# One-Way Vibration Absorber

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}], density ${\rho}_{o}$ [kg/m

^{3}] and wave velocity ${c}_{o}$ [m/s]) has a real impedance with the resistance ${R}_{o}$ [kg/s]

^{2}s], thus they are predestined as waveguide materials. A disadvantage is their low loss factors $\eta $ [-], and therefore the pathway damping has to be increased. Figure 3 shows a railway wheel absorber as a bending waveguide consisting of two metal crescent plates with an intermediate viscous damping layer. In Figure 4 and Figure 5, the pathway damping of a longitudinal waveguide is achieved by alternating layers of metal and plastic plates.

## 2. Derivation of the One-Way Wave Equation and One-Way Horn Equation

#### 2.1. Conversion from the Two-Way Wave Equation to the One-Way Wave Equation

^{3}] and the longitudinal wave velocity:

^{3}] and the divergence force $div\mathrm{T}=\nabla \xb7\nabla E\mathit{s}=\mathrm{\Delta}E\mathit{s}$ of stress tensor $\mathrm{T}$. This second order partial differential Equation (PDE) has two independent solutions, hence the name “two-way wave equation”. Due to this ambiguity, irregular phantom effects occur in numerical (for example seismic) finite-element or also finite-difference wave calculations. For their elimination, a high number of auxiliary equations have been developed; however, none of these approaches prevailed as an accepted standard solution.

^{3}] exists. If the particle velocity $\dot{\mathit{s}}$ propagates with the vectorial wave velocity $\mathbf{c}$ [m/s], the dyadic product $\rho \mathit{c}\dot{\mathit{s}}$ provides a kinetic impulse flow of the dimension [Hy/sm

^{2}], i.e., Huygens[Hy] per unit time and area. The elastic deformation $\mathit{s}$ induces the potential impulse flow $T\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\nabla E\mathit{s}$ [Hy/sm

^{2}]. While Cauchy’s force balance is valid for an infinitesimal volume element $dV\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}dxdydz$, kinetic and potential impulse fulfill a local equilibrium in each field point [16]:

#### 2.2. Introduction of the One-Way Webster Horn Equation

^{2}] is a measure for the sound power $P=i\phantom{\rule{0.166667em}{0ex}}A$ [W] transported across the waveguide cross-section area A. Equal power with $P\left(x\right)=\mathrm{const}.$ requires that the disturbance element on the right side of the equation disappears:

## 3. One-Way Vibration Absorber: Resulting Equations, Solutions and Impedances

^{2}], the characteristic diameter $D:=\sqrt{A}$ [m] is introduced.

^{3}] give the longitudinal wave velocity $c=\sqrt{E/\rho}$ (1). The field variable is the longitudinal displacement $s=s(x,t)$ of a wave with circular frequency $\omega $ [rad/s] and wavelength $\lambda =2\pi c/\omega $ [m]. The stringent 1D condition requires $D/\lambda \to 0$. Reflectionless termination requires $\lambda $ #${}^{\prime}/$# $\to 0$. According to the task, the "one-way" wave equation is used. The equation for the general case is as follows

^{2}]. With #$(x\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0)=\phantom{\rule{0.277778em}{0ex}}$#${}_{o}$ at the waveguide beginning the impedance ${Z}_{o}={\left(zA\right)}_{o}$ [kg/s] with real resistance ${R}_{o}$ and imaginary reactance ${J}_{o}$ follows

## 4. Comparison of Two-Way/One-Way Predictions vs. Measurement

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Acoustic Black Hole (ABH) vibration absorber, technical principle as in Krylov (2004) [2]: 1 = structure to be damped, 2 = base area ${A}_{o}$, 3 = tapered waveguide and 4 = damping layer. Since the mathematical singularity with wave velocity $c\to 0$ is technically not feasible, a damping layer is added for a reflectionless termination.

**Figure 2.**Broadband vibration absorber (BVA) according to Bschorr/Albrecht (1979) [6,7,8]: 1 = structure to be damped, 2 = base area ${A}_{o}$, 3 = tapered waveguide and 4 = damping layer. A one-dimensional absorber with base area ${A}_{o}$ [m

^{2}], density ${\rho}_{o}$ [kg/m

^{3}] and longitudinal wave velocity ${c}_{o}$ [m/s] achieves a maximum resistance ${R}_{o}={\rho}_{o}{c}_{o}{A}_{o}$ [kg/s]. Coupled to a vibrating structure with mean velocity v [m/s] causes the damping power $N\to \phantom{\rule{0.277778em}{0ex}}$

^{1}/

_{2}${R}_{o}{v}^{2}$ [W].

**Figure 3.**Railway wheel absorber (GHH-VALDUNES, Gutehoffnungshütte (GHH), Oberhausen, Germany, reprinted with permission from Ref. [9], Copyright 2012, GHH-Radsatz) consisting of two metal crescent plates with an constrained viscous damping layer as a bending waveguide to reduce wheel-radiated squeaking noises while cornering [10,11].

**Figure 4.**Railway wheel absorber (VICON RASA RSI, Schrey & Veit GmbH, Sprendlingen, Germany, reprinted with permission from Ref. [12], Copyright 2019, Schrey &Veit) comprising a stack of alternating metal and plastic plates as a longitudinal waveguide absorber. Tests proved a reduction in the noise level LAeq, Tp (ISO 3095) by about 4 dB.

**Figure 5.**Rail absorber (VICON AMSA 60 VS, Schrey & Veit GmbH, Sprendlingen, Germany, reprinted with permission from Ref. [13], Copyright 2022, Schrey &Veit) consisting of a stack of alternating metal and plastic plates acting for the reduction of noise from vibrating rails, which are a further major sound emitter in addition to the rail wheels [14,15].

**Figure 6.**1D waveguide with coordinate x [m] and $0<x<L$ for the description of a traveling longitudinal wave in the $+x$ direction; cross-section area A [m

^{2}] (characteristic diameter $D\left[m\right]\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\sqrt{A}$), elasticity modulus E, density $\rho $, local displacement s and waveguide length L [m].

**Figure 7.**For the semi-infinite exponential horn with the contour $A\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}{A}_{o}\mathrm{exp}(-\alpha x)$ the two-way based “force concept” predicts no resistance below the cut-off frequency ${f}_{c}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\alpha c/4\pi $, for frequencies $f\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}>\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{f}_{c}$, there is asymptotic approximation towards ${R}_{o}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}{\left(\rho cA\right)}_{o}$. In contrast, according to the one-way based “impulse concept” a constant resistance ${R}_{o}={\left(\rho cA\right)}_{o}$ exists over the entire frequency range. For the semi-infinite exponential horn both concepts have the same reactance; however, for frequencies $f<{f}_{c}$ the values differ significantly.

**Figure 8.**Experimental setup: loudspeaker, exponential horn, reflectionless termination with wedge foam absorber (6 m) to avoid backtraveling waves; microphones at $x\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0$, $x\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}L$ [23].

**Figure 9.**Exponential horn with length L = 410 mm, ${D}_{0}$ = 10 mm, ${D}_{L}$ = 45 mm, manufactured with rapid prototyping (accuracy 50 $\mu $), wall thickness 15 mm [23].

**Figure 10.**Exponentional horn measurement [23]. The top graph shows the measured phase angle $\mathrm{\Phi}$ and the bottom graph the calculated wave velocity $c\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}2\pi fL/\mathrm{\Phi}$. The sound propagates with a linear phase and a wave velocity of $c\phantom{\rule{-0.166667em}{0ex}}\approx $ 343 m/s, see black lines. The predictions of the Webster horn equation are displayed with red lines. The conditions for conformity with the Principle of Locality ($c={c}_{o}=\sqrt{E/\rho}$) are shown with blue lines. The measurement indicates, that the sound waves in the exponential horn with reflectionless termination travel with speed of sound at/below cut-off frequency ${f}_{C}$ = 200 Hz. This is in contrast to the predictions of the Webster horn equation.

**Table 1.**Overview: 1D one-way wave equations, solutions; generic case and different waveguide materials and contours (D [m] = $\sqrt{A}$). A real elasticity modulus E describes the lossless case. A complex elasticity modulus $E={E}_{o}(1+j\eta )$ with lossfactor $\eta $ [-] considers absorption.

Reference | Generic | Homogeneous | Cylindrical | |
---|---|---|---|---|

Plane Wave | Waveguide | Waveguide | Waveguide | |

Material | $\rho $, $E=\mathrm{const}.$ | $\rho =\rho \left(x\right)$, $E=E\left(x\right)$ | $\rho ,\phantom{\rule{0.277778em}{0ex}}E=\mathrm{const}$. | $\rho =\rho \left(x\right)$, $E=E\left(x\right)$ |

Contour | $A=\mathrm{const}.$ | $A=A\left(x\right)$ | $A=A\left(x\right)$ | $A=\mathrm{const}.$ |

3D Equation | $\rho \mathit{c}\dot{\mathit{s}}+\nabla E\mathit{s}=\mathbf{0}$ | $\rho \mathit{c}\dot{\mathit{s}}+\nabla E\mathit{s}=\mathbf{0}$ | $\rho \mathit{c}\dot{\mathit{s}}+\nabla E\mathit{s}=\mathbf{0}$ | $\rho \mathit{c}\dot{\mathit{s}}+\nabla E\mathit{s}=\mathbf{0}$ |

1D Equation | $\dot{s}+c{s}^{\prime}=0$ | ${\left(DEs\right)}^{\mathbf{\dot{}}}\phantom{\rule{0.277778em}{0ex}}+c{\left(DEs\right)}^{\prime}=0$ | ${\left(Ds\right)}^{\mathbf{\dot{}}}\phantom{\rule{0.277778em}{0ex}}+c{\left(Ds\right)}^{\prime}=0$ | ${\left(Es\right)}^{\mathbf{\dot{}}}\phantom{\rule{0.277778em}{0ex}}+c{\left(Es\right)}^{\prime}=0$ |

1D Solution | $\frac{s}{{s}_{o}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\mathrm{exp}\left(j\omega \left[t\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\frac{x}{c}\right]\phantom{\rule{-0.166667em}{0ex}}\right)$ | $\frac{DEs}{{\left(DEs\right)}_{o}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\mathrm{exp}\left(\phantom{\rule{-0.166667em}{0ex}}j\omega \left[t\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}{\int}_{0}^{x}\phantom{\rule{-0.166667em}{0ex}}\frac{dx}{c}\right]\phantom{\rule{-0.166667em}{0ex}}\right)$ | $\frac{Ds}{{\left(Ds\right)}_{o}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\mathrm{exp}\left(j\omega \left[t\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\frac{x}{c}\right]\phantom{\rule{-0.166667em}{0ex}}\right)$ | $\frac{Es}{{\left(Es\right)}_{o}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\mathrm{exp}\left(\phantom{\rule{-0.166667em}{0ex}}j\omega \left[t\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}{\int}_{0}^{x}\phantom{\rule{-0.166667em}{0ex}}\frac{dx}{c}\right]\phantom{\rule{-0.166667em}{0ex}}\right)$ |

Impedance | ${Z}_{o}=\phantom{\rule{-0.166667em}{0ex}}{\left(\rho cA\right)}_{o}$ | ${Z}_{o}=\phantom{\rule{-0.166667em}{0ex}}{\left(\rho cA\right)}_{o}\phantom{\rule{-0.166667em}{0ex}}\left(1\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}j\frac{c{\left(DE\right)}_{o}^{\prime}}{\omega {\left(DE\right)}_{o}}\right)$ | ${Z}_{o}=\phantom{\rule{-0.166667em}{0ex}}{\left(\rho cA\right)}_{o}\phantom{\rule{-0.166667em}{0ex}}\left(\phantom{\rule{-0.166667em}{0ex}}1\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}j\frac{c{\left(D\right)}_{o}^{\prime}}{\omega {\left(D\right)}_{o}}\phantom{\rule{-0.166667em}{0ex}}\right)$ | ${Z}_{o}=\phantom{\rule{-0.166667em}{0ex}}{\left(\rho cA\right)}_{o}\phantom{\rule{-0.166667em}{0ex}}\left(1\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}j\frac{c{\left(E\right)}_{o}^{\prime}}{\omega {\left(E\right)}_{o}}\right)$ |

Resistance | ${R}_{o}=\phantom{\rule{-0.166667em}{0ex}}{\left(\rho cA\right)}_{o}$ | ${R}_{o}=\phantom{\rule{-0.166667em}{0ex}}{\left(\rho cA\right)}_{o}$ | ${R}_{o}=\phantom{\rule{-0.166667em}{0ex}}{\left(\rho cA\right)}_{o}$ | ${R}_{o}=\phantom{\rule{-0.166667em}{0ex}}{\left(\rho cA\right)}_{o}$ |

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**MDPI and ACS Style**

Bschorr, O.; Raida, H.-J.
One-Way Vibration Absorber. *Acoustics* **2022**, *4*, 554-563.
https://doi.org/10.3390/acoustics4030034

**AMA Style**

Bschorr O, Raida H-J.
One-Way Vibration Absorber. *Acoustics*. 2022; 4(3):554-563.
https://doi.org/10.3390/acoustics4030034

**Chicago/Turabian Style**

Bschorr, Oskar, and Hans-Joachim Raida.
2022. "One-Way Vibration Absorber" *Acoustics* 4, no. 3: 554-563.
https://doi.org/10.3390/acoustics4030034