þ ds α 4α 2 3 " # dκ 4 ϕ 4 ds 5−P 0 sinðϕ0 ÞΩð2ωÞ 4 dκ 5 ϕ5 ds " # 2ϕ1 ϕ2 cosðϕ0 Þ þ 2ϕ2 sinðϕ0 Þ P0 Ωð2ωÞ ¼ 2 2 ðϕ2 −ϕ1 Þcosðϕ0 Þ−2ϕ1 sinðϕ0 Þ 4

0 2 3 ds sinðϕ0 Þ 6 dy 7 6 17 6 7 6 ds 7 6 ϕ1 cosðϕ0 Þ 7 6 7 6 7 6 dy2 7 6 7 ϕ2 cosðϕ0 Þ 6 ds 7 6 7 6 7¼6 6 dy3 7 6 ϕ3 cosðϕ0 Þ−14ðϕ1 2 þ ϕ2 2 Þsinðϕ0 Þ 7 7 6 ds 7 6 7 6 7 6 7 ϕ4 cosðϕ0 Þ−12ϕ1 ϕ2 sinðϕ0 Þ 6 dy4 7 4 5 6 ds 7 2 2 4 5 1 ϕ5 cosðϕ0 Þ−4ðϕ2 −ϕ1 Þsinðϕ0 Þ dy

ð38Þ

5

ð31Þ

ds

ð32Þ

ds

The angles are obtained from the response of the system of Eqs. (30)–(34), which requires the definition of boundary conditions. 2.2.4. Boundary conditions Eq. (29) is written in its dimensionless form: xi ð0Þ ¼ yi ð0Þ ¼ ϕi ð0Þ ¼ κi ð1Þ ¼ 0

∀

i ¼ ½0−5

ð39Þ

ð33Þ 3. Case studies

where

"

1 0

2

00

G ðωÞ þ G ðωÞ

2

0

00

G ðωÞ

G ðωÞ

−G ðωÞ

G ðωÞ

00

#

0

and Eq. (28), which describes the relationship between the bending moment and curvature, is similar in its dimensionless form: ∂ϕi ¼ κi ∂s

and cosine functions, equal forms may be used for expressions of x and y: ( x0 ðsÞ þ ε½x1 ðsÞsinðω tÞ þ x2 ðsÞcosðω tÞ   xðs; ωtÞ ¼ ð35Þ þε2 ½x3 ðsÞ þ x4 ðsÞsin 2ω t þ x5 ðsÞcosð2ω tÞ

2 dy 3

ð1−αÞω 1 þ ω2

The dimensionless Eqs. (24)–(27) can be rewritten in the following form:

ΩðωÞ ¼

47

∀

i ¼ ½0−5

ð34Þ

To develop this system of non-linear differential equations, two boundary conditions are necessary (Eq. (29)). Firstly, order zero is solved (Eqs. (30) and (34)). Then, the angle of rotation ϕ0 is substituted into Eq. (31), thus solving the system composed of Eqs. (31) and (34) (i.e., i¼1, 2), followed by (32) and (34) (i.e., i ¼3), and finally (33) and (34) (i.e., i¼4, 5), thus obtaining the angle and curvature coefficients. Observe that Eq. (30) is non-linear but remaining equations are linear. 2.2.3. Configuration of the beam When presenting the approximation of the curvature, angle and bending moment, given by Eqs. (17)–(19) in the form of sine

In this section, different case studies are presented with numerical comparisons of the curvature at the built-in end in the time and frequency domains. To develop the solution in the time domain, Eqs. (14), (15) and the boundary conditions (16) are considered. For the frequency domain, Eqs. (30)–(34) and the boundary conditions (39) are used. The acquisition of the curvature and angle responses in the time domain requires repeated solutions of these equations, while the accuracy of the response depends principally on the time step, which should be very small to avoid error propagation. The time interval necessary for an accurate response is roughly Δt ¼[0.001–0.1], and the final analysis time is considered in the interval of t¼[50–5000]. Hence, approximately 5  104 systems of differential equations are solved for each time domain simulation. The computational time is therefore very high in order to obtain a solution for a single frequency. The curvature response in the frequency domain compares well with the analysis in the time domain, and the computational time taken to develop these equations is significantly lower because equations are developed only once for a given frequency. For the case studies, a unitary static load (P 0 ¼1) and a sinusoidal load with a magnitude of [10–50%] of the static load (Eq. (3)) are

48

M.A. Vaz, A.J. Ariza / International Journal of Non-Linear Mechanics 54 (2013) 43–54

Fig. 4. Average curvature relative error at the encastré for each time step increment, at different excitation frequencies.

α =0.25

ε =30%

considered. Three values of the relaxation modulus, α¼ [0.25, 0.50 and 0.75], are used to find results for the average curvature and the variation of the maximum curvature. Fig. 4 shows the average curvature relative error at the beam's encastré point. This response is obtained using Eqs. (14), (15) and boundary condition (16), at different time steps and frequencies. The load conditions assumed are P 0 ¼1, ε ¼ 50% and α¼ 0.25. A relative error of 0.01% is considered reasonable in this work, therefore for frequencies 0.03, 0.3 and 1 and 10 the maximum time steps 0.1 and 0.01 must be respectively used. Fig. 5 presents the variation of the maximum curvature with position for α¼0.25, ε ¼0.3 and for three frequencies, ω¼ 0.03, 1 and 10. The curves corresponding to ω¼0.03 and 10 respectively present the greatest and lowest values of curvature variation. The exact results are obtained with time domain formulation. Fig. 6 shows the maximum curvature variation levels. The most representative values are encountered at the built-in end. At low frequencies, the curvature variation increases. To perform a more detailed study, the responses are investigated as a function of the frequency and time. In Fig. 7, five values of the oscillatory load amplitude and a relaxation stress of α¼0.25 are considered. The average curvature at the built-in end is presented, showing that the average curvature does not remain constant with

⎯

α=0.25

⎯ω

⎯ω

⎯ω

Fig. 5. Maximum curvature variation vs. position.

α=0.25

ε

Fig. 6. Curvature variation levels.

Fig. 7. Average curvature at the fixed end vs. frequency.

⎯

α=0.25

Fig. 8. Variation of curvature at the fixed end vs. frequency.

M.A. Vaz, A.J. Ariza / International Journal of Non-Linear Mechanics 54 (2013) 43–54

frequency because the beam is submitted to a static load combined with an oscillatory load. In other words, the application of harmonic load slightly reduces the beam average response. This effect is more pronounced for lower frequencies and higher load variations. For high frequencies, the points of curvature calculated in the time domain present good agreement with the curve obtained by perturbation theory. Even for low frequencies and large oscillatory loads, the data points fit well the curve obtained via perturbation theory. In general, the average curvature obtained by the perturbation theory offers a good approximation. For example, if ε¼50% and frequency 0.03, the average curvature error is less than 0.5%. Fig. 8 shows the variation of the maximum curvature as a function of frequency. This maximum curvature variation represents the difference between the maximum and minimum curvature peaks, which decreases with increasing frequency for the different loadings. The solutions for both theories nearly coincide. When the frequency decreases and the load variation is in the order of 10–30%, the responses are very similar. However, for greater loadings, the “exact” response tends to diverge from the approximate response. For a load variation of 50%, the error between the obtained values is less than 5% for a dimensionless frequency of 0.03, that can be considered acceptable given the large variation in the load.

α=0.5

⎯

Fig. 9. Average curvature at the fixed end vs. frequency.

⎯

α=0.5

Fig. 10. Variation of curvature at the fixed end vs. frequency.

49

Fig. 9 presents the case study results for α¼0.5. The points of average curvature calculated in the time domain and by perturbation theory (frequency) overlap due to an increase in the relaxation modulus. The average curvature increases for the high frequencies and then remains constant. The variation in the average curvature is primarily encountered in the frequency range of [0.1–10] for the case studies presented. Fig. 10 shows the variation of curvature, which has the same behavior as that shown in Fig. 7, and decreases with increasing frequency. The values obtained for the time domain coincide for the different loading levels. Fig. 11 presents the average curvature for α ¼0.75 and different loading levels. A similar but less pronounced behavior as for α¼0.25 and 0.5 is experienced. Fig. 12 shows the variation of the curvature, and the responses in the time and frequency domains coincide perfectly. Due to the high value of the dimensionless relaxation modulus (α¼ 0.75), the maximum curvature variations do not change significantly with frequency, but the results exhibit similar behavior as for α¼0.25 and 0.5. Figs. 13–18 present the average angle and the maximum angle variation for the beam at its free end, for several case studies (P 0 ¼ 1 and α ¼0.25, 0.5 and 0.75). As the beam material behaves as a standard linear solid model, for low (high) frequencies the storage modulus and the loss modulus respectively tends to α (1) ⎯

α=0.75

Fig. 11. Average curvature at the fixed end vs. frequency.

⎯

α=0.75

Fig. 12. Variation of curvature at the fixed end vs. frequency.

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M.A. Vaz, A.J. Ariza / International Journal of Non-Linear Mechanics 54 (2013) 43–54

⎯

α=0.25

α=0.5

⎯

Fig. 13. Average angle at the free end vs. frequency, α ¼ 0.25. Fig. 16. Angle variation at the free end vs. frequency.

α=0.25

⎯

Fig. 14. Angle variation at the free end vs. frequency.

⎯

α=0.5

Fig. 15. Average angle at the free end vs. frequency, α ¼0.5.

⎯

α=0.75

Fig. 17. Average angle at the free end vs. frequency, α ¼ 0.75.

⎯

α=0.75

Fig. 18. Angle variation at the free end vs. frequency.

M.A. Vaz, A.J. Ariza / International Journal of Non-Linear Mechanics 54 (2013) 43–54

⎯

α=0.25

Fig. 19. Average tip y-deflection vs. frequency.

α=0.25

⎯

Fig. 20. Tip y-deflection variation vs. frequency.

⎯

α=0.5

Fig. 21. Average tip y-deflection vs. frequency.

⎯

51

α=0.5

Fig. 22. Tip y-deflection variation vs. frequency.

⎯

α=0.75

Fig. 23. Average tip y-deflection vs. frequency.

⎯

α=0.75

Fig. 24. Tip y-deflection variation vs. frequency.

52

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and 0 (0), therefore the material is softer (stiffer) and higher (smaller) rotations are expected. Another important output when studying the response of a cantilever beam is the transversal displacement of its free end. Figs. 19–24 show its average and maximum variation for P 0 ¼1 and α¼0.25, 0.5 and 0.75. The analysis is developed by imposing a frequency dependent excitation force, and the results also show same frequency dependency, as expected. The calculation of the displacement is straightforward once the angle distribution is known. For the time and frequency domain solutions Eqs. (13), (16) and (34), (38) are respectively employed, and the results are then compared. The frequency domain solution is solved using the perturbation formulation, and the results are considered accurate for ε o40%. The extreme values for α¼0 and 1 are not calculated because the material respectively behaves as a Maxwell model (similar to a viscous fluid) and a linear elastic body (mechanical properties are frequency independent). Figs. 25–30 show the average and maximum variation of the free end longitudinal displacement for P 0 ¼1 and α¼0.25, 0.5 and 0.75 as a function of the frequency. When comparing two extreme conditions in Fig. 25 and 29 (for α¼ 0.25 and 0.75) for ε ¼ 50% and ω¼0.03, the errors between the approximate and exact solutions are very small and of similar order of magnitude. The average

⎯

Fig. 27. Average tip-x vs. frequency α ¼0.5.

⎯

⎯

α=0.5

α=0.5

α=0.25

Fig. 28. Tip-x variation at the free end vs. frequency. Fig. 25. Average tip-x vs. frequency α ¼0.25.

⎯

α=0.25

Fig. 26. Tip-x variation at the free end vs. frequency.

⎯

α=0.75

Fig. 29. Average tip-x vs. frequency α¼ 0.75.

M.A. Vaz, A.J. Ariza / International Journal of Non-Linear Mechanics 54 (2013) 43–54

⎯

α=0.75

ε=20 %

⎯

α=

α=

53

α= ⎯ω ⎯ω

( → ∞)

⎯ω ⎯ω

⎯

( → ∞)

( → ∞)

Fig. 32. Deformed configuration for ε¼ 20%.

Fig. 30. Tip-x variation at the free end vs. frequency.

ε=10%

⎯ α=

α=

α= ⎯ω ⎯ω

( → ∞)

⎯ω ⎯ω

⎯

( → ∞)

( → ∞)

and at steady state. The solution of the equations via a perturbation method shows an advantage over the mathematical model in the time domain, primarily when considering the computational time necessary to develop the solution for an ample domain of frequencies. The values of beam curvature used in the models in the time and frequency domains present nearly same results for high frequencies and different levels of loading, while for low frequencies, the responses obtained by the time and frequency models are also very close, showing better precision for smaller loads and greater values of the relaxation modulus. The average curvature at the built-in end always diminishes with the increase of the amplitude of load variation and increases for higher frequencies. The curvature variation also increases with the load variation amplitude but it decreases with frequency. The behavior described for the average curvature and maximum curvature variation are captured in the mathematical model using perturbation theory because second-order terms are retained during expansion of the variables, hence the asymmetric response is depicted.

Acknowledgments Fig. 31. Deformed configuration for ε¼ 10%.

longitudinal displacement shown in Figs. 25 and 27 follow same tendency, however opposite behavior is seen in Fig. 29. Figs. 31 and 32 show several configurations (maximum end displacements) for the encastré beam for two levels of loading, ε¼10 and 20%. Eqs. (37), (38) and boundary condition (39) are employed. It can be seen that the relaxation coefficient significantly affects the beam response. Similarly the excitation frequency affects the displacement, the lower the frequency the higher are the rotations.

4. Conclusion This work presents a mathematical formulation for characterization of the quasi-static response of a thin cantilever beam composed of a linear viscoelastic material and subjected to harmonic loading at its free extremity. Without sacrificing generality in the formulation, a standard linear solid constitutive model is assumed. A set of non-linear ordinary differential equations describes the response of the system. The curvature, bending moment, angle of rotation and displacement field of the beam depend on the position, time and frequency. A numerical solution was developed to describe the response in the time domain, thus obtaining the system behavior in the transient regime

The authors acknowledge the support from the National Council of Scientific and Technological Development (CNPq) for this study.

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