Consistent approach to describe and evaluate uncertainty in vibration attenuation using resonant piezoelectric shunting and tuned mass dampers

– Undesired vibration may occur in lightweight structures due to low damping and excitation. For the purpose of vibration attenuation, tuned mass dampers (TMD) can be an appropriate measure. A similar approach uses resonantly shunted piezoelectric transducers. However, uncertainty in design and application of resonantly shunted piezoelectric transducers and TMD can be caused by insuﬃcient mathematical modeling, geometric and material deviations or deviations in the electrical and mechanical quantities. During operation, uncertainty may result in detuned attenuation systems and loss of attenuation performance. A consistent and general approach to display uncertainty in load carrying systems developed by the authors is applied to describe parametric uncertainty in vibration attenuation with resonantly shunted piezoelectric transducers and TMD. Mathematical models using Hamilton’s principle and Ritz formulation are set up for a beam, clamped at both ends with resonantly shunted transducers and TMD to demonstrate the effectiveness of both attenuation systems and investigate the eﬀects of parametric uncertainty. Furthermore, both approaches lead to additional masses, piezoelectric material for shunt damping and compensator mass of TMD, in the systems. It is shown that vibration attenuation with TMD is less sensitive to parametric uncertainty and achieves a higher performance using the same additional mass.


Introduction
Structural vibration may occur in several mechanical systems leading to fatigue, reduced durability or undesirable noise. In this context, piezoelectric transducers with a semi-passive inductive resistive electrical network and mechanical tuned mass dampers (TMD) may be an appropriate measure. Both approaches are combined with a host structure and may attenuate its vibration in a limited frequency range, determined by the resonance frequency. Semi-passive vibration attenuation means that no additional energy such as from amplifiers is used for vibration attenuation. However, weak power supply for circuits like a synthetic inductance is necessary. Both attenuation approaches have been subject to research for several decades. A piezoelectric transducer converts mechanical energy into electrical energy. By connecting the electrodes of the transducer to an electrical circuit with a Corresponding author: goetz@szm.tu-darmstadt.de resistor and inductance, an electrical oscillation circuit with the inherent capacitance of the transducer is created. Hence, the vibration energy of the host structure can be reduced with the semi-passive approach by transferring it partly to the electrical vibratory system. The passive TMD achieves vibration attenuation in the same way, but no mechanical energy is converted to electrical energy. Only mechanical energy is transferred from the host structure to the TMD. Usually, a TMD is given by tuned mass, stiffness and damper. Both approaches, semipassive and passive, lead to additional masses that affect adversely the construction of lightweight structures. Especially piezoelectric shunt damping is often mentioned in accordance with lightweight design and weight reduction [1,2]. Furthermore, replacing a classical passive TMD with a piezoelectric patch transducer connected to a resonant shunt could achieve benefits due to the reduced construction space and a more precise tuning of the electrical shunt components compared to mechanical stiffness and damping. In general, systems for passive and semi-passive vibration attenuation are designed for a specific problem. However, deviation of the system parameters may lead to mismatch or detuning and, thus, to loss of vibration attenuation performance. Most of the literature examines passive and semi-passive systems with no regard to effects of uncertainty [1,[3][4][5]. In references [6,7] electrical quantities of a transducer with resonant shunt are considered to be uncertain. However, no investigations regarding different transducer dimensions and the effect on uncertainty are made. There are many ways, approaches and keywords to describe and evaluate uncertainty in technical context. In this paper three simple categories shall be used to describe and evaluate uncertainty in vibration attenuation: unknown, estimated and stochastic uncertainty. The categories are developed in the German Collaborative Research Center SFB 805 "Control of uncertainty in loadcarrying mechanical systems" (see Sect. 3). In this paper, the categories are used to describe and compare the effects of uncertainty on vibration attenuation of an harmonically excited beam clamped at both ends. For that, piezoelectric transducers with semi-passive inductive resistive shunt, in the following referred to as resonant shunt, and passive TMD are used to attenuate the first transversal eigenmode of the beam. In this paper, the additional mass of both approaches, the resonant shunt and TMD, is used to compare the vibration attenuation performance under uncertainty. Only the mass of the piezoelectric ceramic and the oscillating mass of the TMD are considered as additional masses.

Mathematical models
This section outlines the derivation of the complex frequency response function of an Euler-Bernoulli beam with piezoelectric patch transducers and RL-shunt as well as of an beam with TMD. Energy variation with Hamilton's principle is used to derive and Ritz formulation is used to discretize and solve the equation of motion. Linear time-invariant state space formulations of the beam with resonant shunt and of the beam with TMD are derived and transformed into frequency domain to calculate the frequency response function. Figure 1 shows the clamped beam with piezoelectric transducers and shunt design with length l b of the beam, bending stiffness EI b , density ρ b , thickness h b and width a b . The length of one piezoelectric transducer is l p , with thickness h p and width a p . The piezoelectric material properties are represented by the piezoelectric constant d 31 ,"3" indicating the direction of the electric field in zdirection, and "1" indicating the direction of mechanical strain in the piezoelectric transducer in x-direction. β S r,33 is the relative permittivity at constant strain. The bending stiffness of one symmetric transducer couple, one on the top and one on the bottom of the beam, is given by EI p . The properties of the resistive inductive shunt are the resistance R and the inductance L. For an one dimensional Euler-Bernoulli beam, w(x, t) is the transversal displacement in z-direction. The current flowing in the shunt is the first time derivative of the electrical chargeq(t) and the voltage at the piezoelectric electrodes is v(t). A discrete harmonic force F (t) = F cos(Ωt − φ) with the amplitude F , the excitation angular frequency Ω and phase shift φ is exciting the beam at location x F . Four piezoelectric transducers attached to the beam next to the clamps give good vibration attenuation results for the first eigenmode [8] (see Fig. 1).

Beam with piezoelectric patch transducers
The equation of motion for a conservative system may be derived using Hamilton's principle due to energy variation. A generalized form of Hamilton's principle for an electromechanical system is given in reference [9] with the kinetic energy T , potential energy U and the electrical energy W el of the beam with piezoelectric patch transducers when excited by the external work W . Using Ritz formulation, the lateral displacement w(x, t) of the beam with piezoelectric patch transducers can be approximated by a summation of K trial functions ψ k (x) with k = 1, . . . , K and an associated generalized coordinate r k (t) [10,11]: The trial function ψ k (x) must satisfy the geometric boundary conditions of the beam w(0, t) = w(l b , t) = 0 and w (0, t) = w (l b , t) = 0 and must be differentiable at least to the second space-derivative. In Equation (2), Ψ(x) is the (K × 1) vector of trial functions and r(t) is the (K × 1) vector of the generalized coordinates. Taking into account the mechanical and electrical properties, an actuating equation Equation (3a) and a sensing equation Equation (3b) according to [9] lead to and Both equations Equations (3a) and (3b) describe the dynamic behavior of the beam with attached transducers as a coupled electromechanical system. In Equation (3a),ṙ(t) andr(t) are the first and the second time derivative of the generalized coordinate r(t). M b , M p , K b and K p are the (K × K) generalized mass and the (K × K) stiffness matrices of the beam and the piezoelectric patch transducers. D is the (K × K) Rayleigh damping matrix of the beam with coupled piezoelectric patch transducers assuming proportional damping. The factors δ 1 and δ 2 are chosen to obtain the assumed damping of 0.5% for the first two eigenmodes. The electrical part in Equation (3a) is represented by the electrical voltage v(t) at the transducers electrodes and the (K × 1) electromechanical coupling vector Θ. The vector elements Θ k with k = 1, . . . , K of the coupling vector are with ΔH is the Heaviside step function. The energy converted by the transducer is considered to be proportional to its strain in x-direction due to bending only. For an Euler-Bernoulli beam, the bending strain ε x is The elements of the (K ×K) beam and piezoelectric patch mass matrices are with k = 1, . . . , K and l = 1, . . . , L for K = L, the masses per length of the beam η b = ρ b h b a b and the transducer patch couple η p = 2ρ p h p a p . B F is the (K × 1) forcing vector The elements of the (K × K) beam stiffness matrix are The elements of the (K × K) piezoelectric patch stiffness matrix are with the bending stiffness of one symmetric transducer couple. In Equation (11) I * p is the area moment of inertia of one rectangular patch transducer related to its center of area. In Equation (3b), the capacitance of the piezoelectric patch transducers C p in parallel is given by with the vacuum permittivity β 0 .

Beam with piezoelectric transducers and RL shunt
An inductive resistive shunt is connected to the beam with piezoelectric patch transducers. Figure 1 shows the shunt connected to the piezoelectric patch transducers. From the second Kirchhoff's law, the voltage from R and L is obtained Substituting the voltage v(t) in Equations (3a) and (13) by Equation (3b), the equations of motion of the beam with piezoelectric patch transducers and RL-shunt become

State space model
With the equation of motion of the coupled electromechanical system Equations (3a) and (3b), a state space model will be introduced. The state space model is transferred into frequency domain to calculate the frequency response function of the beam with patch transducers and with resonant shunt. Therefore, Equations (14a) and (14b) are transferred into first order differential equations to obtain a linear time-invariant state space model formulation. The complete model of the shunted (sh) electromechanical system according to [9] can be written aṡ C sh is the [1 × (2K + 2)] output vector and D sh the feedthrough constant: All state variables are contained in the [(2K +2)×1] state vector x sh . The input u sh is the force F (t) and the output y sh is the beam displacement at w(x F , t).
To obtain the displacement w(x F , t), a transformation from generalized coordinates r(t) to physical coordinates x(t) is performed with the vector of trial functions Ψ(x F ).
Using the particular integral approach the complex transfer function of the beam with patch transducers and with resonant shunt for the beam displacement amplitudê w( with its amplitude and phase response |H sh (Ω)| and ϕ sh (Ω) = arg{H sh (Ω)}. With the excitation angular frequency Ω and the phase shift φ and r(t) = r e j(Ωt − φ) . (21) In Equation (20), the phase shift φ is assumed to be zero.
Although |H (Ω)| and ϕ(Ω) are functions of Ω, in all following figures the amplitude and phase response are plotted as functions of frequency f in Hz using the conversion Ω = 2πf . The amplitude responses are normalized to the static compliance of the beam.

Optimal tuning parameters
To attenuate the resonance amplitude of the vibrating host structure, the resonant shunt needs to be tuned according to the first resonance angular eigenfrequency ω b of the host structure with piezoelectric patch transducers. Therefore, the optimal value for the inductance L will be calculated. To achieve a maximum broad band damping, the optimal value for the resistance R will be obtained. The optimal values for L and R may be chosen according to [12]. This approach uses the transfer function criterion and minimizes the amplitude response in the attenuated frequency range. The optimal inductance L opt and the optimal resistance R opt are and (23) In Equation (22), λ opt is the tuning parameter and ω OC is the first open circuit angular eigenfrequency. The actual angular eigenfrequency of the tuned electrical shunt is ω el = 1/ C p L. In Equation (23), D opt is the optimal damping. In Equations (22) and (23), K 31 is the generalized electromechanical coupling coefficient. It characterizes the energy exchanged between the mechanical and the electrical domains. The coupling coefficient K 31 is influenced by the piezoelectric material properties, patch transducer dimensions and location on the beam. A high coupling coefficient will lead to high vibration attenuation. K 31 for the first angular eigenfrequency may be obtained from the first open circuit angular eigenfrequency ω OC and the first short circuit angular eigenfrequency ω SC according to Assuming only minor damping, damped angular eigenfrequencies may be neglected. Thus the short circuit and the open circuit angular eigenfrequencies are obtained solving the eigenvalue problem of the undamped system Figure 2a illustrates the amplitude response |H sh (Ω)| = |H RLopt | and phase response ϕ sh (Ω) = ϕ RLopt , (see Eq. (20)). The optimal shunt tuning (solid line) attenuates -22.52 dB compared to the amplitude response |H OC | with open circuit (dashed line). Furthermore, Figure 2a shows the response function |H b | of the beam without attached piezoelectric transducers, its mathematical derivation has been neglected in this paper. Due to the transducers the structure stiffness increases and the first eigenfrequencies f OC,1 and f SC,1 are higher than the original first eigenfrequency f b,1 = 66.2 Hz of the beam. At f b,1 the reduction of the amplitude response is -24.38 dB compared to the resonance amplitude |H b | of the beam. With regard to experimental applications, usually L needs to be in the range of several Henry. This is realized with electrical networks containing operational amplifiers, resistances and a capacitance, also know as gyrators. The circuit simulates the impedance of an inductance by inverting the impedance of the capacitance C 4 . Figure 2b shows a realization of such a circuit. Simplified, the circuit shows the same characteristic like an ideal inductor [13], with the value The required values for L are realized with R 2 = R 3 = 1 kOhm, R 5 = 20 kOhm, C 4 = 1 μF and R 1 for adjusting L.

Beam with tuned mass damper TMD
To obtain the best vibration attenuation for the first eigenmode, the TMD is placed at The TMD is modeled as one degree of freedom system with mass m T and its displacement z T (t) that is connected to the beam with a linear damper and a linear spring. The damping coefficient is b T and the spring's stiffness is k T . Equation (27) shows the equation of motion for the TMD.
where F T (t) is the resulting force of the TMD acting on the beam and on the mass m T . With M b and K b from Equations (7) and (9) the equation of motion for the beam only is given by

State space model
Equations (27) and (28) are transferred into first order differential equations to obtain a state space model formulation. The complete model of the coupled mechanical systems (TMD) can be written aṡ All state variables are contained in the [(2K +2)×1] state vector x TMD . The input u TMD is the force F (t) and the output y TMD is the beam displacement at w(x F , t).
To obtain the displacement w(x F , t), a transformation from generalized coordinates r(t) to physical coordinates x(t) is performed with the vector of trial functions Ψ(x F ).
Using the particular integral approach the complex transfer function of the beam with TMD for the beam displacement amplitudeŵ(x F =l b /2) in frequency domain results in with its amplitude and phase response |H TMD (Ω)| and ϕ TMD (Ω) = arg{H TMD (Ω)}.

Optimal tuning parameters
Comparable to the resonant shunt, the TMD needs to be tuned to the vibrating structure. Therefore b T and k T are chosen according to [14] as In Equation (34), μ T is the ratio of m T and the oscillating mass m osc of the beam. The oscillating mass m osc is obtained by equalizing the kinetic energy of a one degree of freedom oscillator and the beam in the mounting point of the TMD. The mass is calculated to be 0.12 kg with where Φ(x) is the normalized first modeshape of the beam. Figure 4 shows the amplitude and phase response |H TMD (f )| = |H TMD | and ϕ TMD (f ) = ϕ TMD , see Equation (33) and the frequency response |H b | of the beam only. The optimal TMD tuning (solid line) attenuates -25.3 dB compared to the amplitude response of the beam (dashed line). The attenuation at f b,1 is -25.65 dB.

Uncertainty quantification
In this paper, the vibration attenuation capability with respect to the used additional masses on the beam for a TMD and a resonant shunt are compared and effects due to parametric uncertainty are presented. Uncertainty in vibration attenuation arising from uncertain parameters in resonant shunt and TMD is discussed in a simple but consistent and transparent way. A general approach to describe and evaluate uncertainty in load carrying systems is presented in reference [15][16][17]. This approach uses a matrix to assign uncertainty categories to all relevant system properties for a certain state at a certain time, e. g. in the design or operating process. According to [16], geometrical, material, electrical, economical and other properties are of interest. Table 1 shows the relevant properties for the beam clamped at both ends with resonant shunt or TMD after completing the definition phase in the design process at the point of time t def subjected to parametric uncertainty. Uncertainty for each property may be divided into three categories:

Stochastic uncertainty
Stochastic uncertainty is given if a non-deterministic value of an arbitrary property is approximated probabilistically, e.g. from a sufficient number of statistically independent experiments or field data. Hence, distribution functions are known or can be calculated and the parameter variation can be described with mean value μ and standard deviation σ. In this paper, no stochastic uncertainty is taken into account.

Estimated uncertainty
Estimated uncertainty is given if a non-deterministic value of an arbitrary property symbolized by is approximated from experience or literature. For each value, a lower limit − and an upper limit + can be specified, For the considered TMD and as an example, estimated uncertainty m T = [m − T ; m + T ] for the mass m T with a lower limit m − T = m T,0 − 0.1 m T,0 and an upper limit m + T = m T,0 + 0.1 m T,0 , with the tolerance of ±10% estimated by the authors. Lower and upper limits of other parameters are calculated accordingly. The stiffness k T is estimated by the authors to vary up to a maximum of ±15%. Damping b T of a TMD may vary up to the maximum of ±30% [18].
As for the resonant shunt, the maximum deviation of the piezoelectric coefficient d 31 varies up to ±10% based on [19]. The capacitance C p is affected by the transducer's geometry and the bonding to the beam. In general, only the relative permittivity at constant strain β S r,33 and at constant stress β T r,33 , i.e. mechanically blocked or free, are given by the manufacturer. However, a piezoelectric transducer bonded to a host structure is not represented correctly by neither of these constants. Uncertainty result-ing form material, bonding and geometry are condensed in the estimated uncertainty C p . The capacitance C p of piezoelectric transducer may vary up to ±20% (manufacturer: Physik Instrument -PI). The capacitance C 4 used in the gyrator circuit in Figure 2b may vary up to the maximum of ±5%, and a resistance R may vary up to the maximum of ±10%, (manufacturer: WIMA, Multicomp). The maximum deviation of inductance L is ±10% calculated with Equation (26) according to [20].

Unknown uncertainty
Unknown uncertainty is given if no declaration regarding any uncertainty is made and, in terms of parametric uncertainty, the parameter is considered to be deterministic. In that case, one assumed single value 0 of an arbitrary property symbolized by is specified and equivalence = 0 is given. For the resonant shunt and TMD approach, Figures 1 and 3, all geometrical parameters, such as the length l b and l p of the beam and the transducer, the width a b and a p of the beam and the transducer as well as the thickness h b and h p of the beam and transducer are considered to be deterministic. Furthermore, no uncertainty in the material parameters density ρ b and ρ p as well as Young's modulus E b and E p is taken into consideration.

Numerical simulation of vibration attenuation
In this section, the effects of estimated parameter uncertainty on the vibration attenuation performance of the clamped beam with resonant shunt and TMD are outlined. For the resonant shunt frequency response functions H sh (f ), (see Eq. (20)), are calculated for the five cases A1 to A5 (Tab. 2). In case of the TMD, frequency response functions H TMD (f ) (see Eq. (33)), are calculated for the four cases B1 to B4. In case A1, only the capacitance C p of the piezoelectric transducers is subject to assumed estimated uncertainty . For all other properties in Table 1, unknown uncertainty is assumed. In case B1, only the stiffness k T = [k − T ; k + T ] = [k T,opt,0 − 0.15 k T,opt,0 ; k T,opt,0 + 0.15 k T,opt,0 ] is subject to assumed estimated uncertainty. In similar ways, cases A2 to A4 for the resonant shunt and cases B2 to B3 for the TMD take into account estimated uncertainty only for the simple properties d 31 , L and R for the resonant shunt and m T and b T for the TMD separately. Cases A5 and B4 examine the influence of estimated uncertainty in C p , d 31 , L and R for the resonant shunt and in k T , m T and b T for the TMD simultaneously. The results are presented in Sections 4.1 and 4.2.
In Section 4.3 the vibration attenuation performance of resonantly shunted transducers for different dimensions and TMD for different masses m T,0 are compared. By varying the transducer length l p,0 and thickness h p,0 , different masses of piezoelectric ceramic m p,0 are given, as seen in Figure 1. The masses m p,0 and m T,0 are considered as additional masses with respect to the beam's mass m osc . Frequency response functions H sh (f ) and H TMD (f ) according to Equation (20) and Equation (33) are calculated for different additional masses. Furthermore, H sh (f ) and H TMD (f ) are calculated for different additional masses for cases A5 and B4. The vibration attenuation performance of both approaches is compared by the amplitude response |H sh (f b,1 )| and |H TMD (f b,1 )| at the beam's first eigenfrequency f b,1 and the maximum amplitude response |H sh | max and |H TMD | max within the range of the first eigenfrequency f OC,1 with open circuit and f b,1 , respectively.

Vibration attenuation with resonant shunt
The vibration attenuation under the influence of estimated uncertainty is described in cases A1 to A5, Table 2. According to Table 1, unknown uncertainty is assumed exemplary for a resonant shunt with l p,0 = 40 mm, C p,0 = 151.8 nF, h p,0 = 0.4 mm, L 0 = L opt = 29.1 H and R 0 = R opt = 3.7 kOhm. For all cases A1 to A5, the amplitude and phase responses for varying C p , d 31 , L and R are compared to amplitude and phase responses for C p,0 , d 31,0 , L 0 and R 0 . For simplification, the amplitude and phase response in Equation (20) are |H sh (f )| = |H sh | and ϕ sh (f ) = ϕ sh . Figure 5a shows the amplitude |H sh | and phase ϕ sh response for C p = [C − p ; C + p ] = [121.44 nF; 182.16 nF] compared to C p,0 = 151.8 nF if uncertainty is unknown. The capacitance C p may have three effects on the vibration attenuation performance. First, it affects the amount of charge induced in the transducers electrodes for a given deformation. Second, it affects the effective structural stiffness and, thus, the open circuit resonance frequencies. Third, it affects the resonance frequency of the shunt circuit and, hence, the tuning of the resonant shunt.

Case A1
C + p decreases the first structural resonance frequency and C − p increases the resonance frequency according to: The higher the piezoelectric coefficient d 31 , the higher the amount of mechanical energy converted to electrical energy is. Compared to the capacitance C p , it does not detune the shunt, but affecst the open circuit frequencies f OC due to stiffness change in K OC according to Equations (5) and (14a). Figure 5b shows an increase of amplitudes for d − 31 and a decrease for d + 31 , the change in the first resonance frequency f OC,d31 is negligible. There is no distinct phase shift in the resonance frequency either. Case A4 Figure 6b shows the amplitude |H sh | and phase ϕ sh response for R = [R − ; R + ] = [3.33 kOhm; 4.07 kOhm] compared to R 0 = 3.7 kOhm if uncertainty is unknown. The amplitudes increase for R + . For R − the amplitudes mainly decreases between the maximum amplitudes of |H sh |. The resistance R does not change the tuning frequency of the shunt, hence, R has comparable little influence on the attenuation performance and the phase response. The detuning to f − el shifts the −90 • crossing frequency in the phase response to a higher frequency, whereas the detuning f + el has minor influence on the phase response. Comparing the maximum amplitude with Figure 5a one can see, the increase of the maximum amplitude is dominated by C + p . The combination C + p , d − 31 , L + , R + represents the worst with the maximal vibration amplitude in the attenuated frequency range.

Case A5
The worst case combination C + p , d − 31 , L + , R + changes the tuning frequency of the shunt to f − el = 0.84f el,0 , resulting in the maximum detuning and amplitude. The detuning to f − el shifts the −90 • crossing frequency in the phase response to a higher frequency, whereas the detuning f + el has minor influence on the phase response. Comparing the maximum amplitude with Figure 5a one can see, the increase of the maximum amplitude is dominated by C + p .

Vibration attenuation with TMD
The vibration attenuation under the influence of estimated uncertainty is described in cases B1 to B4, Table 2. According to Table 1  responses |H sh | max (dashed line), for the worst case A5 with C + p , d − 31 , L + , R + . All maximum amplitude responses for the beam with TMD and varying μ T are shown in dark gray. The dark gray area is bounded by the maximum amplitude responses |H TMD | max (solid line) when unknown uncertainty is assumed for k T,0 , m T,0 and b T,0 and the maximum amplitude responses |H TMD | max (dashed line), for worst case B4 with k − T , m + T , b − T . Both approaches, resonant shunt and TMD, show asymptotic decreasing amplitudes with increasing additional masses. The electromechanical coupling coefficient K 31 (see Eq. (24)), and ,thus, the maximum amplitude response |H sh | max will change for different l p and h p and different mass ratios μ p of piezoelectric transducers, respectively. For each thickness h p one length l p exists with the coupling coefficient K 31 to be maximal and |H sh | max to be minimal that is shown only in Figure 10.
Resonant shunt and TMD (solid lines) achieve maximum amplitudes less than 20 % of the beam's maximum amplitude. However, the TMD achieves a higher vibration attenuation than the resonant shunt with the same additional masses. Usually [22], μ T is recommended to be less than 20%. For μ T ≤ 0.2, the maximum amplitudes |H TMD | max are significantly smaller than the maximum amplitudes |H sh | max . The maximum deviation of the resonant shunt is 2.5/0.83 |H sh | max ≈ 3 |H sh | max at μ p = 0.025 and 0.26/0.16 |H sh | max ≈ 1.6 |H sh | max at μ p = 0.6 in case of unknown uncertainty. For the TMD it is 1.2/0.4 |H TMD | max = 3 |H TMD | max at μ T = 0.025 and 0.14/0.1 |H TMD | max = 1.4 |H TMD | max at μ T = 0.6 in case of unknown uncertainty. The absolute deviations are smaller for the TMD but the relative deviations are comparable. However, vibration attenuation with TMD is less sensitive to the assumed estimated uncertainty. With increasing additional masses, i.e. the attenuation capability of resonant shunt and TMD increase, the uncertainty for both approaches decrease. With higher μ p and μ T the broad band attenuation increases and, thus, both systems get less sensitive to deviating tuning frequencies. Figure 11 shows the amplitude responses |H sh (f b,1 )| for resonantly shunted transducers and |H TMD (f b,1 )| for the TMD at the beam's first eigenfrequency f b,1 . Dashed lines show cases A5 with C − p , d − 31 , L − , R + and cases B4 with k − T , m + T , b − T . Compared to Figure 10, the amplitude attenuation (solid line) for the TMD does hardly change, whereas the amplitude attenuation for the resonant shunt (solid line) increases. This is due to the tuning method presented in Section 2.2, which does not lead to complete equalized amplitudes in the attenuated frequency range. Since the structure's stiffness and mass change with attached piezoelectric transducers the mechanical eigenfrequency f SC,1 increases (see Eq. (25)). Thus, the attenuated frequency range will also shift leading to an additional amplitude attenuation. The maximum deviation of the resonant shunt is 1.41/0.62 |H sh | max ≈ 2.3 |H sh | max at μ p = 0.025 in case of unknown uncertainty. For the TMD 0.62/0.39 |H TMD | max ≈ 1.6 |H TMD | max at μ T = 0.025 in case of unknown uncertainty. |H sh (f b,1 )| and |H TMD (f b,1 )| are similar from μ T = μ p = 0.3. This is still above the recommended maximum μ T = 0.2. Furthermore, the deviations in |H sh (f b,1 )| and |H TMD (f b,1 )| due to estimated uncertainty are smaller compared to the ones from Figure 10. As observed in Figures 7 and 9b, the maximum amplitude responses occur right and left of the tuning frequency, thus, the amplitude responses at f b,1 are less effected by a detuned attenuation systems. This effect is more distinctive for the resonant shunt since it is supported by the shift of f SC,1 and |H sh (f b,1 )| gets less influenced by uncertain parameters.

Conclusion
The effect of parametric estimated uncertainty on vibration attenuation of a beam clamped at both ends with (a) resonantly shunted piezoelectric transducer and (b) tunes mass damper TMD is investigated separately. It is shown, that both approaches, resonant shunt and TMD, achieve remaining amplitudes less than 20 % of the beam's amplitude. A TMD achieves higher vibration attenuation than a resonant shunt with the same mass that has to be attached additionally on the beam, either as mass of piezoelectric material for shunt damping or as seismic mass for the TMD. Furthermore, vibration attenuation using TMD is less sensitive to the assumed estimated uncertainty. With increasing additional masses and, hence, increasing attenuation performance, both systems get less sensitive to detuning due to uncertainty. Comparing the amplitudes at the first beam eigenfrequency, uncertainty decreases. Attaching piezoelectric transducers to the beam, increases the resonance frequency. This leads to further reduction of the amplitude response of the resonant shunt at the beam's eigenfrequency and the amplitude at the beam's first eigenfrequency gets less affected by estimated uncertainty. Further examinations will investigate resonant shunt damping with negative capacitance regarding parametric uncertainty and additional masses.