## 1. Introduction

A haptic actuator is a component in mobile devices used to transmit tactile signals to the users. The main function of this component is to vibrate, when the mobile device receives messages, e-mails, social network service notifications, and so on. High definition haptic technology has been developed in recent years to transmit a host of dynamic expressions to the user. To realize this, a thin spring is required together with a strong electromagnet for achieving a high acceleration and wide frequency band. However, springs that deliver a realistic tactile feeling are less durable, as they are thin and structurally weak. In particular, drop impact-induced deformation is a typical type of damage observed in the springs, and several studies are being conducted in the industry to analyze this aspect. Due to the high velocity and small size, it is difficult to observe the impact behavior and identify the vulnerabilities of the springs. Therefore, a new analytical approach is required for the design that fulfils the reliability criterion of acceleration variation, which is only 10% when falling at 1.8 m.

Prior academic research has mainly focused on the drop impact tests and analysis of the finished product. The product level differs from the parts level, as it considers the key factors, such as the strain rate and damping modeling, in detail. However, the findings of some of these studies are be summarized to understand the research direction and analysis method.

Goyal et al. [

1] designed a new impact tester and analyzed the multiple impacts of portable electronic products. They constructed an automated system for repeatability, which demonstrated that the impact direction could be regulated while fulfilling the free-drop conditions.

Lim et al. [

2] used ABAQUS/Explicit to investigate the drop impact of an electronic pager. They analyzed the strain and impact force of the pager housing and verified the exceptional correlation between the results of the physical experiment and the computer-simulated analysis. Furthermore, they determined the impact direction and height that could result in critical damages.

Zhu [

3] performed the finite element analysis to evaluate the reliability of a mobile phone. In this work, three impact modes (disengagement of the battery snap fit, preload spring failure, and ball grid array (BGA) solder crack) were numerically modeled and their correlation with the experiment was studied. Finally, the strain rate effect, impact contact force, and impact acceleration were individually confirmed in each mode.

Kim et al. [

4] performed the drop impact simulation of a cell phone using the LS-DYNA explicit code. To verify the analytical model experimentally, both the global and local impact responses were confirmed with a high-speed camera. They also predicted the potential damage of the mobile phone using a statistical analysis in their experiment.

Karppinen et al. [

5] compared the product level and board level drop tests, because the drop tests of handheld products demonstrated different results depending on the enclosure, impact orientation, strike surfaces, and mounting of the board. They also compared the mechanical impact responses, which were measured by laser vibrometry, and the acoustic excitation method was used for the analysis. Though the impact load delivered to the board was different in each test, the default failure mode was observed to be the same.

Mattila et al. [

6] evaluated the drop impact responses of eight smartphones from various manufacturers. A small accelerometer and a strain gauge were attached inside the smartphones to measure their drop impact response. The maximum strain, average of the maximum strain rate, frequency of the mode shape, and the maximum deceleration were calculated for each product.

The haptic actuator spring is 9 mm in diameter and 0.3 mm in thickness and weighs only 0.1 g (

Figure 1). The moving mass, which is approximately twice that of the spring mass, is attached to the spring resulting in serious damage after the primary impact due to an internal secondary impact. Choi et al. conducted a study related to the drop impact of haptic actuators [

7]. They developed their finite element model using material properties, such as the microtensile strength and damping ratio, and compared the impact deformation and force using a drop test. However, research on the high strain rate effect and damping modeling, which can be used to improve the accuracy of an analytical model, are limited.

In this study, the Johnson–Cook model considering wave propagation was applied considering a high strain rate. A 2-step analysis was conducted for improving the calculation efficiency, and damping was modeled for each stage. Consequently, an analytical model was proposed after comparing the results from the analysis of a miniature haptic actuator with the respective experimental values.

## 3. 2-Step Analysis Modeling

The impact analysis using the explicit solver required considerable amount of computation time and resources as it involved several iterations. When an electronic device is analyzed, the impact is first applied to the outside and then transmitted to the inside. Therefore, a full-scale analysis requires considerable amount of time to calculate these two impacts in the transmission process. The calculation of a single drop impact of the haptic actuator specimen required 6 h and used five CPU cores. However, dividing the impact into separate external and internal impacts and analyzing them as two steps reduced the calculation time to 0.5 h each. This method can be used for efficient and repeated analysis by changing the 3D model or material properties in the design process.

The drop impact model of the haptic actuator is shown in

Figure 4. The specimen was modeled using a 20° tilt from the sensor surface to simulate an experiential worst case of the impact force. Under this condition, the center of gravity of the dummy phone was perpendicular to the ground. In addition, there was almost no drop impact under perpendicular to the ground. Therefore, the model was modified by titling the specimen sideways by 5°. This combination was applied, as these two angles (20° and 5°) were most frequently observed in a series of repeated impact tests.

#### 3.1. First Step—External Impact

The purpose of this stage was to extract the velocity history of the attached surface of the haptic actuator, when the specimen collided with an impact force sensor (PCB, ICP® quartz force sensor, 200C50).

The impact force for the initial analytical model is shown in

Figure 5. A peak force of 9.69 kN is observed in this figure; however, an undamped residual force vibration was confirmed. The abnormal attenuation was controlled by damping modeling, as energy dissipation could result in an abnormal rise in stress. The residual force vibration can be used to discover the dominant frequencies with high amplitude using fast Fourier transform (FFT), as shown in

Figure 6. The modeling was performed to attenuate the main vibration frequency (14627 Hz) of the residual force vibration.

The impact force measured in the drop test is shown in

Figure 7. The impact force is estimated to be 8.24 kN, when the specimen collides with the impact force sensor. The second and third peak forces in the residual force vibration are observed to be 1.58 and 1.19 kN, respectively. These values are used in the computation of the damping coefficient (

$\mathsf{\xi}$ = 0.045) using the logarithmic decrement method.

The Rayleigh damping C used in the analysis is defined as follows [

12]:

where M is the mass matrix, K is the stiffness matrix, and

$\mathsf{\alpha}$ and

$\mathsf{\beta}$ are constants. As the dominant frequency of the residual force vibration and the damping ratio were measured during the test, the value of the constant

$\mathsf{\alpha}$ can be calculated as follows:

where the value of

$\mathsf{\alpha}$ is 8301. The analyzed impact force obtained by applying this value is shown in

Figure 7. The peak value of the force is observed to be 9.42 kN, which is 3% less than that before the damping modeling. An error of 14% resulted from the test, because the Johnson–Cook model of the aluminum dummy was not measured.

Figure 8 compares the impact strain from the test and plastic strain from the analysis. It is evident from this figure that the type of deformation observed during the test agrees well with that observed during the analysis, and the maximum strain of 0.11 was confirmed by the analysis.

#### 3.2. Second Step—Internal Impact

The purpose of this stage was to identify the stress distribution and plastic strain in the haptic actuator based on the velocity history obtained from the first step.

The x-, y-, and z-axis velocity history of the haptic actuator attached to the surface of the dummy is shown in

Figure 9. All the components initiate the drop at an initial velocity of −5.88 m/s in the y-axis. Therefore, no disturbance was experienced before the impact. The event is initiated at 0.340 ms, and the occurrence of elastic impact is observed over 0.115 ms. After a threshold of −2.63 m/s, the plastic impact is observed at the maximum repulsion rate of +4.27 m/s for a short duration of 0.01 ms. The y-axis velocity converges to an average of 2.57 m/s after the loss of contact, while the starting point of the velocity variation demonstrates a time delay of 0.022 ms from the impact contact of the dummy.

Generally, springs demonstrate a minimal amount of damping, as they induce elastic deformation for mechanical purposes. However, during the analysis of the input velocity history, the undamped spring (shown in

Figure 10) exhibited no attenuation in residual vibration. Therefore, an FFT was performed in the frequency domain for modeling the damping. The results from this FFT are shown in

Figure 11. At low frequencies, a giant wave is generated with high frequency vibrations. The damping was modeled using a high frequency (4410 Hz), which was dominant during the energy dissipation. The damping ratio (

$\mathsf{\xi}$) required for the modeling was 0.02, which was based on authors’ previous measurements [

7]. The constant

$\mathsf{\alpha}$ for the mass matrix is calculated as 1180 using Equation (3). The y-displacement of the damped spring is shown in

Figure 10. The maximum equivalent stress of the spring is estimated to be 1803 MPa while undamped, which declines by 4% to 1724 MPa after the damping modeling. The moving mass attached to the spring collided with the housing in the −y direction, and this moment of stress distribution is shown in

Figure 12. A high stress distribution of approximately 1700 MPa is observed in the upper notch, where tensile stress is applied to the spring. In contrast, a relatively low stress distribution of approximately 1200 MPa is observed in the lower notch. The figure shows four sites, in which the damage is expected to exceed the yield stress of 1377 MPa. The yield stress is observed to be 18% higher than 1170 MPa at a strain rate of 0.001/s (shown in

Figure 3) in the tensile tests.

The effective plastic strain distribution of the spring is shown in

Figure 13. At each position exceeding the yield, a maximum of 0.161 strain is observed for 3–6 elements, each with a size of 0.1 mm. The quantitative plastic strain of each element is shown in

Figure 14. The occurrence of the plastic strain is observed mainly in the spiral notch region, where the upper and lower plates are connected. The upper site (+y direction in

Figure 13) demonstrates a higher strain and wider stress distribution. In addition, it was confirmed that the notch connected to the upper plate demonstrated a higher plastic strain than the one in the lower plate, because the moving mass caused a secondary impact on the inner housing. The concentration of the plastic strain in the local region was confirmed by this analysis. If the impact load was applied repeatedly, the plastic deformation would increase possibly resulting in malfunction.