Study of flexural wave propagation through a cable stayed beam subjected to a moving load

Abstract

A physical perspective of the propagation and attenuation of flexural waves is presented in this paper for the dynamic behaviors of cable stayed beams subjected to a moving load. Based on the method of reverberation-ray matrix (MRRM), the waveform solutions of the wave equations of a simplified beam-cable system subjected to a moving load (hereinafter referred to as a beam-cable system) are given, and the theory is verified by a numerical example. The dynamic response of cable stayed beams is decomposed into nine kinds of flexural waves, including traveling waves, near-field waves, and nondispersive waves, according to the wavenumber characteristics. Numerical examples are analyzed to demonstrate the propagation characteristics of flexural waves through cable stayed beams. Numerical results show that the flexural waves in the cable stayed beams are mainly low-frequency waves whose frequencies are less than 3 times the structural fundamental frequency, which can be used to further improve the computational efficiency of response analysis method based on MRRM, and the proportion of high-frequency components increases gradually with increasing structural stiffness. The near-field wave can be transformed into a traveling shear wave when its frequency is larger than the critical frequency, which decreases with increasing radius of gyration and decreasing elastic modulus of the beam. With the increase in the radius of gyration and the elastic modulus of the beam, the attenuation effect of the near-field wave weakens. The wave velocity and the wave dispersion effect have a positive correlation with the stiffness-related parameters of the beam-cable system. The study of the effect of the beam-cable system parameters on flexural wave propagation characteristics can be applied to achieve a better dynamic design for engineering structures.

Keywords

cable stayed beam moving load flexural wave near-field wave method of reverberation-ray matrix

Introduction

As we all know, dynamic response can be expressed in terms of waves traveling through structure (Renno and Mace, 2013). The attenuation, dispersion, and transformation of flexural waves in a bridge structure have a great impact on the early dynamic response of the structure (Kolsky, 1953), but the existing design codes often ignore this process, which easily causes unexpected structural damage. For long-span beam-cable system bridges, such as cable stayed bridges, suspension bridges, through or half through arch bridges, the time of flexural waves passing through the structure until reaching equilibrium is the same order of magnitude as the duration of vehicle loading; hence, the process of wave propagation in the above structures cannot be ignored (Wang and Zhu, 2005).

Although the published literatures rarely involve the problem of stress wave propagation in beam-cable systems, we can obtain inspiration from previous studies on other engineering structures. Guo et al. (2018) focused on the characteristics of the wavenumber spectrum of flexural wave in coupled periodic multicomponent beams to explore the relationship between the propagation characteristics of flexural wave and structural parameters, so as to optimize structural mechanical characteristics. Lei et al. (2020) used the modal displacement method to study the propagation performance of vibration waves in a composite nonperiodic isolator and investigated the energy transmission and loss performance. Tang et al. (2021) studied the propagation and dispersion characteristics of traveling waves in infinite Timoshenko beams coupled with periodic resonators in series connections from the perspective of the frequency spectrum to provide a reference for structural seismic design. However, all the above studies focused on the propagation characteristics of flexural waves in various kinds of semi-infinite engineering structures, the research object of this paper is a finite structure system.

For the study of wave propagation behavior in finite engineering structures, the investigation about stress wave distribution at nodes (Li et al., 2011) or discontinuities (Dawood et al., 2013) is the basis. In the early years, Lee and Kolsky (1972) investigated the generation of stress pulses at the junction of two noncollinear rods. In recent years, Abadi (2019) deduced the reflection and transmission coefficients of traveling waves at joints and studied the energy general distribution characteristics of elastic waves at joints. On the basis of previous studies, the method of varying amplitudes was employed to study the propagation characteristics of transverse waves in a finite string with a periodically modulated cross-section (Sorokin and Thomsen, 2017). Using the finite element method, Guo and Sheng (2018) investigated the propagation characteristics and attenuation behavior of flexural waves through periodic bilayer beams. The above researches have brought great inspiration, that is the flexural wave scattering relationship which is deduced based on the compatibility of the displacements and the equilibrium of the generalized forces at the joints, can be used to establish the constraint conditions of the displacement functions of the beam and cable.

Most of the previous studies have focused on the propagation of free waves in semi-infinite mechanical models but less on the propagation of flexural waves in forced vibration engineering structures under operating loads. It is of great engineering significance to study the frequency spectrum, types, and attenuation characteristics of stress waves excited by operation loads in a specific structural system. However, the research in this field is relatively insufficient. To investigate the general characteristics of flexural wave propagation through a cable stayed beam subjected to a moving load, the transient response should be studied first and then transformed into a wave expression.

The study of the dynamic response of engineering structures under moving loads originated as early as 1905 (Kryloff, 1905) and has motivated numerous investigations. The difference between the research methods for the transient response of beams under impact loads is the difference in the methods of solving dynamic equations, which include but are not limited to Pestel and Leckie (1963) with the Matrix Method, Song et al. (2016a, 2016b) with the Eigen function expansion approach combined the Galerkin method and the Runge-Kutta-Fehlberg method, Lu and Li (2018) with the method of assumed displacement function, Zhou et al. (2021) with the explicit finite element algorithm. Most of the dynamic response analysis methods (such as the above methods) are poor in directly obtaining the closed-form expression of the waveform solution of the dynamic equation. So finding a dynamic analysis method based on traveling wave solution is the first problem to be solved.

Howard and Pao (1998) proposed the MRRM to analyze the propagation of elastic waves in plane trusses. After the MRRM was proposed, Pao and his colleagues carried out a series of studies and found that the MRRM can be used to accurately predict the transient response of layered structures (Su et al., 2002), continuous beams under moving loads (Jiang et al., 2012), laminated beams (Miao et al., 2013) and cylindrical shell structures (Zhou et al., 2020). A series of studies verify the reliability and applicability of MRRM in solving problems of structural dynamic response. However, the above studies focused on the transient response of single-form structures and did not involve complex structural systems, such as beam-cable systems for which the MRRM is attempting to be utilized in this paper. The MRRM provides a clear physical explanation for the transient response of structures, that is, the propagation and superposition of stress waves. Therefore, the MRRM is employed to derive the waveform solution of the dynamic response of cable stayed beams subjected to moving loads in the present paper.

In this paper, regarding the large flexibility of long-span beam-cable bridges, the propagation properties of flexural waves in cable stayed beams subjected to moving loads are explored. The transient response obtained by the MRRM is analyzed into an expression representing several kinds of flexural wave propagation through cable stayed beams, and the characteristics of various kinds of flexural waves are discussed. The effects of structural stiffness parameters on the wavenumber spectrum are thoroughly clarified, which provides clear physical meaning for structural dynamic behaviors and a basis for structural design.

Theoretical model and formulations

Mechanical model and the governing equations

A simplified model of beams coupled with cables and subjected to moving load q (x,t) is schematically shown in Figure 1(a). The supporting points and the joints where the beams and cables are connected with an angle α are regarded as nodes identified by letters (I, J, O, and K). Each segment of beams is identified by two letters at both ends (say IJ and JK), and each segment of cables is identified by IO and KO. The dual local coordinate system of a segment of the beam-cable system is illustrated in Figure 1(b), in which the origin O is located at the end of each segment with a straight axis passing through cross-section centroids of each segment and pointing to the other end, which is taken as the spatial x-axis. The vertical direction of the x-axis is identified as the corresponding y-axis according to the right-hand spiral rule.

Figure 1.

Model of a cable-beam system: (a) the finite model and the moving load and (b) dual local coordinate system (cm).

The wave equations governing the propagation of flexural waves through the beams and elastic waves through the cables satisfy the assumptions and basic theories as follows:

(1) Considering the additional influence of the initial tension of the stay cable on the transverse vibration of the beam, the wave equation in the beam is established based on Timoshenko beam theory (Sun and Li, 2019).

(2) For the stayed cables, the instantaneous reduction in tensile strain caused by the moving load is far less than the initial tensile strain, so it is appropriate to use one-dimensional rod theory to simulate the propagation of longitudinal waves in the cable caused by the moving load. (Lu and Li, 2018)

(3) The bending stiffness and sag of the cable are ignored (Lu and Li, 2018), which lead to the shear force and bending moment that are related to the transverse displacement, are neglected. Therefore, for the dynamic response study for the beam, it's not necessary to introduce the flexural wave propagation through the stay cable.

The equations (William et al., 1990; Zhang, 2018) governing waves through the beam segment IJ and cable segment IO in a local coordinate system can be written as (superscript IJ, IO omitted)

E I_{Z} \frac{\partial^{2} φ}{\partial x^{2}} + κ A G (\frac{\partial υ}{\partial x} - φ) - ρ I_{Z} \frac{\partial^{2} φ}{\partial t^{2}} = 0

(1)

κ A G (\frac{\partial^{2} υ}{\partial x^{2}} - \frac{\partial φ}{\partial x}) + N_{x} \frac{\partial^{2} υ}{\partial x^{2}} - ρ A \frac{\partial^{2} υ}{\partial t^{2}} - q (x, t)

(2)

\frac{\partial^{2} u}{\partial t^{2}} - c_{c}^{2} \frac{\partial^{2} u}{\partial x^{2}} = 0

(3)

where

υ

and

φ

, respectively, denote the transverse displacement and the rotation angle of the cross-section of the beams,

u

is the axial displacement of cables, α is the angle between the beam and cable, and T is the initial tension of the cable. The shear coefficient

κ

is taken as

π^{2} / 12

(Doyle, 1997). The respective moment of inertia of the beam cross-section and beam cross-sectional area I_Z and A, elastic modulus, shear modulus and density of beams E, G, and

ρ

, cross-sectional area, elastic modulus, and density of cables

A_{C}

E_{C}

, and

ρ_{C}

are all assumed to be constants for each segment. In addition,

c_{c} = \sqrt{E_{c} / ρ_{c}}

is the velocity of the longitudinal wave in the cable.

N_{x} = T \times \cos (α)

is the initial axial force of the beam.

The expression of moving load $q (x, t)$ with a constant speed V_p (Dugush and Eisenberger, 2002) in a dual local coordinate system can be expressed as equation (4)

\begin{array}{l} q^{I J} (x^{I J}, t) = [H (t - T_{I}) - H (t - T_{J})] q δ (x^{I J} + V_{p} T_{I} - V_{p} t) \\ q^{J I} (x^{J I}, t) = - [H (t - T_{I}) - H (t - T_{J})] q δ (x^{J I} - V_{p} T_{J} + V_{p} t) \end{array}

(4)

where T_I and T_J are, respectively, the instants when the load enters and leaves segment IJ;

δ

, and

H

are, respectively, the Dirac function and step function.

Wave form solution of governing equations

The general solution of equations (1) and (2) is the sum of the general solutions of the corresponding homogeneous equations and the special solutions of equations (1) and (2). The solutions of the homogeneous equations can be obtained by assuming expressions of the simple harmonic, such as $υ = \hat{C} e^{i k x} e^{- i ω t}$ , $φ = \hat{D} e^{i k x} e^{- i ω t}$ and $u = \hat{H} e^{i k x} e^{- i ω t}$ (Wu and Han, 2001). By eliminating the time-dependent factor $e^{- i ω t}$ , the solutions of the homogeneous equations are presented in the frequency domain as (superscript IJ, IO omitted)

\begin{array}{l} \hat{υ} (x, ω) = a_{1} e^{i k_{1} x} + d_{1} e^{i k_{2} x} + a_{3} e^{i k_{3} x} + d_{3} e^{i k_{4} x} \\ = a_{1} e^{i k_{1} x} + d_{1} e^{- i k_{1} x} + a_{3} e^{i k_{3} x} + d_{3} e^{- i k_{3} x} \end{array}

(5)

\begin{array}{l} φ (x, ω) = K_{1} a_{1} e^{i k_{1} x} - K_{1} d_{1} e^{i k_{2} x} + K_{3} a_{3} e^{i k_{3} x} - K_{3} d_{3} e^{i k_{4} x} \\ = K_{1} a_{1} e^{i k_{1} x} - K_{1} d_{1} e^{- i k_{1} x} + K_{3} a_{3} e^{i k_{3} x} + K_{3} d_{3} e^{- i k_{3} x} \end{array}

(6)

\hat{u} (x, ω) = a_{6} e^{i k_{6} x} + d_{6} e^{- i k_{6} x}

(7)

where

a_{i}

and

d_{i}

are the wave amplitudes of the arriving waves and the departing waves, respectively. The wavenumbers k for which the physical meanings are detailed in Flexural waves through the beam can be presented as follows

\begin{array}{l} k_{1} = \sqrt{\frac{- β + \sqrt{β^{2} - 4 χ γ}}{2 χ}} \\ k_{1} = - k_{1} = - \sqrt{\frac{- β + \sqrt{β^{2} - 4 χ γ}}{2 χ}} \\ k_{3} = \sqrt{\frac{- β - \sqrt{β^{2} - 4 χ γ}}{2 χ}} \\ k_{4} = - k_{3} = - \sqrt{\frac{- β - \sqrt{β^{2} - 4 χ γ}}{2 χ}} \\ k_{6} = \frac{ω}{c_{c}} \end{array}

(8)

where

\begin{array}{l} χ = E I_{Z} (κ A G + N_{x}) \\ β = κ A G N_{x} - ω^{2} (E I_{Z} ρ A + κ A G ρ I_{Z} + ρ I_{Z} N_{x}) \\ γ = - ω^{2} κ ρ A^{2} G + ω^{4} ρ^{2} A I_{Z} \end{array}

The amplitude ratios of the rotation with respect to the transverse displacement K_i (i= 1, 2, 3, 4) can be written as

K_{i} = \frac{\hat{D}}{\hat{C}} = \frac{- k_{i}^{2} κ A G - N_{X} k_{i}^{2} + ρ A ω^{2}}{i κ A G k_{i}}

The special solutions of equations (1) and (2), which are represented by $υ * (ω, x), φ * (ω, x)$ , can be obtained by quadratic Fourier transformation on t and x. Finally, the general solutions of equations (1) and (2) in the frequency domain can be expressed as equation (9)

\begin{array}{l} {\hat{υ}}^{I J} = a_{1}^{I J} e^{i k_{1} x^{I J}} + d_{1}^{i J} e^{- i k_{1} x^{I J}} \\ + a_{3}^{I J} e^{i k_{3} x^{I J}} + a_{3}^{I J} e^{- i k_{3} x^{I J}} + {\hat{υ}}^{* I J} \\ {\hat{υ}}^{J I} = a_{1}^{I J} e^{i k_{1} x^{I J}} + d_{1}^{J I} e^{- i k_{1} x^{J I}} \\ + a_{3}^{J I} e^{i k_{3} x^{J I}} + a_{3}^{J I} e^{- i k_{3} x^{J I}} + {\hat{υ}}^{* I J} \\ {\hat{φ}}^{I J} = K_{1} a_{1}^{I J} e^{i k_{1} x^{I J}} - K_{1} d_{1}^{i J} e^{- i k_{1} x^{I J}} \\ + K_{3} a_{3}^{I J} e^{i k_{3} x^{I J}} - K_{3} d^{I J} a_{3}^{I J} e^{- i k_{3} x^{I J}} + {\hat{φ}}^{* I J} \\ {\hat{φ}}^{J I} = K_{1} a_{1}^{J I} e^{i k_{1} x^{J I}} - K_{1} d_{1}^{J I} e^{- i k_{1} x^{J I}} \\ + K_{3} a_{3}^{J I} e^{i k_{3} x^{J I}} - K_{3} d^{J I} a_{3}^{J I} e^{- i k_{3} x^{J I}} + {\hat{φ}}^{* I J} \end{array}

(9)

where

υ^{*} (ω, x), φ^{*} (ω, x)

of segment IJ in the dual local coordinate system are expressed in Appendix 1.

The frequency-domain expressions of the axial force in the cable segment (say IO) and the moment and shear force in the beam segment (say IJ) are shown in equation (10).

\begin{array}{l} {\hat{Q}}^{I J} = κ G A (\frac{d {\hat{υ}}^{I J}}{d x^{I J}} - {\hat{φ}}^{IJ}) \\ {\hat{M}}^{I J} = E I_{Z} \frac{d {\hat{u}}^{I O}}{d x^{I J}} \\ {\hat{N}}^{I O} = E A \frac{d {\hat{u}}^{I O}}{d x^{I O}} \end{array}

(10)

The method of the reverberation-ray matrix

With regard to the beam-cable system under a moving load shown in Figure 1, the general solutions of the wave equations of segments IO and IJ are given in equations (7) and (9). To solve the amplitude of the arriving waves and the departing waves of each node, the MRRM is adopted.

First, the local scattering relations of wave propagation at nodes are established according to the compatibility of the displacements and the equilibrium of the forces under local coordinates. All the local scattering matrices are combined into the global scattering matrix. Second, the phase relationship of harmonic propagation is obtained by the relationship of displacements in the dual local coordinate systems. Finally, the reverberation-ray matrices of the beam-cable system are constructed by scattering matrices and phase matrices.

Scattering matrix

The compatibility of the displacements and the equilibrium of the generalized forces at the joints of the beam-cable system shown in Fig. 1 are presented in Table 1:

Table 1.

Compatibility of the displacements and the equilibrium of the generalized forces.

Node	Expressions
I	${\hat{υ}}^{I J} \sin α = {\hat{u}}^{I O}, {\hat{M}}^{I J} = 0, {\hat{Q}}^{I J} = - {\hat{N}}^{I O} \sin α$
J	${\hat{υ}}^{J I} = 0, {\hat{υ}}^{J K} = 0, {\hat{φ}}^{J I} = {\hat{φ}}^{J K}, {\hat{M}}^{J I} + {\hat{M}}^{J K} = 0$
K	${\hat{υ}}^{K J} \sin α = - {\hat{u}}^{K O}, {\hat{M}}^{K J} = 0, {\hat{Q}}^{K J} = {\hat{N}}^{K O} \sin α$
O	${\hat{u}}^{O I} = 0, {\hat{u}}^{O K} = 0$

Substituting solutions of displacements and forces, such as equations (7), (9), and (10), into the expressions in Table 1 yields the local scattering equation (say Node J) as follows:

d^{J} = S^{J} \times a^{J}

(11)

where

d^{J} = {[{(d^{JI})}^{T}, {(d^{JK})}^{T}]}^{T}

and

a^{J} = {[{(a^{JI})}^{T}, {(a^{JK})}^{T}]}^{T}

are, respectively, amplitude vectors of the departing wave and the arriving wave at Node J. S ^J is the local scattering matrix of Node J.

The global scattering matrix is obtained by combining the local scattering matrix, and then the global scattering equation is obtained as follows:

d = Sa

(12)

where

d = {[d^{I}, d^{J}, d^{K}, d^{O}]}^{T}

and

a = {[a^{I}, a^{J}, a^{K}, α^{O}]}^{T}

are respectively amplitude vectors of the departing wave and the arriving wave of the beam-cable system.

S = diag [S^{I}, S^{J}, S^{K}, S^{O}]

is the global scattering matrix.

Phase matrix and permutation matrix

The compatibility of the displacement at the same segment position (say segment IJ) in the dual coordinate systems is shown as follows

\hat{υ} (x^{I J}) = - {\hat{υ}}^{J I} (L_{B} - x^{I J})

(13)

where L_B are the length of beam segment.

Substituting equation (9) into equation (13) yields the phase relation (Shao et al., 2021) as follows

\begin{array}{l} a^{IJ} = P^{IJ} d^{IJ} + Q^{IJ} \\ a^{JI} = P^{JI} d^{IJ} + Q^{JI} \end{array}

(14)

where

P^{IJ}

and

P^{JI}

are phase matrices, then

Q^{IJ}

and

Q^{JI}

are incentive source vectors due to external load. The expressions of phase matrices and incentive source vectors are presented in Appendix 2

According to the phase relation of each segment of the beam, the global phase equation is obtained as follows

a = P \bar{d} + Q

(15)

where P is the global phase matrix. The order of the scalar state variables of amplitude vectors

\bar{d}

is different from d , which are amplitude vectors of the departing wave.

The permutation matrix (Howard and Pao, 1998) U is introduced to adjust the variable sequential orders of the amplitude vectors of the departing wave

a = P \bar{d} + Q = PUd + Q

(16)

The expression of permutation matrix is presented in Appendix 3

Reverberation-ray matrix

The simultaneous equations (12) and (16) provide that

d = {(I - R)}^{- 1} s

(17)

a = S^{- 1} d

(18)

where

R = SPU

is the reverberation-ray matrix of the beam-cable system and

s = SQ

is the source vector. After substituting equations (7) and (9) into equations (17) and (18), the inverse fast Fourier transform (IFFT) is used for the transient response. The following inverse fast Fourier transform is utilized as follows

f (x, t_{k}) = \frac{1}{T} \sum_{n = 0}^{N - 1} F (ω_{n}) e^{i 2 π n k / N}

(19)

When the structure is in free vibration, it means $s = 0$ in equation (17), or in the following format

(I - R) d = 0

(20)

The natural vibration frequency of the structure is obtained by the following equation

| I - R | = 0

(21)

Flexural waves through the beam

The wave solution of the beam segment (transverse displacement in IJ) in the time domain can be obtained by the inverse Fourier transform of equation (9).

\begin{array}{l} υ (x, t) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} \hat{υ} (x, ω) e^{i ω t} d ω \\ = \frac{1}{2 π} \int_{- \infty}^{+ \infty} a_{1} e^{i (k_{1} x + ω t)} d ω + \frac{1}{2 π} \int_{- \infty}^{+ \infty} d_{1} e^{i (k_{2} x + ω t)} d ω \\ = \frac{1}{2 π} \int_{- \infty}^{+ \infty} a_{3} e^{i (k_{3} x + ω t)} d ω + \frac{1}{2 π} \int_{- \infty}^{+ \infty} d_{1} e^{i (k_{4} x + ω t)} d ω \\ + \frac{1}{2 π} \int_{- \infty}^{+ \infty} {\hat{υ}}^{*} (x, ω) e^{i ω t} d ω \end{array}

(22)

The superscript IJ in equation (22) is omitted. Based the Euler’s Formula, equation (22) is evolved as follows

\begin{array}{l} υ (x, t) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} \hat{υ} (x, ω) e^{i ω t} d ω \\ = \frac{1}{2 π} \int_{- \infty}^{+ \infty} a_{1} e^{i (k_{1} x + ω t)} d ω + \frac{1}{2 π} \int_{- \infty}^{+ \infty} d_{1} e^{i (- k_{1} x + ω t)} d ω \\ = \frac{1}{2 π} \int_{- \infty}^{+ \infty} a_{3} e^{i (k_{3} x + ω t)} d ω + \frac{1}{2 π} \int_{- \infty}^{+ \infty} d_{3} e^{i (- k_{3} x + ω t)} d ω \\ + \frac{1}{2 π} \int_{- \infty}^{+ \infty} - \frac{q (E I_{Z} k_{1}^{2} + κ A G - ρ I_{Z} ω^{2})}{2 i (2 α k_{1}^{3} + β k_{1})} \frac{e^{- i ω T_{I}}}{i k_{1} V_{p} + i ω} e^{i (k_{1} x + ω t)} d ω \\ + \frac{1}{2 π} \int_{- \infty}^{+ \infty} - \frac{q (E I_{Z} k_{1}^{2} + κ A G - ρ I_{Z} ω^{2})}{2 i (2 α k_{1}^{3} + β k_{1})} \frac{e^{- i ω T_{I}}}{i k_{1} V_{p} - i ω} e^{i (k_{1} x + ω t)} d ω \\ + \frac{1}{2 π} \int_{- \infty}^{+ \infty} - \frac{q (E I_{Z} k_{3}^{2} + κ A G - ρ I_{Z} ω^{2})}{2 i (2 α k_{3}^{3} + β k_{3})} \frac{e^{- i ω T_{I}}}{i k_{3} V_{p} + i ω} e^{i (k_{1} x + ω t)} d ω \\ + \frac{1}{2 π} \int_{- \infty}^{+ \infty} - \frac{q (E I_{Z} k_{3}^{2} + κ A G - ρ I_{Z} ω^{2})}{2 i (2 α k_{3}^{3} + β k_{3})} \frac{e^{- i ω T_{I}}}{i k_{3} V_{p} + i ω} e^{i (k_{3} x + ω t)} d ω \\ + \frac{1}{2 π} \int_{- \infty}^{+ \infty} [\begin{array}{l} \frac{q (E I_{Z} k_{1}^{2} + κ A G - ρ I_{Z} ω^{2})}{2 i (2 α k_{1}^{3} + β k_{1})} \frac{1}{i k_{1} V_{p} + i ω} \\ + \frac{q (E I_{Z} k_{1}^{2} + κ A G - ρ I_{Z} ω^{2})}{2 i (2 α k_{1}^{3} + β k_{1})} \frac{1}{i k_{1} V_{p} - i ω} \\ + \frac{q (E I_{Z} k_{3}^{2} + κ A G - ρ I_{Z} ω^{2})}{2 i (2 α k_{3}^{3} + β k_{3})} \frac{1}{i k_{3} V_{p} - i ω} \\ + \frac{q (E I_{Z} k_{3}^{2} + κ A G - ρ I_{Z} ω^{2})}{2 i (2 α k_{3}^{3} + β k_{3})} \frac{1}{i k_{3} V_{p} - i ω} \end{array}] e^{i [- k_{5} x + ω (t - T_{I})]} \end{array}

(23)

For simplicity, $- k_{1} = k_{2}, - k_{3} = k_{4}$ is introduced in equation (23). It can be found that there are nine kinds of flexural waves (It can also be considered that there are only three kinds of flexural waves according to the wavenumber relationship) through cable stayed beams under a moving load. The physical meanings of wavenumbers are expressed as follows:

$R e (k) = 0, Im (k) < 0$ : The near-field wave (Fahy and Gardonio, 2007) decays exponentially along the negative direction of the x-axis.

$R e (k) = 0, Im (k) > 0$ : The near-field wave decays exponentially along the positive direction of the x-axis.

$R e (k) < 0, Im (k) = 0$ : The traveling wave along the positive direction of the x-axis.

$R e (k) > 0, Im (k) = 0$ : The traveling wave along the negative direction of the x-axis.

The spectrum relations of the first four terms on the right side of equation (23) are the same as those of free flexural wave propagation in the structure. The spectrum relations are related to the beam parameters, and the amplitudes of waves are determined by the external load and boundary conditions. The 5–8 terms are flexural waves whose spectrum relations are consistent with the first four terms, and the amplitudes of waves are determined by the external load. The amplitude of the ninth wave whose spectrum relation defined as $k_{5} = ω / V_{p}$ , is determined by the external load. It is a nondispersive disturbance with the same direction and velocity as the moving load.

The equation of the phase velocity of a single frequency harmonic can be obtained as follows

c_{p} = | \frac{ω}{real (k)} |

(24)

The flexural traveling waves propagating in the structure are composed of multiple frequency components. According to equation (24), the phase velocities of flexural waves at different frequencies are different. Therefore, the wave packet is distorted during propagation, which is called the dispersion of the traveling wave. To quantify the dispersion characteristics of flexural waves, $d c_{p} / d ω$ (phase velocity derivation from frequency) is used as the dispersion indicator. The larger the value of the dispersion indicator is, the more obvious the dispersion of the flexural waves, and the more favorable the structure is.

Implementation and validation of the program

The calculation program of transient response and flexural wave propagation of a cable stayed beam under a moving load is compiled by MATLAB. Compared with the transient response calculated by the finite element analysis software ABAQUS, the theory in this paper is verified.

Implementation of the program

It is practical to study the transient response and flexural wave propagation of cable stayed beams subjected to a moving load which is shown in Fig. 1 through a specific calculation example with the parameters of the system predefined in Table 2. The calculation process is shown in Figure 2 in which F_S and N mean sampling frequency and number, respectively.

Table 2.

Parameters of the beam-cable system.

Parameters of the beam					Parameters of the cable
Density/(kg·m⁻¹)	Shear modulus/(Pa)	Elastic modulus/(Pa)	Cross-sectional area/(m²)	Radius of Gyration/(m²)	Density/(kg·m⁻¹)	Elastic modulus/(Pa)	Cross-sectional area/ (m²)	Angle/(°)
7850	7.923e+010	2.06 × 10¹¹	0.25	0.0052	8005	1.950 × 10¹¹	7.854e-003	60

Figure 2.

Flow chart of the calculation program.

Validation of the method

To verify the validation of the method presented in this paper, a comparative study is carried out for transient response calculation for the cable stayed beam shown in the previous section by the finite element method (FEM) and the MRRM. The finite element model is established by ABAQUS. The beam is simulated by the B31 element, and the cable is simulated by the truss element. Using the subroutine VDLOAD of ABAQUS, the moving load is defined in size, moving speed, and action position. A sampling frequency of 1000 Hz with a sampling number of 32,768 is adopted while performing the IFFT for the MRRM. The moving load is specified as a constant force of 1500 N and moves from left to right at speed $V_{p} = 20 m / s$ .

The comparison of the transverse displacement curves of the node at the mid span position of segment IJ calculated by the FEM and MRRM is shown in Figure 3. Fig. 3 indicates that the calculation results obtained by the FEM and the MRRM have a high degree of consistency in the wave trend of the displacement curve and have a slight deviation at the numerical value. The deviation at the peak displacement is approximately 8%, 2%, and 6% when angle α takes a value of 45°, 60°, and 90°, respectively. The small deviation between the results calculated by the two methods is mainly due to the following reasons: (1) Although the research object is an undamped system, damping is introduced in the form of volume viscosity in ABAQUS (linear/quadratic bulk viscosity parameters were set to the value of 0.06/1.2), resulting in a difference between the finite element model and ideal model. (2) The FEM uses the finite difference method to solve the dynamic equation, which needs a large number of meshes. The results are approximate solutions, and there are discrete errors. However, the algorithm in this paper takes the real wave solution as the shape function and does not need to divide the continuous segments. The analysis shows that the accuracy of the current method is guaranteed for the subsequent discussion of the flexural wave propagation characteristics in the low damping system.

Figure 3.

Comparison of displacement curves of beam segment IJ at x=0.5 L calculated by the theory of this paper and FEM: (a) $α = 45 °$ ; (b) $α = 60 °$ ; and (c) $α = 90 °$ (unit:mm).

The comparison of the cable force curves of segment OI at the node O calculated by the FEM and MRRM is shown in Figure 4. As shown in Fig. 4, the node force curves oscillate greatly. Therefore, to ensure that the variation trend of the presented force curve is more obvious, the value of the moving load is increased to 150 kN. Fig. 4 indicates that the calculation results obtained by the FEM and the MRRM have a high degree of consistency in the variation trend of the force curve.

Figure 4.

Comparison of time history curve of axial force at node O calculated by MRRM and FEM.

The efficiency of MRRM for the frequency domain solution is much higher than that of FEM. The computational time of FEM increases with the increase of the number of meshes, but for the transient response of the simplified mechanical model discussed in this paper, the computational efficiency of FEM is slightly higher than that of MRRM.

Results and discussion

The flexural wave propagation through the cable stayed beams mentioned in the previous section is investigated.

Frequency spectral analysis

The effects of the elastic modulus, cross-sectional radius of gyration of the beam and angle between the beam and cable on the amplitude spectrum of the cable stayed beam, whose parameters are shown in Table 2, are illustrated in Figure 5, in which three cases of elastic modulus, cross-sectional radius of gyration and angle take the values shown in Figure 5. Decibel scale (DBS) is introduced to show the amplitudes differences. Decibel scale is expressed as follow: $D B S = 10 \times \log_{10} (A M P_{Nor}) + P_{0}$ , where $A M P_{Nor}$ means normalized amplitude and $P_{0} = 70$ is a minimum constant that guarantees that all results are positive. An associated observation of the three subgraphs of Fig. 5 shows that the increase of the elastic modulus, cross-sectional radius of gyration of the beam and the angle between the beam and cable increases the amplitude of the high-frequency component of the structural response. Table 3 shows that the fundamental frequency of the beam-cable system, whose parameters are shown in Table 2, increases with increasing elastic modulus, cross-sectional radius of gyration of the beam and angle $α$ . This is proven by the algorithm in this paper (see equation (21)) and the FEM. It is generally believed that the greater the fundamental frequency is, the greater the dynamic stiffness of the structure. Therefore, it can be concluded that with the increase in the dynamic stiffness of the system, the proportion of the energy carried by the high-frequency wave in the total energy of the cable stayed beams subjected to a moving load increases gradually.

Figure 5.

Decibel scale of the amplitude spectrum of cable stayed beams subjected to a moving load with different parameters: (a) Beam sections with different radii of gyration; (b) Beams with different elastic moduli; and (c) Beams with different angles with cable.

Table 3.

Fundamental frequency of the beam-cable system.

	Elastic modulus E/(Pa)			Radius of gyration Rz/(m²)			Angle α/(°)
	2.06 × 10¹¹	2.56 × 10¹¹	3.56 × 10¹¹	0.0052	0.083	0.3333	45	60	90
MRRM	1.24	1.59	1.87	1.43	4.78	6.40	1.43	1.43	1.44
FEM	1.23	1.57	1.85	1.42	4.62	6.20	1.41	1.42	1.44

Note: MRRM: method of reverberation-ray matrix; FEM: finite element method.

In addition, Fig. 5 also shows that compared with the mechanical discussion, the response frequency range of the research object in this paper belongs to low-order frequency, and the high-frequency wave carries little energy. equation (19) demonstrates that the dynamic response of the beam is composed of N-order frequency components. For the calculation of the structural response under a moving load, if the contribution of higher-order frequency components is insignificant, the appropriate frequency range can be selected to obtain higher calculation efficiency with less precision loss. To make the discussion more reasonable, the following study takes the ratio of the frequency to be selected and the fundamental frequency of the beam-cable system as the variable.

A sampling frequency of 1000 Hz with a sampling number of 32,768 is adopted for transient response calculation of the cable stayed beam while performing the IFFT. The frequency range [0 Hz, $ω_{c}$ ] is selected for the IFFT, in which the upper limit of frequency $ω_{c}$ is called the cutoff frequency. Generally, the highest frequency of the IFFT is $ω_{F F T} = (1000 / 32768 - 1)$ . The fundamental frequencies of the system $ω_{n}$ obtained by the FEM and equation (21) are 1.42 Hz and 1.43 Hz. 1.43 Hz is the values used. The effects of cutoff frequency on the displacement curves of beam segment IJ at x=0.5 L are illustrated in Figure 6, in which four cases of cutoff frequency whose value is equal to $ω_{FFT}$ and the ratio with respect to $ω_{n}$ , called f, takes the values of 1, 3, and 5, are considered. It can be found that there is a large deviation in the calculation results when $ω_{c} / ω_{n} = f = 1$ . When $ω_{c} / ω_{n} = 3$ , the displacement curve is close to that without frequency interception, and there is a slight difference in the local position of the curve (marked in the figure). For the example in this paper, when $ω_{c} = ω_{FFT}$ , the calculation time is only 7% of that when $ω_{c} / ω_{n} = 3$ . When $ω_{c} / ω_{n} = 5$ , the displacement curve is basically consistent with that when $ω_{c} = ω_{FFT}$ . It can be inferred that for the example in this paper, the energy transmitted in the form of flexural waves is mainly concentrated in low-frequency waves whose frequency is less than 3 times the fundamental frequency. Therefore, for the transient response analysis, the calculation efficiency can be improved by intercepting the low-frequency component.

Figure 6.

Comparison of displacement curves: method of reverberation-ray matrix means the calculation result when $ω_{c} = ω_{FFT}$ and f=1/3/5 means the results when. $ω_{c} / ω_{n} = 1 / 3 / 5$

Discussion of the propagation characteristics of flexural waves

Wavenumber spectral characteristics of the flexural wave

Figure 7(a) shows that with increasing frequency, the traveling wavenumber increases rapidly in the low-frequency range. With the shortening of wavelength, the wavenumber growth slows down gradually due to the interaction between shear and flexural deformation until the high-frequency range [500 Hz, +∞] growth rate remains stable.

Figure 7.

Wavenumber spectrum curves of the traveling wave (a), the near-field wave (b) and the nondispersive disturbance (c) of the cable stayed beams subjected to moving load, which the real part and the imaginary part of the wavenumber spectrum curves are indicated by the red lines and the black lines, respectively.

With respect to the near-field wave, there is a critical frequency in the wavenumber spectrum curves of the beam. According to equation (8), the critical frequency is expressed as follows

ω_{c} = \sqrt{\frac{κ G A}{ρ I_{Z}}}

(25)

As demonstrated in Figure 7(b), when the frequency is lower than 6360 Hz, which is very close to the critical frequency, $ω_{c} = 6355 Hz$ calculated by equation (25), there are a pair of traveling dispersive waves (k₁ and -k₁), a pair of near-field waves (k₃ and -k₃) and a nondispersive wave (say k₅) in the cable stayed beam. It can be seen from Section 3.1 that the flexural wave energy in the cable stayed bridge is mainly concentrated in the low-frequency range, so the discussion of the near-field wave cannot be ignored. It should be pointed out that the near-field wavenumber is transformed from imaginary to real, which means that the near-field flexural wave is transformed into a traveling shear wave when the frequency is higher than the critical frequency (Zhang, 2018). Figure 7(b) shows that when the frequency is less than 3060 Hz, the imaginary part of the wavenumber spectrum increases monotonically, which means that the near-field wave attenuation becomes more obvious with increasing frequency, but the behavior is opposite when the frequency is higher than 3060 Hz.

This is particularly evident in Figure 7(c), which shows that the wavenumber of nondispersive waves increases linearly with increasing frequency and is much larger than the other eight types of elastic waves (Flexural waves through the beam). Even in the low-order frequency range, the wavelength $λ = 2 π / k_{5}$ is still larger than the cross-section geometry, so Timoshenko beam theory is applicable.

It can be observed from Figure 8(a) that the larger the radius of gyration is, the easier it is for the near-field wave to turn into a traveling wave, which is caused by the critical frequency becoming smaller and the amplitude of the high-frequency wave becoming larger. In addition, the maximum value of the imaginary part of the near-field wavenumber curve and the imaginary part of the low-frequency near-field wavenumber decrease with increasing radius of gyration, which means that the attenuation of the near-field wave is less significant according to equation. (22). It can be seen from Figure 8(b) that the larger the elastic modulus of the beam is, the larger the critical frequency, and the smaller the wavenumber in the low-frequency range. The elastic modulus has no significant effect on the maximum value of the imaginary part of the wavenumber curve. An associated observation of Figure 8(a) and 8(b) shows that the effects of the elastic modulus of the beam on the near-field wavenumber spectral characteristics are different from those of the cross-sectional radius of gyration. That is why the two parameters cannot be grouped into the same type of parameters, which are termed stiffness-related parameters. The angle between the beam and cable has no significant effect on the near-field wavenumber spectral characteristics. From the above discussion, we can see that the existence of a near-field wave at the joint is an important reason for the stress concentration near the joint in the cable stayed beam, and the previous dynamic design concept may cause insufficient strength reserve at the joint. A simple increase in stiffness-related parameters weakens the attenuation effect of near-field waves, which results in shear connectors and stiffeners of orthotropic plates being more prone to fatigue failure. For a high-speed railway bridge, the cross-sectional radius of gyration should not be increased too much at the node position to avoid unexpected shear stress in the structure.

Figure 8.

The effect of radius of gyration (a), elastic modulus (b) or angle between beam and cable (c) on wavenumber spectrum curves of the near-field wave of the cable stayed beams subjected to moving load, where the real part and the imaginary part of the wavenumber spectrum curves are indicated by the solid lines and the dotted lines, respectively.

Dispersive characteristics of flexural waves

As shown in Figure 9, the phase velocity of the traveling dispersion wave increases with increasing frequency and trends to a constant value in the high-frequency band. In addition, the increase in the cross-section radius of gyration, the elastic modulus of the material and the angle between the beam and cable, which are positively correlated with the fundamental frequency of the structure, as shown in Table 2, lead to an increase in the phase velocity of the harmonic wave. Meanwhile, the angle between the beam and cable has an insignificant effect on the phase velocity of the traveling waves and the fundamental frequency, which is illustrated in Figure 9(c) and Table 3. Combined with the basic principle of structural dynamics, which is that the higher the fundamental frequency of the structure is, the greater the stiffness of the structure, it can be inferred that the higher the stiffness of the structure is, the faster the phase velocity of the same frequency flexural wave is. Combined with the conclusion in Frequency spectral analysis that the greater the structural stiffness is, the higher the proportion of energy carried by the high-frequency harmonic traveling wave is, it can be concluded that the greater the structural dynamic stiffness is, the faster the energy transmission speed in the form of wave is, and the higher the stress wave transmission efficiency is.

Figure 9.

Phase velocity of traveling wave of cable stayed beam subjected to moving load with different parameters: (a) Beams with different cross-section radius of gyration; (b) Beams with different elastic modulus; and (c) Beams with different angle with cable.

As the value of the dispersion indicator, expressed as $d c_{p} / d ω$ , increases, the sensitivity of the phase velocity to the frequency increases, which indicates that the dispersion phenomenon of the traveling wave is more obvious. The first two subgraphs of Figure 10 show that in the low-frequency range where the flexural wave energy through the cable stayed beam subjected to moving load is mainly concentrated (the frequency range of 0–100 Hz is discussed in this paper), the larger the stiffness-related parameters are, the more obvious the dispersion effect of flexural wave propagation. For structures with the same energy input, the more significant the dispersion effect is, the smaller the stress peak value is. Figure 10(c) shows that the increase in the angle between the beam and cable has no significant effect on the dispersion indicator. Combined with the conclusion in Frequency spectral analysis that is, the increase of structural stiffness will increase the proportion of high-frequency flexural wave, it is found that the excessive increase of stiffness will weaken the optimization effect of large stiffness on structural stress. This has caused a waste of materials.

Figure 10.

The effect of the cross-sectional radius of gyration (a), elastic modulus (b) or angle between the beam and cable (c) on the dispersion indicator of the traveling dispersion wave through the beam combined with cables subjected to a moving load.

Conclusion

A detailed analysis of the general frequency spectral and propagation characteristics of flexural waves in cable stayed beams subjected to moving loads is presented to provide a better physical understanding of the dynamic mechanical behaviors of cable stayed bridges in operation and a new perspective on antifatigue design for bridge designers. The research can be concluded as follows:

(1) The comparative study for the transient response calculation of the cable stayed beams by the FEM and the MRRM verifies the validation of the formulation presented in this paper.

(2) It is found that the energy carried by the flexural wave through the cable stayed beam under a moving load is mainly concentrated in the low-frequency wave and the energy of high-frequency wave increases gradually with increasing structural stiffness. By appropriately intercepting the frequency range for the IFFT, the computational efficiency of response analysis method based on MRRM is improved significantly.

(3) As the cross-sectional radius of gyration increases, the attenuation effect of the near-field wave is weakened, which means that the propagation capacity of the flexural wave increases. The above conclusions are consistent with engineering common sense and provide a clear physical explanation for the macro law of structural dynamic response. The increase in the elastic modulus of the beam increases the critical frequency of the near-field wave, and the wavenumber of the near-field wave decreases in the low-frequency range.

(4) The larger the dynamic stiffness of the beam-cable system, the higher efficiency of stress wave propagation. The influence of the increase in the stiffness-related parameters makes the dispersion effect of traveling flexural wave propagation more obvious.

(5) The characteristics of flexural waves can be applied to achieve better dynamic performance for engineering structures. For high-speed railway bridges, it is better to use a high-elastic modulus material to strengthen the joints than to increase the section because the increase in the elastic modulus reduces the possibility of propagating shear waves. Meanwhile, the local component near the discontinuous region should be strengthened when the beam stiffness-related parameters increase, which weakens the attenuation effect of the near-field wave and leads to the transformation of the near-field wave into a shear propagation wave.

Footnotes

Acknowledgments

The authors gratefully acknowledge the support of Natural Science Foundation of Guangdong Province (No.2021A1515012405), Pearl River Science and Technology Nova Program of Guangzhou (No.201906010009, No.201806010162), Guangdong Provincial Key Laboratory of Modern Civil Engineering Technology (No. 2021B1212040003) and National Natural Science Foundation of China (No.51878295).

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Science Foundation of Guangdong Province (No.2021A1515012405), Pearl River Science and Technology Nova Program of Guangzhou (No.201906010009, No.201806010162), Guangdong Provincial Key Laboratory of Modern Civil Engineering Technology (No. 2021B1212040003) and National Natural Science Foundation of China (No.51878295).

ORCID iDs

Jianyi Ji

Xiang Zhang

Appendix 1. υ ∗ ( ω, x ), φ ∗ ( ω, x ) of segment IJ in dual local coordinate system

\begin{array}{l} {\hat{υ}}^{* I J} = - \frac{(E I_{z} k_{1}^{2} + κ AG - ρ I_{z} ω^{2})}{2 α k_{1}^{3} + β k_{1}} \int_{T_{I}}^{\frac{X^{I J}}{V_{P}}} q \sin [k_{1} (x^{I J} - V_{P} (t - T_{I}))] \exp (- i ω t) d t \\ - \frac{(E I_{z} k_{3}^{2} + κ AG - ρ I_{z} ω^{2})}{2 α k_{3}^{3} + β k_{3}} \int_{T_{I}}^{\frac{X^{I J}}{V_{P}}} q \sin [k_{3} (x^{I J} - V_{P} (t - T_{I}))] \exp (- i ω t) d t \\ {\hat{φ}}^{* I J} = - \frac{κ AG}{α (k_{1}^{2} - k_{3}^{2})} \int_{T_{I}}^{\frac{X^{I J}}{V_{P}} + T_{I}} q \cos [k_{1} (x^{I J} - V_{P} (t - T_{I}))] \exp (- i ω t) d t \\ + \frac{κ AG}{α (k_{1}^{2} - k_{3}^{2})} \int_{T_{I}}^{\frac{X^{I J}}{V_{P}} + T_{I}} q \cos [k_{3} (x^{I J} - V_{P} (t - T_{I}))] \exp (- i ω t) d t \\ \hat{υ} *^{J I} = \frac{(E I_{z} k_{1}^{2} + κ AG - ρ I_{z} ω^{2})}{2 α k_{1}^{3} + β k_{1}} \int_{T_{J} - \frac{X^{J I}}{V_{P}}}^{T_{J}} q \sin [k_{1} (x^{J I} - V_{P} (T_{J} - t))] \exp (- i ω t) d t \\ + \frac{(E I_{z} k_{3}^{2} + κ AG - ρ I_{z} ω^{2})}{2 α k_{3}^{3} + β k_{3}} \int_{T_{J} - \frac{X^{J I}}{V_{P}}}^{T_{J}} q \sin [k_{1} (x^{J I} - V_{P} (T_{J} - t))] \exp (- i ω t) d t \\ \hat{φ} *^{J I} = - \frac{κ AG}{α (k_{1}^{2} - k_{3}^{2})} \int_{T_{J} \frac{X^{J I}}{V_{P}}}^{T_{J}} q \sin [k_{1} (x^{J I} - V_{P} (T_{J} - t))] \exp (- i ω t) d t \\ - \frac{κ AG}{α (k_{1}^{2} - k_{3}^{2})} \int_{T_{J} - \frac{X^{J I}}{V_{P}}}^{T_{J}} q \sin [k_{2} (x^{J I} - V_{P} (T_{J} - t))] \exp (- i ω t) d t \end{array}

Appendix 2. The matrices related to phase relation

Phase matrices of segment IJ in dual local coordinate system

P^{IJ} = [\begin{matrix} - e^{i k_{1} L_{B}} & 0 \\ 0 & - e^{i k_{3} L_{B}} \end{matrix}]

P^{JI} = [\begin{matrix} - e^{i k_{1} L_{B}} & 0 \\ 0 & - e^{i k_{3} L_{B}} \end{matrix}]

Incentive source vectors of segment IJ in dual local coordinate system

Q^{IJ} = [\begin{array}{l} \frac{E I_{Z} k_{1}^{2} + κ A G - ρ I_{Z} ω^{2}}{2 i (2 α k_{1}^{3} + β k_{1})} \int_{T_{I}}^{T_{J}} q e^{i k_{1} V_{P} T_{I}} e^{- i (k_{1} V_{P} + ω) t} dt \\ \frac{E I_{Z} k_{3}^{2} + κ A G - ρ I_{Z} ω^{2}}{2 i (2 α k_{3}^{3} + β k_{1})} \int_{T_{I}}^{T_{J}} q e^{i k_{3} V_{P} T_{I}} e^{- i (k_{3} V_{P} + ω) t} dt \end{array}]

Q^{JI} = [\begin{array}{l} \frac{(E I_{Z} k_{1}^{2} + κ A G - ρ I_{Z} ω^{2}) e^{- i k_{1} L_{B}}}{2 i (2 α k_{1}^{3} + β k_{1})} \int_{T_{I}}^{T_{J}} q e^{- i k_{1} V_{P} T_{I}} e^{i (k_{1} V_{P} + ω) t} dt \\ - \frac{(E I_{Z} k_{3}^{2} + κ A G - ρ I_{Z} ω^{2}) e^{- i k_{3} L_{B}}}{2 i (2 α k_{3}^{3} + β k_{3})} \int_{T_{I}}^{T_{J}} q e^{- i k_{3} V_{P} T_{I}} e^{i (k_{3} V_{P} + ω) t} dt \end{array}]

Appendix 3. Permutation matrix

U = [\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \end{matrix}]

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