# Investigation of vibration modes of a double-lap bonded joint

**Part of the following topical collections:**

## Abstract

The authors were concerned in this work in examining the influence of many mechanical and geometrical parameters on the mode shapes of vibration of a double-lap bonded joint. The parameters varied in this study were: adhesive Young’s modulus, adhesive and adherents’ thicknesses and overlap length. The substrates were made from steel; the adhesive is an epoxy resin. The study was carried out using ANSYS Finite Element software where the first ten modes were extracted. The results obtained from an experimental test conducted by the same workgroup were used to validate the numerical results. Based on the numerical parametric study, the results have shown a dominant influence of the substrates’ thickness and the overlap length: the natural frequency increases remarkably with those two parameters. Moreover, the frequencies of the first ten modes were found to be very sensitive in increasing with the increment of either the adherents’ thicknesses or the overlap length. On the other hand, the influence of the adhesive Young’s modulus was found to be very slight on increasing the natural frequencies for all modes while the adhesive thickness was found to have quite no influence for the first couple of modes, with a slight decrement of frequency for higher modes. Finally, by setting the latter parameters to reference values, a unified parameter function of overlap length and adherent thickness was defined and approximated, and analytical relations for natural frequencies of the first ten modes were established.

## Keywords

Modal analysis Adhesive Double-lap joint Finite element## List of symbols

*a*Overlap length (mm)

*L*Length of plates (mm)

*E*Adhesive Young’s modulus (GPa)

*t*_{a}Adhesive thickness (mm)

*t*_{c}Central plate’s thickness (mm)

*t*_{e}Exterior plates’ thickness (mm)

*W*Structure width (mm)

*ω*Natural frequency of the structure (Hz)

## 1 Introduction

Adhesive bonding is a joining technique that is becoming widespread in the world of industry. Actually, it offers many advantages over traditional means of assembling such as bolting, riveting or welding. Structures are light; assemblies’ preparation is simple and time-saving and stresses are distributed on a large surface of contact. Such assemblies have many applications especially in automotive engineering and aeronautics where they are permanently subjected to many types of loading, one of them is vibration. This explains the interest of many engineers and researchers to study those structures under vibration and this is where the current work is useful for application.

Saito and Tani [1] established an analytical model to study the natural frequencies and the loss factor of single-lap joint beams under coupled axial and bending vibration. Khalil and Kagho [2] developed a non-destructive vibrational approach to detect defects, such as voids and disjoints, in a single lap joint through resonant frequencies’ measurements. The analytical governing equations of single lap jointed beams with viscoelastic adhesive under transverse and longitudinal vibration constituted the main concern of He and Rao [3]. They presented in [4] the numerical solution of their model. A similar job was done by Rao and Zhou [5] on tubular joints; they investigated, in addition, the effect of structural parameters and material properties of the adhesive on the modal loss factor and the resonant frequencies. Numerically, Ko et al. [6] have established a finite element formulation based on isoparametric adhesive interface element where responses of each substrate and the adhesive layer were found separately. Later on, Lin and Ko [7] extended the latter analysis and applied it to a cantilevered stepped bonded plate. Yeh and You [8] have studied experimentally and analytically single-stepped lap composite laminated joints in order to extract the fundamental frequency. They examined the effect of adhesive thickness and fibers orientations. A 3D FEM technique through ABAQUS was applied by He and Oyadiji [9] on a cantilevered single lap adhesively jointed beams where the effect of Young’s modulus and Poisson’s ratio of the adhesive on the natural frequencies and mode shapes of transverse vibration were investigated; in addition they studied the effect of the adhesive strength. Vaziri et al. [10] evaluated analytically the dynamic response of single lap bonded joint subjected to out-of-plane harmonic force; they found that the system is less sensitive to a certain margin of adhesive loss factor and also to the void’s size in the adhesive layer while the location of this void had a remarkable influence. Shear and peel stresses in the overlap region were also obtained. In the same context, Vaziri and Nayeb-Hashemi [11] repeated almost a similar study but for tubular joint under an axial dynamic load; they investigated also properties and geometries for the elastic substrates and viscoelastic adhesive. The same authors studied in [12] theoretically and experimentally the dynamic response of a composite adhesively-repaired beam under harmonic peel loading, a FEM validation was also carried out. Gunes et al. [13] investigated, using two numerical approaches: finite elements and artificial neural network, the effect of geometrical parameters on the free vibration of functionally graded single lap joint. They found also that the first ten modes were insensitive to Young’s modulus, Poisson’s ratio and density of the substrates. An optimal sizing of the joint was also developed. In [14], they carried out similar study but for substrates composed of ceramic (Al_{2}O_{3}) and metal (Ni) varying them through the thickness. Torsional vibration study of single lap joints was the main concern of He [15] using both numerical and experimental approaches: he found that the natural frequencies increase with Young’s modulus of the adhesive and are insensitive to the Poisson’s ratio. The stiffness of the adhesive has a huge influence on the torsional mode shapes. Garcia-Baruetabeña and Cortés [16] developed an experimental procedure to study the influence of geometrical parameters of a bonded metallic beam on its vibrating behavior. They determined resonant frequencies, amplitudes and loss factors. The relationship between vibration and fatigue was tackled by Du and Shi [17] using steel-aluminum single lap joint: they found that when vibration fatigue cycles increase the modal frequencies decrease. They investigated numerically the effect of adhesive Young’s modulus and the contact area of bonding. Samaratunga et al. [18] developed an analytical model called wave spectral finite element to examine the wave propagation in a single lap adhesively composite beam; this model was numerically validated by a finite element method through ABAQUS. However, one can easily remark that most of the works carried out in vibration study have already used the single lap geometry (SLJ) as a default specimen; the double-lap geometry (DLJ) was rarely investigated under such phenomenon. Nevertheless, this latter geometry offers a remarkable advantage over SLJ by exhibiting two planes of symmetry which lead to avoid coupling between axial and bending modes. Maybe the analytical model developed by Challita and Othman [19] is one of the rare works where the shear response was determined for a DLJ subjected to harmonic axial loading using the improved shear lag model. This geometry was studied under impact but not under vibration. In [20], the influence of some adherents’ properties on natural vibration of a double-lap joint was examined. One can cite many numerical works carried out in this field [21, 22, 23, 24, 25] and experimental works [26, 27]. In those works, both metallic and composite substrates were considered. In this paper, a parametric numerical finite element study will be carried out on ANSYS to investigate the natural frequencies (ω) and mode shapes of a DLJ structure made of steel as substrates and an epoxy resin as adhesive. Impulse Excitation Technique (IET) was used to measure experimentally the natural frequencies of DLJ specimens; it has shown good agreement with numerical simulations with a maximum error of 8%. The parameters to be examined are mechanical: the adhesive Young’s modulus, and geometrical: the adhesive thickness, the adherent thickness, and the overlap length. The study is carried out systematically such that only one parameter changes while all the others are set to reference values. This will allow investigating the effect of each parameter independently from the others and finally, a unified parameter involving all the studied parameters was established; it is useful to estimate the natural frequencies upon a defined configuration of the DLJ. In other words, this parameter could be helpful for design purposes.

## 2 Numerical model

### 2.1 Specimen description

_{a}’, thickness of central plates ‘t

_{c}’ and overlap length ‘a’. It should be noticed that the reference value of ‘a’ was chosen to be 30 mm in order to define a percentage of coverage with respect to the total length of the adherent (100 mm) and hence an overlap ratio of 30% was set. This ratio is an indicator of the overall stiffness of the assembly. In addition, the choice of 0.2 mm for the adhesive thickness was found to be a compromise between much lower values which might create very high peaks of stresses at the edges and lead to crack appearance during vibration and much higher values which weaken the resistance and imply a drop in the overall stiffness of the structure.

Dimensions of the simulation reference model

Parameters | Thickness of exterior plates t | Thickness of central plate t | Length of each of the 3 plates L | Overlap length a | Adhesive thickness t | Plate width W |
---|---|---|---|---|---|---|

Dimensions (mm) | 2 | 4 | 100 | 30 | 0.5 | 30 |

Materials’ characteristics for the simulation reference model

Characteristics | Materials | |
---|---|---|

Steel | Epoxy resin | |

Young’s modulus (GPa) | 205 | 3 |

Poisson’s ratio | 0.29 | 0.33 |

Density (kg/m | 7750 | 1186 |

Shear modulus (GPa) | 79.5 | 11.3 |

### 2.2 Finite element model

Converging study for mesh size

Element size (mm) | 0.036125 | 0.06125 | 0.125 | 0.25 |

Simulation time (min) | 30 | 23 | 20 | 17 |

Min % of time-saving with respect to maximum mesh size | 77% | 35% | 18% | N/A |

Max % of frequency difference with respect to maximum mesh size | 0.1% | 0.02% | 0.0% | N/A |

## 3 Reference model

Bonded assemblies are influenced by a huge number of parameters. To establish a criterion, it is worth as a first step to examine the effect of each parameter separately on the vibrating behavior. This was performed in the present work in order to understand deeply the influence of each parameter and later establish a unified parameter taking into consideration all the analyzed parameters together.

The reference model is a set of geometrical and mechanical characteristics of the specimen. This model will be the base of the parametric study: only one value for one parameter per simulation will be changed, all the other values will remain as set in the reference model (Tables 1 and 2). This avoids getting coupling effects.

Reported natural frequency of the structure for the first ten mode shapes

Modes | Reported frequency (Hz) |
---|---|

Mode 1 | 182.9 |

Mode 2 | 593.97 |

Mode 3 | 811.16 |

Mode 4 | 1079.5 |

Mode 5 | 1817.8 |

Mode 6 | 2026.7 |

Mode 7 | 2948.2 |

Mode 8 | 3307.7 |

Mode 9 | 3702.8 |

Mode 10 | 4214.9 |

Parametric study’s variations

Parameter | Variation | ||||||
---|---|---|---|---|---|---|---|

Adhesive Young’s modulus (GPa) | 0.5 | 1 | 2 | 3 | 5 | 7 | 10 |

Adhesive thickness (mm) | 0.2 | 0.3 | 0.5 | 0.6 | 0.8 | 1 | |

Central adherent thickness (mm) | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

Overlap length (cm) | 1 | 2 | 3 | 5 | 6 | 8 |

In fact, a group of practical values for each parameter was chosen in Table 5. However, it was also gone beyond those values by adding some theoretical values to stay on the safe side in covering a wider range of values; this does not mean that all the below mentioned values are applied in practice.

## 4 Experimental validation

Characteristics of the specimen model tested

Specimen characteristics | Young’s modulus (MPa) adherent | Poisson’s ratio | Density (kg/m | t |
---|---|---|---|---|

Adherent | 200,000 | 0.3 | 7850 | 0.3 |

Adhesive | 500 | 0.35 | 1595 |

## 5 Parametric study

### 5.1 Adhesive Young’s modulus

### 5.2 Adhesive thickness

### 5.3 Central adherent thickness

### 5.4 Overlap length

## 6 Unified parameter

After examining the influence of each of the previous parameters separately on the natural frequencies of the first ten modes, it is worth to define a unified parameter *λ* involving all those parameters and to investigate the evolution of the natural frequencies of each mode in terms of this parameter. Later, analytical expressions of the natural frequencies of the first ten modes could be established and valid just in the studied margin of the above parameters.

*t*

_{c}and overlap length a, and decreasing with the adhesive thickness

*t*

_{a}, one may define the unified parameter according to the equation:

*t*

_{c}and

*t*

_{a}in mm.

*t*

_{c}in Table 5 will be applied value by value (one is set as a reference and the other will be varied) to calculate a group of values for

*λ*. Knowing the corresponding values of the natural frequencies related to different modes, one may plot graphs: frequency vs

*λ*for each mode alone, then drawing the closest fitting line representing the evolution with a corresponding analytical equation. Graphs of Fig. 17 show the evolution of the frequencies in terms of the unified parameter. One may remark that the global shape of the evolution is quite similar for all modes.

Coefficients of the analytical expressions of parabolas corresponding to parameter *λ* for the first ten modes

Modes | Coefficient α | Coefficient β | Coefficient γ |
---|---|---|---|

Mode 1 | 9Ε−0.5 | 0.0427 | 101.3 |

Mode 2 | 0.0002 | 0.1524 | 346.64 |

Mode 3 | 0.0005 | − 0.5168 | 815.28 |

Mode 4 | 0.0002 | 0.4756 | 663.42 |

Mode 5 | 0.0007 | 0.5729 | 949.99 |

Mode 6 | 0.0018 | − 0.1532 | 1134.9 |

Mode 7 | 0.0016 | − 0.0046 | 1985.2 |

Mode 8 | 0.0016 | 0.1694 | 2136 |

Mode 9 | 0.0016 | 0.4461 | 2250 |

Mode 10 | 0.0026 | − 0.9128 | 3157.1 |

Indeed, one may mention many comments on this latter study. First, by observing the expression of the unified parameter with the corresponding units of each quantity, one can say that the unit of *λ* is MN m − 1 which reflects a stiffness. All graphs of Fig. 17 show a significant increase of the natural frequencies with this parameter, which is in line with the physical aspect of the stiffness influence on the natural frequencies.

Moreover, it should be noticed that the established analytical equations are valid for the discussed range of *λ* and thus for the studied ranges of the overlap length and the adherent thickness. Since the influence of the adhesive Young’s modulus and thickness is too slight, one may change their values in a restraint margin without affecting significantly the values of the frequencies.

In addition, the above-established equations, allow designing the geometry of a DLJ structure for the desired frequency, by varying either the overlap length or the adherent thickness or both, depending on the case of study. A compromise could be applied since high overlap lengths increase indeed the natural frequency. But their negative effect is that high-stress concentration will appear in the neighborhood of the adhesive layer, while on the other hand, increasing the adherent thickness will lead to an increase in natural frequency and simultaneously, a more homogeneous stress field in the adhesive layer, however the resulting increase in mass and cost of the structure will play a role in the design.

## 7 Conclusion

A parametric study of a double-lap bonded joint structure was carried out to investigate the effect of many parameters on the natural frequencies and mode shapes of the first ten modes. Firstly, an experimental test was conducted for the same DLJ structure and was validated numerically for the model. Then, four parameters were varied: adhesive Young’s modulus, adhesive thickness, adherent thickness, and overlap length.

A unified parameter is elaborated to evaluate analytically approximate values of the natural frequencies for the first ten modes. This unified parameter could be useful for design purposes. In this phase and before manufacturing, modifications can be brought to the structure based on the frequencies according to the targeted application of a similar assembly.

## Notes

### Compliance with ethical standards

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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