One-Way Vibration Absorber

Abstract

A vibration absorber consisting of a one-dimensional waveguide with a reflectionless termination extracts vibrational energy from a structure that is to be damped. An optimum energy dissipation occurs for the so-called power adjustment, i.e, the same level of resistance and the opposite reactance of structure and absorber. The dimensioning of these impedance parameters on the base of the classic second order “two-way” wave equation provides analytical solutions for a few simple waveguide shapes; solutions for all other waveguides are only accessible via numerical finite-element computation. However, the competing first order “one-way” wave equation allows for an analytical conception of both the known broadband vibration absorber and the “Acoustic Black Hole” absorber. For example, for an exponential waveguide, the two-way calculation shows no resistance (and hence no real wave propagation) below a cut-off frequency, while the one-way wave equation predicts absorption in the whole frequency range.

1. Introduction

Mironov’s title “Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval” [1] describes a two-dimensional circular plate with a decreasing local bending wave velocity c towards its center due to a strength reduction. The extreme case with $c\to 0$ signifies a singularity that deflects bending waves to the center and dissipates the energy without reflections.

The article remained unnoticed for a long time—until Krylov [2], in 2004, described such an energy sink as an “Acoustic Black Hole” (ABH). After this, an exponentially growing interest arose: in 2021, Google Scholar listed 56 scientific articles regarding ABH and vibration topics. By this year, numerous ABH related patents were filed worldwide: 62 in China, four in USA/Canada, three in Europe and one in South Korea [3]. An overview of the ABH research can be found in [4].

By using a one-dimensional analytical model for a inhomogeneous lossless fluid, Maysenhölder showed that the ABH effect is not restricted to bending waves [5]. From his point of view, energy conservation is still valid because the wave energy does not disappear in the ABH’s energy sink but is stored: “The wave velocity decreases towards the singularity $c\to 0$. So, the wave energy is transported with decreasing speed, it is accumulated up and stored”.

To use Mironov’s mathematical model in a more robust representation, Krylov took, instead of the two-dimensional circular ABH plate, a compact one-dimensional ABH rod element and added a damping layer on the tapered end [2], see Figure 1.

The comparison of Figure 1 with Figure 2 reveals, that Krylov’s approach resulted in the known “broadband vibration absorber” (BVA) [6,7,8]. This one-dimensional tapered waveguide is coupled at one end, in a force-fitting manner, to the vibrating structure, and at the other end, a reflectionless termination is achieved by pathway damping.

Generally, a phase shift $\varphi$ [rad] of the pressure vs. the velocity amplitude $v_o$ [m/s], extracts damping power N [W] from the vibrating structure according to the formula

$$N=\frac{1}{2}{R}_o{v}_o^{2}cos\left(\varphi \right)$$

(1)

A cylindrical waveguide with constant parameters (cross section area $A_o$ [m²], density ${\rho }_o$ [kg/m³] and wave velocity $c_o$ [m/s]) has a real impedance with the resistance $R_o$ [kg/s]

$$R_o={\rho }_o{c}_o{A}_o=z_o{A}_o$$

(2)

The maximum damping occurs in the case of power adjustment, i.e., for equal resistance $R_o$ and opposite reactance $J_o$ of structure and absorber. Metals have a high specific impedance $z_o={\rho }_o{c}_o$ [kg/m²s], 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.

Conventionally, the design of vibration absorbers is based on the classical Cauchy equation of motion. This is a second order partial differential Equation (PDE), which has two equivalent solutions for describing waves in opposite directions, hence, the name “two-way wave equation”. The calculation of vibrations within an inhomogeneous rod material with a variable rod cross-section is only analytically possible for a few specific contours. In numerical FD calculations, irregular phantom effects occur due to an inherent ambiguity (no unique wave propagation direction).

The aim of this investigation is the calculation of a one-dimensional one-way vibration absorber. For this purpose, the existing “two-way” wave equation has to be rewritten as a “one-way” wave equation. Then, the Webster horn equation, also based on the two-way wave theory, will be adapted to the one-way approach as well. The two-way/one-way predictions are compared with horn waveguide measurements.

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

An inhomogeneous solid with cartesian coordinates $\mathit{x}=\{\mathit{x}\mathit{i},\mathit{y}\mathit{j},\mathit{z}\mathit{k}\}$ [m] has the local isotropic elastic modulus $E=E\left(\mathit{x}\right)$ [Pa], the local density $\rho =\rho \left(\mathit{x}\right)$ [kg/m³] and the longitudinal wave velocity:

$$c=c\left(\mathit{x}\right)=\sqrt{E/\rho }$$

(3)

A longitudinal wave field characterized by the vectorial elastic displacement $\mathit{s}=\mathit{s}(\mathit{x},\mathit{t})$ [m] and the stress tensor $\mathrm{T}=\nabla E\mathit{s}$ [Pa] is calculated according to the first Cauchy law of motion ($\dot{\mathit{s}}=\left(\mathit{s}\right)\mathbf{\dot{}}=\partial \mathit{s}/\partial t$, $\ddot{\mathit{s}}=\left(\mathit{s}\right)\mathbf{\dot{}}\mathbf{\dot{}}={\partial }^2\mathit{s}/\partial t^2$):

$$\rho \ddot{\mathit{s}}-\mathrm{\Delta }E\mathit{s}=\mathbf{0}$$

(4)

This vectorial equation describes a local force equilibrium of d’Alembert’s inertia force $\rho \ddot{\mathit{s}}$ [N/m³] and the divergence force $div\mathrm{T}=\nabla ·\nabla E\mathit{s}=\mathrm{\Delta }E\mathit{s}$ 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.

Due to these difficulties, the conventional force equilibrium was hypothetically replaced by an impulse equilibrium, more precisely: impulse flow equilibrium [16]. This hypothesis can be confirmed by two independent derivations: impedance theorem [17] and factorization [18]. Force and impulse are related and differ by one differentiation level. For this purpose, the impulse unit Huygens Hy = mass times velocity [Hy = mkg/s] was used after being introduced by the Karlsruhe KPK school.

Thus, for a particle velocity $\dot{\mathit{s}}$ [m/s], the local impulse $\rho \dot{\mathit{s}}$ [Hy/m³] 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/sm²], i.e., Huygens[Hy] per unit time and area. The elastic deformation $\mathit{s}$ induces the potential impulse flow $T=\nabla E\mathit{s}$ [Hy/sm²]. While Cauchy’s force balance is valid for an infinitesimal volume element $dV=dxdydz$, kinetic and potential impulse fulfill a local equilibrium in each field point [16]:

$$\rho \mathit{c}\dot{\mathit{s}}+\nabla E\mathit{s}=\mathbf{0}$$

(5)

This tensor equation can be scalarly multiplied with the wave velocity vector $\mathit{c}$. With the material law $E=\rho c^2$, Equation (1), and the time independence of the elasticity modulus with $E\dot{\mathit{s}}={\left(E\mathit{s}\right)}^{\mathbf{\dot{}}}$ results the first order vectorial partial differential equation:

$${\left(E\mathit{s}\right)}^{\mathbf{\dot{}}}+\mathit{c}·\nabla \left(E\mathit{s}\right)=\mathbf{0}$$

(6)

To calculate longitudinal waves in a one-dimensional vibration absorber, the scalar form is sufficient. With $\mathit{s}=s\mathit{i}$ and $\mathit{c}=\mathit{c}\mathit{i}$, the equation ${\left(\mathit{E}\mathit{s}\right)}^{\mathbf{\dot{}}}+\mathit{c}{\left(\mathit{E}\mathit{s}\right)}^{\prime }=0$ is obtained.

2.2. Introduction of the One-Way Webster Horn Equation

A waveguide with a cross-section area $\mathit{A}=\mathit{A}\left(\mathit{x}\right)$ is given. The pressure $\mathit{p}=\mathit{p}(\mathit{x},\mathit{t})$ of a longitudinal wave with wave velocity c is governed by the Webster horn equation:

$$\ddot{\mathit{p}}-{\mathit{c}}^2\frac{{\left(\mathit{A}{\mathit{p}}^{\prime }\right)}^{\prime }}{\mathit{A}}=0$$

(7)

Pierce [19] transformed Equation (5) from the field variable pressure p to the combined field variable $\left(p\sqrt{A}\right)$ [Pa m] (#$=\{p,A,\left(p\sqrt{A}\right)\}$, #${}^{\prime }=\partial$#$/\partial x$, #${}^{\prime \prime }={\partial }^2$#$/\partial x^2$):

$$\frac{1}{c^2}\left(p\sqrt{A}\right)\mathbf{\dot{}}\mathbf{\dot{}}-{\left(p\sqrt{A}\right)}^{\prime \prime }=\left(p\sqrt{A}\right)\left[{\left(A^{\prime }\right)}^2-2A{A}^{\prime \prime }\right]/4A^2$$

(8)

The left side consists of the wave equation for the combined field variable ($p\sqrt{A}$). The squared expression ($p\sqrt{A}{)}^2=p^{2}A$ with the specific sound intensity defined as $i=p^2/\rho c$ [W/m²] is a measure for the sound power $P=iA$ [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:

$${\left(A^{\prime }\right)}^2-2A{A}^{\prime \prime }=0.$$

(9)

The solution of this differential equation is:

$$A=\mathrm{\Omega }{x}^2,$$

(10)

and this represents a conical waveguide with the opening angle $\mathrm{\Omega }$ [sterad]. For $\mathrm{\Omega }=4\pi$, this corresponds to a spherical wave. All the other cross-section profiles, including the often used exponential horn $A=A_o\exp \left(\alpha x\right)$ do not satisfy the condition $P\left(x\right)=\mathrm{const}.$ Due to this discrepancy, the one-way Webster equation was derived in [20]:

$$\dot{s}-c\frac{{\left(s\sqrt{A}\right)}^{\prime }}{\sqrt{A}}=0$$

(11)

Since waveguide cross section area A is time-independent, identity $\left(\dot{s}\sqrt{A}\right)={\left(s\sqrt{A}\right)}^{\begin{array}{c}\hfill (-1,1)(-1,-2)*1.5\end{array}}$ is valid, and a one-way wave equation with the combined field variable results:

$${\left(s\sqrt{A}\right)}^{\mathbf{\dot{}}}-c{\left(s\sqrt{A}\right)}^{\prime }=0.$$

(12)

In contrast to (5), there is a constancy of ($s\sqrt{A}$).

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

The waveguide parameters # $=\{A,D,E,\rho ,c\}$ of the 1D vibration absorber shown in Figure 6 depend on x in the general case, i.e., # = #$\left(x\right)$. Continuity of the functions # and #${}^{\prime }=d$#$/dx$ is assumed. In addition to the cross-section A [m²], the characteristic diameter $D:=\sqrt{A}$ [m] is introduced.

Elastic modulus E [Pa] and density $\rho$ [kg/m³] 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

$${\left(DEs\right)}^{\mathbf{\dot{}}}+c{\left(DEs\right)}^{\prime }=0$$

(13)

With the restriction to wave motions of the angular frequency $\omega$ [rad/s] one obtains the time-independent notation with $\dot{s}=j\omega s$

$$j\omega \left(DEs\right)+c{\left(DEs\right)}^{\prime }=0$$

(14)

According to $(u^{\prime }/u)={\left(\ln \left(u\right)\right)}^{\prime }$ Equation (14) can also be directly integrated for the general case of an inhomogeneous x-dependent wave velocity $c=c\left(x\right)$ and for the combined field variable $\left(DEs\right)$ the respective Helmholtz equation is obtained as solution (with ${\left(DEs\right)}_o=$ integration constant = amplitude at [$x=0$])

$$\left(DEs\right)={\left(DEs\right)}_o\exp \left(-j\omega {\int }_0^x\frac{dx}{c}\right)$$

(15)

As can be verified by substitution, this corresponds to a wave of angular frequency $\omega$ traveling in $+x$-direction

$$\left(DEs\right)={\left(DEs\right)}_o\exp \left(j\omega \left[t-{\int }_0^x\frac{dx}{c}\right]\right)$$

(16)

Displacement $s(x,t)$, particle velocity $\dot{s}$ and gradient $s^{\prime }$ are calculated

$$\begin{array}{cc}\hfill s& =\frac{{\left(DEs\right)}_o}{\left(DE\right)}\exp \left(j\omega \left[t-{\int }_0^x\frac{dx}{c}\right]\right)\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill \dot{s}& =j\omega s\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill s^{\prime }& =-\left(\frac{{\left(DE\right)}^{\prime }}{\left(DE\right)}+j\frac{\omega }{c}\right)s\hfill \end{array}$$

(19)

Therefore, it is possible to calculate further characteristic values of the waveguide. The quotient of compressive stress $p=-E{s}^{\prime }$ and particle velocity $\dot{s}$, gives the specific impedance $z=(p/\dot{s})$ [kg/sm²]. With #$(x=0)=$#${}_o$ 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

$$\begin{array}{cc}\hfill Z_o& =R_o+j{J}_o={\left(A\frac{p}{\dot{s}}\right)}_o\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill R_o& =Re\left(Z_o\right)={\left(\rho cA\right)}_o\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill J_o& =Im\left(Z_o\right)={\left(\rho cA\right)}_o\left(-\frac{c{\left(DE\right)}_o^{\prime }}{\omega {\left(DE\right)}_o}\right)\hfill \end{array}$$

(22)

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	$\rho$, $E=\mathrm{const}.$	$\rho =\rho \left(x\right)$, $E=E\left(x\right)$	$\rho ,E=\mathrm{const}$.	$\rho =\rho \left(x\right)$, $E=E\left(x\right)$
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	$\dot{s}+c{s}^{\prime }=0$	${\left(DEs\right)}^{\mathbf{\dot{}}}+c{\left(DEs\right)}^{\prime }=0$	${\left(Ds\right)}^{\mathbf{\dot{}}}+c{\left(Ds\right)}^{\prime }=0$	${\left(Es\right)}^{\mathbf{\dot{}}}+c{\left(Es\right)}^{\prime }=0$
1D Solution	$\frac{s}{s_o}=\exp \left(j\omega \left[t-\frac{x}{c}\right]\right)$	$\frac{DEs}{{\left(DEs\right)}_o}=\exp \left(j\omega \left[t-{\int }_0^x\frac{dx}{c}\right]\right)$	$\frac{Ds}{{\left(Ds\right)}_o}=\exp \left(j\omega \left[t-\frac{x}{c}\right]\right)$	$\frac{Es}{{\left(Es\right)}_o}=\exp \left(j\omega \left[t-{\int }_0^x\frac{dx}{c}\right]\right)$
Impedance	$Z_o={\left(\rho cA\right)}_o$	$Z_o={\left(\rho cA\right)}_o\left(1-j\frac{c{\left(DE\right)}_o^{\prime }}{\omega {\left(DE\right)}_o}\right)$	$Z_o={\left(\rho cA\right)}_o\left(1-j\frac{c{\left(D\right)}_o^{\prime }}{\omega {\left(D\right)}_o}\right)$	$Z_o={\left(\rho cA\right)}_o\left(1-j\frac{c{\left(E\right)}_o^{\prime }}{\omega {\left(E\right)}_o}\right)$
Resistance	$R_o={\left(\rho cA\right)}_o$	$R_o={\left(\rho cA\right)}_o$	$R_o={\left(\rho cA\right)}_o$	$R_o={\left(\rho cA\right)}_o$

shows the relations for general and specific waveguide properties.

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

For a quantitative comparison, a 1D waveguide consisting of homogeneous material $E\left(x\right)=E_o=\mathrm{const}.$ and $\rho \left(x\right)={\rho }_o=\mathrm{const}.$ and with an exponential cross-sectional area is considered ($\alpha$ [1/m] = exponent)

$$A=A_o\exp (-\alpha x)$$

(23)

According to the second order two-way PDE based on the classic force equilibrium (“force concept”), the semi-infinite exponential horn waveguide with $L\to \infty$ has the input impedance $Z_o$ and the cut-off frequency $f_C$ [1/s] $=\alpha c/4\pi$ [21]. Below cut-off frequency $f_C$ the impedance $Z_o$ is imaginary and no real power is transmitted.

$$\begin{array}{cc}\hfill & \mathrm{Two}-\mathrm{Way}/\mathrm{Force}\mathrm{Concept}:Z_o=R_o+j{J}_o={\left(\rho cA\right)}_o\left(\sqrt{1-{\left(\frac{f_C}{f}\right)}^2}+j\frac{f_C}{f}\right)\hfill \end{array}$$

(24)

In contrast to this, the first order one-way PDE based on the impulse flow equilibrium (“impulse concept”) does not have a cut-off frequency and the impedance follows as [22]

$$\begin{array}{cc}\hfill & \mathrm{One}-\mathrm{Way}/\mathrm{Impulse}\mathrm{Concept}:Z_o=R_o+j{J}_o={\left(\rho cA\right)}_o\left(1+j\frac{f_C}{f}\right)\hfill \end{array}$$

(25)

In Figure 7, the resistances and reactances of the two-way and the one-way approach are compared. The resistances and the reactances significantly differ for low frequencies.

It can be proven that real wave propagation also occurs below cut-off frequency $f_C$. The results of an measurement [23] (see Figure 8 and Figure 9), with an acoustical exponential waveguide are in line with the one-way horn equation, i.e., the waves propagate with constant local wave velocity $c=c_o=c_{⊚}$ (speed of sound $c_o\approx$ 343 m/s), see Figure 10.

5. Conclusions

The conventional broadband vibration (BVA) absorber by Bschorr/Albrecht and the acoustic black hole (ABH) absorber by Kyrov work according to the same principle: both are rod-shaped waveguides, one end is force-fitted to the vibrating structure to be damped, and the other end is reflectionless due to pathway damping.

The characteristic criteria for the damping performance is the absorber’s resistance at the coupling base area. The maximum damping occurs for power adjustment, i.e., the vibration absorber and the structure have equal resistance and equal but opposite reactance.

The calculation of the vibration absorber according to the conventional two-way wave equation only provides analytical solutions for a few waveguide contours; in the general case, numerical methods have to be used. Using two independent methods, the impedance theorem [17] and the factorization [18], it was possible to reduce the two-way wave equation to the one-way wave equation. Mathematically, this corresponds to a factorization of second-order PDE, which is generally only numerically solvable, into a first-order PDE. The first-order PDEs can be solved via quadrature. The first-order approach also aligns with the waveguide measurements at a reflectionless terminated exponential horn.