A Novel Structural Modification Method for Vibration Reduction: Stiffness Sensitivity Analysis with Principal Strain Application

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Introduction
For enhancing the passenger ride comfort of passenger vehicle, the requirement for reducing noise and vibration has increased gradually in recent years. Lightweight design is advocated for automobile production. Joost [1] showed that it can improve passenger vehicle fuel efciency by 6-8% for each 10% reduction in weight. In pursuit of further light weighting, a great number of thin plates need to be used in vehicle components. Out-of-plane vibration of these thin plates is the main source of sound radiation [2,3]. Vibrationproof and soundproof material can efectively reduce vibration and noise by attaching them to the thin plates, but they cannot be used for certain components, such as engine and drive components. Terefore, it is desirable to explore an efcient method to strike a balance between the lightweight design and noise and vibration performance of the structures. Experimental modal analysis (EMA) is an effective instrument for describing, understanding, and modelling the dynamic behaviour of a structure. EMA is considered reliable because it is based on input-output system identifcation, which allows validation of the estimated frequency response functions (FRFs) by coherence functions [4]. To date, considerable research studies based on EMA have been conducted for structure modifcation. Kim proposed a practical method to reduce a medium size test car's interior noise by using the experimental structuralacoustic modal coupling coefcient [5]. Terada and Yoshimura proposed a power spectrum sensitivity analysis for the noise reduction under operational condition without using input identifcation [6]. Ye et al. studied a systematic analysis methodology based on classical transfer path analysis for analyzing and reducing the low-frequency vibration of steering wheel [7]. Nakamura et al. proposed a stifness sensitivity analysis on a panel by using the angular displacement response estimated by a scanning laser doppler vibrometer [8]. However, it is hard to use the scanning laser Doppler vibrometer on a complex surface of general structures. So, we considered the possibility of replacing the scanning laser Doppler vibrometer. Te piezoelectric strain sensor is widely used in structural damage detection for measuring modal strain. Tsurumi et al. proposed a method to detect structural damage based on modal strain energy in 1993 [9]. Doebling et al. used piezoelectric strain sensors in methods of implementing state estimate feedback to aid in damage detection in smart structures [10]. Given its small size, excellent performance, and the ability to cope with complex surfaces, we thought of replacing the Doppler vibrometer with a piezoelectric strain sensor. In the previous research, Yamada et al. tried to conduct a sensitivity analysis based on strain measurement efectively to reduce the vibration of the structure. Te results show the efectiveness of Yamada's work in bending modes, but it is difcult to apply in twisting modes [11]. In this work, a stifness sensitivity analysis with principal strain measurement is proposed to decrease the out-of-plane vibration of structure in both bending and twisting modes. Te piezoelectric strain sensors were used to widen the applicable scope of sensitivity analysis. FEM simulation and experiment approach were conducted for confrming the proposed stifness analysis method. Te results verifed the efectiveness of the method.

Sensitivity Analysis Theory
Calculation of the derivatives of the target response with respect to the design variables is called sensitivity analysis. Te plus-minus sign of the sensitivity value represents the increment-decrement efect of the target response by changing the design variables. It enables us to determine the optimal modifcation locations on the target mechanical structure. By changing the design variables, e.g., stifness or mass, at an appropriate location, efective reduction can be achieved on the target response at a specifc frequency.

Stifness Sensitivity Analysis with Strain Measurement.
Te FRF between the excitation point f and the response point r will be changed by adding the local additional stifness Δk between points i and j on the structure as shown in Figure 1.
Te change in FRF G rf by local additional stifness is where Δk is the additional stifness, G is the compliance FRF, and the subscript "rf" indicates the response point "r" and the excitation point "f" relatively. When the addition stifness Δk is small enough, the stifness sensitivity S k can be estimated by When the research target is a plate-like structure as shown in Figure 2, the compliance FRFs for estimating sensitivity can be approximately replaced by angular displacement FRF, and stifness sensitivity can be expressed as where t is the distance from stifness modifcation surface to neutral surface of the structure.
Since it is difcult to measure angular displacement on the structure, application of strain measurement is explored in this work. Since the distance from the stifness modifcation surface to the neutral surface does not change during deformation, vertical strain can be ignored throughout the plate. It is assumed that the strains between points i and j are uniformly distributed and can be expressed as where b is the distance between points i and j, namely, the length of the stifener attached to the structure. Additional stifness Δk due to stifener is calculated by where a is the width of the stifener; ΔA is the added stifener section area; and Δt is the added stifener thickness. Substituting (4) and (5) into (3), the sensitivity analysis with respect to bending stifness by using the strain measurement can be formulated as In the condition of structure under dominant bending mode, since the modal maximum principal strains distribute along the direction vertical to the bending line on the surface and the minimum principal strains are too small to be ignored, Equation (6) is accessible (this will be verifed in the numerical simulation part). But when structure under dominant twisting mode, both the principal maximum and minimum strains cannot be ignored, so it is necessary to consider X and Y-direction strains since the principal strains are criss-cross distributed on the surface. Te sensitivity formulation is expressed as where the subscripts "x," "y" indicate the strain in x-direction and y-direction relatively. By the way, the strain value that can most accurately refect the sensitivity characteristics of a certain point is the principal strain. Te detailed calculation formula will be described in detail below. In fact, the sensitivity estimated by the above equations cannot directly evaluate the increment-decrement efect since its value is a complex number. To evaluate the increase and decrease of the FRF amplitude by sensitivity, the vector projection of sensitivity onto FRF is used to convert sensitivity value to a real number at a specifc frequency [12]. Te sensitivity projection is estimated by the following equation: where Re() indicates real parts, superscript "conj" indicates complex conjugates, and S is the sensitivity estimated by equation (6).

Mass Sensitivity.
It needs to be considered that as additional stifness increases, additional mass efects will also arise. To properly evaluate the proposed sensitivity analysis by using the strain measurement, mass sensitivity analysis should also be carried out. By calculating the derivatives of the out-of-plane compliance FRF with respect to additional mass, adding the stifener to the location with large mass sensitivity can be avoided. Mass sensitivity is estimated by [13] where ω is the angular frequency.

Modal Strains.
In this study, the strains of surfaces and the out-of-plane displacement w(x, y) of a thin plate with small defection can be expressed by equation (9) [14].
where z is the distance from the stifness modifcation surface to the neutral surface of the structure. Fourth-order polynomial expressions are conveniently used to defne the shape functions with 12 parameters as shown in equation (10).
Substituting coordinate values into (10), the constants α 1 to α 12 can be evaluated by the 12 simultaneous equations linking the values of out-of-plane displacement and rotations (w, θ x , θ y ) of each element.
List all 12 equations, and (10) can be written in matrix form as Shock and Vibration where C is a 12 × 12 matrix depending on nodal coordinates. Substituting (11) into (10), out-of-plane displacement can be expressed as where u e is a vector consisting of the nodal displacements and rotations of the element. Substituting (12) into (9), the strain vector at the element surface can be expressed by where B is the strain matrix of the element containing the second derivatives of the shape functions. Terefore, the modal strains ϕ e ε on the surface of the element can be expressed as where ϕ d e is modal displacement of the element.

Numerical Modal Principal Strains.
In the numerical approach, the r th modal principal (maximum and minimum) strains of the n th element (ϕ 1,2 ε ) nr can be expressed as [15] where (ϕ x ε ) nr , (ϕ y ε ) nr , and (ϕ xy ε ) nr are the r th modal strains of the n th element in the x-direction, y-direction, and shear strain. Tey can be achieved by equation (14).
Principal angle (a counterclockwise direction from x axis to principal strain axis) can be expressed as

Experimental Modal Principal Strains.
In the experimental approach, triaxial rosette strain gauge arrangement can be used. Te arrangement of three strain gauges is shown in Figure 3 [16]. Instead of conventional strain gauges, piezoelectric strain sensors are used for modal response measurement.
Te r th modal principal (maximum and minimum) strains of the n th element (ϕ 1,2 ε ) nr can be expressed as where (ϕ 1 ε ) nr , (ϕ 2 ε ) nr , and (ϕ 3 ε ) nr are the r th modal strains of the n th element in the 1, 2, and 3 directions, respectively.
Principal angle (counterclockwise direction from 1 axis to principal strain axis) can be expressed as In this work, three piezoelectric strain sensors are surface bonded to each candidate location on the structure. Te modal principal strains are calculated for sensitivity analysis to predict the increment-decrement efect of the target response by changing the stifener thickness. Also, the most efective arrangement for stifener is decided by the principal angle.

Numerical Simulation
To examine the validity of the proposed method, fnite element analysis is carried out, and the increment-decrement tendency acquired by sensitivity analysis with strain measurement is compared with the FRF changes by local change of thickness. Te numerical simulation is based on a thin plate model shown in Figure 4. Te length, height, and width of the plate are specifed as 400 mm × 500 mm × 3 mm. Te analysis model consists of 546 nodes with 500 elements. Te boundary condition is set to free-free. Te material of the plate and stifener is stainless steel SUS304. Young's modulus and Poisson's ratio are taken as 197 GPa and 0.3, respectively. Te density is taken as 8000 kg/m 3 . Te principal strain FRFs on each node are used to estimate stifness sensitivities. Te compliance FRFs on each node are used to estimate mass sensitivities. Te covering area of stifener is 20 mm × 20 mm. Te stifener thickness is set as 10 −1 mm.
Te acceleration FRF between the excitation point f and the response point r is shown in Figure 5. Based on the natural mode analysis results shown in Figure 6, a simple twist mode (mode order 7 at 23.98 Hz), a simple bending mode (mode order 8 at 30.19 Hz), a complex twist mode (mode order 14 at 114.13 Hz), and a complex bending mode (mode order 15 at 135.26 Hz) are chosen as target.
Te principal strain distributions of four natural modes are shown in Figure 7. Te maximum principal strains are shown in red, and the minimum principal strains are shown in blue. It can be seen that on the free edge of plate or at the node of some mode, there is a  smaller value of principal strain since the local transformation is smaller. Also, on the anti-node of some modes, there is a greater value since the local transformation is larger. In the case of twisting mode, in the greatest deformation region, the maximum principal strain is almost the same with the minimum principal strain. In the case of bending mode, in the greatest deformation region, the maximum principal strain is along the bending direction, and the minimum principal strain is almost zero. Stifness sensitivity analysis with principal strain measurement is carried out around the target peak frequency. Te selected frequency is taken at 23 Hz (simple twisting mode), 29.2 Hz (simple bending mode), 113.1 Hz (complex twisting mode), and 133.2 Hz (complex bending mode). Tey are 1 Hz less than each peak   Shock and Vibration frequency since the peak of FRF is close to the pole, the tendency and value of sensitivity are extremely changing, and the prediction accuracy of the sensitivity is limited. Te increment-decrement tendencies acquired by stifness sensitivity analysis are shown in Figure 8. Te decrement tendencies are shown in blue, which means that the target FRF will decrease by attaching stifener at each quad element. Te FRF changes at four chosen frequencies by local thickening are shown in Figure 9. Whether the chosen peak frequency corresponds the bending modes or twisting modes, the FRF changes are essentially consistent with the tendency acquired by sensitivity analysis at four chosen frequencies. Te increment region as shown in red cannot be seen in the stifness sensitivity results due to the increment caused by mass attachment. It can also be confrmed by the mass sensitivity results shown in Figure 10.

Experimental Validation
Te validity of the proposed method is also examined by an experimental approach. We used the same stainless plate model in numerical simulation as shown in Figure 11. Te length, height, and width of the plate are specifed as 400 mm × 500 mm × 3 mm. Te boundary condition is assumed to be free-free: the specimen is softly suspended. FFT spectrum analyzer is used to obtain the FRFs between the strain response and force excitation in the frequency range of 0-800 Hz with frequency resolution of 0.6125 Hz. An impact hammer (086C01, PCB) is used to apply the impact force to the plate at excitation point, and a piezoelectric strain sensor (740B02, PCB) and accelerometer (352C65, PCB) measure the response at 5 candidate locations (shown in green dot) and one response point (on the bottom-right corner). Limited by the amount of strain sensors, the 6 Shock and Vibration experiment was carried out in two arrangements. Te detailed layouts are shown in Figure 12. Te experiment data in two layouts were used to calculate the maximum principal strain FRFs on 5 locations and then used to calculate the stifness sensitivity. Stifness sensitivity analysis with principal strain measurement was also carried out at 1 Hz less than each peak frequency. Te selected frequency was taken at 22.9 Hz (simple twisting mode), 29.8 Hz (simple bending mode), 113.5 Hz (complex twisting mode), and 141.2 Hz (complex bending mode). Te experimental results are shown in Figure 13. It can be seen that the results are basically consistent with the simulation results. To have a better performance in all four modes and considering the infuence of mass attachment, candidate location 3 is chosen to add a stifener.
Te stifener size is 80 mm × 40 mm × 0.8 mm, and it is made by stainless steel SUS304. It was attached to the plate as shown in Figure 14 by metal adhesive (Devcon A). Target FRF before and after modifcation is shown in Figure 15. Te result shows an overall peak reduction at 4 selected frequencies, which is consistent with sensitivity analysis results. But it also should be pointed out that the metal adhesives might cause a considerable attenuation efect on the peaks over 70 Hz. In order to minimize the attenuation efect, the target FRF before and after modifcation is processed by Shock and Vibration curve ftting with same modal damping ratio as shown in Figure 16. Te result shows an overall peak shift to higher frequency range by stifener attachment, and this also makes a downward trend at 4 selected frequencies marked at frequency axis. Te most remarkable peak shift and decrement happened at third selected frequency (113 Hz) since  the stifener efectively increases local stifness. Te frst selected frequency (22.9 Hz) and fourth selected frequency (141.3 Hz) also show an obvious peak shift and decrement, while there is no appreciable peak shift and decrement at the second chosen frequency (29.8 Hz) because the efects of additional stifness and additional mass cancel each other out at location 3. Overall, the experimental results are basically consistent with the sensitivity results. It can be efectively applied to fnd the modifcation location on the structure to make a remarkable decrement of response at the target frequency. Tere will be a further reduction efect if consider the damping efect of metal adhesives.

Conclusion
Tis paper introduced a stifness sensitivity analysis with principal strain measurement to decrease the out-of-plane vibration, which is the main cause of the sound radiation of mechanical structures. Te validity of the proposed method was examined through numerical simulation with a FEM model of the plate structure. Te modal principal strain distribution based on twisting mode and bending mode was discussed. Stifness sensitivities were calculated and checked with FRF changes by local thickening at 4 selected frequencies. It was also examined by the experimental approach. Te expected reduction of the response is attained by adding the stifener (a thin stainless plate) to the appropriate location on the plate. In summary, the simulations and experiments validated the correctness of applying principal modal strain to sensitivity analysis. Terefore, it can be used as a quick optimization design tool for mechanical structures with plate components.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest. Shock and Vibration 9