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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (

This work presents a hybrid (experimental-computational) application for improving the vibration behavior of structural components using a lightweight multilayer composite. The vibration behavior of a flat steel plate has been improved by the gluing of a lightweight composite formed by a core of polyurethane foam and two paper mats placed on its faces. This composite enables the natural frequencies to be increased and the modal density of the plate to be reduced, moving about the natural frequencies of the plate out of excitation range, thereby improving the vibration behavior of the plate. A specific experimental model for measuring the Operating Deflection Shape (ODS) has been developed, which enables an evaluation of the goodness of the natural frequencies obtained with the computational model simulated by the finite element method (FEM). The model of composite + flat steel plate determined by FEM was used to conduct parametric study, and the most influential factors for 1st, 2nd and 3rd mode were identified using a multifactor analysis of variance (Multifactor-ANOVA). The presented results can be easily particularized for other cases, as it may be used in cycles of continuous improvement as well as in the product development at the material, piece, and complete-system levels.

The vibration behavior of a structural panel is an important factor in the operation of a machine or structure. A design that does not consider the vibration behavior can cause malfunctioning, noise, discomfort, and/or injuries to the user and even failure of the ensemble. Experimental modal analysis (EMA) is a technique that enables the determination of the resonances of a machine, identification of its modal parameters, and prediction of its vibration behavior [

ODS analysis has already been successfully used for detecting damage in beams, bridges, turbines, machines, and various types of structures [

In the present work, a hybrid (instrumental-computational) approximation based on a finite element computational model (FEM) is proposed, along with an instrumental technique that applies ODS analysis. This approximation allows for experimentally estimating the modal response of the system (mainly the natural frequencies) under determined test conditions, thereby validating the resonance frequencies obtained by the FEM simulation model. To demonstrate the validity of this approximation, the improvement of the vibration behavior of a thin steel sheet is proposed. Thin steel sheets are of great interest to the design of engineering structures such as car bodies, ships, aerospace structures and building structures [

Generally, an undesired vibration level is corrected by decreasing the modal density and displacing the natural frequencies, taking them out of the operation range, increasing the damping and/or the mass, or increasing the rigidity of the ensemble. In the presented practical application, the goal is to reduce the modal density to the interval of 0 to 150 Hz and to increase the first resonance frequency to values above 70 Hz. To increase the resonance frequencies, rigid elements can be added to the vibrating surface, or the geometry can be modified by moldings or curvature changes. Different studies present the results of these strategies, which, although providing a good solution to the vibration problem, can involve increased weight and cost and may even cause manufacturing problems or lead to important modifications in the design [

The present study differs from previous ODS studies [

This section describes on the one hand the methodological approach and on the other hand the tested samples, the experimental test method, and the finite element calculations performed for the practical case.

The applied methodology (

First (step 1), the material of the system to be improved and its vibration behavior is characterized at a test piece level. The material characterization will be used as input data in the computational model. The results of the experimental test will serve to evaluate the validity of the computational model simulated by finite elements (FEM). In this manner, two objectives are reached. On the one hand, the hybrid experimental-computational work methodology (ODS-FEM) is validated; on the other hand, there is a reference for the vibration behavior of the initial system (see step 1: current state,

With this reference, different proposals are given for the improvement of its vibration behavior. With the chosen proposal/s, a computational model is generated, and the experimental tests are performed in the same manner as in the previous step (see step 2: improvement proposal in

In the presented practical application, the aim is to improve the vibration behavior of a steel sheet with a thickness of 0.8 mm, a density of 7700 Kg/m^{3}, and dimensions of 540 × 340 mm^{2}. The plate is fixed in a frame by 16 bolts to a rigid cabin (

To improve the vibration behavior of this plate, a multilayer composite with a thickness of 14 mm, a density of 69.8 Kg/m^{3}, and dimensions of 540 × 340 mm^{2} is glued to the plate. This element is composed of an open-cell polyurethane foam core and two paper mats on its extremes. To glue the plate to the composite, five strips of a polyurethane adhesive commonly used in the manufacturing of vehicles have been used. The plate and composite assembly is fixed in a frame to a rigid cabin under the same conditions as those of the test conducted with the previously described plate. This composite enables the natural frequencies to be increased and the modal density of the plate to be reduced, moving about the natural frequencies of the plate out of excitation range, thereby improving the vibration behavior of the plate.

Instead of performing an EMA, an ODS analysis has been used to estimate the modal characteristics of the tested samples. The modal deformations and ODS are different, but the relationship between both can be used to obtain the modal characteristics (resonance frequencies, damping factors, and modal deformations) by measuring the ODS without the necessity of knowing the excitation force [

Out of all the existing techniques used to obtain the ODS, transmissibility measurements can be the easiest to perform due to their similarity with the Frequency Response Function (FRF). This advantage is limited by the fact that it is generally not possible to identify modal parameters with transmissibility measurements [

To be able to apply transmissibility measurements to estimating modal parameters with some guarantee of success, the transmissibility coherence must be high [

A rigid cabin was used to perform the ODS experimental measurement, in which the samples were fixed with the aid of a rigid frame. Using 16 bolts, the sample was jointly fixed to the frame and to the structure that transmits the excitation force. The movement of the sample to be tested in the frame was completely restricted (

Transmissibility measurements were performed with two acceleration signals at 45 points of each test sample, using as a reference the central spot of the frame in which the plate is screwed to the rigid cabin (

The force was applied in the vertical direction (perpendicular to the test samples) by an electrodynamic shaker (excitation signal: Pink Noise); the measured magnitude was the acceleration in the force direction.

The instruments used for the experimental ODS measurement were as follows: Dual-Channel Signal Analyzer Brüel & Kjær (B&K, Nærum, Denmark) Type 2032, Reference Signal Sensor: Piezoelectric Accelerometer B&K 4374, Response Signal Sensor: Piezoelectric Accelerometer B&K 4384, Power Supply B&K ES 5001, Signal Conditioners B&K 2635, Electrodynamic Shaker B&K 4809, and Accelerometer Calibrator B&K 4294.

The finite element computational model was performed using the software ANSYS^{®}. The procedure began by generating the model of the naked steel plate to enable the results obtained from the simulation to be correlated with the experimental data and thereby yield a simulation reference model. Two models of finite elements were used for the naked plate: a first model in which the mounting holes of the plate were not taken into account, and a second model in which these holes were considered. The type of element used was SHELL63, which is an element developed for low-thickness plates operating under bending [

The mechanical properties (including the damping properties) can be obtained by different experimental tests [

The obtained results enable the development of a simulation model for the composite-metallic plate system that was used in a parametric analysis of the system to examine the influence of the Polyurethane Core Thickness, Polyurethane Core Type, Outer Layers and Adhesive Type (see

As a practical application of the methodology, it is proposed that the modal density of the steel sheet plate described in Section 2.2 be decreased to the frequency range of 0 to 150 Hz, also displacing the first mode of the plate to a frequency higher than 70 Hz. It is intended to achieve these goals without changes in the plate geometry and without a significant increase in the plate mass. For this purpose, a composite is adhered to the plate, as described in Section 2.2. The results obtained for the situation to be improved (Naked Metal Sheet) and the improvement proposal (metal sheet plus the lightweight multilayer composite) are described in this section.

In

The ANSYS^{®}

A thinner meshing of the model generates results slightly higher in frequency, although when reaching a limit, the results do not vary considerably.

The Block-Lanczos method generates a solution with higher frequencies than the Subspace method (mainly for frequencies above 200 Hz).

Including the drillings in the model makes the solution frequency closer to the experimental results (ODS). In general, adding boundary-condition restrictions to the same model increases the frequencies. The FEM model that most closely approximates the experimental results has the edges of the drills recessed and restrictions of xyz turning and z displacement in the plate-rigid frame contact zone. These boundary conditions are the most similar to those present in the experimental test. Performing a comparison with this method and the measured ODS, and always using the experimental ODS as a reference, resonance frequencies for the naked metallic plate are obtained; these values are: 50.8; 73.7; 113.2; 127.5; 148.8; 168.3; 185.6; 243.4 and 238.9 Hz

The following section presents the correlation between the natural frequencies obtained with the modal analysis by finite elements and the experimentally measured ODS.

It is intended to achieve the goals without changes in the plate geometry and without a significant increase in the plate mass. For this purpose, a composite is adhered to the plate, as described in Section 2.2.

Following the same procedure as that described for the galvanized steel sheet plate, ODS measurements were performed for the composite glued to the steel sheet plate. Forty-five transmissibility measurements were performed, generating a fictitious cure “H-Sum (black color in

Observing this curve, we can detect the peaks: 85.6; 118.1; 156.9; 166.3; 189.4; 221.9; 238.1; 288.1 and 301.3, considered as resonances (for the naked metallic plate plus lightweight multilayer composite (for the objectives marked, we are interested in frequencies below 150 Hz).

The model described in Section 2.4 was used in the modal analysis calculation for the metallic plate plus composite using the same boundary conditions and resolution method as in the simulation of the naked metallic plate. The natural frequencies for the FEM simulation model are 85.9; 120.0; 165.9; 180.5; 205.0; 220.4; 243.3; 285.1 and 301.6.

Adding the composite stiffens the naked plate, causing the natural frequencies of the naked metallic plate to be displaced toward higher values (see

The obtained results enable the development of a simulation model for the composite-metallic plate system that was used in a parametric analysis of the system to examine the influence of the adhesive (

The ANOVA analysis (

In the analysis of 2nd and 3rd Mode (see

Finally, it should be noted that adhesive type has a negligible influence on 1st, 2nd and 3rd Mode. Although the model of composite + flat steel plate determined by FEM suits the aim of this investigation, in order to study a more complicated system using the presented approach, an accurate modelling of Metal Sheet and Multilayer Composite as shown in

The proposed methodological approach allows for obtaining the resonance frequencies of the test sample by measuring the transmissibility functions without knowing the excitation force. To relate an ODS to the resonance frequency of the test sample with a modal deformation at that frequency, it is necessary to determine the location of the force and to restrict the movement of the system to the rigid frame where the force is applied. To achieve this objective, the sample must be strongly attached to a rigid frame screwed to a rigid cabin, applying the force to the frame periphery. For the ODS measurement, it is required that all the measurement points are collected simultaneously and that the mass of the sensors does not influence the measurement. In the discussed case, the force is stationary, which allows measurements to be performed point by point, and a specific low mass sensor for the modal analysis was used. The experimental tests have been performed with low-complexity instrumentation, and these tests can be generalized as necessary. Furthermore, a computational model is generated for obtaining the reference parameters to be improved and for developing better solution proposals. Thus, it was possible to correlate this model with the experimentally measured ODS, which in turn facilitated the tuning of a solution by only performing the necessary experimental measurements.

The practical application yielded an extremely small frequency difference between the resonance frequency obtained by the computational modal analysis (FEM) and that obtained by the ODS measurement (less than 9% for the steel plate plus multilayer composite, and 4% for the naked steel plate). Thus, it is concluded that the experimental test design enables the identification of the natural frequencies of the system, overcoming the difficulties encountered when performing this identification with transmissibility measurements.

In addition, the utilized composite helped to overcome the problem, increasing the first resonance of the naked plate by almost 70% and reducing the modal density (in the range of 0–150 Hz) by 40%. The computational model enables a parametric analysis of the variables than can affect the vibration behavior of the assembly (rigidity, weight, geometry,

The authors declare no conflict of interest.

Hybrid approach to improve vibrational behavior of structural panel.

ODS measurements: measurement points, sensors and sample location.

Metal Sheet and Multilayer Composite: 3D FEM.

(

Naked Metal Sheet: ODS and Mode Shape (FEM).

Transmissibility Naked Metal Sheet plus Lightweight Multilayer Composite.

Naked Metal Sheet plus Lightweight Multilayer Composite: ODS (

Modes: Naked Metal Sheet

(

(

(

(

Mode Shape and ODS comparative (adapted from Schwarz

Mode Shape | Natural | Linear | No | Relative motion of one DOF |

ODS | Any | Non linear | Yes | Actual motion of one DOF |

ODS Measurement setup.

Channel | 1 | 2 |

Frequency Range | 0–500 Hz | 0–500 Hz |

Number of spectral lines | 800 | 800 |

Resolution | 1.6 Hz | 1.6 Hz |

Number of Averages | 25 | 25 |

Window | Hannig | Hannig |

Sensor Sensitivity | Reference: 1 pc/m^{2} |
Response: 0.114 pc/m^{2} |

Factors of computational parametric analysis.

Polyurethane Core Thickness | 2 | Core composite thickness: 6 mm–12 mm |

Polyurethane Type | 2 | Density of Polyurethane Foam: a: Low; b: High |

Outer Layers Type | 3 | Outer layers Stiffness: Low-Medium-High |

Adhesive Type | 2 | Density of Adhesive: aa: Low; bb: High |

Naked Metal Sheet FEM

1 | 50.8 | 50.6 | 0.4 | (1, 1) |

2 | 73.7 | 73.8 | 0.1 | (2, 1) |

3 | 113.2 | 114 | 0.7 | (3, 1) |

4 | 127.5 | 123 | 3.5 | (1, 2) |

5 | 148.8 | 148 | 0.5 | (2, 2) |

6 | 168.3 | 168 | 0.2 | (4, 1) |

7 | 185.6 | 186 | 0.2 | (2, 3) |

8 | 243.4 | 233 | 2.2 | (1, 3) |

9 | 238.9 | 238 | 0.4 | (5, 1) |

Naked Metal Sheet plus Lightweight Multilayer Composite FEM

1 | 85.9 | 85.6 | 0.37 | (1, 1) |

2 | 120.0 | 118.1 | 1.58 | (2, 1) |

3 | 165.9 | 156.9 | 5.75 | (3, 1) |

4 | 180.5 | 166.3 | 8.54 | (1, 2) |

5 | 205.0 | 189.4 | 8.26 | (2, 2) |

6 | 220.4 | 221.9 | 0.66 | (4, 1) |

7 | 243.3 | 238.1 | 2.16 | (2, 3) |

8 | 285.1 | 288.1 | 1.06 | (1, 3) |

9 | 301.6 | 301.3 | 0.12 | (5, 1) |

1st Mode: analysis of variance.

Main factors | A: Thickness | 331.5270 | 1 | 331.5270 | 6938.9300 | <0.0001 | 22.98 |

B: Foam | 1.1267 | 1 | 1.1267 | 23.5800 | 0.0009 | 0.08 | |

C: Outer layers | 1092.5400 | 2 | 546.2720 | 11433.5900 | <0.0001 | 75.74 | |

D: Adhesive | 0.1067 | 1 | 0.1067 | 2.2300 | 0.1693 | 0.01 | |

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Interactions | AB | 0.6017 | 1 | 0.6017 | 12.5900 | 0.0062 | 0.04 |

AC | 16.2233 | 2 | 8.1117 | 169.7800 | <0.0001 | 1.12 | |

AD | 0.0017 | 1 | 0.0017 | 0.0300 | 0.8560 | 0.00 | |

BC | 0.0033 | 2 | 0.0017 | 0.0300 | 0.9658 | 0.00 | |

BD | 0.0017 | 1 | 0.0017 | 0.0300 | 0.8560 | 0.00 | |

CD | 0.0133 | 2 | 0.0067 | 0.1400 | 0.8716 | 0.00 | |

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Error | 0.4300 | 9 | 0.0478 | - | - | 0.03 | |

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Corrected Total | 1442.5800 | 23 | - | - | - | - |

2nd Mode: Analysis of Variance.

Main factors | A: Thickness | 495.0420 | 1 | 495.0420 | 24,525.000 | <0.0001 | 22.03 |

B: Foam | 0.0267 | 1 | 0.0267 | 1.3200 | 0.2800 | 0.00 | |

C: Outer layers | 1738.0400 | 2 | 869.0200 | 43,052,390 | <0.0001 | 77.35 | |

D: Adhesive | 0.0600 | 1 | 0.0600 | 2.9700 | 0.1188 | 0.00 | |

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Interactions | AB | 0.0017 | 1 | 0.0017 | 0.0800 | 0.7804 | 0.00 |

AC | 13.2708 | 2 | 6.6354 | 328.7300 | <0.0001 | 0.59 | |

AD | 0.0817 | 1 | 0.0817 | 4.0500 | 0.0752 | 0.00 | |

BC | 0.1258 | 2 | 0.0629 | 3.1200 | 0.0936 | 0.01 | |

BD | 0.0600 | 1 | 0.0600 | 2.9700 | 0.1188 | 0.00 | |

CD | 0.1425 | 2 | 0.0713 | 3.5300 | 0,.738 | 0.01 | |

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Error | 0.1817 | 9 | 0.0202 | - | - | 0.01 | |

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Corrected Total | 2247.0300 | 23 | - | - | - | - |

3rd Mode: Analysis of Variance.

Main Factors | A: Thickness | 624.2400 | 1 | 624.2400 | 1074.2200 | <0.0001 | 12.33 |

B: Foam | 1.1267 | 1 | 1.1267 | 1.9400 | 0.1972 | 0.02 | |

C: Outer Layers | 4405.9900 | 2 | 2203.0000 | 3791.0100 | <0.0001 | 87.01 | |

D: Adhesive | 0.0067 | 1 | 0.0067 | 0.0100 | 0.9171 | 0.00 | |

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Interactions | AB | 3.6817 | 1 | 3.6817 | 6.3400 | 0.0329 | 0.07 |

AC | 14.1175 | 2 | 7.0588 | 12.1500 | 0.0028 | 0.28 | |

AD | 0.0417 | 1 | 0.0417 | 0.0700 | 0.7949 | 0.00 | |

BC | 9.2808 | 2 | 4.6404 | 7.9900 | 0.0101 | 0.18 | |

BD | 0.0417 | 1 | 0.0417 | 0.0700 | 0.7949 | 0.00 | |

CD | 0.0408 | 2 | 0.0204 | 0.0400 | 0.9656 | 0.00 | |

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Error | 5.2300 | 9 | 0.5811 | - | - | 0.10 | |

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Corrected Total | 5063.8000 | 23 | - | - | - | - |