On the Forced Vibration of Bending-Torsional-Warping Coupled Thin-Walled Beams Carrying Arbitrary Number of 3-DoF Spring-Damper-Mass Subsystems

Abstract

This paper presents an analytical transfer matrix modeling framework for the forced vibration of a bending-torsional-warping coupling Euler-Bernoulli thin-walled beam carrying an arbitrary number of three degree-of-freedom (DOF) spring-damper-mass (SDM) subsystems. The thin-walled beam is divided into a series of distinct sub-beams whose ends are connected to the SDM subsystems. The transfer matrix for each sub-beam is developed based on the exact shape functions of the bending-torsional-warping coupling Euler-Bernoulli theory. Each SDM system is modelled by a set of effective springs based on the dynamic condensation method. The governing matrix equation is formulated based on the compatibility conditions of the placement and the force at the common interfaces of two adjacent sub-beams. Then, a closed-form expression for the frequency response function of the thin-walled beam system is proposed. The results computed by the proposed method achieve good agreement with those obtained by the conventional finite-element method, which shows the accuracy and reliability of the proposed method. The effects of the system parameters on the vibration transmission and vibration isolation properties of the thin-walled beam system are studied. The presented method can simultaneously consider arbitrary number of SDM subsystems and boundary conditions. Furthermore, none of the associated matrices are larger than 12 × 12, which provides a significant computational advantage.

1. Introduction

Thin-walled structures are commonly found in most branches of structural engineering such as aircraft wings, vehicle bodies, ship-hull structures, wind-turbine towers, construction machinery arms and bridge decks. The inconsistency of the shear center and mass center leads to severe bending-torsional-warping coupling dynamic behaviors of the thin-walled structures. The evaluation of these dynamic behaviors of thin-walled structures has long been regarded as a significant task for designers in order to avoid dangerous resonance situations and improve dynamics performance and structural robustness.

A large numbers of researchers have made efforts to deal with the dynamic models of thin-walled structures. Narayanan et al. [1] investigated the free vibration characteristics of a thin-walled open section beam with unconstrained damping layers at the flanges. Bank and Kao [2] studied the forced and free vibration of a thin-walled composite material Timoshenko beam. Rao and Jin [3] proposed a universal grey number-based approach and an interval-discretization method to analyze the coupled bending-torsional vibration of thin-walled beams involving uncertainties. To give a more accurate description to the dynamic behavior of thin-walled structures, tarping is further considered in the modeling of thin-walled structures. Li et al. [4,5,6] employed the dynamic transfer matrix method to investigate the free and stochastic vibration of the bending-torsion of coupled thin-walled structures, and their method took the effect of warping stiffness into account and allowed the presence of axial force. Lezgy-Nazargah et al. [7] proposed a quasi-3D finite element model to study the coupled bending and torsional–warping dynamics of thin-walled beams. Jrad et al. [8] analytically study the free and forced vibration of warping and flexural–torsional coupling thin-walled beams and discussed the dynamic behavior of thin-walled beams in the design of buildings and bridges. Thin-walled structures with complex shape section were also extensively studied for practical applications. Based on the d’Alembert principle, Chen and Hsiao [9] derived the governing equations of the axial-torsional coupled vibration of thin-walled Z-section beam. Prokić [10,11] derived the governing differential equations for triply coupled and fivefold coupled vibration of thin-walled with arbitrary cross-section by using the principle of virtual displacements. Chen and Hsiao [12] presented a finite element formulation for the quadruply coupled vibration analysis of thin-walled beams with a generic open section. By introducing Vlasov’s assumptions, Kim et al. [13,14] investigated the free vibration and spatial stability of the circular curved thin-walled beams with non-symmetric sections. Xu et al. [15] analyzed the high-order vibration modes of thin-walled beams with complex open cross-sections using the framework of Carrera Unified Formulation.

The studies in Refs. [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] concentrated on the dynamic behavior of uniform beams, with no attachments or subsystems mounted along the span. However, in engineering practice, thin-walled structures such as aircraft wings, vehicle bodies, ship-hulls and wind-turbine towers are often mounted with lumped mass oscillations. Hence, study of the dynamic behaviors of thin-walled structures with elastically mounted masses is desperately needed for the structural design of these systems. Many investigations have been focused on the dynamic behavior of classical Bernoulli-Euler and/or Timoshenko beams with single or multiple lumped masses, including the free vibration (beams with attachments [16,17,18], beams with single-degree of freedom SDM subsystems [19,20,21,22,23], beams with two-degree of freedom SDM subsystems [24,25,26,27,28]) and forced vibration (beams with attachments [17,29], beams with single-degree of freedom SDM subsystems [30,31,32,33,34,35], and beams with two-degree of freedom SDM subsystem [36,37]) dynamics. In general, for a large number of thin-walled beams used in practical engineering, the shear center and the mass centroid are not coincident, which results in a coupling of the bending and torsional modes of the oscillation that are independent in classical beams. Therefore, the coupled vibration between rigid motions of subsystems and thin-walled beams becomes more complicated than classical beams and needs to be clarified [38,39]. Oguamanam et al. studied the coupled flexural–torsional free vibration of thin-walled beams with tip mass attachments [38]. Gökdağ and Kopmaz [39] analyzed the influences of tip mass and distributed mass on the coupled bending and torsional free vibration characteristic of cantilever thin-walled beams with different cross sections. Wang et al. [40] investigated the natural frequencies and mode shapes of a thin-walled turbine tower with different elastic attachments, and in his model the variations of material and cross section are considered using D’Alembert’s principle. Wu and Titurus [41] studied the damping enhancement of thin-walled a helicopter blades system by internally distributed spring-damper elements; in this research, the closed-form governing equation of the thin-walled systems are established by Lagrange’s equation. Hoffmeyer and Høgsberg [42] studied the vibration absorber characteristic of multiple tuned mass absorbers acting on bending-torsion coupled thin-walled beams. In the above studies, the dimensions of the governing equations are increased with the number of subsystems, which result in a heavy burden of computation. By employing the complex modal analysis approach, Burlon et al. obtained the closed analytical solutions for the frequency response of bending-torsional-warping coupled thin-walled beams with arbitrary attached masses, springs [43,44,45], dampers [46] and sub-beams [47], and the free, forced and stochastic vibration dynamics of these models were systematically discussed. Furthermore, all of the associated matrices of their modal are less than 6 × 6, resulting in a significant computational advantage.

It is noticed, however, that, the solutions in [43,44,45,46,47] are unable to compute the frequency response of bending-torsional-warping coupled thin-walled beams with multiple DOF spring-damper-mass subsystems. Hance, closed analytical solutions for the bending-torsional-warping coupled frequency response of thin-walled beams with multiple DOF spring-damper-mass subsystems is significant for the parameter design of multiple DOF absorbance and elastic suspension in vibration reduction application. Furthermore, despite the valuable insights provided by transform methods [48,49], no analytical expressions are available for the bending-torsional-warping coupled frequency response of thin-walled beams with multiple DOF spring-damper-mass subsystems; this requires a specific complex transformation approach with dynamic condensation.

This paper presents a new closed analytical approach to obtaining the forced vibration of bending-torsional-warping coupled thin-walled beams with an arbitrary number of 3-DoF spring-damper-mass subsystems and boundary conditions, based on the transfer matrix approach and dynamic condensation methods. The size of all associated matrices is always less than 12 × 12 regardless of the number of subsystems, which leads to great computational efficiency. The results computed by the proposed method achieve good agreement with those obtained by the conventional finite-element method. The effects of the system parameters on the vibration transmission and vibration isolation properties of the thin-walled beam system are studied.

2. Derivation of the Frequency Response Function

Figure 1 illustrates the physical model of the thin-walled beam system under investigation, which consists of a bending-torsional coupled thin-walled beam with N three-degree-of-freedom spring-damper-mass systems attached to it. The whole beam is subdivided into N + 1 segments with length l_i (i = 1,2⋯N + 1) at the positions xi where (i = 1,2⋯N) the subsystems locate.

The primary assumptions of the system are as follows: (a) the beam is a Euler-Bernoulli thin-walled beam with only the transverse in z and y directions and torsional vibration considered; (b) the cross-section is perfectly rigid in its own plane; (c) all the springs are idealized as general points connecting to the thin-walled beam.

Figure 2a illustrate the physical model for the cross section of the thin-walled beam system at the connection point. As shown in the figure, the variables are defined as follows: m_i the lumped mass; J_i the mass moment of inertia; $k_{z,i}^{\mathrm{L}}$, $k_{z,i}^{\mathrm{R}}$, $k_{y,i}^{\mathrm{L}}$, $k_{z,i}^{\mathrm{R}}$ and $c_{z,i}^{\mathrm{L}}$, $c_{z,i}^{\mathrm{R}}$, $c_{y,i}^{\mathrm{L}}$, $c_{z,i}^{\mathrm{R}}$ denote the stiffness and the damping of the left and right springs of the SDM system in z and y directions, respectively; $l_i^{\mathrm{L}}$ and $l_i^{\mathrm{R}}$ are the distance between the lumped mass and the left and right springs; y_i and θ_i, the translational displacement and the rotational angle of the lumped mass respectively; $v_{z,i}^{\mathrm{L}}$, $v_{z,i}^{\mathrm{R}}$, $v_{y,i}^{\mathrm{L}}$, and $v_{y,i}^{\mathrm{R}}$ the transverse displacements of the beam in z and y directions at the node connected with the two springs, respectively; $u_{z,i}^{\mathrm{L}}$, $u_{z,i}^{\mathrm{R}}$, $u_{y,i}^{\mathrm{L}}$, and $u_{y,i}^{\mathrm{R}}$ denote the corresponding displacements of the lump mass. S and C are the shearing center and mass center of the cross section, respectively. v_i and w_i denote the displacement in z and y direction of the thin-walled beam, ψ_i is rotational displacement in x direction. z_c is the distance between S and C. a and b is the length and wide of the cross section.

2.1. Dynamic Condensation of the Three-DOF SDM Subsystems

The internal force of springs $k_{y,i}^{\mathrm{L}}$, $k_{y,i}^{\mathrm{R}}$, $k_{z,i}^{\mathrm{L}}$ and $k_{z,i}^{\mathrm{R}}$ can be reasonably given as

$$F_{y,i}^{\mathrm{L}}=k{e}_{y,i}^{11}{v}_{y,i}^{\mathrm{L}}+k{e}_{y,i}^{12}{v}_{y,i}^{\mathrm{R}}\hspace{1em}\hspace{1em}{F}_{y,i}^{\mathrm{R}}=k{e}_{y,i}^{21}{v}_{y,i}^{\mathrm{L}}+k{e}_{y,i}^{22}{v}_{y,i}^{\mathrm{R}}$$

(1)

$$F_{z,i}^{\mathrm{L}}=k{e}_{z,i}^{11}{v}_{z,i}^{\mathrm{L}}+k{e}_{z,i}^{12}{v}_{z,i}^{\mathrm{R}}\hspace{1em}\hspace{1em}{F}_{z,i}^{\mathrm{R}}=k{e}_{z,i}^{21}{v}_{z,i}^{\mathrm{L}}+k{e}_{z,i}^{22}{v}_{z,i}^{\mathrm{R}}$$

(2)

where

$$k{e}_{y,i}^{11}=\frac{(-m_i{\overline{\omega }}^2-\overline{\omega }{c}_{y,i}^{\mathrm{R}}-k_{y,i}^{\mathrm{R}})(\overline{\omega }{c}_{y,i}^{\mathrm{L}}+k_{y,i}^{\mathrm{L}})}{m_i{\overline{\omega }}^2+\overline{\omega }(c_{y,i}^{\mathrm{L}}+c_{y,i}^{\mathrm{R}})+(k_{y,i}^{\mathrm{L}}+k_{y,i}^{\mathrm{R}})}$$

(3)

$$k{e}_{y,i}^{12}=\frac{(\overline{\omega }{c}_{y,i}^{\mathrm{R}}+k_{y,i}^{\mathrm{R}})(\overline{\omega }{c}_{y,i}^{\mathrm{L}}+k_{y,i}^{\mathrm{L}})}{m_i{\overline{\omega }}^2+\overline{\omega }(c_{y,i}^{\mathrm{L}}+c_{y,i}^{\mathrm{R}})+(k_{y,i}^{\mathrm{L}}+k_{y,i}^{\mathrm{R}})}$$

(4)

$$k{e}_{y,i}^{21}=\frac{(\overline{\omega }{c}_{y,i}^{\mathrm{L}}+k_{y,i}^{\mathrm{L}})(\overline{\omega }{c}_{y,i}^{\mathrm{R}}+k_{y,i}^{\mathrm{R}})}{m_i{\overline{\omega }}^2+\overline{\omega }(c_{y,i}^{\mathrm{L}}+c_{y,i}^{\mathrm{R}})+(k_{y,i}^{\mathrm{L}}+k_{y,i}^{\mathrm{R}})}$$

(5)

$$k{e}_{y,i}^{22}=\frac{(-m_i{\overline{\omega }}^2-\overline{\omega }{c}_{y,i}^{\mathrm{L}}-k_{y,i}^{\mathrm{L}})(\overline{\omega }{c}_{y,i}^{\mathrm{R}}+k_{y,i}^{\mathrm{R}})}{m_i{\overline{\omega }}^2+\overline{\omega }(c_{y,i}^{\mathrm{L}}+c_{y,i}^{\mathrm{R}})+(k_{y,i}^{\mathrm{L}}+k_{y,i}^{\mathrm{R}})}$$

(6)

$$k{e}_{z,i}^{11}=W_{11}{X}_1{}^2-W_{12}{l}_i^{\mathrm{L}}{X}_1{}^2-W_{21}{l}_i^{\mathrm{L}}{X}_1{}^2\hspace{0.17em}+W_{22}{l}_i^{\mathrm{L}}{}^2{X}_1{}^2-X_1$$

(7)

$$k{e}_{z,i}^{12}=W_{11}{X}_1{X}_2+W_{12}{l}_i^{\mathrm{R}}{X}_1{X}_2-W_{21}{l}_i^{\mathrm{R}}{X}_1{X}_2-W_{22}{l}_i^{\mathrm{L}}{l}_i^{\mathrm{R}}{X}_1{X}_2$$

(8)

$$k{e}_{z,i}^{21}=W_{11}{X}_1{X}_2-W_{12}{l}_i^{\mathrm{L}}{X}_1{X}_2+W_{21}{l}_i^{\mathrm{L}}{X}_1{X}_2-W_{22}{l}_i^{\mathrm{L}}{l}_i^{\mathrm{R}}{X}_1{X}_2$$

(9)

$$k{e}_{z,i}^{22}=W_{11}{X}_2{}^2+W_{12}{l}_i^{\mathrm{R}}{X}_2{}^2+W_{21}{l}_i^{\mathrm{R}}{X}_2{}^2\hspace{0.17em}+W_{22}{l}_i^{\mathrm{R}}{}^2{X}_2{}^2-X_2$$

(10)

and

$$X_1=k_{z,i}^{\mathrm{L}}+\overline{\omega }{c}_{z,i}^{\mathrm{L}}\hspace{1em}{X}_2=k_{z,i}^{\mathrm{R}}+\overline{\omega }{c}_{z,i}^{\mathrm{R}}$$

$$\begin{array}{l}{W}_{11}=\left[\begin{array}{l}{J}_i{\overline{\omega }}^2+\overline{\omega }(c_{z,i}^{\mathrm{L}}{l}_{z,i}^{\mathrm{L}}{}^2+c_{z,i}^{\mathrm{R}}{l}_{z,i}^{\mathrm{R}}{}^2)\\ +(k_{z,i}^{\mathrm{L}}{l}_{z,i}^{\mathrm{L}}{}^2+k_{z,i}^{\mathrm{R}}{l}_{z,i}^{\mathrm{R}}{}^2)\end{array}\right]/\Delta \\ W_{12}=\left[(k_{z,i}^{\mathrm{L}}{l}_{z,i}^{\mathrm{L}}-k_{z,i}^{\mathrm{R}}{l}_{z,i}^{\mathrm{R}})+\overline{\omega }(c_{z,i}^{\mathrm{L}}{l}_{z,i}^{\mathrm{L}}-c_{z,i}^{\mathrm{R}}{l}_{z,i}^{\mathrm{R}})\right]/\Delta =W_{21}\\ W_{22}=\left[m_i{\overline{\omega }}^2+\overline{\omega }(c_{z,i}^{\mathrm{L}}+c_{z,i}^{\mathrm{R}})+(k_{z,i}^{\mathrm{L}}+k_{z,i}^{\mathrm{R}})\right]/\Delta \end{array}$$

with

$$\Delta =\left|\begin{array}{cc}{m}_i{\overline{\omega }}^2+\overline{\omega }(c_{z,i}^{\mathrm{L}}+c_{z,i}^{\mathrm{R}})+(k_{z,i}^{\mathrm{L}}+k_{z,i}^{\mathrm{R}})& -(k_{z,i}^{\mathrm{L}}{l}_{z,i}^{\mathrm{L}}-k_{z,i}^{\mathrm{R}}{l}_{z,i}^{\mathrm{R}})-\overline{\omega }(c_{z,i}^{\mathrm{L}}{l}_{z,i}^{\mathrm{L}}-c_{z,i}^{\mathrm{R}}{l}_{z,i}^{\mathrm{R}})\\ -(k_{z,i}^{\mathrm{L}}{l}_{z,i}^{\mathrm{L}}-k_{z,i}^{\mathrm{R}}{l}_{z,i}^{\mathrm{R}})-\overline{\omega }(c_{z,i}^{\mathrm{L}}{l}_{z,i}^{\mathrm{L}}-c_{z,i}^{\mathrm{R}}{l}_{z,i}^{\mathrm{R}})& J_i{\overline{\omega }}^2+\overline{\omega }(c_{z,i}^{\mathrm{L}}{l}_{z,i}^{\mathrm{L}}{}^2+c_{z,i}^{\mathrm{R}}{l}_{z,i}^{\mathrm{R}}{}^2)+(c_{z,i}^{\mathrm{L}}{l}_{z,i}^{\mathrm{L}}{}^2+c_{z,i}^{\mathrm{R}}{l}_{z,i}^{\mathrm{R}}{}^2)\end{array}\right|$$

From Equations (1) and (2), one can see that the four springs, four dampers, lumped mass and mass movement of inertia of the three-DOF SDM system (see Figure 2a) are, respectively, condensed into eight effective springs ${ke}_{z,i}^{11}$, ${ke}_{z,i}^{12}$, ${ke}_{y,i}^{21}$, ${ke}_{y,i}^{22}$ and ${ke}_{z,i}^{11}$, ${ke}_{z,i}^{12}$, ${ke}_{y,i}^{21}$, ${ke}_{y,i}^{22}$ as shown in Figure 2b; the derivations of Equations (1)–(10) are given in Appendix A.

2.2. The Frequency Response Function for the Thin-Walled Beam System

The governing equations of motion of Euler-Bernoulli thin-walled beams illustrated in Figure 2a on which the external loads is zero can be written as follows:

$$E{I}_z\frac{{\partial }^{4}v}{\partial x^4}+\rho A\frac{{\partial }^{2}v}{\partial t^2}=0$$

(11)

$$E{I}_y\frac{{\partial }^{4}w}{\partial x^4}+\rho A\frac{{\partial }^{2}w}{\partial t^2}+\rho A{z}_c\frac{{\partial }^2\psi }{\partial t^2}=0$$

(12)

$$E{\mathsf{\Gamma }}_0\frac{{\partial }^4\psi }{\partial x^4}-GJ\frac{{\partial }^2\psi }{\partial x^2}+\rho A{z}_c\frac{{\partial }^{2}w}{\partial t^2}+\rho I_0\frac{{\partial }^2\psi }{\partial t^2}=0$$

(13)

where E is Young’s modulus of elasticity and G is the shear modulus of the material, EI_z and EI_y are the bending stiffness of the thin-walled beam about the centroidal principal axes, which are parallel to the y and z axes, respectively. GJ and EΓ₀ are, respectively, the torsional stiffness and warping stiffness of the thin-walled beam. ρA is the mass of the thin-walled beam per unit length and I₀ denotes the polar mass moment of inertia of per unit length beam cross section about the x axis.

For forced vibration of the constrained thin-walled beam (beam with subsystems), the harmonic responses take the form:

$$v=V(x)e^{\overline{\omega }t}\hspace{1em}w=W(x)e^{\overline{\omega }t}\hspace{1em}\psi =\Psi (x)e^{\overline{\omega }t}$$

(14)

where $\overline{\omega }=\Omega \overline{i}$ is the frequency of the excitation force and $\overline{i}=\sqrt{-1}$.

Substituting Equation (14) into Equations (11)–(13) yields

$$E{I}_z{V}^{(4)}(x)+\rho A{\overline{\omega }}^{2}V(x)=0$$

(15)

$$E{I}_y{W}^{(4)}(x)+\rho A{\overline{\omega }}^{2}W(x)+\rho A{z}_c{\overline{\omega }}^2\Psi (x)=0$$

(16)

$$E{\Gamma }_0{\mathsf{\Psi }}^{(4)}(x)-GJ{\Psi }^{(2)}(x)+\rho A{z}_c{\overline{\omega }}^{2}W(x)+\rho I_0{\overline{\omega }}^2\Psi (x)=0$$

(17)

From Equation (16):

$$\Psi (x)=-\frac{E{I}_y}{\rho A{z}_c{\overline{\omega }}^2}{W}^{(4)}(x)-\frac{1}{z_c}W(x)$$

(18)

Substituting Equation (18) into Equation (17) yields:

$$\left\{{\lambda }_4{D}^8+{\lambda }_3{D}^6+{\lambda }_2{D}^4+{\lambda }_1{D}^2+{\lambda }_0\right\}\hspace{0.17em}W(x)=0$$

(19)

where

$$\begin{array}{l}D=d/dx\hspace{1em}{\lambda }_4=E^2{\mathsf{\Gamma }}_0{I}_y\hspace{1em}{\lambda }_3=-GJE{I}_y\hspace{1em}{\lambda }_2=E{\mathsf{\Gamma }}_0\rho A{\overline{\omega }}^2+\rho I_0{\overline{\omega }}^{2}E{I}_y\hspace{1em}\\ {\lambda }_1=-GJ\rho A{\overline{\omega }}^2\hspace{1em}{\lambda }_0=-(\rho A{z}_c{\overline{\omega }}^2{)}^2+{\rho }^2{I}_{0}A{\overline{\omega }}^4\end{array}$$

The solution of Equation (19) can be obtained by substituting the trial solution W(x) = w₀e^κx to give the characteristic equation

$${\lambda }_4{\kappa }^8+{\lambda }_3{\kappa }^6+{\lambda }_2{\kappa }^4+{\lambda }_1{\kappa }^2+{\lambda }_0=0$$

(20)

Let h = κ², the Equation (20) can be rewritten as

$${\lambda }_4{h}^4+{\lambda }_3{h}^3+{\lambda }_2{h}^2+{\lambda }_1{h}^1+{\lambda }_0=0$$

(21)

For Equation (21), all of its four roots are real when the vibration frequency is within the practical range [4,5], two of them are positive (h₁ and h₂) and the other two are negative (−h₃ and −h₄). Then the eight roots of the characteristic Equation (21) are β₁, −β₁, β₂, −β₂, β₃$\overline{i}$, −β₃$\overline{i}$, β₄i, −β₄$\overline{i}$, where $\overline{i}$ =$\sqrt{-1}$ and β₁ = $\sqrt{h_1}$, β₂ = $\sqrt{h_2}$, β₃ = $\sqrt{-h_3}$, β₄ = $\sqrt{-h_4}$. It follows that the solutions of Equations (15)–(17) are in the following forms:

$$V(x)=c_9\sin {\beta }_{5}x+c_{10}\cos {\beta }_{5}x+c_{11}\sinh {\beta }_{5}x+c_{12}\cosh {\beta }_{5}x$$

(22)

$$W(x)=c_1\sinh {\beta }_{1}x+c_2\cosh {\beta }_{1}x+c_3\sinh {\beta }_{2}x+c_4\cosh {\beta }_{2}x+c_5\sinh {\beta }_{3}x+c_6\cosh {\beta }_{3}x+c_7\sinh {\beta }_{4}x+c_8\cosh {\beta }_{4}x$$

(23)

$$\Psi (x)=t_1{c}_1\sinh {\beta }_{1}x+t_1{c}_2\cosh {\beta }_{1}x+t_2{c}_3\sinh {\beta }_{2}x+t_2{c}_4\cosh {\beta }_{2}x+t_3{c}_5\sinh {\beta }_{3}x+t_3{c}_6\cosh {\beta }_{3}x+t_4{c}_7\sinh {\beta }_{4}x+t_4{c}_8\cosh {\beta }_{4}x$$

(24)

where ${\beta }_5{}^4=\frac{\rho A}{E{I}_z}{\overline{\omega }}^2$ and c₁ to c₁₂ is a set of constants, and

$$\begin{array}{l}{t}_1=-(E{I}_y{\beta }_1{}^4+\rho A{\overline{\omega }}^2)/\rho A{z}_c{\overline{\omega }}^2\hspace{1em}\hspace{1em}{t}_2=-(E{I}_y{\beta }_2{}^4+\rho A{\overline{\omega }}^2)/\rho A{z}_c{\overline{\omega }}^2\\ t_3=-(E{I}_y{\beta }_3{}^4+\rho A{\overline{\omega }}^2)/\rho A{z}_c{\overline{\omega }}^2\hspace{1em}\hspace{1em}{t}_4=-(E{I}_y{\beta }_4{}^4+\rho A{\overline{\omega }}^2)/\rho A{z}_c{\overline{\omega }}^2\end{array}$$

Following the sign convention as shown in Figure 2, the expressions of bending moment M(x), shear force S(x), torque T(x) and bi-moment B(x) can be obtained as

$$\begin{array}{l}{M}_{}^v(x)=E{I}_z{V}^{″}(x)=E{I}_z{\beta }_5{}^2(-c_9\sin {\beta }_{5}x-c_{10}\cos {\beta }_{5}x+c_{11}\sinh {\beta }_{5}x+c_{12}\cosh {\beta }_{5}x)\\ M_{}^w(x)=E{I}_y{W}^{″}(x)=E{I}_y(c_1{\beta }_1{}^2\sinh {\beta }_{1}x+c_2{\beta }_1{}^2\cosh {\beta }_{1}x+c_3{\beta }_2{}^2\sinh {\beta }_{2}x+c_4{\beta }_2{}^2\cosh {\beta }_{2}x\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+c_5{\beta }_3{}^2\sinh {\beta }_{3}x+c_6{\beta }_3{}^2\cosh {\beta }_{3}x+c_7{\beta }_4{}^2\sinh {\beta }_{4}x+c_8{\beta }_4{}^2\cosh {\beta }_{4}x)\end{array}$$

(25)

$$\begin{array}{l}{S}_{}^v(x)=E{I}_z{V}^{‴}(x)=E{I}_z{\beta }_5{}^3(-c_9\cos {\beta }_{5}x+c_{10}\sin {\beta }_{5}x+c_{11}\cosh {\beta }_{5}x+c_{12}\sinh {\beta }_{5}x)\\ S_{}^w(x)=E{I}_y{W}^{‴}(x)\hspace{0.17em}=E{I}_y(c_1{\beta }_1{}^3\cosh {\beta }_{1}x+c_2{\beta }_1{}^3\sinh {\beta }_{1}x+c_3{\beta }_2{}^3\cosh {\beta }_{2}x+c_4{\beta }_2{}^3\sinh {\beta }_{2}x\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+c_5{\beta }_3{}^3\cosh {\beta }_{3}x+c_6{\beta }_3{}^3\sinh {\beta }_{3}x+c_7{\beta }_4{}^3\cosh {\beta }_{4}x+c_8{\beta }_4{}^3\sinh {\beta }_{4}x)\end{array}$$

(26)

$$\begin{array}{l}T(x)=E{\mathsf{\Gamma }}_0{\Psi }^{‴}(x)-GJ{\Psi }^{\prime }(x)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=E{\mathsf{\Gamma }}_0(c_1{\beta }_1{}^3{t}_1\cosh {\beta }_{1}x+c_2{\beta }_1{}^3{t}_1\sinh {\beta }_{1}x+c_3{\beta }_2{}^3{t}_2\cosh {\beta }_{2}x+c_4{\beta }_2{}^3{t}_2\sinh {\beta }_{2}x\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+c_5{\beta }_3{}^3{t}_3\cosh {\beta }_{3}x+c_6{\beta }_3{}^3{t}_3\sinh {\beta }_{3}x+c_7{\beta }_4{}^3{t}_4\cosh {\beta }_{4}x+c_8{\beta }_4{}^3{t}_4\sinh {\beta }_{4}x)\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-GJ(t_1{c}_1{\beta }_1\cosh {\beta }_{1}x+t_1{c}_2{\beta }_1\sinh {\beta }_{1}x\hspace{0.17em}+t_2{c}_3{\beta }_2\cosh {\beta }_{2}x+t_2{c}_4{\beta }_2\sinh {\beta }_{2}x\\ \hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+t_3{c}_5{\beta }_3\cosh {\beta }_{3}x+t_3{c}_6{\beta }_3\sinh {\beta }_{3}x+t_4{c}_7{\beta }_4\cosh {\beta }_{4}x+t_4{c}_8{\beta }_4\sinh {\beta }_{4}x)\end{array}$$

(27)

$$\begin{array}{l}B(x)=E{\mathsf{\Gamma }}_0{\Psi }^{″}(x)=E{\mathsf{\Gamma }}_0(t_1{c}_1{\beta }_1{}^2\sinh {\beta }_{1}x+t_1{c}_2{\beta }_1{}^2\cosh {\beta }_{1}x+t_2{c}_3{\beta }_2{}^2\sinh {\beta }_{2}x+t_2{c}_4{\beta }_2{}^2\cosh {\beta }_{2}x\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+t_3{c}_5{\beta }_3{}^2\sinh {\beta }_{3}x+t_3{c}_6{\beta }_3{}^2\cosh {\beta }_{3}x+t_4{c}_7{\beta }_4{}^2\sinh {\beta }_{4}x+t_4{c}_8{\beta }_4{}^2\cosh {\beta }_{4}x)\end{array}$$

(28)

The spatial solution of Equations (15)–(17) can be expressed in terms of both sinusoidal and hyperbolic terms, and can be written in vector form as

$$V(x)=\mathit{V}(x){\mathit{C}}_i^v\hspace{1em}W(x)=\mathit{W}(x){\mathit{C}}_i^w\hspace{1em}\Psi (x)=\mathsf{\Psi }(x){\mathit{C}}_i^{\psi }$$

(29)

where ${\mathit{C}}_i^v$, ${\mathit{C}}_i^w$, ${\mathit{C}}_i^{\Psi }$ are column vector of integration constants associated with the sub-beam i. V(x), W(x), Ψ(x) are row vectors of mode shape terms. And

$${\mathit{C}}_i^v={\left[c_9\hspace{1em}{c}_{10}\hspace{1em}{c}_{11}\hspace{1em}{c}_{12}\right]}^T{\mathit{C}}_i^v={\left[c_9\hspace{1em}{c}_{10}\hspace{1em}{c}_{11}\hspace{1em}{c}_{12}\right]}^T\mathit{V}\left(x\right)=[\sin {\beta }_{5}x\hspace{1em}\cos {\beta }_{5}x\hspace{1em}\sinh {\beta }_{5}x\hspace{1em}\cosh {\beta }_{5}x]$$

$$\mathit{W}\left(x\right)=[\sinh {\beta }_{1}x\hspace{1em}\cosh {\beta }_{1}x\hspace{1em}\sinh {\beta }_{2}x\hspace{1em}\cosh {\beta }_{2}x\hspace{1em}\hspace{0.17em}\hspace{0.17em}\sinh {\beta }_{3}x\hspace{1em}\cosh {\beta }_{3}x\hspace{1em}\sinh {\beta }_{4}x\hspace{1em}\cosh {\beta }_{4}x]$$

$$\mathsf{\Psi }\left(x\right)=[t_1\sinh {\beta }_{1}x\hspace{1em}{t}_1\cosh {\beta }_{1}x\hspace{1em}{t}_2\sinh {\beta }_{2}x\hspace{1em}\hspace{0.17em}{t}_2\cosh {\beta }_{2}x\hspace{1em}{t}_3\sinh {\beta }_{3}x\hspace{1em}{t}_3\cosh {\beta }_{3}x\hspace{1em}{t}_4\sinh {\beta }_{4}x\hspace{1em}{t}_4\cosh {\beta }_{4}x]$$

From Figure 2a, the relationship between the displacement of connection points and the shear center can be given as follows

$$v_{y,i}^L=w_i+z_b{\psi }_i\hspace{1em}{v}_{y,i}^R=w_i+z_b{\psi }_i\hspace{1em}{v}_{z,i}^L=v_i-y_u{\psi }_i\hspace{1em}{v}_{z,i}^R=v_i+y_u{\psi }_i$$

(30)

where z_b and y_u = a/2 are the distances between the shearing center and spring connection point in z direction and y direction, respectively.

Substituting Equation (30) into Equations (1) and (2), one can obtain

$$F_{y,i}^L=(k{e}_{y,i}^{11}+k{e}_{y,i}^{12})(w_i+z_b{\psi }_i)\hspace{1em}{F}_{y,i}^R=(k{e}_{y,i}^{21}+k{e}_{y,i}^{22})(w_i+z_b{\psi }_i)$$

(31)

$$F_{z,i}^L=(k{e}_{z,i}^{11}+k{e}_{z,i}^{12})v_i+(-k{e}_{z,i}^{11}+k{e}_{z,i}^{12})y_u{\psi }_i\hspace{1em}{F}_{z,i}^R=(k{e}_{z,i}^{21}+k{e}_{z,i}^{22})v_i+(-k{e}_{z,i}^{21}+k{e}_{z,i}^{22})y_u{\psi }_i$$

(32)

Then the external force and torque applied at the shear center can be written as

$$F_{y,i}=F_{y,i}^L+F_{y,i}^R=(k{e}_{y,i}^{11}+k{e}_{y,i}^{12}+k{e}_{y,i}^{21}+k{e}_{y,i}^{22})(w_i+z_b{\psi }_i)=\overline{K}{y}_i(w_i+z_b{\psi }_i)$$

(33)

$$F_{z,i}=F_{z,i}^L+F_{z,i}^R=(k{e}_{z,i}^{11}+k{e}_{z,i}^{12}+k{e}_{z,i}^{21}+k{e}_{z,i}^{22})v_i+(-k{e}_{z,i}^{11}+k{e}_{z,i}^{12}-k{e}_{z,i}^{21}+k{e}_{z,i}^{22})y_u{\psi }_i=\overline{K}{z}_i{v}_i+\widehat{K}{z}_i{y}_u{\psi }_i$$

(34)

$$\begin{array}{l}{T}_i=T_{yi}+T_{zi}=(F_{y,i}^L+F_{y,i}^R)z_b-F_{z,i}^L{y}_u+F_{z,i}^R{y}_u\\ \hspace{1em}=(k{e}_{y,i}^{11}+k{e}_{y,i}^{12}+k{e}_{y,i}^{21}+k{e}_{y,i}^{22})(w_i+z_b{\psi }_i)z_b+(-k{e}_{z,i}^{11}-k{e}_{z,i}^{12}+k{e}_{z,i}^{21}+k{e}_{z,i}^{22})v_i{y}_u+(k{e}_{z,i}^{11}-k{e}_{z,i}^{12}-k{e}_{z,i}^{21}+k{e}_{z,i}^{22})y_u{}^2{\psi }_i\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\overline{K}{y}_i(w_i+z_b{\psi }_i)z_b+\stackrel{⌢}{K}{z}_i{v}_i{y}_u+\tilde{K}{z}_i{y}_u{}^2{\psi }_i\end{array}$$

(35)

The continuity condition at the cross section where the subsystem mounts enforces that the displacements and slopes at the common interface of two adjacent sub-beams must be equal, hence

$$W_i(l_i)=W_{i+1}(0),\hspace{1em}{V}_i(l_i)=V_{i+1}(0)\hspace{1em}{\Psi }_i(l_i)={\Psi }_i(0)\hspace{1em}{W}_i^{\prime }(l_i)=W_{i+1}^{\prime }(0),\hspace{1em}{V}_i^{\prime }(l_i)=V_{i+1}^{\prime }(0)\hspace{1em}{\Psi }_i^{\prime }(l_i)={\Psi }_i^{\prime }(0)$$

(36)

Similarly, the moment, force, torque and bi-moment at interface must be equal to zero and the equilibrium conditions are illustrated in Figure 3. Hence, one can obtain:

$$M_i^w(l_i)=M_{i+1}^w(0)\hspace{1em}{M}_i^v(l_i)=M_{i+1}^v(0)$$

(37)

$$S_i^w(l_i)+F_{y,i}=S_{i+1}^w(0)\hspace{1em}{S}_i^v(l_i)+F_{z,i}=S_{i+1}^v(0)$$

(38)

$$T_i(l_i)+T_{Ti}=T_{i+1}(0)$$

(39)

$$B_i(l_i)=B_{i+1}(l_{i+1})$$

(40)

Substituting Equations (22)–(29) and (33)–(35) into Equations (36)–(40), and writing the compatibility conditions across the spring connection point in matrix form, one obtains

$${\mathbf{T}}_{Ri}^w{\mathit{C}}_{i+1}^w={\mathbf{T}}_{Li}^w{\mathit{C}}_i^w$$

(41)

$${\mathbf{T}}_{Ri}^{\psi }{\mathit{C}}_{i+1}^w={\mathbf{T}}_{Li}^{\psi }{\mathit{C}}_i^w+{\mathbf{T}}_{Li}^{\psi v}{\mathit{C}}_i^v$$

(42)

$${\mathbf{T}}_{Ri}^v{\mathit{C}}_{i+1}^v={\mathbf{T}}_{Li}^v{\mathit{C}}_i^v+{\mathbf{T}}_{Li}^{v\psi }{\mathit{C}}_i^{\psi }$$

(43)

where T^w, T^ψ and T^v are the derivative matrices, the components of which are given in Appendix B.

Combining Equations (41)–(43) and organizing them in matrix form, one obtains

$$\underset{12\times 12}{\underbrace{\left[\begin{array}{cc}{\mathbf{T}}_{Ri}^w& 0_{4\times 4}\\ {\mathbf{T}}_{Ri}^{\psi }& 0_{4\times 4}\\ 0_{4\times 8}& {\mathbf{T}}_{Ri}^v\end{array}\right]}}{\left[\begin{array}{c}{\mathit{C}}_{i+1}^w\\ {\mathit{C}}_{i+1}^v\end{array}\right]}_{12\times 1}=\underset{12\times 12}{\underbrace{\left[\begin{array}{cc}{\mathbf{T}}_{Li}^w& 0_{4\times 4}\\ {\mathbf{T}}_{Li}^{\psi }& {\mathbf{T}}_{Li}^{\psi v}\\ {\mathbf{T}}_{Li}^{v\psi }& {\mathbf{T}}_{Li}^v\end{array}\right]}}{\left[\begin{array}{c}{\mathit{C}}_i^w\\ {\mathit{C}}_i^v\end{array}\right]}_{12\times 1}$$

(44)

Then, the transitive relationship between the integration constants of the ith and the i + 1th sub-beam is obtained:

$$\left[\begin{array}{c}{\mathit{C}}_{i+1}^w\\ {\mathit{C}}_{i+1}^v\end{array}\right]={\left[\begin{array}{cc}{\mathbf{T}}_{Ri}^w& 0_{4\times 4}\\ {\mathbf{T}}_{Ri}^{\psi }& 0_{4\times 4}\\ 0_{4\times 8}& {\mathbf{T}}_{Ri}^v\end{array}\right]}^{-1}\left[\begin{array}{cc}{\mathbf{T}}_{Li}^w& 0_{4\times 4}\\ {\mathbf{T}}_{Li}^{\psi }& {\mathbf{T}}_{Li}^{\psi v}\\ {\mathbf{T}}_{Li}^{v\psi }& {\mathbf{T}}_{Li}^v\end{array}\right]\left[\begin{array}{c}{\mathit{C}}_i^w\\ {\mathit{C}}_i^v\end{array}\right]={\mathbf{H}}_i{}_{12\times 12}{\mathit{C}}_i^{}$$

(45)

By repeating Equation (45), the transitive relationship between the first sub-beam and the last sub-beam can be written as

$${\mathit{C}}_{N+1}=\left({\displaystyle \prod _{i=1}^N{\mathbf{H}}_{N-i+1}}\right){\mathit{C}}_1={\mathbf{H}}_{12\times 12}{\mathit{C}}_1$$

(46)

where N is the total number of three-DOF SDM systems.

For the derivation of frequency response function of the thin-walled beam system, the basic idea is to treat the excitation force as an external discontinuity and involve it in the boundary or compatibility conditions associated with the shear force and torque of the thin walled beam.

2.2.1. Force Excitation at the Ends

Case 1. With a harmonic force ($F_{\mathrm{L}}^{\mathrm{w}}$e^iΩt, $T_{\mathrm{L}}^{\mathsf{\psi }}$e^iΩt, $F_{\mathrm{L}}^{\mathrm{v}}$e^iΩt) at the left-hand end of a thin-walled beam, the boundary conditions can be written as follows:

$$\underset{12\times 12}{\underbrace{\left[\begin{array}{c}{B}_{\mathrm{L}\hspace{0.17em}6\times 12}^{}\\ B_{\mathrm{R}\hspace{0.17em}6\times 12}\left({\displaystyle \prod _{i=1}^N{H}_{N-i+1}}\right)\end{array}\right]}}\left[\begin{array}{c}{\mathit{C}}_1^w\\ {\mathit{C}}_1^v\end{array}\right]=\underset{12\times 1}{\underbrace{\left[\begin{array}{c}{\mathit{F}}_{\mathrm{L}}{}_{6\times 1}\\ 0_{6\times 1}\end{array}\right]}}={\overline{\mathit{F}}}_{\mathrm{L}}$$

(47)

where $\stackrel{\_}{F}$_L = [0 $F_{\mathrm{L}}^w$ 0 $T_{\mathrm{L}}^{\psi }$ 0 $F_{\mathrm{L}}^v$ 0 0 0 0 0 0]^T is excitation force vector; B_L and B_R are the coefficient matrices associated with the left and right ends; the expressions of these matrices are given in Appendix C. For any other boundaries, the boundary constraint equations are all in the form of Equation (47). Through changing the elements of matrices B_L, B_R, one can formulate the constraint equation for arbitrary boundary conditions easily.

By substituting Equation (46) into Equation (47), the integration constant parameters for the first component can be obtained as

$$\left[{\mathit{C}}_1\right]=\left[\begin{array}{c}{\mathit{C}}_1^w\\ {\mathit{C}}_1^v\end{array}\right]={\underset{12\times 12}{\underbrace{\left[\begin{array}{c}{B}_{\mathrm{L}\hspace{0.17em}6\times 12}\\ B_{\mathrm{R}\hspace{0.17em}6\times 12}\left({\displaystyle \prod _{i=1}^N{H}_{N-i+1}}\right)\end{array}\right]}}}^{-1}{\overline{\mathit{F}}}_{\mathrm{L}}$$

(48)

Next, introducing Equation (48) into Equation (31) lead to the constant parameters for the nth sub-beam C_n and the corresponding FRFs can be expressed as [28]:

$$Y(\overline{x})={\displaystyle \sum _{n=1}^{N+1}{\mathsf{\Psi }}_n^{}(\overline{x}-{\overline{x}}_{n-1}){\mathbf{C}}_n^{}[H(\overline{x}-{\overline{x}}_{n-1})-H(\overline{x}-{\overline{x}}_n)]}$$

(49)

where $\stackrel{\_}{x}$ is the global coordinate with $\stackrel{\_}{x}$₀ = 0 and $\stackrel{\_}{x}$_N+1 = L, H($\stackrel{\_}{x}$−x₀) is the Heaviside function, which increases from zero to unity at location x_0.

Case 2. With a harmonic force (${\mathrm{F}}_{\mathrm{R}}^{\mathrm{w}}$e^iΩt, ${\mathrm{T}}_{\mathrm{R}}^{\mathsf{\psi }}$e^iΩt, ${\mathrm{F}}_{\mathrm{R}}^{\mathrm{v}}$e^iΩt) applied at the right-hand end of the thin-walled beam, through similar matrix manipulations as in Equation (47), the integration constant parameters of the first sub-beam are

$$\left[{\mathit{C}}_1\right]=\left[\begin{array}{c}{\mathit{C}}_1^w\\ {\mathit{C}}_1^v\end{array}\right]={\underset{12\times 12}{\underbrace{\left[\begin{array}{c}{B}_{\mathrm{L}\hspace{0.17em}6\times 12}\\ B_{\mathrm{R}\hspace{0.17em}6\times 12}\left({\displaystyle \prod _{i=1}^N{H}_{N-i+1}}\right)\end{array}\right]}}}^{-1}{\overline{\mathit{F}}}_{\mathrm{R}}$$

(50)

where $\stackrel{\_}{\mathit{F}}$_R = [ 0 0 0 0 0 0 0 $F_{\mathrm{R}}^w$ 0 $T_{\mathrm{R}}^{\psi }$ 0 $F_{\mathrm{R}}^v$]^T. Similarly, the FRFs can be expressed as Equation (49).

2.2.2. Force Excitation at an Arbitrary Position of the Thin-Walled Beam

In most conditions, the excitation force F_ue^iΩt is applied at arbitrary positions of the system. The force equilibrium conditions at the excitation point of the mth sub-beam are illustrated in Figure 3 and can be written as

$${\mathbf{T}}_{m,2}^w{\mathit{C}}_{m,2}^w={\mathbf{T}}_{m,1}^w{\mathit{C}}_{m,1}^w+{\mathit{F}}^w$$

(51)

$${\mathbf{T}}_{m,2}^{\psi }{\mathit{C}}_{m,2}^w={\mathbf{T}}_{m,1}^{\psi }{\mathit{C}}_{m,1}^w+{\mathbf{T}}^{\psi }$$

(52)

$${\mathbf{T}}_{m,2}^v{\mathit{C}}_{m,2}^v={\mathbf{T}}_{m,1}^v{\mathit{C}}_{m,1}^v+{\mathit{F}}^v$$

(53)

where the subscripts (m, 1) and (m, 2) correspond to the 1th (left) and 2th (right) component of the mth sub-beam; F^w= [0 0 0 F^w]^T, T^ψ= [0 0 0 T^ψ]^T, F^v= [0 0 0 F^v]^T. the components of the derivative matrices are given in Appendix D.

The combination of Equations (51)–(53) leads to

$$\underset{12\times 12}{\underbrace{\left[\begin{array}{cc}{\mathbf{T}}_{m,2}^{w,1}& 0_{4\times 4}\\ {\mathbf{T}}_{m,2}^{\psi ,1}& 0_{4\times 4}\\ 0_{4\times 8}& {\mathbf{T}}_{m,2}^{v,1}\end{array}\right]}}{\left[\begin{array}{c}{\mathit{C}}_{m,2}^w\\ {\mathit{C}}_{m,2}^v\end{array}\right]}_{8\times 1}=\underset{12\times 12}{\underbrace{\left[\begin{array}{cc}{\mathbf{T}}_{m,1}^{w,2}& 0_{4\times 4}\\ {\mathbf{T}}_{m,1}^{\psi ,2}& 0_{4\times 4}\\ 0_{4\times 8}& {\mathbf{T}}_{m,1}^{v,2}\end{array}\right]}}{\left[\begin{array}{c}{\mathit{C}}_{m,1}^w\\ {\mathit{C}}_{m,1}^v\end{array}\right]}_{8\times 1}+\left[\begin{array}{c}{\mathit{F}}^w\\ {\mathbf{T}}^{\psi }\\ {\mathit{F}}^v\end{array}\right]$$

(54)

Then, the transitive relationship between the integration constants of the 1th and the 2th component of the mth sub-beam is obtained:

$$\left[\begin{array}{c}{\mathit{C}}_{m,2}^w\\ {\mathit{C}}_{m,2}^v\end{array}\right]={\left[\begin{array}{cc}{\mathbf{T}}_{m,2}^{w,1}& 0_{4\times 4}\\ {\mathbf{T}}_{m,2}^{\psi ,1}& 0_{4\times 4}\\ 0_{4\times 8}& {\mathbf{T}}_{m,2}^{v,1}\end{array}\right]}^{-1}\left[\begin{array}{cc}{\mathbf{T}}_{m,1}^{w,2}& 0_{4\times 4}\\ {\mathbf{T}}_{m,1}^{\psi ,2}& 0_{4\times 4}\\ 0_{4\times 8}& {\mathbf{T}}_{m,1}^{v,2}\end{array}\right]\left[\begin{array}{c}{\mathit{C}}_{m,1}^w\\ {\mathit{C}}_{m,1}^v\end{array}\right]+{\left[\begin{array}{cc}{\mathbf{T}}_{m,2}^{w,1}& 0_{4\times 4}\\ {\mathbf{T}}_{m,2}^{\psi ,1}& 0_{4\times 4}\\ 0_{4\times 8}& {\mathbf{T}}_{m,2}^{v,1}\end{array}\right]}^{-1}\left[\begin{array}{c}{\mathit{F}}^w\\ {\mathbf{T}}^{\psi }\\ {\mathit{F}}^v\end{array}\right]={\widehat{\mathbf{H}}}_m\left[\begin{array}{c}{\mathit{C}}_{m,1}^w\\ {\mathit{C}}_{m,1}^v\end{array}\right]+{\mathbf{H}}_F\widehat{\mathit{F}}$$

(55)

Case 1.m = 1

The substitution of Equation (55) into Equation (46) leads to

$$\left[\begin{array}{c}{\mathit{C}}_{N+1}^w\\ {\mathit{C}}_{N+1}^v\end{array}\right]=\left({\displaystyle \prod _{i=1}^N{\mathbf{H}}_{N-i+1}}\right){\widehat{\mathbf{H}}}_1\left[\begin{array}{c}{\mathit{C}}_1^w\\ {\mathit{C}}_1^v\end{array}\right]+\left({\displaystyle \prod _{i=1}^N{\mathbf{H}}_{N-i+1}}\right){\mathbf{H}}_F\widehat{\mathit{F}}$$

(56)

Then the boundary conditions can be written as follows:

$$\underset{12\times 12}{\underbrace{\left[\begin{array}{c}{B}_{\mathrm{L}\hspace{0.17em}6\times 12}\\ B_{\mathrm{R}}^{}{}_{\hspace{0.17em}6\times 12}\left({\displaystyle \prod _{i=1}^N{\mathbf{H}}_{N-i+1}}\right){\widehat{\mathbf{H}}}_1\end{array}\right]}}\left[\begin{array}{c}{\mathit{C}}_1^w\\ {\mathit{C}}_1^v\end{array}\right]=\left[\begin{array}{c}0\\ -B_{\mathrm{R}}^{}\left({\displaystyle \prod _{i=1}^N{\mathbf{H}}_{N-i+1}}\right){\mathbf{H}}_F\widehat{\mathit{F}}\end{array}\right]=\overline{\mathit{F}}$$

(57)

Next, the integration constant parameters of the first sub-beam can be written as

$$\left[\begin{array}{c}{\mathit{C}}_1^w\\ {\mathit{C}}_1^v\end{array}\right]={\left[\begin{array}{c}{B}_{\mathrm{L}\hspace{0.17em}6\times 12}\\ B_{\mathrm{L}\hspace{0.17em}6\times 12}\left({\displaystyle \prod _{i=1}^N{\mathbf{H}}_{N-i+1}}\right){\widehat{\mathbf{H}}}_m\end{array}\right]}^{-1}\overline{\mathit{F}}$$

(58)

Therefore, the integration constants of the n + 1 (n = 1, 2⋯N) sub-beam can be expressed as

$$\left[\begin{array}{c}{\mathit{C}}_{n+1}^w\\ {\mathit{C}}_{n+1}^v\end{array}\right]=\left({\displaystyle \prod _{i=1}^n{\mathbf{H}}_{n-i+1}}\right){\widehat{\mathbf{H}}}_1\left[\begin{array}{c}{\mathit{C}}_1^w\\ {\mathit{C}}_1^v\end{array}\right]+\left({\displaystyle \prod _{i=1}^n{\widehat{\mathbf{H}}}_{n-i+1}}\right){\mathbf{H}}_F\widehat{\mathit{F}}$$

(59)

Case 2.m = 2, 3⋯N

Similarly, the relationship between the integration constants of the N + 1th and the 1th sub-beam can be expressed as

$$\left[\begin{array}{c}{\mathit{C}}_{N+1}^w\\ {\mathit{C}}_{N+1}^v\end{array}\right]=\left({\displaystyle \prod _{i=1}^{N-m+1}{\mathbf{H}}_{N-i+1}}\right){\widehat{\mathbf{H}}}_m\left({\displaystyle \prod _{i=N-m+2}^N{\mathbf{H}}_{N-i+1}}\right)\left[\begin{array}{c}{\mathit{C}}_1^w\\ {\mathit{C}}_1^v\end{array}\right]+\left({\displaystyle \prod _{i=1}^{N-m+1}{\mathbf{H}}_{N-i+1}}\right){\mathbf{H}}_F\widehat{\mathit{F}}$$

(60)

Then, the boundary conditions can be expressed as

$$\underset{12\times 12}{\underbrace{\left[\begin{array}{c}{B}_{\mathrm{L}\hspace{0.17em}6\times 12}\\ B_{\mathrm{R}\hspace{0.17em}6\times 12}\left({\displaystyle \prod _{i=1}^{N-m+1}{\mathbf{H}}_{N-i+1}}\right){\widehat{\mathbf{H}}}_m\left({\displaystyle \prod _{i=N-m+2}^N{\mathbf{H}}_{N-i+1}}\right)\end{array}\right]}}\left[\begin{array}{c}{\mathit{C}}_1^w\\ {\mathit{C}}_1^v\end{array}\right]=\underset{12\times 1}{\underbrace{\left[\begin{array}{c}{0}_{6\times 1}\\ -B_{\mathrm{R}\hspace{0.17em}6\times 12}\left({\displaystyle \prod _{i=1}^{N-m+1}{\mathbf{H}}_{N-i+1}}\right){\mathbf{H}}_F\widehat{\mathit{F}}\end{array}\right]}}=\overline{\mathit{F}}$$

(61)

Next, substituting Equation (61) into Equation (60), the integration constant parameters for the first component can be obtained:

$$\left[\begin{array}{c}{\mathit{C}}_1^w\\ {\mathit{C}}_1^v\end{array}\right]={\left[\begin{array}{c}{B}_{\mathrm{L}\hspace{0.17em}6\times 12}\\ B_{\mathrm{R}\hspace{0.17em}6\times 12}\left({\displaystyle \prod _{i=1}^{N-m+1}{\mathbf{H}}_{N-i+1}}\right){\widehat{\mathbf{H}}}_m\left({\displaystyle \prod _{i=N-m+2}^N{\mathbf{H}}_{N-i+1}}\right)\end{array}\right]}^{-1}\overline{\mathit{F}}$$

(62)

Introducing Equation (62) into Equation (60) leads to the integration constants of the n + 1 (n = 1, 2⋯N) sub-beam:

$$\left[\begin{array}{c}{\mathit{C}}_{n+1}^w\\ {\mathit{C}}_{n+1}^v\end{array}\right]=\{\begin{array}{l}\left({\displaystyle \prod _{i=1}^n{\mathbf{H}}_{n-i+1}}\right)\left[\begin{array}{c}{\mathit{C}}_1^w\\ {\mathit{C}}_1^v\end{array}\right]\hspace{1em}\hspace{1em}n=1,2\cdots m-1\\ \{\left({\displaystyle \prod _{i=1}^{n-m+1}{\mathbf{H}}_{n-i+1}}\right){\widehat{\mathbf{H}}}_m\left({\displaystyle \prod _{i=n-m+2}^n{\mathbf{H}}_{n-i+1}}\right)\left[\begin{array}{c}{\mathit{C}}_1^w\\ {\mathit{C}}_1^v\end{array}\right]+\left({\displaystyle \prod _{i=1}^{n-m+1}{\mathbf{H}}_{n-i+1}}\right){\mathbf{H}}_F\widehat{\mathit{F}}\}\hspace{1em}\hspace{0.17em}n=m,2\cdots N\end{array}$$

(63)

Case 3.m = N + 1

For the similar derivations, the expressions of these constants are

$$\left[\begin{array}{c}{\mathit{C}}_{n+1}^w\\ {\mathit{C}}_{n+1}^v\end{array}\right]=\{\left({\displaystyle \prod _{i=1}^n{\mathbf{H}}_{n-i+1}}\right)\left[\begin{array}{c}{\mathit{C}}_1^w\\ {\mathit{C}}_1^v\end{array}\right]\hspace{1em}n=1,2\cdots N$$

(64)

where

$$\left[\begin{array}{c}{\mathit{C}}_1^w\\ {\mathit{C}}_1^v\end{array}\right]={\left[\begin{array}{c}{B}_{\mathrm{L}\hspace{0.17em}6\times 12}\\ B_{\mathrm{R}\hspace{0.17em}6\times 12}{\widehat{\mathbf{H}}}_{N+1}\left({\displaystyle \prod _{i=1}^N{\mathbf{H}}_{N-i+1}}\right)\end{array}\right]}^{-1}\overline{\mathit{F}}$$

(65)

with

$$\overline{\mathit{F}}=\underset{12\times 1}{\underbrace{\left[\begin{array}{c}{0}_{6\times 1}\\ B_{\mathrm{R}\hspace{0.17em}6\times 12}{\mathbf{H}}_F\widehat{\mathit{F}}\end{array}\right]}}$$

(66)

2.2.3. Force Excitation at the Subsystem

If the excitation force is applied at the mth subsystem, the compatibility conditions across the spring connection point can be expressed as

$${\mathbf{T}}_{Rm}^w{\mathit{C}}_{m+1}^w={\mathbf{T}}_{Lm}^w{\mathit{C}}_m^w+\mathit{F}{e}^w$$

(67)

$${\mathbf{T}}_{Ri}^{\psi }{\mathit{C}}_{m+1}^w={\mathbf{T}}_{Lm}^{\psi }{\mathit{C}}_m^w+{\mathbf{T}}_{Lm}^{\psi v}{\mathit{C}}_m^v+\mathit{T}{\mathit{e}}^{\psi }$$

(68)

$${\mathbf{T}}_{Rm}^v{\mathit{C}}_{m+1}^v={\mathbf{T}}_{Lm}^v{\mathit{C}}_m^v+{\mathbf{T}}_{Lm}^{v\psi }{\mathit{C}}_m^{\psi }+\mathit{F}{e}^v$$

(69)

where Fe^w= [0 0 0 Fe^w]^T, Te^ψ= [0 0 0 Te^ψ]^T, Fe^v= [0 0 0 Fe^v]^T.

Combining Equations (67)–(69) and organizing them in matrix form, one obtains:

$$\underset{12\times 12}{\underbrace{\left[\begin{array}{cc}{\mathbf{T}}_{Rm}^w& 0_{4\times 4}\\ {\mathbf{T}}_{Rm}^{\psi }& 0_{4\times 4}\\ 0_{4\times 8}& {\mathbf{T}}_{Rm}^v\end{array}\right]}}\underset{8\times 1}{\underbrace{\left[\begin{array}{c}{\mathit{C}}_{m+1}^w\\ {\mathit{C}}_{m+1}^v\end{array}\right]}}=\underset{12\times 12}{\underbrace{\left[\begin{array}{cc}{\mathbf{T}}_{Lm}^w& 0_{4\times 4}\\ {\mathbf{T}}_{Lm}^{\psi }& {\mathbf{T}}_{Lm}^{\psi v}\\ {\mathbf{T}}_{Lm}^{v\psi }& {\mathbf{T}}_{Lm}^v\end{array}\right]}}\underset{8\times 1}{\underbrace{\left[\begin{array}{c}{\mathit{C}}_m^w\\ {\mathit{C}}_m^v\end{array}\right]}}+\underset{12\times 1}{\underbrace{\left[\begin{array}{c}\mathit{F}{\mathit{e}}^w\\ \mathit{T}{\mathit{e}}^{\psi }\\ \mathit{F}{\mathit{e}}^v\end{array}\right]}}$$

(70)

Then, the transitive relationship between the integration constants of the ith and the i + 1th sub-beam is

$$\left[\begin{array}{c}{\mathit{C}}_{m+1}^w\\ {\mathit{C}}_{m+1}^v\end{array}\right]={\left[\begin{array}{cc}{\mathbf{T}}_{Rm}^w& 0_{4\times 4}\\ {\mathbf{T}}_{Rm}^{\psi }& 0_{4\times 4}\\ 0_{4\times 8}& {\mathbf{T}}_{Rm}^v\end{array}\right]}^{-1}\left[\begin{array}{cc}{\mathbf{T}}_{Lm}^w& 0_{4\times 4}\\ {\mathbf{T}}_{Lm}^{\psi }& {\mathbf{T}}_{Lm}^{\psi v}\\ {\mathbf{T}}_{Lm}^{v\psi }& {\mathbf{T}}_{Lm}^v\end{array}\right]\left[\begin{array}{c}{\mathit{C}}_m^w\\ {\mathit{C}}_m^v\end{array}\right]+{\left[\begin{array}{cc}{\mathbf{T}}_{Rm}^w& 0_{4\times 4}\\ {\mathbf{T}}_{Rm}^{\psi }& 0_{4\times 4}\\ 0_{4\times 8}& {\mathbf{T}}_{Rm}^v\end{array}\right]}^{-1}\left[\begin{array}{c}\mathit{F}{\mathit{e}}^w\\ \mathit{T}{\mathit{e}}^{\psi }\\ \mathit{F}{\mathit{e}}^v\end{array}\right]={\mathbf{H}}_m\left[\begin{array}{c}{\mathit{C}}_m^w\\ {\mathit{C}}_m^v\end{array}\right]+{\mathbf{H}}_F\widehat{\mathit{F}}\mathit{e}$$

(71)

By repeating Equation (71), the transitive relationship between the first sub-beam and the last sub-beam can be written as

$$\left[\begin{array}{c}{\mathit{C}}_{N+1}^w\\ {\mathit{C}}_{N+1}^v\end{array}\right]=\left({\displaystyle \prod _{i=1}^N{\mathbf{H}}_{N-i+1}}\right)\left[\begin{array}{c}{\mathit{C}}_1^w\\ {\mathit{C}}_1^v\end{array}\right]+\left({\displaystyle \prod _{i=1}^{N-m}{\mathbf{H}}_{N-i+1}}\right)\widehat{\mathit{F}}\mathit{e}$$

(72)

Next, the boundary conditions can be written as follows:

$$\underset{12\times 12}{\underbrace{\left[\begin{array}{c}{B}_{6\times 12}^{\mathrm{L}}\\ B_{6\times 12}^{\mathrm{R}}\left({\displaystyle \prod _{i=1}^N{\mathbf{H}}_{N-i+1}}\right)\end{array}\right]}}\hspace{0.17em}{\mathit{C}}_1=\left[\begin{array}{c}{0}_{6\times 1}\\ -B_{6\times 12}^{\mathrm{R}}\left({\displaystyle \prod _{i=1}^{N-m}{\mathbf{H}}_{N-i+1}}\right)\widehat{\mathit{F}}\mathit{e}\end{array}\right]=\overline{\mathit{F}}\mathit{e}$$

(73)

The substitution of Equation (73) into Equation (72) obtains the integration constants of the n + 1 (n = 1, 2⋯N) sub-beam:

$${\mathit{C}}_{n+1}=\{\begin{array}{l}\left({\displaystyle \prod _{i=1}^n{\mathbf{H}}_{n-i+1}}\right){\mathit{C}}_1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}n=1\dots m-1\\ \left({\displaystyle \prod _{i=1}^n{\mathbf{H}}_{n-i+1}}\right){\mathit{C}}_1+\left({\displaystyle \prod _{i=1}^{n-m}{\mathbf{H}}_{n-i+1}}\right)\widehat{\mathit{F}}\mathit{e}\hspace{0.17em}\hspace{0.17em}n=m\dots N\end{array}$$

(74)

where C₁ is the integration constants of the 1th sub-beam and

$${\mathit{C}}_1={\left[\begin{array}{c}{B}_{\mathrm{L}\hspace{0.17em}6\times 12}^{}\\ B_{\mathrm{R}\hspace{0.17em}6\times 12}^{}\left({\displaystyle \prod _{i=1}^N{\mathbf{H}}_{N-i+1}}\right)\end{array}\right]}^{-1}\overline{\mathit{F}}\mathit{e}$$

(75)

From the derivations above, it is obvious that all of the computing matrices (H_i, B_L, B_R and H_F) associated with C_i are not larger than 12 × 12, which will lead to a small-dimension matrix operation, and hence a significant computational advantage.

3. Validation of the Proposed Method

To validate the reliability of the proposed method, some of the results in this work are checked with those obtained from the conventional FEM. The motion equation of thin-walled beam system can be written as

$$[\mathbf{M}]\left\{\ddot{x}\right\}+[\mathbf{C}]\left\{\dot{x}\right\}+[\mathbf{K}]\left\{x\right\}=\left\{\mathit{F}\right\}$$

(76)

where [M], [K] and [C] denote the mass, stiffness and damping matrix for the thin-walled beam system, {v} represent the node displacement vector of beam, and {F} is the force vector. The overall matrixes are obtained by imposing the prescribed boundary condition and assembling the associated mass matrix [Me] stiffness matrix [Ke] and the damping matrix [Ce] of the constrained beam element, which are given in Appendix E.

Introducing harmonic solutions in the form

$$\left\{x\right\}=\left\{\overline{x}\right\}{e}^{i\hspace{0.17em}\Omega \hspace{0.17em}t}$$

(77)

The frequency response functions can be obtained as

$$\{\overline{x}\}={(-{\Omega }^2\mathbf{M}+i\Omega \mathbf{C}\hspace{0.17em}+\mathbf{K})}^{-1}\{\mathit{F}\}$$

(78)

The forced vibration of a elastically constraint thin-walled beam carrying three spring-damper-mass subsystem (See in Figure 4) is solved both by the presented method and conventional finite-element method. The geometrical and physical properties of the thin-walled beam system are given as: E = 2.1 × 10¹¹ Pa, G = 80 × 10¹⁰ Pa, ρ = 7850 kg/m³, I_z = 1.3745 × 10⁻⁴ m⁴, I_y = 1.6625 × 10⁻⁴ m⁴, J = 4.7657 × 10⁵ m⁴, Γ₀ = 2.2783 × 10⁻⁸ m⁶, I₀ = 5.9992 × 10⁻⁴ m⁴, A = 0.0116 m², L = 3 m, z_c = 0.0619 m, z_b = −0.0550 m, a = 0.26 m, b = 0.32 m, $K_z^{\mathrm{L}}$ = $K_z^{\mathrm{R}}$ = $K_y^{\mathrm{L}}$ = $K_y^{\mathrm{R}}$ = 40,000 kN/m $C_z^{\mathrm{L}}$ = $C_z^{\mathrm{R}}$ = $C_y^{\mathrm{L}}$ = $C_y^{\mathrm{R}}$ = 800 N∙s/m, m_i = 100 kg, J_i = 53.33 kg∙m², $K_{z,i}^{\mathrm{L}}$ = $K_{z,i}^{\mathrm{R}}$ = $K_{y,i}^{\mathrm{L}}$ = $K_{y,i}^{\mathrm{R}}$ = 16,666.67 kN/m, $c_{z,i}^{\mathrm{L}}$ = $c_{z,i}^{\mathrm{R}}$ = $c_{y,i}^{\mathrm{L}}$ = $c_{y,i}^{\mathrm{R}}$ = 33.33 N∙s/m, l₁ = 0.8 m, l₂ = 0.8 m, l₃ = 0.8 m, l₄ = 0.8, $l_1^{\mathrm{L}}$ = 0.08 m, $l_1^{\mathrm{R}}$ = 0.36 m, $l_2^{\mathrm{L}}$ = 0.36 m, $l_2^{\mathrm{R}}$ = 0.08 m, $l_3^{\mathrm{L}}$ = 0.08 m, $l_3^{\mathrm{R}}$ = 0.36 m, i = 1,2,3.

Without loss of generality, the forced vibration of the thin-walled beam system with an excitation force (F^we^iΩt, F^ψe^iΩt, F^νe^iΩt ) applied at P₁ (representing the end point of thin-walled beam), P₂ (representing the arbitrary position of the thin-walled beam) and P₃ (representing the subsystem) are calculated by the proposed method and FEM, respectively. The frequency responses of P₁, P₂, P₄, P₅ (right end of the beam) in each calculation are illustrated in Figure 5. It can be seen that the curves calculated by the proposed method are greatly consistent with FEM. Hence, it is believed that the analytical method and FEM are accurate and reliable.

The frequency responses of the bare beam at P₁, P₂, P₄, and P₅ with force applying at P₁ are illustrate in Figure 6. The comparison of Figure 5 and Figure 6 reveals that the introduction of a subsystem can generate several anti-resonance bands, in which the three-DOF SDM systems act as absorbers for the thin-walled beams, leading to a high suppression of structural vibration transmission. Moreover, a special frequency bandwidth is observed, in which the magnitude of the frequency response at the excitation points is the largest and, the further the detection point deviates from the excitation points, the smaller the response is. This phenomenon indicates that the elastic vibration in the thin-walled beam is increasingly and dramatically decayed when moving from the right end to the left end. Therefore, this structure can be applied in engineering practice to restrain the vibration transmission in the thin-walled beam based on these results.

4. Parameter Study and Discussion

4.1. Effect of Subsystem Parameters on Vibration Transmission of the Thin-Walled Beam

This subsection investigates the influence of the subsystem parameters on the frequency response (FR) at the right end of the beam. In this subsection, the stiffness and damping of the elastic constraints are set to 0, and the excitation force (F^we^iΩt, F^ψe^iΩt, F^νe^iΩt) is applied at the left end of the beam.

Figure 7 illustrates the Frequency Response at the right end of the beam with symmetrical $l_i^{\mathrm{L}}$ = $l_i^{\mathrm{L}}$ = 0.22 m) and asymmetric ($l_1^{\mathrm{L}}$ = 0.08 m, $l_1^{\mathrm{R}}$= 0.36 m, $l_2^{\mathrm{L}}$ = 0.36 m, $l_2^{\mathrm{R}}$ = 0.08 m, $l_3^{\mathrm{L}}$ = 0.08 m, $l_3^{\mathrm{R}}$ = 0.36 m) subsystems. The introduction of asymmetric subsystems is able to make the bending vibration in W direction and the torsional vibration of the thin-walled beam couple with the bending vibration of it in V direction, which are independent in the thin-walled beam with symmetrical subsystems. The vibration of subsystem in z direction and θ direction will couple strongly when the centroidal deviation of subsystem is large, which leads to all the degrees of freedom of the whole system coupling with each other. Notice that (1) when the excitation force is applied in W direction, the force transmission path is W-y, W-Ψ-θ-z-V; (2) when the excitation force is applied in Ψ direction, the force transmission path is Ψ-W-y, Ψ-θ-z-V; (3) when the excitation force is applied in V direction, the force transmission path is V-z-θ-Ψ-W-y. Moreover, the vibration coupling is very strong at low frequencies (Ω < 2200 Hz) since the natural frequencies of low order modes of the thin-walled are close to the natural frequencies of subsystems; thus, the bending vibration in W directions and the torsional vibration of the thin-walled beam strongly couple with the bending vibration of it in V direction via the force transmission of subsystem; conversely, the natural frequencies of high order modes are much higher than the subsystems, which result in a very slight force transmission between the three directions of thin-walled beam, and lead to a significant weak vibration coupling between them.

The effects of the subsystems’ stiffness k_si, damping c_si, mass m_i, moment of inertia J_i on the vibration transmission characteristics of the thin-walled beam are illustrated in Figure 8 and Figure 9, respectively. Figure 8 shows that the location of the anti-resonance bands can be adjusted by the subsystems’ stiffness to suppress the vibration with a specific frequency, since their move to a higher frequency area with the increase of k_si. Figure 9 reveals that a larger damping of the subsystem is helpful for the suppression of vibration transmission in the thin-walled beams, since c_si can significantly decrease the peak values of FR. It can be observed in Figure 10 that, when the mass is small, the resonance peaks and anti-resonance points generated by the subsystems are very weak, since the translational motion mode of subsystems (J_i is relatively small) are not strong enough to affect the vibration characteristic of the whole system; by increasing the value of m_i, the effect of subsystems on vibration characteristic of the whole thin-walled system become stronger, resulting in more remarkable resonance peaks and anti-resonance points. It is worth noting that a higher m_i corresponds to a decrease in the frequency of the potential well in Ψ direction, accompanying an increase in the width of the frequency band. Figure 11 shows the moment of inertia of the subsystem has a great effect on the vibration transmission potential well in all three directions. Generally, a higher J_i corresponds to a decrease in the frequency of the anti-resonance band.

Apparently, a relatively deeper and wider anti-resonance band can guarantee effective restraint of the vibration transmission in the beam, hence the parameter design of the subsystem should follow the following principle: (1) For an excitation force with a relatively high frequency, the stiffness of subsystem should be large; (2) For an excitation force with a relatively low frequency, larger mass and moment of inertia are required.

4.2. Effect of Parameters on the Vibration Isolation Characteristic of the Thin-Walled Beams Systems

For the system, consisting of power machines and thin-walled structures, the isolation of the excitation force of the machines’ transfer to the structures is a main measure to control the structure vibration and noise radiation of the system. Obviously, the parameters of the elastic suspensions and the position of the machines are believed to significantly influence the dynamic response of the thin-walled beam. This subsection studies the relationship between these parameters and the dynamic response of the thin-walled beam. The system is illustrated in Figure 12, and the parameters of the system were arbitrarily selected as follows: $K_z^{\mathrm{L}}$ = $K_z^{\mathrm{R}}$ = $K_y^{\mathrm{L}}$ = $K_{\mathrm{y}}^{\mathrm{R}}$ = 40,000 kN/m, $C_z^{\mathrm{L}}$ = $C_z^{\mathrm{R}}$ = $C_y^{\mathrm{L}}$ = $C_{\mathrm{y}}^{\mathrm{R}}$ = 800 N∙s/m, $K_{\mathrm{T}}^{\mathrm{L}}$ = $K_{\mathrm{T}}^{\mathrm{R}}$ = 1500 kN∙m/rad; $K_{\mathrm{T}}^{\mathrm{L}}$ = $K_{\mathrm{T}}^{\mathrm{R}}$ = 300 N∙m∙s/rad; m = 300 kg, J = 160 kg∙m², $k_z^{\mathrm{L}}$ = $k_z^{\mathrm{R}}$ = $k_y^{\mathrm{L}}$ = $k_{\mathrm{y}}^{\mathrm{R}}$ = k_s = 40,000 kN/m, $c_z^{\mathrm{L}}$ = $c_z^{\mathrm{R}}$ = $c_y^{\mathrm{L}}$ = $c_{\mathrm{y}}^{\mathrm{R}}$ = 100 N∙s/m, l₁ = 1.6 m, l₂ = 1.6 m, $l_1^{\mathrm{L}}$ = 0.08 m, $l_1^{\mathrm{R}}$ = 0.36 m.

In order to accurately describe the intensity of the structural vibration, the vibration energy of the thin-walled beam is defined as follows:

$$EV=EK/{\Omega }^2={\displaystyle \int \rho A(W}{(x)}^2+V{(x)}^2)+\rho I_0\Psi {(x)}^{2}dx$$

(79)

where EK is the kinetic energy of the thin-walled beam.

The influences of the subsystems’ stiffness k_si, damping c_si, location d (the distance between the subsystem and the symmetry center of the thin-walled beam), elastic constraint stiffness K and elastic constraint damping C on the vibration isolation characteristics of the thin-walled beam systems are illustrated in Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17, respectively. It can be observed for Figure 13 that the high frequency magnitude of EV increases with the increase in k_si due to the increasing coupling degree of the rigid motion of the subsystem and the higher order mode of the thin-walled beam. Figure 14 shows that a suitable lager c_si is helpful to suppress the resonance peaks values of EV, while the excessive damping will deteriorate the isolation performance of the thin-walled beam systems due to the bonding effect between the subsystem and beam. It can be seen from Figure 15 that the resonance peak values of EV decrease along with the increase of d, which also leads to an increased number of resonance peaks, since the stronger asymmetry of the system will cause more resonances mode. Figure 16 shows that the frequency of the resonance peaks increases with K due to the increase of the mode stiffness of the thin-walled beams. However, the excessive stiffness will decrease the mode damping coefficient of the thin-walled beam, resulting in a dramatic increase of the magnitude of the resonance peak of EV. Figure 17 reveals that the EV curves of the thin-walled for a higher C have a smaller peak value compared to that for a lower C due to the increase of the mode damping coefficient of the thin-walled beam.

Based on the above discussions, relatively smaller stiffnesses and larger damping of the subsystem and elastic constraint are beneficial to isolate the excitation force of the machines from the base structures.

5. Conclusions

This paper presents a new closed analytical approach to obtaining the forced vibration of bending-torsional-warping coupled thin-walled beams with an arbitrary number of 3-DoF spring-damper-mass subsystems and boundary conditions based on the transfer matrix approach and dynamic condensation methods. The thin-walled beam is divided into a series of distinct sub-beams whose ends are connected to the SDM subsystems. The transfer matrix for each sub-beam is developed based on the exact shape functions of the bending-torsional-warping coupling Euler-Bernoulli theory. Each SDM system is modeled by a set of effective springs based on the dynamic condensation method. The governing matrix equation is formulated based on the compatibility conditions of the placement and the force at the common interfaces of two adjacent sub-beams.

The proposed method is very convenient to apply and can be conveniently used to obtain exact close form expression of the frequency response function; it enables simultaneous consideration of arbitrary boundary conditions and an arbitrary number of SDM systems; furthermore, the proposed method reasonably yields the associate matrixes with size never larger than 12 × 12 independent of the number of subsystems, thus leading to a significant computational advantage.

The results computed by the proposed method achieve good agreement with those obtained by the conventional finite-element method. The effects of the system parameters on the vibration transmission and vibration isolation properties of the thin-walled beam system are studied. The results provide useful information for the analysis and design of thin-walled structures with power machines for the purposes of elastic vibration transmission reduction and vibration isolation.