Hybrid material and foundation damping of Timoshenko beams

Abstract

In this paper, vibration suppression of shear deformable beams with hybrid material-foundation viscoelastic damping is studied. The precise behavior of viscoelastic polymeric structures is taken into consideration by Boltzmann superposition integral and utilizing dynamic mechanical analysis (DMA) results while foundation viscoelasticity is adapted by Kelvin-Voigt model. The integro-partial differential equations of motion are derived based on the Timoshenko theory via Hamilton principle. Assessment of the solution procedure is carried out by comparison with elastic Timoshenko and viscoelastic Euler-Bernoulli cases. Transient response, natural frequency and modal loss factor are attained by Fourier transform, weighted residual method and numerical iterative algorithm. Influences of hybrid viscoelastic damping, foundation properties, geometrical parameters and boundary conditions on vibration characteristics and dynamic response are investigated via comprehensive parametric study. Due to the absence of similar results in the literature, this paper is likely to fill a gap in the state of the art of this problem.

Keywords

Damped vibration DMA results viscoelastic damping Galerkin method viscoelastic foundation

1. Introduction

Time dependency of some materials like polymers and plastics emerges in some specific mechanical responses such as stress relaxation and creep while this treatment has noticeable physical effects. Energy dissipation, vibration suppression, mitigation of disturbance and noise abatement are just a few part of viscoelastic materials merits. Frequency/time dependency in these materials, which is known viscoelasticity, has important role on dynamic treatment, stability improvement and vibration isolation of viscoelastic structures under loading circumstances. This role has encouraged researchers to scrutinize viscoelastic materials more precisely from various theoretical and functional points of view.

Phenomenological constitutive modeling has been one the most interesting topics in viscoelasticity field; accordingly, various surveys are performed on linear and nonlinear constitutive models (Schapery, 1999; Wineman, 2009). Among several patterns for phenomenological simulation of viscoelasticity, superposition assumption is an accurate model, which adapts time-history of all material properties on constitutive relation of viscoelastic materials. This assumption is the base of Boltzmann superposition integral that is reasonable in both linear and nonlinear regimes. It is worth mentioning that mostly, researchers preferred to apply time-independent constants for some viscoelastic properties, while an accurate study should consider time-dependency for all material properties (Tschoegl et al., 2002; Lakes and Wineman, 2006; Ilyasov, 2007).

From constitutive modeling point of view, Boltzmann superposition integral has a reasonable definition, which has significant influences on dynamic simulation, particularly on vibration damping of viscoelastic structures. Generally, damping simulation is divided to active and passive branches so that viscoelasticity as an inherent property locates on the latter. Several studies with various approaches are performed for modeling of viscoelastic damping (Park, 2001; Ibrahim, 2008). Park (2001) devoted a comprehensive review on viscoelastic dampers modeling, based on various mathematical sources and their different performances. Restrictions of power-law and fractional derivative models that lead to some digression from accurate consideration of viscoelastic treatment are discussed thoroughly. However, some merits of Prony series in computational viscoelasticity, such as mathematical modeling, data fitting, Laplace transformation and frequency modeling of material was represented. Ibrahim (2008) carried out a complete investigation on vibration isolation by various nonlinear passive damping. The behavior of viscoelastic materials were examined based on the mutual relations between modulus/loss factor and frequency-temperature variations. These relations were analyzed from two separate standpoints, one specific model, fractional derivative, and phenomenological modeling of dynamic mechanical analysis (DMA) results. In the latter, strain rate, temperature, excitation frequency, memory effect, type of material and adapted model for material treatment have undeniable effects on the mutual relations. Finally, the survey approved feasibility, practicality and accuracy of the DMA results for analytical modeling.

Although viscoelastic damping appertains to the passive damping, it can apply as an active damping in specific fields, such as railway vehicles, seismic isolation, automobile industry and aviation, see for instance (Fan et al., 2009; Rao, 2003; De Rosa, 1995). On the railway vehicles, interior and exterior viscoelastic dampers are taken into consideration by researchers widely. Fan et al. (2009) investigated noise reduction and vibration control within railway vehicles by viscoelastic dampers. They used bitumen-based material, water-based damping coating and butyl rubber separately in the carriage of train for noise abatement. Also, they carried out experimental approaches for damping analysis of appointed carriages. Exterior viscoelastic damper known as the viscoelastic foundation is a great factor of energy dissipating in the railway machines. The viscoelastic foundation describes elastic stiffness and viscous dampers simultaneously by various combinations of linear/nonlinear springs and dashpots. Kelvin-Voigt foundation is one the most common and accurate models for the viscoelastic foundation that is used for material modeling (Sapountzakis and Kampitsis, 2013; Kampitsis et al., 2013; Sapountzakis and Kampitsis, 2011; Çalim, 2009; Chen et al., 2001). This viscoelastic model is implemented for damped vibration of elastic beams and plates resting on foundation. In other words, foundation viscoelasticity is the main factor of vibration suppression of elastic structures resting on viscoelastic foundation. Sapountzakis and Kampitsis (2011) studied dynamic response of Timoshenko beams resting on tensionless viscoelastic foundation under moving load. Also, they used boundary element method to derive geometrical nonlinear response of beams for the transverse and axial displacements. Moreover, the effects of shear and axial compression on the dynamic response of the systems are investigated thoroughly. The same authors (Sapountzakis and Kampitsis, 2013) also, examined a more developed study with three-dimensional nonlinear displacement functions for elastic Timoshenko beam-columns on nonlinear three-parameter viscoelastic foundation. In the same procedure, the coupled problem is solved by boundary element method. (Kampitsis et al., 2013) carried out an investigation on the soil– pile–structure kinematic and inertial interaction. They also utilized a hybrid spring configuration includes of a nonlinear spring and Kelvin-Voigt model where the earlier refers to the near-field plastification and the latter refers to the far-field viscoelastic character of the soil. For solving the coupled nonlinear equations, Open Sees code for 2-dimensional finite element model of beam and a three-dimensional continuum finite element scheme materialized in the ABAQUS code are applied. Among various procedures for solving vibration of elastic Timoshenko beams on viscoelastic foundation, extended dynamic stiffness matrices is taken into consideration by researchers. (Chen et al., 2001) used this method for an infinite and semi-infinite Timoshenko beams to derive resonant frequency and critical velocity of moving loads. This approach also is employed by (Çalim, 2009) in the Laplace domain using the complementary functions method while the computed solutions are transformed to the time domain by the Durbin inverse Laplace transform method. Furthermore, similar approach is applied by (Kim and McCullough, 2003) for elastic plates resting on viscoelastic foundation. Although semi-analytical and close–form approaches result in some mathematical complexity, some studies are performed by researchers (Ansari et al., 2011; Sun, 2001). Ansari et al. (2011) utilized Galerkin weighted residual method in conjunction with multiple scales method to obtain the dynamic response of Euler-Bernoulli beams resting on viscoelastic foundation. Furthermore, Sun (2001) represented a close-form solution of Euler-Bernoulli beams resting on viscoelastic medium under moving load based on Green functions.

Numerous research have taken a further step in energy dissipation of structures. Much interest is received by hybrid damping due to implementation of two or more active and passive damping sources simultaneously such as viscoelasticity, pseudoelasticity and piezoelectricity (Chen and Levy, 1999; Trindade and Benjeddou, 2002; Zamani et al., 2015). Chen and Levy (1999) analyzed hybrid viscoelastic-pseudoelastic damping of flexible beams with constrained viscoelastic layer and shape memory alloy layer. They studied temperature effects on frequency, loss factor, dynamic response and control of the flexible beam by passive and active-passive approaches. The vibration time, especially in the case of small damping and a slender beam is reduced obviously by the latter approach. Trindade and Benjeddou (2002) surveyed geometric patterns, different models and control algorithms for hybrid piezoelectric-viscoelastic damping of beams. Also, they examined the treatment length and viscoelastic layer thickness on modal damping, modes controllability and transient response of sandwich beams. Zamani et al. (2015) studied active-passive damping of thin monoclinic viscoelastic structures on Kelvin-Voigt foundation. An accurate approach, which considered time dependency of all material properties, was employed. Moreover, material and foundation viscoelasticity was adapted simultaneously for damping analysis of polymeric based composite beams and plates.

In accordance with literature review, few studies are dedicated to accurate approach of viscoelastic structures under mechanical loading (Zamani et al., 2015; Ilyasov, 2007), whereas many researchers preferred simplified model (Afshin et al., 2010; Falahatgar et al., 2009). Furthermore, hybrid viscoelasticity of material and foundation seems rarely in the literature, see for instance (Zamani et al., 2015; Sheng et al., 2004). To the best of authors’ knowledge, there is no study on hybrid material-foundation viscoelastic damping of Timoshenko beams with accurate approach for all material properties.

In the present study, in order to accurate scrutiny of dynamic behavior of Timoshenko beams, a direct viscoelastic approach is employed for viscoelasticity of materials. It is worth mentioning that simplifying assumptions such as constant Poisson ratio, shear and bulk moduli are developed by researchers while in this study all the mechanical properties are time-dependent and accurate behavior of each stiffness coefficient is considered. For viscoelastic phenomenological simulation, two models are prepared, Kelvin-Voigt for viscoelastic foundation and Boltzmann superposition integral in conjunction with DMA results for material treatment. The beams governing equations of motion are derived based on Timoshenko displacement hypothesizes via Hamilton principle. The Galerkin weighted residual method is applied for degradation of integro-partial differential equations to nonlinear algebraic equations and therewith, solving the resulted equations with iterative QZ algorithm to obtain frequencies of Timoshenko beams with hybrid material-foundation viscoelastic damping. The accuracy of this approach is assessed by comparison with exact solution of elastic Timoshenko and viscoelastic Euler-Bernoulli beams. Then, the effects of geometrical parameters, boundary conditions (BCs), hybrid viscoelasticity, stiffness and viscosity of foundations are studied via parametric study on the dynamic treatment of Timoshenko beams.

2. Mathematical formulation

Based on Figure 1, beam with rectangular cross section on viscoelastic foundation can be considered with dimensions L, b and h in the x, y and z directions, respectively. Also, the Cartesian coordinate system (x, y, z) on the middle surface of the plate is displayed.

2.1. Kinematic relations

The small deformation of shear deformable beam is considered by first order shear deformation theory. The displacement field in the Timoshenko beam theory is expressed as (Wang et al., 2000):

u (x, z, t) = z ϕ x (x, t)

(1a)

w (x, z, t) = w 0 (x, t)

(1b)

where u and w are the displacement components of a point along x and z directions, respectively. Also,

ϕ x

is the rotation of a transverse normal about the y axis with positive sign in the counterclockwise direction. The strain-displacement kinematic equations related to the displacement field (1) in terms of displacement components can be written as:

ɛ i = [ɛ_{x}^{0} ɛ_{xz}^{0}] + z [κ_{x}^{0} κ_{xz}^{0}] = [0 w 0, x + ϕ x] + z [ϕ x, x 0]

(2)

where

ɛ_{i}^{0}

and

κ_{i}^{0} (i = x, xz)

denote the membrane and flexural strains, respectively. Also, comma and the latter variable subscripts denote derivative operator and derivative variable, respectively.

2.2. Constitutive relations

Based on the Boltzmann superposition integral, linear viscoelastic constitutive relations for isotropic shear deformable beam can be represented as (Brinson and Brinson, 2007):

{σ x (t) σ xz (t)} = \int_{- \infty}^{t} [E (t - ξ) 00 G (t - ξ)] {ɛ x, ξ (ξ) ɛ xz, ξ (ξ)} d ξ

(3)

where

σ i, ɛ i (i, j = x, xz)

and ξ denote the components of stress, strain and Boltzmann integral variable, respectively. Also, E and G denote time-dependent Young and shear moduli, respectively, which can be expressed based on the DMA results in elastic case as:

C ij = [\frac{9 KG}{3 K + G} 00 G]

(4)

where K denote bulk modulus. Implementation of Alfrey’s corresponding principle and some mathematical manipulations together with inverse Laplace transform lead to time-dependent components of stiffness matrix in terms of DMA results as:

C 11 (t) = E (t) = L - 1 {\frac{9 \bar{K} (s) \bar{G} (s)}{3 \bar{K} (s) + \bar{G} (s)}}

(5)

where s and

L - 1

denote Laplace variable and Laplace inverse operator, respectively. Also, superior bar represents average value of the quantity in the Laplace domain.

As mentioned previously, in this study the Poisson ratio is a time-dependent property, which leads to mathematical complexity, whereas many researchers have avoided this by simplifying the Poisson ratio as a constant value.

2.3. Motion equations

In accordance with Timoshenko beam theory, Hamilton principle and the displacement field (1), the governing equations of motion for Timoshenko beams on viscoelastic medium can be written as (Wang et al., 2000):

Q xz, x - kw 0 - cw 0, t = I 0 w 0, tt

(6a)

M x, x - Q xz = I 2 ϕ x, tt

(6b)

where k and c are the stiffness and damping coefficient of Kelvin-Voigt foundation, respectively. Additionally,

Q xz, M x

and

I i (i = 0, 2)

indicate transverse force resultants, moment resultants and inertia terms, respectively, which are expressed as:

{Q xz M x} = \int_{- h / 2}^{h / 2} {k s σ xz z σ x} d z

(7a)

(I 0, I 2) = \int_{- h / 2}^{h / 2} ρ (1, z 2) d z

(7b)

where

ρ, h

and k_s are mass density, thickness and constant shear correction coefficient, respectively. Substitutions of strain-displacement equation (2), into the inegro constitutive relation, then the resulted equation into the equations (7) lead to the integro-partial differential equations of motion for viscoelastic Timoshenko beams resting on viscoelastic medium as:

\int_{- \infty}^{t} {k s hG (t - ξ) (w 0, ξ xx + ϕ x, ξ x)} d ξ - kw 0 - cw 0, t = I 0 w 0, tt

(8a)

\int_{- \infty}^{t} {\frac{h 3}{12} E (t - ξ) ϕ x, ξ xx - k s hG (t - ξ) (w 0, ξ x + ϕ x, ξ)} d ξ = I 2 ϕ x, tt

(8b)

3. Solution procedure

The Galerkin weighted residual method with proper shape functions is the main mean for solving the integro-partial differential equations (8) in this section. Based on the harmonic assumption besides the appropriate shape functions for vibrational modes, separable solutions are considered as (Wang et al., 2000):

w 0 (x, y, t) = W (x, y) e i ω t

(9a)

ϕ x (x, y, t) = φ x (x, y) e i ω t

(9b)

where ω is the natural frequency, additionally, W and

φ x

denote spatial functions of viscoelastic Timoshenko beams, which are represented in Table 1 for simply supported (SS), fully clamped (CC) and clamped simply supported (CS) boundary conditions. Implementation of equations (9) into the equations (8) with some mathematical manipulations, lead to the governing equations of motion in terms of spatial functions and frequency as:

k s h \hat{G} (W, xx + φ x, x) + i (k + ic ω - I 0 ω 2) W = 0

(10a)

\frac{h 3}{12} \hat{E} φ x, xx - k s h \hat{G} (W, x + φ x) - iI 2 ω 2 φ x = 0

(10b)

where

\hat{E}

and

\hat{G}

stand for the frequency-dependent moduli, with definition as:

(\hat{E}, \hat{G}) = ω \int_{0}^{\infty} (E (ϕ), G (ϕ)) e - ω ϕ d ϕ

(11)

Using proper modal shape functions in conjunction with Galerkin weighted residual method degrades the reformed partial differential equation of motion to a system of nonlinear algebraic equations. Therewith, the equations are solved to attain the vibrational characteristics- complex frequency, modal loss factor and natural frequency, which are all defined respectively, as:

ω = ω Re + i ω Im

(12a)

η = 2 ω Re ω Im / {(ω Re) 2 - (ω Im) 2}

(12b)

ω n = ω Re

(12c)

where real and imaginary parts of complex frequency, are respectively,

ω_{n}^{Re}

and

ω_{n}^{Im}

for mode number, n. Eventually, the transient response of viscoelastic Timoshenko beams in terms of spatial components is obtained using complex frequency in displacement fields as:

θ (x, t) = \sum_{n = 1}^{\infty} Θ n (x) \exp (- ω_{n}^{Im} t) (A n \sin ω_{n}^{Re} t + B n \cos ω_{n}^{Re} t), θ = w 0, ϕ x, Θ = W, φ x

(13)

Also, Fourier transform with initial conditions are applied for deriving the final form of dynamic response as:

θ (x, t) = \sum_{n = 1}^{\infty} Θ n (x) \exp (- ω_{n}^{Im} t) (\frac{\overset{\cdot}{θ} 00 + ω_{n}^{Im} θ 00}{ω_{n}^{Re}} \sin ω_{n}^{Re} t + θ 00 \cos ω_{n}^{Re} t)

(14)

where

θ 00

and

\overset{\cdot}{θ} 00

stand for the initial conditions of

θ 0 (x, t)

and

\overset{\cdot}{θ} 0 (x, t)

t = 0

Table 1.

Beam spatial displacement function for different BCs.

BCs	W(x)	$φ x (x)$
CC	$\cos (2 m π x / L) - 1$	$1 - \cos (2 m π x / L)$
CS	$\cos (3 m π x / 2 L) - \cos (m π x / 2 L)$	$1 - \cos (2 m π x / L)$
SS	$\sin (m π x / L)$	$\cos (m π x / L)$

4. Numerical results and discussion

For examination of the present procedure, non-dimensional natural frequency of elastic Timoshenko beams is compared with counterparts reported by Thomas and Abbas (1975). In addition, complex frequencies of viscoelastic beam with Timoshenko theory (TT) and Euler-Bernoulli theory (EBT) are compared. The effects of material viscoelasticity, geometry, BCs, foundation properties and hybrid viscoelastic damping on the dynamic treatment of Timoshenko beams are studied via a comprehensive parametric study and eventually, relevant consequences are extracted. Moreover, simply supported and clamped BCs can be written respectively, as:

M x = w = 0

(15a)

w = ϕ x = 0

(15b)

Also, the first and second symbols of BCs are denoted by the capital letter, which rely on the edge conditions at x = 0 and x = L, respectively.

4.1. Numerical examples with comparative study

In order to examine the present procedure, the non-dimensional natural frequency,

\sqrt{ρ AL 4 ω 2 / EI}

, is compared with exact solutions, which were obtained by finite element method (Thomas and Abbas, 1975). Non-dimensional properties for the first set of properties, M1, are G/E = 3/8, k_s = 2/3, I/AL²= 0.08² and for the second set, M2, are G/E = 5/13, k_s = 0.85, I/A L²= 0.08². The first forth non-dimensional natural frequencies of Timoshenko beams are obtained by Galerkin method and they are compared with exact solutions as presented in Table 2 for SS beams. As presented in Table 2, the resulted frequency parameters have excellent correlation with results of exact solution. As expected, the frequency parameter of Galerkin method has higher value than exact solution for all modes and different sets of materials, because weighted residual methods such as Galerkin method represent higher value of natural frequency or critical buckling loading than exact approaches. In these sets of materials, the higher values of shear correction factors lead to higher frequency parameters. This remark is obvious both in the Galerkin and in exact approaches. Although higher values are obtained by Galerkin method, the discrepancies between two methods are not exceeded than 0.005 for the third mode of M1 materials and 0.004 for the first mode of M2 materials.

Table 2.

The frequency parameter of SS Timoshenko beam.

Mode	M1		M2
	Present	Exact	Present	Exact
1	8.644	8.645	8.841	8.8397
2	26.961	26.960	28.461	28.461
3	47.685	47.680	51.499	51.498
4	68.727	68.726	75.366	75.364

In order to assess the present procedure in viscoelastic domain, the natural frequencies and modal loss factors of viscoelastic Timoshenko and Euler-Bernoulli beams are compared based on viscoelastic properties in Tables 3. Moreover, the beam mass density, length, width and thickness are given as:

ρ = 1200 kg / m 3, L = 10 m, b = 0.05 m, H = 0.2 m

Table 3.

Prony-Dirichlet series coefficients.

i	τ_i(s)	G_i (MPa)	K_i (MPa)
1	0.017	85	197.5
2	0.142	85	197.5
3	1.202	83	197.5
4	10.206	83	175
5	86.67	82	46
6	735.99	82	30
7	6250	80	7.5
8	∞	180	1500

In Table 4, the first five natural frequencies and modal loss factors for various BCs are presented. As mentioned by Wang et al. (2000), TT with shear deformation and rotary inertia effects has less natural frequency than EBT while these values lead to less value of modal loss factor for TT. In other words, more flexibility reduces the real parts of complex frequency and therefore reduces the natural frequency. In Figures 2 and 3, respectively, the ratio of Euler-Bernoulli to Timoshenko natural frequency and loss factor are presented. The fully clamped beams has higher values of both modal loss factors and natural frequencies, while CS and SS edge conditions have lower and the lowest values of ratios, respectively. Also, every BCs approximately, have linear variation with mode numbers increment. In other words, by mode number promotion, the natural frequency and modal loss factor maintain their linear manner, additionally, the modal loss factor ratios represent smoother variation in comparison with natural frequency ratios.

Table 4.

Dynamic characteristics of SS viscoelastic EBT and TT.

BC	TT		EBT
BC	$ω n$	η	$ω n$	η
CC	185	0.38	271	0.44
	372	0.87	1087	1.97
	558	1.35	2446	4.52
	745	1.84	4349	8.11
	931	2.33	6795	12.72
CS	175	0.34	187	0.28
	256	0.76	753	1.34
	510	1.23	1695	3.11
	721	1.78	3015	5.59
	884	2.21	4710	8.79
SS	161	0.32	166	0.15
	322	0.74	470	0.81
	484	1.16	1059	1.92
	645	1.58	1883	3.47
	807	2.00	2942	5.46

As can be seen in Figure 2, the fundamental ratios approximately represent similar values so that the ratios of CS and SS edge conditions seem coincide while in reality they have different values. Fundamental natural frequencies ratio of all the mentioned BCs have similar values with negligible difference, while values of fundamental loss factors are obviously distinctive. Finally, Figures 2 and 3 reveal that in spite of higher values of Euler-Bernoulli natural frequency and loss factor, the ratios for both dynamic characteristics have linear variations with mode increment.

4.2. Numerical example with parametric study

In this section, the influences of geometric parameters and also, BCs, foundation properties such as viscosity and stiffness on dynamic response, natural frequency and loss factor of viscoelastic Timoshenko beams with 0.05 m width and 0.2 m height are investigated. Moreover, the high values of foundation properties are given as (Ansari et al., 2011): $k = 138 MN / m 2, c = 1732.5 KN . s / m 2$

4.2.1. Beam length

Influence of beam length variation on the natural frequency and modal loss factor of viscoelastic Timoshenko beams are scrutinized in Table 5 for CC, CS and SS edge conditions. In addition, beams length are 2 m, 10 m, and 20 m, which are adapted from moderately thick to very thin hypothesis for the studied beams. Based on the data in Table 5, as expected, the natural frequencies of beams with more constraints of degree of freedom (DOF) have higher values. Moreover, the modal loss factors of CC are higher for a specific length value. In addition, for a specific value of beams length, the loss factors of CS and SS are very similar to each other while the CC loss factors are greater than other edge conditions. As a result, the beams with more constraints of DOF and those that vibrate in higher modes have more ability for energy dissipation. By length increment for a specific BC, the both natural frequency and modal loss factor decrease. However, the modal loss factors nonlinearity decrease. In order to distinguish the length effects on dynamic characteristics, Figures 4 and 5 are illustrated for beams with CS edge condition. As can be seen, the slope of natural frequency and modal loss factor for beams with 2 m length is greater than dynamic characteristics of beams with 10 m and 20 m length. In order to demonstration of physical effects of material viscoelasticity, midpoint displacement of CC, CS and SS viscoelastic Timoshenko beam are demonstrated by Figure 6, so that the initial displacement and initial velocity are 1 mm and 1 mm/s, respectively.

Table 5.

The natural frequencies and loss factors of viscoelastic beams.

L	2		10		20
BC	$ω n$	η	$ω n$	η	$ω n$	η
CC	932	2.33	185	0.38	91	0.15
	1863	4.78	372	0.87	185	0.38
	2795	7.23	558	1.35	279	0.62
	3726	9.68	745	1.84	372	0.87
	4658	12.13	931	2.33	465	1.11
CS	808	2.01	175	0.29	88	0.38
	1712	4.37	256	0.56	174	0.39
	2676	6.92	510	1.23	239	0.51
	3608	9.37	721	1.78	360	0.83
	4493	11.70	884	2.21	433	1.02
SS	806	2.00	161	0.32	79	0.12
	1613	4.13	322	0.74	161	0.32
	2665	6.25	484	1.16	242	0.53
	3475	8.37	645	1.58	322	0.74
	4283	10.49	807	2.00	403	0.95

4.2.2. Foundation stiffness

As expected, beams resting on elastic foundation exhibit higher stiffness. Based on Figures 7 -9 for beams with 2 m length, the natural frequencies increase by foundation stiffness increment, while the modal loss factors decrease significantly. For beams on foundation with 1 MN/m² stiffness, the fundamental natural frequencies are very similar to those without foundation while the fundamental loss factors are larger than those loss factors of beams without foundation. As can be seen in Figures 7 and 8, when the foundation stiffness reaches to the 10 MN/m², the fundamental natural frequencies increase slightly whereas the fundamental loss factors reduce extremely. In other words, the fundamental loss factors are very sensitive to the stiffness promotion. For higher modes, the natural frequencies and modal loss factors represent a few changes in their values in comparison with beam resting on elastic foundation with 1 MN/m² stiffness. By increment of foundation stiffness to 50 MN/m², the modal loss factors decrease significantly whereas the natural frequencies have no considerable changes in their values. This manner approximately continues for higher values of stiffness, 138 MN/m², additionally the modal loss factors approach to the specific values. As can be observed in Figure 8, the modal loss factors present extreme reduction for the third modes; additionally, the values in this mode approximately represent approached values for the following loss factor. In other words, significant reduction of the third loss factor arises from this point that the imaginary part of complex natural frequency has a considerable reduction in comparison with stiffness gradation. Comparison of beams with various edge conditions and equal value of foundation modulus is an attractive issue that is illustrated in Figure 9. As can be observed, the edge condition has minor effect on the natural frequency of viscoelastic beams resting on elastic foundations. In other words, the foundation stiffness is the most prominent factor on dynamic manner of beams on elastic foundation with material damping. In order to grasp physical effect of foundation stiffness, fundamental natural frequency and fundamental loss factor versus K-L parameters are demonstrated in Figures 10 and 11 for CC beams, respectively. The effect of foundation stiffness on the central deflection time history of CC viscoelastic Timoshenko beams with 2 m length on elastic foundation is illustrated in Figure 12. As can be seen, damping time for viscoelastic beams resting on elastic foundation with 50 MN/m², 100 MN/m² and 138 MN/m² stiffnesses are 76 s, 132 s and 158 s respectively.

4.2.3. Foundation viscosity

In Table 6, the natural frequencies and modal loss factors of Timoshenko beams with 1 m length and hybrid material-foundation viscoelastic damping is highlighted. Foundation has 138MN/m² stiffness and various viscosities. As expected, foundation viscosity enhances damping capability significantly. However, the main effect of foundation viscosity is great reduction of natural frequencies. In other words, the natural frequency is reduced by foundation viscosity as a specific factor, which is the main reason of damping capability enhancement. Increment of viscosity up to 100 KN.s/m² has negligible effect on natural frequency while increment of viscosity up to 500 KN.s/m² leads meaningful reduction and tiny variations of fundamental natural frequency and natural frequencies of higher modes, respectively. In spite of predictable variations of natural frequency, there is no specific manner for modal loss factor variations with increment of foundation viscosity. From a mathematical point of view, changes of natural frequency means that viscosity increment lead to reduction of real part of complex frequency, although the imaginary part of complex frequency shows minor promotion. Also, natural frequency and modal loss factor increase by mode number increment for a specific set of foundation parameters. Vibrating in the higher modes has more capability to reduce the vibration amplitude and damping times. In other words, the fundamental natural frequencies are dominant for transient response of vibrating beams in this case. As can be found in Table 6, edge conditions with more constraints of DOF represent higher values of natural frequency and loss factor. It means that CC, CS and SS respectively, have higher values of natural frequency and modal loss factors. Importance of viscosity effects on natural frequencies and modal loss factors are illustrated in Figures 13 and 14, respectively. As can be observed in Figure 14, the modal loss factors show irregular variation whereas the fundamental natural frequencies represent regular treatment with variations of foundation viscosity. Furthermore, the effects of foundation parameters on dynamic characteristics of vibrating CC beams with hybrid material-foundation viscoelastic damping are highlighted in Table 7. The viscoelastic beams have 1 m length with similar cross section area to those beams in the previous parts. It is observed that the increment of stiffness leads to smooth increment of natural frequency. However, a specific manner for loss factors variations cannot be observed. In other words, in this case, natural frequencies have less sensitivity to stiffness of foundation. In addition, higher modes, specially the fifth modes have tiny increment by promotion of stiffness. Viscosity of foundation leads to the different manner of natural frequency. The lower modes show more reduction by viscosity increment for a specific stiffness, while the higher modes maintain their values with negligible variations. Furthermore, for all the fundamental natural frequencies, promotion of foundation viscosity up to 100 KN.s/m² leads to negligible variation whereas for higher values, the natural frequencies represent great decrement. However, the modal loss factors have unknown variation with either stiffness or viscosity variations. In spite this manner, the viscosity development leads to unimportant effects on modal loss factor while this increment affects natural frequencies prominently. Moreover, edge conditions with more constraints of DOF have more natural frequencies and loss factors. Due to similarity of CC and other edge conditions, the interpretations of CS edge condition in Table 8 is omitted, but the fundamental dynamic characteristics versus variations of foundation properties are depicted in Figures 15 and 16.

Table 6.

The effects of foundation viscosity on dynamic characteristics of beams.

c	10		50		100		500		1000		1732.5
BC	$ω n$	η	$ω n$	η	$ω n$	η	$ω n$	η	$ω n$	η	$ω n$	η
CC	1941	0.14	1940	0.44	1936	0.28	1797	0.02	1439	0.007	951	0.003
	3765	0.71	3765	1.17	3762	0.27	3693	0.04	3484	0.02	3000	0.008
	5616	2.10	5615	0.98	5614	0.35	5568	0.05	5425	0.027	5065	0.014
	7473	5.38	7473	1.07	7472	0.43	7437	0.07	7329	0.031	7049	0.019
	9333	15.11	9332	1.21	9332	0.51	9304	0.09	9217	0.04	8990	0.025
CS	1798	0.12	1796	0.31	1792	0.32	1640	0.02	1267	0.006	811	0.002
	3559	0.62	3558	1.28	3556	0.26	3483	0.03	3262	0.018	2760	0.007
	5410	1.88	5409	0.98	5408	0.34	5360	0.05	5212	0.02	4840	0.013
	7237	4.78	7237	1.05	7236	0.42	7200	0.07	7088	0.03	6800	0.018
	9020	12.44	9019	1.18	9018	0.50	8990	0.09	8900	0.04	8670	0.023
SS	1731	0.10	1930	0.07	1925	0.05	1786	0.01	1427	0.005	941	7 × 10^-4
	3520	0.31	3515	0.18	3513	0.12	3439	0.03	3216	0.01	2714	0.007
	5120	0.63	5119	0.31	5118	0.19	5067	0.04	4911	0.02	4522	0.012
	6730	0.96	6728	0.44	6727	0.26	6689	0.06	6569	0.03	6261	0.017
	8340	1.38	8339	0.58	8338	0.34	8307	0.07	8210	0.04	7957	0.021

Table 7.

Dynamic characteristics of CC beam with hybrid viscoelastic damping.

c	10		50		100		500		1000		1732.5
K	$ω n$	η	$ω n$	η	$ω n$	η	$ω n$	η	$ω n$	η	$ω n$	η
1	1864	0.79	1862	0.17	1858	0.09	1725	0.02	1389	0.007	932	0.002
	3727	1.55	3726	0.35	3724	0.18	3655	0.03	3449	0.016	2972	0.008
	5590	2.30	5590	0.52	5589	0.26	5542	0.05	5400	0.026	5042	0.014
	7454	3.06	7454	0.69	7453	0.35	7418	0.07	7310	0.035	7031	0.019
	9317	3.82	9317	0.86	9317	0.44	9289	0.09	9202	0.044	8976	0.025
10	1869	1.37	1867	0.19	1863	0.09	1730	0.02	1392	0.007	933	0.002
	3730	1.95	3729	0.36	3727	0.18	3658	0.03	3452	0.016	2974	0.008
	5592	2.67	5592	0.54	5590	0.27	5544	0.05	5402	0.026	5043	0.014
	7455	3.41	7455	0.71	7454	0.36	7419	0.07	7312	0.035	7032	0.019
	9319	4.16	9318	0.88	9317	0.44	9290	0.09	9203	0.044	8976	0.025
50	1891	0.61	1890	0.35	1885	0.12	1750	0.02	1406	0.007	939	0.002
	3741	11.69	3740	0.46	3738	0.20	3669	0.03	3462	0.017	2982	0.008
	5599	9.23	5599	0.62	5598	0.29	5552	0.05	5409	0.026	5050	0.014
	7461	6.97	7461	0.79	7459	0.38	7425	0.07	7317	0.03	7038	0.019
	9323	6.91	9323	0.96	9322	0.46	9294	0.09	9208	0.044	8981	0.025
100	1919	0.22	1918	2.87	1914	0.18	1777	0.02	1425	0.007	945	0.002
	3755	1.21	3754	0.71	3752	0.24	3683	0.04	3474	0.017	2993	0.008
	5609	4.46	5608	0.79	5607	0.32	5561	0.05	5418	0.03	5058	0.014
	7468	22.93	7468	0.93	7466	0.40	7432	0.07	7324	0.035	7044	0.019
	9329	40.23	9329	1.09	9327	0.49	9300	0.09	9213	0.044	8986	0.025
138	1941	0.14	1940	0.44	1936	0.28	1797	0.02	1439	0.007	951	0.003
	3765	0.71	3765	1.17	3762	0.27	3693	0.04	3484	0.02	3000	0.008
	5616	2.10	5615	0.98	5614	0.35	5568	0.05	5425	0.027	5065	0.014
	7473	5.38	7473	1.07	7472	0.43	7437	0.07	7329	0.031	7049	0.019
	9333	15.11	9332	1.21	9332	0.51	9304	0.09	9217	0.04	8990	0.025

Table 8.

Dynamic characteristics of CS beam with hybrid viscoelastic damping.

c	10		50		100		500		1000		1732.5
K	$ω n$	η	$ω n$	η	$ω n$	η	$ω n$	η	$ω n$	η	$ω n$	η
1	1713	0.73	1712	0.16	1707	0.08	1564	0.015	1218	0.006	795	0.002
	3518	1.46	3518	0.33	3515	0.17	3443	0.033	3226	0.015	2736	0.007
	5383	2.22	5383	0.50	5381	0.25	5334	0.051	5187	0.025	4817	0.013
	7218	2.96	7217	0.67	7216	0.34	7180	0.069	7069	0.034	6872	0.019
	9004	3.69	9003	0.83	9003	0.42	8974	0.08	8885	0.043	8650	0.024
10	1719	1.36	1717	0.18	1712	0.08	1568	0.015	1221	0.006	796	0.002
	3521	1.88	3520	0.34	3518	0.17	3445	0.033	3228	0.015	2737	0.007
	5385	2.59	5384	0.52	5383	0.26	5335	0.051	5188	0.025	4818	0.013
	7218	3.32	7218	0.69	7217	0.34	7181	0.069	7071	0.034	6783	0.019
	9005	4.03	9004	0.85	9004	0.43	8975	0.086	8886	0.043	8651	0.024
50	1743	0.48	1741	0.35	1737	0.11	1591	0.016	1235	0.006	800	0.002
	3533	7.28	3532	0.45	3530	0.19	3457	0.034	3239	0.016	2746	0.008
	5393	10.07	5392	0.61	5391	0.28	5343	0.052	5196	0.025	4825	0.013
	7224	7.05	7224	0.77	7223	0.36	7187	0.069	7076	0.034	6789	0.019
	9010	6.88	9009	0.93	9008	0.45	8980	0.087	8890	0.043	8656	0.024
100	1774	0.18	1772	1.72	1768	0.17	1619	0.017	1253	0.006	806	0.002
	3548	1.02	3547	0.71	3545	0.23	3472	0.035	3252	0.016	2756	0.008
	5402	3.86	5402	0.77	5401	0.31	5352	0.053	5205	0.025	4833	0.013
	7231	7.33	7231	0.91	7230	0.39	7194	0.071	7083	0.034	6795	0.019
	9015	8.46	9014	1.06	9014	0.48	8985	0.088	8896	0.043	8661	0.024
138	1798	0.12	1796	0.31	1792	0.32	1640	0.02	1267	0.006	811	0.002
	3559	0.62	3558	1.28	3556	0.26	3483	0.03	3262	0.018	2760	0.007
	5410	1.88	5409	0.98	5408	0.34	5360	0.05	5212	0.02	4840	0.013
	7237	4.78	7237	1.05	7236	0.42	7200	0.07	7088	0.03	6800	0.018
	9020	12.44	9019	1.18	9018	0.50	8990	0.09	8900	0.04	8670	0.023

The influence of edge conditions on the transient response of Timoshenko beams with hybrid material-foundation viscoelastic damping is depicted in Figure 17. In this case, the viscoelastic beams with 1 m length, are located on foundation with 138 MN/m² stiffness and 1732.5 KN.s/m² viscosity. It is seen that the fully clamped, CS and SS beams respectively, fail initial amplitude up to 98, 94 and 72% at t = 150 sec. In addition, the deflection amplitude attenuates to zero within 170, 242 and 645 sec for CC, CS and SS edge conditions, respectively. In other words, edge conditions with more constraints of DOF lose more initial amplitude at a specific time and have smaller damping time.

5. Conclusions

In this work, an investigation on dynamic treatment of Timoshenko beams with hybrid viscoelastic damping of material and foundation is studied. The Boltzmann superposition integral in conjunction with DMA results is adapted for constitutive relation of viscoelastic materials. The description of viscoelastic foundation is carried out by Kelvin-Voigt model. The integro-partial differential equations of motion are extracted based on Timoshenko beam theory by implementation of Hamilton principle. The dynamic characteristics and transient response are attained by Fourier transform, Galerkin weighted residual method and iterative QZ algorithm. In order to examine the present procedure, the natural frequencies of elastic Timoshenko beams and viscoelastic Euler-Bernoulli beams are obtained and compared with those reported in the literature. Several case studies are presented for demonstrating the effects of foundation parameters, geometrical dimensions, edge conditions and hybrid viscoelastic damping on transient response and dynamic characteristics of Timoshenko beams with hybrid viscoelastic damping of material and foundation.

As depicted in the previous sections, viscoelasticity of material has obvious effects on dynamic and vibration treatment of Timoshenko beams. Also, the viscoelasticity of foundation can be added as a source of damping in the system of beam-foundations. This combination of damping sources has prominent effects on vibrating behavior of the system with relative conclusions as follow:

Edge conditions with more constraints of DOF have higher values of natural frequencies and loss factors for a specific value of beam length. While the longer beams shows lower natural frequencies and modal loss factors for a specific edge conditions.

In higher modes of vibration, energy dissipation ability increases so that reduces the vibration amplitude and damping times excessively.

Ascending of foundation stiffness leads to increment of natural frequencies, prominent reduction of fundamental loss factors and approaching loss factors to the specific values.

For viscoelastic beams resting on elastic foundation, the edge condition has minor effect on natural frequency whereas the foundation stiffness is the most prominent factor on dynamic behavior.

For hybrid viscoelastic damping of material and foundation, the main influence of foundation damping is great reduction of natural frequencies while natural frequencies have less sensitivity to the stiffness of foundation in this case.

The lower modes of vibrations depict more decrement by viscosity ascending for a specific stiffness, while the higher modes almost maintain their values.

In comparison with thinner structures, Timoshenko beams represent less sensitivity to the foundation properties and resist remaining in harmonic motion.

The consequences of this work are considered to comprehensive insight to physical and mathematical aspects of hybrid material-foundation viscoelastic damping of Timoshenko beams. Practically, these consequences can be implemented for dynamic treatment of railroads with hybrid viscoelasticity of beam structures and foundations.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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