Torsional vibrations and waves in a radially inhomogeneous cylinder

Abstract

This paper studies the problem of torsional vibration of a radially inhomogeneous isotropic cylinder. It is assumed that the elastic moduli and the density of the cylinder material are power functions of the cylinder radius. Cases where the lateral surface of the cylinder is free from stresses and fixed are studied. After satisfying homogeneous boundary conditions specified on the lateral surfaces of the cylinder, dispersion equations are obtained. Exact solutions to the problem are constructed. For a cylinder of small thickness, an analysis of the roots of the dispersion equations with respect to a small parameter characterizing the cylinder thickness is performed. Asymptotic solutions are constructed that allow calculating the three-dimensional stress–strain state of a radially inhomogeneous isotropic cylinder of small thickness. The propagation of torsional elastic waves in a radially layered cylinder consisting of alternating hard and soft layers is investigated. A theorem on the stratification of the thickness resonance frequency is obtained. In the vicinity of the origin, in the vicinity of the thickness resonance frequency, for sufficiently large values of the wavenumber and frequency, when their ratio is finite, asymptotic curves for dispersion curves are determined. Using a combination of asymptotic and numerical analysis, dispersion curves for a three-layer cylinder are constructed.

Keywords

Torsional vibrations dispersion equation asymptotic solution a radially layered cylinder dispersion curves

1. Introduction

The problem of propagation of torsional waves in a cylinder belongs to the classical problems of elasticity theory. This problem has important applied significance from an engineering point of view and has been studied by many researchers. In [1,2], the propagation of torsional waves was studied in composite infinite rods and in a two-layer cylinder. In [3], the propagation of torsional waves in a finite hollow cylinder made of piezoelectric material was investigated. In [4], torsional vibrations of a magneto-orthotropic hollow circular cylinder were studied. The phase velocities of perturbations for various wave numbers are calculated. In [5], dispersion curves were constructed for a torsional wave propagating in a thick transversely isotropic cylinder of infinite length. In [6], the propagation of torsional waves in an elastic solid cylinder subjected to an initial stress was investigated. In [7,8], using Biot’s theory, torsional vibrations of a poroelastic cylinder and a poroelastic composite cylinder were investigated. In [9], torsional vibrations in an initially stressed composite poroelastic cylinder were investigated within the framework of Biot’s theory. In [10], the influence of a magnetic field on torsional vibrations in poroelastic hollow cylinders was studied. A significant influence of a magnetic field on dispersion curves was shown. In [11 –14], the propagation of torsional waves in a prestressed multilayer cylinder was studied within the framework of a piecewise homogeneous body model. In [15], the problem of torsional vibrations of an anisotropic hollow cylinder with different boundary conditions at the ends was investigated using a numerical-analytical method. In [16], an analytical and numerical study of torsional vibrations of a three-layer transversely isotropic cylinder filled with a viscous fluid is presented. In [17], the problem of torsional vibration of an isotropic cylinder is analyzed using the method of homogeneous solutions.

Despite the above-mentioned multidirectional studies, exact and asymptotic solutions to the problem of torsional vibration of a radially inhomogeneous cylinder have not been constructed; asymptotic expressions for determining the stress–strain state have not been established; and the problem of propagation of elastic torsional waves in a radially multilayered cylinder consisting of rigid and soft layers has not been investigated by analytical methods.

In this study, the problems of torsional vibration of a radially inhomogeneous cylinder and the propagation of torsional waves in a radially layered cylinder are considered. Torsional vibrations of a cylinder with free lateral surfaces and with fixed lateral surfaces are investigated, where the shear modulus and material density vary continuously (according to a power law) as functions of the radius. These problems mathematically and physically complement each other by examining the influence of different boundary conditions prescribed on the lateral surfaces of the cylinder on the same physical processes. For these problems, exact and asymptotic solutions are constructed. New classes of solutions with boundary-layer properties, absent in applied theories, are obtained.

Furthermore, the propagation of torsional waves in a radially layered cylinder composed of alternating rigid and soft layers is investigated, in which the material properties are piecewise continuous functions of the radius. These problems are chosen as successive steps from a radially inhomogeneous cylinder, where the material properties are continuous functions of the radius, to a radially layered cylinder, where the material properties are piecewise continuous functions of the radius. It should be noted that when the material properties are piecewise continuous functions of the radial coordinate, this gives rise to certain difficulties in the analysis. The study is structured as a successive complication of the physical model and the mathematical apparatus.

To achieve this goal, the following tasks have been set:

construction of exact solutions to the problem of torsional oscillation of a radially inhomogeneous cylinder;

construction of asymptotic solutions to the problem of torsional oscillation of a radially inhomogeneous cylinder of small thickness;

studying the behavior of constructed asymptotic solutions;

construction of asymptotic formulas for displacements and stresses;

study of the propagation of elastic torsional waves in a radially layered cylinder;

determination of possible asymptotics of dispersion curves; and

construction of dispersion curves based on asymptotic and numerical analysis.

2. Torsional vibrations of a cylinder with free lateral surfaces

2.1. Mathematical formulation of the problem

We study torsional vibrations of a radially inhomogeneous isotropic cylinder. We assume that in the cylindrical coordinate system $r, ϕ, z$ the cylinder occupies a volume $Γ = {r_{1} \leq r \leq r_{2}, 0 \leq ϕ \leq 2 π, - l_{0} \leq z \leq l_{0}}$ .

The equations describing torsional vibrations of a cylinder in a cylindrical coordinate system $r, ϕ, z$ are of the form [18]:

\frac{\partial σ_{r ϕ}}{\partial r} + \frac{\partial σ_{ϕ z}}{\partial z} + \frac{2}{r} σ_{r ϕ} = g \frac{\partial^{2} υ_{ϕ}}{\partial t^{2}} .

(1)

The components of the stress tensor $σ_{r ϕ}, σ_{ϕ z}$ are expressed through the components of the displacement vector $υ_{ϕ} = υ_{ϕ} (r, z, t)$ as follows [18]:

σ_{r ϕ} = G (\frac{\partial υ_{ϕ}}{\partial r} - \frac{υ_{ϕ}}{r}),

(2)

σ_{ϕ z} = G \frac{\partial υ_{ϕ}}{\partial z} .

(3)

It is assumed that $G -$ the shear modulus and $g -$ the density of the cylinder material change along the radius according to power laws:

G (r) = E_{*} r^{n}, g = m_{*} r^{n} .

(4)

Substituting (2), (3) into (1) taking into account (4), we obtain:

\frac{\partial^{2} u_{ϕ}}{\partial ρ^{2}} + \frac{(n + 1)}{ρ} \frac{\partial u_{ϕ}}{\partial ρ} - \frac{(n + 1)}{ρ^{2}} u_{ϕ} + \frac{\partial^{2} u_{ϕ}}{\partial ξ^{2}} = \frac{g_{0}}{G_{0}} \frac{\partial^{2} u_{ϕ}}{\partial τ^{2}} .

(5)

Here $ρ = \frac{r}{r_{0}}, ξ = \frac{z}{r_{0}}$ are new dimensionless variables; $g_{0} = \frac{m_{*} r_{0}^{n}}{g_{*}}, G_{0} = \frac{E_{*} r_{0}^{n}}{G_{*}},$ $u_{ϕ} = \frac{υ_{ϕ}}{r_{0}}, τ = \frac{t}{r_{0}} \sqrt{\frac{G_{*}}{g_{*}}}$ are dimensionless quantities; $G_{*}$ is some characteristic parameter having the dimension of the modulus of elasticity; $g_{*}$ is some characteristic parameter having the dimension of density; $r_{0} = \frac{r_{1} + r_{2}}{2};$ $ξ \in [- l; l],$ $ρ \in [ρ_{1}; ρ_{2}],$ $l = \frac{l_{0}}{r_{0}}$ .

It is assumed that the lateral surface of the cylinder is free from stress:

σ_{ρ ϕ} = G_{0} ρ^{n} {(\frac{\partial u_{ϕ}}{\partial ρ} - \frac{u_{ϕ}}{ρ}) |}_{ρ = ρ_{s}} = 0, (s = 1, 2)

(6)

and boundary conditions are specified at the ends of the cylinder

σ_{ϕ ξ} = G_{0} ρ^{n} {\frac{\partial u_{ϕ}}{\partial ξ} |}_{ξ = \pm l} = f_{1}^{\pm} (ρ) e^{i λ τ} .

(7)

Here $σ_{ρ ϕ} = \frac{σ_{r ϕ}}{G_{*}}, σ_{ϕ ξ} = \frac{σ_{ϕ z}}{G_{*}}$ are dimensionless quantities, λ is oscillation frequency.

2.2. Construction of exact solutions

We seek the solution (5), (6) in the form:

u_{ϕ} (ρ, ξ, τ) = b (ρ) a (ξ) e^{i λ τ} .

(8)

Substituting (8) into (5), (6), we have:

b^{″} (ρ) + \frac{(n + 1)}{ρ} b^{'} (ρ) + (β^{2} + \frac{g_{0}}{G_{0}} λ^{2} - \frac{(n + 1)}{ρ^{2}}) b (ρ) = 0,

(9)

G_{0} ρ^{n} {(b^{'} (ρ) - \frac{b (ρ)}{ρ}) |}_{ρ = ρ_{s}} = 0, (s = 1, 2) .

(10)

The function $a (ξ)$ included in (8) satisfies the equation

a^{″} (ξ) - β^{2} a (ξ) = 0 .

(11)

Here β is spectral parameter.

Solution (9) has the form:

b (ρ) = ρ^{- \frac{n}{2}} (C_{1} J_{1 + \frac{n}{2}} (α ρ) + C_{2} Y_{1 + \frac{n}{2}} (α ρ)) .

(12)

Here $J_{1 + \frac{n}{2}} (α ρ), Y_{1 + \frac{n}{2}} (α ρ) -$ Bessel functions; $α = {(β^{2} + \frac{g_{0}}{G_{0}} λ^{2})}^{\frac{1}{2}}$ .

Based on (12) we have:

u_{ϕ} (ρ, ξ, τ) = ρ^{- \frac{n}{2}} (J_{1 + \frac{n}{2}} (α ρ) C_{1} + Y_{1 + \frac{n}{2}} (α ρ) C_{2}) a (ξ) e^{i λ τ},

(13)

\begin{matrix} σ_{ρ ϕ} = G_{0} ρ^{\frac{n}{2} - 1} {[α ρ J_{1 + \frac{n}{2}}^{'} (α ρ) - (1 + \frac{n}{2}) J_{1 + \frac{n}{2}} (α ρ)] \times \\ \times C_{1} + [α ρ Y_{1 + \frac{n}{2}}^{'} (α ρ) - (1 + \frac{n}{2}) Y_{1 + \frac{n}{2}} (α ρ)] C_{2}} a (ξ) e^{i λ τ}, \end{matrix}

(14)

σ_{ϕ ξ} = G_{0} ρ^{\frac{n}{2}} [C_{1} J_{1 + \frac{n}{2}} (α ρ) + C_{2} Y_{1 + \frac{n}{2}} (α ρ)] a (ξ) e^{i λ τ} .

(15)

We substitute (12) into (10):

{\begin{cases} G_{0} ρ_{1}^{\frac{n}{2} - 1} {[α ρ_{1} J_{1 + \frac{n}{2}}' (α ρ_{1}) - (1 + \frac{n}{2}) J_{1 + \frac{n}{2}} (α ρ_{1})] C_{1} + \\ + [α ρ_{1} Y'_{1 + \frac{n}{2}} (α ρ_{1}) - (1 + \frac{n}{2}) Y_{1 + \frac{n}{2}} (α ρ_{1})] C_{2}} = 0, \\ G_{0} ρ_{2}^{\frac{n}{2} - 1} {[α ρ_{2} J_{1 + \frac{n}{2}}' (α ρ_{2}) - (1 + \frac{n}{2}) J_{1 + \frac{n}{2}} (α ρ_{2})] C_{1} + \\ + [α ρ_{2} Y_{1 + \frac{n}{2}}' (α ρ_{2}) - (1 + \frac{n}{2}) Y_{1 + \frac{n}{2}} (α ρ_{2})] C_{2}} = 0 . \end{cases}

(16)

For the existence of a non-trivial solution of the system of homogeneous linear equations (16), the determinant of this system must be equal to zero:

Δ_{1} = | \begin{matrix} α ρ_{1} J_{1 + \frac{n}{2}}^{'} (α ρ_{1}) - (1 + \frac{n}{2}) J_{1 + \frac{n}{2}} (α ρ_{1}) α ρ_{1} Y_{1 + \frac{n}{2}}^{'} (α ρ_{1}) - (1 + \frac{n}{2}) Y_{1 + \frac{n}{2}} (α ρ_{1}) \\ α ρ_{2} J_{1 + \frac{n}{2}}^{'} (α ρ_{2}) - (1 + \frac{n}{2}) J_{1 + \frac{n}{2}} (α ρ_{2}) α ρ_{2} Y_{1 + \frac{n}{2}}^{'} (α ρ_{2}) - (1 + \frac{n}{2}) Y_{1 + \frac{n}{2}} (α ρ_{2}) \end{matrix} | = 0 .

From the last equality, the following dispersion equation is obtained:

\begin{matrix} Δ_{1} (β, λ, ρ_{1}, ρ_{2}) = α^{2} ρ_{1} ρ_{2} L_{1 + \frac{n}{2}}^{(1; 1)} (α) - α (1 + \frac{n}{2}) \times \\ \times [ρ_{1} L_{1 + \frac{n}{2}}^{(1; 0)} (α) + ρ_{2} L_{1 + \frac{n}{2}}^{(0; 1)} (α)] + {(1 + \frac{n}{2})}^{2} L_{1 + \frac{n}{2}}^{(0; 0)} (α) = 0 . \end{matrix}

(17)

Here $L_{1 + \frac{n}{2}}^{(s_{1}; s_{2})} (α) = J_{1 + \frac{n}{2}}^{(s_{1})} (α ρ_{1}) Y_{1 + \frac{n}{2}}^{(s_{2})} (α ρ_{2}) - J_{1 + \frac{n}{2}}^{(s_{2})} (α ρ_{2}) Y_{1 + \frac{n}{2}}^{(s_{1})} (α ρ_{1}) (s_{1}; s_{2} = 0, 1)$ .

The equation (17) has a countable set of roots $β_{k}$ . The constants $C_{1 k}, C_{2 k}$ , corresponding to the roots $β_{k}$ of the equation (17) are proportional to the algebraic complements of the elements of the first row of the main determinant of the system (16):

C_{1 k} = (α_{k} ρ_{2} Y_{1 + \frac{n}{2}}^{'} (α_{k} ρ_{2}) - (1 + \frac{n}{2}) Y_{1 + \frac{n}{2}} (α_{k} ρ_{2})) D_{k},

(18)

C_{2 k} = - (α_{k} ρ_{2} J_{1 + \frac{n}{2}}^{'} (α_{k} ρ_{2}) - (1 + \frac{n}{2}) J_{1 + \frac{n}{2}} (α_{k} ρ_{2})) D_{k} .

(19)

Substituting (18), (19) into (13)-(15) and carrying out summations over a countable set of roots (17), we have:

u_{ϕ} (ρ, ξ, τ) = \sum_{k = 1}^{\infty} ρ^{- \frac{n}{2}} [α_{k} ρ_{2} L_{1 + \frac{n}{2}}^{(0; 1)} (α_{k} ρ; α_{k} ρ_{2}) - (1 + \frac{n}{2}) L_{1 + \frac{n}{2}}^{(0; 0)} (α_{k} ρ; α_{k} ρ_{2})] t_{k} (ξ) e^{i λ τ},

(20)

\begin{matrix} σ_{ρ ϕ} = \sum_{k = 1}^{\infty} G_{0} ρ^{\frac{n}{2} - 1} [α_{k}^{2} ρ ρ_{2} L_{1 + \frac{n}{2}}^{(1; 1)} (α_{k} ρ; α_{k} ρ_{2}) - α_{k} (1 + \frac{n}{2}) (ρ L_{1 + \frac{n}{2}}^{(1; 0)} (α_{k} ρ; α_{k} ρ_{2}) + \\ + ρ_{2} L_{1 + \frac{n}{2}}^{(0; 1)} (α_{k} ρ; α_{k} ρ_{2})) + {(1 + \frac{n}{2})}^{2} L_{1 + \frac{n}{2}}^{(0; 0)} (α_{k} ρ; α_{k} ρ_{2})] t_{k} (ξ) e^{i λ τ}, \end{matrix}

(21)

σ_{ϕ ξ} = \sum_{k = 1}^{\infty} G_{0} ρ^{\frac{n}{2}} [α_{k} ρ_{2} L_{1 + \frac{n}{2}}^{(0; 1)} (α_{k} ρ; α_{k} ρ_{2}) - (1 + \frac{n}{2}) L_{1 + \frac{n}{2}}^{(0; 0)} (α_{k} ρ; α_{k} ρ_{2})] t_{k}^{'} (ξ) e^{i λ τ},

(22)

where $L_{1 + \frac{n}{2}}^{(s_{1}; s_{2})} (α_{k} ρ; α_{k} ρ_{2}) = J_{1 + \frac{n}{2}}^{(s_{1})} (α_{k} ρ) Y_{1 + \frac{n}{2}}^{(s_{2})} (α_{k} ρ_{2}) - J_{1 + \frac{n}{2}}^{(s_{2})} (α_{k} ρ_{2}) Y_{1 + \frac{n}{2}}^{(s_{1})} (α_{k} ρ);$ $(s_{1}; s_{2} = 0, 1)$ , $t_{k} (ξ) = D_{k} a_{k} (ξ),$ $t_{k} (ξ) = A_{1 k} e^{- β_{k} ξ} + A_{2 k} e^{β_{k} ξ}$ ; $A_{1 k}, A_{2 k}$ are arbitrary constants.

We will represent boundary value problems (9), (10) in the form of an operator equation

Fb = α^{2} b

(23)

Fb = {- (\frac{d^{2} b}{d ρ^{2}} + \frac{(n + 1)}{ρ} \frac{db}{d ρ} - \frac{(n + 1)}{ρ^{2}} b); G_{0} ρ^{n} {(\frac{db}{d ρ} - \frac{b}{ρ}) |}_{ρ = ρ_{s}} = 0; (s = 1; 2)} .

Let us introduce a Hilbert space $H (ρ_{1}; ρ_{2}) -$ with a scalar product

(b_{k}, b_{j}) = \int_{ρ_{1}}^{ρ_{2}} b_{k} (ρ) b_{j} (ρ) ρ^{n + 1} d ρ

(24)

Lemma 1: $F : H \to H$ is a non-negative operator.

Proof: Based on (24) for arbitrary functions $b_{1} (ρ), b_{2} (ρ) \in D_{F}$ we have

\begin{matrix} (F b_{1}, b_{2}) - (b_{1}, F b_{2}) = \int_{ρ_{1}}^{ρ_{2}} F b_{1} \cdot b_{2} (ρ) G_{0} ρ^{n + 1} d ρ - \int_{ρ_{1}}^{ρ_{2}} b_{1} (ρ) \cdot F b_{2} \cdot G_{0} ρ^{n + 1} d ρ = \\ = \int_{ρ_{1}}^{ρ_{2}} [- b_{1}^{″} (ρ) - \frac{(n + 1)}{ρ} b_{1}^{'} (ρ) + \frac{(n + 1)}{ρ^{2}} b_{1} (ρ)] b_{2} (ρ) G_{0} ρ^{n + 1} d ρ - \\ - \int_{ρ_{1}}^{ρ_{2}} [- b_{2}^{″} (ρ) - \frac{(n + 1)}{ρ} b_{2}^{'} (ρ) + \frac{(n + 1)}{ρ^{2}} b_{2} (ρ)] b_{1} (ρ) G_{0} ρ^{n + 1} d ρ = \\ = G_{0} \int_{ρ_{1}}^{ρ_{2}} [b_{2}^{″} (ρ) b_{1} (ρ) - b_{1}^{″} (ρ) b_{2} (ρ) + \frac{(n + 1)}{ρ} (b_{2}^{'} (ρ) b_{1} (ρ) - b_{1}^{'} (ρ) b_{2} (ρ))] ρ^{n + 1} d ρ . \end{matrix}

(25)

In (25), after integration by parts and taking into account the conditions (10), we finally have:

(F b_{1}, b_{2}) - (b_{1}, F b_{2}) = 0,

i.e. $(F b_{1}, b_{2}) = (b_{1}, F b_{2})$ and $F : H \to H$ is a symmetric operator [19].

For any $b (ρ) \in D_{F}$ we have:

\begin{array}{l} (F b, b) = \int_{ρ_{1}}^{ρ_{2}} F b \cdot b (ρ) G_{0} ρ^{n + 1} d ρ = \\ = G_{0} \int_{ρ_{1}}^{ρ_{2}} [- b^{''} (ρ) - \frac{(n + 1)}{ρ} b^{'} (ρ) + \frac{(n + 1)}{ρ^{2}} b (ρ)] b (ρ) ρ^{n + 1} d ρ = \\ = - G_{0} \int_{ρ_{1}}^{ρ_{2}} b (ρ) ρ^{n + 1} d (b^{'} (ρ)) - \\ - G_{0} \int_{ρ_{1}}^{ρ_{2}} (n + 1) b^{'} (ρ) b (ρ) ρ^{n} d ρ + G_{0} \int_{ρ_{1}}^{ρ_{2}} (n + 1) b^{2} (ρ) ρ^{n + 1} d ρ = \\ = - G_{0} ρ^{n + 1} b (ρ) b^{'} (ρ) |_{ρ_{1}}^{ρ_{2}} + G_{0} \int_{ρ_{1}}^{ρ_{2}} {(b^{'} (ρ))}^{2} ρ^{n + 1} d ρ + G_{0} \int_{ρ_{1}}^{ρ_{2}} (n + 1) b^{2} (ρ) ρ^{n - 1} d ρ = \\ = G_{0} \int_{ρ_{1}}^{ρ_{2}} {(b^{'} (ρ) - \frac{b (ρ)}{ρ})}^{2} ρ^{n + 1} d ρ + G_{0} \int_{ρ_{1}}^{ρ_{2}} (\frac{2}{ρ} b (ρ) b^{'} (ρ) - \frac{b^{2} (ρ)}{ρ^{2}}) ρ^{n + 1} d ρ - \\ - G_{0} ρ^{n + 1} b (ρ) b^{'} (ρ) |_{ρ_{1}}^{ρ_{2}} + G_{0} \int_{ρ_{1}}^{ρ_{2}} (n + 1) b^{2} (ρ) ρ^{n - 1} d ρ + 2 G_{0} \int_{ρ_{1}}^{ρ_{2}} b (ρ) b^{'} (ρ) ρ^{n} d ρ - \\ - G_{0} \int_{ρ_{1}}^{ρ_{2}} b^{2} (ρ) ρ^{n - 1} d ρ = \\ = G_{0} \int_{ρ_{1}}^{ρ_{2}} ρ^{n + 1} {(b^{'} (ρ) - \frac{b (ρ)}{ρ})}^{2} d ρ - G_{0} . ρ^{n + 1} b (ρ) b^{'} (ρ) |_{ρ_{1}}^{ρ_{2}} + . G_{0} ρ^{n} b^{2} (ρ) |_{ρ_{1}}^{ρ_{2}} - \\ - G_{0} \int_{ρ_{1}}^{ρ_{2}} ρ^{n} d (b^{2} (ρ)) + 2 G_{0} \int_{ρ_{1}}^{ρ_{2}} ρ^{n} b^{'} (ρ) b (ρ) d ρ = G_{0} \int_{ρ_{1}}^{ρ_{2}} ρ^{n + 1} {(b^{'} (ρ) - \frac{b (ρ)}{ρ})}^{2} d ρ - \\ - G_{0} . ρ^{n + 1} b (ρ) (b^{'} (ρ) - \frac{b (ρ)}{ρ}) |_{ρ_{1}}^{ρ_{2}} - 2 G_{0} \int_{ρ_{1}}^{ρ_{2}} ρ^{n} b (ρ) b^{'} (ρ) d ρ + 2 G_{0} \int_{ρ_{1}}^{ρ_{2}} ρ^{n} b^{'} (ρ) b (ρ) d ρ = \\ = \int_{ρ_{1}}^{ρ_{2}} G_{0} ρ^{n + 1} {(b^{'} (ρ) - \frac{b (ρ)}{ρ})}^{2} d ρ \geq 0 . \end{array}

(26)

For any $b (ρ) \in D_{F}$ the inequality $(Fb, b) \geq 0$ holds. ■

The eigenfunctions ${b_{k} (ρ)}$ of the non-negative operator F form an orthogonal basis in the space $H (ρ_{1}; ρ_{2})$ [19]:

{(b_{k}, b_{j})}_{H} = \int_{ρ_{1}}^{ρ_{2}} b_{k} (ρ) b_{j} (ρ) ρ^{n + 1} d ρ = 0 (k \neq j)

(27)

The eigenvalues $α^{2}$ of the non-negative operator F are real and $α^{2} \geq 0$ .

Since the eigenvalues of $α^{2} = β^{2} + \frac{g_{0}}{G_{0}} λ^{2}$ are real, $β^{2}$ is a real number. This means that the values of β are real or purely imaginary numbers.

We substitute expression (22) into (7):

{\sum_{k = 1}^{\infty} G_{0} ρ^{n} b_{k} (ρ) {t^{'}}_{k} (ξ) |}_{ξ = \pm l} = f_{1}^{\pm} (ρ) .

(28)

Multiplying both parts of (28) by $ρ b_{j} (ρ)$ and integrating the resulting expression over the $[ρ_{1}; ρ_{2}]$ , we have:

{\sum_{k = 1}^{\infty} (\int_{ρ_{1}}^{ρ_{2}} G_{0} ρ^{n + 1} b_{j} (ρ) b_{k} (ρ) d ρ) t_{k}^{'} (ξ) |}_{ξ = \pm l} = \int_{ρ_{1}}^{ρ_{2}} f_{1}^{\pm} (ρ) b_{j} (ρ) ρ d ρ .

(29)

We take into account the relation (27) in (29):

{{t^{'}}_{j} (ξ) |}_{ξ = \pm l} = q_{1 j}^{\pm},

(30)

where $q_{1 j}^{\pm} = \frac{\int_{ρ_{1}}^{ρ_{2}} f_{1}^{\pm} (ρ) b_{j} (ρ) ρ d ρ}{G_{0} \int_{ρ_{1}}^{ρ_{2}} b_{j}^{2} (ρ) ρ^{n + 1} d ρ}$ .

Based on (30) we obtain

{\begin{matrix} - β_{j} e^{β_{j} l} A_{1 j} + β_{j} e^{- β_{j} l} A_{2 j} = q_{1 j}^{-}, \\ - β_{j} e^{- β_{j} l} A_{1 j} + β_{j} e^{β_{j} l} A_{2 j} = q_{1 j}^{+} . \end{matrix}

(31)

From (31) we have:

A_{1 j} = \frac{q_{1 j}^{+} e^{- β_{j} l} - q_{1 j}^{-} e^{β_{j} l}}{2 β_{j} sh (2 β_{j} l)}, A_{2 j} = \frac{q_{1 j}^{+} e^{β_{j} l} - q_{1 j}^{-} e^{- β_{j} l}}{2 β_{j} sh (2 β_{j} l)},

(32)

2.3. Construction of asymptotic solutions

We assume that the cylinder has a small thickness. We will construct an asymptotic solution (5), (6) for a cylinder of small thickness. Let us introduce a small parameter $ε = \frac{r_{2} - r_{1}}{2 r_{0}}$ , which characterizes the thickness of the cylinder.

We note that

ρ_{1} = 1 - ε, ρ_{2} = 1 + ε

(33)

We substitute expressions (33) into (17):

Δ_{1} (β, λ, ρ_{1}, ρ_{2}) = Δ_{1} (β, λ, ε) = 0 .

(34)

We expand the expression $Δ_{1} (β, λ, ε)$ in terms of the small parameter ε:

\begin{matrix} Δ_{1} (β, λ, ε) = \frac{4 ε α^{2}}{π} {1 + \frac{2}{3} [- α^{2} + {(1 + \frac{n}{2})}^{2} + 2 (1 + \frac{n}{2})] ε^{2} + \\ + \frac{2}{15} [α^{4} + (1 - 2 {(1 + \frac{n}{2})}^{2} - 4 (1 + \frac{n}{2})) α^{2} + {(1 + \frac{n}{2})}^{4} + \\ + 4 {(1 + \frac{n}{2})}^{3} + 6 {(1 + \frac{n}{2})}^{2} + 4 (1 + \frac{n}{2})] ε^{4} + \dots} = 0 . \end{matrix}

(35)

We note that

Δ_{1} (β, λ, ε) = α^{2} Δ_{0} (β, λ, ε)

(36)

and

lim_{α \to 0} Δ_{0} (β, λ, ε) \neq 0 .

(37)

Based on (36), (37) we obtain that the number $α^{2} = 0$ is the zero of the function $Δ_{1} (β, λ, ε)$ .

From the equality $α^{2} = 0$ we have

β^{2} + \frac{g_{0}}{G_{0}} λ^{2} = 0,

i.e.

β = \pm i λ \sqrt{\frac{g_{0}}{G_{0}}} .

(38)

In case $λ = O (1)$ when $ε \to 0$ , the equation (17) has a countable set of roots $β_{k}$ , having order $O (ε^{- 1})$ :

β_{k} = \frac{δ_{1 k}}{ε} + O (ε) .

(39)

Substituting (39) into (17) taking into account $λ = O (1)$ for $ε \to 0$ and using asymptotic expansions of Bessel functions for large values of the argument [20] to determine $δ_{1 k}$ , we obtain:

\sin 2 δ_{1 k} = 0 .

(40)

Let us consider the case $λ \to \infty$ at $ε \to 0$ . Such oscillations are called ultra-high-frequency oscillations [17,21]. The following cases are possible:

a) $ε λ \to 0$ ; $b) ε λ \to const$ ; $c) ε λ \to \infty$

Here only the case $ε λ \to const$ holds. In the case $λ = λ_{0} ε^{- 1}$ at $ε \to 0$ equation (17) has a countable set of roots $β_{k}$ , having the order $O (ε^{- 1})$ :

β_{k} = \frac{δ_{2 k}}{ε} + O (ε) .

(41)

Substituting (41) into (17) taking into account $λ = λ_{0} ε^{- 1}$ when $ε \to 0$ and using asymptotic expansions of Bessel functions for large values of the argument[20] to determine $δ_{2 k}$ , we obtain:

\sin (2 \sqrt{δ_{2 k}^{2} + \frac{g_{0}}{G_{0}} λ_{0}^{2}}) = 0 .

(42)

For the solution corresponding to the root $α^{2} = 0$ of the dispersion equation, we obtain:

u_{ϕ} (ρ, ξ, τ) = ρ [B_{1} \cos (λ \sqrt{\frac{g_{0}}{G_{0}}} ξ) + B_{2} \sin (λ \sqrt{\frac{g_{0}}{G_{0}}} ξ)] e^{i λ τ},

(43)

σ_{ϕ ξ} = - \sqrt{g_{0} G_{0}} ρ^{n + 1} λ [B_{1} \sin (λ \sqrt{\frac{g_{0}}{G_{0}}} ξ) - B_{2} \cos (λ \sqrt{\frac{g_{0}}{G_{0}}} ξ)] e^{i λ τ},

(44)

σ_{ρ ϕ} = 0 .

(45)

In (19)-(21), assuming $ρ = 1 + ε η (- 1 \leq η \leq 1)$ and expanding in terms of the small parameter, we obtain asymptotic formulas for the corresponding roots (39):

u_{ϕ} (η, ξ, τ) = \sum_{k = 1}^{\infty} (- δ_{1 k} \cos (δ_{1 k} (1 + η)) + O (ε)) t_{k} (ξ) e^{i λ τ},

(46)

σ_{ρ ϕ} = \frac{G_{0}}{ε} \sum_{k = 1}^{\infty} (δ_{1 k}^{2} \sin (δ_{1 k} (1 + η)) + O (ε)) t_{k} (ξ) e^{i λ τ},

(47)

σ_{ϕ ξ} = G_{0} \sum_{k = 1}^{\infty} (- δ_{1 k} \cos (δ_{1 k} (1 + η)) + O (ε)) {t^{'}}_{k} (ξ) e^{i λ τ} .

(48)

The asymptotic formulas of the corresponding roots having asymptotics (41) have the form:

u_{ϕ} (η, ξ, τ) = \sum_{k = 1}^{\infty} (\cos (\sqrt{δ_{2 k}^{2} + \frac{g_{0}}{G_{0}} λ_{0}^{2}} (1 - η)) + O (ε)) t_{k} (ξ) e^{i λ τ},

(49)

σ_{ρ ϕ} = \frac{G_{0}}{ε} \sum_{k = 1}^{\infty} \sqrt{δ_{2 k}^{2} + \frac{g_{0}}{G_{0}} λ_{0}^{2}} (\sin (\sqrt{δ_{2 k}^{2} + \frac{g_{0}}{G_{0}} λ_{0}^{2}} (1 - η)) + O (ε)) t_{k} (ξ) e^{i λ τ},

(50)

σ_{ϕ ξ} = G_{0} \sum_{k = 1}^{\infty} (\cos (\sqrt{δ_{2 k}^{2} + \frac{g_{0}}{G_{0}} λ_{0}^{2}} (1 - η)) + O (ε)) {t^{'}}_{k} (ξ) e^{i λ τ} .

(51)

Note that (43)-(45) is a penetrating solution. With increasing distance from the ends of the cylinder, solutions (46)-(48), (49)-(51) decrease exponentially. These solutions have the character of a boundary layer and are absent from applied theories. The first terms of solution (46)-(48) are equivalent to the Saint-Venin edge effect of a plate with non-uniform thickness [17,21].

3. Torsional vibrations of a cylinder with a fixed lateral surface

3.1. Construction of exact solutions

Let us assume that the lateral surface of the cylinder is fixed

{u_{ϕ} |}_{ρ = ρ_{s}} = 0, (s = 1, 2)

(52)

and at the ends of the cylinder the boundary conditions

σ_{ϕ ξ} = G_{0} ρ^{n} {\frac{\partial u_{ϕ}}{\partial ξ} |}_{ξ = \pm l} = f_{2}^{\pm} (ρ) e^{i λ τ}

(53)

are specified.

We seek the solution to the boundary value problem (9), (52) in the form (8):

b^{″} (ρ) + \frac{(n + 1)}{ρ} b^{'} (ρ) + (β^{2} + \frac{g_{0}}{G_{0}} λ^{2} + \frac{(n + 1)}{ρ^{2}}) b (ρ) = 0,

(54)

{b (ρ) |}_{ρ = ρ_{s}} = 0 .

(55)

Substituting the solution of equation (54), determined by formulas (12), into the boundary conditions (55), we obtain:

{\begin{matrix} ρ_{1}^{- \frac{n}{2}} J_{1 + \frac{n}{2}} (α ρ_{1}) C_{1} + ρ_{1}^{- \frac{n}{2}} Y_{1 + \frac{n}{2}} (α ρ_{1}) C_{2} = 0, \\ ρ_{2}^{- \frac{n}{2}} J_{1 + \frac{n}{2}} (α ρ_{2}) C_{1} + ρ_{2}^{- \frac{n}{2}} Y_{1 + \frac{n}{2}} (α ρ_{2}) C_{2} = 0 . \end{matrix}

(56)

From the condition of existence of non-trivial solutions of system (56) we obtain the dispersion equation:

Δ_{2} (β, λ, ρ_{1}, ρ_{2}) = L_{1 + \frac{n}{2}}^{(0, 0)} (α) = 0 .

(57)

Carrying out summations over a countable set of roots $β_{k}$ of equation (57), we obtain:

u_{ϕ} (ρ, ξ, τ) = \sum_{k = 1}^{\infty} ρ^{- \frac{n}{2}} L_{1 + \frac{n}{2}}^{(0, 0)} (α_{k} ρ, α_{k} ρ_{2}) \cdot t_{k} (ξ) e^{i λ τ},

(58)

σ_{ρ ϕ} = G_{0} ρ^{\frac{n}{2} - 1} \sum_{k = 1}^{\infty} [α_{k} ρ L_{1 + \frac{n}{2}}^{(1, 0)} (α_{k} ρ, α_{k} ρ_{2}) - (1 + \frac{n}{2}) L_{1 + \frac{n}{2}}^{(0, 0)} (α_{k} ρ, α_{k} ρ_{2})] t_{k} (ξ) e^{i λ τ},

(59)

σ_{ϕ ξ} = G_{0} ρ^{\frac{n}{2}} \sum_{k = 1}^{\infty} L_{1 + \frac{n}{2}}^{(0, 0)} (α_{k} ρ, α_{k} ρ_{2}) \cdot {t^{'}}_{k} (ξ) e^{i λ τ} .

(60)

We represent the boundary value problem (54), (55) in the form

\begin{array}{l} Mb = α^{2} b, \\ Mb = {- (\frac{d^{2} b}{d ρ^{2}} + \frac{(n + 1)}{ρ} \frac{db}{d ρ} - \frac{(n + 1)}{ρ^{2}} b), {b |}_{ρ = ρ_{s}} = 0; (s = 1, 2)} . \end{array}

(61)

Lemma 2: $M : H \to H$ is a positive operator.

Proof: Based on (25) for arbitrary functions $b_{1} (ρ), b_{2} (ρ) \in D_{M}$ we have:

\begin{array}{l} (M b_{1}, b_{2}) - (b_{1}, M b_{2}) = \int_{ρ_{1}}^{ρ_{2}} M b_{1} \cdot b_{2} (ρ) G_{0} ρ^{n + 1} d ρ - \int_{ρ_{1}}^{ρ_{2}} b_{1} (ρ) \cdot M b_{2} \cdot G_{0} ρ^{n + 1} d ρ = \\ = G_{0} \int_{ρ_{1}}^{ρ_{2}} [b_{2}^{''} (ρ) b_{1} (ρ) - b_{1}^{''} (ρ) b_{2} (ρ) + \frac{(n + 1)}{ρ} ({b^{'}}_{2} (ρ) b_{1} (ρ) - b_{1}^{'} (ρ) b_{2} (ρ))] ρ^{n + 1} d ρ . \end{array}

(62)

After integrating by parts in (62) and taking into account the boundary conditions (55), we have:

(M b_{1}, b_{2}) - (b_{1}, M b_{2}) = 0,

that is, $(M b_{1}, b_{2}) = (b_{1}, M b_{2})$ and $M : H \to H$ is a symmetric operator [19].

Based on (26) for an arbitrary function $b (ρ) \in D_{M}$ we have:

(Mb, b) = \int_{ρ_{1}}^{ρ_{2}} Mb \cdot b (ρ) \cdot G_{0} ρ^{n + 1} d ρ = \int_{ρ_{1}}^{ρ_{2}} G_{0} ρ^{n + 1} {(b^{'} (ρ) - \frac{b (ρ)}{ρ})}^{2} d ρ \geq 0 .

(63)

From (63) for the case $(Mb, b) = 0$ we obtain:

b^{'} (ρ) - \frac{b (ρ)}{ρ} = 0,

i.e.

b (ρ) = D ρ .

(64)

Taking into account (64) and (55), we have $D = 0$ . In (63) the equality $(Mb, b) = 0$ is satisfied only when $b (ρ) = 0$ . This means that $M : H \to H$ is a positive operator [19].

The eigenfunctions ${b_{k} (ρ)}$ of a positive operator M constitute an orthogonal basis in the space $H (ρ_{1}; ρ_{2})$ [19]. The eigenvalues $α^{2}$ of the positive operator M are real and $α^{2} > 0$ [19].

We substitute relation (60) into (53):

\sum_{k = 1}^{\infty} G_{0} ρ^{n} b_{k} (ρ) {t_{k}^{'} (ξ) |}_{ξ = \pm l} = f_{2}^{\pm} (ρ) .

(65)

Multiplying both parts of (65) by $ρ b_{j} (ρ)$ and integrating the resulting expression over the $[ρ_{1}, ρ_{2}]$ taking into account (27), we have:

{t_{j}^{'} (ξ) |}_{ξ = \pm l} = q_{2 j}^{\pm},

(66)

where

q_{2 j}^{\pm} = \frac{\int_{ρ_{1}}^{ρ_{2}} f_{2}^{\pm} (ρ) b_{j} (ρ) ρ d ρ}{G_{0} \int_{ρ_{1}}^{ρ_{2}} b_{j}^{2} (ρ) ρ^{n + 1} d ρ} .

From system (66) the constants $A_{1 j}, A_{2 j}$ are determined.

3.2. Construction of asymptotic solutions

Let us construct an asymptotic solution to the problem (9), (52) for a cylinder of small thickness.

Substituting (33) into (58), we have:

Δ_{2} (β, λ, ρ_{1}, ρ_{2}) = Δ_{2} (β, λ, ε) = 0 .

(67)

The dispersion equation (67) for $λ = O (1)$ with $ε \to 0$ has a countable set of roots $β_{k}$ , which have asymptotics (39). Equation (67) for $λ \to \infty$ $(ε λ \to const)$ when $ε \to 0$ has a countable set of roots $β_{k}$ , which have asymptotics (41).

For the asymptotic solutions corresponding to the indicated roots of the dispersion equation (67), we obtain:

a) λ=O(1) (ε→0)

u_{ϕ} (η, ξ, τ) = \sum_{k = 1}^{\infty} [\sin (δ_{1 k} (1 - η)) + O (ε)] t_{k} (ξ) e^{i λ τ},

(68)

σ_{ρ ϕ} = \frac{G_{0}}{ε} \sum_{k = 1}^{\infty} [- δ_{1 k} \cos (δ_{1 k} (1 - η)) + O (ε)] t_{k} (ξ) e^{i λ τ},

(69)

σ_{ϕ ξ} = G_{0} \sum_{k = 1}^{\infty} [\sin (δ_{1 k} (1 - η)) + O (ε)] {t^{'}}_{k} (ξ) e^{i λ τ} .

(70)

b) λ = λ₀ε⁻¹→ ∞, ελ→const (ε→0)

u_{ϕ} (η, ξ, τ) = \sum_{k = 1}^{\infty} [\sin (\sqrt{δ_{2 k}^{2} + \frac{g_{0}}{G_{0}} λ_{0}^{2}} \cdot (1 - η)) + O (ε)] t_{k} (ξ) e^{i λ τ},

(71)

σ_{ϕ ξ} = G_{0} \sum_{k = 1}^{\infty} [\sin (\sqrt{δ_{2 k}^{2} + \frac{g_{0}}{G_{0}} λ_{0}^{2}} \cdot (1 - η)) + O (ε)] {t^{'}}_{k} (ξ) e^{i λ τ},

(72)

σ_{ρ ϕ} = \frac{G_{0}}{ε} \sum_{k = 1}^{\infty} \sqrt{δ_{2 k}^{2} + \frac{g_{0}}{G_{0}} λ_{0}^{2}} \cdot [\cos (\sqrt{δ_{2 k}^{2} + \frac{g_{0}}{G_{0}} λ_{0}^{2}} (1 - η)) + O (ε)] t_{k} (ξ) e^{i λ τ} .

(73)

Solutions (68)-(70), (71)-(73) have the character of a boundary layer and are absent from applied theories.

Based on the obtained asymptotic solutions, it is possible to construct a new refined applied theory for inhomogeneous cylindrical shells.

4. Propagation of elastic torsional waves in a radially layered cylinder

4.1. Statement of the boundary value problem

Let us consider the propagation of stationary elastic torsional waves in a radially layered cylinder consisting of alternating hard and soft layers of the number $n = 2 m - 1$ . We assume that the outer layers are rigid. We assign each rigid layer an odd number, $j = 1, 3, \dots, n$ , and each soft layer an even number, $i = 2, 4, \dots, n - 1$ (numbering is ordered from the center of the cylinder). For simplicity, we assume that the shear moduli and densities of all hard and all soft layers are the same:

\begin{array}{l} {\tilde{G}}_{1} = {\tilde{G}}_{3} = \dots = {\tilde{G}}_{n} = {\tilde{G}}_{h}, {\tilde{g}}_{1} = {\tilde{g}}_{3} = \dots = {\tilde{g}}_{n} = {\tilde{g}}_{h} \\ {\tilde{G}}_{2} = {\tilde{G}}_{4} = \dots = {\tilde{G}}_{n - 1} = {\tilde{G}}_{s}, {\tilde{g}}_{2} = {\tilde{g}}_{4} = \dots = {\tilde{g}}_{n - 1} = {\tilde{g}}_{s} \end{array}

We will take the small parameter $p = \frac{{\tilde{G}}_{s}}{{\tilde{G}}_{h}}$ as a relative characteristic of rigidity.

The inner radius of the «k»th layer will be designated as $ρ_{1 k}$ , the outer one $ρ_{2 k}$ .

The equation of motion of the «k»th layer in a cylindrical coordinate system has the form [18]:

\frac{\partial σ_{ρ ϕ}^{(k)}}{\partial ρ} + \frac{\partial σ_{ϕ ξ}^{(k)}}{\partial ξ} + \frac{2 σ_{ρ ϕ}^{(k)}}{ρ} = g_{k} \frac{\partial^{2} u_{ϕ}^{(k)}}{\partial τ^{2}},

(74)

where $u_{ϕ}^{(k)} = u_{ϕ}^{(k)} (ρ, ξ, τ)$ are components of the displacement vector of the «k»th layer; $σ_{ρ ϕ}^{(k)}$ , $σ_{ϕ ξ}^{(k)}$ are components of the stress tensor of the «k»th layer, $g_{k} = \frac{{\tilde{g}}_{k}}{g_{*}}$ ( $k = \bar{1, n}$ ) are dimensionless quantities.

The components of the stress tensor of the «k»th layer $σ_{ρ ϕ}^{(k)}$ , $σ_{ϕ ξ}^{(k)}$ are expressed through the components of the displacement vector as follows [18]:

σ_{ρ ϕ}^{(k)} = G_{k} (\frac{\partial u_{ϕ}^{(k)}}{\partial ρ} - \frac{u_{ϕ}^{(k)}}{ρ}),

(75)

σ_{ϕ ξ}^{(k)} = G_{k} \frac{\partial u_{ϕ}^{(k)}}{\partial ξ},

(76)

where $G_{k} = \frac{{\tilde{G}}_{k}}{G_{*}}$ is dimensionless quantity.

The connection of layers will be considered rigid, which means that the following conjugation conditions are met:

{\begin{matrix} σ_{ρ ϕ}^{(q)} (ρ_{2 q}, ξ, τ) = σ_{ρ ϕ}^{(q + 1)} (ρ_{1 q + 1}, ξ, τ) \\ u_{ϕ}^{(q)} (ρ_{2 q}, ξ, τ) = u_{ϕ}^{(q + 1)} (ρ_{1 q + 1}, ξ, τ), \end{matrix}

(77)

where $q = \bar{1, n - 1} .$

We assume that the lateral surface of the cylinder is stress-free, i.e.

σ_{ρ ϕ}^{(1)} (ρ_{1 1}, ξ, τ) = σ_{ρ ϕ}^{(n)} (ρ_{2 n}, ξ, τ) = 0 .

(78)

After substituting (75), (76) into (74), (77), (78) we obtain respectively:

{\begin{matrix} Δ u_{ϕ}^{(k)} - \frac{1}{ρ^{2}} u_{ϕ}^{(k)} = \frac{g_{k}}{G_{k}} \frac{\partial^{2} u_{ϕ}^{(k)}}{\partial τ^{2}}, \\ G_{1} {(\frac{\partial u_{ϕ}^{(1)}}{\partial ρ} - \frac{u_{ϕ}^{(1)}}{ρ}) |}_{ρ_{11}} = 0, \\ {u_{ϕ}^{(q)} |}_{ρ_{2 q}} = {u_{ϕ}^{(q + 1)} |}_{ρ_{1 q + 1}}, \\ G_{q} {(\frac{\partial u_{ϕ}^{(q)}}{\partial ρ} - \frac{u_{ϕ}^{(q)}}{ρ}) |}_{ρ_{2 q}} = G_{q + 1} {(\frac{\partial u_{ϕ}^{(q + 1)}}{\partial ρ} - \frac{u_{ϕ}^{(q + 1)}}{ρ}) |}_{ρ_{1 q + 1}}, \\ G_{n} {(\frac{\partial u_{ϕ}^{(n)}}{\partial ρ} - \frac{u_{ϕ}^{(n)}}{ρ}) |}_{ρ_{2 n}} = 0 \end{matrix}

(79)

where $Δ = \frac{\partial^{2}}{\partial ρ^{2}} + \frac{1}{ρ} \frac{\partial}{\partial ρ} + \frac{\partial^{2}}{\partial ξ^{2}}$ ; $k = \bar{1, n}$ ; $q = \bar{1, n - 1}$ .

4.2. Asymptotic analysis of spectral problems

We seek the solution to the problems (79) in the form:

u_{ϕ}^{(k)} (ρ, ξ, τ) = b_{k} (ρ) e^{i (γ ξ - λ τ)} .

(80)

After substituting (80) into (79) we have:

{\begin{matrix} b_{k}^{″} + \frac{1}{ρ} b_{k}^{'} + [(\frac{λ^{2}}{e_{k}^{2}} - γ^{2}) - \frac{1}{ρ^{2}}] b_{k} = 0, \\ G_{1} {(b_{1}^{'} - \frac{1}{ρ} b_{1}) |}_{ρ_{11}} = 0, \\ {b_{q} |}_{ρ_{2 q}} = {b_{q + 1} |}_{ρ_{1 q + 1}}, \\ G_{q} {(b_{q}^{'} - \frac{1}{ρ} b_{q}) |}_{ρ_{2 q}} = G_{q + 1} {(b_{q + 1}^{'} - \frac{1}{ρ} b_{q + 1}) |}_{ρ_{1 q + 1}}, \\ {G_{n} (b_{n}^{'} - \frac{1}{ρ} b_{n}) |}_{ρ_{2 n}} = 0 \end{matrix}

(81)

where $e_{k} = {(\frac{G_{k}}{g_{k}})}^{\frac{1}{2}}$ ; $k = \bar{1, n}$ ; $q = \bar{1, n - 1}$ .

The boundary value problem (81) contains two spectral parameters λ and γ. The determination of the wave pattern in a radially layered cylinder is associated with the construction of dispersion curves $γ_{t} = γ_{t} (λ)$ . For any real λ, problem (81) has a finite number of real eigenvalues and a countable set of purely imaginary eigenvalues $γ_{t}$ . The real eigenvalues $γ_{t}$ correspond to homogeneous waves propagating along the axis of the cylinder and carrying out energy transfer. The imaginary eigenvalues $γ_{t}$ determine inhomogeneous waves that do not carry energy.

For a radially layered cylinder, the construction of dispersion curves can be fully accomplished using a combination of analytical and numerical methods. To improve the efficiency of the numerical analysis, it is necessary to find the possible asymptotic behavior of the dispersion curves $γ_{t} = γ_{t} (λ)$ . Let us study problem (81) for $λ = 0$ and $γ = 0$ . The first problem for $p \to 0$ was studied in [22]. Problem (81) for $γ = 0$ describes thickness resonances [17,23]. In the $(γ, λ)$ plane, each curve intersects the frequency axis at the point $(0, λ_{t})$ , where $λ_{t}$ is the thickness resonance frequency and is the beginning of the dispersion curves.

Let us study the set of thickness resonant frequencies. Setting $γ = 0$ , we investigate problem (81):

{\begin{matrix} b_{k}^{″} + \frac{1}{ρ} b_{k}^{'} + (\frac{λ^{2}}{e_{k}^{2}} - \frac{1}{ρ^{2}}) b_{k} = 0, \\ {(b_{1}^{'} - \frac{1}{ρ} b_{1}) |}_{ρ_{11}} = 0, \\ {b_{q} |}_{ρ_{2 q}} = {b_{q + 1} |}_{ρ_{1 q + 1}}, \\ G_{q} {(b_{q}^{'} - \frac{1}{ρ} b_{q}) |}_{ρ_{2 q}} = G_{q + 1} {(b_{q + 1}^{'} - \frac{1}{ρ} b_{q + 1}) |}_{ρ_{1 q + 1}}, \\ {(b_{n}^{'} - \frac{1}{ρ} b_{n}) |}_{ρ_{2 n}} = 0 \end{matrix}

(82)

where $k = \bar{1, n}$ ; $q = \bar{1, n - 1}$ .

Let us analyze the spectral problem (82) for $p \to 0$ . Two cases are distinguished:

a) $p \to 0$ is equivalent to $G_{s} \to 0$ , where $G_{h}$ is fixed;

b) $p \to 0$ is equivalent to $G_{h} \to \infty$ , where $G_{s}$ is fixed.

The spectral problem (82) is reduced to the study of some homogeneous algebraic systems with a matrix whose elements analytically depend on the spectral parameter γ and linearly depend on the parameter $p = 1$ . Applying perturbation theory [24] to the indicated algebraic systems, we obtain the following theorem.

Theorem: The set of eigenvalues $R (p)$ of problem (82) for $p \to 0$ is a countable set and is represented in the form:

R (p) = R_{1} (p) ⋃ R_{2} (p) ⋃ R_{3} (p),

(83)

at that

1. $R_{1} (p)$ consists of a double eigenvalue $λ_{0} = 0$ and $2 (m - 1)$ eigenvalues of the form

λ_{k} = p^{\frac{1}{2}} μ_{0 k} + O (p^{3 / 2}) .

(84)

Where $μ_{0 k}$ are nonzero eigenvalues of the Jacobian algebraic system

{\tilde{D}}_{*} {\bar{X}}_{k} = μ_{0 k}^{2} {\bar{X}}_{k} .

(85)

Here ${\tilde{D}}_{*} = {\tilde{D}}_{1}^{- 1} {\tilde{D}}_{0}$ is Jacobian matrix; ${\tilde{D}}_{1}$ is diagonal, ${\tilde{D}}_{0}$ is symmetric Jacobian matrix:

\begin{matrix} {\tilde{D}}_{1} = ∥ \begin{matrix} \begin{matrix} {\tilde{d}}_{11} & 0 & \dots & 0 & 0 \\ 0 & {\tilde{d}}_{13} & \dots & 0 & 0 \end{matrix} \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \\ \begin{matrix} 0 & 0 & \dots & 0 & {\tilde{d}}_{1 n} \end{matrix} \end{matrix} ∥, {\tilde{D}}_{0} = ∥ \begin{matrix} \begin{matrix} {\tilde{d}}_{01} & - {\tilde{d}}_{01} & 0 & \dots & 0 & 0 \\ - {\tilde{d}}_{01} & {\tilde{d}}_{01} + {\tilde{d}}_{03} & - {\tilde{d}}_{03} & \dots & 0 & 0 \end{matrix} \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots . . \\ \begin{matrix} 0 & 0 & 0 & \dots & - {\tilde{d}}_{0 n - 2} & {\tilde{d}}_{0 n - 2} \end{matrix} \end{matrix} ∥, \\ {\tilde{d}}_{1 j} = \frac{ρ_{2 j}^{4} - ρ_{1 j}^{4}}{4 e_{j}^{2}}, {\tilde{d}}_{0 j} = \frac{2 ρ_{2 j}^{2} ρ_{1 j + 2}^{2}}{ρ_{1 j + 2}^{2} - ρ_{2 j}^{2}}, {\bar{X}}_{k} = {(x_{1 k}, x_{3 k}, \dots, x_{nk})}^{T} . \end{matrix}

2. $R_{2} (p)$ consists of $″ m ″$ sets of eigenvalues of the form

λ_{jt} = λ_{0 jt} + O (p^{δ}),

(86)

where $λ_{0 jt}$ is the root of the equation

L_{00} (λ_{0 j}) - \frac{2 e_{j}}{λ_{0 j} ρ_{2 j}} L_{01} (λ_{0 j}) - \frac{2 e_{j}}{λ_{0 j} ρ_{1 j}} L_{10} (λ_{0 j}) + \frac{4 e_{j}^{2}}{λ_{0 j}^{2} ρ_{1 j} ρ_{2 j}} L_{11} (λ_{0 j}) = 0 .

(87)

3. $R_{3} (p)$ consists of $″ m - 1 ″$ sets of eigenvalues of the form

λ_{it} = λ_{0 it} + O (p^{δ}),

(88)

where $λ_{0 it}$ —is the root of the equation

L_{00} (λ_{0 i}) = 0 .

(89)

Here $L_{s_{1} s_{2}} (λ_{0 k}) = J_{s_{1}} (λ_{0 k} ρ_{1 k}) Y_{s_{2}} (λ_{0 k} ρ_{2 k}) - J_{s_{2}} (λ_{0 k} ρ_{2 k}) Y_{s_{1}} (λ_{0 k} ρ_{1 k})$ ; $J_{s}$ , $Y_{s}$ are Bessel functions; $δ = 1$ , if $λ_{0 jt} \neq λ_{0 j + 1 t}$ , $λ_{0 jt} \neq λ_{0 j - 1 t}$ and $δ = \frac{1}{2}$ , if at least one of the equalities $λ_{0 jt} = λ_{0 j + 1 t}$ , $λ_{0 jt} = λ_{0 j - 1 t}$ holds.

Considering the cumbersome nature of the proof of the theorem, we note the general outline of the proof. The first limiting variant $G_{s} \to 0$ , $G_{h}$ is fixed, corresponds to a system of “m” cylinders not connected to each other (rigid layers), the cylindrical surfaces of which are free from stress. The spectrum of the limit problem will be the union of the “m” sets of eigenvalues of the spectral problems corresponding to individual cylinders. Each such set consists of a double eigenvalue $λ_{0 j} = 0$ and a countable set of values $λ_{0 jt}$ ( $j = 1, 3, \dots, n$ ), which in (86) are the first terms of the analytical expansions in terms of the parameter p. For small $p \neq 0$ , the numbers $λ_{0 j}$ generate $R_{2} (p)$ .

The second limiting variant $G_{h} \to \infty$ , $G_{s}$ is fixed, corresponds to a system of “ $m - 1$ ” cylinders not connected to each other (soft layers), on the cylindrical surfaces of which the displacements are equal to zero. The spectrum of the limit problem will also be the union “ $m - 1$ ” of the sets of eigenvalues of the spectral problems corresponding to individual cylinders. Each such set consists of a countable set of values $λ_{0 it}$ ( $i = 2, 4, \dots, n - 1$ ), which in (88) are the first terms of the analytical expansions in terms of the parameter p. For small $p \neq 0$ , the numbers $λ_{0 i}$ generate $R_{3} (p)$ .

Let us consider the construction of asymptotic approximations in the neighborhood of $(γ; λ) = (0; 0)$ . When $λ = 0$ , the point $γ = 0$ is a double point of the spectrum (81). Taking this into account, the solution to the problem (81) in the vicinity of $(γ; λ) = (0; 0)$ , in accordance with the methods of branching theory [25], is found in the form

γ^{2} = y_{1}^{2} λ^{2} + y_{2}^{2} λ^{4} + \dots, b_{k} = b_{0 k} + λ b_{1 k} + \dots

(90)

Substituting (90) into (81) we have:

y_{1} = \pm {(\frac{\sum_{k = 1}^{n} g_{k} (ρ_{2 k}^{4} - ρ_{1 k}^{4})}{\sum_{k = 1}^{n} G_{k} (ρ_{2 k}^{4} - ρ_{1 k}^{4})})}^{1 / 2} .

In the vicinity of $(γ; λ) = (0; 0)$ , the formula

γ = y_{1} λ (1 + O (λ))

(91)

describes the beginning of the first dispersion curve.

In the vicinity of the resonant frequency $λ_{t}$ we determine the asymptotics $γ_{t} = γ_{t} (λ)$ , taking the difference $θ_{t} = λ^{2} - λ_{t}^{2}$ as a small parameter. We find $b_{kt}$ and $γ_{t}$ in the following form

b_{kt} = b_{0 kt} + γ_{t} b_{1 kt} + \dots, γ_{t} = z_{t 0} θ_{t}^{1 / 2} + z_{t 1} θ_{t} + \dots

(92)

Substituting (92) into (81) after a certain transformation, we have:

z_{t 0} = {(\frac{\sum_{k = 1}^{n} g_{k} \int_{ρ_{1 k}}^{ρ_{2 k}} b_{0 kt}^{2} ρ d ρ}{\sum_{k = 1}^{n} G_{k} \int_{ρ_{1 k}}^{ρ_{2 k}} b_{0 kt}^{2} ρ d ρ})}^{1 / 2}

(93)

In the vicinity of the resonant frequency $λ_{t}$ , the dispersion curves are determined by the relations:

γ_{t} = z_{t 0} θ_{t}^{1 / 2} + O (θ_{t}) .

(94)

At low frequencies, $b_{0 kt}$ is determined by the formula

b_{0 jt} (ρ) = ρ x_{jt}, b_{0 it} (ρ) = \frac{1}{ρ_{2 i}^{2} - ρ_{1 i}^{2}} [x_{i + 1, t} ρ_{2 i}^{2} (ρ - \frac{ρ_{1 i}^{2}}{ρ}) - x_{i - 1, t} ρ_{1 i}^{2} (ρ - \frac{ρ_{2 i}^{2}}{ρ})] .

(95)

Let us consider the behavior of dispersion curves for $γ \to \infty$ , $λ \to \infty$ , provided that the limit of their ratio $lim \frac{λ}{γ} = c$ is some finite value. This corresponds to the case where, with increasing frequency, the wavelength becomes significantly smaller than the thickness of the cylinder in question.

Let us transform (81) to the form:

{\begin{matrix} γ_{*}^{2} (b_{k}^{″} + \frac{1}{ρ} b_{k}^{'} - \frac{1}{ρ^{2}} b_{k}) + (\frac{c^{2}}{e_{k}^{2}} - 1) b_{k} = 0, \\ {(b_{1}^{'} - \frac{1}{ρ} b_{1}) |}_{ρ_{11}} = 0, \\ {b_{q} |}_{ρ_{2 q}} = {b_{q + 1} |}_{ρ_{1 q + 1}}, \\ G_{q} {(b_{q}^{'} - \frac{b_{q}}{ρ}) |}_{ρ_{2 q}} = G_{q + 1} {(b_{q + 1}^{'} - \frac{1}{ρ} b_{q + 1}) |}_{ρ_{1 q + 1}}, \\ {(b_{n}^{'} - \frac{1}{ρ} b_{n}) |}_{ρ_{2 n}} = 0 \end{matrix}

(96)

where $γ_{*} = \frac{1}{γ}$ , $c = \frac{λ}{γ}$ is phase velocity of wave propagation, $k = \bar{1, n}$ , $q = \bar{1, n - 1}$ .

It is clear that $γ_{*} \to 0$ when $γ \to \infty$ , that is, $γ_{*}$ is a small parameter. Considering «c» to be a spectral parameter, we study the problem (96) for $γ_{*} \to 0$ . From the structure (96) it is clear that the small parameter stands before the highest derivative. In accordance with the Vishik–Lyusternik method [26], which determines the solution away from the boundary surfaces $ρ = ρ_{11}$ , $ρ = ρ_{2 n}$ and the interface between the media $ρ_{2 q}$ ( $q = \bar{1, n - 1}$ ), we find the solution in the form:

b_{k} = b_{k 0} + γ_{*} b_{k 1} + \dots, c^{2} = c_{0}^{2} + γ_{*} c_{1}^{2} + \dots

(97)

Substituting (97) into (96) we obtain:

c_{0 k} = {(\frac{G_{k}}{g_{k}})}^{1 / 2}, c_{1 k} = 0 .

The asymptotes of the dispersion curves at $γ \to \infty$ , $λ \to \infty$ are obtained from the following considerations. When the curvature of individual layers plays a small role, replacing the layered cylinder with a layered space and based on research [27], one can conclude that at $γ \to \infty$ , $λ \to \infty$ each dispersion curve asymptotically tends to a straight line whose slope tangent is equal to $\frac{e_{i}}{e_{j}}$ ( $\frac{e_{s}}{e_{h}}$ ).

5. Numerical analysis

Let’s plot dispersion curves for a three-layer cylinder with a soft core. Figures 1 and 2 show dispersion curves for the following cases:

1) $p = 0.01$ ; $\frac{g_{s}}{g_{h}} = 0.2$ ; 2) $p = 0.0001$ ; $\frac{g_{s}}{g_{h}} = 0.128$ .

In both cases $ρ_{11} = 0.2$ ; $ρ_{21} = 0.5$ ; $ρ_{13} = 0.6$ ; $ρ_{23} = 1$ .

Figure 1.

Dispersion curves for a three-layer cylinder with a soft core $p = 0.01$ ; $\frac{g_{s}}{g_{h}} = 0.2$

Figure 2.

Dispersion curves for a three-layer cylinder with a soft core $p = 0.0001$ ; $\frac{g_{s}}{g_{h}} = 0.128$ .

Based on (84), for the first frequency of thickness resonance we have:

λ_{1} = p^{\frac{1}{2}} μ_{01} + O (p^{3 / 2})

(98)

where $μ_{01}^{2} = \frac{{\tilde{d}}_{01} ({\tilde{d}}_{11} + {\tilde{d}}_{13})}{{\tilde{d}}_{11} {\tilde{d}}_{13}}$ .

1. Let us conduct an analysis of dispersion curves when $p = 0.01$ (Figure 1). The first dispersion branch, leaving zero λ, increases and approaches a straight line with the tangent of the angle of inclination equal to the velocity of propagation of transverse waves in a rigid layer. At $λ = 5.5$ , a transition zone is observed. With further increase in λ, the dispersion curve asymptotically approaches a straight line corresponding to the phase velocity of transverse waves in a soft layer.

On the second dispersion branch, in the region of imaginary wavenumbers, there is no propagating wave. At $λ \geq 0.87$ , two waves can propagate in the waveguide. As λ increases, this curve approaches a straight line with a slope equal to the velocity of transverse waves in the rigid layer. For sufficiently large values of γ and λ, the asymptote will be a straight line with the tangent of the angle of inclination equal to $\frac{e_{i}}{e_{j}}$ ( $\frac{e_{s}}{e_{h}}$ ). Formula (98) determines the values of the first thickness resonance with an error of 6%.

2. Let us conduct an analysis of dispersion curves when $p = 0.0001$ (Figure 2).

The first branch originates from a point $(γ; λ) = (0; 0)$ and is a straight line, the slope of which is equal to the shear wave velocity in the rigid layer. As $λ = 0.67$ a transition zone is observed. With further growth λ, the curve becomes a straight line, the slope of which is equal to the shear wave velocity in the soft layer. As $λ \geq 0.826$ , three waves can propagate in the cylinder. On the third branch, in the region of purely imaginary wave numbers, there is no propagating wave. With growth λ, this curve also becomes a straight line, the slope of which is determined by the shear wave velocity in the rigid layer. With further growth λ, the curve becomes a straight line, the slope of which is equal to the shear wave velocity in the soft layer. For sufficiently large values of γ and λ the dispersion curve is a straight line with slope $\frac{e_{s}}{e_{h}}$ .

From the research conducted it turns out that in both cases 1) $p = 0.01$ , 2) $p = 0.0001$ in multilayer cylinders with alternating hard and soft layers, even at low frequencies there is more than one propagating wave, in contrast to a homogeneous cylinder of the same thickness.

Conclusion

Exact solutions to the problems of torsional vibrations of a radially inhomogeneous cylinder are constructed. Asymptotic solutions to the problems of torsional vibrations of a radially inhomogeneous cylinder of small thickness are constructed. Based on the analysis performed, the nature of the constructed solutions has been determined. Asymptotic formulas defining the stress–strain state of a radially inhomogeneous cylinder are obtained. The propagation of torsional elastic waves in a radially layered cylinder is studied. A theorem on the stratification of the thickness resonance frequency is obtained. In the vicinity of $(γ; λ) = (0; 0)$ , in the vicinity of the thickness resonance frequency and at $γ \to \infty$ , $λ \to \infty$ (where $lim \frac{λ}{γ} = c$ is a finite value), the asymptotics for the dispersion curves are determined. Using a combination of asymptotic and numerical analysis, dispersion curves are constructed.

Footnotes

ORCID iD

Natiq K Akhmedov

Funding

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

Declaration of conflicting interests

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

References

Armenàkas

. Torsional waves in composite rods. J Acoust Soc Am 1965; 38: 439–446.

Bhattacharya

. On the torsional wave propagation in a two-layered circular cylinder with imperfect bonding. Proc Indian Natn Sci Acad 1975; 41: 613–619.

Sharma

. Torsional wave propagation in a finite inhomogeneous cylindrical shell under time-dependent shearing stress. Proc Indian Acad Sci 1982; 91(3): 201–210.

Abd-Alla

. Torsional wave propagation in an orthotropic magneto-elastic hollow circular cylinder. Appl Math Comput 1994; 63(2–3): 281–293.

Kudlicka

. Dispersion of torsional waves in a thick-walled transversely isotropic circular cylinder of infinite length. J Sound Vib 2006; 294: 368–373.

Selim

. Torsional waves propagation in an initially stressed dissipative cylinder. Appl Math Sci 2007; 1(29): 1419–1427.

Tajuddin

Ahmed, Shah

. On torsional vibrations of infinite hollow poroelastic cylinders. J Mech Mater Struct 2007; 2(1): 189–200.

Ahmed, Shah

Tajuddin

. Torsional vibrations of infinite composite poroelastic cylinders. Int J Eng Sci Technol 2010; 2(6): 150–161.

Sandhya

Rani, B

Malla

Reddy, P

Shamantha, Reddy

. Study of torsional vibrations in composite transversely isotropic poroelastic cylinders. Procedia Eng 2015; 127: 462–468.

10.

Ramagiri

. Investigation of torsional vibrations in magnetic field poroelastic hollow cylinders. Carib J Sci Tech 2022; 10(1): 36–42.

11.

Ozturk

Akbarov

. Propagation of torsional waves in a prestretched compound hollow circular cylinders. Mech Compos Mater 2008; 44(1): 77–86.

12.

Ozturk

Akbarov

. Torsional wave dispersion relations in a pre-stressed bi-material compounded cylinder. ZAMM Z fur Angew Math Mech 2009; 89(9): 754–766.

13.

Kepceler

. Torsional wave dispersion relations in a prestressed bi-material compounded cylinder with an imperfect interface. Appl Math Model 2010; 34(12): 4058–4073.

14.

Akbarov

Kepceler

Egilmez

. Torsional wave dispersion in a finite lypre–strained hollow sandwich circular cylinder. J Sound Vib 2011; 330: 4519–4537.

15.

Efimova

. Solution of problems of free torsional vibrations of thick-walled orthotropic inhomogeneous cylinders. J Math Sci 2010; 168(4): 613–623.

16.

Abassi

El, Baroudi

Arts

. Torsional vibrations of fluid-filled multilayered transversely isotropic finite circular cylinder. Int J Appl Mech 2016; 8(3): 1650032.

17.

Mekhtiev

. Vibrations of hollow elastic bodies. Berlin: Springer, 2018.

18.

Lur’e

. Theory of elasticity. Berlin: Springer, 2005.

19.

Mikhlin

. Partial linear equations (in Russian). Moscow: Visshaya Skola, 1977.

20.

Beitmen

Erdeyi

. Higher transcendental functions (in Russian). Moscow: Nauka, 1965.

21.

Ustinov

. The mathematical theory of transversely non-uniform plates (in Russian). Rostov-on-Don: OOOTsVVR, 2006.

22.

Akhmedov

Ustinov

. On the Saint-Venant principle in the problem of torsion of a glass cylinder. Appl Math Mech 1988; 52(2): 264–268.

23.

Grinchenko

Meleshko

. Harmonic oscillations and waves in elastic bodies. Kyiv: Naukova Dumka, 1981.

24.

Kato

. Perturbation theory for linear operators. Berlin: Springer, 1995.

25.

Vainberg

Trenogin

. Theory of branching of solutions of non-linear equations. Leyden, The Netherlands: Noordhoff, 1974.

26.

Vishik

Lyusternik

. The solution of some perturbation problems for matrices and self adjoint or non-self adjoint differential equations I. Uspekhi Mat Nauk 1960; 15(3): 3–80.

27.

Brekhovskikh

. Waves in layered media. Moscow: Publishing House of the USSR Academy of Sciences, 1957.