Position Optimization of Passive Patch Based on Mode Contribution Factor for Vibration Attenuation of Asymmetric 1D Structure

,


Introduction
1.1.Research Background.In cases where resonance cannot be avoided in models with low loss factors, viscoelastic materials such as rubber are used to reduce mechanical vibration or noise.Passive patches are used when the structure to be designed has weight or space limitations.Tey are currently being used in a variety of felds such as the aerospace, marine, construction, and automotive felds [1].To achieve a light weight, the thickness of the structures is reduced, and the damping efect of passive patches can be clearly observed.Te passive patch also has a simple structure and is cost-efective.Additionally, research on beam structure analysis prior to patch application is being actively conducted.Baran et al. [2] investigate the infuence of the Adomian decomposition method (ADM) and diferential transform method (DTM) on the free vibration of Timoshenko beams and analyse the efects on variables such as axial compressive load and ground reaction force by considering boundary conditions.Te results of DTM and ADM show excellent agreement, utilizing the dynamic stifness method (DSM) to verify the mode shapes and highlighting its applicability to free vibration of beam assembly structures resting on a viscoelastic basis.Additionally, the natural frequencies and harmonic response of the cracked frame were analysed using the transfer matrix method (TMM), single variable shear deformation theory (SVSDT), and Timoshenko beam theory, and it was shown that TMM can be used for simple and efcient analysis [3][4][5].
Te application of passive patches is divided into two categories: constrained attenuation with elastic restraint and viscoelastic layers and nonconstrained attenuation with only viscoelastic layers.In general, the damping efect of the constrained damping technique is large.In this technique, the vibration energy dissipation due to the shear deformation of the viscoelastic layer induces attenuation [6].Te technique provides the greatest attenuation when used over the entire range.However, a passive patch of a reasonable size is more efective in terms of cost and design.Hence, the patch position is mainly determined by considering the mode with a large contribution.However, in order to achieve great efciency using minimal passive patches, vibration analysis must be done to determine patch locations where vibration attenuation for various modes can be obtained.
Passive patches can achieve attenuation efects over a wide frequency range.Although a good attenuation efect can be obtained in the high-frequency band, the attenuation performance is insufcient in the low-frequency band.Various methods, including the Rayleigh-Ritz method, have been investigated for modeling passive patch viscoelastic materials [7,8].In addition, various methods have been studied for modeling viscoelastic materials used in passive patches.Tese include an approach for modeling passive patches for plates and beams using the Rayleigh-Ritz method to estimate the natural frequencies and loss coefcients quickly [9][10][11][12][13].Modeling of harmonic excitations is proposed when a passive patch and an active patch are installed [14].In such a case, it is necessary to optimize the position of a small passive patch before applying the active patch.In the case of passive patches, the result is afected not only by the position of the patch's bending shape function but also by the loss factor of the viscoelastic layer, the shear coefcient, thickness, length, elastic modulus and thickness of the lower object to be controlled, and the length of the passive patch [13].Tus, the efects of the previous parameters on the design of passive patches have been studied [13,15,16].
To optimize the position of a passive patch, Zheng et al. minimized the length and position of the patch by using a genetic algorithm based on the penalty function method [17,18].Lei and Zheng have optimized the passive patch location through topological optimization of the penalization model [19,20], and Fang has used the level set method [21].Araujo et al. performed optimization using the Feasible Arc Interior Point Algorithm (FAIPA) to derive the maximum loss factor [22].El Hafdi et al. optimized patch location and improved algorithm convergence through genetic algorithms and Latin Hypercube Sampling (LHS) algorithms [23].Askar et al. used a genetic algorithm to optimize the position of the circular aluminium patch [24].Te literature review indicates that various algorithms have been used to optimize the position of the passive patch.However, a simple and quick design method is required to determine the optimal position of a passive patch in real industrial felds.

Objectives.
In this paper, a method to optimize the position of a passive patch with a certain length and thickness is introduced to 1D structures, especially for use with cantilevered beams.First, the method of obtaining the frequency response, natural frequency, and mode loss coefcient is described, which is based on the analytical model for beams with passive patches expressed by complex stifness.Next, the change in the natural frequency and the loss factor for each mode of the passive patch of a certain length are examined.
Although the position of the passive patch can be determined using the above method, it would be simpler to use the beam shape function.Considering the wide frequency range, the sum of squares using the bending shape function of the beam is obtained together with the modal contribution by the specifc excitation range.Subsequently, the position of the appropriate patch is determined and then verifed through experiments using the predicted results.
Tis paper is organized as follows.Section 2 describes the approximate model of the existing passive patch, and Section 3 analyses and explains the characteristics of the passive patch through simulation.Ten, the optimal position of the passive patch is determined by using the sum of squares of the proposed bending shape function, and the performance of the designed passive patch is analysed.Finally, Section 4 discusses conclusions and future plans.

Beam with a Patch.
A cantilever beam model with a passive patch is shown in Figure 1.
In a thin Euler beam of fxed length L, a passive patch of short beam shape with length L x consisting of a viscoelastic layer and a confning layer is attached at position x p .Disturbance acts in the form of harmonic function at position x d , which has 5 cm from fxed point, where the length L of the beam is assumed to be sufciently longer than the width L y of the beam.As the beam is thin, the deformation in the thickness direction is neglected and the material of the beam is assumed to be isotropic, homogeneous, and linear elastic with the same properties in all directions.
Here, the lower beam is defned as layer 3, the viscoelastic layer as layer 2, and the constraint layer as layer 1. Figure 2 shows the deformation due to bending of each layer.Te subscript denotes the layer number, w is the transverse displacement, u is the longitudinal displacement, ψ is the rotation angle, and c is the shear angle.

Displacement
Vector.Te motion of each layer was investigated by Kung and Singh [5,6].Te displacement vector r representing the motion of each layer is defned as the following equation (1).Te superscript p represents the number of the layers.
Te displacement vector r is assumed to be temporally and spatially continuous.It is assumed that the lateral displacement w in layers 1 and 2 is equal to the displacement 3 ), the shear angle is assumed to be negligible (c 1 , c 3 ≪ c 2 ) as shown in Figure 2, which is much smaller than the viscoelastic layer.Terefore, the total angle of rotation at layers 1 and 3 is equal to the partial diferential of the bend (ψ 1 � ψ 3 � zw/zx).Te rotation angle in layer 2 is defned as ψ 2 � zw/zx − c 2 , which is the diference between the partial derivative of the bend and the shear angle.
Let us assume that the bending displacement w in the orthogonal coordinate z direction at an arbitrary position xon the vibrating beam can be expressed by the sum of the terms multiplied by the shape function ϕ w,k (x) of the k-th mode and the weighting function q k (t) as follows.Te continuous system has infnite degrees of freedom, but this problem has N degrees of freedom if it is sufcient to consider it as a linear combination of the product of the N shape functions and the weighting factors.At this time, the lateral displacement w is expressed by the following equation (2): Here, q k is a weighting function for each mode.Both the disturbance and the response are assumed to be a harmonic function of a certain size.Φ w is a shape function vector of size 1 × N and q is a weight vector of size N × 1. Trough these two functions, the shape function S can be organized as follows: Φ w (x) � ϕ w,1 (x), . . .ϕ w,k (x) . . .ϕ w,N (x)  , q(t) � q 1 (t), q 2 (t), . . ., q k (t), . . ., q N (t) (3)

Mass Matrix and Stifness Matrix.
In the case of inertia matrix H, it is a matrix derived in the process of defning the mass matrix.First of all, kinetic energy is defned by shape function and weight as follows: In this equation, the generalized mass is m kj �  ϕ w,k (x, y)ϕ w,j (x, y)dm and the integration is performed over the bin length.If it is a plate of ρ with a thickness of h in a continuous system made up of infnitesimal masses, it can be expressed as dm � ρbhdx.For the rotational displacement ψ(x, t), the infnitesimal mass is dm � ρ(bh 3 /12)dx and the generalized mass using integration can be expressed as follows: Undeformed Deformed

Viscoelastic layer
Lower beam

Shock and Vibration
(5) Using the shape function vector, the mass matrix can be expressed as shown, and the inertia matrix H is defned based on this equation: Te mass matrix M of size n × n is in the form of a superposed mass matrix using the inertia matrix H and the shape function matrix S of size 3 × N.
Here, the inertia matrix H is expressed by equation (10).
Tus, the kinetic energy of equation ( 11) is fnally expressed as follows.Te relationship between the displacement vector r and the generalized displacement vector q can be expressed with the matrix of shape functions S(x) [9].
In the case of strain energy (potential energy), the displacement is restored by the deformation in the vibrating beam and converted into motion.In the case of a linear elastic body, the energy is expressed as the product of the force and the strain, such that added by the energy density to the entire beam becomes the total strain energy of the beam.Te integral of each energy density equation is integrated over the length, and the displacement vector r is summarized as follows: Here, the diferential operator D and the elasticity matrix E are as follows [9]: Equation ( 6) is summarized as follows through equation ( 2): Here, the stifness matrix K is as follows: For each layer, both the mass matrix and the inertia matrix are added to obtain the kinetic energy and strain energy of the whole, and the integral range corresponds to the length of each layer.Te mass matrix M and the stifness matrix K for the entire layer are defned as follows: By substituting equation (17) into equations ( 11) and ( 14), the kinetic energy and strain energy for the whole system are calculated.

4
Shock and Vibration 2.4.Approximation.When an object is assumed to have N vibration modes, its degree of freedom is N. Since the beam with several passive patches has a multiple number of subparts including base layer, adhesives, and patches, the total degree of freedom would be increased by the multiple of part numbers.Terefore, to simplify the problem, it is necessary to reduce the degree of freedom for calculation.
Here, it is assumed that the motion of the passive patch is determined with respect to the bending shape function of the base layer.First, in the case of the base layer, the Euler beam is fxed on one side.For the beam, the eigenvalue can be analytically calculated through the characteristic equation and then the normalized shape function can be obtained.
For the remaining shape functions, the Rayleigh-Ritz method using strain energy can be expressed as a linear combination of the permissible function satisfying the boundary condition, or the natural frequency can be approximated without solving the complex eigenvalue problem through the weak-core hypothesis.
First, the relationship of motion between layers is considered a weak-core assumption, [21] is applied as the modulus of elasticity of the middle viscoelastic layer, it is smaller than that of the beam, and the constraint layer in the structure of the laminated layer in which the axial load is applied.Te longitudinal displacement of layer 1 and layer 3 has the following relationship: Integrating equation ( 19) with x yields the following: where d p k is a constant for indicating the relationship of the shape function.Next, the longitudinal displacement u 1 , u 3 and the rotational angle ψ 1 , ψ 3 in layer 2 can be expressed using the longitudinal displacement u 2 and the rotational angle ψ 2 of layers 1 and 3. Te relationship is represented in Figure 3.For each passive patch, the shape function for each mode can be expressed as equations ( 21) and ( 22): 2.5.Natural Frequency and Loss Factor.Te natural frequency and mode loss factor for each mode can be obtained from the strain energy and kinetic energy obtained previously.First, the loss factor is expressed as complex stifness in the stifness matrix.Te complex modulus of elasticity is expressed as E * � E(1 + jη).Here, * denotes a complex value.In the case of the viscoelastic layer, the shear factor and the loss factor η depend on the frequency.Terefore, the modulus of elasticity is expressed as E * 2 (ω).Te total strain energy and kinetic energy written in complex stifness are as follows: At this time, as the response is assumed to be a harmonic response, the complex eigenvalue problem in free oscillation can be written as follows: In this case, as the stifness matrix has a frequency dependent characteristic, the eigenvalue converges through iterative calculation.In equation (24), λ * k represents the complex eigenvalue.Te natural frequency and loss coefcient can be defned from the complex eigenvalue λ * k .

Forced Vibration by Disturbance.
When the disturbance is applied to a point of the beam, it is assumed that only the bending is generated and the disturbance vector Q is defned as follows: Shock and Vibration where F represents the magnitude of the disturbance for each mode.Tis is nonconservative and is added to the right side of the equation of motion in equation (24).At this point, by calculating the weight vector q, the displacement of the beam can be calculated, and the displacement of the remaining layers is also determined.
When disturbance in the form of harmonic function acts on a point x d , F(x, t) is expressed as follows: Ten, the disturbance vector Q is calculated, and equation ( 26) is summarized as follows: Here, the Fourier transform can be taken to calculate the frequency response. (29)

Simulation and Experiment
3.1.Loss Coefcient Variation.First, simulation was performed on the length of a passive patch in order to check how the loss of each mode changes depending on the position of the passive patch.Te material, properties, and shape of the lower beam and passive patches were determined as shown in Table 1, with reference to the viscoelastic layer used in Plattenburg et al. [12].Te natural frequencies calculated for beams without passive patches are given in Table 2. Te frst target is the vibration attenuation for the fve modes of the beam, with the patch designed to attenuate vibrations of 0-1000 [Hz].Te length L of the beam will be normalized, and other parameters will be expressed for the length of the beam.Te normalized mode shape of the lateral vibration of the cantilever beam is shown in Figure 4.As the shape function does not change even when the patch is attached, the design of the patch depends on the mode shape [25].
When the mode shape of the beam is viewed, the shortest distance between nodes is mode 5 and the distance corresponds to a normalized length of 0.2.Terefore, the normalized length corresponding to 1/2 of the distance between nodes is determined as 0.1.
Tis condition enables the simulation of the manner in which the loss factor for each mode is afected by the position of the patch.Te position of the patch has a fnite length, so it shows when the position changes from 0.01 to 0.95 from the normalization position of 0.05. Figure 5 shows the second derivative of the mode shapes.For each mode, the correlation between the magnitude of the second-order diferential absolute value |z 2 w/zx 2 | of the normalized bend shape and the loss factor can be observed by plotting these on the graph simultaneously.
In Figure 6, the loss factor for the 1 st and 2 nd modes and the curvature of the beam are shown for example.Similar trends have been observed for 3 rd , 4 th , and 5 th modes as well.It is confrmed that the magnitude of the absolute value of the second derivative w ″ of the mode shape is proportional to the magnitude of the second derivative w ″ .Te second-order derivative w ″ of the mode shape is the curvature κ of the beam, and the larger the size, the more efective the attenuation of the passive patch.Te reason is that the larger the curvature κ of the beam, the greater the magnitude of the moment acting on the beam.Further, the strain on the beam surface increases and afects the motion of the patch.When the curvature κ is seen, the fxed part of the beam has the largest value.However, as it is difcult to attach a patch having a fnite length to the fxed portion, it is excluded from the attachment position of the passive patch.
Figures 7 and 8 show the variation of the natural frequency according to the position of the passive patch.Additionally, Table 3 summarizes the natural frequency changes according to the patch location.It is observed that the value of the natural frequency changes greatly as the degree of the mode increases.In addition, when the passive patch is installed near the fxed end, the natural frequency is observed to be high in all modes, whereas when it is attached near the free end, it is confrmed that the natural frequency is lowered.

Optimal Patch Location.
Te infuence of the curvature of the beam was confrmed.Tus, the values for the curvature of the beam are utilized to design the patch.Te position x d in the beam does not change the mode shape.6

Shock and Vibration
However, the contribution of each mode changes.As the mode contribution Γ k of each mode represents the degree of contribution of the mode and the attenuation across the beam is aimed at all the frequency bands, the mode contribution is determined only by the exciter, and unlike the weight vector q defned above, it is defned as in equation (30).If a frequency band is mainly used, the mode contribution should be estimated considering only the modes within the frequency band.
If the value of the excitation magnitude F(x) is a constant probability in the range of 0.05 to 0.1, the modal contribution for each mode is calculated as given in Table 4.
For mode contribution, mode 4 and mode 5 are large.Each mode contribution is weighted and multiplied by the second derivative ϕ ″ w,k of the mode shape to be squared and then added.Ss denotes the sum of squares.
Using the mode contributions calculated in Table 4 and the values in Figure 5, the sum of squares of the shape functions can be plotted as shown in Figure 9.In this case, the position of the passive patch can be selected as 0.19, 0.41, 0.59, 0.82 except for the normalized positions 0 and 1 given the length of the patch is fnite.However, as the size is largest at 0.82, the optimum patch position is determined to be 0.82.In order to compare the efect of the patch position, two cases of passive patches with the normalized length of 0.82

Shock and Vibration
and the low value of 0.5 are compared to predict the performance of the passive patch by simulation.In this case, 0.5 is selected because passive patches cannot be installed in the case of 0.08.

Simulation of Attenuation Performance.
Te passive patch attenuation performance at the two selected positions is simulated by the loss factor and the reduced peak size.First, loss factors for each mode at two locations are compared.Te loss factors at each location are given in Table 5.
Next, the frequency spectrum predicted from 0 to 1000 Hz at the observation point is shown.Te observation point was selected as the normalization position of 0.375.
Figure 10 shows the simulation of the accelerance from equation (29) for undamped beams and passive patches in two selected cases.Te loss tends to be the same as the result of the loss factor, and it can be confrmed that the peak size is 0.82 at the resonance frequency except for mode 2. In addition, it is confrmed that a large attenuation is invoked in modes 3, 4, and 5 when a passive patch is attached at a position of 0.82.In the case of the natural frequency, it is confrmed that the stifness and the mass of the system are changed by attaching the passive patch, and the natural frequency is decreased accordingly.Tat is, it can be seen that the efect of mass is greater than that of stifness.Te natural frequency does not drop much, but if the excitation at that frequency is large, the passive patch   Shock and Vibration should be avoided.Te size of the reduced peak produced by attaching the passive patch is given in Table 6.Te data written in the table indicates the decibel size before attachment and at position 2 and the reduction amount at the resonance frequency corresponding to each mode.Terefore, the frequencies at the compared peaks are all diferent.
In Table 6, the reduction of the peak is compared with the resonance frequency in one mode, and it is confrmed that attenuation is large when the loss coefcient is large, except for mode 1 (with the patch at 0.82).In Figure 10, it can be observed that mode 1 has relatively small amplitude compared to the other modes and the amplitude level in the time domain is not much afected to those modes.However, if one passive patch is compared with the other modes, the loss factor is not necessarily large and the peak decrease is not always large.

Experimental Validation.
Next, the efect of the position of the passive patch in the two selected cases was verifed by experiments.Figure 11 shows the experimental equipment.An accelerometer is mounted on a thin beam fxed to the vise, and an impact hammer is used to generate the impulse.Te signal from the accelerometer is received from the data collector and displayed on the monitor through signal processing.Te sampling frequency is 2000 Hz, observed from 0 Hz to 1000 Hz in total, and the frequency resolution is 0.1 Hz.
Figure 12 shows the position of the beam on the beam and the observation position of the response.In the case of excitation, it is assumed that a constant probability distribution is applied in the range of the normalization length of 0.1.Terefore, the excitation position is divided into three parts, and averaging is performed.Figure 13 shows the attached passive patch.
Figure 14 shows the acceleration measured by the accelerometer.In this case, it is possible to observe that a diferent peak appears from the simulation.Tis is because the length is longer than 10 times the width, but other modes such as twist have also appeared in the region above 250 Hz.If the same results with simulation need to be obtained through the experiment, the system should be modelled by a thin plate model.In addition, the clearly appeared modes besides bending modes are because of the contact condition of beam and vise, which is an incomplete supporting condition exciting other structures such as vise, vibration table, etc. Te yellow circles indicate the bending mode, and it is confrmed that the frst to ffth modes are displayed in order.As with the simulation, it is confrmed that the natural frequency of each mode is reduced by attaching the passive patch.
Also, Table 7 lists the peak magnitude at resonance frequency in decibels for each mode and the attenuation performance when attached at the normalization position of 0.5 compared to the normalization position of 0.8 for mode 1 and mode 2. However, for modes 3, 4, and 5, it is confrmed that the passive patch attached to 0.82 shows much better attenuation performance.As the mode contribution is large     10 Shock and Vibration in modes 3, 4, and 5, it is targeted mainly at the high modes, and the result is also the result of the experiment.In mode 4, it is confrmed that large attenuation occurs.Tis is because the resonance frequency of the twist mode near 500 Hz, which is the resonance frequency of mode 4, is reduced by about 30 Hz with the passive patch attachment.
Figure 15 compares the peak-to-peak values obtained through experiments and simulations.For the positions of each passive patch, the experiment and the simulation are similar in the remaining areas except for mode 1 and there is a larger decrease in the value of the peak in the experiment.Tis is presumably because the passive patch reduces not only the bending mode but also the other modes.Further, in mode 1, the attenuation in the simulation was large.In the experiment, the resonance frequency of mode 1, which is a small signal, is not well represented by the noise.Figure 16 shows the correlation between the loss factor and attenuation levels in dB scale obtained from simulation and experiment, which obviously shows the same trend except for mode 1 from the simulation.
Figure 17 shows the frequency spectrum on a linear scale.Tis metric can be used to determine the RMS value for measuring the overall vibration attenuation, and its value is given in Table 8.Comparing the case where the passive patch is not attached and the case where the passive patch is attached at 0.5 position, it is confrmed that the vibration of about 27% is totally reduced.
In addition, it is confrmed that the size of the RMS is further reduced to 41.9% when it is attached to the normalization position 0.82.Amplitude attenuation occurs not only in bending but in other modes as well.When the peak and RMS size are reduced, the design of the passive patch is considered appropriate.Shock and Vibration

Conclusion
To optimize the position of the passive patch, the sum of squares of the bend shape and the position of the passive patch are used.Further, the efect of the passive patch is verifed by using the existing passive patch model.First, the model expression of the passive patch is examined.Te relationship between the passive patch and the beam is investigated in detail, and the loss coefcient and the frequency spectrum due to the natural frequency and the forced vibration in the presence of several passive patches are numerically simulated.
Next, in order to determine the position of the passive patch, the change in the loss coefcient and natural frequency according to the position is examined.It is confrmed that the factor is dependent on the curvature which is the second derivative of the bending mode shape.Te position of the passive patch is determined through the sum of squares value expressed as a linear combination of mode shape and mode contribution.Passive patches of constant length and thickness are designed in this way.Weighting is applied to the case where the mode contribution is large so that the attenuation efect is enhanced in a mode in which a large response is expected.It is confrmed that the natural frequency is lowered for each mode due to the passive patch changing the vibration system.
In order to verify the performance of the positionally determined passive patch, two cases are compared.It is assumed that one passive patch is designed to have a constant length on one cantilevered beam.In this case, the passive patch attenuates the overall vibration, and the peakto-peak value also decreases regardless of position.However, it can be seen that the size of the response is greatly reduced in the case of the passive patch, determined by obtaining the sum of squares using the shape function.Experiments were conducted to verify the efectiveness of passive patches.Experimental results show that the frequency response in mode 1 is small and difcult to observe.Because the experiment involves a three-dimensional mode of vibration, other peaks also appear, which is diferent from the simulation.Te attenuation efect on the bending of the passive patch was not accurately observed, while the natural frequency was shifted by the passive mode such as the twist mode.
Future research should investigate applying the passive patch design method to the actual vibration system by the excitation force of various frequency spectrums.In this study, a certain size is assumed and used to verify the damping efect using the proposed method considering the mode shape and frequency of the actual model.
In addition, it is necessary to study the optimization method of the position design in installing passive patches as well as active patches using the piezoelectric element.Passive patches are efective at high frequencies, so weights can be applied to achieve optimal vibration damping along with active patches.
Te passive patch location design method proposed in this paper is expected to contribute to the improvement of the practical application of passive patches without the need for complicated processes.In addition, if applied to real machine systems with active control, this method will contribute to the research corpus on the topic of next generation vibration reduction systems.

2
Shock and Vibrationw in the lower layer.As the shear rate of layers 1 and 3 is larger than that of layer 2

Figure 2 :Figure 1 :
Figure 2: Before and after deformation of passive patch structure.

Figure 4 :
Figure 4: Five mode shapes of beam.

Figure 5 :
Figure 5: Te second derivative of the beam's fve mode shapes.

Figure 6 :
Figure 6: Te loss factor for the (a) 1st mode and (b) 2nd mode and the curvature of the beam.

Figure 15 :Figure 16 :Figure 17 :
Figure 15: Comparison of peak-to-peak attenuation in the dB scale (simulation vs. experiment) corresponding to the patch location of (a) 0.5 mm and (b) 0.82 mm.

Table 1 :
Te physical properties and shape of each layer.

Table 2 :
Natural frequency of beam.

Table 4 :
Mode contributions for each mode.

Table 3 :
Natural frequency change according to patch location.

Table 5 :
Loss factor for each mode.

Table 6 :
Peak-to-peak of each mode for manual patching in both cases.

Table 7 :
Peak-to-peak and size reduction for each mode for manual patching in both cases.

Table 8 :
RMS reduction for manual patching in both cases.