Structural Damping Analysis of a Vehicle Front Hood: Experimental Modal Parameters Extraction and Simulation Correlation
1
Faculty of Mechanical Engineering, Gheorghe Asachi Technical University, 700050 Iasi, Romania
2
Renault Technologie Roumanie, Preciziei 3g, 062204 Bucharest, Romania
3
Renault S.A.S., 1 Av. du Golf, 78280 Guyancourt, France
*
Authors to whom correspondence should be addressed.
Machines 2024, 12(12), 862; https://doi.org/10.3390/machines12120862
Submission received: 29 October 2024
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Revised: 23 November 2024
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Accepted: 25 November 2024
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Published: 28 November 2024
(This article belongs to the Section Vehicle Engineering)
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Figure 1
<p>Main components of the hood structure: (1) inner panel; (2) outer panel; (3) hinge reinforcers; (4) striker reinforcer; (5) striker wire; (6) hinges; (7) maintain bumper reinforcer.</p> "> Figure 2
<p>Assembled simulation model.</p> "> Figure 3
<p>Strain-energy density plot on the hood inner panel, first bending mode, 38.74 Hz.</p> "> Figure 4
<p>Modal data input for FRF calculations.</p> "> Figure 5
<p>Load, excitation directions, and response input window.</p> "> Figure 6
<p>Displacement response function for a mesh node on Z direction to a Z excitation.</p> "> Figure 7
<p>Modal participation factors for the calculated displacement response.</p> "> Figure 8
<p>Boundary conditions for the hood free-free modal analysis experiment.</p> "> Figure 9
<p>Measurement points locations: (<b>a</b>) TestLab geometry: inner panel -green points, outer panel-blue points (<b>b</b>) outer panel; (<b>c</b>) inner panel.</p> "> Figure 10
<p>Excitation directions (red arrows) on the hood striker: (<b>a</b>) X direction; (<b>b</b>) Y direction; (<b>c</b>) Z direction.</p> "> Figure 11
<p>Power spectral density of the input excitation for all 21 runs.</p> "> Figure 12
<p>Computed coherence functions examples.</p> "> Figure 13
<p>Modal data selection.</p> "> Figure 14
<p>The complex sum of the selected FRFs and peak selection.</p> "> Figure 15
<p>FRF and bandwidth selection applied for Time MDOF and PolyMAX techniques (Testlab workflow).</p> "> Figure 16
<p>Stabilization diagrams and modal indicator functions for modal parameter selection: (<b>a</b>) Least Square Complex Exponential technique; (<b>b</b>) PolyMAX technique.</p> "> Figure 17
<p>Calculated mode shapes for mode 1—1.573 Hz and mode 2—3.411 Hz, LSCE method (hood outer panel-blue; hood inner panel-red).</p> "> Figure 18
<p>Modal assurance criterion matrix: (<b>a</b>) LSCE technique; (<b>b</b>) PolyMAX technique.</p> "> Figure 19
<p>Modal participation factors: (<b>a</b>) LSCE technique; (<b>b</b>) PolyMAX technique.</p> "> Figure 20
<p>Re-calculated modal assurance criterion: (<b>a</b>) LSCE technique; (<b>b</b>) PolyMAX technique.</p> "> Figure 21
<p>Detailed analysis of the high damping mode at 57.297 Hz.</p> "> Figure 22
<p>Export options for the experimental mode set.</p> "> Figure 23
<p>Maximum displacement plot for the numerical derived mode shapes from 11 to 18 and the accelerometer positions on physical structure (the yellow nodes).</p> "> Figure 24
<p>Metapost (BETA CAE Systems) modal/FRF correlation window.</p> "> Figure 25
<p>Modal assurance criterion matrix for numerical and experimental sets of data.</p> "> Figure 26
<p>Frequency differences between the two mode sets.</p> "> Figure 27
<p>Displacement response function for a mesh node on the Z direction to a Z excitation with initially assumed damping ratios (red curve) and experimentally derived damping ratios (green curve).</p> "> Figure 28
<p>Examples of potential design improvements based on strain-energy density plots.</p> ">
Versions Notes
<p>Main components of the hood structure: (1) inner panel; (2) outer panel; (3) hinge reinforcers; (4) striker reinforcer; (5) striker wire; (6) hinges; (7) maintain bumper reinforcer.</p> "> Figure 2
<p>Assembled simulation model.</p> "> Figure 3
<p>Strain-energy density plot on the hood inner panel, first bending mode, 38.74 Hz.</p> "> Figure 4
<p>Modal data input for FRF calculations.</p> "> Figure 5
<p>Load, excitation directions, and response input window.</p> "> Figure 6
<p>Displacement response function for a mesh node on Z direction to a Z excitation.</p> "> Figure 7
<p>Modal participation factors for the calculated displacement response.</p> "> Figure 8
<p>Boundary conditions for the hood free-free modal analysis experiment.</p> "> Figure 9
<p>Measurement points locations: (<b>a</b>) TestLab geometry: inner panel -green points, outer panel-blue points (<b>b</b>) outer panel; (<b>c</b>) inner panel.</p> "> Figure 10
<p>Excitation directions (red arrows) on the hood striker: (<b>a</b>) X direction; (<b>b</b>) Y direction; (<b>c</b>) Z direction.</p> "> Figure 11
<p>Power spectral density of the input excitation for all 21 runs.</p> "> Figure 12
<p>Computed coherence functions examples.</p> "> Figure 13
<p>Modal data selection.</p> "> Figure 14
<p>The complex sum of the selected FRFs and peak selection.</p> "> Figure 15
<p>FRF and bandwidth selection applied for Time MDOF and PolyMAX techniques (Testlab workflow).</p> "> Figure 16
<p>Stabilization diagrams and modal indicator functions for modal parameter selection: (<b>a</b>) Least Square Complex Exponential technique; (<b>b</b>) PolyMAX technique.</p> "> Figure 17
<p>Calculated mode shapes for mode 1—1.573 Hz and mode 2—3.411 Hz, LSCE method (hood outer panel-blue; hood inner panel-red).</p> "> Figure 18
<p>Modal assurance criterion matrix: (<b>a</b>) LSCE technique; (<b>b</b>) PolyMAX technique.</p> "> Figure 19
<p>Modal participation factors: (<b>a</b>) LSCE technique; (<b>b</b>) PolyMAX technique.</p> "> Figure 20
<p>Re-calculated modal assurance criterion: (<b>a</b>) LSCE technique; (<b>b</b>) PolyMAX technique.</p> "> Figure 21
<p>Detailed analysis of the high damping mode at 57.297 Hz.</p> "> Figure 22
<p>Export options for the experimental mode set.</p> "> Figure 23
<p>Maximum displacement plot for the numerical derived mode shapes from 11 to 18 and the accelerometer positions on physical structure (the yellow nodes).</p> "> Figure 24
<p>Metapost (BETA CAE Systems) modal/FRF correlation window.</p> "> Figure 25
<p>Modal assurance criterion matrix for numerical and experimental sets of data.</p> "> Figure 26
<p>Frequency differences between the two mode sets.</p> "> Figure 27
<p>Displacement response function for a mesh node on the Z direction to a Z excitation with initially assumed damping ratios (red curve) and experimentally derived damping ratios (green curve).</p> "> Figure 28
<p>Examples of potential design improvements based on strain-energy density plots.</p> ">
Abstract
:Structural damping is a type of energy dissipation that occurs within the structure of a mechanical system. Unlike other forms of damping that rely on external devices or materials, structural damping is intrinsic to the construction and assembly of the structure itself. This study focuses on the experimental determination of the structural damping ratios for a vehicle front hood fabricated from steel, with the main objective being to accurately identify these damping characteristics. To achieve this, the modal parameter extraction process utilized both the Least Squares Complex Exponential (LSCE) and PolyMAX methods, providing a robust and comprehensive approach to identifying dynamic properties of a hood structure. The hood was subjected to free vibration decay in a free-free condition, with dynamic properties—such as natural frequencies, mode shapes, and damping coefficients—extracted. Additionally, a correlation study was performed between numerical and experimental results, evaluating the Modal Assurance Criterion (MAC) and frequency differences to validate the numerical model’s accuracy. The findings underscore the damping capacity of a standard steel front hood structure and highlight the relationship between damping coefficients and mode shapes, resulting in a well-correlated model for frequency response functions that can be used in transient response calculations.
1. Introduction
Damping in mechanical structures refers to the mechanisms by which energy is dissipated in a vibrating system. It is a critical aspect of mechanical design because it influences the dynamic behavior of structures, including their response to vibrations, noise, and potential fatigue failure. Damping helps in reducing the amplitude of vibrations over time, thus enhancing the stability and longevity of mechanical systems. Besides viscous and Coulomb (friction) damping which may co-exist inside vehicle systems or sub-systems, structural damping has a major contribution to the overall damping mechanism, especially when considering the body in white structures such as front hoods, doors, trunk doors etc. In structural damping, energy is dissipated by the internal friction within material [1] but also through internal friction mechanisms within the structure, including interactions at joints, interfaces, and boundaries. In the present article, the experimental determination of the structural damping coefficient is investigated for a vehicle front hood made of steel along with a correlation study between numerical-derived and experimentally obtained modal parameters. The reason of choosing a steel hood structure for this study is its wide use among serial car manufacturers. A benchmark performed on a A2MAC-global benchmarking online platform for professionals [2], by selecting B, C, D segment cars, no older than 2014, shows that almost 64.5% of the vehicles have steel hood structures.
When considering vehicle front hood structures, several studies have been performed through the years, related to main hood performances such as static stiffness, dynamic behavior, or pedestrian impact [3,4,5,6,7,8].
Thanks to its ability to reduce weight and enhance fuel efficiency, aluminum is a strong candidate for hood structures. Although the static stiffness of an aluminum hood is lower compared to that of a steel hood, it can withstand approximately 20% more load before experiencing plastic deformation [3]. The issue of reduced stiffness in aluminum structures, compared to steel, can be addressed by designing hybrid constructions. By using a steel inner panel and an aluminum outer panel for the hood, it is possible to achieve stiffness values close to those of an all-steel structure, while reducing the overall weight of the assembly [4]. Although composite hoods have not achieved widespread use in vehicle production due to high costs and a complex manufacturing process, they can offer better stiffness performance than steel hoods, depending on factors such as fiber orientation, layer arrangement, and stacking sequence [5], as well as improved dynamic behavior related to the stiffness-to-mass ratio [6].
Besides the materials, the manufacturing and assembly processes can affect both the static and dynamic stiffness of a hood structure. During the stamping process, the hood’s elements undergo a reduction in thickness and experience stress accumulation. While residual stress has minimal impact on the dynamic performance, the change in thickness significantly influences the hood’s dynamic response, generating a shift in the natural frequencies [7].
A passive safety study has shown that the modal analysis technique can be used not only for the determination of the dynamic properties of the structure but also for designing front hoods that perform well in pedestrian impact performances. By identifying key frequencies related to deformation and adding holes in the areas where maximum deformation occurs, a significant reduction in Head Injury Criterion can be achieved, compared to the original design [8].
Sandwich structures used in vehicle bodies may offer better dynamic behavior. Using sandwich panels with a thin viscoelastic material core, instead of standard damping treatments on the body in white panels, can reduce the weight by up to 70% while maintaining the same damping performances [9]. Vehicle hoods made from sandwich structures exhibit higher natural frequencies compared to bonnets made entirely of metal, particularly at higher modes. Additionally, the deformation is reduced, as confirmed by the modal frequency response analysis, due to the damping effect provided by the viscoelastic polymer foam [10].
Viscoelastic materials have a wide range of applications in vehicle body-in-white structures due to their ability to dissipate a significant amount of energy per vibration cycle [11]. The performance of viscoelastic materials has also been demonstrated in other fields such as aerospace [12] and railway [13]. Several studies have shown that viscoelastic pads are very effective in damping vibrations in vehicle body-in-white floor panels [14]. By applying the pads in the right positions on the body-in-white structure, a significant reduction in both vibration and noise levels can be achieved [14]. In addition to the placement of the viscoelastic pads, their thickness and material are also crucial when seeking a high-energy dissipation solution [15]. Liquid-applied sprayable damper materials can be used as an alternative to bitumen pads and may help reduce both mass and application surface area, as they have a lower density than standard bitumen materials [15].
As another alternative to bitumen pads mounted on panels, such as lateral doors, high-dissipative foams can be used [16]. A study conducted on a Peugeot 207 front door panel showed that high-dissipative foams can be as effective as standard bitumen pads used in serial production in terms of damping capacity, but at a lower weight [16].
Apart from vehicle hood articles, the structural damping phenomenon has also been studied in simple structures such as structural beams [17] and thin plates [18]. Previous experiments on structural beams showed that when density is increased, the damping ratio also increases, meaning that a structural beam made of brass will have a higher damping ratio than one made of aluminum [17]. A detailed analysis of the damping sources acting in a thin suspended aluminum plate highlighted the existence of multiple dissipation mechanisms, such as thermoelastic and viscoelastic damping within the material itself, acoustic damping, and air (fluid) damping [18]. According to the authors, acoustic damping accounts for the vibrational energy lost due to noise radiation in the air, while fluid damping results from the airflow along the structure’s edges [18]. The study revealed that the effect of air viscosity on the plate’s overall damping can be neglected, while the low acoustic damping values increase with frequency and become substantial for a few modes in the higher part of the considered frequency domain [18]. Similar studies have been performed by different authors, such as [19]—damping in wood and metal plates, and damping sources in strings, including thermoelastic damping, viscoelastic material damping, and air flow damping [20], in which similar conclusions were drawn.
Besides different damping solutions, experimental methods for damping characterization have also been developed over the past years. In most cases, modal analysis is used to obtain frequency response functions of structures, followed by the application of methods such as the half-power bandwidth method [21] or logarithmic decrement method [22] for estimating the damping ratio. Authors like [23] propose damping treatments for a vehicle body based on scanning particle velocity measurements. The direct visualization of normal acoustic particle velocity distribution helps to identify key acoustic radiation zones and the optimal position for damping pads, improving the vibration and acoustic levels of a vehicle body-in-white [23]. A method of damping localization based on modal projection has been developed and implemented at PSA Peugeot Citroën [24]. By projecting the component free-free modes onto the basis of global body-in-white modes, it is possible to identify critical parts of the structure to be damped [24]. According to [25], damping can also be used as a parameter for damage identification within structures. Typically, natural frequencies and mode shapes of structures are used for damage characterization by associating damage with a reduction in local stiffness; however, studies have shown that, in certain applications, damping has a greater sensitivity for damage identification than natural frequencies and mode shapes [25].
When speaking about modal parameter extraction from measured frequency response functions (FRFs), various techniques have been developed over the years. These techniques can generally be categorized into two approaches: time-domain methods and frequency-domain methods. Time-domain methods, such as the Ibrahim Time Domain Method [26], the Eigensystem Realization Algorithm (ERA) [27], and the Least Squares Complex Exponential (LSCE) method [28], focus on directly analyzing the time response of a system. In contrast, frequency-domain methods, including Peak Peaking [29] and PolyMAX [30,31], operate by analyzing the system’s behavior in the frequency domain, often using resonance peaks or parametric models to extract modal information.
Considering the state of the art of the research on structural damping and vehicle hood structure topics, this article presents the experimental determination of the structural damping ratio along with natural frequencies and mode shapes of a steel vehicle front hood using the free-free modal analysis method.
In the context of the automotive industry, the importance of accurate damping values cannot be overstated, as they directly influence noise, vibration, and the dynamic behavior of the hood structure under operating conditions. Typically, in numerical analysis of the transient response of body-in-white structures, such as front hoods, lateral doors, and tailgates, damping values are adopted based on previous recommendations or standards. However, these approximations often introduce errors in predicting the actual dynamic behavior of the structures under real-world conditions. The experimentally obtained damping values, as presented in this study, can offer a more precise prediction of the dynamic response of the hood structure under various excitations, contributing to better optimization of both structural integrity and NVH performance. This has significant implications for the design of lightweight, high-performance vehicle components that meet both durability and comfort standards in a competitive automotive market. Furthermore, as vehicle designs evolve toward greater lightweighting, improved pedestrian safety, electrification, and reduced time-to-market, accurately predicting the dynamic response of vehicle systems and structures from the early design phases becomes increasingly crucial.
Overall, the presented research makes significant contributions to the modal analysis of hood structures, including the creation of a detailed numerical model that enables accurate simulations and reliable correlation with experimental data. It compares several methods for extracting modal parameters from physical measurements, focusing on effective curve-fitting techniques and optimal accelerometer placement. This study also highlights the relation between the structural damping, natural frequencies, and mode shapes. Additionally, a step-by-step workflow for correlating numerical and experimental results is provided, along with design improvement recommendations for the hood structure, aimed at optimizing vibrational performance and structural integrity.
In the following chapters, the numerical and the experimental methods, the results, and the conclusion of the study are presented.
2. Numerical Methods
Free-free modal analysis is a technique used in structural dynamics to study the natural frequencies and mode shapes of a structure when it is not restrained or subjected to external forces. In this context, “free-free” refers to the condition where the structure is free from any boundary constraints—meaning it is floating in space without any fixed supports. This form of analysis is important for understanding the intrinsic dynamic characteristics of a structure and has wide applications in different domains, such as aerospace or automotive engineering, starting from the design phase. It allows for designing components that can withstand dynamic loads and fatigue and avoid resonance phenomena or dynamic coupling within a system that may affect the integrity and reduce the operational lifetime.
The first six natural frequencies in this type of analysis are typically zero, corresponding to rigid body motions—three translational and three rotational. Higher frequencies, however, reveal the structure’s deformation behavior and are the ones of interest when designing mechanical components.
Among a big variety of solvers available on the market that are able to perform modal analysis simulations, Nastran V2021.4 (MSC Software, Newport Beach, California, USA ) SOL 103 solver was used in the present research due to several advantages, such as excellent compatibility with various design software, streamlining the workflow from design to analysis and enabling quick adjustments to the simulation model based on design changes; multiple eigenvalue extraction methods allowing the user to choose the most efficient method for their specific problem; highly accurate results for natural frequencies and mode shapes and high efficiency for large models.
The Nastran SOL 103 solver uses the eigenvalue extraction method to solve the system of equations governing the structure’s dynamic behavior in the absence of external loads [32].
The process begins with the creation of a stiffness matrix and a mass matrix based on the material properties and geometry of the structure. The solver then solves the generalized eigenvalue problem:
where is the stiffness matrix, is the mass matrix, are the eigenvalues (which correspond to the square of the natural frequencies), and are the eigenvectors (mode shapes) [32], Equation (1) being deduced from the general equation of motion for a free vibration system, without damping.
The SOL 103 method, which offers a range of eigenvalue extraction techniques such as Lanczos or Householder, was selected based on the model’s scale and intricacy to optimize the computation of natural frequencies and corresponding mode shapes [32]. This adaptability ensures the method’s efficiency across varying structural configurations.
Considering this, a numerical simulation was performed using a standard steel hood structure. The principal components of this structure, illustrated in Figure 1, were designed using the Catia V6 software (Dassault Systèmes, Vélizy-Villacoublay, France), which facilitated precise modeling and integration of all constituent parts.
Given the linear nature of the modal analysis simulation, which is confined to the elastic behavior of the material, only a select set of material and geometric properties were used as input parameters for the simulation. Specifically, the thickness of the components, along with the material’s Young’s Modulus and Poisson’s Ratio, were employed to define the stiffness matrix, while the material’s density was essential for the accurate formulation of the mass matrix. Detailed information about the mentioned parameters is captured in Table 1.
The designed geometries were pre-processed using ANSA V23.1.1.1 (BETA CAE Systems, Root, Switzerland), employing meshing elements specifically designed for NASTRAN (MSC Sofware) to ensure precise simulation preparation. Triangular and quadrilateral shell elements were used to mesh the inner panel, the outer panel, and the hood structure reinforcers, Figure 2. Shell elements are widely used to model thin structures, where one dimension (typically the thickness) is significantly smaller than the other two [33]. These elements discretize the structure along its mid-plane, with the thickness being defined by the element’s cross-sectional properties. This aspect facilitates the iteration process when thickness adjustment is needed [33].
Solid hexahedral and pentahedral elements were utilized to mesh the hinges, striker wire, and various connection elements, including welding points that join the reinforcements to the inner panel, Figure 2. Additionally, these elements were applied to the adhesive lines located between the inner and outer panels, particularly in the middle region around the cut-outs and in the hemming area. Furthermore, the glue lines connecting the striker reinforcement to the outer panel at the front end were also modeled using solid mesh. For the hemming glue lines, an elastic modulus of 205 MPa was considered while for the middle glue lines, an elastic modulus of 5 MPa was applied.
For both type of elements, shell and solid, a standard 5 mm mesh size was considered by the authors to be adequate for this type of linear dynamic analysis.
The link between the shell elements, the solid welding spots, and adhesive lines was modeled by using RBE3 (Rigid Body Element, Form 3) elements which allow the load to be distributed across multiple nodes while still permitting flexibility and deformation in the connected nodes [33].
RBE2 (Rigid Body Element, Form 2) elements were used to model the screws that connect the hinges to the hood structure and the four welding seams between the striker wire and the striker reinforcer. Compared to RBE3, this type of element rigidly connects a dependent node to an independent node meaning that the motion of the dependent node is completely controlled by the master node [33]. RBE2 elements can inadvertently introduce excessive stiffness into the model, which may lead to inaccurate results, particularly in bending scenarios, but are considered by the authors’ experience as adequate in the context of the actual simulation.
The overall synthesis of the meshing elements is presented in Table 2.
The final assembly of the simulation model, as shown in Figure 2, yielded a total calculated mass of 14.9 kg for the designed hood structure, including the mobile hinges.
Given the free-free nature of the selected modal analysis, no boundary conditions or external load were imposed to the simulation model. In this configuration, the model is allowed to move freely without any restraints, meaning no fixed supports or constraints were applied to any of the nodes. This approach ensures that the natural vibrational characteristics of the structure, such as its inherent natural frequencies and mode shapes, can be identified independently of any external influences.
The post-processing of the NASTRAN SOL 103 results was carried out using METAPOST V23.1.1.1 (BETA CAE Systems, Root, Switzerland) software, which was selected due to its robust capabilities in handling large datasets and providing detailed visualization and analysis tools for dynamic simulations. Natural frequencies of the hood structure were identified, and the associated mode shapes were analyzed and compared within the 0 to 100 Hz frequency range. This range is a standard in structural dynamics, where significant dynamic solicitations occur [34], and it captures the realistic vibration sources encountered in typical vehicle conditions. Road-induced vibrations generally fall below 50 Hz, while engine vibrations at idle and moderate speeds, as well as aerodynamic forces, predominantly remain within the 0 to 100 Hz range. Additionally, deformation amplitudes are typically highest within this frequency range, where structural resonances are most prominent. For frequencies above 100 Hz, the amplitudes of deformation are generally lower, presenting minimal structural risk. By focusing on these frequencies, the analysis accurately assesses dynamic behavior and structural performance without excessive computational demands.
Furthermore, node-to-node analysis was conducted at specific points of interest, where the local Frequency Response Functions (FRFs) were calculated using METAPOST advanced features. These nodes were strategically chosen based on the strain-energy density distribution, which highlighted critical regions within the structure, ensuring that the most significant areas were thoroughly investigated.
3. Numerical Results
Usually when performing free-free modal analysis using Nastran or other solvers, the first six modes are typically considered rigid body modes. This is because free-free modal analysis simulates the structure as if it is unconstrained, meaning it has no fixed supports or boundary conditions. In such a scenario, the structure can move freely in space without any external restrictions, which leads to six degrees of freedom, three translations X, Y, Z, and three rotations around the same axes. These rigid body modes have a natural frequency close to zero (or exactly zero in ideal cases), as there is no stiffness or restoring force resisting rigid body motion. The small numerical values of these frequencies arise due to rounding errors or numerical inaccuracies in the solution process. The presence of the first six modes that have near-zero frequencies can also be used as a quality indicator for the finite element model. If more than six modes have natural frequencies near zero, this may indicate an improper model set up.
The analysis performed on the hood structure, Figure 2, highlighted the existence of the first six rigid body modes as per Table 3:
The first thing to notice is that all rigid body modes from Table 3 have natural frequency values close to zero, and they all have balanced movement along or around the reference axes, which is a direct indicator of the robustness of the prepared simulation model. The second thing to mention is that these six modes are not associated with the physical deformation of the hood structure, but simply with its capacity to move and rotate as a whole body. In the equipped vehicle, when the hood is properly constrained on the body-in-white, these modes are eliminated, and higher modes representing physical deformation become dominant.
A total number of 12 deformation modes were identified through the analysis in the 0–100 Hz frequency range, Table 4.
A first torsion mode (Y axis) was observed at 22.89 Hz. The hood corners, both front and rear, exhibited the largest amplitude, describing out-of-phase vertical motion along the diagonals. The total displacement plot (see Table 4) indicates that the central area of the hood remains stationary, effectively acting as a nodal zone where no vibration occurs. This mode shape engages the entire hood structure and should not coincide with the torsion mode of the vehicle’s body-in-white. If this occurs, dynamic coupling could amplify hood deformation when assembled on the body. Given the low natural frequency, this mode may be excited during vehicle roll on rough roads. To increase the natural frequency, modifications could be made to the inner panel’s structural ring section, such as adding ribs or reinforcements.
At 38.74 Hz, the first bending mode (around the X axis) appears, also known as hood flutter, combined with a vertical local pumping in the central area between the hood hinges, which is out-of-phase with the overall bending motion. The localized pumping significantly contributes to deformations in the hood outer panel. Even though it is a local deformation, it amplifies the overall hood response at this particular frequency. To mitigate the amplitude of this pumping effect, a glued connection can be applied between the inner and outer panels in this area, providing additional structural stiffness.
A second torsion mode around the X-axis emerges at 55.77 Hz and exhibits a significant vertical pumping motion on both sides of the hood’s symmetry axis. This motion is out-of-phase between the left and right sides. As shown in Table 4, the highest amplitude occurs near the same area as the first bending mode, suggesting a potential low-stiffness region in this part of the hood.
The same conclusion can be drawn for the second bending mode, 58.87 Hz, and the third torsion mode at 68.63 Hz. In both cases, overall motions are combined with local pumping areas similar to those presented above.
Next, in the frequency spectrum, two low-amplitude bending modes, 70.04 Hz and 76.27 Hz, can be observed with similar central pumping patterns. The only difference between these two mode shapes is that in the case of the 70.04 Hz mode, the hinges are in phase with the outer panel pumping while for the 76.27 Hz mode, they have an out-of-phase movement.
The remaining modes from 86.36 Hz to 98.13 Hz represent local torsion and pumping modes which are not related to global deformation of the hood structure and usually have no big influence on the overall dynamic behavior. This type of local mode is difficult to be identified during an experimental modal analysis due to the frequency response functions averaging process.
An interesting thing to notice is that for almost all deformation modes, a common pumping area exists on the hinge fixation line which is a clear indicator of a weak local design. A strain-energy density plot can help provide a better understanding on the adjustments needed to reduce the amplitude of a particular vibrational mode; for example, Figure 3.
In the case of the first bending mode at 38.74 Hz, a high concentration of deformation energy is observed around the holes in the hood inner panel, highlighted in the red box from Figure 3. These cut-outs are typically made to reduce mass and improve pedestrian impact performance, as a more flexible hood helps to lower the Head Injury Criterion [35]. However, the downside of these cut-outs is a reduction in stiffness in the affected areas, which can negatively impact the vibrational modes. Thus, a compromise is needed between these conflicting performance requirements. One way to enhance this particular mode is by closing some of the holes or reducing their size, which would increase stiffness and, in turn, improve the natural frequency.
Strain-energy density plots should be used in this type of study instead of total strain energy because they provide a detailed, localized measure of energy concentration within the structure, allowing for precise identification of weak spots [36]. Total strain energy, on the other hand, is a global measure and cannot reveal such local variations in energy distribution, making it less effective for identifying areas of structural weakness. This approach can be applied to all structural modes, or at least to those that contribute most significantly to overall deformation.
Another method to assess the contribution of vibrational modes to structural deformations, potential damage, and fatigue is by calculating Frequency Response Functions (FRFs) at different nodes of the hood mesh. The most critical nodes can be selected based on areas with the highest strain-energy density. Alternatively, engineering experience may guide the selection, especially in high-stress zones such as hinge interfaces or the hood striker region.
The Metapost (BETA CAE Systems) post-processor includes a module that allows FRF calculations using modal analysis results as the input [37]. The node-to-node calculation is valid because, in a modal analysis, the equations of motion are decoupled, enabling each node in the mesh to be treated as a single-degree-of-freedom system.
As a first step, the modal results are loaded in the calculation window and a standard structural damping ratio is assigned to each natural frequency, Figure 4, the first rigid body modes being excluded.
In the next window, the nodes of interest, excitation directions, loads, and the type o responses are selected, Figure 5.
Figure 6 presents an example of the displacement response in the Z direction for a Z- excitation at a selected node on the hood’s inner panel surface. The curve shows that only three out of the twelve vibrational modes have a significant contribution to the Z-axis motion in the selected area. Therefore, if amplitude reduction is required in this specific region, attention should be focused on these vibrational modes. The fact that in the graph a phase difference can be seen between load and response, it is a clear indicator of a structural deformation and not a simple displacement.
A clearer understanding can be obtained by calculating modal participation factors, which measure how strongly a given mode contributes to the structural response when subjected to force or displacement excitation in a specific direction as shown in Figure 7.
The calculation shows that the first bending mode at 38.74 Hz contributes 57.08% to the total pumping deformation in the selected area, followed by the fourth bending mode at 76.27 Hz and the second bending mode at 58.87 Hz. This analysis method can be used not only to identify weak areas in the hood structure, but also to perform a relative comparison between different designed geometries.
Knowing the natural frequencies of the free-free hood structure within a specific range will help identify the critical frequencies that lead to high deformation amplitudes when excited. By comparing these values with the natural frequencies of other vehicle systems and subsystems, potential issues like dynamic coupling can be avoided. Dynamic coupling can cause vibrations to combine when parts are assembled, increasing the risk of component failure.
Ideally, natural frequencies should be outside the excitation range, but this is not always possible. Therefore, the main objective of the modal analysis becomes the reduction of the deformation amplitudes of the associated mode shapes.
Usually, deformation modes of complex structures like a vehicle’s front hood have complicated shapes, making them difficult to describe with standard deformation patterns such as pure bending, torsion, or twisting. Some of these shapes may have a global character, while others may be localized, and combinations of global and local deformations can coexist within a front hood. The resulting shape is mainly influenced by mass distribution, stiffness distribution, and parts geometry. Geometry elements such as holes, ribs, and radiuses can generate localized responses in the hood during dynamic solicitation because they alter these distributions. Overall, identifying the deformation shapes can help pinpoint weak areas within the hood design and apply adjustments to the parts’ geometry so that the entire assembly can better withstand dynamic solicitations and fatigue.
Besides providing guidance for design adjustments to the hood structure starting from the early design phases, numerical modal analysis can offer a better understanding of the experimental modal results. Identifying the dynamic properties of the structure during physical measurements can be very challenging, especially when dealing with continuous systems such as a vehicle hood. Poor data measurements affected by noise, insufficient measurement points, closely spaced vibrational modes, imprecise modal parameter extraction, and other factors may lead to incorrect decisions. Having numerical results before starting the experiments can help determine the optimum number of measurement points and filter out erroneous results affected by the issues mentioned above.
In the next chapters, the experimental methods, results, and discussions are presented, and a correlation study with a modal parameter validation workflow is highlighted.
4. Experimental Methods
From general modal theory, it is clear that measured frequency response functions are needed in order to extract the parameters of interest, such as natural frequencies, mode shapes, and damping ratios [38]. In this context, several excitation techniques exist that allow for measuring the response characteristics, such as the impact hammer technique or shaker technique [38]. For this experiment, the impact hammer technique was used as the excitation method because of its advantages, including the ability to provide a controlled, repeatable force input, its ease of use, portability, and cost-effectiveness. Additionally, the technique allows for rapid data collection and real-time analysis, making it ideal for conducting modal analysis in a non-destructive manner across a wide range of materials and structural configurations.
The hood structure assembly, depicted in Figure 1, was fabricated from high-strength steel. Upon completion of the manufacturing process, the assembly was subjected to precision weighing, revealing a total mass of 14.1 kg. The gap of 0.8 kg between the weighted mass and the calculated one can be explained by the thickness reduction phenomenon that occurs during the stamping process [7]. By the authors’ experience, it can vary between 6% and 12% for a standard steel hood structure and will be considered during the analysis phase of the experimental modal analysis results.
To be close to a free-free condition, the tested hood structure was suspended by the hinges using a negligible stiffness rope as per Figure 8. This type of setup allows the behavior of the studied part to remain unaffected.
The geometry for the experiment consists of 118 points in total: 59 on the inner panel and 59 on the outer panel, as shown in Figure 4. The grid point configuration was created in the Simcenter TestLab software V2021.2.1 (Siemens Digital Industries Software, Plano, TX, USA) to enhance visualization and facilitate a comprehensive analysis, as depicted in Figure 9a.
The selection of the number and spatial positioning of measurement points was chosen by considering multiple criteria. As a first step, hood structure corners and edges were selected as measurement points. The corners and edges of the hood are likely to have distinct dynamic behavior due to their stiffness characteristics. Including these regions in the measurements can reveal bending or twisting modes. Additionally, to highlight global bending and twisting patterns, several accelerometers were positioned on the X and Y symmetry axis of the hood structure and near the center of gravity.
Furthermore, both stiff and flexible areas were selected for the measurements. Stiff areas included hinge fixation points, hinge reinforcer, striker reinforcer, and bumper reinforcer, while flexible areas included unsupported outer panel regions, cut-outs, and glued connections. Placing accelerometers on both stiff and flexible zones enables a comparison of their vibrational responses, helping to detect any localized resonance or mode shapes unique to these areas.
Practical considerations were also taken into account, such as avoiding interference between the accelerometers and structural ribs and addressing mounting challenges like the lack of flat surfaces. The outer and inner panels feature complex shapes, making some areas unsuitable for accelerometer placement.
Overall, for an accurate representation of the mode shapes across the entire hood, a uniform and symmetric distribution of the measurement points was considered.
The responses at each selected point were measured using 18 tri-axial Integrated Circuit Piezoelectric (ICP) accelerometers, model PCB 356A32 (PCB Piezotronics, Depew, NY, USA). These accelerometers were mounted within the vehicle’s reference frame along the X, Y, and Z axes. They were attached to the structure using Loctite 454 adhesive and moved through all predefined measurement points to capture the complete set of responses by making multiple runs. The key specifications of the utilized accelerometer model are provided in Table 5.
As the excitation point, the latching point on the striker was chosen due to the fact that it is rigid enough and it is one of the entry effort points of the hood structure solicitated in many performances such as slamming durability, vibrational durability, or even passive safety. The excitation was applied on all three directions, X, Y, Z, vehicle reference frame, to excite all possible natural modes of the hood, Figure 10.
A PCB impact hammer, model 086D05 (PCB Piezotronics) equipped with a rubber tip, Table 6, was used to excite the structure and an average of four impacts per direction was performed in the excitation points for each experiment run. The reason for making multiple impacts in a single run is to improve the accuracy and reliability of the measurements and reduce noise. Averaging multiple responses increases the signal-to-noise ratio, making it easier to identify the system’s true dynamic characteristics.
For data acquisition, the LMS SCADAS Mobile model SCM09 (Siemens Digital Industries Software, Plano, Texas, USA) front-end hardware system was utilized in conjunction with a workstation equipped with the LMS TestLab Impact Testing software module V2021.2.1 (Siemens Digital Industries Software, Plano, Texas, USA). The SC 316 hardware typically supports a large number of input channels, allowing for simultaneous data acquisition from multiple sensors and ensuring precise measurement of data, which is essential for reliable analysis. The integration with LMS TestLab provides an intuitive interface for setup and data visualization, making it easier for engineers to configure tests and interpret results effectively. Although the frequency band of interest for structural modal analysis is considered to be 0 to 100 Hz, acquisition was performed for a frequency band up to 512 Hz, all acquisition parameters being mentioned in Table 7.
The exponential decay response window was used in order to isolate relevant data and filter noise for the measured signal. Usually, this type of window is utilized in impact testing experiments to also force the response of the system to be periodic within a sample interval, allowing an FFT algorithm to be applied [38].
After acquiring the raw time-domain acceleration data from the tri-axial accelerometers, post-treatment of the data was performed using the LMS TestLab (SIEMENS) analysis chain. The post-treatment process included several steps:
- Data cleaning and filtering used to remove noise and irrelevant frequencies through passband and application of windowing.
- Converting data from the time domain to frequency domain using the Fast Fourier Transform (FFT) algorithm and calculating the Frequency Response Functions as the ratio between response and excitation.
- Computing of power spectrum density, cross-power spectrum density for both input and output and the coherence function [38].
- Frequency Response Functions (FRFs) averaging. Averaging measurements from multiple spatial points gives a more global picture of the system’s response. This helps ensure that the calculated modal parameters (natural frequencies, damping ratios, mode shapes) are representative of the entire structure, rather than being biased by a single point.
Following the extraction and rigorous validation of the structural modal parameters—namely the natural frequencies, mode shapes, and damping ratios—of the vehicle front hood, a comprehensive correlation study was conducted between the experimentally derived modal characteristics and those obtained from finite element simulations.
A key metric used in the correlation process was the Modal Assurance Criterion (MAC) [43], which quantifies the similarity between experimental and simulated mode shapes. MAC values range from 0 to 1, with values closer to 1 indicating high consistency between the modal deformation patterns, thereby confirming that the experimentally captured modes are well-represented in the simulation. Conversely, a value near 0 suggests little to no similarity in the deformation shapes.
Additionally, a frequency difference graph was used to compare the natural frequencies obtained from both datasets. By plotting frequency discrepancies across various modes, the graph highlights deviations, indicating potential differences in mass and stiffness distributions between the numerical model and the physical structure. Together, the MAC and frequency difference graph provided a comprehensive validation framework, supporting conclusions about the accuracy of the numerical model and experimental measurements.
This correlation analysis aimed to assess the fidelity of the numerical model in capturing the dynamic behavior of the front hood under real-world conditions. The experimentally derived damping ratios, reflective of the inherent material and structural dissipation in a standard steel front hood, were compared to the assumptions used in the finite element model. Insights were drawn regarding the model’s capability to replicate the actual energy dissipation mechanisms, which are critical for predicting vibratory response and fatigue behavior.
5. Experimental Results: Discussions
For the acquisition of all measurement points, 21 experimental runs were performed, resulting in a total number of 354 responses for the three references, the references representing the excitation points. Once the input and output signals were captured using the FFT analyzer, the auto-power spectral density was computed for both, multiplying the signals with their respective complex conjugates. Compared to the linear spectra, the power spectrum is a real valued function with no phase information [38].
A first check on the quality level of the excitation was performed by using input power spectral density, Figure 11.
As can be seen, Figure 11 indicates a good level of excitation for the 0 to 100 Hz frequency range, with a drop of less than 10 dB, which is considered reasonable for the structural studies [38]. Therefore, it can be concluded that the excitation levels achieved in all 21 experimental runs are sufficient to excite the existing structural modes within the frequency band of interest.
Following this, the cross-power spectral density was calculated to analyze the relationship between the input and output signals through the coherence function ensuring that the measured responses are due to the intended excitation and not influenced by noise or errors [38]. By definition, the coherence function is the ratio between the cross-power spectral density and the auto-power spectral densities of two signals. It provides a measure of how well the two signals are correlated at each frequency, ranging from 0 (no correlation) to 1 (perfect correlation) [38]. In Figure 12, three examples of computed coherence functions are presented: high correlation (green curve), medium correlation (orange curve), and low correlation (red curve).
The green curve, Figure 12, has a function value close to 1 which is a clear indicator of the robust relation between the excitation and the response. The drops that can be seen at 17.69 Hz, 58.38 Hz, and 80.79 Hz are related to anti-resonances or points where the system’s response is weak or minimal. These frequency drops indicate regions where the signals are less correlated, which is expected in such scenarios where the system exhibits a low output. The coherence value for the orange curves, Figure 12, have a fluctuation between 0.63 and 1 in the low frequency band up to 50 Hz. This highlights a presence of a possible noise in the measurements which may indicate a point with low vibration levels or some bad accelerometer cable mounting. Low coherence on a specific frequency band can also be found when computing for different input/output directions. For example, an excitation on the Y direction will contribute less to the X response of the structure and so, by the authors’ experience, in the case of continuous complex structures such as the front hood, a coherence value between 0.7 and 1 can be good enough [38].
The red curve, Figure 12, indicates a real noisy measurement that can be generated by improper accelerometer mounting. In this case, the accelerometer should be checked, and the measurement should be repeated.
The coherence functions were calculated for all 354 responses and measurements were repeated for those where the coherence value was outside the 0.7 to 1 range.
After the complete validation of the acquired data, the LMS TestLab (Siemens) modal analysis standard workflow was used to extract and validate the hood structural modal parameters. The process started with a modal data selection window, Figure 13, where all the responses were selected for all three references resulting in a total number of 1062 frequency response functions. Sometimes, it is interesting to take an overall look on all the frequency response functions at once on a linear amplitude scale, because it will help to identify if all of them merge at the same time into resonant peaks or if there is any localized response of individual points. This can provide indications of the existence of a local mode or a potential accelerometer problem during the experiment. The right-hand graph from Figure 13 highlights the existence of merged peaks starting with 1.73 Hz frequency. The first three peaks are well separated, while starting from 45.31 Hz, the presence of closely spaced modes can be spotted. A strange phenomenon can be observed around the 69 Hz frequency value, where only three FRFs have clear resonant peaks which may be explained by the presence of a local mode, some accelerometer problem, or some hood structure non-conformity and should be considered in the further analysis.
After the data selection, the complex sum of the selected FRFs was performed to obtain one curve that reflects the overall behavior of the hood structure, Figure 14, the red curve. The complex sum of multiple frequency response functions is performed by summing their complex value at each frequency point. Since FRFs have both real and imaginary components, the summation involves adding the corresponding real and imaginary parts and so no information is lost concerning the amplitude and the phase [38].
Next, various modal parameter extraction techniques were applied to identify a meaningful set of modes and their corresponding natural frequencies, damping ratios, and mode shapes. As an initial approach, the classic peak-picking method was tested, with the results plotted in Figure 14.
Using this initial approach, a total of 11 modes were identified, and the corresponding damping ratios ζ % were calculated for each selected peak using the half-power bandwidth method [44]. The upper table in Figure 14 presents significant variation in the calculated damping ratios across the modes. Notably, the damping ratio for the mode at 78.69 Hz reached an unusually high value of 56.37%, which is inconsistent with typical structural damping values observed in metallic structures [17]. Additionally, slight variations are visible on the summed FRF curve in Figure 14, particularly around 1.50 Hz, 32.18 Hz, and between the peaks at 60.74 Hz and 73.09 Hz. These features may suggest the presence of closely spaced modes, which are difficult to accurately distinguish using this method.
In conclusion, it can be stated that for complex structures, such as vehicle hoods, the peak-picking method faces challenges in distinguishing closely spaced or overlapping modes, making it difficult to accurately assess the damping ratio for each mode. Furthermore, this method primarily identifies natural frequencies based on the peak amplitudes of the FRF but does not provide direct information about the decay rate, which is essential for determining damping ratios. To estimate damping, additional approximations, such as the half-power bandwidth method, are often used. However, these methods can introduce errors, particularly when the structure does not exhibit ideal behavior.
Next, the process continued with the application of two curve-fitting methods available in TestLab: the Time Multi Degree of Freedom (MDOF) method, based on Least Squares Complex Exponential technique [28] and the PolyMAX method. based on a different approach that leverages rational function-fitting in the frequency domain [30,31]. Both represent curve-fitting techniques in which modal parameters like natural frequencies, damping ratios, and mode shapes are obtained by minimizing the error between the measured system response and a model. The Least-Squares Complex Exponential (LSCE) method is using complex exponentials while the PolyMAX method employes a polynomial fitting approach to accurately extract modal information from frequency response data.
In both methods, the process started with the selection of frequency response functions and the analyzed frequency range, Figure 15. To ensure no modal data are lost, it is recommended to place the cursors in regions of minimal response, or at anti-resonant points, where the system exhibits low vibrational activity [45].
As a second step, the stabilization diagram was used to select stable modes by analyzing the behavior of the identified poles across different iterations. In Figure 16, each mode is plotted as a track of letters on the diagram, showing how its natural frequency and damping ratio evolve across the analysis: o—new pole; f—stable frequency; d—frequency and damping; v—frequency and modal participation vector; s—all. In addition, to help in the selection of the meaningful modes, a modal indicator function (MIF) was overlapped on the stability diagram, Figure 16 [41]. Usually, the number of the modal indicator functions is equal to the number of the references, three in this case, and the dips observed on the green, blue, and purple curves in Figure 16 indicate the presence of a vibrational mode [41]. If multiple function drops are observed at close frequencies, this may indicate closely spaced vibrational modes or repeated roots.
A total number of 17 modes and the associated damping ratios were initially selected through the LSCE method while with PolyMAX, 16 vibrational modes were identified, Table 8. The first four modes show a slight difference on frequency values while a significant difference can be observed on the damping ratios. LSCE shows much higher damping ratios on mode 1 and mode 2 compared to PolyMAX, 3.02% compared to 0.4% on the first mode and 21.49% compared to 11.50% for the second one. The LSCE method appears to produce higher damping ratios than PolyMAX, particularly for lower frequency modes. This could suggest that LSCE is more sensitive to damping effects in these modes or that there are inherent differences in how the two methods interpret the data. The damping ratio of the second identified mode appears to fall outside the typical range for metallic structures, suggesting that this mode may be a rigid body mode. Additionally, the boundary conditions of the hood could be influencing the results. Since the hood structure is not freely floating in space but rather supported by two ropes, this setup may introduce additional damping, particularly for rigid body modes. Therefore, careful consideration should be given to the selection of the hanging ropes during modal analysis experiments. For higher modes (e.g., Mode 5 and above), the damping ratios are more comparable, though still show some variation.
For modes 3 through 17, generally closed value frequencies can be observed within the two methods; however, the LSCE method highlighted two closely spaced modes at 53.419 Hz and 53.622 Hz that have similar damping ratios and a completely different one at 69.167 which will be further investigated.
Overall, the LSCE method appears to produce higher damping ratios than PolyMAX, particularly for lower frequency modes. This could suggest that LSCE is more sensitive to damping effects in these modes or that there are inherent differences in how the two methods interpret the data. While there are discrepancies in the frequency values for some modes, overall, the two methods yield consistent results, particularly at higher frequencies. This suggests that the fundamental characteristics of the system are captured by both methods.
The differences in damping ratios, especially in the context of structural analysis or system response, can have significant implications for design and evaluation. In the next steps, the validation of the obtained data was performed by using modal assurance criterion, which compare the mode shape vectors [38]. Modal participation factors (MPs) indicate the contribution of each mode to the global deformation [46] and frequency response functions synthesis, which compares the recalculated FRF curve based on selected modal parameters with the actual measured FRFs [4].
The mode shapes for both sets were calculated in real form rather than complex form. From the authors’ experience, visualizing and understanding mode shapes in real form—focusing solely on the amplitude of motion and assuming a uniform phase—is generally easier than interpreting complex forms, which include phase data. The latter can sometimes complicate visual understanding and comparison with shapes obtained from numerical simulations.
The calculations highlighted that the first two extracted modes around 1.5 Hz and 3.4 Hz are, as expected, rigid body modes, rotation around the Y axis, and rotation around the Z axis, Figure 17.
The fact that the values of the identified rigid body modes are not close to zero as those identified in the numerical simulation is due to the boundary conditions utilized in the experiment. These modes were not considered in the modal participation factor calculations.
Figure 18 presents the computed modal assurance criterion for both mode sets, LSCE, Figure 18a, and PolyMAX, Figure 18b, where the modes were compared with themselves inside the set of data (Auto MAC).
The diagonal elements of an MAC matrix should ideally be close to 1, represented by the red color here, Figure 18. The strong red diagonal suggests that the modes are well captured and consistently identified for the two datasets. The off-diagonal elements ideally should be close to zero, which indicates that modes are distinct from each other. However, a few off-diagonal elements show cyan and green, indicating some correlation between modes that ideally should not correlate. This could suggest some mode mixing, closely spaced modes that are harder to distinguish, or some computational error. Additionally, values of MAC inferior to 1 may be caused by non-linearities in the tested hood structure.
On the MAC matrix calculated for the LSCE set of modes, Figure 18a, a correlation of 59.48% can be spotted between the 31.867 Hz mode and the 33.445 Hz mode, which may indicate some errors in the computation or modal data selection through the stability diagram. To identify which of these two modes is the correct one, a visual check of the mode shapes and comparison with the shapes obtained in the numerical simulation was performed, and the modal participation factors were computed. Both datasets showed that the 33.445 Hz mode is more of a transient response than a real vibrational mode, with very low participation in the overall deformation pattern. Therefore, it was decided to exclude this mode from the dataset.
The same conclusion was made for mode 4 and mode 5 from the PolyMAX dataset, Figure 18b, and thus the 34.233 Hz mode was also excluded. Next, on Figure 18a matrix, a similarity of 97.43% was identified between the 53.419 Hz mode and 53.622 Hz which is a clear indicator of a mode selection error. In addition, it can be seen that in the matrix, the 53.419 Hz has a 24.015% correlation with mode 9 (57.704) while the 53.622 Hz mode has only 16% of correlation with the same mode. Considering the aspects above and the fact that mode 53.419 has less participation in the overall deformation, it was decided to exclude it. The modal participation factors computed for both datasets, Figure 19, show that mode 8 from Figure 19a and mode 10 from Figure 19b have the lowest participation in the overall deformation of the structure, around 2%, indicating some local vibrations on the hood inner panel that may be related to some structure non-conformity like cracked glue lines and for this reason it was decided to remove them from both sets. This local effect for the 69.167 Hz mode was also observed on few emerging peaks at this frequency on the plot of all frequency response functions, Figure 13. These findings suggest that free-free modal analysis can be used not only to determine the dynamic properties of the structure but also to identify structure non-conformity in serial production. Reference FRFs can be measured on a good-condition front hood and after compared to those measured in a serial production surveil plan without damaging it.
The values from Figure 19a,b reveal that for both datasets, the modes from 1 to 5, respectively, up to 60 Hz range, have the biggest participation to the overall deformation than the others, pointing to real global deformation modes rather than localized ones.
After eliminating the rigid body modes, spurious modes, and local modes mentioned above, 12 validated deformation modes were retained. This is consistent with the number of deformation modes identified in the simulation. The recalculated Auto MAC for the validated mode sets is illustrated in Figure 20.
Both matrices have strong diagonal elements (in red), meaning both methods have captured the mode shapes well, with high correlation values close to 1. The LSCE method in Figure 20a shows strong correlations for the first few modes (1 to 7), but some deviation starts appearing in modes 10, 11, and 12 where the diagonal becomes less dominant, suggesting slight similarities between the modes. The PolyMAX method, Figure 20b, seems to offer a better consistency, and less similar mode shapes within the matrix, suggesting cleaner modal extraction for this case.
The refinement of the mode extraction process may be conducted by adjusting the frequency resolution or applying more sophisticated curve-fitting techniques. The similarities found between the modes within the same set of data may be removed and a cleaner Auto MAC matrix can be obtained by increasing the number of measurement points.
In the last phase of data validation, the synthesized FRFs were computed and then compared with the measured FRFs, and global correlation and error were calculated, Table 9.
With both methods, a quite-high global error was obtained, with PolyMAX performing slightly better than LSCE, respectively, 40.59% compared to 51.85%. On the global correlation, less difference was observed between the methods of only 3.31%. This relatively low level of correlation does not necessarily indicate poor modal parameter extraction. It could be attributed to the presence of local modes or measurement points with low vibrational levels. Additionally, having multiple excitation and response directions may contribute to this error level. For instance, excitation in the X direction has a limited effect on the Z response of the hood, and vice versa, due to the hood’s structural design. Numerical simulations also indicate that the hood’s primary responses occur in the Z direction, which makes sense given that the Z section of the hood is smaller than those in the other two directions, resulting in a lower moment of inertia and thus reduced stiffness.
Given that the PolyMAX technique outperformed the LSCE technique in terms of correlation, the 12 validated deformation modes of the tested hood structure are presented in Table 10, with the associated mode shapes and damping ratios. The damping ratios from Table 10 represent the ratio of the actual damping in a system to the critical damping.
The deformation modes for the vehicle front hood, spanning from 17.688 Hz to 93.175 Hz, appear reasonable for a large, flexible component like this. Such a frequency range aligns with expectations for a front hood, which typically has relatively low stiffness and is subjected to bending and torsional modal deformations at low frequencies due to its size and shape. This range also corresponds well with the frequencies observed in numerical simulations, as noted in Table 4. Only the first four mode shapes exhibit clear deformation patterns that can be associated with global torsion and global bending, respectively. For modes 5 to 12, the bending and torsion patterns are combined with local pumping responses in the Z direction.
The frequency distribution is broader for the first four modes, with a mean frequency difference of around 11 Hz, indicating that these are well-separated modes, mostly related to global deformations rather than localized ones. Starting with mode 5 up to mode 12, closely spaced modes are observed, with a mean frequency difference of around 5 Hz, which may indicate that these modes have similar local deformation patterns. The analysis of the mode shapes from both the simulation (Table 4) and the experiment (Table 10) validates the statement above.
From the design point of the view, the existence of multiple local modes can be explained by the presence of multiple cut-outs in the hood inner panel, Figure 3, which locally weaken the structure. As explained before, these are usually added for pedestrian impact performance but also for the weight optimization process and so a compromise should be found between these conflicting performances.
From a manufacturing process point of view, some non-conformities on the glue lines and some exaggerated thickness reductions during stamping or assembly problems may also create areas with low stiffness which will generate local responses of the structure.
The damping ratios range from 0.76% (mode 1) to 4.47% (mode 5). These are typical values for structural damping in metals like steel [17]. Steel generally exhibits low inherent damping, which is reflected in these small damping ratios.
Mode 1 has a very low damping ratio (0.76%), meaning that the structure has very little inherent energy dissipation in this mode, which clear confirms that this is a global mode with large scale motion of the entire hood structure. The low damping in mode 1 suggests that the hood could resonate at 17.688 Hz for extended periods if excited by an external source, which might lead to unwanted noise, vibration, or even fatigue damage.
Mode 5 has the highest damping ratio (4.47%), which indicates that more energy is dissipated at this mode, potentially due to some specific localized material behavior or structural feature such as a flexible region or a region with glued connection. Glue (or adhesives) introduces viscoelastic behavior, where energy is dissipated through internal friction within the adhesive layer when the structure deforms [47]. This may be one of the reasons for such a relatively high damping ratio on mode 5. Another reason for this particular value can be explained by the computational limitations of the curve-fitting method that was utilized in the selection process. Table 8 shows that with the LSCE method, a much lower damping ratio of 1.58% was obtained for a closely valued mode, respectively, 57.704 Hz. The detailed analysis of the 57.297 Hz mode shape highlighted the existence of a local high displacement point which is positioned exactly on the glue line connecting the outer and the inner panel, Figure 21. Considering the findings above, the higher value of the damping ratio for this particular mode can be explained. As an overall conclusion, increasing the surface of the glued connections between the inner and outer panel may increase the inherent damping capacity of hood assembly especially for local pumping modes.
In general, higher damping ratios, approaching 3%, are observed in modes associated with areas of lower stiffness, such as unsupported outer panel zones (modes 2, 3, 4, 7, and 10). Conversely, modes involving stiffer areas, like the hinge reinforcement zone, show lower damping ratios, typically below 2% (modes 6, 8, 9, 11, and 12).
The variation in damping ratios across modes lacks a consistent pattern related directly to frequency, indicating that local factors influence damping behavior. This suggests that damping might be affected by localized structural differences, such as material composition, structural damping elements, or reinforcements and stiffeners specifically designed into the hood. These design features could contribute to variations in how energy is dissipated across different regions of the hood, leading to mode shape-dependent damping behavior.
6. Numerical and Experimental Results Correlation Study
The correlation study between numerically derived and experimentally obtained modal parameters was conducted using the Metapost post-processor (BETA CAE Systems). The process began with exporting the experimental mode set from TestLab (SIEMENS) in universal format. This export file includes essential data such as geometry, node numbers, mode shapes, and additional parameters, as shown in Figure 22. To prevent scaling issues, the exported geometry should use the same units (e.g., millimeters) as those in the post-processor. Otherwise, the geometries will have mismatched dimensions and will not align properly.
Next, both the numerical and experimental results were imported into Metapost (BETA CAE Systems) in separate active windows. An initial observation, made when overlaying the finite element geometry with the accelerometer positioning geometry (Figure 23), revealed that the areas of maximum displacement in the numerical mode shapes (specifically for modes 11 through 18 corresponding to 5–12 deformation modes), did not have a perfect match with the accelerometer position. Consequently, a low correlation in the modal assurance criterion is expected for these specific modes.
A second observation was that a large number of accelerometers were mounted in areas with low vibration levels across the modes. This positioning likely contributes to the relatively low correlation between the synthesized and measured Frequency Response Functions (FRFs), as shown in Table 9.
For the calculation of the modal assurance criterion and frequency differences of the two sets of data, the Metapost (BETA CAE Systems) modal correlation module was involved. The data were loaded in the calculation window, Figure 24, and the node pairs were selected using the PlotEL nodes option.
The PlotEL nodes option automatically plots the nodes from the mesh and the nodes from the geometry used in the experiment that are closest to each other [37], allowing for the calculation of the modal assurance criterion (MAC).
In Figure 25, the calculated MAC matrix for both sets of data is presented.
A good level of correlation between the two sets of data can be observed on the first four deformation modes, the values varying between 71.9% and 89.58% which seem to be acceptable for such complex structures like a vehicle front hood. However, as it was expected from the preliminary analysis of the accelerometer position compared to maximum displacement areas, for the modes from 5 to 12, low values of correlation were obtained on the diagonal. For mode 6, for example 60.58 Hz (experiment) and 70.04 Hz (simulation), 39.9% was obtained on the mode shape correlation while for the higher modes that have more local deformation patterns, correlation values closer to 20% were obtained. It is important to note that these lower correlation values for the higher modes do not inherently indicate deficiencies in the mode shape extraction process. Instead, they are likely attributable to an insufficient number of measurement points in regions exhibiting high response characteristics. Furthermore, the presence of noise and potential non-conformities in the physical structure may result in certain accelerometers registering vibrations even in areas where the simulations indicate negligible vibrational activity. This discrepancy can further contribute to the observed low correlation between the experimental and simulated data.
This observation underscores the need for an enhanced measurement strategy to capture the dynamic behavior of these complex structures accurately [48].
Despite attaining a robust correlation in mode shapes for the first four vibrational modes, a frequency difference of up to 22.74% was observed between the computationally derived and experimentally extracted vibrational frequencies, as illustrated in Figure 26.
Upon analyzing the diagonals of the frequency difference matrix (see Figure 26), a distinct pattern emerges, indicating that the natural frequencies obtained experimentally are consistently lower than those calculated through simulations. Typically, differences in natural frequencies can be attributed to discrepancies in mass or stiffness. However, given that the weighted mass of the physical hood was slightly lower than the computed mass, this suggests that the observed differences are primarily due to variations in stiffness. Contributing factors may include the thickness distribution of the hood’s physical structure, a slightly lower Young’s modulus of the steel components, and the properties of the adhesive elements. A detailed material analysis of the tested structure may help to identify the proper material parameters, and a better level of correlation can be achieved.
In a final step, the local frequency response function was computed for the same mesh node as in Figure 6, with the updated experimentally obtained damping ratios and the two curves compared, Figure 27.
A pronounced difference in the amplitude of the response is evident in the curves depicted in Figure 27 when employing experimentally derived damping ratios. This observation underscores the critical significance of accurately integrating the true damping characteristics of a structure during the analysis of various transient responses or fatigue assessments, particularly when utilizing the modal participation method. Such considerations are essential for ensuring the reliability and precision of dynamic behavior predictions in engineering applications.
Based on the correlated numerical model, several design strategies can be developed to enhance the dynamic response of the hood structure for the deformation modes of interest, with examples in Figure 28. As discussed in Section 3, strain-energy density plots can be useful for identifying weak areas that require improvement.
The design improvements may involve closing the holes in the hood inner panel to reduce flexibility, adding structural ribs for increased rigidity, adding additional reinforcers the rear area, or increasing the moment of inertia of the hood’s rear beam by deepening the stamping profile. Together, these adjustments help distribute stiffness more effectively across the hood structure, resulting in a significantly improved dynamic response and better resistance to deformation under various loading conditions.
However, all of these modifications should also be evaluated in terms of other performance aspects of the hood structure, which can sometimes conflict with structural requirements. For instance, pedestrian impact safety often requires a more flexible hood. Therefore, a multidisciplinary optimization approach is necessary to balance these competing objectives, potentially serving as a foundation for future work.
7. Conclusions
Free-free modal analysis is essential for understanding the dynamic behavior of complex structures like vehicle hoods, as it identifies key vibrational properties such as natural frequencies, mode shapes, and damping characteristics. Due to challenges from complex geometry and material inconsistencies, precise accelerometer placement is crucial. Positioning sensors in high-response areas and avoiding low-response zones, like glue lines, minimizes noise and improves alignment between experimental and theoretical results.
The choice of modal parameter extraction methods significantly impacts the accuracy and reliability of the results. The study showed that PolyMAX algorithm outperforms Least Square Complex Exponential in terms of global correlation and global error of the synthesized Frequency Response Functions, proving that the PolyMAX technique is more efficient in the identification of the closed-space vibrational modes.
The vehicle hood’s deformation modes, occurring between 17.7 Hz and 93.2 Hz, reflect its low stiffness and tendency toward bending and torsional deformations due to its shallow Z-section and are typical for a large flexible component [6,8,49]. The first four modes, in the 0–50 Hz range, show global deformation patterns critical to structural integrity, while higher modes mainly cause localized deformations with less impact. Hood design should prioritize resistance to these low-frequency global modes, as they are more likely to be excited by engine vibrations or road impacts.
It was shown that the structural damping of the hood varies across modes, without a clear pattern related to frequency. The findings suggest that structural damping of a front hood structure is more related to the mode shape, and it might vary locally across the hood, possibly due to stiffeners, reinforcement, or specific design features. Lower values of damping ratio, closer to 1%, were observed on the modes that describe global deformation patterns and higher values of damping on modes that have localized deformation between 2% and 3% in more flexible areas. Notably, the first torsional mode at 17.688 Hz exhibited a low damping ratio of 0.76%, indicating that the structure may resonate strongly at this frequency. This susceptibility to low-frequency excitations, such as road-induced vibrations, implies that design adjustments may be needed to address resonance at this mode.
Correlation analysis between numerical and experimental results showed a strong match in mode shapes for the first four global modes (72–89%). In contrast, local modes (5 to 12) had lower correlation values, likely due to limited measurement points in high-response areas rather than issues with modal extraction. Noise and minor structural inconsistencies may also cause some accelerometers to capture unexpected vibrations. These discrepancies highlight the importance of precise sensor placement in high-response areas to capture accurate modal responses and reduce noise.
The frequency difference between the numerically derived and experimentally extracted vibration modes up to 22% on the first mode indicates the presence of some discrepancies between the stiffness distributions of the two sets. Factors affecting correlation may include thickness variations across the hood, a slightly lower Young’s modulus for the steel, and the adhesive properties in bonded areas. A detailed material analysis, with attention to these parameters, could help refine the model and enhance correlation.
Based on the obtained data, several design strategies, such as closing holes in the hood’s inner panel, adding structural ribs, reinforcing the rear area, and increasing the rear beam’s moment of inertia, can be implemented to improve the hood structure’s dynamic response and deformation resistance. Strain-energy density plots are valuable in identifying areas that need reinforcement. However, since structural improvements may conflict with other performance requirements, such as pedestrian impact safety, a multidisciplinary optimization approach is essential to balance these objectives and could serve as a foundation for future work.
Author Contributions
Conceptualization, V.P.; Methodology, V.P. and C.B.; Software, V.P.; Validation, V.P. and O.B.; Formal analysis, C.B.; Investigation, V.P. and O.B.; Writing—original draft, V.P.; Writing—review & editing, C.B.; Visualization, C.B.; Supervision, C.B. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
Data are contained within the article.
Conflicts of Interest
The authors declare no competing interests. The author Olivier Barat was employed by the company Renault S.A.S. There is no conflict of interest between any of the authors and the company Renault S.A.S.
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Figure 1.
Main components of the hood structure: (1) inner panel; (2) outer panel; (3) hinge reinforcers; (4) striker reinforcer; (5) striker wire; (6) hinges; (7) maintain bumper reinforcer.
Figure 1.
Main components of the hood structure: (1) inner panel; (2) outer panel; (3) hinge reinforcers; (4) striker reinforcer; (5) striker wire; (6) hinges; (7) maintain bumper reinforcer.
Figure 2.
Assembled simulation model.
Figure 3.
Strain-energy density plot on the hood inner panel, first bending mode, 38.74 Hz.
Figure 4.
Modal data input for FRF calculations.
Figure 5.
Load, excitation directions, and response input window.
Figure 6.
Displacement response function for a mesh node on Z direction to a Z excitation.
Figure 7.
Modal participation factors for the calculated displacement response.
Figure 8.
Boundary conditions for the hood free-free modal analysis experiment.
Figure 9.
Measurement points locations: (a) TestLab geometry: inner panel -green points, outer panel-blue points (b) outer panel; (c) inner panel.
Figure 9.
Measurement points locations: (a) TestLab geometry: inner panel -green points, outer panel-blue points (b) outer panel; (c) inner panel.
Figure 10.
Excitation directions (red arrows) on the hood striker: (a) X direction; (b) Y direction; (c) Z direction.
Figure 10.
Excitation directions (red arrows) on the hood striker: (a) X direction; (b) Y direction; (c) Z direction.
Figure 11.
Power spectral density of the input excitation for all 21 runs.
Figure 12.
Computed coherence functions examples.
Figure 13.
Modal data selection.
Figure 14.
The complex sum of the selected FRFs and peak selection.
Figure 15.
FRF and bandwidth selection applied for Time MDOF and PolyMAX techniques (Testlab workflow).
Figure 15.
FRF and bandwidth selection applied for Time MDOF and PolyMAX techniques (Testlab workflow).
Figure 16.
Stabilization diagrams and modal indicator functions for modal parameter selection: (a) Least Square Complex Exponential technique; (b) PolyMAX technique.
Figure 16.
Stabilization diagrams and modal indicator functions for modal parameter selection: (a) Least Square Complex Exponential technique; (b) PolyMAX technique.
Figure 17.
Calculated mode shapes for mode 1—1.573 Hz and mode 2—3.411 Hz, LSCE method (hood outer panel-blue; hood inner panel-red).
Figure 17.
Calculated mode shapes for mode 1—1.573 Hz and mode 2—3.411 Hz, LSCE method (hood outer panel-blue; hood inner panel-red).
Figure 18.
Modal assurance criterion matrix: (a) LSCE technique; (b) PolyMAX technique.
Figure 19.
Modal participation factors: (a) LSCE technique; (b) PolyMAX technique.
Figure 20.
Re-calculated modal assurance criterion: (a) LSCE technique; (b) PolyMAX technique.
Figure 21.
Detailed analysis of the high damping mode at 57.297 Hz.
Figure 22.
Export options for the experimental mode set.
Figure 23.
Maximum displacement plot for the numerical derived mode shapes from 11 to 18 and the accelerometer positions on physical structure (the yellow nodes).
Figure 23.
Maximum displacement plot for the numerical derived mode shapes from 11 to 18 and the accelerometer positions on physical structure (the yellow nodes).
Figure 24.
Metapost (BETA CAE Systems) modal/FRF correlation window.
Figure 25.
Modal assurance criterion matrix for numerical and experimental sets of data.
Figure 26.
Frequency differences between the two mode sets.
Figure 27.
Displacement response function for a mesh node on the Z direction to a Z excitation with initially assumed damping ratios (red curve) and experimentally derived damping ratios (green curve).
Figure 27.
Displacement response function for a mesh node on the Z direction to a Z excitation with initially assumed damping ratios (red curve) and experimentally derived damping ratios (green curve).
Figure 28.
Examples of potential design improvements based on strain-energy density plots.
Table 1.
Input parameters used in the simulation.
| Nr. | Part | Material | Density | Thickness mm | Young’s Modulus MPa | Poisson’s Ratio |
|---|---|---|---|---|---|---|
| 1 | Inner panel | Steel | 7850 | 0.55 | 210,000 | 0.3 |
| 2 | Outer panel | Steel | 7850 | 0.65 | 210,000 | 0.3 |
| 3 | Hinge reinforcer | Steel | 7850 | 1.2 | 210,000 | 0.3 |
| 4 | Striker reinforcer | Steel | 7850 | 1.2 | 210,000 | 0.3 |
| 5 | Striker wire | Steel | 7850 | 8 | 210,000 | 0.3 |
| 6 | Hinge | Steel | 7850 | 3 | 210,000 | 0.3 |
| 7 | Bumper reinforcer | Steel | 7850 | 0.95 | 210,000 | 0.3 |
Table 2.
The type of meshing elements utilized in the pre-processing of the hood structure.
| Component | Element Type |
|---|---|
| Inner panel | Shell
|
| Outer panel | |
| Hinge reinforcer | |
| Striker reinforcer | |
| Bumper reinforcer | |
| Striker wire | Solid
|
| Hinge | |
| Adhesive lines | |
| Welding spots | |
| Welding seams | RBE2 (Rigid Body Element, Form 2) 71 elements |
| Screws | |
| Adhesive/shell connections | RBE3 (Rigid Body Element, Form 3) 3588 elements |
Table 3.
The first six rigid body modes of the hood structure.
Mode 1—Rotation 0.0013 Hz | Mode 2—Translation X 0.00096 Hz |
Mode 3—Rotation 0.00023 Hz | Mode 4—Rotation 0.00033 Hz |
Mode 5—Translation Y 0.00059 Hz | Mode 6—Translation Z 0.00069 Hz |
Table 4.
Deformation modes of the hood structure, scale 7:1.
Mode 7—First torsion 22.89 Hz | Mode 8—First bending 38.74 Hz |
Mode 9—Second torsion 55.77 Hz | Mode 10—Second bending 58.87 Hz |
Mode 11—Third torsion 68.63 Hz | Mode 12—Third bending 70.04 Hz |
Mode 13—Fourth bending 76.27 Hz | Mode 14—Local torsion 86.36 Hz |
Mode 15—Local pumping 93.08 Hz | Mode 16—Local pumping 94.07 Hz |
Mode 17—Local pumping 97.19 Hz | Mode 18—Local pumping 98.13 Hz |
Table 5.
PCB 356A32 accelerometer key specifications [39].
Table 5.
PCB 356A32 accelerometer key specifications [39].
| Sensitivity (±10%) | 10.2 mV/(m/s2) |  |
| Measurement Range | ±491 m/s2 pk | |
| Frequency Range (±5%) | 1.0 to 4000 Hz | |
| Size—Height | 11.4 mm | |
| Size—Length | 11.4 mm | |
| Size—Width | 11.4 mm | |
| Weight | 5.4 g |
Table 6.
PCB 086D05 impact hammer key specifications [40].
Table 6.
PCB 086D05 impact hammer key specifications [40].
| Sensitivity (±10%) | 0.23 mV/N |  |
| Measurement Range | ±22,240 N pk | |
| Resonant Frequency | ≥22 kHz | |
| Head Diameter | 25 mm | |
| Tip Diameter | 6.3 mm | |
| Hammer Length | 227 mm | |
| Weight | 320 g |
Table 7.
Acquisition parameters.
| Useful frequency band | 0–512 Hz |
| Sampling frequency | 1024 Hz |
| Passband | 512 Hz |
| Number of spectral lines | 2048 |
| Frequency resolution | 0.25 Hz |
| Exponential decay response window | 100% |
| Number of averages | 4 |
| Perpendicular excitation to the impact surface, brief and non-rebounding | |
Table 8.
Initially extracted modal parameters using both methods.
| Nr. | LSCE | PolyMAX | ||
|---|---|---|---|---|
| Frequency Hz | Damping Ratios% | Frequency Hz | Damping Ratios % | |
| Mode 1 | 1.573 | 3.02 | 1.551 | 0.40 |
| Mode 2 | 3.411 | 21.49 | 3.354 | 11.50 |
| Mode 3 | 17.696 | 1.00 | 17.688 | 0.76 |
| Mode 4 | 31.867 | 4.32 | 31.835 | 2.69 |
| Mode 5 | 33.445 | 3.33 | 34.233 | 2.12 |
| Mode 6 | 45.446 | 1.52 | 45.102 | 2.20 |
| Mode 7 | 53.419 | 2.84 | 53.438 | 2.20 |
| Mode 8 | 53.622 | 2.18 | 57.297 | 4.47 |
| Mode 9 | 57.704 | 1.58 | 60.577 | 1.65 |
| Mode 10 | 60.987 | 1.63 | 64.552 | 2.56 |
| Mode 11 | 65.330 | 1.85 | 72.628 | 1.09 |
| Mode 12 | 69.167 | 1.70 | 77.023 | 1.54 |
| Mode 13 | 73.004 | 1.65 | 78.424 | 0.9 |
| Mode 14 | 78.005 | 1.54 | 83.591 | 2.22 |
| Mode 15 | 83.314 | 1.66 | 86.033 | 1.84 |
| Mode 16 | 86.820 | 1.72 | 93.175 | 1.09 |
| Mode 17 | 93.707 | 1.31 | x | x |
Table 9.
Global correlation and global error for the two methods: LSCE and PolyMAX.
| LSCE | PolyMAX | |
|---|---|---|
| Global correlation | 60.66% | 63.97% |
| Global error | 51.85% | 40.59% |
Table 10.
The validated deformation modes of the hood structure (Polymax).
| Mode 1—First torsion 17.688 Hz Damping ratio ζ = 0.76%  | Mode 2—First bending 31.835 Hz Damping ratio ζ = 2.69%  |
| Mode 3—Second torsion 45.102 Hz Damping ratio ζ = 2.20%  | Mode 4—Second bending 53.438 Hz Damping ratio ζ = 2.20%  |
| Mode 5—Third torsion 57.297 Hz Damping ratio ζ = 4.47%  | Mode 6—Third bending 60.577 Hz Damping ratio ζ = 1.65%  |
| Mode 7—Fourth bending 64.552 Hz Damping ratio ζ = 2.56%  | Mode 8—Local torsion 72.628 Hz Damping ratio ζ = 1.09%  |
| Mode 9—Local pumping 77.023 Hz Damping ratio ζ = 1.54%  | Mode 10—Local pumping 83.591 Hz Damping ratio ζ = 2.22%  |
| Mode 11—Local pumping 86.033 Hz Damping ratio ζ = 1.84%  | Mode 12—Local pumping 93.175 Hz Damping ratio ζ = 1.09%  |
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Pinzaru, V.; Bujoreanu, C.; Barat, O. Structural Damping Analysis of a Vehicle Front Hood: Experimental Modal Parameters Extraction and Simulation Correlation. Machines 2024, 12, 862. https://doi.org/10.3390/machines12120862
AMA Style
Pinzaru V, Bujoreanu C, Barat O. Structural Damping Analysis of a Vehicle Front Hood: Experimental Modal Parameters Extraction and Simulation Correlation. Machines. 2024; 12(12):862. https://doi.org/10.3390/machines12120862
Chicago/Turabian StylePinzaru, Valerian, Carmen Bujoreanu, and Olivier Barat. 2024. "Structural Damping Analysis of a Vehicle Front Hood: Experimental Modal Parameters Extraction and Simulation Correlation" Machines 12, no. 12: 862. https://doi.org/10.3390/machines12120862
APA StylePinzaru, V., Bujoreanu, C., & Barat, O. (2024). Structural Damping Analysis of a Vehicle Front Hood: Experimental Modal Parameters Extraction and Simulation Correlation. Machines, 12(12), 862. https://doi.org/10.3390/machines12120862
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