On the Compressed Elastic Rod with Rotary Inertia on a Viscoelastic Foundation

1. Formulation of the Problem Consider an elastic rod simply supported at both ends. Let L be the length of the rod. We assume that the rod is loaded by an axial force P (t) that is a known function of time and has a fixed direction coinciding with the rod axis in the initial (undeformed) state. In this work we shall generalize our previous analysis [1] in two directions. First, we assume here that the 8 B. Stanković, T. M. Atanacković axial force is given by generalized functions and second we assume that the rod has a (not negligible) rotary inertia. Thus we allow for impulsive (described by a Dirac distribution) axial loading of the rod. The rod is positioned on a viscoelastic foundation described by a fractionl derivative type constitutive equation (see Figure 1). Figure 1. Coordinate system and load configuration Such foundations are important for vibration damping and have been recently analyzed in the context of the railpad in the railway track model (see [2] and [3] for the physical explanation of the model). The equilibrium equations of active and inertial forces are (see [4] p. 338) ∂H ∂S = ρ0 ∂2x ∂t2 − qx, ∂V ∂S = ρ0 ∂2y ∂t2 − qy, ∂M ∂S = −V cosθ+H sinθ− J ∂ 2θ ∂t2 , ∂x ∂S = cosθ, ∂y ∂S = sinθ, ∂θ ∂S = M EI , S ∈ (0, L) , t ≥ 0, (1) where x and y are coordinates of an arbitrary point on the rod axis, S is the arc length of the rod axis in the undeformed state so that S ∈ (0, L), t is the time, H and V are components of the force in an arbitrary cross–section of the rod along the x̄ and ȳ axes of a rectangular Cartesian coordinate system x̄ − B − ȳ, respectively, M is the bending moment, and qx, qy are the intensities of the distributed forces per unit length of the rod axis in the undeformed state, J is moment of inertia of a part of the rod of unit length, θ is the angle between the tangent to the rod axis and the x̄ axis, ρ0 is the line density of the rod, EI is the bending rigidity of the rod. For the rod On the compressed elastic rod with rotary inertia 9 shown in Figure 1 the boundary conditions are y (0, t) = 0, x (0, t) = 0; y (L, t) = 0, M (0, t) = 0, M (L, t) = 0; H (L, t) = −P , t ≥ 0. (2) Suppose that the rod is positioned on a viscoelastic foundation. We assume that the foundation is of the fractional derivative type. If the foundation is made of a fractional type viscoelastic material, then the force in the foundation Q and deformation Δ of the foundation (in our case Δ = y) are connected as Q+ τQQ = Ep ( y + τyy ) , (3) with 0 < β < 1. In (3) we used (·) to denote the β-th derivative of a function (·) taken in Riemann-Liouville form as (see [5], and [6]) g ≡ d β dtβ g (t) ≡ d dt 1 Γ (1− β) ∫ t 0 g (ξ) dξ (t− ξ) . (4) The dimension of the constants τy and τQ is [time] α . The constants Ep, τQ and τy in (3) are called the instantaneous modulus of the pad and the relaxation times, respectively. In Figure 1 the rheological model of the foundation is presented, as given in [7], for example. We assume that the following inequality, as a consequence of the second law of thermodynamics, is satisfied (see [9] and [8]) E > 0, τQ > 0, τy > τQ. (5) We assume that the pads are positioned under the rod so that qx = 0, qy = −bQ, (6) where b is a constant depending on the part of the rod’s width that is supported by pads. Note that in the case β = 1 the foundation becomes a standard viscoelastic solid. The trivial solution to the system (1),(2),(3) and (5) in which the rod axis is straight reads H (S, t) = −P , V 0 (S, t) = 0, M (S, t) = 0, x (S, t) = S, y (S, t) = 0, θ (S, t) = 0, Q (S, t) = 0. (7) Let the solution to (1),(2),(3) and (5) be written in the form H = H0 + ΔH, ...Q = Q0 + ΔQ, where ΔH, ...,ΔQ are perturbations, assumed to be 10 B. Stanković, T. M. Atanacković small. By substituting this in (7) and neglecting the higher order terms in the perturbations ΔH, ...,ΔQ, we obtain ∂ΔH ∂S = ρ0 ∂2Δx ∂t2 , ∂ΔV ∂S = ρ0 ∂2Δy ∂t2 + bΔQ, ∂ΔM ∂S = −ΔV − PΔθ− J ∂ 2Δθ ∂t2 , ∂Δx ∂S = 0, ∂Δy ∂S = Δθ, ∂Δθ ∂S = ΔM EI , ΔQ+ τQΔQ = Ep ( Δy + τyΔy ) , (8)