Free Vibration Characteristic of Multilevel Beam Based on Transfer Matrix Method of Linear Multibody Systems

Abstract

In this paper, an approach based on transfer matrix method of linear multibody systems (MS-TMM) is developed to analyze the free vibration of a multilevel beam, coupled by spring/dashpot systems attached to them in-span. The Euler-Bernoulli model is used for the transverse vibration of the beams, and the spring/dashpot system represents a simplified model of a viscoelastic material. MS-TMM reduces the dynamic problem to an overall transfer equation which only involves boundary state vectors. The state vectors at the boundaries are composed of displacements, rotation angles, bending moments, and shear forces, which are partly known and partly unknown, and end up with reduced overall transfer matrix. Nontrivial solution requires the coefficient matrix to be singular to yield the required natural frequencies. This paper implements two novel algorithms based on the methodology by reducing the zero search of the reduced overall transfer matrix's determinate to a minimization problem and demonstrates a simple and robust algorithm being much more efficient than direct enumeration. The proposal method is easy to formulate, systematic to apply, and simple to code and can be extended to complex structures with any boundary conditions. Numerical results are presented to show the validity of the proposal method against the published literature.

1. Introduction

The vibration problem of beam-type structures is of particular urgent issue in many branches of modern aerospace, mechanical, and civil engineering. Natural vibration frequencies and modes are one of the most important dynamic characteristics of these kinds of systems. For example, the precision in manufacturing can be highly influenced by vibrations. If the vibration characteristics cannot be solved or preestimated exactly when designing a mechanism system, it is often hard to obtain a good dynamic performance of the mechanism system and consequently hard to control its vibration.

There are different types of beam models. One of the well-known models is the Euler-Bernoulli beam theory that works well for slender beams. According to the Euler-Bernoulli beam theory, the length of each beam section is much greater than the height of each section and the shear and rotary inertia effects are ignored.

The vibration theory of single-beam systems is well developed and studied in detail in hundreds of contributions. The vibration of systems composed of uniform double-beam coupled by translational springs or elastic layers have been studied extensively in the literature. Inceoğlu and Gürgöze [1] studied the bending vibrations of a combined system consisting of two clamped-free beams carrying tip masses to which several double spring-mass systems are attached across the span. Using Green's function method, the frequency equation of the system is established. Kukla [2] solved the problem of free vibration of two axially loaded beams which are connected by many translational springs. The solution contains possible combinations of the classical boundary conditions. The technique of the solution consists of developing a Green function. In [3], the vibrations of uniform beams connected by homogeneous elastic layer are devoted. Oniszczuk [4] discussed the free transverse vibrations of two parallel simply supported beams continuously joined by a Winkler elastic layer. The motion of the system was described by a homogeneous set of two partial differential equations, which was solved by using the classical Bernoulli-Fourier method. Oniszczuk [5] is devoted to analyze the undamped forced transverse vibrations of an elastically connected complex double-beam system in the case of simply supported beams. The classical modal expansion method was applied to determine the dynamic responses of the beams due to arbitrarily distributed continuous loads. Vu et al. [6] presented an exact method for solving the vibration problem of a damped double-beam system subjected to harmonic excitation. The double-beam system consists of two identical beams with the same boundary conditions on both sides. The beams are connected by a viscoelastic layer. Natural frequencies and mode shapes of vibration of the system are determined and the forced vibrations are investigated. Gürgöze and Erol [7] determined the natural frequencies of a clamped-free double-beam system carrying tip masses to which several spring mass systems are attached across the span. However, there are only few contributions dealing with the vibration of multibeam systems. That is, probably, the general vibration analyses of an elastically connected multibeam system are complicated and laborious in view of a large variety of possible combinations of boundary conditions, and thus, the solution of the governing coupled partial differential equations is difficult [8].

Multibody system dynamics (MSD) has become an important theoretical tool for wide engineering problems analysis in the world. Lots of methods of MSD have been studied by many authors on theory and computational method [9–15]. Professor Rui Xiaoting and his students have been enlightened by the method of letting state vectors (SVs) be transferred into classical transfer matrix method and built up a new multibody dynamics method called “Transfer Matrix Method of Linear Multibody Systems-MS-TMM” [16]. Using MS-TMM, the eigenvalue of linear multi-rigid-flexible-body system is computed easily, the computational ill-condition is overcome, and the computational efficiency is increased. Over 20 years, MS-TMM has been developed and used widely in engineering applications.

Motivated by the interesting study by Kukla [2] which was published on the problem of the natural longitudinal vibrations of two rods coupled by many translational springs and by the two novel algorithms based on the new recursive scanning approach presented by Bestle et al. [17], this paper presents a unique yet simple scenario of obtaining the exact free vibration characteristics of undamped/damped multilevel beam coupled elastically. The scenario developed in this paper is based on MS-TMM and Euler-Bernoulli beam theory.

The text is organized as follows. The problem statement is presented in Section 2. In Sections 3 and 4, the general theorem brief of MS-TMM and problem solution scenario are shown. In Section 5, some results calculated by MS-TMM and the other method are given which can validate the proposed method. The conclusions are presented in Section 6.

2. Problem Statement

The transversely vibrating system in a plane under consideration consists of multi-level parallel, elastic, and homogeneous Euler-Bernoulli beam with general boundary conditions. Beam bending stiffness is EI, where E is the elastic modulus and I is the area moment of inertia. Beam mass per unit length $\stackrel{-}{m}=\rho A$ , where ρ is the material density and A is the beam cross-section area. Of course, any different materials, different dimensions, or different boundary conditions could be considered in MS-TMM. The beams have the same length L and are joined by the j-system of spring/dashpot located at different positions (e.g., x₁ and x _j , where j is the number of spring/dashpot systems between two levels). k _y and c are linear translational spring constant and damping coefficient, respectively, as shown in Figure 1.

Figure 1:

A system of a multi-level beam with a distributed spring/dashpot system(s).

3. MS-TMM Strategy in the Context of Free Vibration Characteristics

3.1. MS Topologies

According to the natural attribute of bodies, a complicated MS can be represented by various bodies (e.g., rigid bodies, elastic bodies, lumped masses, etc.) interconnected by hinges (e.g., spherical joints, sliding joints, cylindrical joints, dampers, springs, etc.). In MS-TMM, there are different topologies based on a certain set of modeling variables to formulate the dynamic equations of MS. Such topologies are chain, tree, closed loop (as illustrated in Figure 2 for reference), network, and so forth.

Figure 2:

Multibody systems (a) chain, (b) tree, and (c) closed loop topology.

3.2. State Vector, State Variables, and Transfer Direction

The state vector (SV) at a connection point of MS is a column vector denoting the mechanics state of this point. It includes the displacements of the point (including angular displacements) and the corresponding internal forces (including internal moments). Therefore, the SV is given by kinematics (displacements) and kinetics (internal forces) quantities, called state variables. For convention in this paper, z with bold lowercase represents the SV in the physical coordinates and Z with bold capital represents the SV in the modal coordinates. Vibrations in space are described by displacement coordinates x,y,z along the Cartesian axes and angular rotations θ _x ,θ _y ,θ _z about Cartesian axes. Cutting forces and moments are given by q _x ,q _y ,q _z and m _x ,m _y ,m _z , respectively. Positive directions at input points are shown in Figure 3(a). Positive directions of forces and moments at output points (Figure 3(b)) are opposite due to the principle of action equals reaction. In 3D case with n _s = 12 (n _s is the number of state variables in the SV), the SVs in physical and modal coordinates at the connection point p _i,k (where the first subscript i is the serial number of element for boundary end and the second subscript k is the serial number of the hinge element and k = 0 for boundary end) are summarized in a vector, receptively:

$\begin{array}{c}{{\mathbf{z}}_{i,k}|}_{\text{physical}\hspace{0.17em}\hspace{0.17em}\text{coordinates}}={\left[x,y,z,{\theta }_x,{\theta }_y,{\theta }_z,m_x,m_y,m_z,q_x,q_y,q_z\right]}_{i,k}^T,\\ {{\mathbf{Z}}_{i,k}|}_{\text{modal}\hspace{0.17em}\hspace{0.17em}\text{coordinates}}={\left[X,Y,Z,{\mathrm{\Theta }}_x,{\mathrm{\Theta }}_y,{\mathrm{\Theta }}_z,M_x,M_y,M_z,Q_x,Q_y,Q_z\right]}_{i,k}^T.\end{array}$

(1)

Figure 3:

Sign convention at (a) input; (b) output.

In case of 1D or 2D applications, the SV will be reduced as shown later. Defining a boundary point of the MS as the transfer end, the direction from all other boundary points to the transfer end is called transfer direction. Along the transfer direction, the nodes entering into elements are called inputs denoted by I and the nodes leaving from elements are called outputs O.

3.3. Transfer Equation, Transfer Matrix, Overall System Transfer Matrix, and Overall System State Vector

A vibrating MS comprised of n-components, see Figure 4, is used as an example to show how to deduce the overall transfer equation and overall transfer matrix of the system. In order to describe conveniently the idea, the chain topology is considered in the following. This vibrating system is comprised of n components and n + 1 connection points. The SVs of the boundary right extremity and other boundary left extremity of the system are expressed as z_n,0 and z_1,0, respectively. Transfer direction of the system is always from another boundary end to may call it the root. Following the transfer direction shown in Figure 3, the transfer equation between the component input and output is

$\begin{array}{c}{\mathbf{z}}_{n,\mathrm{0}}={\mathbf{U}}_n{\mathbf{z}}_{n-\mathrm{1},n}\\ {\mathbf{z}}_{n-\mathrm{1},n}={\mathbf{U}}_{n-\mathrm{1}}{\mathbf{z}}_{n-\mathrm{2},n-\mathrm{1}}\\ {\mathbf{z}}_{n-\mathrm{2},n-\mathrm{1}}={\mathbf{U}}_{n-\mathrm{2}}{\mathbf{z}}_{n-\mathrm{3},n-\mathrm{2}}\\ ⋮\\ {\mathbf{z}}_{\mathrm{1,2}}={\mathbf{U}}_{\mathrm{1}}{\mathbf{z}}_{\mathrm{1,0}}.\end{array}$

(2)

Figure 4:

TMM-MS in the sense of chain topology. A vibrating system comprised of n-components with n + 1 connections.

The constant matrix U _k is the transfer matrix of the kth component. Transfer matrices of basic components are considered as building blocks, which can be assembled together to provide the transfer matrix of the whole system according to the chain MS-TMM topology as follows:

${\mathbf{z}}_{n,\mathrm{0}}=\mathbf{T}{\mathbf{z}}_{\mathrm{1,0}},\hspace{1em}\text{where}\hspace{0.17em}\hspace{0.17em}\mathbf{T}=\prod _{k=\mathrm{0}}^{n-\mathrm{1}}{\mathbf{U}}_{n-k}.$

(3)

Rewrite (2) as

${{\mathbf{U}}_{\text{all}}|}_{n_s\times (\mathrm{2}\times n_s)}{{\mathbf{z}}_{\text{all}}|}_{(\mathrm{2}\times n_s)\times \mathrm{1}}=\mathbf{0},$

(4a)

where

$\mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{\hspace{0.17em}\hspace{0.17em}}{\mathbf{z}}_{\text{all}}^T={\left[\begin{array}{cc}{\mathbf{z}}_{\mathrm{1,0}}^T& {\mathbf{z}}_{n,\mathrm{0}}^T\end{array}\right]}^T,$

(4b)

${\mathbf{U}}_{\text{all}}=\left[\begin{array}{cc}\mathbf{T}& -{\mathbf{I}}_{n_s}\end{array}\right].$

(4c)

Herein, U_all is the overall system transfer matrix and z_all is the overall system state vector.

3.4. Eigenfrequency Equation of the Whole MS

The overall transfer equation (4a) only involves the boundary SVs, and the SVs at all other connection points do not appear. The SVs at the boundary are composed of displacements, rotation angles, moments, and shears, which are partly known and partly unknown. For common boundary conditions, half of state variables of z_all (4b) are zeros due to known constraints. Thus, (4a) reduces to ${\stackrel{-}{\mathbf{U}}}_{\text{all}}{\stackrel{-}{\mathbf{z}}}_{\text{all}}=\mathbf{0}$ , where ${\stackrel{-}{\mathbf{z}}}_{\text{all}}$ is composed of the unknown state variables and ${\stackrel{-}{\mathbf{U}}}_{\text{all}}$ is a square matrix resulting from elimination of all columns of U_all (4c) associated zeros in z_all. For harmonic vibrations, solutions maybe written as z _i,j = Z _i,j e ^λt where λ = − λ ^r ±iλ ⁱ , λ ^r ,λ ⁱ ∈ ℝ are the eigenvalues. The real part (− λ ^r ) is related to the magnitude of damping, where the imaginary part (λ ⁱ ) is related to the vibration frequency of the damped system. For undamped systems, λ ^r = 0 and λ ⁱ = ω. Finally, ${\stackrel{-}{\mathbf{U}}}_{\text{all}}$ is only a function of the unknown λ _i of the system. For nontrivial solutions, the Eigenfrequency equation

$\mathrm{\Delta }\left(\lambda \right)=\det {\stackrel{-}{\mathbf{U}}}_{\text{all}}\stackrel{\mathrm{\hspace{0.17em}\hspace{0.17em}}!\mathrm{\hspace{0.17em}\hspace{0.17em}}}{=}\mathrm{0}$

(5)

has to be fulfilled. The natural frequencies of the system can now be computed.

3.5. Beam Transfer Matrix

The full derivation of transfer matrices for the Timoshenko and Euler-Bernoulli beams vibrating in a plane (with kinematics and kinetics’ SV defined as z = [y,θ _z ,m _z ,q _y ] ^T ) may be found in [18], which is an open access article and the reader may download it from the Internet. However, for completeness, only the transfer matrix for the Euler-Bernoulli beam will be presented, Figure 5. The differential equation of a Euler-Bernoulli beam is

$\begin{array}{c}EI\frac{{\partial }^{\mathrm{4}}y}{\partial x^{\mathrm{4}}}+\stackrel{-}{m}\frac{{\partial }^{\mathrm{2}}y}{\partial t^{\mathrm{2}}}=\mathrm{0}\mathrm{\hspace{0.17em}\hspace{0.17em}}\underset{\stackrel{\phantom{y\left(x,t\right)\mathrm{00000}=Y\left(x\right){\text{e}}^{\lambda t}}}{\to }}{y\left(x,t\right)=Y\left(x\right)e^{\lambda t}}\frac{{\partial }^{\mathrm{4}}Y\left(x\right)}{\partial x^{\mathrm{4}}}+\frac{\stackrel{-}{m}{\lambda }^{\mathrm{2}}}{EI}Y\left(x\right)=\mathrm{0},\\ \text{where}\hspace{0.17em}\hspace{0.17em}Y\left(x\right)=A_{\mathrm{1}}\cosh \beta x+A_{\mathrm{2}}\sinh \beta x+A_{\mathrm{3}}\cos \beta x+A_{\mathrm{4}}\sin \beta x,\end{array}$

(6)

A₁, A₂, A₃, and A₄ are arbitrary constants, and $\beta =\sqrt[\mathrm{4}]{-\stackrel{-}{m}{\lambda }^{\mathrm{2}}/\left(EI\right)}$ . For the Euler-Bernoulli beam, the linearized relations in modal coordinates Θ _z = Y′, M _z = EIY″, and Q _y = M _z ′ maybe added to end up with the transfer relation:

$\mathbf{Z}\left(x\right)=\mathbf{B}\left(x\right)\mathbf{a}\hspace{1em}\text{or}\hspace{1em}{\left[\begin{array}{c}Y\\ {\mathrm{\Theta }}_z\\ M_z\\ Q_y\end{array}\right]}_x=\left[\begin{array}{cccc}\cosh \beta x& \sinh \beta x& \cos \beta x& \sin \beta x\\ \beta \sinh \beta x& \beta \cosh \beta x& -\beta \sin \beta x& \lambda \cos \beta x\\ EI{\beta }_{}^{\mathrm{2}}\cosh \beta x& EI{\beta }_{}^{\mathrm{2}}\sinh \beta x& -EI{\beta }_{}^{\mathrm{2}}\cos \beta x& -EI{\beta }_{}^{\mathrm{2}}\sin \beta x\\ EI{\beta }_{}^{\mathrm{3}}\sinh \beta x& EI{\beta }_{}^{\mathrm{3}}\cosh \beta x& EI{\beta }_{}^{\mathrm{3}}\sin \beta x& -EI{\beta }_{}^{\mathrm{3}}\cos \beta x\end{array}\right]\left[\begin{array}{c}{A}_{\mathrm{1}}\\ A_{\mathrm{2}}\\ A_{\mathrm{3}}\\ A_{\mathrm{4}}\end{array}\right].$

(7)

Figure 5:

The direction of the state vector of Euler-Bernoulli beam component.

The coefficient vector a = [A₁,A₂,A₃,A₄] ^T summarizes the unknown constants to be adopted to boundary conditions. At input end Z _I (x = 0), we get Z _I = [B(0)]a. Thus, the coefficient vector can be expressed as a = [B(0)]₋₁Z _O and substituting it into (7) for the beam output end at x = l, one gets

${\mathbf{Z}}_O=\left[\mathbf{B}\left(l\right)\right]\mathbf{a}=\left[\mathbf{B}\left(l\right)\right]{\left[\mathbf{B}\left(\mathrm{0}\right)\right]}^{-1}{\mathbf{Z}}_I=\mathbf{U}{\mathbf{Z}}_I,$

(8a)

where

$\begin{array}{c}\mathbf{U}=\mathbf{B}\left(l\right){\mathbf{B}}^{-\mathrm{1}}\left(\mathrm{0}\right)=\left[\begin{array}{cccc}S& \frac{T}{\beta }& \frac{U}{EI{\beta }^{\mathrm{2}}}& \frac{V}{EI{\beta }^{\mathrm{3}}}\\ \beta V& S& \frac{T}{EI\beta }& \frac{U}{EI{\beta }^{\mathrm{2}}}\\ EI{\beta }^{\mathrm{2}}U& EI\beta V& S& \frac{T}{\beta }\\ EI{\beta }^{\mathrm{3}}T& EI{\beta }^{\mathrm{2}}U& \beta V& S\end{array}\right],\\ S=\frac{ch+c}{\mathrm{2}},\hspace{1em}T=\frac{sh+s}{\mathrm{2}},\\ U=\frac{ch-c}{\mathrm{2}},\hspace{1em}V=\frac{sh-s}{\mathrm{2}},\\ ch=\cosh \left(\beta l\right),\hspace{1em}sh=\sinh \left(\beta l\right),\\ c=\cos \left(\beta l\right),\hspace{1em}s=\sin \left(\beta l\right)\end{array}$

(8b)

is the transfer matrix of the Euler-Bernoulli beam component.

3.6. Summary

In the context of free vibration characteristic, the general strategy of linear MS-TMM, in summary, is as follows.

(1) Break up the complicated MS into components with simple dynamic properties, which can be expressed in matrix form and SVs (for each component, it is possible to obtain the close form expression of the transfer matrix giving the displacements and the forces applied to one extremity to the displacements and forces applied to the other extremity). In other words, on the component level, the governing partial and ordinary differential or algebraic equations are transformed to algebraic transfer equations, where the output state results from a product of the input state and an element specific transfer matrix. These component matrices are considered as building blocks. In fact, the transfer matrix of such components needs not to be rededuced but may be taken directly from a transfer matrix library.

(2) Following the transfer direction that has been designed already by the analyst and according to the topology of the MS, these component transfer matrices are then assembled and end up with a system of linear algebraic equations called the overall transfer equation.

(3) Substitute the boundary conditions into the overall transfer equation to construct the eigenfrequency equation. Consequently, the vibration characteristics such as frequencies can be deduced as the roots of a transcendental equation. Due to narrow couples of natural frequencies, the classical zero search method is likely to fail. However, a new recursive scanning approach for minima of the absolute values of the determinant shows much more efficiency and reliability than direct enumeration.

4. Problem Solution Scenario

Figure 6 illustrates the suggested scenario for the solution of the problem statement and as follows. There are (1: m) multi-level beam. Each beam level is divided into (1: n) components, which have (0: n + 1) connection points and coupled with another beam level by a system (say 1: j) of a viscoelastic material (modeled as a spring/dashpot system). The massless dummy body as shown in Figure 6 (e.g., A,B,C, and so on) is a connection point between the two beam segment components and spring/dashpot system(s). The main key of MS-TMM is transferring the SV from one component to another following the general transfer equation (2). For the beam segment component, the transfer matrix is available (8b), while it is not for spring/dashpot system at the connection point between two or multi-level beam in this paper. However, it is based on the kinematics and kinetics of the spring/dashpot to formulate the transfer matrix and it needs two steps. First, as an example, let us consider a connection point j between two beam segment components and system of spring/dashpot may be grounded or connected to another connection point. Figure 7(a) sketches the first step. According to the continuity, j has identical displacements, angles, and moments at input and output:

$Y_O=Y_I,\hspace{1em}\hspace{1em}{\mathrm{\Theta }}_{z_O}={\mathrm{\Theta }}_{z_I},\hspace{1em}\hspace{1em}{M}_{z_O}=M_{z_I}.$

(9a)

And from the force analysis shown in Figure 7(b), the spring/dashpot force F changes the shear forces:

$Q_{y_O}=Q_{y_I}+F.$

(9b)

Equations (9a) and (9b) in the view of the SV as a matrix form

${\mathbf{Z}}_O={\mathbf{Z}}_I+{\mathbf{e}}_{\mathrm{4}}F.$

(9c)

Figure 6:

Structure of a multi-level beam with the distribution of spring/dashpot.

Figure 7:

(a) Two beam segments and spring/dashpot system connected at massless dummy body, (b) state vectors (SV) at the massless dummy body j-connection point, (c) force analysis of a spring/dashpot system, (d) a system 1, 1 connected two levels 1–2 through two connection points A and B shown in Figure 6, and (e) two systems j, 1 and 1, m − 1 connected with multi-level (1–2–3) through three connection points G,C, and V shown in Figure 6.

${\mathbf{e}}_{\mathrm{4}}={\left[\begin{array}{cccc}\mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right]}^T$ is a unit vector assigning F to the transfer equation of the massless connection body. The spring compression and damping forces are given as (see Figure 7(c))

$\begin{array}{c}{F}_{\text{spring}}=k_y(Y_I-Y_O)=k_y\mathrm{\Delta }Y=k_y{\mathbf{e}}_{\mathrm{1}}^T\mathrm{\Delta }\mathbf{Z}\\ F_{\text{dashpot}}=c({\dot{y}}_I-{\dot{y}}_O)\underset{\stackrel{\phantom{y\mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{000}={Y\text{e}}^{\lambda t}}}{\to }}{y={Ye}^{\lambda t}}=c\lambda (Y_I-Y_O)=c\lambda {\mathbf{e}}_{\mathrm{1}}^T\mathrm{\Delta }\mathbf{Z}\\ \to F_{\text{sys}}=\left(k_y+c\lambda \right){\mathbf{e}}_{\mathrm{1}}^T\mathrm{\Delta }\mathbf{Z}.\end{array}$

(10)

Herein, ${\mathbf{e}}_{\mathrm{1}}^T=\left[\begin{array}{cccc}\mathrm{1}& \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right]$ is a unit vector, which extracts Y from the transfer SV. Substitute (10) into (9c) and knowing that F = ± F_sys, we ended the first step with

$\begin{array}{c}{\mathbf{Z}}_O={\mathbf{Z}}_I\pm \mathbf{D}\mathrm{\Delta }\mathbf{Z},\\ \text{where}\hspace{0.17em}\hspace{0.17em}\mathbf{D}=\left(k_y+c\lambda \right){\mathbf{e}}_{\mathrm{4}}{\mathbf{e}}_{\mathrm{1}}^T=\left[\begin{array}{cccc}\mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ k_y+c\lambda & \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right].\end{array}$

(11)

For system 1, 1 connected two levels 1–2 through two connection points A and B shown in Figure 6, the second step is how to deduce the transfer equation between the input and output related to these two levels. For the connection point A (see Figure 7(d)) and using (11) with F = F_sys,

$\begin{array}{c}{\mathbf{Z}}_{\mathrm{1},O}={\mathbf{Z}}_{\mathrm{1},I}+{\mathbf{D}}_{\mathrm{1,1}}({\mathbf{Z}}_{\mathrm{2},I}-{\mathbf{Z}}_{\mathrm{1},I})=\underset{{\mathbf{U}}_{\mathrm{1,1}}^d}{\underbrace{(\mathbf{I}-{\mathbf{D}}_{\mathrm{1,1}})}}{\mathbf{Z}}_{\mathrm{1},I}+\underset{{\mathbf{U}}_{\mathrm{1,1}}^c}{\underbrace{{\mathbf{D}}_{\mathrm{1,1}}}}{\mathbf{Z}}_{\mathrm{2},I}\\ {\mathbf{Z}}_{\mathrm{1},O}={\mathbf{U}}_{\mathrm{1,1}}^d{\mathbf{Z}}_{\mathrm{1},I}+{\mathbf{U}}_{\mathrm{1,1}}^c{\mathbf{Z}}_{\mathrm{2},I}.\end{array}$

(12a)

Similarly, for connection point B and using (11) with F = − F_sys,

$\begin{array}{c}{\mathbf{Z}}_{\mathrm{2},O}={\mathbf{Z}}_{\mathrm{2},I}-{\mathbf{D}}_{\mathrm{1,1}}\mathrm{\hspace{0.17em}\hspace{0.17em}}\left({\mathbf{Z}}_{\mathrm{2},I}-{\mathbf{Z}}_{\mathrm{1},I}\right),\\ {\mathbf{Z}}_{\mathrm{2},O}={\mathbf{U}}_{\mathrm{1,1}}^c{\mathbf{Z}}_{\mathrm{1},I}+{\mathbf{U}}_{\mathrm{1,1}}^d{\mathbf{Z}}_{\mathrm{2},I}.\end{array}$

(12b)

Combining (12a) and (12b) in one matrix form, we ended up the second step by writing the transfer equation and transfer matrix of a system located between two levels as

${\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{1}}\\ {\mathbf{Z}}_{\mathrm{2}}\end{array}\right\}}_O=\underset{\text{transfer}\hspace{0.17em}\hspace{0.17em}\text{matrix}}{\underbrace{\left[\begin{array}{cc}{\mathbf{U}}_{\mathrm{1,1}}^d& {\mathbf{U}}_{\mathrm{1,1}}^c\\ {\mathbf{U}}_{\mathrm{1,1}}^c& {\mathbf{U}}_{\mathrm{1,1}}^d\end{array}\right]}}\mathrm{\hspace{0.17em}\hspace{0.17em}}{\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{1}}\\ {\mathbf{Z}}_{\mathrm{2}}\end{array}\right\}}_I.$

(12c)

The transfer equation for two systems j, 1 and 1, m − 1 connected with multi-level (1–2–3) through three connection points G, C, and V shown in Figures 6 and 7(e) can be obtained similarly:

$\begin{array}{c}{\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{1}}\\ {\mathbf{Z}}_{\mathrm{2}}\\ {\mathbf{Z}}_{\mathrm{3}}\end{array}\right\}}_O=\left[\begin{array}{ccc}{\mathbf{U}}_{j,\mathrm{1}}^d& {\mathbf{U}}_{j,\mathrm{1}}^c& \mathbf{0}\\ {\mathbf{U}}_{j,\mathrm{1}}^c& {\mathbf{U}}_{j,\mathrm{1}/\mathrm{1},m-\mathrm{1}}^d& {\mathbf{U}}_{\mathrm{1},m-\mathrm{1}}^c\\ \mathbf{0}& {\mathbf{U}}_{\mathrm{1},m-\mathrm{1}}^c& {\mathbf{U}}_{\mathrm{1},m-\mathrm{1}}^d\end{array}\right]\mathrm{\hspace{0.17em}\hspace{0.17em}}{\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{1}}\\ {\mathbf{Z}}_{\mathrm{2}}\\ {\mathbf{Z}}_{\mathrm{3}}\end{array}\right\}}_I,\\ \text{where}\hspace{0.17em}\hspace{0.17em}{\mathbf{U}}_{j,\mathrm{1}}^d\equiv \mathbf{I}-{\mathbf{D}}_{j,\mathrm{1}},\hspace{1em}{\mathbf{U}}_{j,\mathrm{1}}^c\equiv {\mathbf{D}}_{j,\mathrm{1}}\\ {\mathbf{U}}_{\mathrm{1},m-\mathrm{1}}^d\equiv \mathbf{I}-{\mathbf{D}}_{\mathrm{1},m-\mathrm{1}},\hspace{1em}{\mathbf{U}}_{\mathrm{1},m-\mathrm{1}}^c\equiv {\mathbf{D}}_{\mathrm{1},m-\mathrm{1}}\\ {\mathbf{U}}_{j,\mathrm{1}/\mathrm{1},m-\mathrm{1}}^d\equiv \mathbf{I}-{\mathbf{D}}_{j,\mathrm{1}/\mathrm{1},m-\mathrm{1}},\\ {\mathbf{D}}_{j,\mathrm{1}/\mathrm{1},m-\mathrm{1}}=\left({\left(k_y+c\lambda \right)}_{j,\mathrm{1}}+{\left(k_y+c\lambda \right)}_{\mathrm{1},m-\mathrm{1}}\right){\mathbf{e}}_{\mathrm{4}}{\mathbf{e}}_{\mathrm{1}}^T.\end{array}$

(12d)

Now we are in the position to end the solution scenario. From the MS-TMM topology point of view, the system shown in Figure 1 or equivalently Figure 6 can be modeled as a multichain, following the similar procedures mentioned above in Section 3.3 to determine the overall transfer equation and ending up with determinant solution of the reduced overall transfer matrix.

5. Numerical Examples

Before the free vibration analysis of a multi-level elastic beam coupled by a spring/dashpot system(s) is performed, the reliability of the suggested scenario and the Matlab computer program developed for this paper are confirmed by comparing the present results with those obtained from the existing literature.

5.1. Reliability of the Suggested Scenario

Figure 8(a) consists of two free-free and clamped-clamped undamped beams connected to each other with two symmetrically distributed linear translational springs. The hybrid system is vibrating transversely in the x − y plane. The beams are supposed to have the same length, material, and geometrical data, that is, (L = 1 m, ${\stackrel{-}{m}}_{\mathrm{1}}={\stackrel{-}{m}}_{\mathrm{2}}=\stackrel{-}{m}$ and (EI)₁ = (EI)₂ = EI). It is worth mentioning here that the proposal scenario can be extended to a system consisting of any number of uniform/nonuniform beams coupled with any number of spring/dashpot systems. However, the two springs k_{y1, 1} and k_{y2, 1} are located at distances x₁ and x₂ = L − x₁, respectively, and are moving as a function of x₁ from 0 to 0.5 m as shown in Figure 8(a). This example is presented by Kukla [2], which studied the dynamics of the longitudinal vibrations of two rods coupled by several translational springs using Green's functional method. Figure 8(b) shows a vibrating two-level beam comprised of 10 components with 4 boundaries, namely, Z_1,0, Z_6,0 on the system left side and Z_5,0, Z_10,0 on the right side. Herein, Z _i,k = [Y,Θ _z ,M _z ,Q _y ] _i,k ^T . Chain MS-TMM topology is systematic to apply. Following the transfer direction from the left to right and applying (2) for two-level, the transfer equation can be written as

$\begin{array}{c}{\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{5,0}}\\ {\mathbf{Z}}_{\mathrm{8,0}}\end{array}\right\}}_O=\left[\begin{array}{cc}{\mathbf{U}}_{\mathrm{5}}& \mathbf{0}\\ \mathbf{0}& {\mathbf{U}}_{\mathrm{8}}\end{array}\right]{\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{4,5}}\\ {\mathbf{Z}}_{\mathrm{4,8}}\end{array}\right\}}_I\\ \downarrow \\ {\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{4,5}}\\ {\mathbf{Z}}_{\mathrm{4,8}}\end{array}\right\}}_O=\left[\begin{array}{cc}{\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},d}& {\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},c}\\ {\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},c}& {\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},d}\end{array}\right]{\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{3,4}}\\ {\mathbf{Z}}_{\mathrm{7,4}}\end{array}\right\}}_I\\ \downarrow \\ {\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{3,4}}\\ {\mathbf{Z}}_{\mathrm{7,4}}\end{array}\right\}}_O=\left[\begin{array}{cc}{\mathbf{U}}_{\mathrm{3}}& \mathbf{0}\\ \mathbf{0}& {\mathbf{U}}_{\mathrm{7}}\end{array}\right]{\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{2,3}}\\ {\mathbf{Z}}_{\mathrm{2,7}}\end{array}\right\}}_I\\ \downarrow \\ {\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{2,3}}\\ {\mathbf{Z}}_{\mathrm{2,7}}\end{array}\right\}}_O=\left[\begin{array}{cc}{\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},d}& {\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},c}\\ {\mathbf{U}}_{\mathrm{1,1}}^c& {\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},d}\end{array}\right]{\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{1,2}}\\ {\mathbf{Z}}_{\mathrm{6,2}}\end{array}\right\}}_I\\ \downarrow \\ {\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{1,2}}\\ {\mathbf{Z}}_{\mathrm{6,2}}\end{array}\right\}}_O=\left[\begin{array}{cc}{\mathbf{U}}_{\mathrm{1}}& \mathbf{0}\\ \mathbf{0}& {\mathbf{U}}_{\mathrm{6}}\end{array}\right]{\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{1,0}}\\ {\mathbf{Z}}_{\mathrm{6,0}}\end{array}\right\}}_I\\ \Downarrow \\ {\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{5,0}}\\ {\mathbf{Z}}_{\mathrm{8,0}}\end{array}\right\}}_O=\left[\begin{array}{cc}{\mathbf{U}}_{\mathrm{5}}& \mathbf{0}\\ \mathbf{0}& {\mathbf{U}}_{\mathrm{8}}\end{array}\right]\left[\begin{array}{cc}{\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},d}& {\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},c}\\ {\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},c}& {\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},d}\end{array}\right]\left[\begin{array}{cc}{\mathbf{U}}_{\mathrm{3}}& \mathbf{0}\\ \mathbf{0}& {\mathbf{U}}_{\mathrm{7}}\end{array}\right]\times \left[\begin{array}{cc}{\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},d}& {\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},c}\\ {\mathbf{U}}_{\mathrm{1,1}}^c& {\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},d}\end{array}\right]\left[\begin{array}{cc}{\mathbf{U}}_{\mathrm{1}}& \mathbf{0}\\ \mathbf{0}& {\mathbf{U}}_{\mathrm{6}}\end{array}\right]{\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{1,0}}\\ {\mathbf{Z}}_{\mathrm{6,0}}\end{array}\right\}}_I\\ \downarrow \\ {\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{5,0}}\\ {\mathbf{Z}}_{\mathrm{8,0}}\end{array}\right\}}_O=\left[\begin{array}{cc}\left(\begin{array}{c}{\mathbf{U}}_{\mathrm{5}}{\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},d}{\mathbf{U}}_{\mathrm{3}}{\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},d}{\mathbf{U}}_{\mathrm{1}}\\ +{\mathbf{U}}_{\mathrm{5}}{\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},c}{\mathbf{U}}_{\mathrm{7}}{\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},c}{\mathbf{U}}_{\mathrm{1}}\end{array}\right)& \left(\begin{array}{c}{\mathbf{U}}_{\mathrm{5}}{\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},d}{\mathbf{U}}_{\mathrm{3}}{\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},c}{\mathbf{U}}_{\mathrm{6}}\\ +{\mathbf{U}}_{\mathrm{5}}{\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},c}{\mathbf{U}}_{\mathrm{7}}{\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},d}{\mathbf{U}}_{\mathrm{6}}\end{array}\right)\\ \left(\begin{array}{c}{\mathbf{U}}_{\mathrm{8}}{\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},c}{\mathbf{U}}_{\mathrm{3}}{\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},d}{\mathbf{U}}_{\mathrm{1}}\\ +{\mathbf{U}}_{\mathrm{8}}{\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},d}{\mathbf{U}}_{\mathrm{7}}{\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},c}{\mathbf{U}}_{\mathrm{1}}\end{array}\right)& \left(\begin{array}{c}{\mathbf{U}}_{\mathrm{8}}{\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},c}{\mathbf{U}}_{\mathrm{3}}{\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},c}{\mathbf{U}}_{\mathrm{6}}\\ +{\mathbf{U}}_{\mathrm{8}}{\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},d}{\mathbf{U}}_{\mathrm{7}}{\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},d}{\mathbf{U}}_{\mathrm{6}}\end{array}\right)\end{array}\right]\times {\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{1,0}}\\ {\mathbf{Z}}_{\mathrm{6,0}}\end{array}\right\}}_I\\ \downarrow \\ {\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{5,0}}\\ {\mathbf{Z}}_{\mathrm{8,0}}\end{array}\right\}}_O=\left[\begin{array}{cc}{\mathbf{T}}_{\mathrm{1,1}}& {\mathbf{T}}_{\mathrm{1,2}}\\ {\mathbf{T}}_{\mathrm{2,1}}& {\mathbf{T}}_{\mathrm{2,1}}\end{array}\right]{\left\{\begin{array}{c}{\mathbf{Z}}_{\mathrm{1,0}}\\ {\mathbf{Z}}_{\mathrm{6,0}}\end{array}\right\}}_I.\end{array}$

(13)

U₁, U₃, U₅, U₆, U₇, and U₈ represent the beam segment transfer matrices (8b). U_1,1^2,d, U_1,1^2,c, and U_2,1^4,d, U_2,1^4,c are the transfer matrices of a system between two connection points located between two beams (12c). The overall transfer equation, overall SV, and overall transfer matrix are, respectively,

${{\mathbf{U}}_{\text{all}}|}_{\mathrm{8}\times \mathrm{16}}\mathrm{\hspace{0.17em}\hspace{0.17em}}{{\mathbf{Z}}_{\text{all}}|}_{\mathrm{16}\times \mathrm{1}}=\mathbf{0},$

(14a)

$\text{where}\hspace{0.17em}\hspace{0.17em}{\mathbf{Z}}_{\text{all}}^T={\left[\begin{array}{cccc}\underset{\mathrm{4}\times \mathrm{1}}{\underbrace{{{\mathbf{Z}}_{\mathrm{1,0}}}^T}}& \underset{\mathrm{4}\times \mathrm{1}}{\underbrace{{{\mathbf{Z}}_{\mathrm{6,0}}}^T}}& \underset{\mathrm{4}\times \mathrm{1}}{\underbrace{{{\mathbf{Z}}_{\mathrm{5,0}}}^T}}& \underset{\mathrm{4}\times \mathrm{1}}{\underbrace{{{\mathbf{Z}}_{\mathrm{8,0}}}^T}}\end{array}\right]}^T,$

(14b)

${\mathbf{U}}_{\text{all}}=\left[\begin{array}{cccc}{{\mathbf{T}}_{\mathrm{1,1}}|}_{\mathrm{4}\times \mathrm{4}}& {{\mathbf{T}}_{\mathrm{1,2}}|}_{\mathrm{4}\times \mathrm{4}}& -{\mathbf{I}}_{\mathrm{4}\times \mathrm{4}}& {\mathbf{O}}_{\mathrm{4}\times \mathrm{4}}\\ {{\mathbf{T}}_{\mathrm{2,1}}|}_{\mathrm{4}\times \mathrm{4}}& {{\mathbf{T}}_{\mathrm{2,2}}|}_{\mathrm{4}\times \mathrm{4}}& {\mathbf{O}}_{\mathrm{4}\times \mathrm{4}}& -{\mathbf{I}}_{\mathrm{4}\times \mathrm{4}}\end{array}\right].$

(14c)

Figure 8:

(a) Free-free and clamped-clamped beams connected with two linear translational springs, (b) chain TMM-MS: State vectors and transfer direction of the suggested solution scenario.

Applying the boundary conditions listed in Table 1, half of state variables of Z_all (14b) are zeros due to known constraints. Thus, the overall transfer equation reduced to ${{\stackrel{-}{\mathbf{U}}}_{\text{all}}|}_{\mathrm{8}\mathrm{\hspace{0.17em}\hspace{0.17em}}\times \mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{8}}{{\stackrel{-}{\mathbf{Z}}}_{\text{all}}|}_{\mathrm{8}\mathrm{\hspace{0.17em}\hspace{0.17em}}\times \mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{1}}=\mathbf{0}$ that is ready for eigenproblem (5).

Table 1:

Common boundary conditions for a beam vibrating in a plane.

Support type	Zero terms	Nonzero terms
Fixed	Y,Θ z	Q y ,M z
Pinned	Y,M z	Q y ,Θ z
Free	Q y ,M z	Y,Θ z
Guided	Q y ,Θ z	Y,M z

The natural frequencies of the system can now be computed by zero search of the determinant, which is based on sign change of Δ(ω) (in case of real numbers) or Δ(λ) (in case of complex numbers) during a scanning of an interesting frequency range. This procedure can be cumbersome for several reasons. Therefore, reliable and efficient algorithms called recursive scanning approach are applied (see [17] for more details of the proposed algorithm) by switching from zero search for Δ to minimization of the absolute value |Δ| of the determinant, which is equally well applicable to both the real and the complex cases. The algorithm general idea is as follows. In a first iteration step, it divides an interesting band of frequencies into a number of sample points and searches for lower peaks. Each region having a lower peak is then divided again into small intervals to find more narrow regions of lower peaks as second iteration step. The algorithm proceeds until the required precision of roots is achieved. MS-TMM natural frequency results are obtained using fMin1D algorithm [17] (frequency ranges $\underset{{\lambda }^i=\omega }{\underbrace{\left[\mathrm{1,3000}\right]}}\hspace{0.17em}\text{rad}/\text{sec}$ , scanning grids N_x0 = 500, absolute precision tolerance ε = 10⁻⁶). Figure 9 shows log₁₀|Δ| obtained from the fMin1D algorithm versus the first 10 dimensionless frequency values ( $\left(\overline{\omega }=\sqrt[4]{{\left({\lambda }^i\equiv \omega \right)}^2\overline{m}{L}^4/EI}\right)$ ) for k_y1,1 = k_y2,1 = k _y = 100(EI/L³) and x₁ = 0.3 m. $\stackrel{-}{\omega }$ values are plotted versus the location of the two springs in Figures 10(a) and 10(b) for k _y = 100(EI/L³) and k _y = 1000(EI/L³), respectively. The first and second springs are moving opposite to each other from x₁ = 0 at the beams support ends and coincide together at the middle of the beams (x₁ = 0.5 m). Reference [2] does not provide the results as Tables or specific data but only showed Figures. The reader may download the reference from the library or from the Internet because the authors of the present paper cannot include the figures to maintain the journal publication rights. However, the MS-TMM results are in very good agreement with Figures 2(a) and 2(b) [2, page 131].

Figure 9:

fMin1D function determinant of a system consisting of free-free and clamped-clamped beams coupled with two springs. k _y = 100(EI/L³) and x₁ = 0.3 m.

Figure 10:

Dimensionless frequency parameter $\stackrel{-}{\omega }$ values as a function of the spring locations (x₁ = 0 → 0.5 m) for a system consisting of free-fee and clamped-clamped beams coupled by two springs. (a) k _y = 100(EI/L³); (b) k _y = 1000(EI/L³).

5.2. Free Vibration of Damped Elastically Coupled Triple Beams

The physical model of the transversely vibrating system under consideration is composed of three parallel uniform rectangular Euler-Bernoulli beams of homogenous properties. Each two-level beam is joined (connected) together by two spring/dashpot systems located at x₁ and x₂, respectively (see Figure 1). The beams have the same length and are pinned at their ends. The small damped vibrations of the system are considered. In the sense of the chain MS-TMM (see Figure 11), the overall transfer equation, overall transfer matrix, and overall SV are, respectively,

${\mathbf{U}}_{\text{all}}{\mathbf{Z}}_{\text{all}}=\mathbf{0},$

(15a)

$\text{where}\hspace{0.17em}\hspace{0.17em}{\mathbf{U}}_{\text{all}}={\left[\begin{array}{cccccc}{{\mathbf{T}}_{\mathrm{1,1}}|}_{\mathrm{4}\times \mathrm{4}}& {{\mathbf{T}}_{\mathrm{1,2}}|}_{\mathrm{4}\times \mathrm{4}}& {{\mathbf{T}}_{\mathrm{1,3}}|}_{\mathrm{4}\times \mathrm{4}}& -{\mathbf{I}}_{\mathrm{4}\times \mathrm{4}}& {\mathbf{O}}_{\mathrm{4}\times \mathrm{4}}& {\mathbf{O}}_{\mathrm{4}\times \mathrm{4}}\\ {{\mathbf{T}}_{\mathrm{2,1}}|}_{\mathrm{4}\times \mathrm{4}}& {{\mathbf{T}}_{\mathrm{2,2}}|}_{\mathrm{4}\times \mathrm{4}}& {{\mathbf{T}}_{\mathrm{2,3}}|}_{\mathrm{4}\times \mathrm{4}}& {\mathbf{O}}_{\mathrm{4}\times \mathrm{4}}& -{\mathbf{I}}_{\mathrm{4}\times \mathrm{4}}& {\mathbf{O}}_{\mathrm{4}\times \mathrm{4}}\\ {{\mathbf{T}}_{\mathrm{3,1}}|}_{\mathrm{4}\times \mathrm{4}}& {{\mathbf{T}}_{\mathrm{3,2}}|}_{\mathrm{4}\times \mathrm{4}}& {{\mathbf{T}}_{\mathrm{3,3}}|}_{\mathrm{4}\times \mathrm{4}}& {\mathbf{O}}_{\mathrm{4}\times \mathrm{4}}& {\mathbf{O}}_{\mathrm{4}\times \mathrm{4}}& -{\mathbf{I}}_{\mathrm{4}\times \mathrm{4}}\end{array}\right]}_{\mathrm{12}\mathrm{\hspace{0.17em}\hspace{0.17em}}\times \mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{24}},$

(15b)

${\mathbf{Z}}_{\text{all}}^T={\left[\begin{array}{cccccc}{{\mathbf{Z}}_{\mathrm{1,0}}}^T& {{\mathbf{Z}}_{\mathrm{6,0}}}^T& {{\mathbf{Z}}_{\mathrm{11,0}}}^T& {{\mathbf{Z}}_{\mathrm{5,0}}}^T& {{\mathbf{Z}}_{\mathrm{10,0}}}^T& {{\mathbf{Z}}_{\mathrm{13,0}}}^T\end{array}\right]}_{\mathrm{24}\times \mathrm{1}}^T,$

(15c)

$\begin{array}{c}{\mathbf{T}}_{\mathrm{1,1}}={\mathbf{U}}_{\mathrm{5}}{\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},d}{\mathbf{U}}_{\mathrm{3}}{\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},d}{\mathbf{U}}_{\mathrm{1}}+{\mathbf{U}}_{\mathrm{5}}{\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},c}{\mathbf{U}}_{\mathrm{8}}{\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},c}{\mathbf{U}}_{\mathrm{1}},\\ {\mathbf{T}}_{\mathrm{1,2}}={\mathbf{U}}_{\mathrm{5}}{\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},d}{\mathbf{U}}_{\mathrm{3}}{\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},c}{\mathbf{U}}_{\mathrm{6}}+{\mathbf{U}}_{\mathrm{5}}{\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},c}{\mathbf{U}}_{\mathrm{8}}{\mathbf{U}}_{\mathrm{1,1}/\mathrm{1,2}}^{\mathrm{2}/\mathrm{7},d}{\mathbf{U}}_{\mathrm{6}},\\ {\mathbf{T}}_{\mathrm{1,3}}={\mathbf{U}}_{\mathrm{5}}{\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},c}{\mathbf{U}}_{\mathrm{8}}{\mathbf{U}}_{\mathrm{1,2}}^{\mathrm{7},c}{\mathbf{U}}_{\mathrm{11}},\\ {\mathbf{T}}_{\mathrm{2,1}}={\mathbf{U}}_{\mathrm{10}}{\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},c}{\mathbf{U}}_{\mathrm{3}}{\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},d}{\mathbf{U}}_{\mathrm{1}}+{\mathbf{U}}_{\mathrm{10}}{\mathbf{U}}_{\mathrm{2,1}/\mathrm{2,2}}^{\mathrm{4}/\mathrm{9},d}{\mathbf{U}}_{\mathrm{8}}{\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},c}{\mathbf{U}}_{\mathrm{1}},\\ {\mathbf{T}}_{\mathrm{2,2}}={\mathbf{U}}_{\mathrm{10}}{\mathbf{U}}_{\mathrm{2,1}}^{\mathrm{4},c}{\mathbf{U}}_{\mathrm{3}}{\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},c}{\mathbf{U}}_{\mathrm{6}}+{\mathbf{U}}_{\mathrm{10}}{\mathbf{U}}_{\mathrm{2,1}/\mathrm{2,2}}^{\mathrm{4}/\mathrm{9},d}{\mathbf{U}}_{\mathrm{8}}{\mathbf{U}}_{\mathrm{1,1}/\mathrm{1,2}}^{\mathrm{2}/\mathrm{7},d}{\mathbf{U}}_{\mathrm{6}}+{\mathbf{U}}_{\mathrm{10}}{\mathbf{U}}_{\mathrm{2,2}}^{\mathrm{9},c}{\mathbf{U}}_{\mathrm{12}}{\mathbf{U}}_{\mathrm{1,2}}^{\mathrm{7},c}{\mathbf{U}}_{\mathrm{6}},\\ {\mathbf{T}}_{\mathrm{2,3}}={\mathbf{U}}_{\mathrm{10}}{\mathbf{U}}_{\mathrm{2,1}/\mathrm{2,2}}^{\mathrm{4}/\mathrm{9},d}{\mathbf{U}}_{\mathrm{8}}{\mathbf{U}}_{\mathrm{1,2}}^{\mathrm{7},c}{\mathbf{U}}_{\mathrm{11}}+{\mathbf{U}}_{\mathrm{10}}{\mathbf{U}}_{\mathrm{2,2}}^{\mathrm{9},c}{\mathbf{U}}_{\mathrm{12}}{\mathbf{U}}_{\mathrm{1,2}}^{\mathrm{7},d}{\mathbf{U}}_{\mathrm{11}},\\ {\mathbf{T}}_{\mathrm{3,1}}={\mathbf{U}}_{\mathrm{13}}{\mathbf{U}}_{\mathrm{2,2}}^{\mathrm{9},c}{\mathbf{U}}_{\mathrm{8}}{\mathbf{U}}_{\mathrm{1,1}}^{\mathrm{2},c}{\mathbf{U}}_{\mathrm{1}},\\ {\mathbf{T}}_{\mathrm{3,2}}={\mathbf{U}}_{\mathrm{13}}{\mathbf{U}}_{\mathrm{2,2}}^{\mathrm{9},c}{\mathbf{U}}_{\mathrm{8}}{\mathbf{U}}_{\mathrm{1,1}/\mathrm{1,2}}^{\mathrm{2}/\mathrm{7},d}{\mathbf{U}}_{\mathrm{6}}+{\mathbf{U}}_{\mathrm{13}}{\mathbf{U}}_{\mathrm{2,2}}^{\mathrm{9},d}{\mathbf{U}}_{\mathrm{12}}{\mathbf{U}}_{\mathrm{1,2}}^{\mathrm{7},c}{\mathbf{U}}_{\mathrm{6}},\\ {\mathbf{T}}_{\mathrm{3,3}}={\mathbf{U}}_{\mathrm{13}}{\mathbf{U}}_{\mathrm{2,2}}^{\mathrm{9},c}{\mathbf{U}}_{\mathrm{8}}{\mathbf{U}}_{\mathrm{1,2}}^{\mathrm{7},c}{\mathbf{U}}_{\mathrm{11}}+{\mathbf{U}}_{\mathrm{13}}{\mathbf{U}}_{\mathrm{2,2}}^{\mathrm{9},d}{\mathbf{U}}_{\mathrm{12}}{\mathbf{U}}_{\mathrm{1,2}}^{\mathrm{7},d}{\mathbf{U}}_{\mathrm{11}}.\end{array}$

(15d)

Herein, U₁, U₃, U₅, U₆, U₈, U₁₀, U₁₁, U₁₂, and U₁₃ represent the beam segment transfer matrices; U_1,1^2,d, U_1,1^2,c, U_2,1^4,d, U_2,1^4,c, U_1,2^7,d, U_1,2^7,c, U_2,2^9,d, and U_2,2^9,c are the transfer matrices of a system between two connection points located between two beams (12c). U_1,1/1,2^2/7,d and U_2,1/2,2^4/9,d are the transfer matrices for two systems connected with three-level through three connection points (12d). Applying the boundary conditions, that is, Z_1,0 = [0,Θ _z ,0,Q _y ]_1,0 ^T ; Z_6,0 = [0,Θ _z ,0,Q _y ]_6,0 ^T ; Z_11,0 = [0,Θ _z ,0,Q _y ]_11,0 ^T , Z_5,0 = [0,Θ _z ,0,Q _y ]_5,0 ^T , Z_10,0 = [0,Θ _z ,0,Q _y ]_10,0 ^T , and Z_13,0 = [0,Θ _z ,0,Q _y ]_13,0 ^T , the overall transfer equation (15a) reduced to $\underset{\mathrm{12}\mathrm{\hspace{0.17em}\hspace{0.17em}}\times \mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{12}}{\underbrace{{\stackrel{-}{\mathbf{U}}}_{\text{all}}}}\underset{\mathrm{12}\mathrm{\hspace{0.17em}\hspace{0.17em}}\times \mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{1}}{\underbrace{{\stackrel{-}{\mathbf{Z}}}_{\text{all}}}}=\mathbf{0}$ . The geometric and material properties (steel) of the multi-level beam system are given as follows: L = 1 m, b₁ = b₂ = b₃ = 0.02 m, h₁ = h₂ = h₃ = 0.01 m, E₁ = E₂ = E₃ = E = 2.069 × 10¹¹ N/m², and ρ₁ = ρ₂ = ρ₃ = ρ = 7850 kg/m³. Here, b _i and h _i represent the width and height of the beam, respectively. The cross-sectional area and the moment of inertial of the beam cross-section are A _i = b _i × h _i and I _i = b _i × h _i ³/12, respectively. For comparison, the lowest three “exact” dimensionless frequency values $\stackrel{-}{\omega }$ for the single pinned-pinned beam that might be found in structural or vibration text books are 3.1416, 6.2832 and 9.4248. MS-TMM results for undamped, uncoupled pinned-pinned three-beam are 3.141592, 6.28318, and 9.424777 as shown in Figure 12 using fMin1D algorithm. For undamped elastically coupled three-beam, Figure 13 shows that $\stackrel{-}{\omega }$ values were plotted versus the location movement of spring systems. Systems 2 and 4 (k_y1,1 = k_y1,2 = k _y = 25(EI/L³)) and systems 7 and 9 (k_y1,2 = k_y2,2 = k _y = 25(EI/L³)) are moving opposite to each other from (x₁ = 0 → 0.5 m). The system shows symmetric and antisymmetric vibrations. For symmetric vibration, the frequency parameters ${\stackrel{-}{\omega }}_{\mathrm{1,1}}$ , ${\stackrel{-}{\omega }}_{\mathrm{1,2}}$ , and ${\stackrel{-}{\omega }}_{\mathrm{1,3}}$ (where l and p are the beam number and vibration mode, resp.) are constant throughout the spring systems movement as seen in Figure 13(a). The antisymmetric of the first mode vibration ${\stackrel{-}{\omega }}_{\mathrm{2,1}}$ and ${\stackrel{-}{\omega }}_{\mathrm{3,1}}$ started with equally values as ${\stackrel{-}{\omega }}_{\mathrm{1,1}}$ at x₁ = 0 (spring systems are rigidly mounted at the support ends) and increased significantly at x₁ = 0.5 m (both spring systems are in the same position). However, it is evident that ${\stackrel{-}{\omega }}_{\mathrm{2,1}}$ and ${\stackrel{-}{\omega }}_{\mathrm{3,1}}$ are sensitive to the variation of spring positions. The results presented of the other antisymmetric vibration modes, ${\stackrel{-}{\omega }}_{\mathrm{2,2}}$ , ${\stackrel{-}{\omega }}_{\mathrm{3,2}}$ , ${\stackrel{-}{\omega }}_{\mathrm{2,3}}$ , and ${\stackrel{-}{\omega }}_{\mathrm{3,3}}$ , indicate that as the order of the frequency increases, the shape of the frequency vibration versus the spring systems location resembles a half or full sine wave with insignificant variation in the frequency ranges. The strong ability of fMin1D algorithm is shown in Figure 13(b) by capturing the very narrow regions of lower peaks.

Figure 11:

Chain TMM-MS: state vectors and transfer direction of pinned-pinned multi-level beam connected by spring/dashpot systems.

Figure 12:

fMin1D function determinant to evaluate the lowest three $\stackrel{-}{\omega }$ for undamped, uncoupled pinned-pinned three-beam.

Figure 13:

(a) The lowest three $\stackrel{-}{\omega }$ values as a function of the spring systems locations (x₁ = 0 → 0.5 m) for undamped, coupled pinned-pinned three-beam; (b) fMin1D function determinant to evaluate the dimensionless frequency parameter $\stackrel{-}{\omega }$ for k _y = 25(EI/L³) and x₁ = 0.25 m.

The effect of the dashpot is included in the system making the model more complicated. The two spring/dashpot systems coupled with three-beam are located at position x₁ = 0.25 m. The spring/dashpot system in parallel is similar to viscoelastic material model. The given values of damping are $c_{\mathrm{1,1}}=c_{\mathrm{2,1}}=c_{\mathrm{1,2}}=c_{\mathrm{2,2}}=c=\mathrm{5}\mathrm{\hspace{0.17em}\hspace{0.17em}}\sqrt{EI\mathrm{\hspace{0.17em}\hspace{0.17em}}\stackrel{-}{m}/L^{\mathrm{2}}}$ . Next, it is the turn to implement fMin2D algorithm [17] for complex eigenproblem solution. For undamped given system as it is computed above, $\stackrel{-}{\omega }$ values do not exceed more than 10, which is equivalent to (λ ⁱ = ω ≤ 1500 rad/sec). However, the damping (λ ^r = δ) is unknown in range. Therefore, within the required system frequency band, a couple scanning grids are demanded to understand and explore the damping ranges of the system. Figure 14 shows different scanning grids for damping ranges as follows: (a) [− 2000,2000], (b) [− 1000,1000], (c) [− 500,500], and (d) [− 100,600]. It is obviously seen that there are 9 promising regions distributed randomly on the surface. These regions represent the complex eigenvalues of the system. In the following searching complex roots, the damping range (Figure 14(d)) is considered. MS-TMM for damped, coupled three-beam results are shown in Figure 15 and Table 2 after using fMin2D algorithm coded under Matlab environment with the following input data: $(\underset{{\lambda }^i}{\underbrace{\left[-\mathrm{1,1500}\right]}}$ , $\underset{{\lambda }^r}{\underbrace{\left[-\mathrm{100,600}\right]}}$ , N_x0 = 35, N_y0 = 20, ε _x = 10⁻⁸, ε _y = 10⁻⁸).

Table 2:

Chain MS-TMM eigenvalues results of damped, coupled pinned-pinned three-beam.

λ = − λ r ±iλ i	Chain MS-TMM results using fMin2D λ i (rad/sec)
λ1	− 2.463512 × 102 + 4.010226 × 101i
λ2	− 2.074369 × 10−9 + 1.462698 × 102i
λ3	− 7.409510 × 101 + 1.652322 × 102i
λ4	− 4.577982 × 102 + 4.615502 × 102i
λ5	− 2.074369 × 10−9 + 5.850790 × 102i
λ6	− 1.484833 × 102 + 5.861790 × 102i
λ7	− 2.159398 × 102 + 1.243120 × 103i
λ8	− 7.448404 × 101 + 1.311750 × 103i
λ9	− 2.074369 × 10−9 + 1.316428 × 103i

Figure 14:

fMin2D first step: damping (λ ^r = δ) range scanning of a specific frequency band (λ ⁱ = ω ≤ 1500 rad/sec) for damped, coupled pinned-pinned three-beam, (a) [− 2000, 2000], (b) [− 1000, 1000], (c) [− 500, 500], and (d) [− 100, 600].

Figure 15:

fMin2D second step: function determinant to evaluate the eigenvalues (λ = − λ ^r ± iλ ⁱ ) for a damped, coupled pinned-pinned three-beam.

Chain MS-TMM for undamped/damped coupled multilevel beam with the suggested scenario and the two novel recursive scanning algorithms provide a closed-form solution, not only presents the principles of the vibration problem but also shed light on practical applications. Since the solution is almost exact, it allows a complete understanding of a problem.

6. Conclusions

Starting from the principle of mechanics and the elementary formulations for the flexible beam, the free vibration analysis of laterally vibrating system made up of a multi-level Euler-Bernoulli beam to which spring/dashpot systems are attached across the span is performed using one of the Transfer Matrix Method of Linear Multibody Systems (MS-TMM) scenarios. Although the number of coupling springs or spring/dashpot systems considered in the examples given was limited to three, there is no inherent difficulty in extending the current method to solve the problems of vibration of systems consisting of any number of uniform/nonuniform beams with different boundary conditions and coupled with any number of spring/dashpot systems. The numerical results obtained to reveal that the eigenfrequencies calculated by this method are in very good agreement with those obtained by the published literature. Moreover, MS-TMM is encouraging for further investigations of more complex multibody systems of this type with rigid bodies due to simplicity in the formulation of the transfer equation, being systematic to apply, and being easy to program.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.