Theoretical analysis of the energy conversion and vibration control characteristics of a slanted beam termination

Abstract

The fundamental beam structure is often regarded as a wave or energy carrier in a wide range of research topics for structural engineering. Nevertheless, in the related literature the beam is positioned either horizontally or vertically, which may limit its application flexibility. Very few studies have investigated the energy conversion and vibration control characteristics induced by the flexural and axial waves coupling at a slanted angle discontinuity. This research aims to investigate the dynamic characteristics of a slanted beam termination (SBT), a finite beam with one end attached to a host structure and the other end free in a slant configuration. A generic wave-based formulation is developed to obtain both the waveguides distribution and the point impedance of the SBT taking the flexural and axial waves coupling into account. The semi-infinite beam (SIB) with the proposed SBT case is compared with the classic two SIBs case in terms of the energy conversion phenomena influenced by the connection angle and frequency. In the SIB with the SBT case, a certain connection angle will enable the SBT to achieve a substantial energy conversion at its resonance. From the vibration control perspective, a benchmark cantilevered beam is adopted to examine the SBT’s vibration control performance theoretically and is verified experimentally. This research lays the foundation for the design of the beam-like device for energy conversion and vibration suppression by the variation of connection angle rather than the conventional tuning method based on the stiffness, mass and damping.

Appendices

Appendix A: Formulation of Reflection matrix and Transmission matrix

In Fig. 1a, suppose the incident waveguide ${\mathbf{q}}_{\mathbf{A}}^{+}$ is towards the connection point, a reflected waveguide ${\mathbf{q}}_{\mathbf{A}}^{-}$ in the horizontal beam and a transmitted waveguide ${\mathbf{q}}_{\mathbf{B}}^{+}$ are generated in the slanted beam respectively. Note that the two beams are semi-infinite, so ${\mathbf{q}}_{\mathbf{B}}^{-}$ does not exist in this case. The components in the waveguide vector comprise the wave magnitudes of both flexural and axial waves. In the horizontal section, the incident flexural wave ${W}_{in}$ and axial wave ${U}_{in}$ as well as the reflected flexural wave ${W}_{rf}$ and axial wave ${U}_{rf}$ coexist and they can be represented in the wave formulation.

$${W}_{in}={A}^{+}{e}^{-i{k}_{b1}{x}_{1}}+{A}_{N}^{+}{e}^{-{k}_{b1}{x}_{1}}$$

(15)

$${W}_{rf}={A}^{-}{e}^{i{k}_{b1}{x}_{1}}+{A}_{N}^{-}{e}^{{k}_{b1}{x}_{1}}$$

(16)

$${U}_{in}={B}^{+}{e}^{-i{k}_{a1}{x}_{1}}$$

(17)

$${U}_{rf}={B}^{-}{e}^{i{k}_{a1}{x}_{1}}$$

(18)

where ${k}_{b1}\,\text{and}\,{k}_{a1}$ are the wavenumbers for the flexural and axial waves in the horizontal beam, respectively. The incident waveguide and the reflected waveguide are therefore given as:

$${\mathbf{q}}_{\mathbf{A}}^{+}={\left[{A}^{+}{A}_{N}^{+}{B}^{+}\right]}^{T};$$

(19)

$${\mathbf{q}}_{\mathbf{A}}^{-}={\left[{A}^{-}{A}_{N}^{-}{B}^{-}\right]}^{T}.$$

(20)

In the slanted beam part, the transmitted flexural and axial waves ${W}_{tm}$ and ${U}_{tm}$ are given by:

$${W}_{tm}={C}^{+}{e}^{-i{k}_{b2}{x}_{1}}+{C}_{N}^{+}{e}^{-{k}_{b2}{x}_{1}};$$

(21)

$${U}_{tm}={D}^{+}{e}^{-i{k}_{a2}{x}_{1}},$$

(22)

where ${k}_{b2}$ and ${k}_{a2}$ are the wavenumbers for the flexural and axial waves in the slanted beam, respectively.

The transmitted waveguide is then represented as:

$${\mathbf{q}}_{\mathbf{B}}^{+}=\left[{C}^{+}{C}_{N}^{+}{D}^{+}\right].$$

(23)

The reflection and transmission matrices from A to B satisfy the following equilibrium:

$${\mathbf{q}}_{\mathbf{A}}^{-}={\mathbf{R}}_{\mathbf{A}\mathbf{B}}{\mathbf{q}}_{\mathbf{A}}^{+};$$

(24)

$${\mathbf{q}}_{\mathbf{B}}^{+}={\mathbf{T}}_{\mathbf{A}\mathbf{B}}{\mathbf{q}}_{\mathbf{A}}^{+}.$$

(25)

To derive the reflection and transmission matrices at the connection point in Fig. 1a, the force and displacement equilibriums are established.

$$\left\{\begin{array}{l}{E}_{1}{I}_{1}\frac{{\partial }^{2}{W}_{1}}{\partial {x}_{1}^{2}}={E}_{2}{I}_{2}\frac{{\partial }^{2}{W}_{2}}{\partial {x}_{2}^{2}};\\ {E}_{1}{S}_{1}\frac{\partial {U}_{1}}{\partial {x}_{1}}={E}_{2}{S}_{2}\frac{\partial {U}_{2}}{\partial {x}_{2}}\mathrm{cos}\beta +{E}_{2}{I}_{2}\frac{{\partial }^{3}{W}_{2}}{\partial {x}_{2}^{3}}\mathrm{sin}\beta ;\\ {E}_{1}{S}_{1}\frac{{\partial }^{3}{W}_{1}}{\partial {x}_{1}^{3}}=-{E}_{2}{S}_{2}\frac{\partial {U}_{2}}{\partial {x}_{2}}\mathrm{sin}\beta +{E}_{2}{I}_{2}\frac{{\partial }^{3}{W}_{2}}{\partial {x}_{2}^{3}}\mathrm{cos}\beta ;\\ {U}_{1}={U}_{2}\mathrm{cos}\beta -{W}_{2}\mathrm{sin}\beta \\ {W}_{1}={U}_{2}\mathrm{sin}\beta +{W}_{2}\mathrm{cos}\beta \\ \frac{\partial {W}_{2}}{\partial {x}_{2}}=\frac{\partial {W}_{1}}{\partial {x}_{1}}.\end{array}\right.$$

(26)

where ${W}_{1}={W}_{in}+{W}_{rf}$; ${W}_{2}={W}_{tm}$; ${U}_{1}={U}_{in}+{U}_{rf};\,{U}_{2}={U}_{tm}.$ Each column of ${\mathbf{R}}_{\mathbf{A}\mathbf{B}}$ and ${\mathbf{T}}_{\mathbf{A}\mathbf{B}}$ correspond to the transmission coefficient from the incident wave magnitude to the transmitted and reflected wave magnitude respectively. For instance, let ${\mathbf{q}}_{\mathbf{A}}^{+}=\left[{A}^{+}\,0\,0\right]$, the first column of ${\mathbf{R}}_{\mathbf{A}\mathbf{B}}$ and ${\mathbf{T}}_{\mathbf{A}\mathbf{B}}$ could be derived by solving the above equations in Eq. (26). Similarly, when the incident wave is from the inverse direction, i.e., ${\mathbf{q}}_{\mathbf{B}}^{-}$ the incident wave, ${\mathbf{q}}_{\mathbf{A}}^{-}$ the transmitted wave and ${\mathbf{q}}_{\mathbf{B}}^{+}$ the reflected wave, the reflected and transmitted matrices ${\mathbf{R}}_{\mathbf{B}\mathbf{A}}$ and ${\mathbf{T}}_{\mathbf{B}\mathbf{A}}$ could be derived through the same process. The derivation of the transmission and reflection matrices when there is only a unidirectional incident waveguide is the same as presented in the study (Horner and White [32]). But when the bidirectional incident waveguides, i.e., ${\mathbf{q}}_{\mathbf{B}}^{-}$ and ${\mathbf{q}}_{\mathbf{A}}^{+}$ coexist, Eq. (1) in this paper should be employed to present the more general case.

After obtaining the waveguide magnitude through Eq. (26), the reflected and transmitted waves’ energy ratios to the incident wave could be derived. Take the incident propagating flexural wave as an example.

The energy ratio of the reflected flexural wave to the incident flexural wave is:

$${\eta }_{ff}={({A}^{-}/{A}^{+})}^{2}$$

(27)

The energy ratio of the reflected axial wave to the incident flexural wave is:

$${\eta }_{af}={E}_{1}{S}_{1}{k}_{a1}{({B}^{-}/{A}^{+})}^{2}/\left(2{E}_{1}{I}_{1}{k}_{b1}^{3}\right)$$

(28)

The energy ratio of the transmitted flexural wave to the incident flexural wave is

$${\delta }_{ff}={(E}_{2}{I}_{2}{k}_{b2}^{3}{{{C}^{+})}^{2}/({E}_{1}{I}_{1}{k}_{b1}^{3}{A}^{+})}^{2}$$

(29)

The energy ratio of the transmitted flexural wave to the incident flexural wave is

$${\delta }_{af}={(E}_{2}{S}_{2}{k}_{a2}{{{D}^{+})}^{2}/(2{E}_{1}{I}_{1}{k}_{b1}^{3}{A}^{+})}^{2}$$

(30)

Appendix B: Wave approach formulation

Suppose there is a finite beam of length $L$. The origin of the reference coordinate system is assumed to start at the left end. The internal forces and velocities of the beam can be represented by the shear forces ${\text{S}}_{\text{i}}$, rotational moments ${\text{M}}_{\text{i}}$, lateral velocity ${\dot{\text{W}}_{\text{i}}}$ and angular velocity ${{\dot{\theta}}}_{\text{i}}$. The subscript $\text{i}=1$ denotes the left end ($\text{x}=0$), while $\text{i}=2$ denotes the right end ($\text{x}=\text{L}$). The force and velocity vectors, $\mathbf{f}$ and $\text{v}$, are composed of the shear forces and moments, and the flexural and rotational velocities at the two ends, i.e., $\text{f}={{{\text{S}}_{{1}} {\text{M}}_{{1}} {\text{S}}_{{2}} {\text{M}}_{{2}}}}^{{\rm T}}$ and $\text{v}={\left\{\begin{array}{ll}\begin{array}{ll}{\dot{W}}_{1}& {\dot{\theta }}_{1}\end{array}& \begin{array}{ll}{\dot{W}_{2}}& {\dot{\theta }}_{2}\end{array}\end{array}\right\}}^{\text{T}}$ as shown in Fig.

Fig. 9

a The shear forces, rotational moments, lateral displacement and rotational angle at the two ends of a finite uniform Euler–Bernoulli beam; b axial forces and displacements at the two ends of a finite uniform rod structure

9a.

The lateral displacement at any point $x$ along a beam can be interpreted as the superposition of positive and negative waves (Fahy and Gardonio 2007):

$$W\left(x\right)={A}_{1}{e}^{-i{k}_{b}x}{+A}_{1N}{e}^{-{k}_{b}x}+{A}_{2}{e}^{i{k}_{b}x}+{A}_{2N}{e}^{{k}_{b}x};$$

(31)

where ${k}_{b}={\left(\rho S/EI\right)}^{1/4}{\omega }^{1/2}$ is the wavenumber of the flexural wave and ${A}_{1},\,{A}_{2,}{A}_{1\mathrm{N}}{,\,A}_{2\mathrm{N}}$ are the complex magnitudes of the propagating and evanescent waves in both directions. Assuming the wave component vector $\mathbf{q}={\left\{{A}_{1}\,{A}_{1\mathrm{N}}\,{A}_{2}{\,A}_{2\mathrm{N}}\right\}}^{T}$, the internal force vector $\text{f}$ and the velocity vector $\text{v}$ satisfy the following relationship:

$$\mathbf{f}={\mathbf{T}}_{1}\mathbf{q};\mathbf{v}={\mathbf{T}}_{2}\mathbf{q}$$

(32)

where transfer matrices ${\mathbf{T}}_{1}{,\mathbf{T}}_{2}\in {\mathbb{C}}^{4\times 4}$ and the detailed expressions for ${\mathbf{T}}_{1}$ and ${\mathbf{T}}_{2}$ can be derived by the relationship between general forces, velocities, and wave magnitudes based on the waveform in Eq. (31). Consequently, the impedance matrix ${\mathbf{Z}}_{\mathbf{b}\mathbf{e}\mathbf{a}\mathbf{m}}\in {\mathbb{C}}^{4\times 4}$ can be derived by ${\mathbf{Z}}_{\mathbf{b}\mathbf{e}\mathbf{a}\mathbf{m}}={\mathbf{T}}_{1}{{\mathbf{T}}_{2}}^{-1}$ as follows:

$${\mathbf{f}=\mathbf{Z}}_{\mathbf{b}\mathbf{e}\mathbf{a}\mathbf{m}}\mathbf{v}=\frac{EI{k}_{b}^{3}}{j\omega N}\left[\begin{array}{ll}\begin{array}{ll}-{K}_{11}& -P\\ -P& {Q}_{11}\end{array}& \begin{array}{ll}{K}_{12}& V\\ -V& {Q}_{12}\end{array}\\ \begin{array}{ll}{K}_{12}& -V\\ V& {Q}_{12}\end{array}& \begin{array}{ll}{-K}_{11}& P\\ P& {Q}_{11}\end{array}\end{array}\right]\mathbf{v};$$

(33)

where

$$\begin{aligned} & K_{{11}} = \cos \left( {k_{b} L} \right)\sinh \left( {k_{b} L} \right) + \sin \left( {k_{b} L} \right)\cosh \left( {k_{b} L} \right), \\ & K_{{12}} = \sinh \left( {k_{b} L} \right) + \sin \left( {k_{b} L} \right), \\ & P = \sinh \left( {k_{b} L} \right)\sin \left( {k_{b} L} \right)/k_{b} , \\ & V = \left[ {\cos \left( {k_{b} L} \right) - \cosh \left( {k_{b} L} \right) } \right]/k_{b} \\ & Q_{{11}} = \left[ {\cos \left( {k_{b} L} \right)\sinh \left( {k_{b} L} \right) - \sin \left( {k_{b} L} \right)\cosh \left( {k_{b} L} \right) } \right]/k_{b}^{2} \\ & Q_{{12}} = \left[ {\sin \left( {k_{b} L} \right) - \sinh \left( {k_{b} L} \right)} \right]/k_{b}^{2} , \\ & N = \cos \left( {k_{b} L} \right)\cosh \left( {k_{b} L} \right) - 1. \\ \end{aligned}$$

(34)

Identically, the longitudinal direction’s impedance ${\mathbf{Z}}_{\mathbf{r}\mathbf{o}\mathbf{d}}\in {\mathbb{C}}^{2\times 2}$ can be derived by the wave formulation of a finite rod with length $L$ in Fig. 9b. The axial waves existing in the rod are non-dispersive, so the axial displacement at any point $x$ comprises only the propagating waves:

$$U\left(x\right)={B}_{1}{e}^{-i{k}_{l}x}+{B}_{2}{e}^{i{k}_{l}x}$$

(35)

where ${k}_{l}={\left(\rho /E\right)}^{1/2}\omega$ is the axial wavenumber and ${B}_{1}$ and ${B}_{2}$ are the complex magnitudes of the propagating waves in both directions. The force and velocity vectors ${\mathbf{f}}_{\mathbf{x}}={{{\text{F}}_{{\rm x}1} {\text{F}}_{{\rm x}2}}}^{{\rm T}}$ and ${\mathbf{v}}_{\mathbf{x}}={{{\dot{U}}_{\text{1}}{\dot{U}}_{\text{2}}}}^{\text{T}}$ comprise the axial forces and velocities at the two ends.

$${\mathbf{f}}_{\mathbf{x}}={\mathbf{Z}}_{\mathbf{r}\mathbf{o}\mathbf{d}}{\mathbf{v}}_{\mathbf{x}}=\left[\begin{array}{ll}{Z}_{11}& {Z}_{12}\\ {Z}_{21}& {Z}_{22}\end{array}\right]\left\{\begin{array}{l}{\dot{U}}_{\text{1}}\\ {\dot{U}}_{\text{2}}\end{array}\right\}$$

(36)

where ${Z}_{11}=-{Z}_{22}=-jS\sqrt{E\rho }\mathrm{cot}\left({k}_{l}L\right),{Z}_{12}={-Z}_{21}=jS\sqrt{E\rho }/\mathrm{sin}\left({k}_{l}L\right)$.

The expressions for the impedance matrices are the same as those directly given by (Gardonio and Brennan [28]), where the procedure of obtaining these impedance matrices was not presented. This section validates the consistency between the wave approach and impedance matrix formulation.

Appendix C: Beam mobility formulation

The mobility formulation of the cantilevered host beam will be introduced with both axial and flexural motions being considered. The mobility matrix representing the transmission from the force vector at ${x}_{i}$ to the velocity vector at ${x}_{j}$ is given through the modal superposition (Gardonio and Brennan [28]) as:

$${\mathbf{Y}}_{\mathbf{i}\mathbf{j}}=\left[\begin{array}{lll}{Y}_{{\dot{W}}_{j}{S}_{i}}& {Y}_{{\dot{W}}_{j}{M}_{i}}& 0\\ {Y}_{{\dot{\theta }}_{j}{S}_{i}}& {Y}_{{\dot{\theta }}_{j}{M}_{i}}& 0\\ 0& 0& {Y}_{{\dot{U}}_{j}{M}_{i}}\end{array}\right]$$

(37)

$$\begin{aligned} & Y_{{\dot{\theta }_{j} S_{i} }} = \mathop \sum \limits_{{n = 1}}^{\infty } \frac{{i\omega \Psi _{n}^{{\text{'}}} \left( {x_{j} } \right)\Psi _{n} \left( {x_{i} } \right)}}{{\rho SL\left[ {\omega _{n}^{2} \left( {1 + j\eta } \right) - \omega ^{2} } \right]}},\quad Y_{{\dot{\theta }_{j} M_{i} }} = \mathop \sum \limits_{{n = 1}}^{\infty } \frac{{i\omega \Psi _{n}^{{\text{'}}} \left( {x_{j} } \right)\Psi _{n}^{{\text{'}}} \left( {x_{i} } \right)}}{{\rho SL\left[ {\omega _{n}^{2} \left( {1 + j\eta } \right) - \omega ^{2} } \right]}} \\ & Y_{{\dot{W}_{j} S_{i} }} = \mathop \sum \limits_{{n = 1}}^{\infty } \frac{{i\omega \Psi _{n} \left( {x_{i} } \right)\Psi _{n} \left( {x_{j} } \right)}}{{\rho SL\left[ {\omega _{n}^{2} \left( {1 + j\eta } \right) - \omega ^{2} } \right]}},\quad Y_{{\dot{W}_{j} M_{i} }} = \mathop \sum \limits_{{n = 1}}^{\infty } \frac{{i\omega \Psi _{n}^{'} \left( {x_{i} } \right)\Psi _{n} \left( {x_{j} } \right)}}{{\rho SL\left[ {\omega _{n}^{2} \left( {1 + j\eta } \right) - \omega ^{2} } \right]}}, \\ & Y_{{\dot{U}_{j} F_{{xi}} }} {\text{ = }}\mathop \sum \limits_{{m = 1}}^{\infty } \frac{{i\omega \varphi _{m} \left( {x_{i} } \right)\varphi _{m} \left( {x_{j} } \right)}}{{\rho SL\left[ {{{\Omega }}_{m}^{2} \left( {1 + j\eta } \right) - \omega ^{2} } \right]}}, \\ \end{aligned}$$

(38)

where ${\Psi }_{n}\left(x\right)$ and ${\omega }_{n}$ are the ${n}^{th}$ modal shape function and the natural frequency of a cantilevered beam, respectively, and ${\varphi }_{m}\left(x\right)$ and ${{\Omega }}_{m}$ are the ${m}^{th}$ modal shape function and the natural frequency of a fixed-free rod, respectively. The hysteresis damping ratio $\eta$ is assumed to be material damping. It is set to zero in the main content.