Three alternative versions of the theory for a Timoshenko–Ehrenfest beam on a Winkler

Abstract

This paper analyzes the free vibration frequencies of a beam on a Winkler–Pasternak foundation via the original Timoshenko–Ehrenfest theory, a truncated version of the Timoshenko–Ehrenfest equation, and a new model based on slope inertia. We give a detailed comparison between the three models in the context of six different sets of boundary conditions. In particular, we analyze the most common combinations of boundary conditions deriving from three typical end constraints, namely the simply supported end, clamped end, and free end. An interesting intermingling phenomenon is presented for a simply-supported (S-S) beam together with proof of the ‘non-existence’ of zero frequencies for free-free (F-F) and simply supported-free (S-F) beams on a Winkler–Pasternak foundation.

Keywords

Timoshenko–Ehrenfest beam theory Winkler foundation Pasternak foundation shear deformation rotary inertia

1. Introduction

Beams resting on elastic foundations and their dynamic behavior have been subject of much interest since the pioneering studies of Winkler [1] and Pasternak [2]. In the beginning, Winkler set his model as a continuous layer of independent linear springs attached to the beam that exert a force $F_{w}$ on the beam that is proportional to beam’s displacement $F_{w} = k_{W} w (x, t)$ . Due to its simplicity this model has been used in many engineering applications, but it comes short of describing structures in continuous media, such as soil, especially at the free ends. Thus, Pasternak advanced another model in which a foundation exerted a force $F_{p}$ , which is proportional to the beam’s curvature, $F_{p} = - k_{P} \partial^{2} w / \partial x^{2}$ .

Vibrations of Timoshenko–Ehrenfest beams on an elastic foundation have been treated in several works, namely by Abohadima et al. [3], Auciello [4], Avramidis and Morfidis [5], Atternejad et al. [6], Calio and Greco [7], de Rosa [8], Hassan and Nassar [9], Manevich [10], Matsunaga [11], Obara [12], Sadeghian and Ekhterai Toussi [13], Wang and Gagnon [14], Wu [15], Yin [16], Yokoyama [17,18], and others. As example, in the work by de Rosa [8], the free vibration frequencies of Timoshenko–Ehrenfest beams on a two-parameter elastic foundation (Winkler and Pasternak) are examined; two variants of the equations of motion are deduced, and both axial flexibilities and rotational flexibilities of the constraints are taken in to account. Yokoyama [17,18], indeed, worked on a finite element technique for determining the vibration characteristics of a uniform Timoshenko–Ehrenfest beam-column supported on a two-parameter elastic foundation. The governing equations are derived using Hamilton’s principle. Calio and Greco [7] analyzed the free vibration and the stability of axially loaded Timoshenko–Ehrenfest beams on an elastic foundation by considering an elastic soil containing two parameters having the characteristics of both Winkler and Pasternak foundations.

In the present paper, we conduct the vibration analysis of a beam on a Winkler–Pasternak foundation via traditional the full, original Timoshenko–Ehrenfest beam theory and two of its truncated versions. Indeed Timoshenko [19] wrote in 1916 about his collaboration in the development of the theory jointly with Paul Ehrenfest (1880–1933). He stressed the cooperation in his later paper [20]. He refrained from mentioning Bresse’s [21] contribution on considering rotary inertia, later incorporated also by Rayleigh [22], apparently independently of Bresse [21]. Later on, Timoshenko [20,23] re-published the results of the joint work with Ehrenfest as two scientific papers almost coinciding with each other. However, in 1972, Timoshenko [24] reiterated about his cooperation with Ehrenfest by re-publishing the book in [19]. It appears, therefore, that the beam theory incorporating the effects of rotary inertia and shear deformation ought to be called the Bresse–Rayleigh–Timoshenko–Ehrenfest theory. Hereinafter we will utilize a more concise name of the Timoshenko–Ehrenfest theory, to make justice to the joint work of Timoshenko with Ehrenfest. Detailed comparison between the three presented models is conducted in the context of the titles. In this paper we extend the studies by Cazzani et al. [25,26] to investigate the effect of boundary conditions (BCs) in the beam on an elastic foundation; an analogous effort for a beam without an elastic foundation was conducted by Kocatürk and Şimşek [27 –38].

2. Governing Timoshenko–Ehrenfest equation

The original Timoshenko–Ehrenfest equation for a beam on a Winkler–Pasternak foundation has been obtained by incorporating the effect of the foundations into the translational and rotational equilibrium equations

\frac{\partial V}{\partial x} = ρ A (\frac{\partial^{2} w (x, t)}{\partial t^{2}}) + k_{W} w (x, t) - k_{P} \frac{\partial^{2} w (x, t)}{\partial x^{2}}

(1)

Vdx - \frac{\partial M}{\partial x} dx - ρ I \frac{\partial^{2} ψ (x, t)}{\partial t^{2}} dx = 0

(2)

where V is the shear force, M is the bending moment, $ρ$ is density expressed as mass over volume $(ρ = m / V)$ , $w$ is the transversal displacement, $ψ$ is the rotation angle, $k_{W}$ is the stiffness of the Winkler foundation, $k_{P}$ is the stiffness of the Pasternak foundation, and I is the moment of inertia.

Substitution of constitutive equations for the bending moment and shear force is

M = - EI \frac{\partial ψ (x, t)}{\partial x} V = KGA η = KGA (\frac{\partial w (x, t)}{\partial x} - ψ)

(3)

where E is the Young’s modulus, G is the shear modulus, K is the shear correction factor, and $η$ is the shear deformation angle, substituting Equation (3) into Equations (1) and (2) results in

KGA (\frac{\partial^{2} w (x, t)}{\partial x^{2}} - \frac{\partial ψ (x, t)}{\partial x}) - ρ A \frac{\partial^{2} w (x, t)}{\partial t^{2}} - k_{w} w + k_{P} \frac{\partial^{2} w (x, t)}{\partial x^{2}} = 0

(4)

EI \frac{\partial^{2} ψ (x, t)}{\partial x^{2}} + KGA (\frac{\partial w (x, t)}{\partial x} - ψ (x, t)) - ρ I \frac{\partial^{2} ψ (x, t)}{\partial t^{2}} = 0

(5)

We are making use of Equations (4) and (5) to eliminate $ψ (x, t) . Thus,$ we obtain the original Timoshenko–Ehrenfest equation in terms of displacement $w (x, t)$

\begin{matrix} EI (1 + \frac{k_{p}}{KGA}) \frac{\partial^{4} w (x, t)}{\partial x^{4}} - ρ I (1 + \frac{E}{KG} + \frac{k_{p}}{KGA}) \frac{\partial^{4} w (x, t)}{\partial x^{2} \partial t^{2}} \\ - (k_{p} + \frac{EI k_{W}}{KGA}) \frac{\partial^{2} w (x, t)}{\partial x^{2}} + (m + \frac{ρ I k_{W}}{KGA}) \frac{\partial^{2} w (x, t)}{\partial t^{2}} + \frac{ρ^{2} I}{KG} \frac{\partial^{4} w (x, t)}{\partial t^{4}} + k_{W} w (x, t) = 0 \end{matrix}

(6)

Since we are looking for a solution of the form

w (x, t) = W (x) e^{i ω t}

(7)

where $i = \sqrt{- 1}$ , we substitute Equation (7) into Equation (6). Then we consider the non-dimensional coordinate $ξ = x / L$ , obtaining

\frac{d^{4} W (ξ)}{d ξ^{4}} + L^{2} \frac{ρ I ω^{2} (1 + \frac{E}{KG} + \frac{k_{p}}{KGA}) - k_{p} - \frac{EI k_{W}}{KGA}}{EI (1 + \frac{k_{p}}{KGA})} \frac{d^{2} W (ξ)}{d ξ^{2}} + L^{4} \frac{k_{w} + \frac{ρ^{2} I}{KG} ω^{4} - (\frac{ρ I k_{W}}{KGA} + m) ω^{2}}{EI (1 + \frac{k_{p}}{KGA})} W (ξ) = 0

(8)

3. Truncated version of the Timoshenko–Ehrenfest equation

The truncated version of the original Timoshenko–Ehrenfest equation that we discuss was introduced by Elishakoff [28] in 2010 and later on by Elishakoff et al. [29] (see also Elishakoff and Livshits [30]). The formula has been obtained by replacing Equation (5) with

EI \frac{\partial^{2} ψ}{\partial x^{2}} + KGA (\frac{\partial w}{\partial x} - ψ) - ρ I (\frac{\partial^{3} w}{\partial x \partial t^{2}}) = 0

(9)

which takes into account the rotary inertia effect. Making use of Equations (4) and (9) to eliminate ψ and introduction of the non-dimensional coordinate brings the next equation

\frac{d^{4} W (ξ)}{d ξ^{4}} + L^{2} \frac{ρ I ω^{2} (1 + \frac{E}{KG}) - k_{p} - \frac{EI k_{W}}{KGA}}{EI (1 + \frac{k_{p}}{KGA})} \frac{d^{2} W (ξ)}{d ξ^{2}} - L^{4} \frac{m ω^{2} - k_{w}}{EI (1 + \frac{k_{p}}{KGA})} W (ξ) = 0

(10)

which differs from the original Timoshenko–Ehrenfest by the lack of some terms. As a consequence, we can introduce three artificial parameters, named $λ$ , $γ$ , and $ϱ$ , that govern the transition from the original Timoshenko–Ehrenfest equation to its truncated version. They can be either equal to one or zero at the same time. Thus, we obtain

\frac{d^{4} W (ξ)}{d ξ^{4}} + L^{2} \frac{ρ I ω^{2} (1 + \frac{E}{KG} + λ \frac{k_{p}}{KGA}) - k_{p} - \frac{EI k_{W}}{KGA}}{EI (1 + \frac{k_{p}}{KGA})} \frac{d^{2} W (ξ)}{d ξ^{2}} + L^{4} \frac{k_{w} + ϱ \frac{ρ^{2} I}{KG} ω^{4} - (γ \frac{ρ I k_{W}}{KGA} + m) ω^{2}}{EI (1 + \frac{k_{p}}{KGA})} W (ξ) = 0

(11)

4. Derivation of the slope inertia version of the Timoshenko–Ehrenfest equation

Another version of the original Timoshenko–Ehrenfest that we discuss considers the slope inertia effect and was introduced by Elishakoff et al. [31] in 2015. This theory is based on the application of Hamilton’s principle, which reads

\int_{t_{1}}^{t_{2}} δ (T - V_{s} - V_{b}) = 0

(12)

to obtain the differential equation of motion of a general beam where T is the kinetic energy related to a beam on a Winkler–Pasternak foundation

T = \frac{1}{2} \int_{0}^{L} ρ A {(\frac{\partial w}{\partial t})}^{2} d x + \frac{1}{2} \int_{0}^{L} ρ I {(\frac{\partial w}{\partial x \partial t})}^{2} d x + \frac{1}{2} \int_{0}^{L} k_{w} w^{2} d x + \frac{1}{2} \int_{0}^{L} k_{p} {(\frac{\partial w}{\partial x})}^{2} d x

(13)

and $V_{s}$ and $V_{b}$ are respectively the strain energy due to the shear effect and bending moment

V_{s} = \frac{1}{2} \int_{0}^{L} K G A {(\frac{\partial v}{\partial x} - ψ)}^{2} d x

(14)

V_{b} = \frac{1}{2} \int_{0}^{L} E I {(\frac{\partial ψ}{\partial x})}^{2} d x

(15)

Application of Hamilton’s principle brings the following equation

\begin{matrix} EI (1 + \frac{k_{p}}{KGA}) \frac{\partial^{4} w (x, t)}{\partial x^{4}} - ρ I (1 + \frac{E}{KG}) \frac{\partial^{4} w (x, t)}{\partial x^{2} \partial t^{2}} - (k_{p} + \frac{EI k_{W}}{KGA}) \frac{\partial^{2} w (x, t)}{\partial x^{2}} \\ + m \frac{\partial^{2} w (x, t)}{\partial t^{2}} + \frac{ρ E I^{2}}{KGA} \frac{\partial^{6} w (x, t)}{\partial x^{4} \partial t^{2}} + k_{W} w (x, t) = 0 \end{matrix}

(16)

which contains a sixth-order derivative for the displacement $w (x, t)$ absent in other formulas. As before, Equation (14) can be reduced to

\frac{d^{4} W (ξ)}{d ξ^{4}} + L^{2} \frac{ρ I ω^{2} (1 + \frac{E}{KG}) - k_{p} - \frac{EI k_{W}}{KGA}}{EI (1 + \frac{k_{p}}{KGA} - \frac{ρ I}{KGA} ω^{2})} \frac{d^{2} W (ξ)}{d ξ^{2}} - L^{4} \frac{m ω^{2} - k_{w}}{EI (1 + \frac{k_{p}}{KGA} - \frac{ρ I}{KGA} ω^{2})} W (ξ) = 0

(17)

5. Unified formulation of the problem

The three differential equations can be re-written in a unified manner as

\frac{d^{4} W (ξ)}{d ξ^{4}} + β_{n} \frac{d^{2} W (ξ)}{d ξ^{2}} + α_{n} W (ξ) = 0

(18)

where $n = 1$ refers to the original Timoshenko–Ehrenfest equation, $n = 2$ to the truncated version, and $n = 3$ to the model based on slope inertia.

We remark here that same results can be obtained by reducing the equation of motion for a beam in terms of cross-section rotation $ψ (x, t)$ instead of in terms of transversal displacement $w (x, t)$ .

We set $n = 1$ to deal with the original Timoshenko–Ehrenfest equation. Thus, parameters $α_{n}$ and $β_{n}$ become

α_{1} = L^{4} \frac{N_{α 1}}{EI (1 + \frac{k_{p}}{KGA})}

(17)

β_{1} = L^{2} \frac{N_{β 1}}{EI (1 + \frac{k_{p}}{KGA})}

(18)

Where

N_{α_{1}} = k_{w} + \frac{ρ^{2} I}{KG} ω^{4} - (\frac{ρ I k_{W}}{KGA} + m) ω^{2}

(19)

N_{β_{1}} = ρ I ω^{2} (1 + \frac{E}{KG} + \frac{k_{p}}{KGA}) - k_{p} - \frac{EI k_{W}}{KGA}

(20)

Coefficients $α_{1}$ and $β_{1}$ may be positive, null, or negative. The denominators of both parameters are positive. Thus, we set $N_{α_{1}} = 0$ , obtaining

ω_{A}^{2} = \frac{k_{w}}{m}

(21)

ω_{B}^{2} = \frac{KGA}{{mr}_{g}^{2}}

(22)

where $r_{g}$ is the radius of gyration of the beam cross-section, which is $r_{g} = \sqrt{I / A}$ . Note that frequency $ω_{B}^{2}$ appeared for the first time in the literature in Downs [32]. This frequency is referred as the ‘transition frequency’ by Cazzani et al. [33,34].

Thus, $α_{1}$ is negative for ω² belonging to $[ω_{A}^{2}, ω_{B}^{2}]$ , it is positive for ω² outside the interval $[ω_{A}^{2}, ω_{B}^{2}]$ , and it is zero when either $ω^{2} = ω_{A}^{2}$ or $ω^{2} = ω_{B}^{2}$ . Then we set $N_{β_{1}} = 0$ , obtaining

ω_{C}^{2} = \frac{k_{p} + \frac{EI k_{W}}{KGA}}{ρ I (1 + \frac{E}{KG} + \frac{k_{p}}{KGA})}

(23)

Thus, $β_{1}$ is positive for $ω^{2} > ω_{C}^{2}$ , it is negative for $ω^{2} < ω_{C}^{2}$ , and it is zero when $ω^{2} = ω_{C}^{2}$ .

We set $n = 2$ to deal with the truncated version of the Timoshenko–Ehrenfest equation. Thus, parameters $α_{n}$ and $β_{n}$ become

α_{2} = L^{4} \frac{N_{α_{2}}}{EI (1 + \frac{k_{p}}{KGA})}

(24)

β_{2} = L^{2} \frac{N_{β_{2}}}{EI (1 + \frac{k_{p}}{KGA})}

(25)

where

N_{α_{2}} = k_{w} - m ω^{2}

(26)

N_{β_{2}} = ρ I ω^{2} (1 + \frac{E}{KG}) - k_{p} - \frac{EI k_{W}}{KGA}

(27)

We study the sign of coefficients $α_{2}$ and $β_{2}$ . We observe that the denominator of both parameters is positive. Thus, we set $N_{α_{2}} = 0$ , obtaining the root in Equation (21). As a consequence, $α_{2}$ is positive for $ω^{2} < ω_{A}^{2}$ , it is negative for $ω^{2} > ω_{A}^{2}$ , and it is zero when $ω^{2} = ω_{A}^{2}$ . Then we set $N_{β_{2}} = 0$ , obtaining

ω_{D}^{2} = \frac{k_{p} + \frac{EI k_{W}}{KGA}}{ρ I (1 + \frac{E}{KG})}

(28)

Thus, $β_{2}$ is positive for $ω^{2} > ω_{D}^{2}$ , it is negative for $ω^{2} < ω_{D}^{2}$ , and it is null for $ω^{2} = ω_{D}^{2}$ .

We finally deal with the model based on slope inertia by setting $n = 3$ . Thus, parameters $α_{n}$ and $β_{n}$ become

α_{3} = L^{4} \frac{N_{α 3}}{EI (1 + \frac{k_{p}}{KGA} - \frac{ρ I}{KGA} ω^{2})}

(29)

β_{3} = L^{2} \frac{N_{β 3}}{EI (1 + \frac{k_{p}}{KGA} - \frac{ρ I}{KGA} ω^{2})}

(30)

where

N_{α_{3}} = k_{w} - m ω^{2}

(31)

N_{β_{3}} = ρ I ω^{2} (1 + \frac{E}{KG}) - k_{p} - \frac{EI k_{W}}{KGA}

(32)

Once again, we study the sign of parameters $α_{3}$ and $β_{3}$ . In this case the sign of the denominators depends on $ω$ . Thus, we set its expression equal to zero, obtaining the root

ω_{E}^{2} = \frac{1 + \frac{k_{p}}{KGA}}{\frac{ρ I}{KGA}} = \frac{KGA}{{mr}_{g}^{2}} + \frac{k_{p}}{ρ I}

(33)

As a consequence, the denominator is positive for $ω^{2} < ω_{E}^{2}$ and it is negative for $ω^{2} > ω_{E}^{2}$ . Obviously, $ω^{2}$ cannot be equal to $ω_{E}^{2}$ for mathematical reasons. We then set $N_{α_{3}} = 0$ , obtaining the root in Equation (21). Thus, $N_{α_{3}}$ is positive for $ω^{2} < ω_{A}^{2}$ , it is negative for $ω^{2} > ω_{A}^{2}$ , and it is zero when $ω^{2} = ω_{A}^{2}$ . Finally, we set $N_{β_{3}} = 0$ . We obtain the root in Equation (28). Thus, it is positive for $ω^{2} > ω_{D}^{2}$ , it is negative for $ω^{2} < ω_{D}^{2}$ , and it is null for $ω^{2} = ω_{D}^{2}$ .

Here we present a comparison between the found values of the natural frequencies. With the help of some simple mathematical calculus, we prove that the following relationships hold

ω_{C}^{2} = \frac{k_{p} + \frac{EI k_{W}}{KGA}}{ρ I (1 + \frac{E}{KG} + \frac{k_{p}}{KGA})} < \frac{k_{p} + \frac{EI k_{W}}{KGA}}{ρ I (1 + \frac{E}{KG})} = ω_{D}^{2}

(34)

ω_{B}^{2} = \frac{KGA}{{mr}_{g}^{2}} < \frac{KGA}{{mr}_{g}^{2}} + \frac{k_{p}}{ρ I} = ω_{E}^{2}

(35)

Hereinafter we assume that the modulus of the elastic foundations will be varying; however, in the first case we consider that $k_{w} = k_{p} = 5$ . In these circumstances, $ω_{A}^{2} < ω_{C}^{2} < ω_{D}^{2} < ω_{B}^{2} < ω_{E}^{2}$ . Note that this relationship is true for all $k_{w} = k_{p} < 4 . 10^{9}$ .

Also, by varying $k_{w}$ and $k_{p}$ we can obtain different relationships between $ω_{A}^{2}$ , $ω_{B}^{2}$ , $ω_{C}^{2}$ , $ω_{D}^{2}$ , and $ω_{E}^{2}$ . Indeed, the following cases can be distinguished:

for $4 . 10^{9} < k_{w} = k_{p} < 1 . 10^{12}$ we obtain $ω_{A}^{2} < ω_{C}^{2} < ω_{B}^{2} < ω_{D}^{2} < ω_{E}^{2}$ ;

for $1 . 10^{12} < k_{w} = k_{p} < 2 . 10^{12}$ we obtain $ω_{C}^{2} < ω_{A}^{2} < ω_{B}^{2} < ω_{D}^{2} < ω_{E}^{2}$

for $k_{w} = k_{p} > 2 . 10^{12}$ we obtain $ω_{B}^{2} < ω_{C}^{2} < ω_{A}^{2} < ω_{D}^{2} < ω_{E}^{2}$ .

6. Solution of the equation of motion

The characteristic equation associated to Equation (16) reads

r^{4} + β_{n} r^{2} + α_{n} = 0

(36)

the solutions to which are

r_{I}^{2} = \frac{- β_{n} + \sqrt{β_{n}^{2} - 4 α_{n}}}{2}; r_{II}^{2} = \frac{- β_{n} - \sqrt{β_{n}^{2} - 4 α_{n}}}{2}

(37)

The nature of these solutions has been investigated by following the next procedure (Figure 1):

identify the sign of the discriminant $Δ = β_{n}^{2} - 4 α_{n}$ ;

verify whether $r_{I}^{2}$ is positive, zero, or negative;

verify whether $r_{II}^{2}$ is positive, zero, or negative.

6.1. Original Timoshenko–Ehrenfest equation

We re-write Equation (36) as

r^{4} + β_{1} r^{2} + α_{1} = 0

(38)

for which the solutions are

r_{I}^{2} = \frac{- β_{1} + \sqrt{β_{1}^{2} - 4 α_{1}}}{2}; r_{II}^{2} = \frac{- β_{1} - \sqrt{β_{1}^{2} - 4 α_{1}}}{2}

(39)

where $r_{I}^{2} and r_{II}^{2}$ depend on the yet unknown $ω$ . Thus, we distinguish the following cases.

Case 1. $ω^{2} < ω_{A}^{2} : α_{1} > 0 and β_{1} < 0$ .

Possibilities and solutions for roots in case $ω^{2} < ω_{A}^{2} are discussed in$ Table 1.

Case 2. $ω^{2} = ω_{A}^{2} : α_{1} = 0 and β_{1} < 0$ .

Table 1.

Possibilities and solutions for roots in case $ω^{2} < ω_{A}^{2}$ .

Sequence of possibilities	Possibilities	Solution for roots
1	$β_{1}^{2} - 4 α_{1} > 0$ ; $- β_{1} - \sqrt{β_{1}^{2} - 4 α_{1}} > 0$	$r_{1, 2} = \pm r_{I}; r_{3, 4} = \pm r_{II}$
2	$β_{1}^{2} - 4 α_{1} > 0$ ; $- β_{1} - \sqrt{β_{1}^{2} - 4 α_{1}} < 0$	$r_{1, 2} = \pm r_{I}; r_{3, 4} = \pm i r_{II}$
3	$β_{1}^{2} - 4 α_{1} < 0$	$r_{1, 2} = \pm i r_{I}; r_{3, 4} = \pm i r_{II}$
4	$β_{1}^{2} - 4 α_{1} = 0$	$r_{1, 2} = \pm r_{I} = r_{3, 4} = \pm r_{II}$

The solution consists of a couple of real roots and a null real root

r_{1, 2} = \pm r_{I}; r_{3, 4} = 0

Case 3. $ω_{A}^{2} < ω^{2} < ω_{C}^{2} : α_{1} < 0 and β_{1} < 0$ .

The solution consists of a couple of real roots and a couple of imaginary conjugated roots

r_{1, 2} = \pm r_{I}; r_{3, 4} = \pm i r_{II}

Case 4. $ω^{2} = ω_{C}^{2} : α_{1} < 0 and β_{1} = 0$ .

The solution consists of a couple of real roots and a couple of imaginary conjugated roots

r_{1, 2} = \pm r_{I}; r_{3, 4} = \pm i r_{II}

Case 5. $ω_{C}^{2} < ω^{2} < ω_{B}^{2} : α_{1} < 0 and β_{1} > 0$ .

The solution consists of a couple of real roots and a couple of imaginary conjugated roots

r_{1, 2} = \pm r_{I}; r_{3, 4} = \pm i r_{II}

Case 6. $ω^{2} = ω_{B}^{2} : α_{1} = 0 and β_{1} > 0$ .

The solution consists of a null real root and a couple of imaginary conjugated roots

r_{1, 2} = 0; r_{3, 4} = \pm i r_{II}

Case 7. $ω^{2} > ω_{B}^{2} : α_{1} > 0 and β_{1} > 0$ .

The possibilities and solutions for the roots in case $ω^{2} > ω_{B}^{2}$ are shown in Table 2.

Table 2.

Possibilities and solutions for roots in case $ω^{2} > ω_{B}^{2}$ .

Sequence of possibilities	Possibilities	Solution for roots
1	$β_{1}^{2} - 4 α_{1} > 0$ ; $- β_{1} + \sqrt{β_{1}^{2} - 4 α_{1}} > 0$	$r_{1, 2} = \pm r_{I}; r_{3, 4} = \pm i r_{II}$
2	$β_{1}^{2} - 4 α_{1} > 0$ ; $- β_{1} + \sqrt{β_{1}^{2} - 4 α_{1}} < 0$	$r_{1, 2} = \pm i r_{I}; r_{3, 4} = \pm i r_{II}$
3	$β_{1}^{2} - 4 α_{1} < 0$	$r_{1, 2} = \pm i r_{I}; r_{3, 4} = \pm i r_{II}$
4	$β_{1}^{2} - 4 α_{1} = 0$	$r_{1, 2} = \pm i r_{I} = r_{3, 4} = \pm i r_{II}$

6.2. Elishakoff’s truncated version

We re-write Equation (36) as

r^{4} + β_{2} r^{2} + α_{2} = 0

(40)

for which the solutions are

r_{I}^{2} = \frac{- β_{2} + \sqrt{β_{2}^{2} - 4 α_{2}}}{2}; r_{II}^{2} = \frac{- β_{2} - \sqrt{β_{2}^{2} - 4 α_{2}}}{2}

(41)

where $r_{I}^{2} and r_{II}^{2}$ depend on the yet unknown $ω$ . Thus, we distinguish the following cases.

Case 1. $ω^{2} < ω_{A}^{2} : α_{2} > 0 and β_{2} < 0$ .

The possibilities and solutions for the roots in case $ω^{2} < ω_{A}^{2}$ are demonstrated in Table 3.

Case 2. $ω^{2} = ω_{A}^{2} : α_{2} = 0 and β_{2} < 0$ .

Table 3.

Possibilities and solutions for roots in case $ω^{2} < ω_{A}^{2}$ .

Sequence of possibilities	Possibilities	Solution for roots
1	$β_{2}^{2} - 4 α_{2} > 0$ ; $- β_{2} - \sqrt{β_{2}^{2} - 4 α_{2}} > 0$	$r_{1, 2} = \pm r_{I}; r_{3, 4} = \pm r_{II}$
2	$β_{2}^{2} - 4 α_{2} > 0$ ; $- β_{2} - \sqrt{β_{2}^{2} - 4 α_{2}} < 0$	$r_{1, 2} = \pm r_{I}; r_{3, 4} = \pm i r_{II}$
3	$β_{2}^{2} - 4 α_{2} < 0$	$r_{1, 2} = \pm i r_{I}; r_{3, 4} = \pm i r_{II}$
4	$β_{2}^{2} - 4 α_{2} = 0$	$r_{1, 2} = \pm r_{I} = r_{3, 4} = \pm r_{II}$

The solution consists of a couple of real roots and a null real root

r_{1, 2} = \pm r_{I}; r_{3, 4} = 0

Case 3. $ω_{A}^{2} < ω^{2} < ω_{D}^{2} : α_{2} < 0 and β_{2} < 0$ .

The solution consists of a couple of real roots and a couple of imaginary conjugated roots

r_{1, 2} = \pm r_{I}; r_{3, 4} = \pm i r_{II}

Case 4. $ω^{2} = ω_{D}^{2} : α_{2} < 0 and β_{2} = 0$ .

The solution consists of a couple of real roots and a couple of imaginary conjugated roots

r_{1, 2} = \pm r_{I}; r_{3, 4} = \pm i r_{II}

Case 5. $ω^{2} > ω_{D}^{2} : α_{2} < 0 and β_{2} > 0$ .

The solution consists of a couple of real roots and a couple of imaginary conjugated roots

r_{1, 2} = \pm r_{I}; r_{3, 4} = \pm i r_{II}

6.3. Slope inertia version

We re-write Equation (36) as

r^{4} + β_{3} r^{2} + α_{3} = 0

(42)

for which the solutions are

r_{I}^{2} = \frac{- β_{3} + \sqrt{β_{3}^{2} - 4 α_{3}}}{2}; r_{II}^{2} = \frac{- β_{3} - \sqrt{β_{3}^{2} - 4 α_{3}}}{2}

(43)

where $r_{I}^{2} and r_{II}^{2}$ depend on the yet unknown $ω$ . Thus, we distinguish the following cases.

Case 1. $ω^{2} < ω_{A}^{2} : α_{3} > 0 and β_{3} < 0$ .

The possibilities and solutions for the roots in case $ω^{2} < ω_{A}^{2}$ are discussed in Table 4.

Case 2. $ω^{2} = ω_{A}^{2} : α_{3} = 0 and β_{3} < 0$ .

Table 4.

Possibilities and solutions for roots in case $ω^{2} < ω_{A}^{2}$ .

Sequence of possibilities	Possibilities	Solution for roots
1	$β_{3}^{2} - 4 α_{3} > 0$ ; $- β_{3} - \sqrt{β_{3}^{2} - 4 α_{3}} > 0$	$r_{1, 2} = \pm r_{I}; r_{3, 4} = \pm r_{II}$
2	$β_{3}^{2} - 4 α_{3} > 0$ ; $- β_{3} - \sqrt{β_{3}^{2} - 4 α_{3}} < 0$	$r_{1, 2} = \pm r_{I}; r_{3, 4} = \pm i r_{II}$
3	$β_{3}^{2} - 4 α_{3} < 0$	$r_{1, 2} = \pm i r_{I}; r_{3, 4} = \pm i r_{II}$
4	$β_{3}^{2} - 4 α_{3} = 0$	$r_{1, 2} = \pm r_{I} = r_{3, 4} = \pm r_{II}$

The solution consists of a couple of real roots and a null real root

r_{1, 2} = \pm r_{I}; r_{3, 4} = 0

Case 3. $ω_{A}^{2} < ω^{2} < ω_{D}^{2} : α_{3} < 0 and β_{3} < 0$ .

The solution consists of a couple of real roots and a couple of imaginary conjugated roots

r_{1, 2} = \pm r_{I}; r_{3, 4} = \pm i r_{II}

Case 4. $ω^{2} = ω_{D}^{2} : α_{3} < 0 and β_{3} = 0$ .

The solution consists of a couple of real roots and a couple of imaginary conjugated roots

r_{1, 2} = \pm r_{I}; r_{3, 4} = \pm i r_{II}

Case 5. $ω_{D}^{2} < ω^{2} < ω_{E}^{2} : α_{3} < 0 and β_{3} > 0$ .

The solution consists of a couple of real roots and a couple of imaginary conjugated roots

r_{1, 2} = \pm r_{I}; r_{3, 4} = \pm i r_{II}

Case 6. $ω^{2} > ω_{E}^{2} : α_{3} > 0 and β_{3} < 0$ .

The possibilities and solutions for the roots in case $ω^{2} \leq ω_{C}^{2}$ are discussed in Table 5.

Table 5.

Possibilities and solutions for roots in case $ω^{2} \leq ω_{C}^{2}$ .

Sequence of possibilities	Possibilities	Solution for roots
1	$β_{3}^{2} - 4 α_{3} > 0$ ; $- β_{3} - \sqrt{β_{3}^{2} - 4 α_{3}} > 0$	$r_{1, 2} = \pm r_{I}; r_{3, 4} = \pm r_{II}$
2	$β_{3}^{2} - 4 α_{3} > 0$ ; $- β_{3} - \sqrt{β_{3}^{2} - 4 α_{3}} < 0$	$r_{1, 2} = \pm r_{I}; r_{3, 4} = \pm i r_{II}$
3	$β_{3}^{2} - 4 α_{3} < 0$	$r_{1, 2} = \pm i r_{I}; r_{3, 4} = \pm i r_{II}$
4	$β_{3}^{2} - 4 α_{3} = 0$	$r_{1, 2} = \pm r_{I} = r_{3, 4} = \pm r_{II}$

7. Vibration mode of the Timoshenko–Ehrenfest beam

We write the solutions for the three differential equations for $w (x, t)$ and $ψ (x, t)$ as

W (ξ) = C_{1} e^{r_{1} ξ} + C_{2} e^{r_{2} ξ} + C_{3} e^{r_{3} ξ} + C_{4} e^{r_{4} ξ}

(44)

Ψ (ξ) = D_{1} e^{r_{1} ξ} + D_{2} e^{r_{2} ξ} + D_{3} e^{r_{3} ξ} + D_{4} e^{r_{4} ξ}

(45)

According to the analysis, only the following forms of mode shapes appear by varying ω:

Form 1 of mode shape: four real solutions;

Form 2 of mode shape: two real solutions and two purely imaginary conjugated roots;

Form 3 of mode shape: four purely imaginary conjugate roots;

Form 4 of mode shape: two imaginary conjugate solutions and a null real root;

Form 5 of mode shape: two real solutions and a null real root.

Hereinafter, we present results separately for each form. Note that $Ψ (ξ)$ and $W (ξ)$ are interconnected by the following relationship

\frac{\partial Ψ}{\partial x} = (1 + \frac{k_{p}}{KGA}) \frac{\partial^{2} W}{\partial x^{2}} + (\frac{ρ}{KG} ω^{2} - \frac{k_{w}}{KGA}) W

(46)

Thus, coefficients {D₁,…,D₄} and {C₁,…,C₄} are interdependent.

7.1. Form 1 of the mode shape

We re-write Equations (44) and (45) as

W (ξ) = C_{1} \cosh (r_{1} ξ) + C_{2} \sinh (r_{2} ξ) + C_{3} \cosh (r_{3} ξ) + C_{4} \sinh (r_{4} ξ)

(47)

Ψ (ξ) = D_{1} \cosh (r_{1} ξ) + D_{2} \sinh (r_{2} ξ) + D_{3} \cosh (r_{3} ξ) + D_{4} \sinh (r_{4} ξ)

(48)

The relationships between the above parameters are

D_{1} = - \frac{1}{r_{1}} (r_{1}^{2} + \frac{ρ}{KG} ω^{2} + \frac{k_{p}}{KGA} r_{1}^{2} - \frac{k_{w}}{KGA}) C_{2}

(49)

D_{2} = - \frac{1}{r_{1}} (r_{1}^{2} + \frac{ρ}{KG} ω^{2} + \frac{k_{p}}{KGA} r_{1}^{2} - \frac{k_{w}}{KGA}) C_{1}

(50)

D_{3} = - \frac{1}{r_{3}} (r_{3}^{2} + \frac{ρ}{KG} ω^{2} + \frac{k_{p}}{KGA} r_{3}^{2} - \frac{k_{w}}{KGA}) C_{4}

(51)

D_{4} = - \frac{1}{r_{3}} (r_{3}^{2} + \frac{ρ}{KG} ω^{2} + \frac{k_{p}}{KGA} r_{3}^{2} - \frac{k_{w}}{KGA}) C_{3}

(52)

hence

Ψ (ξ) = - \frac{a_{1}}{r_{1}} [C_{2} \cosh (r_{1} ξ) - C_{1} \sinh (r_{1} ξ)] - \frac{b_{1}}{r_{3}} [C_{4} \cosh (r_{3} ξ) - C_{3} \sinh (r_{3} ξ)]

(53)

where

a_{1} = r_{1}^{2} + \frac{ρ}{KG} ω^{2} + \frac{k_{p}}{KGA} r_{1}^{2} - \frac{k_{w}}{KGA}

(54)

b_{1} = r_{3}^{2} + \frac{ρ}{KG} ω^{2} + \frac{k_{p}}{KGA} r_{3}^{2} - \frac{k_{w}}{KGA}

(55)

7.2. Form 2 of the mode shape

We re-write Equations (44) and (45) as

W (ξ) = C_{1} \cosh (r_{1} ξ) + C_{2} \sinh (r_{2} ξ) + C_{3} \cos (r_{3} ξ) + C_{4} \sin (r_{4} ξ)

(56)

Ψ (ξ) = D_{1} \cosh (r_{1} ξ) + D_{2} \sinh (r_{2} ξ) + D_{3} \cos (r_{3} ξ) + D_{4} \sin (r_{4} ξ)

(57)

The relationships between the above parameters are

D_{1} = - \frac{1}{r_{1}} (r_{1}^{2} + \frac{ρ}{KG} ω^{2} + \frac{k_{p}}{KGA} r_{1}^{2} - \frac{k_{w}}{KGA}) C_{2}

(58)

D_{2} = - \frac{1}{r_{1}} (r_{1}^{2} + \frac{ρ}{KG} ω^{2} + \frac{k_{p}}{KGA} r_{1}^{2} - \frac{k_{w}}{KGA}) C_{1}

(59)

D_{3} = \frac{1}{r_{3}} (- r_{3}^{2} + \frac{ρ}{KG} ω^{2} - \frac{k_{p}}{KGA} r_{3}^{2} - \frac{k_{w}}{KGA}) C_{4}

(60)

D_{4} = - \frac{1}{r_{3}} (- r_{3}^{2} + \frac{ρ}{KG} ω^{2} - \frac{k_{p}}{KGA} r_{3}^{2} - \frac{k_{w}}{KGA}) C_{3}

(61)

hence

Ψ (ξ) = - \frac{a_{2}}{r_{1}} [C_{2} \cosh (r_{1} ξ) - C_{1} \sinh (r_{1} ξ)] + \frac{b_{2}}{r_{3}} [C_{4} \cos (r_{3} ξ) + C_{3} \sin (r_{3} ξ)]

(62)

where

a_{2} = r_{1}^{2} + \frac{ρ}{KGA} ω^{2} + \frac{k_{p}}{KGA} r_{1}^{2} - \frac{k_{w}}{KGA}

(63)

b_{2} = - r_{3}^{2} + \frac{ρ}{KGA} ω^{2} - \frac{k_{p}}{KGA} r_{3}^{2} - \frac{k_{w}}{KGA}

(64)

7.3. Form 3 of the mode shape

We re-write Equations (44) and (45) as

W (ξ) = C_{1} \cos (r_{1} ξ) + C_{2} \sin (r_{2} ξ) + C_{3} \cos (r_{3} ξ) + C_{4} \sin (r_{4} ξ)

(65)

Ψ (ξ) = D_{1} \cos (r_{1} ξ) + D_{2} \sin (r_{2} ξ) + D_{3} \cos (r_{3} ξ) + D_{4} \sin (r_{4} ξ)

(66)

The relationships between the above parameters are

D_{1} = \frac{1}{r_{1}} (- r_{1}^{2} + \frac{ρ}{KG} ω^{2} - \frac{k_{p}}{KGA} r_{1}^{2} - \frac{k_{w}}{KGA}) C_{2}

(67)

D_{2} = - \frac{1}{r_{1}} (- r_{1}^{2} + \frac{ρ}{KG} ω^{2} - \frac{k_{p}}{KGA} r_{1}^{2} - \frac{k_{w}}{KGA}) C_{1}

(68)

D_{3} = \frac{1}{r_{3}} (- r_{3}^{2} + \frac{ρ}{KG} ω^{2} - \frac{k_{p}}{KGA} r_{3}^{2} - \frac{k_{w}}{KGA}) C_{4}

(69)

D_{4} = - \frac{1}{r_{3}} (- r_{3}^{2} + \frac{ρ}{KG} ω^{2} - \frac{k_{p}}{KGA} r_{3}^{2} - \frac{k_{w}}{KGA}) C_{3}

(70)

hence

Ψ (ξ) = \frac{a_{3}}{r_{1}} [C_{2} \cos (r_{1} ξ) + C_{1} \sin (r_{1} ξ)] + \frac{b_{3}}{r_{3}} [C_{4} \cos (r_{3} ξ) + C_{3} \sin (r_{3} ξ)]

(71)

where

a_{3} = - r_{1}^{2} + \frac{ρ}{KG} ω^{2} - \frac{k_{p}}{KGA} r_{1}^{2} - \frac{k_{w}}{KGA}

(72)

b_{3} = - r_{3}^{2} + \frac{ρ}{KG} ω^{2} - \frac{k_{p}}{KGA} r_{3}^{2} - \frac{k_{w}}{KGA}

(73)

7.4. Form 4 of the mode shape

We re-write Equations (44) and (45) as

W (ξ) = C_{1} + C_{2} ξ + C_{3} \cos (r_{3} ξ) + C_{4} \sin (r_{4} ξ)

(74)

Ψ (ξ) = D_{1} + D_{2} ξ + D_{3} \cos (r_{3} ξ) + D_{4} \sin (r_{4} ξ)

(75)

The relationships between the above parameters are

C_{2} = 0

(76)

D_{2} = (\frac{ρ}{KG} ω^{2} - \frac{k_{w}}{KGA}) C_{1}

(77)

D_{3} = \frac{1}{r_{3}} (- r_{3}^{2} + \frac{ρ}{KG} ω^{2} - \frac{k_{p}}{KGA} r_{3}^{2} - \frac{k_{w}}{KGA}) C_{4}

(78)

D_{4} = - \frac{1}{r_{3}} (- r_{3}^{2} + \frac{ρ}{KG} ω^{2} - \frac{k_{p}}{KGA} r_{3}^{2} - \frac{k_{w}}{KGA}) C_{3}

(79)

hence

W (ξ) = C_{1} + C_{3} \cos (r_{3} ξ) - C_{4} \sin (r_{3} ξ)

(80)

Ψ (ξ) = D_{1} + b_{4} C_{1} ξ + \frac{a_{4}}{r_{3}} [C_{4} \cos (r_{3} ξ) + C_{3} \sin (r_{3} ξ)]

(81)

where

a_{4} = - r_{3}^{2} + \frac{ρ}{KG} ω^{2} - \frac{k_{p}}{KGA} r_{3}^{2} - \frac{k_{w}}{KGA}

(82)

b_{4} = \frac{ρ}{KG} ω^{2} - \frac{k_{w}}{KGA}

(83)

7.5. Form 5 of the mode shape

We re-write Equations (44) and (45) as

W (ξ) = C_{1} \cosh (r_{1} ξ) + C_{2} \sinh (r_{2} ξ) + C_{3} + C_{4} ξ

(84)

Ψ (ξ) = D_{1} \cosh (r_{1} ξ) + D_{2} \sinh (r_{2} ξ) + D_{3} + D_{4} ξ

(85)

The relationships between the above parameters are

D_{1} = - \frac{1}{r_{1}} (r_{1}^{2} + \frac{ρ}{KG} ω^{2} + \frac{k_{p}}{KGA} r_{1}^{2} - \frac{k_{w}}{KGA}) C_{2}

(86)

D_{2} = - \frac{1}{r_{1}} (r_{1}^{2} + \frac{ρ}{KG} ω^{2} + \frac{k_{p}}{KGA} r_{1}^{2} - \frac{k_{w}}{KGA}) C_{1}

(87)

C_{4} = 0

(88)

D_{4} = (\frac{ρ}{KG} ω^{2} - \frac{k_{w}}{KGA}) C_{3}

(89)

hence

W (ξ) = C_{1} \cosh (r_{1} ξ) - C_{2} \sinh (r_{1} ξ) + C_{3}

(90)

Ψ (ξ) = - \frac{a_{5}}{r_{1}} [C_{2} \cosh (r_{1} ξ) - C_{1} \sinh (r_{1} ξ)] + D_{3} + b_{5} C_{3} ξ

(91)

where

a_{5} = r_{1}^{2} + \frac{ρ}{KG} ω^{2} + \frac{k_{p}}{KGA} r_{1}^{2} - \frac{k_{w}}{KGA}

(92)

b_{5} = \frac{ρ}{KG} ω^{2} - \frac{k_{w}}{KGA}

(93)

8. Application of the three models for a beam on a Winkler–Pasternak foundation

The modal analysis of a beam on Winkler–Pasternak foundation has been developed by using the three analyzed models in order to depict the complete spectrum of the beam.

8.1. Material properties

We focus attention on a particular beam whose characteristics are in agreement with Cazzani et al. [33,34], except for the length of the beam, which has been set equal to $0.8 m$ . Thus, the chosen material properties are:

straight uniform and homogeneous beam whose length is $L = 0.8 m$ ;

square cross-sectional area with side length $b = 0.1 m$ ;

cross-sectional area $A = b^{2} = 0.01 m^{2}$ ;

area moment of inertia $I = b^{4} / 12 = 1 / 120, 000 m^{4}$ ;

material density $ρ = 8000 kg / m^{3}$ ;

Young’s modulus $E = 260 GPa = 260 10^{3} N / m m^{2}$ ;

Poisson’s ratio $ν = 0.3$ ;

shear modulus $G = 100 GPa = 100 10^{3} N / m m^{2}$ ;

shear correction factor $K = 5 / 6$ ;

modulus of the Winkler and Pasternak foundations $k_{w} = k_{p} = 5$ .

In these circumstances, the following results for the natural frequencies are obtained

ω_{A} = 0.25 \frac{rad}{s}; ω_{B} = 111803.39 \frac{rad}{s}; ω_{C} = 4.27 \frac{rad}{s}; ω_{D} = 4.27 \frac{rad}{s}; ω_{E} = 111803.39 \frac{rad}{s}

We observe that the difference between $ω_{B}$ and $ω_{E}$ and between $ω_{C}$ and $ω_{D}$ is null. Hence, in the following, we assume that $ω_{C} = ω_{D}$ and $ω_{B} = ω_{E}$ .

8.2. Case study 1: beam simply supported at both ends

The BCs for a simply supported beam are

W (0) = 0; M (0) = 0; W (1) = 0; M (1) = 0

(94)

where the homogeneous condition $M (ξ) = 0$ is equivalent to imposing $Ψ^{'} (ξ) = 0$ . When plugging the above BCs into the relative equations we obtain a system of simultaneous linear algebraic equations of the form

Δ C = 0

(95)

where Δ is a squared matrix composed of the coefficients in front of $C_{1}, \dots, C_{4}$ , while C and 0 are two column vectors.

8.2.1. Form 1 of the mode shape

We substitute the BCs in Equation (94) into the relative equations coming from (47) and (53), obtaining four different equations in four unknowns

C_{1} + C_{3} = 0

(96)

a_{1} C_{1} + b_{1} C_{3} = 0

(97)

C_{1} \cosh (r_{1}) - C_{2} \sinh (r_{1}) + C_{3} \cosh (r_{3}) - C_{4} \sinh (r_{3}) = 0

(98)

a_{1} C_{1} \cosh (r_{1}) - a_{1} C_{2} \sinh (r_{1}) + b_{1} C_{3} \cosh (r_{3}) - b_{1} C_{4} \sinh (r_{3}) = 0

(99)

We reduce the number of equations by comparing (96) and (97). Indeed, we obtain

- a_{1} C_{3} + b_{1} C_{3} = C_{3} (b_{1} - a_{1}) = 0

(100)

Since

b_{1} - a_{1} = (r_{3}^{2} - r_{1}^{2}) (1 + \frac{k_{p}}{KGA}) \neq 0

(101)

we say that $C_{1}$ and $C_{3}$ are zero. Hence, the reduced system of equations reads

[\begin{matrix} \sin (r_{1}) & \sin (r_{3}) \\ a_{1} \sin (r_{1}) & b_{1} {\sin (r_{3}) \end{matrix}] {\begin{matrix} C_{2} \\ C_{4} \end{matrix}} = {\begin{matrix} 0 \\ 0 \end{matrix}}

(102)

A non-trivial solution exists when the determinant of Δ is equal to zero

(a_{1} - b_{1}) \sinh (r_{1}) \sinh (r_{3}) = 0

(103)

Since $a_{1} - b_{1} \neq 0$ , the following transcendental equation must be satisfied

\sinh (r_{1}) \sinh (r_{3}) = 0

(104)

for which the solutions are

r_{1} = 0 or r_{3} = 0

(105)

8.2.2. Form 2 of the mode shape

We substitute the BCs in Equation (94) into the relative equations coming from (56) and (62), obtaining four different equations in four unknowns

C_{1} + C_{3} = 0

(106)

a_{2} C_{1} + b_{2} C_{3} = 0

(107)

C_{1} \cosh (r_{1}) - C_{2} \sinh (r_{1}) + C_{3} \cos (r_{3}) - C_{4} \sin (r_{3}) = 0

(108)

a_{2} C_{1} \cosh (r_{1}) - a_{2} C_{2} \sinh (r_{1}) + b_{2} C_{3} \cos (r_{3}) - b_{2} C_{4} \sin (r_{3}) = 0

(109)

We reduce the number of equations by comparing (106) and (107). Indeed, we obtain

- a_{2} C_{3} + b_{2} C_{3} = C_{3} (b_{2} - a_{2}) = 0

(110)

Since

b_{2} - a_{2} = - (r_{3}^{2} + r_{1}^{2}) (1 + \frac{k_{p}}{KGA}) \neq 0

(111)

we say that $C_{1}$ and $C_{3}$ are zero. Hence, the reduced system of equations reads

[\begin{matrix} \sinh (r_{1}) \\ a_{2} \sinh (r_{1}) \end{matrix} \begin{matrix} \sin (r_{3}) \\ b_{2} \sin (r_{3}) \end{matrix}] {\begin{matrix} C_{2} \\ C_{4} \end{matrix}} = {\begin{matrix} 0 \\ 0 \end{matrix}}

(112)

A non-trivial solution exists when the determinant of Δ is equal to zero

(a_{2} - b_{2}) \sinh (r_{1}) \sin (r_{3}) = 0

(113)

Since $a_{2} - b_{2} \neq 0$ , the following transcendental equation must be satisfied

\sinh (r_{1}) \sin (r_{3}) = 0

(114)

for which the solutions are

r_{1} = 0 or r_{3} = n π, (n \in N)

(115)

8.2.3. Form 3 of the mode shape

We substitute the BCs in Equation (94) into the relative equations coming from (65) and (71), obtaining four different equations in four unknowns

C_{1} + C_{3} = 0

(116)

a_{3} C_{1} + b_{3} C_{3} = 0

(117)

C_{1} \cos (r_{1}) - C_{2} \sin (r_{1}) + C_{3} \cos (r_{3}) - C_{4} \sin (r_{3}) = 0

(118)

a_{3} C_{1} \cos (r_{1}) - a_{3} C_{2} \sin (r_{1}) + b_{3} C_{3} \cos (r_{3}) - b_{3} C_{4} \sin (r_{3}) = 0

(119)

We reduce the number of equations by comparing (116) and (117). Indeed, we obtain

- a_{3} C_{3} + b_{3} C_{3} = C_{3} (b_{3} - a_{3}) = 0

(120)

Since

b_{3} - a_{3} = (r_{1}^{2} - r_{3}^{2}) (1 + \frac{k_{p}}{KGA}) \neq 0

(121)

we say that $C_{1}$ and $C_{3}$ are zero. Hence, the reduced system of equations reads

[\begin{matrix} \sin (r_{1}) \\ a_{3} \sin (r_{1}) \end{matrix} \begin{matrix} \sin (r_{3}) \\ b_{3} \sin (r_{3}) \end{matrix}] {\begin{matrix} C_{2} \\ C_{4} \end{matrix}} = {\begin{matrix} 0 \\ 0 \end{matrix}}

(122)

A non-trivial solution exists when the determinant of Δ is equal to zero

(a_{3} - b_{3}) \sin (r_{1}) \sin (r_{3}) = 0

(123)

Since $a_{3} - b_{3} \neq 0$ , the following trigonometric equation must be satisfied

\sin (r_{1}) \sin (r_{3}) = 0

(124)

for which the roots are

r_{1} = n π or r_{3} = n π, (n \in N)

(125)

8.2.4. Form 4 of the mode shape

We substitute the BCs in Equation (94) into the relative equations coming from (80) and (81), obtaining four different equations in four unknowns

C_{1} + C_{3} = 0

(126)

b_{4} C_{1} + a_{4} C_{3} = 0

(127)

C_{1} + C_{3} \cos (r_{3}) - C_{4} \sin (r_{3}) = 0

(128)

b_{4} C_{1} + a_{4} C_{3} \cos (r_{3}) - a_{4} C_{4} \sin (r_{3}) = 0

(129)

In this case we cannot reduce the system of equations. Hence, it reads

[\begin{matrix} 1 \\ b_{4} \\ 1 \\ b_{4} \end{matrix} \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \begin{matrix} 1 \\ a_{4} \\ \cos (r_{3}) \\ a_{4} \cos (r_{3}) \end{matrix} \begin{matrix} 0 \\ 0 \\ - \sin (r_{3}) \\ - a_{4} \sin (r_{3}) \end{matrix}] {\begin{matrix} C_{1} \\ D_{1} \\ C_{3} \\ C_{4} \end{matrix}} = {\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix}}

(130)

where matrix Δ never has $rank (Δ) > 3$ . As a consequence, the system of equations is defective. Therefore, if $rank (Δ) = 3$ , we obtain

(b_{4} - a_{4}) \sin (r_{3}) = r_{3}^{2} (1 + \frac{k_{p}}{KGA}) \sin (r_{3}) \neq 0

(131)

for which only the non-trivial solution is given by $C_{1} = 0$ , $C_{3} = 0$ , $C_{4} = 0$ , while $D_{1} \neq 0$ . Meanwhile, if $rank (Δ) < 3$ , we obtain

r_{3}^{2} (1 + \frac{k_{p}}{KGA}) \sin (r_{3}) = 0

(132)

for which the non-trivial solution is given by $C_{1} = 0$ and $C_{3} = 0$ , while $C_{4} \neq 0$ and $D_{1} \neq 0$ .

8.2.5. Form 5 of the mode shape

We substitute the BCs in Equation (94) into the relative equations coming from (80) and (81), obtaining four different equations in four unknowns

C_{1} + C_{3} = 0

(133)

a_{5} C_{1} + b_{5} C_{3} = 0

(134)

C_{1} \cosh (r_{1}) - C_{2} \sinh (r_{1}) + C_{3} = 0

(135)

a_{5} C_{1} \cosh (r_{1}) - a_{5} C_{2} \sinh (r_{3}) + b_{5} C_{3} = 0

(136)

Again, we cannot reduce the system of equations. Hence, it reads

[\begin{matrix} 1 \\ a_{5} \\ \cosh (r_{1}) \\ a_{5} \cosh (r_{1}) \end{matrix} \begin{matrix} 0 \\ 0 \\ - \sinh (r_{1}) \\ - a_{5} \sinh (r_{1}) \end{matrix} \begin{matrix} 1 \\ b_{5} \\ 1 \\ b_{5} \end{matrix} \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix}] {\begin{matrix} C_{1} \\ C_{2} \\ C_{3} \\ D_{3} \end{matrix}} = {\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix}}

(137)

where matrix Δ never has $rank (Δ) > 3$ . As a consequence, also this system of equations is defective. Therefore, if $rank (Δ) = 3$ , we obtain

(a_{5} - b_{5}) \sinh (r_{1}) = r_{1}^{2} (1 + \frac{k_{p}}{KGA}) \sinh (r_{1}) \neq 0

(138)

for which only the non-trivial solution is given by $C_{1} = 0$ , $C_{2} = 0$ , $C_{3} = 0$ , while $D_{1} \neq 0$ . Meanwhile, if $rank (Δ) < 3$ , we obtain

r_{1}^{2} (1 + \frac{k_{p}}{KGA}) \sinh (r_{1}) = 0

(139)

for which the non-trivial solution is given by $C_{1} = 0$ and $C_{3} = 0$ , while $C_{2} \neq 0$ and $D_{3} \neq 0$ .

8.3. Construction of the spectrum for a simply supported beam

We give here an analytical derivation of the spectrum for a beam on a Winkler–Pasternak foundation. Note that the so-called ‘Form 1 of the mode shape’ is of no interest for future studies since its occurrence can be disregarded. Indeed, there is no value of natural frequencies capable of bringing that kind of solution. For this reason, hereinafter, the related results are not presented. Also, we ought to remark that the so-called ‘Form 4 of the mode shape’ appears only in the original Timoshenko–Ehrenfest equation and it refers to the case $ω^{2} = ω_{B}^{2}$ .

8.3.1. Original Timoshenko–Ehrenfest model

8.3.1.1. Form 2 of the mode shape

Solutions for $r_{1}$ and $r_{3}$ are given by Equation (115). We disregard the former solution since it happens only when $ω$ is either equal to $ω_{A}$ or $ω_{B}$ , which does not correspond to the present form of the mode shape. The latter occurrence brings to following equation in terms of natural frequency, which can be written as

ω^{4} + g_{1} ω^{2} + h_{1} = 0

(140)

Where

g_{1} = - [{(\frac{n π}{L})}^{2} \frac{KG}{ρ} (1 + \frac{E}{KG} + \frac{k_{p}}{KGA}) + \frac{mKG}{ρ^{2} I} + \frac{k_{w}}{m}]

(141)

h_{1} = {(\frac{n π}{L})}^{4} \frac{KGE}{ρ^{2}} (1 + \frac{k_{p}}{KGA}) + {(\frac{n π}{L})}^{2} (\frac{KG k_{p}}{ρ^{2} I} + \frac{E k_{w}}{ρ^{2} A}) + \frac{KG k_{w}}{ρ^{2} I}

(142)

whose roots are

ω_{I}^{2} = \frac{1}{2} (- g_{1} + \sqrt{g_{1}^{2} - 4 h_{1}}); ω_{II}^{2} = \frac{1}{2} (- g_{1} - \sqrt{g_{1}^{2} - 4 h_{1}})

(143)

Since only admissible solutions are the positive ones

ω_{n} = \sqrt{ω_{II}^{2} (n)}

(144)

The corresponding normalized form of the mode shape reads

W_{n} (ξ) = \sin (r_{3} ξ), Ψ_{n} (ξ) = \frac{b_{2}}{r_{3}} \cos (r_{3} ξ)

(145)

8.3.1.2. Form 3 of the mode shape

Solutions for $r_{1}$ and $r_{3}$ are given by Equation (125). The admissible solution for Equation (140) turns out to be given by $ω_{n} = ω_{I}$ and $ω_{n} = ω_{II}$ . The normalized mode shape for $r_{1} = n π$ is

W_{n} (ξ) = \sin (r_{1} ξ), Ψ_{n} (ξ) = \frac{a_{3}}{r_{1}} \cos (r_{1} ξ)

(146)

while the normalized mode shape for $r_{3} = n π$ reads

W_{n} (ξ) = \sin (r_{3} ξ), Ψ_{n} (ξ) = \frac{b_{3}}{r_{3}} \cos (r_{3} ξ)

(147)

8.3.1.3. Form 4 of the mode shape

The Solution of system (130) depends on the rank of matrix $Δ$ . If $rank (Δ) = 3$ , root $r_{3}$ is given by Equation (131) and the normalized form of the mode shape is such that $D_{1} = 1$ and reads

W_{n} (ξ) = 0, Ψ_{n} (ξ) = 1

(148)

while if $rank (Δ) < 3$ , root $r_{3}$ is given by (132) and the normalized form of the mode shape depends on two arbitrary chosen amplitudes, named $C_{4} = K_{1}$ and $D_{1} = H_{1}$ , which results in

W (ξ) = - K_{1} \sin (r_{3} ξ), Ψ (ξ) = H_{1} + \frac{a_{4}}{r_{3}} K_{1} \cos (r_{3} ξ)

(149)

8.3.1.4. Form 5 of the mode shape

The solution of system (139) depends on the rank of matrix $Δ$ . In particular, the same observation is made when dealing with the so-called ‘Form 4 of the mode shape,’ which can be applied to the present case. Hence, if $rank (Δ) = 3$ , the root for $r_{3}$ is given by Equation (138) and the normalized form of the mode shape by Equation (149). Meanwhile, if $rank (Δ) < 3$ , the root for $r_{3}$ is given by Equation (139) and the normalized form of the mode shape depends on two arbitrary chosen amplitudes, named $C_{2} = K_{2}$ and $D_{3} = H_{2}$ , which correspond to

W (ξ) = - K_{2} \sinh (r_{1} ξ), Ψ (ξ) = H_{2} + \frac{a_{5}}{r_{1}} K_{2} \cosh (r_{1} ξ)

(150)

8.3.1.5. Results for the natural frequency

The results for the natural frequency are shown in Table 6. The value of the natural frequency is expressed in rad/s, while all the other parameters are dimensionless. Note that two branches of frequencies appear when dealing with the original Timoshenko–Ehrenfest equation. Indeed, the first 10 natural frequencies appearing in Table 6 belong to the first branch of frequencies. The next, 11th, and 12th natural frequencies belong to the second branch, whereas the 13th belongs again to the first spectrum. Such an alternating behavior has been called the ‘intermingling phenomenon.’ Also, we report values of $ω_{A}$ , $ω_{B}$ , and $ω_{C}$ with the only purpose to show their position with respect to other natural frequencies.

Table 6.

Natural frequencies predicted by the original Timoshenko–Ehrenfest equation for a simply-supported (S-S) beam on a Winkler–Pasternak foundation.

ω value	Original Timoshenko–Ehrenfest equation
	n	n ₁	n ₂	g ₇	h ₇	ω
ω = ω_A	-	-	-	-	-	0.25
ω = ω_C	-	-	-	-	-	4.27
ω_C < ω < ω_B	1	-	1	−1.32E+10	8.05E+16	2473.824
	2	-	2	−1.51E+10	1.29E+18	9247.994
	3	-	3	−1.85E+10	6.52E+18	18,983.488
	4	-	4	−2.31E+10	2.06E+19	30,497.858
	5	-	5	−2.90E+10	5.03E+19	43,014.780
	6	-	6	−3.63E+10	1.04E+20	56,076.635
	7	-	7	−4.49E+10	1.93E+20	69,421.520
	8	-	8	−5.49E+10	3.30E+20	82,900.033
	9	-	9	−6.61E+10	5.28E+20	96,426.971
	10	-	10	−7.87E+10	8.05E+20	109,954.147
ω = ω_B	-	-	-	-	-	111,803.40
ω > ω_B	11	1	-	−1.32E+10	8.05E+16	114,698.340
	12	2	-	−1.51E+10	1.29E+18	122,726.486
	13	-	11	−9.26E+10	1.18E+21	123,454.992
	14	3	-	−1.85E+10	6.52E+18	134,521.700
	15	-	12	−1.08E+11	1.67E+21	136,915.672
	16	4	-	−2.31E+10	2.06E+19	148,859.477
	17	-	13	−1.24E+11	2.30E+21	150,329.872
	18	-	14	−1.42E+11	3.09E+21	163,695.691
	19	5	-	−2.90E+10	5.03E+19	164,910.437
	20	-	15	−1.61E+11	4.08E+21	177,013.761
	21	6	-	−3.63E+10	1.04E+20	182,157.220
	22	-	16	−1.82E+11	5.28E+21	190,286.121
	23	7	-	−4.49E+10	1.93E+20	200,275.487
	24	-	17	−2.04E+11	6.72E+21	203,515.522
	25	-	18	−2.27E+11	8.45E+21	216,705.015

8.3.2. Elishakoff’s truncated version

8.3.2.1. Form 2 of the mode shape

Solutions for $r_{1}$ and $r_{3}$ are given by Equation (115). The former occurrence has to be disregarded since $r_{1} = 0$ never happens. The latter occurrence, which corresponds to $r_{3} = n π$ , brings the following equation in terms of natural frequencies, which can be written as

g_{2} ω^{2} - h_{2} = 0

(151)

where

g_{2} = {(\frac{n π}{L})}^{2} ρ I (1 + \frac{E}{KG}) + m

(152)

h_{2} = {(\frac{n π}{L})}^{4} EI (1 + \frac{k_{p}}{KGA}) + {(\frac{n π}{L})}^{2} (k_{p} + \frac{EI k_{w}}{KGA}) + k_{w}

(153)

whose roots are simply

ω^{2} (n) = \frac{h_{2}}{g_{2}}

(154)

Since the only admissible solutions for $ω$ are positive ones, Equation (154) reduces to

ω_{n} = \sqrt{ω^{2} (n)} = \sqrt{\frac{h_{2}}{g_{2}}}

(155)

The definition of the corresponding normalized form of mode shape is given by Equation (145).

8.3.2.2. Form 3 of the mode shape

Solutions for $r_{1}$ and $r_{3}$ are given by Equation (125). An analogous result has been obtained by considering the latter occurrence. As a consequence, Equation (153) admits as a solution the roots $ω_{n} = ω_{I}$ and $ω_{n} = ω_{II}$ . The normalized forms of the mode shape for $r_{1}$ and $r_{3}$ are given by Equations (146) and (147), respectively.

8.3.2.3. Form 5 of the mode shape

The solution of the system (139) depends on the rank of matrix $Δ$ . If $rank (Δ) = 3$ , the root for $r_{3}$ is given by Equation (138). Meanwhile, if $rank (Δ) < 3$ , the root for $r_{3}$ is given by Equation (139) and the normalized form of the mode shape by Equation (150).

8.3.2.4. Results for the natural frequency

The results for the natural frequency are shown in Table 7. Note that we report values of $ω_{A}$ and $ω_{D}$ with the only purpose of showing their position with respect to other natural frequencies. Also, we ought to remark that only one branch of frequency appears by using the truncated model.

Table 7.

Natural frequencies predicted by the truncated model for a simply-supported (S-S) beam on a Winkler–Pasternak foundation.

ω value	Truncated model
	n	g ₈	h₈	ω
ω = ω_A	-	-	-	0.25
ω = ω_D	-	-	-	4.27
ω > ω_D	1	8.42E+01	5.15E+08	2473.248
	2	9.69E+01	8.24E+09	9221.849
	3	1.18E+02	4.17E+10	18,797.242
	4	1.48E+02	1.32E+11	29,877.261
	5	1.86E+02	3.22E+11	41,622.173
	6	2.32E+02	6.68E+11	53,594.533
	7	2.88E+02	1.24E+12	65,592.707
	8	3.51E+02	2.11E+12	77,533.534
	9	4.23E+02	3.38E+12	89,388.855
	10	5.04E+02	5.15E+12	101,154.609

8.3.3. Slope inertia version

8.3.3.1. Form 2 of the mode shape

The solutions for $r_{1}$ and $r_{3}$ are given by Equation (115). The former occurrence has to be disregarded since $r_{1} = 0$ never happens. The latter occurrence, which corresponds to $r_{3} = n π$ , brings the following equation in terms of natural frequency

[ρ I {(\frac{n π}{L})}^{2} (1 + \frac{E}{KG} + {(\frac{n π}{L})}^{2} \frac{EI}{KGA}) + m] ω^{2} - [EI {(\frac{n π}{L})}^{4} (1 + \frac{k_{p}}{KGA}) + {(\frac{n π}{L})}^{2} (k_{p} + \frac{EI k_{w}}{KGA}) + k_{w}] = 0

(156)

which can be re-written as

g_{3} ω^{2} - h_{3} = 0

(157)

where

g_{3} = ρ I {(\frac{n π}{L})}^{2} (1 + \frac{E}{KG} + {(\frac{n π}{L})}^{2} \frac{EI}{KGA}) + m

(158)

h_{3} = EI {(\frac{n π}{L})}^{4} (1 + \frac{k_{p}}{KGA}) + {(\frac{n π}{L})}^{2} (k_{p} + \frac{EI k_{w}}{KGA}) + k_{w}

(159)

whose roots are simply

ω^{2} = \frac{h_{3}}{g_{3}}

(160)

Since the only admissible solutions for $ω$ are positive ones, Equation (160) reduces to

ω_{n} = \sqrt{ω^{2} (n)} = \sqrt{\frac{h_{3}}{g_{3}}}

(161)

The normalized form of the mode shape is given by Equation (145).

8.3.3.2. Form 3 of the mode shape

Equation (146) admits as a solution the roots $ω_{n} = \sqrt{ω_{I}^{2}}$ or $ω_{n} = \sqrt{ω_{II}^{2}}$ . The normalized form of the mode shape for $r_{1}$ and $r_{3}$ is given by Equations (146) and (147), respectively.

8.3.3.3. Form 5 of the mode shape

The solution of system (139) depends on the rank of matrix $Δ$ . If $rank (Δ) = 3$ , the root for $r_{3}$ is given by Equation (138) and the normalized form of the mode shape by Equation . Meanwhile, if $rank (Δ) < 3$ , the root for $r_{3}$ is given by Equation (139) and the normalized form of the mode shape by Equation (150).

8.3.3.4. Results for the natural frequency

The results for the natural frequency are shown in Table 8. Note that we report values of $ω_{A}$ , $ω_{D}$ , and $ω_{E}$ with the only purpose of showing their position with respect to other natural frequencies. Also, we remark that $ω_{D}$ does not divide the natural frequencies into two branches. Indeed, only one branch of frequency appears when using the model based on slope inertia.

Table 8.

Natural frequencies predicted by the model based on slope inertia for a simply-supported (S-S) beam on a Winkler–Pasternak foundation.

ω value	Based on slope inertia model
	n	g ₉	h ₉	ω
ω = ω_A	-	-	-	0.250
ω = ω_D	-	-	-	4.27
ω_D < ω < ω_E	1	8.43E+01	5.15E+08	2472.644
	2	9.76E+01	8.24E+09	9190.638
	3	1.21E+02	4.17E+10	18,537.075
	4	1.58E+02	1.32E+11	28,864.399
	5	2.12E+02	3.22E+11	39,006.825
	6	2.86E+02	6.68E+11	48,328.691
	7	3.87E+02	1.24E+12	56,575.039
	8	5.20E+02	2.11E+12	63,712.461
	9	6.94E+02	3.38E+12	69,817.374
	10	9.16E+02	5.15E+12	75,010.045
ω = ω_E	-	-	-	111,803.399

A graphical representation of the obtained natural frequencies for a simply-supported (S-S) beam on a Winkler–Pasternak foundation by using all three models is given in Figure 2. The intermingling phenomenon appears after the 10th frequency when using the original Timoshenko–Ehrenfest equation.

9. Discussion of results

The presented study refers to the case in which the length of the beam is eight times bigger than its height. The analysis reveals that the intermingling phenomenon happens only when dealing with the original Timoshenko–Ehrenfest equation of a S-S beam. In this situation, two branches of natural frequencies exist, while, for all other cases, only one branch of frequency appears.

According to the exposed tables, the truncated version of the Timoshenko-Ehrenfest equation and the model based on slope inertia are reliable only for the very first values of natural frequency when a beam on a Winkler–Pasternak foundation is considered. Indeed, the difference between the numerical results is very small only in the first region of natural frequencies.

Particular attention has been focused on the S-S beam. Indeed, the use of a stiffer foundation could reduce the number of frequencies prior to the beginning of the so-called intermingling phenomenon. As a consequence, depending on the material properties, there exist a values of $k_{w}$ and $k_{p}$ for which only one natural frequency appears before the beginning of the intermingling phenomenon. In this situation, only if it is possible to approximate the beam as possessing one degree of freedom (DoF) would it be possible to resort to truncated or slope inertia models. Otherwise, full Timoshenko–Ehrenfest beam equations may be needed. As an example, if we consider $k_{w} = 10^{10}$ and $k_{p} = 5 . 10^{10}$ we obtain the results for natural frequencies listed in Table 9 in which intermingling appears after the very first natural frequency.

Table 9.

Intermingling phenomenon for a simply-supported (S-S) beam using the original Timoshenko–Ehrenfest equation with $k_{p} = 3 . 10^{12}$ .

n₁	n₂	ω
-	1	96,802.990
1	-	116,422.917
-	2	119,101.716
-	3	129,453.340
-	4	142,421.557
-	5	157,486.186
-	6	174,122.863
-	7	191,928.290
2	-	199,098.566
-	8	210,608.041
-	9	229,949.890
-	10	249,800.440
-	11	270,047.717
-	12	290,608.911
3	-	297,603.534

The study shows also that when we consider BCs that differ from the simply supported one, it has not been possible to arrive at a closed-form solution for the governing equations of all the treated cases. Indeed, the values of natural frequencies have been obtained by using a Computer Algebra System, such as Maple.

In what follows we conduct two comparisons for the obtained frequencies with those reported in other studies. We start observing that when parameters $k_{w}$ and $k_{p}$ are set equal to zero, and the length of the beam is set equal to 2 m, the same results presented by Cazzani et al. [33,34] are obtained.

We also confirm that the solutions presented in this paper coincide with the results reported by Wang et al. [35], except for a very small difference. Indeed, by following their paper, we introduce the following parameters

Ω = \frac{EI}{KGA L^{2}}, {\bar{k}}_{w} = \frac{k_{w} L^{4}}{EI}, {\bar{k}}_{p} = \frac{k_{p} L^{2}}{EI}, \bar{ω} = \frac{ω^{2} ρ A L^{4}}{EI}

where the value $Ω = 0$ is associated with the Euler–Bernoulli solution, and we obtain Table 10.

Table 10.

Comparison of the first natural frequency with that of Wang et al. [33].

Boundary conditions	${\bar{k}}_{w}$	$\frac{{\bar{k}}_{p}}{π^{2}}$	$Ω$	From paper of Wang et al. [33]	Results
				$\sqrt{\bar{ω}}$	Original Timoshenko–Ehrenfest equation	Elishakoff’s truncated model	Slope inertia model
Simply-supported (S-S)	100	2.5	$0$	4.582	4.582	4.582	4.581
			$0.1$	3.972	3.972	3.971	3.971
			$0.2$	3.266	3.265	3.265	3.264
Clamped-clamped (C-C)			$0$	5.477	5.477	5.477	5.476
			$0.1$	5.121	5.121	5.120	5.120
			$0.2$	4.204	4.203	4.203	4.202
Clamped-simply supported (C-S)			$0$	4.983	4.983	4.983	4.982
			$0.1$	3.812	3.812	3.811	3.811
			$0.2$	2.803	2.802	2.802	2.801
Free-free (F-F)			$0$	6.043	6.043	6.043	6.042
			$0.1$	5.675	5.675	5.674	5.674
			$0.2$	4.711	4.710	4.710	4.709
Clamped-free (C-F)			$0$	6.040	6.040	6.040	6.039
			$0.1$	5.278	5.278	5.277	5.277
			$0.2$	5.041	5.040	5.040	5.039
Simply supported-free (S-F)			0	5.577	5.577	5.577	5.576
			0.1	5.524	5.524	5.523	5.523
			0.2	4.460	4.459	4.459	4.458

10. Conclusion

This paper presents an analysis of the original Timoshenko–Ehrenfest equation and two of its truncated versions for a beam on a Winkler–Pasternak foundation. It reveals that some differences exist between the three models. Indeed, the so-called ‘Elishakoff’s truncated model’ differs from the original Timoshenko–Ehrenfest equation by the absence of one term related to $k_{p}$ , one term related to $k_{w}$ , and one term related to the fourth-order derivative of the displacement with respect of time. Instead, the model based on slope inertia contains an additional term related to the sixth-order derivative of the displacement $w (x, t)$ , which is absent in both of the other equations.

The study reports some important differences for the analyzed models when dealing with different BCs applied to a Timoshenko–Ehrenfest beam on a Winkler–Pasternak foundation. It shows that two branches of frequencies exist only when dealing with the original Timoshenko–Ehrenfest equation for a simply supported beam. This distinction disappears when we consider the two truncated models. Indeed, these two theories represent a correction of only the first branch of natural frequencies. Thus, we can use these theories only when the contribution of the second branch of natural frequencies is negligible with respect to the first one. We remark that all the other discussed cases of BCs present only one branch of natural frequencies indiscriminately from the used model.

For the given material properties, the first 25 natural frequencies predicted by the original Timoshenko–Ehrenfest beam theory have been evaluated for a simply supported beam in order to illustrate an interesting intermingling phenomenon appearing after the 10th natural frequency in a beam in which its length is eight times longer than its height. For the same kind of BC only the first 10 natural frequencies out of the above set of 25 natural frequencies are reported numerically by using the two truncated models, since they are able to predict only the first branch of frequencies. However, the behavior of these frequencies is shown in detail in Figure 2 for the first 25 frequencies. As a consequence, if the excitation spectrum covers the first 10 or fewer frequencies, the use of the original Timoshenko–Ehrenfest theory is not needed, as it turns out to be overcomplicated.

Footnotes

Acknowledgements

This work was conducted as the part of Master’s Thesis of Mr Giulio Tonzani when he served as a Visiting Scholar at Florida Atlantic University. We are appreciative of the helpful comments of the reviewers.

Funding

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

ORCID iD

Isaac Elishakoff

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Three alternative versions of the theory for a Timoshenko–Ehrenfest beam on a Winkler–Pasternak foundation

Abstract

Keywords

1. Introduction

2. Governing Timoshenko–Ehrenfest equation

3. Truncated version of the Timoshenko–Ehrenfest equation

4. Derivation of the slope inertia version of the Timoshenko–Ehrenfest equation

5. Unified formulation of the problem

6. Solution of the equation of motion

6.1. Original Timoshenko–Ehrenfest equation

6.2. Elishakoff’s truncated version

6.3. Slope inertia version

7. Vibration mode of the Timoshenko–Ehrenfest beam

7.1. Form 1 of the mode shape

7.2. Form 2 of the mode shape

7.3. Form 3 of the mode shape

7.4. Form 4 of the mode shape

7.5. Form 5 of the mode shape

8. Application of the three models for a beam on a Winkler–Pasternak foundation

8.1. Material properties

8.2. Case study 1: beam simply supported at both ends

8.2.1. Form 1 of the mode shape

8.2.2. Form 2 of the mode shape

8.2.3. Form 3 of the mode shape

8.2.4. Form 4 of the mode shape

8.2.5. Form 5 of the mode shape

8.3. Construction of the spectrum for a simply supported beam

8.3.1. Original Timoshenko–Ehrenfest model

8.3.1.1. Form 2 of the mode shape

8.3.1.2. Form 3 of the mode shape

8.3.1.3. Form 4 of the mode shape

8.3.1.4. Form 5 of the mode shape

8.3.1.5. Results for the natural frequency

8.3.2. Elishakoff’s truncated version

8.3.2.1. Form 2 of the mode shape

8.3.2.2. Form 3 of the mode shape

8.3.2.3. Form 5 of the mode shape

8.3.2.4. Results for the natural frequency

8.3.3. Slope inertia version

8.3.3.1. Form 2 of the mode shape

8.3.3.2. Form 3 of the mode shape

8.3.3.3. Form 5 of the mode shape

8.3.3.4. Results for the natural frequency

9. Discussion of results

10. Conclusion

Footnotes

Acknowledgements

Funding

ORCID iD

References