# Out-of-Plane Free Vibration and Forced Harmonic Response of a Curved Beam

Authors: Hancheng Mao; Guangbin Yu; Wei Liu; Tiantian Xu
Journal: Shock and Vibration
 (2020)

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## Abstract

Based on the governing differential equation of out-of-plane curved beam, the wave propagation behavior, free vibration, and transmission properties are presented theoretically in this paper. Firstly, harmonic wave solutions are given to investigate the dispersion relation between frequency and wave number, cut-off frequency, displacement, amplitude ratio, and phase diagram. The frequency spectrum results are obtained to verify the work by Kang and Lee. Furthermore, natural frequencies of the single and composite curved beam are calculated through solving the characteristic equation in the case of free-free, clamped-clamped, and free-clamped boundaries. Finally, the transfer matrices of the out-of-plane curved beam are derived by combining the continuity between the different interfaces. The transmissibility curves of the single and composite curved beam are compared to find the vibration attention band. This work will be valuable to extend the study of the out-of-plane vibration of curved beams.


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## Body

## 1. Introduction

Curved beams are widely used in many built-up structures and arch structures because of their valuable engineering applications. Compared with straight beams, plate, and shafts, the curved beams are extremely complex, and the vibration problems of these curved beam structures are still serious, especially the out-of-plane vibration.The free vibration and transmission response of curved members have been deeply investigated. Mace has made a detail research on the vibration behavior of beam systems by dividing waves into positive-going and negative-going waves using wave approach. He also deduced the reflection matrix in the case of simply supported, clamped, and free boundaries. From the wave point, the characteristic equation of the beam is obtained theoretically for the analysis of free vibration [1]. Huang investigated the free vibration of rotating thin rings using the wave approach. The harmonic wave solutions, frequency spectra, displacement amplitude ratio, and cut-off frequencies are also analyzed theoretically [2].Using the governing equation, Kang analyzed the free vibration of a planar circular curved beam system through considering multiple point discontinuities such as elastic support, attached mass, and curvature changes. The dispersion equations and cut-off frequencies are determined [3]. Based on Flugge’s theory, Lee considered the wave motion of in-plane curved beams with constant curvature. The displacement propagation matrix reflection matrix is derived numerically. The free vibration and energy flow associated with waves in the curved beam is also discussed in detail [4, 5]. Howson presented the exact out-of-plane natural frequencies of the composed curved Timoshenko beams. The natural frequencies of a continuous curved beam are calculated when the effects of shear deflection and rotary inertia are considered [6]. Liu presented the radial vibration of the circular plate using the wave propagation approach and the classical method containing Bessel solution and Hankel solution for calculating the natural frequency theoretically [7]. Based on the in-plane and out-of-plane governing equation of the curved beam, Liu analyzed the transmission response of periodic curved beams by using the transfer matrix method and Bloch theorem. Their theoretical calculations are also simulated by FEM method [8]. Lee decoupled the governing characteristic differential equations for out-of-plane vibrations of curved nonuniform beams of constant radius, and the influence of taper ratio, center angle, and arc length on the natural frequencies of the curved beams is illustrated [9]. Tufekci investigated the out-of-plane free vibration of a circular arch with uniform cross-section by taking into account the effects of transverse shear and rotatory inertia. The results showed that flexural and torsional rotatory inertia and shear deformation have essential effects on resonance frequency, even if slender shallow arches are considered [10]. Wu proposed a new approach for free vibration analysis of arches with the effects of shear deformation and rotary inertia considered. He calculated the natural frequencies for the clamped-clamped and free-free boundaries [11]. To obtain the natural frequencies and mode shapes of curved beams with hinged-hinged, hinged-clamped, and clamped-clamped boundaries, Lee derived the differential equations governing out-of-plane free vibrations of the elastic curved beams with variable curvature. Their experimental measures show that the natural frequencies are in good agreement with the theoretical results [12]. Through adopting the classical Kirchhoff thin plate theory, Zadeh summarizes the effects of dispersion relation curves on the out-of-plane wave propagation in planar elastic plates with constant curvature [13]. Walsh studied the vibration power transmission in curved beams, considering the moment of inertia and shear deformation [14]. Chidambaram summarized the extensive published literature on the vibrations of curved bars, beams, rings, and arches of arbitrary shape. He deduced the characteristic differential equations of the in-plane, out-of-plane, and coupled vibrations theoretically. His researches mainly focus on the natural frequencies for the extensional modes and inextensional modes with different boundary conditions [15]. Wang investigated the in-plane vibration of a curved beam by considering the moment of inertia and shear effect. He obtained the dispersion relation of wave number and frequency and analyzed the radial and tangential coupled band gaps of a periodic curved beam [16].The above cited literatures mainly focus on in-plane vibration analysis of curved beams, while the out-of-plane vibration analysis of curved beams is rare. Additionally, to the authors’ knowledge, lots of out-of-plane free vibration and transmission response of curved members are investigated in the case of not considering the inertia moment and shear force. To accurately show the dynamic behavior, this paper attempts to make a detailed investigation of curved beams including two factors for analyzing the dispersion relation, free vibration, and transmission response.The paper is organized into five sections.Section 1 is a brief introduction. In Section 2, the characteristic equation of the out-of-plane vibration for the curved beam is calculated theoretically by considering the moment of inertia and shear effect. In Section 3, the natural frequencies of single and composite curved beam models with clamped-clamped, free-free, and clamped-free boundaries are calculated using the classical method. In Section 4, the transmission response of the single and periodic curved beam model is obtained. Section 5 is the conclusion.


## 2. Theoretical Analysis

### 2.1. Governing Equations

Consider a small segment of an out-of-plane curved beam model, shown in Figure1, where w is the transverse displacement, ψ is the bending slope, ϕ is the rotation angle, S is the torsional moment, M is the bending moment about a radial axis, and Q is the shear force. After considering the shear force and inertia moment, the coupled governing equation of out-of-plane curved beam can be written as follows [6]:(1)1R∂Q∂θ=ρA∂2w∂t2,(2)1R∂M∂θ+SR−Q=ρI∂2ψ∂t2,(3)1R∂S∂θ−MR=ρIp∂2ϕ∂t2,where ρ denotes the density, A the cross-sectional area, I the second moment of area, Ip the polar second moment, and R the radius of the center.Figure 1
Schematic of out-of-plane vibration of a curved beam.


### 2.2. Harmonic Wave Solution

To solve equations (1)–(3), w, ψ, and ϕ are given by a harmonic form:(4)wθ,t=Wθejωt,(5)ψθ,t=Ψθejωt,(6)ϕθ,t=Φθejωt,where ω denotes the circle frequency, j=−1 is the imaginary unit, Wθ, Ψθ, and Φθ are the amplitude of the transverse displacement, bending slope, and rotation angle, respectively. Submitting equations (4)–(6) into equations (1)–(3), one can have(7)κAGR2d2Wdθ2+κAGRdΨdθ=−ω2ρAW,(8)−κAGRdWdθ+EI+GJR2dΦdθ+EIR2d2Ψdθ2−GJR2+κAGΨ=−ω2ρIΨ,(9)GJR2d2Φdθ2−EIR2Φ−EI+GJR2dΨdθ=−ω2ρIpΦ.Equations (7)–(9) can be arranged as the following matrix form:(10)κAGR2D2+ω2ρAκAGRD0−κAGRDEIR2D2−GJR2+κAG+ω2ρIEI+GJR2D0−EI+GJR2DGJR2D2−EIR2+ω2ρIpWΨΦ=0.Through making the determinant of the coefficient matrix zero, it can be simplified to(11)μβ6+2μ+ηγ2+μr2+s2γ2λ2β4+μ1−γ2η+λ2−r21−ηγ2−μs2γ2λ2γ2λ2+s2ηγ2+2μγ2λ2β2+1+μs21−ηγ2+r2s2ηγ2−1γ2λ2γ2λ2=0.Here, the specific expression of parameters gives as follows:(12a)D=ddθ,(12b)r2=IAR2,(12c)s2κR2=EI,(12d)κ=k′AG,(13a)μEI=GJ,(13b)ηI=Ip,(13c)λ2I=AR2,(13d)γ2E=ρω2R2.According to the definition of harmonic motion, one can obtain the following relationsW, Ψ, Φ, S, M, and Q:(14)Wθ=∑j=16H1iCieβiθ,(15)Ψθ=∑j=16H2iCieβiθ,(16)Φθ=∑j=16H3iCieβiθ,(17)Sθ=∑j=16H4iCieβiθ(18)Mθ=∑j=16H5iCieβiθ,(19)Qθ=∑j=16H6iCieβiθ.βii=1,2,3,4,5,6 then can be determined as the root of the characteristic equations (2)–(19), and g=β2. Thus, g satisfies the following relation:(20)μg3+2μ+ηγ2+μr2+s2γ2λ2g2+μ1−γ2η+λ2−r21−ηγ2−μs2γ2λ2γ2λ2+s2ηγ2+2μγ2λ2g+1+μs21−ηγ2+r2s2ηγ2−1γ2λ2γ2λ2=0,where g1, g2, and g3 are the roots of equation (20). Therefore, β can be rewritten as follows:(21)β1=−β4=g1,β2=−β5=g2,β3=−β6=g3.To simplify conveniently, hereH3i=1, the following relations can be obtained:(22)H1i=WθΦθ=H1iH3i=1+μβi2+s2γ2λ2R1−μβi2−ηγ2,(23)H2i=Ψθ=βi1+s2βi2+s4γ2λ2−Rs2βi1+μH1iRs2r2γ2λ2−μ−1,(24)H4i=Sθ=EIxμβiH1i−H2iR,(25)H5i=Mθ=EIxβiH2i+H1i,(26)H6i=Qθ=EIxβi1+μRH1i−βi2+μ−r2+s2γ2λ2R31+s2μ−r2γ2λ2.


#### 2.2.1. Cut-Off Frequency

Equation (11) describes the dispersion frequency spectrum relations for the wave propagation in the curved beam. The cut-off frequency, denoted by fc=ωc/2π, defines this additional frequency spectrum for the curved beam. When the limit of β tends to zero, equation (11) can be further simplified as(27)limβ⟶014=1+μs21−ηγ2+r2s2ηγ2−1γ2λ2γ2λ2=0.Through solving equation (27), the three roots for γ are given by(28)γ1=0,γ2=1η,γ3=1+μs2λrs.From equation (13d), one can have(29)ωc=γREρ.Here,γ gives the following three roots:(i)
Considering the first rootγ1=0, namely, ωc=0, from equations (29) and (30), obtain Wθ=0, Ψθ=0, and Φθ≠0. The abovementioned characteristics reveal that there is only rotation motion, but no bending motion and no torsional motion.(ii)
Considering the second rootγ2=1/η, namely, ωc=1/RE/ρη, from equations (29) and (30), obtain Wθ≠0, Ψθ≠0, and Φθ=0. It reveals that the curved beam only has bending motion and torsional motion, but no rotation motion.(iii)
Consider the third rootγ3=1+μs2/λrs, namely, ωc=1/λrsRE1+μs2/ρ. According to equations (29) and (30), it indicates Wθ≠0, Ψθ=0, and Φθ≠0, resulting in only bending motion and rotation motion, but no torsional motion.


#### 2.2.2. Frequency Spectrum

Waves can be divided into positive-going waves and negative-going wave while the wave propagates in the curved beam. These wave motion behaviors are essential for the characteristics of the curved beam. Here, the material parameter is given in Table1, and the structural parameter is selected as follows.Table 1
Material parameters.
MaterialDensity,ρ (kg/m3)Young’s modulus,E (Pa)Shear modulus,G (Pa)A(PMMA)10620.32 × 10100.12 × 1010Structural parameter: radiusR=0.075m, length b1=0.012m, width b2=0.012m, thickness h=0.012m, and shape factor κ=6/5.Through calculating the roots of characteristic equation (20), the dispersion relations of the curved beam can be obtained theoretically. Shown in Figure 2 are the real wavenumber and imaginary wavenumber motion behaviors of the frequency spectrum for the curved beam model.Figure 2
Dispersion relations of out-of-plane vibration of a curved beam. (a) Real wavenumber values. (b) Imaginary wavenumber values.
(a)(b)Figure2 indicates that these waves divide the frequency spectrum into four regions while the wave propagates in this model, namely, region I, region II, region III, and region IV. There are three bifurcation points fa, fb, and fc. It can then be observed that these bifurcation points play an important role in determining the wave motion behavior. Here, the value of cut-off frequency can be calculated as fc=γE/ρ/2πR=2605Hz, depicted in Figure 2. At this frequency, the circumference of the curved beam happens to hold a radial wavelength, and it is called the ring frequency [4].Equation (20) can be considered as a cubic equation containing a variable parameter g. Due to g=β2, equation (20) has six roots, namely, three forward propagating waves and three negative propagating waves. As is indicated in Figure 2, the frequency spectra are symmetrical for the extensional curved beam model [2]. In region I 0<f<fa, the wave modes are all pure real so that the overall wave can propagate along the curved beam without attenuation. In region II fb>f>fa, due to the complex wave, there are real and imagined coupled wave modes, which indicate the waves are positive-going and negative-going. In region III fc>f>fb, it occurs in the similar cases that the waves are both positive-going and negative-going. In region IV f>fc, only one wave mode is purely real, the corresponding propagating wave, and the other wave will be dissipated gradually.Based on Love’s theory, Kang derived the in-plane equation of the curved beam and analyzed the dispersion relations of the frequency spectrum, shown in Figure3. Similarly, based on Flugge’s theory, Lee also obtained the dispersion relations shown in Figures 4 and 5, and the present results verify the correctness of Kang’s work. These works are very significant for the analysis of curved beam models, while it is a simplified model with neglecting the effects of rotary inertia shear deformation. Actually, these factors are essential and cannot be ignored. Now, the researches on out-of-plane are rare. To make a further research, this paper not only emphasizes the dispersion relations but also further focuses on the free vibration and forced response.Figure 3
Frequency spectrum of a curved beam withk=0.0289. (a) Real wavenumber value. (b) Imaginary wavenumber values (extensional by Kang).
(a)(b)Figure 4
Frequency spectra (by Lee).Figure 5
Amplitude radio (by Lee).


#### 2.2.3. Amplitude Ratio of Displacement

The transverse displacementw, bending slope ψ, and rotation angle ϕ of out-of-plane vibration of the curved beam are not independent of each other. That is to say, they are coupled with each other. From equations (22) and (23), the amplitude ratio and phase of the transverse displacement and bending slope can be obtained theoretically for the three positive-going waves through assuming H31=H32=H33=1.Figure6 shows the amplitude ratio and phase of the transverse displacement to the rotation angle for the three positive-going waves for the curved beam. Similarly, Figure 7 shows the amplitude ratio and phase of the bending slope to the rotation angle for the three positive-going waves for the curved beam. From Figures 6 and 7, it can be seen that the four regions shown in Figure 2 are not marked apparently, but the cut-off frequency fc is marked. Additionally, it can be observed that the phases of the transverse displacement and bending slope are between −π/2 and π/2.Figure 6
Amplitude ratio and phase of the transverse displacement to the rotation angle for the curved beam (a) and amplitude ratio (b) phase.
(a)(b)Figure 7
Amplitude ratio and phase of the bending slope to the rotation angle for the curved beam (a) and amplitude ratio (b) phase.
(a)(b)


## 2.1. Governing Equations

Consider a small segment of an out-of-plane curved beam model, shown in Figure1, where w is the transverse displacement, ψ is the bending slope, ϕ is the rotation angle, S is the torsional moment, M is the bending moment about a radial axis, and Q is the shear force. After considering the shear force and inertia moment, the coupled governing equation of out-of-plane curved beam can be written as follows [6]:(1)1R∂Q∂θ=ρA∂2w∂t2,(2)1R∂M∂θ+SR−Q=ρI∂2ψ∂t2,(3)1R∂S∂θ−MR=ρIp∂2ϕ∂t2,where ρ denotes the density, A the cross-sectional area, I the second moment of area, Ip the polar second moment, and R the radius of the center.Figure 1
Schematic of out-of-plane vibration of a curved beam.


## 2.2. Harmonic Wave Solution

To solve equations (1)–(3), w, ψ, and ϕ are given by a harmonic form:(4)wθ,t=Wθejωt,(5)ψθ,t=Ψθejωt,(6)ϕθ,t=Φθejωt,where ω denotes the circle frequency, j=−1 is the imaginary unit, Wθ, Ψθ, and Φθ are the amplitude of the transverse displacement, bending slope, and rotation angle, respectively. Submitting equations (4)–(6) into equations (1)–(3), one can have(7)κAGR2d2Wdθ2+κAGRdΨdθ=−ω2ρAW,(8)−κAGRdWdθ+EI+GJR2dΦdθ+EIR2d2Ψdθ2−GJR2+κAGΨ=−ω2ρIΨ,(9)GJR2d2Φdθ2−EIR2Φ−EI+GJR2dΨdθ=−ω2ρIpΦ.Equations (7)–(9) can be arranged as the following matrix form:(10)κAGR2D2+ω2ρAκAGRD0−κAGRDEIR2D2−GJR2+κAG+ω2ρIEI+GJR2D0−EI+GJR2DGJR2D2−EIR2+ω2ρIpWΨΦ=0.Through making the determinant of the coefficient matrix zero, it can be simplified to(11)μβ6+2μ+ηγ2+μr2+s2γ2λ2β4+μ1−γ2η+λ2−r21−ηγ2−μs2γ2λ2γ2λ2+s2ηγ2+2μγ2λ2β2+1+μs21−ηγ2+r2s2ηγ2−1γ2λ2γ2λ2=0.Here, the specific expression of parameters gives as follows:(12a)D=ddθ,(12b)r2=IAR2,(12c)s2κR2=EI,(12d)κ=k′AG,(13a)μEI=GJ,(13b)ηI=Ip,(13c)λ2I=AR2,(13d)γ2E=ρω2R2.According to the definition of harmonic motion, one can obtain the following relationsW, Ψ, Φ, S, M, and Q:(14)Wθ=∑j=16H1iCieβiθ,(15)Ψθ=∑j=16H2iCieβiθ,(16)Φθ=∑j=16H3iCieβiθ,(17)Sθ=∑j=16H4iCieβiθ(18)Mθ=∑j=16H5iCieβiθ,(19)Qθ=∑j=16H6iCieβiθ.βii=1,2,3,4,5,6 then can be determined as the root of the characteristic equations (2)–(19), and g=β2. Thus, g satisfies the following relation:(20)μg3+2μ+ηγ2+μr2+s2γ2λ2g2+μ1−γ2η+λ2−r21−ηγ2−μs2γ2λ2γ2λ2+s2ηγ2+2μγ2λ2g+1+μs21−ηγ2+r2s2ηγ2−1γ2λ2γ2λ2=0,where g1, g2, and g3 are the roots of equation (20). Therefore, β can be rewritten as follows:(21)β1=−β4=g1,β2=−β5=g2,β3=−β6=g3.To simplify conveniently, hereH3i=1, the following relations can be obtained:(22)H1i=WθΦθ=H1iH3i=1+μβi2+s2γ2λ2R1−μβi2−ηγ2,(23)H2i=Ψθ=βi1+s2βi2+s4γ2λ2−Rs2βi1+μH1iRs2r2γ2λ2−μ−1,(24)H4i=Sθ=EIxμβiH1i−H2iR,(25)H5i=Mθ=EIxβiH2i+H1i,(26)H6i=Qθ=EIxβi1+μRH1i−βi2+μ−r2+s2γ2λ2R31+s2μ−r2γ2λ2.


### 2.2.1. Cut-Off Frequency

Equation (11) describes the dispersion frequency spectrum relations for the wave propagation in the curved beam. The cut-off frequency, denoted by fc=ωc/2π, defines this additional frequency spectrum for the curved beam. When the limit of β tends to zero, equation (11) can be further simplified as(27)limβ⟶014=1+μs21−ηγ2+r2s2ηγ2−1γ2λ2γ2λ2=0.Through solving equation (27), the three roots for γ are given by(28)γ1=0,γ2=1η,γ3=1+μs2λrs.From equation (13d), one can have(29)ωc=γREρ.Here,γ gives the following three roots:(i)
Considering the first rootγ1=0, namely, ωc=0, from equations (29) and (30), obtain Wθ=0, Ψθ=0, and Φθ≠0. The abovementioned characteristics reveal that there is only rotation motion, but no bending motion and no torsional motion.(ii)
Considering the second rootγ2=1/η, namely, ωc=1/RE/ρη, from equations (29) and (30), obtain Wθ≠0, Ψθ≠0, and Φθ=0. It reveals that the curved beam only has bending motion and torsional motion, but no rotation motion.(iii)
Consider the third rootγ3=1+μs2/λrs, namely, ωc=1/λrsRE1+μs2/ρ. According to equations (29) and (30), it indicates Wθ≠0, Ψθ=0, and Φθ≠0, resulting in only bending motion and rotation motion, but no torsional motion.


### 2.2.2. Frequency Spectrum

Waves can be divided into positive-going waves and negative-going wave while the wave propagates in the curved beam. These wave motion behaviors are essential for the characteristics of the curved beam. Here, the material parameter is given in Table1, and the structural parameter is selected as follows.Table 1
Material parameters.
MaterialDensity,ρ (kg/m3)Young’s modulus,E (Pa)Shear modulus,G (Pa)A(PMMA)10620.32 × 10100.12 × 1010Structural parameter: radiusR=0.075m, length b1=0.012m, width b2=0.012m, thickness h=0.012m, and shape factor κ=6/5.Through calculating the roots of characteristic equation (20), the dispersion relations of the curved beam can be obtained theoretically. Shown in Figure 2 are the real wavenumber and imaginary wavenumber motion behaviors of the frequency spectrum for the curved beam model.Figure 2
Dispersion relations of out-of-plane vibration of a curved beam. (a) Real wavenumber values. (b) Imaginary wavenumber values.
(a)(b)Figure2 indicates that these waves divide the frequency spectrum into four regions while the wave propagates in this model, namely, region I, region II, region III, and region IV. There are three bifurcation points fa, fb, and fc. It can then be observed that these bifurcation points play an important role in determining the wave motion behavior. Here, the value of cut-off frequency can be calculated as fc=γE/ρ/2πR=2605Hz, depicted in Figure 2. At this frequency, the circumference of the curved beam happens to hold a radial wavelength, and it is called the ring frequency [4].Equation (20) can be considered as a cubic equation containing a variable parameter g. Due to g=β2, equation (20) has six roots, namely, three forward propagating waves and three negative propagating waves. As is indicated in Figure 2, the frequency spectra are symmetrical for the extensional curved beam model [2]. In region I 0<f<fa, the wave modes are all pure real so that the overall wave can propagate along the curved beam without attenuation. In region II fb>f>fa, due to the complex wave, there are real and imagined coupled wave modes, which indicate the waves are positive-going and negative-going. In region III fc>f>fb, it occurs in the similar cases that the waves are both positive-going and negative-going. In region IV f>fc, only one wave mode is purely real, the corresponding propagating wave, and the other wave will be dissipated gradually.Based on Love’s theory, Kang derived the in-plane equation of the curved beam and analyzed the dispersion relations of the frequency spectrum, shown in Figure3. Similarly, based on Flugge’s theory, Lee also obtained the dispersion relations shown in Figures 4 and 5, and the present results verify the correctness of Kang’s work. These works are very significant for the analysis of curved beam models, while it is a simplified model with neglecting the effects of rotary inertia shear deformation. Actually, these factors are essential and cannot be ignored. Now, the researches on out-of-plane are rare. To make a further research, this paper not only emphasizes the dispersion relations but also further focuses on the free vibration and forced response.Figure 3
Frequency spectrum of a curved beam withk=0.0289. (a) Real wavenumber value. (b) Imaginary wavenumber values (extensional by Kang).
(a)(b)Figure 4
Frequency spectra (by Lee).Figure 5
Amplitude radio (by Lee).


### 2.2.3. Amplitude Ratio of Displacement

The transverse displacementw, bending slope ψ, and rotation angle ϕ of out-of-plane vibration of the curved beam are not independent of each other. That is to say, they are coupled with each other. From equations (22) and (23), the amplitude ratio and phase of the transverse displacement and bending slope can be obtained theoretically for the three positive-going waves through assuming H31=H32=H33=1.Figure6 shows the amplitude ratio and phase of the transverse displacement to the rotation angle for the three positive-going waves for the curved beam. Similarly, Figure 7 shows the amplitude ratio and phase of the bending slope to the rotation angle for the three positive-going waves for the curved beam. From Figures 6 and 7, it can be seen that the four regions shown in Figure 2 are not marked apparently, but the cut-off frequency fc is marked. Additionally, it can be observed that the phases of the transverse displacement and bending slope are between −π/2 and π/2.Figure 6
Amplitude ratio and phase of the transverse displacement to the rotation angle for the curved beam (a) and amplitude ratio (b) phase.
(a)(b)Figure 7
Amplitude ratio and phase of the bending slope to the rotation angle for the curved beam (a) and amplitude ratio (b) phase.
(a)(b)


## 2.2.1. Cut-Off Frequency

Equation (11) describes the dispersion frequency spectrum relations for the wave propagation in the curved beam. The cut-off frequency, denoted by fc=ωc/2π, defines this additional frequency spectrum for the curved beam. When the limit of β tends to zero, equation (11) can be further simplified as(27)limβ⟶014=1+μs21−ηγ2+r2s2ηγ2−1γ2λ2γ2λ2=0.Through solving equation (27), the three roots for γ are given by(28)γ1=0,γ2=1η,γ3=1+μs2λrs.From equation (13d), one can have(29)ωc=γREρ.Here,γ gives the following three roots:(i)
Considering the first rootγ1=0, namely, ωc=0, from equations (29) and (30), obtain Wθ=0, Ψθ=0, and Φθ≠0. The abovementioned characteristics reveal that there is only rotation motion, but no bending motion and no torsional motion.(ii)
Considering the second rootγ2=1/η, namely, ωc=1/RE/ρη, from equations (29) and (30), obtain Wθ≠0, Ψθ≠0, and Φθ=0. It reveals that the curved beam only has bending motion and torsional motion, but no rotation motion.(iii)
Consider the third rootγ3=1+μs2/λrs, namely, ωc=1/λrsRE1+μs2/ρ. According to equations (29) and (30), it indicates Wθ≠0, Ψθ=0, and Φθ≠0, resulting in only bending motion and rotation motion, but no torsional motion.


## 2.2.2. Frequency Spectrum

Waves can be divided into positive-going waves and negative-going wave while the wave propagates in the curved beam. These wave motion behaviors are essential for the characteristics of the curved beam. Here, the material parameter is given in Table1, and the structural parameter is selected as follows.Table 1
Material parameters.
MaterialDensity,ρ (kg/m3)Young’s modulus,E (Pa)Shear modulus,G (Pa)A(PMMA)10620.32 × 10100.12 × 1010Structural parameter: radiusR=0.075m, length b1=0.012m, width b2=0.012m, thickness h=0.012m, and shape factor κ=6/5.Through calculating the roots of characteristic equation (20), the dispersion relations of the curved beam can be obtained theoretically. Shown in Figure 2 are the real wavenumber and imaginary wavenumber motion behaviors of the frequency spectrum for the curved beam model.Figure 2
Dispersion relations of out-of-plane vibration of a curved beam. (a) Real wavenumber values. (b) Imaginary wavenumber values.
(a)(b)Figure2 indicates that these waves divide the frequency spectrum into four regions while the wave propagates in this model, namely, region I, region II, region III, and region IV. There are three bifurcation points fa, fb, and fc. It can then be observed that these bifurcation points play an important role in determining the wave motion behavior. Here, the value of cut-off frequency can be calculated as fc=γE/ρ/2πR=2605Hz, depicted in Figure 2. At this frequency, the circumference of the curved beam happens to hold a radial wavelength, and it is called the ring frequency [4].Equation (20) can be considered as a cubic equation containing a variable parameter g. Due to g=β2, equation (20) has six roots, namely, three forward propagating waves and three negative propagating waves. As is indicated in Figure 2, the frequency spectra are symmetrical for the extensional curved beam model [2]. In region I 0<f<fa, the wave modes are all pure real so that the overall wave can propagate along the curved beam without attenuation. In region II fb>f>fa, due to the complex wave, there are real and imagined coupled wave modes, which indicate the waves are positive-going and negative-going. In region III fc>f>fb, it occurs in the similar cases that the waves are both positive-going and negative-going. In region IV f>fc, only one wave mode is purely real, the corresponding propagating wave, and the other wave will be dissipated gradually.Based on Love’s theory, Kang derived the in-plane equation of the curved beam and analyzed the dispersion relations of the frequency spectrum, shown in Figure3. Similarly, based on Flugge’s theory, Lee also obtained the dispersion relations shown in Figures 4 and 5, and the present results verify the correctness of Kang’s work. These works are very significant for the analysis of curved beam models, while it is a simplified model with neglecting the effects of rotary inertia shear deformation. Actually, these factors are essential and cannot be ignored. Now, the researches on out-of-plane are rare. To make a further research, this paper not only emphasizes the dispersion relations but also further focuses on the free vibration and forced response.Figure 3
Frequency spectrum of a curved beam withk=0.0289. (a) Real wavenumber value. (b) Imaginary wavenumber values (extensional by Kang).
(a)(b)Figure 4
Frequency spectra (by Lee).Figure 5
Amplitude radio (by Lee).


## 2.2.3. Amplitude Ratio of Displacement

The transverse displacementw, bending slope ψ, and rotation angle ϕ of out-of-plane vibration of the curved beam are not independent of each other. That is to say, they are coupled with each other. From equations (22) and (23), the amplitude ratio and phase of the transverse displacement and bending slope can be obtained theoretically for the three positive-going waves through assuming H31=H32=H33=1.Figure6 shows the amplitude ratio and phase of the transverse displacement to the rotation angle for the three positive-going waves for the curved beam. Similarly, Figure 7 shows the amplitude ratio and phase of the bending slope to the rotation angle for the three positive-going waves for the curved beam. From Figures 6 and 7, it can be seen that the four regions shown in Figure 2 are not marked apparently, but the cut-off frequency fc is marked. Additionally, it can be observed that the phases of the transverse displacement and bending slope are between −π/2 and π/2.Figure 6
Amplitude ratio and phase of the transverse displacement to the rotation angle for the curved beam (a) and amplitude ratio (b) phase.
(a)(b)Figure 7
Amplitude ratio and phase of the bending slope to the rotation angle for the curved beam (a) and amplitude ratio (b) phase.
(a)(b)


## 3. Free Vibration of the Curved Beam

In this section, the free vibrations of out-of-plane vibrations for the single curved beam are given theoretically by adopting the classical method. According to the classical boundary condition, the characteristic equation of the curved beam can be obtained.


### 3.1. For a Single Model

The natural frequencies of out-of-plane vibration for the single curved beam shown in Figure8 are given theoretically by adopting the classical method. According to the classical boundary condition, the characteristic equation of the curved beam can be obtained.Figure 8
A single curved beam model.For a curved beam with clamped-clamped boundaries, one can obtain(30)W0=∑j=16H1iCieβi∗0=0,Ψ0=∑j=16H2iCieβi∗0=0,Φ0=∑j=16H3iCieβi∗0=0,Wπ=∑j=16H1iCieβi∗π=0,Ψπ=∑j=16H2iCieβi∗π=0,Φπ=∑j=16H3iCieβi∗π=0.For the free-free boundaries, one has the following relations:(31)S0=∑j=16H4iCieβi∗0=0,M0=∑j=16H5iCieβi∗0=0,Q0=∑j=16H6iCieβi∗0=0,Sπ=∑j=16H4iCieβi∗π=0,Mπ=∑j=16H5iCieβi∗π=0,Qπ=∑j=16H6iCieβi∗π=0.For the clamped-free boundaries, it gives(32)W0=∑j=16H1iCieβi∗0=0,Ψ0=∑j=16H2iCieβi∗0=0,Φ0=∑j=16H3iCieβi∗0=0,Sπ=∑j=16H4iCieβi∗π=0,Mπ=∑j=16H5iCieβi∗π=0,Qπ=∑j=16H6iCieβi∗π=0.In this case, equation (26) can be further reduced to(33)Cω=∑j=16H1ieβi∗0∑j=16H2ieβi∗0∑j=16H3ieβi∗0∑j=16H4ieβi∗π∑j=16H5ieβi∗π∑j=16H6ieβi∗πC1C2C3C4C5C6=0.Equation (33) is the characteristic equation of out-of-plane vibration for the curved beam with clamped-free boundaries. Similarly, the characteristic equation with clamped-clamped and free-free boundaries also can be obtained theoretically. Figure 9 depicts the characteristic curve of natural frequency for the curved beam in the case of clamped-clamped, free-free, and clamped-free boundaries. By solving the characteristic equation, the natural frequency of the curved beam can be obtained. It can be found that the intersection point of the characteristic curve in X-axis is the natural frequency. Table 2 shows the natural frequencies calculated at clamped-clamped and free-free boundaries. Figure 10 describes the transmission response of out-of-plane vibration for the curved beam model. It can be found that the values of natural frequency in Figure 10 are equal to the values of clamped-free case in Table 2, which illustrates the correctness of the numerical results.Figure 9
Characteristic curves of natural frequency of out-of-plane vibration for the curved beam, (a) clamped-clamped boundary, (b) free-free boundary, and (c) clamped-free boundary.
(a)(b)(c)Table 2
Natural frequencies (Hz) with clamped-clamped and free-free boundaries for the single model.
Boundary1st mode2nd mode3rd modeClamped-clamped47305859Free-free47645929Clamped-free64218726Figure 10
Transmission response of out-of-plane vibration for the single curved beam.


### 3.2. For a Composite Model

Similarly, the natural frequencies of out-of-plane vibration for the composite curved beam are also calculated numerically by adopting the classical method. According to the classical boundary condition, the characteristic equation of the curved beam can be obtained.The solving process of clamped-clamped and free-free boundaries for the composite curved beam model shown in Figure11 are similar to the clamped-free case. Here, the clamped-free case is presented to calculate the natural frequencies. For the clamped-free boundaries, one has(34)W0=∑j=16H4iCieβi∗0=0,Ψ0=∑j=16H5iCieβi∗0=0,Φ0=∑j=16H6iCieβi∗0=0,Sπ=∑j=16H4iCieβi∗πT1=0,Mπ=∑j=16H5iCieβi∗πT1=0,Qπ=∑j=16H6iCieβi∗πT1=0.Figure 11
A composite curved beam model.In this case, by combining equations (32) and (33), the following set of relations can be established:(35)Cω=∑j=16H1ieβi∗0∑j=16H2ieβi∗0∑j=16H3ieβi∗0∑j=16H4ieβi∗πT1∑j=16H5ieβi∗πT1∑j=16H6ieβi∗πT1C1C2C3C4C5C6=0.Through solving the root in equation (35), the natural frequencies of out-of-plane vibration for the composite curved beam can be obtained. Figure 12 gives the characteristic curves of natural frequency of out-of-plane vibration for the composite curved beam in the case of three classical boundaries. It can be found that the intersection point of the characteristic curve in X-axis is the natural frequency. The values of natural frequency obtained for these three cases are given in Table 3. Figure 13 gives the transmission response of out-of-plane vibration for the composite curved beam, and the natural frequencies described in Figure 13 are the same in Table 3. That is to say, the resonance points in the transmission curves obtained by impulse response are the natural frequencies.Figure 12
Characteristic curves of natural frequency of out-of-plane vibration for the curved beam, (a) clamped-clamped boundary, (b) free-free boundary, and (c) clamped-free boundary.
(a)(b)(c)Table 3
Natural frequencies (Hz) with clamped-clamped and free-free boundaries for the composite model.
Boundary1st mode2nd mode3rd mode4th modeClamped-clamped4771278545Free-free5698248568Clamped-free34107262586Figure 13
Transmission response of out-of-plane vibration for the curved beam.


## 3.1. For a Single Model

The natural frequencies of out-of-plane vibration for the single curved beam shown in Figure8 are given theoretically by adopting the classical method. According to the classical boundary condition, the characteristic equation of the curved beam can be obtained.Figure 8
A single curved beam model.For a curved beam with clamped-clamped boundaries, one can obtain(30)W0=∑j=16H1iCieβi∗0=0,Ψ0=∑j=16H2iCieβi∗0=0,Φ0=∑j=16H3iCieβi∗0=0,Wπ=∑j=16H1iCieβi∗π=0,Ψπ=∑j=16H2iCieβi∗π=0,Φπ=∑j=16H3iCieβi∗π=0.For the free-free boundaries, one has the following relations:(31)S0=∑j=16H4iCieβi∗0=0,M0=∑j=16H5iCieβi∗0=0,Q0=∑j=16H6iCieβi∗0=0,Sπ=∑j=16H4iCieβi∗π=0,Mπ=∑j=16H5iCieβi∗π=0,Qπ=∑j=16H6iCieβi∗π=0.For the clamped-free boundaries, it gives(32)W0=∑j=16H1iCieβi∗0=0,Ψ0=∑j=16H2iCieβi∗0=0,Φ0=∑j=16H3iCieβi∗0=0,Sπ=∑j=16H4iCieβi∗π=0,Mπ=∑j=16H5iCieβi∗π=0,Qπ=∑j=16H6iCieβi∗π=0.In this case, equation (26) can be further reduced to(33)Cω=∑j=16H1ieβi∗0∑j=16H2ieβi∗0∑j=16H3ieβi∗0∑j=16H4ieβi∗π∑j=16H5ieβi∗π∑j=16H6ieβi∗πC1C2C3C4C5C6=0.Equation (33) is the characteristic equation of out-of-plane vibration for the curved beam with clamped-free boundaries. Similarly, the characteristic equation with clamped-clamped and free-free boundaries also can be obtained theoretically. Figure 9 depicts the characteristic curve of natural frequency for the curved beam in the case of clamped-clamped, free-free, and clamped-free boundaries. By solving the characteristic equation, the natural frequency of the curved beam can be obtained. It can be found that the intersection point of the characteristic curve in X-axis is the natural frequency. Table 2 shows the natural frequencies calculated at clamped-clamped and free-free boundaries. Figure 10 describes the transmission response of out-of-plane vibration for the curved beam model. It can be found that the values of natural frequency in Figure 10 are equal to the values of clamped-free case in Table 2, which illustrates the correctness of the numerical results.Figure 9
Characteristic curves of natural frequency of out-of-plane vibration for the curved beam, (a) clamped-clamped boundary, (b) free-free boundary, and (c) clamped-free boundary.
(a)(b)(c)Table 2
Natural frequencies (Hz) with clamped-clamped and free-free boundaries for the single model.
Boundary1st mode2nd mode3rd modeClamped-clamped47305859Free-free47645929Clamped-free64218726Figure 10
Transmission response of out-of-plane vibration for the single curved beam.


## 3.2. For a Composite Model

Similarly, the natural frequencies of out-of-plane vibration for the composite curved beam are also calculated numerically by adopting the classical method. According to the classical boundary condition, the characteristic equation of the curved beam can be obtained.The solving process of clamped-clamped and free-free boundaries for the composite curved beam model shown in Figure11 are similar to the clamped-free case. Here, the clamped-free case is presented to calculate the natural frequencies. For the clamped-free boundaries, one has(34)W0=∑j=16H4iCieβi∗0=0,Ψ0=∑j=16H5iCieβi∗0=0,Φ0=∑j=16H6iCieβi∗0=0,Sπ=∑j=16H4iCieβi∗πT1=0,Mπ=∑j=16H5iCieβi∗πT1=0,Qπ=∑j=16H6iCieβi∗πT1=0.Figure 11
A composite curved beam model.In this case, by combining equations (32) and (33), the following set of relations can be established:(35)Cω=∑j=16H1ieβi∗0∑j=16H2ieβi∗0∑j=16H3ieβi∗0∑j=16H4ieβi∗πT1∑j=16H5ieβi∗πT1∑j=16H6ieβi∗πT1C1C2C3C4C5C6=0.Through solving the root in equation (35), the natural frequencies of out-of-plane vibration for the composite curved beam can be obtained. Figure 12 gives the characteristic curves of natural frequency of out-of-plane vibration for the composite curved beam in the case of three classical boundaries. It can be found that the intersection point of the characteristic curve in X-axis is the natural frequency. The values of natural frequency obtained for these three cases are given in Table 3. Figure 13 gives the transmission response of out-of-plane vibration for the composite curved beam, and the natural frequencies described in Figure 13 are the same in Table 3. That is to say, the resonance points in the transmission curves obtained by impulse response are the natural frequencies.Figure 12
Characteristic curves of natural frequency of out-of-plane vibration for the curved beam, (a) clamped-clamped boundary, (b) free-free boundary, and (c) clamped-free boundary.
(a)(b)(c)Table 3
Natural frequencies (Hz) with clamped-clamped and free-free boundaries for the composite model.
Boundary1st mode2nd mode3rd mode4th modeClamped-clamped4771278545Free-free5698248568Clamped-free34107262586Figure 13
Transmission response of out-of-plane vibration for the curved beam.


## 4. Transmission Response for Periodic Curved Beam

### 4.1. Base Model

In this section, to further study the transmission response, the periodic curved beam models for the out-of-plane vibration are taken as a numerical example for the analysis of the transmissibility through adopting the transfer matrix method. Herein, Figure14 is the base model which is composited of eight periods, which consist of material I resin and material II steel. The angles of resin and steel are θ1 and θ2, respectively. Table 4 gives the material parameters. Shear modulus, density, and Poisson’s ratio of resin are E1, ρ1, and δ1, respectively. Shear modulus, density, and Poisson’s ratio of steel are E2, ρ2, and δ2.Figure 14
Eight period composite curved beam model.Table 4
Material parameters.
MaterialDensity,ρ (kg/m3)Young’s modulus,E (Pa)Poisson’s ratioI (PMMA)10620.32 × 10100.333II (Al)27997.21 × 10100.345


### 4.2. Transfer Matrix

To obtain the transmission response of the curved beam model, the transfer matrix for the out-of-plane vibration is derived theoretically. The connection ways between the two different materials are selected with glue. Material and structural parameters have been given in Table1. According to equations (14)–(19), the expression of six parameters W, Ψ, Φ, S, M, and Q in terms of the harmonic motion can be rewritten as follows:(36)Wθ=H11C1a+eg1θ+H12C1b+eg2θ+H13C1c+eg3θ+H14C1a−eg4θ+H15C1b−eg5θ+H16C1c−eg6θ,Ψθ=H21C1a+eg1θ+H22C1b+eg2θ+H23C1c+eg3θ+H24C1a−eg4θ+H25C1b−eg5θ+H26C1c−eg6θ,Φθ=H31C1a+eg1θ+H32C1b+eg2θ+H33C1c+eg3θ+H34C1a−eg4θ+H35C1b−eg5θ+H36C1c−eg6θ,Sθ=H41C1a+eg1θ+H42C1b+eg2θ+H43C1c+eg3θ+H44C1a−eg4θ+H45C1b−eg5θ+H46C1c−eg6θ,Mθ=H51C1a+eg1θ+H52C1b+eg2θ+H53C1c+eg3θ+H54C1a−eg4θ+H55C1b−eg5θ+H56C1c−eg6θ,Qθ=H61C1a+eg1θ+H62C1b+eg2θ+H63C1c+eg3θ+H64C1a−eg4θ+H65C1b−eg5θ+H66C1c−eg6θ.The continuity condition on the rotation angleθ=π/8 between material I and material II is given by(37)W1π6=W2π6,Ψ1π6=Ψ2π6,Φ1π6=Φ2π6,S1π6=S2π6,M1π6=M2π6,Q1π6=Q2π6.The following matrix equation can be established:(38)K1Ψn2=H1Ψn1,where Ψn1=Pn1a+Pn1b+Pn1c+Pn1a−Pn1b−Pn1c−T and Ψn2=Pn2a+Pn2b+Pn2c+Pn2a−Pn2b−Pn2c−T.Similarly, imposing the continuity condition atθ1=0 and θ2=π/4, one has(39)W10=W2π3,Ψ10=Ψ2π3,Φ10=Φ2π3,S10=S2π3,M10=M2π3,Q10=Q2π3.Then, these parameters at the continuity position lead to the following matrix relations:(40)K2ψn2=H2ψn+11,where Ψn+11=Pn+11a+Pn+11b+Pn+11c+Pn+11a−Pn+11b−Pn+11c−T.Equations (38) and (40) can be combined to give(41)Ψn+11=H2−1K2K1−1H1Ψn1=TaΨn1,where Ta=H2−1K2K1−1H1 is the transfer matrix [13] of the periodic curved beam model.To obtain the transmission response of this curved beam model, then suppose that the left end is used to load displacement and the right end is used to pick the displacement. Thus, the transverse displacement at the left end is considered as one unit, and the other parameters seem to be zero. Based on these definitions, the following relations can be established:(42)W0=H11C1a+eg1∗0+H12C1b+eg2∗0+H13C1c+eg3∗0+H14C1a−eg4∗0+H15C1b−eg5∗0+H16C1c−eg6∗0=0,(43)Ψ0=H21C1a+eg1∗0+H22C1b+eg2∗0+H23C1c+eg3∗0+H24C1a−eg4∗0+H25C1b−eg5∗0+H26C1c−eg6∗0=0,(44)Φ0=H31C1a+eg1∗0+H32C1b+eg2∗0+H33C1c+eg3∗0+H34C1a−eg4∗0+H35C1b−eg5∗0+H36C1c−eg6∗0=0,(45)Sπ4=H41C1a+eg1∗π4+H42C1b+eg2∗π4+H43C1c+eg3∗π4+H44C1a−eg4∗π4+H45C1b−eg5∗π4+H46C1c−eg6∗π4=0,(46)Mπ4=H51C1a+eg1∗π4+H52C1b+eg2∗π4+H53C1c+eg3∗π4+H54C1a−eg4∗π4+H55C1b−eg5∗π4+H56C1c−eg6∗π4=0,(47)Qπ4=H61C1a+eg1∗π4+H62C1b+eg2∗π4+H63C1c+eg3∗π4+H64C1a−eg4∗π4+H65C1b−eg5∗π4+H66C1c−eg6∗π4=0.The above equations can be rewritten into a matrix form:(48)C1a+C1b+C1c+C1a−C1b−C1c−=H11eg1∗0H12eg2∗0H13eg3∗0H14eg40H15eg50H16eg60H21eg1∗0H22eg2∗0H23eg3∗0H24eg40H25eg50H26eg60H31eg1∗0H32eg2∗0H33eg3∗0H34eg40H35eg50H36eg60H41eg1∗π4H42eg2∗π4H43eg3∗π4H44eg4∗π4H45eg5∗π4H46eg6∗π4H51eg1∗π4H52eg2∗π4H53eg3∗π4H54eg4∗π4H55eg5∗π4H56eg6∗π4H61eg1∗π4H62eg2∗π4H63eg3∗π4H64eg4∗π4H65eg5∗π4H66eg6∗π4−1100000.Combining equations (46)–(48), the transverse displacements are given by(49)WI=H11C1a+eg10H12C1b+eg20H13C1c+eg30H14C1a−eg40H15C1b−eg50H16C1c−eg60TaC1a+C1b+C1c+C1a−C1b−C1c−−1.Therefore, the vibration transmission response can be defined as(50)dB=20logWI.


### 4.3. Transmission Response

Considering the effects of inertia and shear deformation, the transmission response of the out-of-plane vibration for the curved beam model is calculated numerically by using the transfer matrix and boundary loads. The transmission response curves are plotted in Figure14. To facilitate this investigation, the transmission responses for the case of single model and eight-periodic models are compared.Regarding the single model, Figure15 clearly shows that the vibration attenuation region of the transmission response is approximately 20 dB. Actually, there is no vibration attenuation when the elastic wave propagates in a single model, such as rods, beams, and shaft. Thus, this case reveals that the vibration attenuation is significantly caused by the curvature radius of the curved beam model, and the resonance frequency is given in Table 2.Figure 15
Transmission response for the curved beam with single model and eight-periodic models.Regarding the eight-periodic model, Figure15 clearly indicates that there are three vibration attention regions, marked as band A 113 Hz–272 Hz, band B 281 Hz–553 Hz, and band C 559 Hz–800 Hz. These low frequency bands are very significant for the vibration reduction of the engineering field, such as bridges. The resonance point in the transmission curf obtained by impulse response is the natural frequency. Thus, in many practical engineering fields, LMS Test.Lab or Pulse can be used to obtain the transmission response curves easily by adopting the hammering method for finding the natural frequency instead of testing model.By comparing the single model and eight-periodic models, it can be observed that they have the similar vibration attention amplitude. This phenomenon indicates that the periodic model is not very efficient for the vibration attention, which is one point different from the previous conclude.As is known for us, the curvature radius is a key factor for the curved beam and has an important influence on the transmission response. Furthermore, Figure16 clearly gives the transmission response of the curved beam with different curvature radius. It can be seen that the region of the vibration reduction moves towards the low frequency with the curvature radius increasing, but the amplitude of vibration attention almost has no change. Thus, we can conclude that one can select a big curvature radius of the curved beam while one wants to have a low frequency band. Moreover, it should be mentioned that the attention region marked as band D is 57 Hz–140 Hz, which is very useful for avoiding resonance.Figure 16
Transmission response for the curved beam with different curvature radius.


## 4.1. Base Model

In this section, to further study the transmission response, the periodic curved beam models for the out-of-plane vibration are taken as a numerical example for the analysis of the transmissibility through adopting the transfer matrix method. Herein, Figure14 is the base model which is composited of eight periods, which consist of material I resin and material II steel. The angles of resin and steel are θ1 and θ2, respectively. Table 4 gives the material parameters. Shear modulus, density, and Poisson’s ratio of resin are E1, ρ1, and δ1, respectively. Shear modulus, density, and Poisson’s ratio of steel are E2, ρ2, and δ2.Figure 14
Eight period composite curved beam model.Table 4
Material parameters.
MaterialDensity,ρ (kg/m3)Young’s modulus,E (Pa)Poisson’s ratioI (PMMA)10620.32 × 10100.333II (Al)27997.21 × 10100.345


## 4.2. Transfer Matrix

To obtain the transmission response of the curved beam model, the transfer matrix for the out-of-plane vibration is derived theoretically. The connection ways between the two different materials are selected with glue. Material and structural parameters have been given in Table1. According to equations (14)–(19), the expression of six parameters W, Ψ, Φ, S, M, and Q in terms of the harmonic motion can be rewritten as follows:(36)Wθ=H11C1a+eg1θ+H12C1b+eg2θ+H13C1c+eg3θ+H14C1a−eg4θ+H15C1b−eg5θ+H16C1c−eg6θ,Ψθ=H21C1a+eg1θ+H22C1b+eg2θ+H23C1c+eg3θ+H24C1a−eg4θ+H25C1b−eg5θ+H26C1c−eg6θ,Φθ=H31C1a+eg1θ+H32C1b+eg2θ+H33C1c+eg3θ+H34C1a−eg4θ+H35C1b−eg5θ+H36C1c−eg6θ,Sθ=H41C1a+eg1θ+H42C1b+eg2θ+H43C1c+eg3θ+H44C1a−eg4θ+H45C1b−eg5θ+H46C1c−eg6θ,Mθ=H51C1a+eg1θ+H52C1b+eg2θ+H53C1c+eg3θ+H54C1a−eg4θ+H55C1b−eg5θ+H56C1c−eg6θ,Qθ=H61C1a+eg1θ+H62C1b+eg2θ+H63C1c+eg3θ+H64C1a−eg4θ+H65C1b−eg5θ+H66C1c−eg6θ.The continuity condition on the rotation angleθ=π/8 between material I and material II is given by(37)W1π6=W2π6,Ψ1π6=Ψ2π6,Φ1π6=Φ2π6,S1π6=S2π6,M1π6=M2π6,Q1π6=Q2π6.The following matrix equation can be established:(38)K1Ψn2=H1Ψn1,where Ψn1=Pn1a+Pn1b+Pn1c+Pn1a−Pn1b−Pn1c−T and Ψn2=Pn2a+Pn2b+Pn2c+Pn2a−Pn2b−Pn2c−T.Similarly, imposing the continuity condition atθ1=0 and θ2=π/4, one has(39)W10=W2π3,Ψ10=Ψ2π3,Φ10=Φ2π3,S10=S2π3,M10=M2π3,Q10=Q2π3.Then, these parameters at the continuity position lead to the following matrix relations:(40)K2ψn2=H2ψn+11,where Ψn+11=Pn+11a+Pn+11b+Pn+11c+Pn+11a−Pn+11b−Pn+11c−T.Equations (38) and (40) can be combined to give(41)Ψn+11=H2−1K2K1−1H1Ψn1=TaΨn1,where Ta=H2−1K2K1−1H1 is the transfer matrix [13] of the periodic curved beam model.To obtain the transmission response of this curved beam model, then suppose that the left end is used to load displacement and the right end is used to pick the displacement. Thus, the transverse displacement at the left end is considered as one unit, and the other parameters seem to be zero. Based on these definitions, the following relations can be established:(42)W0=H11C1a+eg1∗0+H12C1b+eg2∗0+H13C1c+eg3∗0+H14C1a−eg4∗0+H15C1b−eg5∗0+H16C1c−eg6∗0=0,(43)Ψ0=H21C1a+eg1∗0+H22C1b+eg2∗0+H23C1c+eg3∗0+H24C1a−eg4∗0+H25C1b−eg5∗0+H26C1c−eg6∗0=0,(44)Φ0=H31C1a+eg1∗0+H32C1b+eg2∗0+H33C1c+eg3∗0+H34C1a−eg4∗0+H35C1b−eg5∗0+H36C1c−eg6∗0=0,(45)Sπ4=H41C1a+eg1∗π4+H42C1b+eg2∗π4+H43C1c+eg3∗π4+H44C1a−eg4∗π4+H45C1b−eg5∗π4+H46C1c−eg6∗π4=0,(46)Mπ4=H51C1a+eg1∗π4+H52C1b+eg2∗π4+H53C1c+eg3∗π4+H54C1a−eg4∗π4+H55C1b−eg5∗π4+H56C1c−eg6∗π4=0,(47)Qπ4=H61C1a+eg1∗π4+H62C1b+eg2∗π4+H63C1c+eg3∗π4+H64C1a−eg4∗π4+H65C1b−eg5∗π4+H66C1c−eg6∗π4=0.The above equations can be rewritten into a matrix form:(48)C1a+C1b+C1c+C1a−C1b−C1c−=H11eg1∗0H12eg2∗0H13eg3∗0H14eg40H15eg50H16eg60H21eg1∗0H22eg2∗0H23eg3∗0H24eg40H25eg50H26eg60H31eg1∗0H32eg2∗0H33eg3∗0H34eg40H35eg50H36eg60H41eg1∗π4H42eg2∗π4H43eg3∗π4H44eg4∗π4H45eg5∗π4H46eg6∗π4H51eg1∗π4H52eg2∗π4H53eg3∗π4H54eg4∗π4H55eg5∗π4H56eg6∗π4H61eg1∗π4H62eg2∗π4H63eg3∗π4H64eg4∗π4H65eg5∗π4H66eg6∗π4−1100000.Combining equations (46)–(48), the transverse displacements are given by(49)WI=H11C1a+eg10H12C1b+eg20H13C1c+eg30H14C1a−eg40H15C1b−eg50H16C1c−eg60TaC1a+C1b+C1c+C1a−C1b−C1c−−1.Therefore, the vibration transmission response can be defined as(50)dB=20logWI.


## 4.3. Transmission Response

Considering the effects of inertia and shear deformation, the transmission response of the out-of-plane vibration for the curved beam model is calculated numerically by using the transfer matrix and boundary loads. The transmission response curves are plotted in Figure14. To facilitate this investigation, the transmission responses for the case of single model and eight-periodic models are compared.Regarding the single model, Figure15 clearly shows that the vibration attenuation region of the transmission response is approximately 20 dB. Actually, there is no vibration attenuation when the elastic wave propagates in a single model, such as rods, beams, and shaft. Thus, this case reveals that the vibration attenuation is significantly caused by the curvature radius of the curved beam model, and the resonance frequency is given in Table 2.Figure 15
Transmission response for the curved beam with single model and eight-periodic models.Regarding the eight-periodic model, Figure15 clearly indicates that there are three vibration attention regions, marked as band A 113 Hz–272 Hz, band B 281 Hz–553 Hz, and band C 559 Hz–800 Hz. These low frequency bands are very significant for the vibration reduction of the engineering field, such as bridges. The resonance point in the transmission curf obtained by impulse response is the natural frequency. Thus, in many practical engineering fields, LMS Test.Lab or Pulse can be used to obtain the transmission response curves easily by adopting the hammering method for finding the natural frequency instead of testing model.By comparing the single model and eight-periodic models, it can be observed that they have the similar vibration attention amplitude. This phenomenon indicates that the periodic model is not very efficient for the vibration attention, which is one point different from the previous conclude.As is known for us, the curvature radius is a key factor for the curved beam and has an important influence on the transmission response. Furthermore, Figure16 clearly gives the transmission response of the curved beam with different curvature radius. It can be seen that the region of the vibration reduction moves towards the low frequency with the curvature radius increasing, but the amplitude of vibration attention almost has no change. Thus, we can conclude that one can select a big curvature radius of the curved beam while one wants to have a low frequency band. Moreover, it should be mentioned that the attention region marked as band D is 57 Hz–140 Hz, which is very useful for avoiding resonance.Figure 16
Transmission response for the curved beam with different curvature radius.


## 5. Conclusion

This paper concerns out-of-plane vibration of curved beams, considering the shear force and inertia moment. Based on the coupled governing equation of the curved beam, the dispersion relations, cut-off frequency, and displacement amplitude ratio are investigated in detail. Furthermore, the free vibration and transmission properties for the single and composite curved beam models are investigated, and the following conclusions are drawn:(1)
From the dispersion relation of the out-of-plane curved beam, it can be found that the elastic wave can be divided into three forward propagating waves and three negative propagating waves, and these waves can be divided into four regions.(2)
Natural frequencies of the single and composite curved beam model with three boundaries are calculated through using the harmonic wave solutions. It can be found that the resonance frequency can be obtained from the impulse response.(3)
Curvature radius is a key factor for the curved beam and has an important influence on the transmission response. It can be seen that the region of the vibration reduction moves towards the low frequency with the curvature radius increasing, but the amplitude of vibration attention almost has no change.Our research manifests that the natural frequencies of composite TCP structures can be obtained easily using the wave approach, which can be very meaningful for many engineering fields. For example, rotating parts are often coupled with transverse vibrations (such as the gear driving systems). Apparently, analyzing the transmission properties and parameter’s effect is very significant for transverse vibration suppression.



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*Source: 8891585-2020-12-30.xml*