Parametric analysis of viscoelastic hyperboloidal helical rod

Abstract

The objective of this study is to perform a pioneering research about a viscoelastic hyperboloidal helical rod having a standard type of distortional behavior and a Kelvin type of bulk compressibility. Field equations are based on the Timoshenko beam theory, and the exact curvatures of the hyperboloidal geometry are considered through the formulation. The numerical analysis is carried out by the mixed finite element method, considering the rotary inertia, in the Laplace space, and the results are transformed back to time space numerically using the modified Durbin’s algorithm. A cantilevered hyperboloidal helical rod having solid circular, hollow circular, and thin-walled hollow circular cross sections is handled, and the rod is loaded by rectangular and triangular impulsive types of point load at the tip. Through the analysis, different values of retardation time, three different relaxation functions associated with shear modulus, and three different creep functions associated with bulk modulus are handled. Finally, a benchmark example is presented, and the influence of the loading and the material parameters on the helix geometry is discussed.

Keywords

hyperboloidal helix Laplace space mixed finite element method Timoshenko beam theory viscoelasticity

Introduction

The behavior of some materials, such as polymers, can be simulated by various viscoelastic mechanical models. Some remarkable engineering applications of the viscoelastic helical rods may be cited as viscoelastic dampers, the viscoelastic suspension system of construction vehicles, and motor valve springs. Many experimental studies have shown that composite materials possess viscoelastic properties (Badalov et al., 1991; Bogdanovich, 1993; Éshmatov, 2006; Sun and Zhang, 2001). Although many works exist in the literature that examine the dynamic analysis of viscoelastic straight rods, the number of researches concerning dynamic analysis of viscoelastic helical rods is limited. The free vibrations of viscoelastic rods were investigated in Yamada et al. (1974). Chen and Lin (1982) investigated viscoelastic straight beams by incremental finite element method without any integral transformation to deal with viscoelastic structures. Chen (1995) used hybrid Laplace transform/finite element method for solving Maxwell fluid and three parameter solid-type Timoshenko beams. The exact relationships between the bending solutions of linear viscoelastic Timoshenko and Euler–Bernoulli beams under quasi-static load are presented in Wang et al. (1997). Enelund et al. (1999) analyzed the capability of linear viscoelastic model for describing creep, relaxation, and damped dynamic responses with numerical and analytical methods. In Kocatürk and Şimşek (2006a, 2006b), the dynamic analysis of the viscoelastic Euler–Bernoulli and Timoshenko beams under an eccentric compressive force and a moving harmonic load is studied using the Lagrange equations. The expressions for the viscoelastic Timoshenko shear functions in terms of stresses and material properties are derived by Hilton (2009). The weak form finite element formulation for determining nonlinear quasi-static response for linear viscoelastic Euler–Bernoulli and Timoshenko beams is developed in Payette and Reddy (2010). Quasi-static solutions of a linear viscoelastic cantilevered beam subjected to the harmonic loading at the free end are given in Vaz and Ariza (2013). An analytical solution based on the Euler–Bernoulli theory for the response of beams on discrete viscoelastic supports under dynamic loading is presented in Li et al. (2014). Temel (2004) and Temel et al. (2004) investigated quasi-static and dynamic analysis of viscoelastic helical rods subjected to time-dependent loads in the Laplace domain using the complementary functions method and the ordinary differential equations based on the Timoshenko beam theory. The dynamic behavior of viscoelastic helixes with noncircular cross sections based on the Timoshenko beam theory with the inclusion of the rotary inertia using the mixed finite element method is first investigated by Eratlı et al. (2014). The mixed finite element analysis based on the approximated curvature of the conical and hyperboloidal helices with hollow elliptical cross section is performed by Eratlı and Kutlu (2015). Based on the exact curvatures of the noncylindrical helicoidal bars, a dynamic viscoelastic analysis via mixed finite elements is performed by Ermis (2015).

In this study, the dynamic characteristics of linear viscoelastic hyperboloidal helical rod having solid circular, hollow circular, and thin-walled hollow circular cross sections are examined. This study improves the previous study of Eratlı et al. (2014) by employing exact differential length and curvatures (instead of interpolating curvatures) in the formulation, for a precise description of the three-dimensional (3D) curved rod geometry. The Kelvin and standard models considered in Eratlı et al. (2014) are modified by a complex bulk modulus. A mixed finite element formulation based on Timoshenko beam theory is developed in Laplace space. The system matrix generated by two-nodded curvilinear space rod elements reflects the exact geometry of the hyperboloidal helix. A viscoelastic material that exhibits a standard type of distortional behavior and a Kelvin type of bulk compressibility is considered. The viscoelastic material properties are accounted based on the correspondence principle (Shames and Cozarelli, 1997) and implemented into the finite element algorithm. The solution of the system matrices is carried out in Laplace space. Then, the results obtained in Laplace space are transformed back to time space with the use of modified Durbin’s algorithm (Dubner and Abate, 1968; Durbin 1974; Narayanan, 1980). The influence of the section geometry, the helix geometry, the density of the material, the impulsive load type, the effect of viscous bulk compressibility (viscous dilatational behavior), and the effects of viscous shear modulus (viscous distortional behavior) are investigated on the dynamic response of a viscoelastic cantilevered hyperboloidal helical rod.

Helix geometry

The geometrical properties of the helix in the Cartesian coordinate system are

\begin{array}{l} x = R (φ) \cos φ, y = R (φ) \sin φ, z = p (φ) φ, \\ p (φ) = R (φ) \tan α \end{array}

(1)

where $α$ denotes the pitch angle and $R (φ)$ and $p (φ)$ signify the centerline radius and the step for unit angle of the helix as a function of the horizontal angle $φ$ , respectively. The horizontal radius of any point on the centroidal axis of hyperboloidal helix is defined using the following formula

\begin{matrix} Hyperboloidal helix : \\ R (φ) = R_{\min} + (R_{\max} - R_{\min}) {(1 - \frac{φ}{n π})}^{2} \end{matrix}

(2)

where n is the number of active turns of helix and $R_{\min}$ and $R_{\max}$ are the minor and major radii of the hyperboloidal helix geometry, respectively. The infinitesimal arc length becomes $d s = c (φ) d φ$ where

c (φ) = \sqrt{R^{2} (φ) + p^{2} (φ)} d φ

(3)

In the hyperboloidal helix geometry

χ (φ) = \frac{R (φ)}{c^{2} (φ)}, τ (φ) = \frac{p (φ)}{c^{2} (φ)}

(4)

are the curvature and the torsion of the helix axis, respectively, and they are determined using equations (1) and (2) (Ermis, 2015). The helix geometry is defined by referring to Frenet coordinate system $t, n, b$ , where t is the tangential vector to the helix centerline, n is the unit normal vector, and $b = t \times n$ is the binormal vector.

Field equations and functional in Laplace space

The field equations are based on Timoshenko beam theory, and for the dynamic analysis, they are given in Laplace space as follows (Eratlı et al., 2014)

Equations of motion

\begin{matrix} - {\bar{T}}_{, s} - \bar{q} + ρ A z^{2} \bar{u} = 0 \\ - {\bar{M}}_{, s} - t \times \bar{T} - \bar{m} + ρ I z^{2} \bar{Ω} = 0 \end{matrix}

(5)

Constitutive equations

\begin{matrix} {\bar{u}}_{, s} + t \times \bar{Ω} - {\bar{C}}_{γ} \bar{T} = 0 \\ {\bar{Ω}}_{, s} - {\bar{C}}_{κ} \bar{M} = 0 \end{matrix}

(6)

where the over bars denote the Laplace transformed terms, comma as a subscript designates the differentiation with respect to arc length coordinate s, and z is the Laplace transformation parameter. In equations (5) and (6), $\bar{u} ({\bar{u}}_{t}, {\bar{u}}_{n}, {\bar{u}}_{b})$ is the displacement vector; $\bar{Ω} ({\bar{Ω}}_{t}, {\bar{Ω}}_{n}, {\bar{Ω}}_{b})$ is the rotation vector; $\bar{T} ({\bar{T}}_{t}, {\bar{T}}_{n}, {\bar{T}}_{b})$ is the force vector; $\bar{M} ({\bar{M}}_{t}, {\bar{M}}_{n}, {\bar{M}}_{b})$ is the moment vector; $ρ$ is the density of homogeneous material; A is the area of the cross section; $I (I_{t}, I_{n}, I_{b})$ is the moment of inertia of the cross section; $\bar{q}$ and $\bar{m}$ are the distributed external force and moment vectors in the Laplace space, respectively; and ${\bar{C}}_{γ}$ and ${\bar{C}}_{κ}$ are the compliance matrices in the Laplace space. Field equations are derived in Frenet coordinates. The Laplace transformed functional of the helical rod is

\begin{array}{l} I (\bar{y}) = - [\bar{u}, {\bar{T}}_{, s}] + [t \times \bar{Ω}, \bar{T}] - [{\bar{M}}_{, s}, \bar{Ω}] \\ - \frac{1}{2} [{\bar{C}}_{κ} \bar{M}, \bar{M}] - \frac{1}{2} [{\bar{C}}_{γ} \bar{T}, \bar{T}] + \frac{1}{2} ρ A z^{2} [\bar{u}, \bar{u}] \\ + \frac{1}{2} ρ z^{2} [Ι \bar{Ω}, \bar{Ω}] - [\bar{q}, \bar{u}] - [\bar{m}, \bar{Ω}] + {[(\bar{T} - \hat{\bar{T}}), \bar{u}]}_{σ} \\ + {[(\bar{M} - \hat{\bar{M}}), \bar{Ω}]}_{σ} + {[\hat{\bar{u}}, \bar{T}]}_{ε} + {[\hat{\bar{Ω}}, \bar{M}]}_{ε} \end{array}

(7)

It is noted that detailed formulation exists in Eratlı et al. (2014). The terms with hats in equation (7) are the known values on the boundary. The subscripts $ε$ and $σ$ represent the geometric and dynamic boundary conditions, respectively.

Mixed finite element formulation

In this study, the finite element formulation for hyperbolic helical rods is developed by taking the exact helix geometry which is given by equation (2) into account, and the derivation of the curvatures is based on equation (1). Thus, in the formulation of element matrices, for describing $c (φ)$ , the curvature $χ (φ)$ , and the torsion $τ (φ)$ over the elements, exact expressions which are given by equations (3) and (4) are used. Finally, the element matrices for the two-nodded isoparametric curvilinear space rod element with 2 × 12 degrees of freedom (DOF) are developed by means of the linear shape functions $ϕ_{i} = 1 - ξ$ and $ϕ_{j} = ξ$ in terms of the dimensionless coordinate $ξ = (φ - φ_{i}) / Δ φ$ , where $0 < ξ < 1$ and $Δ φ = φ_{j} - φ_{i}$ . Nodal unknowns are the displacements, cross-sectional rotations, forces, and moments. In Eratlı et al. (2014), conventional finite element interpolation is used for describing $c (φ)$ , the curvature $χ (φ)$ , and the torsion $τ (φ)$ over the elements, that is, $c^{e} = c_{i} ϕ_{i} + c_{j} ϕ_{j}$ , $χ^{e} = χ_{i} ϕ_{i} + χ_{j} ϕ_{j}$ , and $τ^{e} = τ_{i} ϕ_{i} + τ_{j} ϕ_{j}$ . However, in this study, the above quantities over the elements are evaluated from their exact expressions.

Viscoelastic model

It is assumed that the material exhibits the standard type of distortional behavior and Kelvin type of bulk compressibility. The form of complex shear modulus can be expressed as

\bar{G} = G [\frac{1 + β^{G} τ_{r}^{G} z}{1 + τ_{r}^{G} z}], β^{G} = \frac{G_{g}}{G} > 1

(8)

where $τ_{r}^{G}$ , G, and $G_{g}$ are the retardation time, the equilibrium value, and the instantaneous value of relaxation function associated with shear modulus, respectively (Baranoglu and Mengi, 2006; Eratlı et al., 2014; Mengi and Argeso, 2006). The proposed form of the complex bulk modulus is expressed by

\bar{K} = K [1 + τ_{r}^{K} z]

(9)

where $τ_{r}^{K}$ and K are the retardation time and the equilibrium value of creep function associated with bulk modulus, respectively. The equations of viscoelastic helical rod are obtained in Laplace space from the equations of elastic helical rod by replacing the elastic constants with the complex moduli according to selected viscoelastic models (correspondence principle; Shames and Cozarelli, 1997). The solution of the system matrices of finite element method is carried out in Laplace space. Then, the results obtained in Laplace space are transformed back to time space with the use of modified Durbin’s algorithm (Narayanan, 1980).

Numerical examples

The objective of this section is to investigate the influences of the shape of the cross section and the helix geometry, the effects of viscous bulk compressibility, and the viscous shear modulus on the dynamic response of a viscoelastic cantilever hyperboloidal helical rod (see Figure 1(a)). The analyses are carried out in the Laplace space, and the results are transformed back to the time space numerically using modified Durbin’s algorithms. In the examples that follow, the time history curves of the displacements ( $u_{x}$ , $u_{z}$ ) and the rotation $Ω_{y}$ at the tip of the rod and the shear force $T_{z}$ and the moment $M_{y}$ at the fixed end of the rod are determined.

Figure 1.

(a) Hyperboloidal helix geometry, (b) rectangular impulsive load, and (c) triangular impulsive load.

Common parameters for the examples

The shear modulus is $G = 80 GPa$ , the bulk modulus is $K = 175 GPa$ , and the ratio of the height of helix-to-maximum radius of helix is $H / R_{\max} = 2$ , where $R_{\max} = 2 m$ . The rod is subjected to a rectangular impulsive type of external dynamic load $P = P (t)$ acting at the tip of the rod (Figure 1(b)), except example 5. The intensity and the duration of the loading are $P = P_{o} = 1 kN$ and $t_{load} = 50 s$ , respectively. The dynamic response of the rod is determined within $0 \leq t \leq 100 s$ . The parameters used in the analysis for inverse Laplace transformation algorithm are $N = 2^{11}$ and $aT = 6$ (Eratlı et al., 2014).

Example 1: convergence analysis

A solid circular cross-sectioned rod has $A = 153.94 c m^{2}$ . For the parameters that describe the helix geometry, the following values are selected: the number of active turns of the rod is $n = 6$ ; the ratio of the minimum radius of helix geometry to maximum radius of helix geometry is $R_{\min} / R_{\max} = 0.5$ (Figure 1(a)). The parameters of the viscoelastic material are $τ_{r}^{G} = 0.01 s$ , $β^{G} = 3$ , and $τ_{r}^{K} = 0.1 s$ . The density of the material is $ρ = 7850 kg / m^{3}$ . The results are determined by discretizing the rod using 40, 60, 80, and 100 finite elements. Figure 2 presents the time histories $u_{x}$ , $u_{z}$ , $Ω_{y}$ , and $T_{z}$ . From the time variation curves (which are evaluated using 80 and 100 elements), the values of first six extrema (see Figure 2) that occur within the forced vibration zone are determined. Absolute percent differences of these extrema values, belonging to the solutions determined using 80 and 100 elements, are evaluated, and they are normalized with respect to the extrema values of 100 elements. The standard deviations of the normalized absolute percent differences for displacements ( $u_{x}$ , $u_{z}$ ), rotation $Ω_{y}$ and shear force $T_{z}$ are $0.782 \pm 0.03$ , $0.642 \pm 0.17$ , $1.005 \pm 0.06$ , and $0.063 \pm 0.07$ , respectively. As a consequence, in the following examples, 100 elements are employed.

Figure 2.

Convergence test for a viscoelastic cantilever hyperboloidal helical rod: (a) the displacement $u_{x}$ at the tip of the rod, (b) the displacement $u_{z}$ at the tip of the rod, (c) the rotation $Ω_{y}$ at the tip of the rod, and (d) the shear force $T_{z}$ at the fixed end of the rod.

Example 2: influence of the parameters of viscoelastic material

The rod materials having different retardation times are investigated. The solid circular cross-sectional area is $A = 153.94 c m^{2}$ . The values of the parameters that describe the helix geometry are the same with those used in the convergence analysis ( $R_{\min} / R_{\max} = 0.5$ and $n = 6$ ). The density of the material is $ρ = 7850 kg / m^{3}$ . The influence of the parameters of the viscoelastic material ( $τ_{r}^{G}$ , $β^{G}$ , and $τ_{r}^{K}$ ) on the dynamic behavior is shown in Figures 3 to 5 by plotting the time histories of $u_{x}$ , $u_{z}$ , $Ω_{y}$ , $T_{z}$ , and $M_{y}$ .

Figure 3.

Time histories of viscoelastic hyperboloidal helical rod for different values of retardation time associated with shear modulus ( $τ_{r}^{G}$ ): (a) the displacement $u_{x}$ at the tip of the rod, (b) the displacement $u_{z}$ at the tip of the rod, (c) the rotation $Ω_{y}$ at the tip of the rod, (d) the shear force $T_{z}$ at the fixed end of the rod, and (e) the moment $M_{y}$ at the fixed end of the rod.

Figure 4.

Time histories of viscoelastic hyperboloidal helical rod for different values of $β^{G}$ associated with shear modulus: (a) the displacement $u_{x}$ at the tip of the rod, (b) the displacement $u_{z}$ at the tip of the rod, (c) the rotation $Ω_{y}$ at the tip of the rod, (d) the shear force $T_{z}$ at the fixed end of the rod, and (e) the moment $M_{y}$ at the fixed end of the rod.

Figure 5.

Time histories of viscoelastic hyperboloidal helical rod for different values of retardation time associated with bulk modulus ( $τ_{r}^{K}$ ): (a) the displacement $u_{x}$ at the tip of the rod, (b) the displacement $u_{z}$ at the tip of the rod, (c) the rotation $Ω_{y}$ at the tip of the rod, (d) the shear force $T_{z}$ at the fixed end of the rod, and (e) the moment $M_{y}$ at the fixed end of the rod.

In Figure 3, time histories are determined for $τ_{r}^{G} = 0.005, 0.01, and 0.1 s$ ; $β^{G} = 3$ ; and $τ_{r}^{K} = 0.1 s$ , whereas in Figure 4, they are calculated for $β^{G} = 1.1, 1.5, and 3.0$ ; $τ_{r}^{G} = 0.01 s$ ; and $τ_{r}^{K} = 0.1 s$ . We recall that parameters $τ_{r}^{G}$ and $β^{G}$ describe the standard type of distortional behavior. From these two figures, it is observed that as the values of $τ_{r}^{G}$ and $β^{G}$ increase, $u_{x}$ , $u_{z}$ , $Ω_{y}$ , $T_{z}$ , and $M_{y}$ dissipate more rapidly, and they also have smaller amplitudes. This behavior is expected since the increase in the values of these parameters signifies that the distortional viscous behavior becomes more dominant in the material. It is also seen that the change in the values of $τ_{r}^{G}$ and $β^{G}$ does not affect the vibration period.

In Figure 5, time histories of the same quantities are plotted for monitoring the effects of viscous bulk compressibility. The results are evaluated by choosing, $β^{G} = 3$ ; $τ_{r}^{G} = 0.01 s$ ; and $τ_{r}^{K} = 0.0, 0.1, and 0.5 s$ . It can be recalled that the parameter $τ_{r}^{K}$ describes Kelvin type of dilatational behavior, and $τ_{r}^{K} = 0.0$ states that the material exhibits elastic bulk compressibility. Examining the results in Figure 5 reveals that as the value of $τ_{r}^{K}$ increases (dilatational viscous behavior becomes more dominant in the material), the time variations of $u_{x}$ , $u_{z}$ , $Ω_{y}$ , and $M_{y}$ dissipate more rapidly, and they also have smaller amplitudes. However, it is seen that the time variation of $T_{z}$ is not affected from the changes of the values of $τ_{r}^{K}$ . This shows that the shear stresses that give rise to $T_{z}$ are not induced by the viscous bulk behavior. From the time variation curves of $u_{x}$ , $u_{z}$ , $Ω_{y}$ , and $M_{y}$ , we also observe that there is a slight decrease in the vibration period as $τ_{r}^{K}$ increases.

Example 3: the effect of the helix geometry and the density of the material

The solid circular cross-sectional area is $A = 153.94 c m^{2}$ , and the values of viscoelastic parameters are $τ_{r}^{G} = 0.01 s$ , $β^{G} = 3$ , and $τ_{r}^{K} = 0.1 s$ .

Example 3.1: the effect of the helix geometry

The density of the material is $ρ = 7850 kg / m^{3}$ . For the rods having different numbers of active turns ( $n = 2, 4, and 6$ ) and $R_{\min} / R_{\max} = 0.5$ , the time histories of $u_{x}$ , $u_{z}$ , $Ω_{y}$ , $T_{z}$ , and $M_{y}$ are given in Figure 6. It is observed that as n increases, $u_{x}$ , $u_{z}$ , and $Ω_{y}$ oscillate about larger values (absolutely) in the forced vibration zone. More clearly, in the forced vibration zone, for the values of $n = 2, 4, and 6$ , the oscillations for $u_{x}$ , $u_{z}$ , and $Ω_{y}$ are given in Table 1. However, $T_{z}$ and $M_{y}$ oscillate about $1 kN$ and 0 in the forced vibration zone for all values of n (e.g. Figure 6(d) and (e)). From the time variation curves, it is also observed that, as n decreases, all the responses damp more rapidly while their amplitude and vibration period decrease.

Figure 6.

Time histories of viscoelastic hyperboloidal helical rod for different values of the number of active turns (n): (a) the displacement $u_{x}$ at the tip of the rod, (b) the displacement $u_{z}$ at the tip of the rod, (c) the rotation $Ω_{y}$ at the tip of the rod, (d) the shear force $T_{z}$ at the fixed end of the rod, and (e) the moment $M_{y}$ at the fixed end of the rod.

Table 1.

Effect of the helix geometry on the dynamic behavior of hyperboloidal helical rod.

n	$u_{x}$ (mm)	$u_{z}$ (mm)	$Ω_{y}$ (rad)	$R_{\min} / R_{\max}$	$u_{x}$ (mm)	$u_{z}$ (mm)	$Ω_{y}$ (rad)
2	−17.5	26.1	−0.01	0.3	−46.5	69.0	−0.02
4	−38.1	59.3	−0.02	0.5	−59.0	91.0	−0.03
6	−60.1	90.7	−0.03	0.7	−70.0	125.0	−0.04

Figure 7 presents the time histories for the rods having different $R_{\min} / R_{\max}$ ratios ( $0.3, 0.5, and 0.7$ ) and $n = 6$ . For the increasing values of $R_{\min} / R_{\max}$ ratios, a similar dynamic behavior is observed as we have encountered for the increasing values of n in the previous analysis. In the forced vibration zone, for the values of $R_{\min} / R_{\max} = 0.3, 0.5, and 0.7$ , the oscillations for $u_{x}$ , $u_{z}$ , and $Ω_{y}$ are given in Table 1.

Figure 7.

Time histories of viscoelastic hyperboloidal helical rod for three different $R_{\min} / R_{\max}$ ratios: (a) the displacement $u_{x}$ at the tip of the rod, (b) the displacement $u_{z}$ at the tip of the rod, (c) the rotation $Ω_{y}$ at the tip of the rod, (d) the shear force $T_{z}$ at the fixed end of the rod, and (e) the moment $M_{y}$ at the fixed end of the rod.

Example 3.2: the influence of the density of the material

The number of active turns of the rod is $n = 6$ and the ratio of the minimum radius of helix geometry to maximum radius of helix geometry is $R_{\min} / R_{\max} = 0.5$ . The investigation is carried on for three different values of the densities of the materials ( $ρ = 6000, 7850, and 9000 kg / m^{3}$ ). For these density values, the time histories of $u_{x}$ and $T_{z}$ are given in Figure 8. It is observed that due to the increase in $ρ$ , the magnitudes and the vibration periods of $u_{x}$ and $T_{z}$ increased. The values of first, third, and fifth extrema of the forced vibration zone corresponding to three different $ρ$ values are determined from Figure 8, and they are compared with the result that corresponds to the quasi-static response. The percent increases are listed in Table 2 where the negative signs indicate that the peak values are greater than the results of quasi-static case.

Figure 8.

Time histories of viscoelastic hyperboloidal helical rod for three different densities of the materials ( $ρ$ ): (a) the displacement $u_{x}$ at the tip of the rod and (b) the shear force $T_{z}$ at the fixed end of the rod.

Table 2.

Effect of the density of the material on the dynamic behavior of hyperboloidal helical rod.

	$t_{i} (s)$	$(u_{x}^{6000})_{i}$	$t_{i} (s)$	$(u_{x}^{7850})_{i}$	$t_{i} (s)$	$(u_{x}^{9000})_{i}$	$t_{i} (s)$	$(T_{z}^{6000})_{i}$	$t_{i} (s)$	$(T_{z}^{7850})_{i}$	$t_{i} (s)$	$(T_{z}^{9000})_{i}$
i	$% Difference = [u_{x}^{q s} - {(u_{x}^{ρ})}_{i}] \times 100 / u_{x}^{q s}$						$% Difference = [T_{z}^{q s} - {(T_{z}^{ρ})}_{i}] \times 100 / T_{z}^{q s}$
First peak	1.11	−79.6	1.28	−80.9	1.40	−81.4	0.82	−101.9	1.00	−102.6	1.05	−103.3
Third peak	5.80	−50.3	6.62	−53.9	7.10	−55.6	3.87	−67.0	4.39	−72.8	4.75	−75.5
Fifth peak	10.4	−32.7	11.9	−37.1	12.8	−39.2	6.91	−36.5	7.97	−42.6	8.50	−45.8

$(u_{x}^{ρ})_{i}$ , $(T_{z}^{ρ})_{i}$ : the extrema values of $u_{x}$ and $T_{z}$ determined from Figure 8 for three different $ρ_{i}$ values where $i = 1, 3, and 5$ . $u_{x}^{qs}$ , $T_{z}^{qs}$ : the values of the quasi-static response in the forced vibration zone.

Example 4: the effect of the cross section

As next analysis, the dynamic behaviors of rods having different cross-sectional geometries are compared with each other. For this purpose, keeping the net cross-sectional area constant ( $A = 153.94 c m^{2}$ , same as above examples), solid circular, hollow circular, and thin-walled hollow circular cross sections are considered (see Figure 9). The radius of the solid circular cross section is $r = 7 cm$ ; for the hollow circular cross section, the thickness-to-cross-sectional average radius ratio is $t / r_{o} = 0.5$ where the average radius is $r_{o} = 7 cm$ ; and, finally, for the thin-walled hollow circular cross section, the thickness-to-cross-sectional average radius ratio is $t / r_{o} = 0.1$ where the average radius is $r_{o} = 15.65 cm$ . The values of viscoelastic parameters are taken as $τ_{r}^{G} = 0.01 s$ , $β^{G} = 3$ , and $τ_{r}^{K} = 0.1 s$ . The density of the material is $ρ = 7850 kg / m^{3}$ . The geometry of the helix is defined by the parameters $R_{\min} / R_{\max} = 0.5$ and $n = 6$ . For the rods having solid circular, hollow circular, and thin-walled hollow circular cross sections, the time histories of $u_{x}$ , $u_{z}$ , $Ω_{y}$ , $T_{z}$ , and $M_{y}$ are given in Figure 8. It is seen that the time variations are significantly affected from the shape of the cross sections. In the forced vibration zone for solid circular, hollow circular, and thin-walled hollow circular rods, the oscillations are about −58.1, −27.0, and $- 5.9 mm$ for $u_{x}$ ; 90.9, 42.5, and $8.8 mm$ for $u_{z}$ ; and −0.03, −0.01, and $- 0.003 rad$ for $Ω_{y}$ , respectively. The responses determined for thin-walled hollow circular rod damp very rapidly, and the vibration period and amplitude of these responses are very low valued when compared with the others. Within the three rods, thin-walled hollow circular rod turns out to be the most stiff, while the opposite behavior is true for the solid circular rod.

Figure 9.

Time histories of viscoelastic hyperboloidal helical rod for three different types of the cross section: (a) the displacement $u_{x}$ at the tip of the rod, (b) the displacement $u_{z}$ at the tip of the rod, (c) the rotation $Ω_{y}$ at the tip of the rod, (d) the shear force $T_{z}$ at the fixed end of the rod, and (e) the moment $M_{y}$ at the fixed end of the rod.

Example 5: the triangular impulsive–type load

The hyperboloidal helicoidal rod is subjected to a triangular impulsive–type dynamic load acting at the tip of it (Figure 1(c)). The cross section, the helix geometry, and the material properties are as given in example 1. The density of the material is $ρ = 7850 kg / m^{3}$ . The intensity and the duration of the loading are $P = 2 P_{o} = 2 kN$ and $t_{load} = 50 s$ , respectively. The area of the forced vibration zone ( $A_{F} = \int P d t$ ) is kept constant for rectangular (Figure 1(b)) and triangular (Figure 1(c)) impulsive loads. The solutions of the helical rod under the triangular impulsive loading are compared with the rectangular impulsive–type load results given by example 1 (for 100 finite elements). In Figure 10, the time histories of the displacements $u_{x}$ and the shear forces $T_{z}$ are given for both the impulsive load cases. The values of $u_{x}$ and $T_{z}$ for the quasi-static cases that occur at $t = 25 s$ are determined using Figure 10(c) and (d), and the results are compared with the ones that correspond to the rectangular impulsive load (see Figure 10(a) and (b)). It is observed that the percent increases in the case of the triangular impulsive load are approximately −99.8% for both values of $u_{x}$ and $T_{z}$ . The dynamic behavior of the viscoelastic hyperboloidal helical rods dissipates within the sampling time interval and approaches to the quasi-static case (see Figure 10). Considering the decay of the dynamic responses, same damping parameters are more influential in the case of triangular impulsive–type of loading (see Figure 10(c) and (d)).

Figure 10.

Time histories of viscoelastic hyperboloidal helical rod for two different impulsive loadings: (a) the displacement $u_{x}$ at the tip of the rod for rectangular impulsive load, (b) the shear force $T_{z}$ at the fixed end of the rod for rectangular impulsive load, (c) the displacement $u_{x}$ at the tip of the rod for triangular impulsive load, and (d) the shear force $T_{z}$ at the fixed end of the rod for triangular impulsive load.

Conclusion

Using a viscoelastic material model which takes the standard type of distortional behavior and Kelvin type of bulk compressibility into account, a cantilevered viscoelastic hyperboloidal helical rod is analyzed via mixed finite element method. The effects of viscous bulk compressibility, viscous shear modulus, helix geometry, density of the material, cross-sectional shape, and the impulsive loading type on the dynamic response of the hyperboloidal helical rods are investigated. As far as the knowledge of the authors, the hyperboloidal helix and its viscoelastic model analyzed in this study is an original example for the literature. In the following comments provided, percents are referring to the mean values of the oscillating responses in the forced vibration zone. Keeping the ratio of $H / R_{\max} = 2$ constant, the results can be summarized as follows:

Increasing the values of the distortional and dilatational viscous behavior of the material ( $τ_{r}^{G}$ and $τ_{r}^{K}$ ) and $β^{G}$ , the selected nodal time histories dissipate. On the results of the displacements ( $u_{x}$ , $u_{z}$ ), rotation ( $Ω_{y}$ ), force ( $T_{z}$ ), and moment ( $M_{y}$ ), a change in the viscous bulk behavior has relatively less influence compared to the variation of the retardation time associated with the shear modulus. The overall variation of $T_{z}$ with respect to time is slightly influenced by the viscous bulk properties (example 2).

The decrease in the number of active turns (n) causes a reduction in the magnitude of the displacements ( $u_{x}$ , $u_{z}$ ) and rotation ( $Ω_{y}$ ; example 3.1). Referring to $u_{x}$ , $u_{z}$ , and $Ω_{y}$ values at the tip of the rod, the percent reductions corresponding to $n = 2$ and $n = 4$ cases with respect to $n = 6$ case are approximately $71 %$ and $37 %$ , $71 %$ and $35 %$ , and $67 %$ and $33 %$ , respectively.

As the taper ratio $R_{\min} / R_{\max}$ decreases, the magnitudes of the displacements ( $u_{x}$ , $u_{z}$ ) and rotations ( $Ω_{y}$ ) become smaller (example 3.1). Referring to $u_{x}$ , $u_{z}$ , and $Ω_{y}$ values at the tip of the rod, the percent reductions corresponding to $R_{\min} / R_{\max} = 0.3$ and $R_{\min} / R_{\max} = 0.5$ cases with respect to $R_{\min} / R_{\max} = 0.7$ case are approximately $34 %$ and $16 %$ , $45 %$ and $27 %$ , and $50 %$ and $25 %$ , respectively.

As values of the density of the material increase, the magnitudes and the vibration periods of the displacement ( $u_{x}$ ) and the shear force ( $T_{z}$ ) increase (example 3.2).

The magnitudes of the displacements ( $u_{x}$ , $u_{z}$ ) and rotation ( $Ω_{y}$ ) determined for the thin-walled hollow circular cross-sectioned rod are significantly lower than the displacements and the rotation obtained for solid circular and hollow circular cross-sectioned rods (example 4). If the $u_{x}$ , $u_{z}$ , and $Ω_{y}$ values of hollow circular and thin-walled hollow circular cross-sectioned rods are normalized with respect to the corresponding values of solid circular cross-sectioned rod, the percent reductions are found approximately $90 %$ and $54 %$ , $90 %$ and $53 %$ , and $90 %$ and $67 %$ , respectively.

The dynamic behavior of the hyperboloidal helical rod is affected from the form of applied dynamic load (example 5).

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by the Scientific and Technological Research Council of Turkey under project no. 111M308 and Research Foundation of ITU under project no. 38078.

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