# Modal Analysis of Ultrasonic Spot Welding for Lightweight Metals Joining

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

#### 2.1. Steady-State Dynamics under Given Frequency

^{NM}is one term of stiffness matrix considering initial stresses and deformation as expressed by Equation (3). $\Omega $ is the the vibration frequency,

**B**is the strain matrix, u denotes the nodal displacement,

**D**is the elasticity matrix,

^{el}**σ**is the stress tensor under base state, M

_{0}^{NM}is the corresponding term of mass matrix, ρ is the material density,

**N**is the array of shape function of the finite element and

**t**is the surface traction force vector.

_{0}is the averaged vibration magnitude, f is the vibration frequency and t is the welding time. Therefore, the sonotrode force can be defined as the function of peak force F

_{0}, vibration frequency f and welding time t. Welding power can also be calculated through the integration of sonotrode force and velocity as shown in Equations (7) and (8).

#### 2.2. Simulation by Coupled Thermo-Mechanical Analysis

^{2}/K, which linearly changes with pressure within range of 0.1–10 MPa [21,22,23], while the heat transfer coefficient was defined as 30 W/m

^{2}/K [24] for all surfaces that are exposed to the environment.

^{3}. The volumetric heat source q

_{v}denotes the heat generation rate during plastic dissipation, which can be calculated through the integration of plastic work at each material point using Equation (10):

**σ**is the stress tensor and ${\dot{\epsilon}}_{p}$ is the plastic strain rate.

_{f}is the heat generated through frictional work at the interface, and it can be written as Equation (11). It is assumed that 100% of friction work is converted into heat.

#### 2.3. Connection of Simulation Approaches and Experiments

## 3. Results and Discussion

#### 3.1. Al Alloy USW Vibration and Joint Strength

**.**Such a welding condition represents an approximate one-dimensional (1D) problem that can be solved analytically and used to facilitate the understanding and comparison with numerical modeling. It is noted that sonotrode vibration direction is along the longitudinal direction to trigger the resonance vibration at a specific coupon geometry and processing condition. The top sheet was welded without any restraint such that the left end could move freely. The extension length L from sonotrode position to the short edge was defined as a variable influencing the vibration and joint quality. The wavelength λ of ultrasonics propagation in the Al material can be calculated using formula $\mathsf{\lambda}=\frac{1}{f}\sqrt{\frac{E}{\rho}}$ as 252 mm. The results in Figure 4 reassembles the stress wave shape of the material under ultrasonics vibration and stress propagation, with the differences in the node and anti-node positions. For L = ¼ λ and L = ¾ λ, the sonotrode tip has only a few microns of vibration amplitude, appearing to be the node position. Taking the case of L = ¼ λ as an example (red curve), the start point of the horizontal axis, which is the center of sonotrode tip, corresponds to a vibration amplitude of less than 5 μm, and the relative motion at the faying interface is expected to be even smaller. On the other hand, the vibration amplitude of the sonotrode is more than 20 μm when L = 1/2 λ or L = λ. For the case with L = λ (green curve), the sonotrode vibration amplitude is nearly 25 μm under the given ultrasonic power, which is favorable for frictional heat generation at the joint interface.

#### 3.2. Hot Spot Phenomenon in Lightweight Material USW

^{®}with a fixed vibrational frequency of 20 kHz was employed in the present study. The welding machine has a maximum power of 3500 W and clamping pressure of 100 psi. The sonotrode tip has a round shape with a diameter of about 10 mm, and it was preset in the holder with the teeth on the contact surface parallel to the long edge of sheets, since the vibration direction is perpendicular to the long edge. The nominal clamp pressure of the USW machine was preset as 0.414 MPa (60 psi), and the actual load as calibrated was about 1300 N. The anvil has a cylinder shape (diameter D = 16mm) with many tiny sharp teeth on the top surface. Two AZ31 thin sheets in the same dimension (25.4 × 101.6 × 1.0 mm) were stacked up with the long edge aligned and overlapped at a distance of 35–45 mm. The surface of the two magnesium sheets was ground using SiC abrasive paper with a grit size of 80 to remove original painting and oxides. The top surface of the upper Mg sheet was sprayed with high-emissivity painting to facilitate the infrared (IR) measurement. The welding location had the same distance to the two long edges of the top sheet. The Mg sheets were welded under a welding power of 2000 W and processing time of 2.5 s with the ultrasonic welding machine and sonotrode. IR imaging was employed to record the overall surface temperature evolution during USW.

#### 3.3. Mg Alloy Multi-Spot Welding

## 4. Concluding Remarks

- (1)
- The vibration predicted from the modal analysis has good correlation with the 3D transient thermomechanical analysis. When the sonotrode is applied near the node position of the global vibration mode, much lower relative motion and heat generation are predicted. On the other hand, a large amount of heat generation and relative motion are produced for the coupon with an extension length of anti-node vibration.
- (2)
- Compared with coupled thermomechanical analysis, modal analysis offers a far more efficient path to evaluate the joining capability and efficiency under the assumption of elasticity and steady-state response. Only a few minutes were needed for modal analysis to complete a prediction.
- (3)
- The developed modal analysis tool also provides detailed information about structural stress concentration during ultrasonic welding. Hot spots have been observed in the USW of Mg alloy and Al alloy at the condition predicted using the numerical model, which can be related to crack occurrence in lightweight metal joining.
- (4)
- With the aid of modal analysis, hot spots can be pre-screened, and the welding condition can be optimized by means of changing overlap length or applying clamping fixtures.
- (5)
- Multi-spot welding of Mg-alloy thin sheets was simulated via modal analysis, which successfully revealed high shear stress in the sheet with shorter weld spacing as validated by the fracture behavior in the existing bond. Weld spacing can be optimized to avoid high impact force to achieve multiple joints on the same work piece.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Notice of Copyright

## References

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**Figure 5.**Dependence of interfacial relative motion (current simulation) and experimental weld strength [16] on extension length.

**Figure 6.**Temperature field predicted through coupled thermomechanical analysis. (

**a**) L = 63 mm. (

**b**) L = 126 mm.

**Figure 8.**Comparison of modeling and experiment results: (

**a**) peak temperature on top Al sheet and (

**b**) relative motion at the mating interface.

**Figure 9.**Predicted stress distribution under harmonic excitation and IR imaging of temperature rise in 1.0 mm AZ31 USW.

**Figure 10.**Modeling and experiment of AA5754 sheet USW: (

**a**) coupon dimension and welding configuration; (

**b**) transient stress distribution predicted using model; (

**c**) IR image during USW in grayscale.

**Figure 11.**Mitigation of hot spots through applying fixtures on the two ends of top sheet: (

**a**) stress prediction via modal analysis; (

**b**) experiment validation of AA5754 USW.

**Figure 12.**Computational vibration mode and shear stress distribution for even-spacing (a = 25.4 mm) ultrasonic spot welding: (

**a**) configuration of weld spots; (

**b**) vibration amplitude of the sheets and interfacial motion; (

**c**) shear stress distribution induced by ultrasonics.

**Figure 13.**Relative motion and maximum shear stress at the faying interface, predicted via the modal analysis: (

**a**) Relative motion (

**b**) Interfacial shear stress.

**Figure 14.**Temperature field and history measured via IR camera: (

**a**) IR image in welding with spacing 25.4 mm; (

**b**) IR image in welding with spacing a = 31.8 mm; (

**c**) welding temperature in case a = 25.4 mm; (

**d**) welding temperature in case a = 31.8 mm.

**Figure 15.**Failure of existing welds due to ultrasonic vibration in multi-spot AZ31 ultrasonic welding.

Temperature (°C) | 25 | 205 | 315 | 370 |

Young’s modulus (GPa) | 68 | 59 | 47 | 38 |

Yield strength (MPa) | 275 | 130 | 34 | 21 |

Heat conductance (W/m/°C) | 167 | 193 | 206 | 217 |

Heat capacity (J/kg/°C) | 896 | 1030 | 1078 | 1104 |

Material | AZ31 Mg Alloy | AA5754 | ||
---|---|---|---|---|

Overlap length | 35 | 40 | 45 | 40 |

Model prediction | No | Yes | No | Yes |

Experiment result | No | Yes | No | Yes |

Material | AZ31 Mg Alloy | AA5754 | ||
---|---|---|---|---|

Hot spot Location | Left | Right | Left | Right |

Model prediction | 32.9 | 31.0 | 27.9 | 31.1 |

Experiment result | 31.3 | 31.8 | 30.9 | 31.5 |

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**MDPI and ACS Style**

Huang, H.; Chen, J.; Feng, Z.; Sun, X.
Modal Analysis of Ultrasonic Spot Welding for Lightweight Metals Joining. *Metals* **2023**, *13*, 1735.
https://doi.org/10.3390/met13101735

**AMA Style**

Huang H, Chen J, Feng Z, Sun X.
Modal Analysis of Ultrasonic Spot Welding for Lightweight Metals Joining. *Metals*. 2023; 13(10):1735.
https://doi.org/10.3390/met13101735

**Chicago/Turabian Style**

Huang, Hui, Jian Chen, Zhili Feng, and Xin Sun.
2023. "Modal Analysis of Ultrasonic Spot Welding for Lightweight Metals Joining" *Metals* 13, no. 10: 1735.
https://doi.org/10.3390/met13101735