1 Introduction

In recent years, the areal recording density of hard disk drives (HDDs) is increasing rapidly for the continual demand of mass storage, which requires ultra-low flying height, more accurate seeking position, and higher disk rotation speed. However, the high speed rotary flow inside the narrow HDDs space inevitably induces vibrations on the Head Gimbals Assembly (HGA), which may lead to some undesirable problems, such as positioning error or even devastating head-disk crash. In order to enhancing the robustness of drives in the operational process, it is very important to mitigate the flow-induced off-plate vibration (FIOPV) on the HGA. Some previous works have been focused on the flow-induced vibration issues inside the drives (Ikegawa et al. 2006; Ng et al. 2011; Shimizu et al. 2003; Tsuda et al. 2003; Aruga et al. 2007; Zhang et al. 2010), including mechanism analysis and structure optimization studies of the vibrations. However, they were passive methods to alleviate the vibrations and would not be controllable due to different fluid fields inside the drives. Recently, Min et al. (2012a, b, 2013a, b, c, 2014a, b) systematically studied the relationship between the flow pressure fluctuations around the HGA and the vibrations of the HGA. They introduced a piezoelectric transducer (PZT) which can be used to transform an electric current to an acoustic pressure field, glued onto the inner surface of the HDD cover to generate acoustic pressure to suppress the HGA flow-induced vibration in their experiments. Their results provided a new method to use acoustic pressure to balance the flow pressure fluctuations. However, there exist some deficiencies in their work. Due to the hole drilled on the drive cover and the slot cut on the side base, the flow field around the HGA may have changed. In addition, the 2-D model they used cannot truly represent the real 3-D flow field inside the HDD, and the pressure redistribution is not clear due to the introduced PZT. Moreover, the experimental investigations are hardly carried out to study the helium-filled HDDs by the similar method because of the difficulty of tight seal (Kil et al. 2012a, b). Yet till now, there are no simulation results in mitigating the vibrations of the HGA inside air-filled and helium-filled drives by the introduced acoustic pressure method.

In this paper, the flow-induced off-plate vibration (FIOPV) of the HGA and its suppression by an acoustic PZT actuator in the air-filled and helium-filled drives are studied respectively, using the fluid–structure and acoustics-piezoelectric-structure interaction simulation method. Firstly, a three-dimensional finite element model of HDD is built with an acoustic PZT actuator. Then, the flow-induced vibration spectrum of the HGA is analyzed using the fast Fourier transform algorithm. After that, by designing an acoustic PZT actuator on the HDD cover inner surface, the acoustic pressure induced vibration (APIV) of the HGA is analyzed. Finally, the suppressed vibration after superposing APIV and FIOPV is obtained by following the superposition principle.

2 Methodology

A geometrical model of a 2.5-inch HDD with the HGA positioned at the middle diameter, which was built by our previous work (Zhang et al. 2015), is shown in Fig. 1. The length of the slider is 0.85 mm and its width and height are 0.70 and 0.23 mm, respectively. The initial gap between slider and disk is 0.17 mm. The disk is rotated at 10,000 rpm. An acoustic PZT actuator with a diameter of 20 mm and thickness of 1 mm was designed on the HDD cover inner surface as shown in Fig. 1b. The investigated filling gases in the HDD include air and helium, and their physical properties are listed in Table 1.

Fig. 1
figure 1

a Geometric model of a 2.5-inch HDD and b acoustic PZT actuator

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Table 1 Physical properties of air and helium used in this work
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The suppression of the FIOPV of the HGA in this work is a typical multi-physics filed interaction issue involving fluid–structure interaction (FSI) and acoustics-piezoelectric-structure interaction. The commercial software ANSYS 15.0 (ANSYS Inc. 2013) is used to perform the simulation. The material properties of the drive model are presented in Table 2. Due to the complexity of the HDD geometry and the interaction simulation, the work is divided into three parts: (1) FSI analysis to obtain the FIOPV; (2) acoustics-piezoelectric-structure interaction analysis to study the APIV; (3) superposition of the FIOPV and APIV.

Table 2 Material properties used in this work
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Part 1 Computational fluid dynamics (CFD) simulation is carried out to calculate the aerodynamic forces acting on the arm and the HGA. In order to simulate the flow perturbations accurately, the large eddy simulation (LES) model is used (Zhang et al. 2010). The important parameters including mesh size, mesh quantity, y + and time step should be properly valued before performing LES simulation. In our simulation, the Kolmogorov length scale η = 0.026 mm is calculated by

$$\eta = \left( {\frac{{\nu^{3} }}{\varepsilon }} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}}$$
(1)

where ν is the kinematic viscosity, ε is the turbulent dissipation rate. The mesh size near the HGA we used is 0.05 mm which is about twice of the Kolmogorov length scale. In addition, there are at least three nodes in the shortest edge. The time step for the CFD simulation is set to 10 μs, which is about one-fifth of the Kolmogorov time scale τ η  = 51 μs calculated by

$$\tau_{\eta } = \left( {\frac{{\nu^{3} }}{\varepsilon }} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}$$
(2)

The equiangle skew of grid cells near the HGA, Q EAS , ranges from 0 to 0.35 calculated by

$$Q_{EAS} = max\left\{ {\frac{{\theta_{max} - \theta_{e} }}{{180 - \theta_{e} }},\frac{{\theta_{e} - \theta_{min} }}{{\theta_{e} }}} \right\}$$
(3)

where, θ max and θ min are the maximum and minimum angles between the edges of the mesh element, and θ e is the characteristic angle corresponding to an equilateral cell of similar form. Boundary layer mesh on surfaces is constructed. The y + of the near-wall mesh cell is reasonable. Therefore, the LES model is considered acceptable in our simulation.

The RNG k − ε turbulent model is performed for the air flow in steady state simulation, while the laminar model is used for the helium flow (Aruga et al. 2007). Then, the LES method is carried out for the transient simulation, and the subgrid-scale flow structures are modeled using a dynamic adaptation of the Smogrinsky-Lilly model (Kazemi 2008). The spatial and time derivatives are approximated using bounded central-differencing and second order upwind discretization schemes. The transient computations are executed for 50 ms so as to allow flow reach to statistically steady. Thereafter, the aerodynamic forces are collected for 200 ms to conduct the subsequent structure vibration analysis. The spectrum of FIOPV is calculated with the fast Fourier transform algorithm.

Part 2 PZT-5H which has good piezoelectric properties is chosen as an acoustic pressure excitating source in this work. Its material properties are as follows:

$$\left[ \varepsilon \right] = \left[ {\begin{array}{*{20}c} {1.503} & 0 & 0 \\ 0 & {1.503} & 0 \\ 0 & 0 & {13} \\ \end{array} } \right]\;10^{ - 8} \;\left( {{{\text{F}} \mathord{\left/ {\vphantom {{\text{F}} {\text{m}}}} \right. \kern-0pt} {\text{m}}}} \right)$$
$$\left[ c \right] = \left[ {\begin{array}{*{20}c} {127.21} & {80.20} & {84.67} & 0 & 0 & 0 \\ 0 & {127.20} & {84.67} & 0 & 0 & 0 \\ 0 & 0 & {117.44} & 0 & 0 & 0 \\ 0 & 0 & 0 & {22.99} & 0 & 0 \\ 0 & 0 & 0 & 0 & {22.99} & 0 \\ 0 & 0 & 0 & 0 & 0 & {23.47} \\ \end{array} } \right]\;\left( {\text{GPa}} \right)$$
$$\left[ e \right] = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & {17} & 0 \\ 0 & 0 & 0 & {17} & 0 & 0 \\ { - 6.5} & { - 6.5} & {23.3} & 0 & 0 & 0 \\ \end{array} } \right]\;\left( {{{\text{C}} \mathord{\left/ {\vphantom {{\text{C}} {{\text{m}}^{2} }}} \right. \kern-0pt} {{\text{m}}^{2} }}} \right)$$

where [ε] is the dielectric constant matrix, [c] is the elastic coefficient matrix, and [e] is the piezoelectric stress matrix. The density of PZT-5H is 7600 kg/m3 and the Poisson’s ratio is 0.28.

The acoustics-piezoelectric-structure interaction in this work comprises three fundamental physics: acoustic pressure, solid mechanics and electrostatics. The wave equations for sound propagation in the HDD filling with gas medium can be described by an equation in the Cartesian coordinate systems as (Yi et al. 2001):

$${\nabla }^{2} p\left( {x,t} \right) - \frac{1}{{c^{2} }} \cdot \frac{{\partial^{2} p\left( {x,t} \right)}}{{\partial t^{2} }}$$
(4)

where ∇ is the Laplace operator, p denotes the acoustic pressure in terms of the receiver position x and time t, and c represents the speed of acoustic wave in the gas medium. The acoustic speed in air and helium are taken as 343 and 923 m/s as listed in Table 1. While the maximum speeds of air flow and helium flow are much <1 Mach at the disk rotation speed of 10,000 RPM, therefore c can be assumed to equal the value in the static flow (Min et al. 2013a).

The solid mechanics and electrostatics physics are coupled via the constitutive equations that correlate stresses and strains to electric displacement and electric field. These physics interfaces solve for the balance of body forces σ and volume charge density D respectively as shown in Eqs. (5) and (6):

$${\nabla } \cdot \sigma = 0$$
(5)
$${\nabla } \cdot D = 0$$
(6)

At the interface between the gas medium and solid domain, the normal acceleration of the solid (PZT) surface is used to drive the gas domain. This is described by the following equation:

$$n \cdot \left( {\frac{1}{\rho }\left( {{\nabla }p} \right)} \right) = a_{n}$$
(7)

where n is denotes the normal direction, ρ is the density of medium, a n is the normal acceleration.

The acoustic pressure at the interface between the gas medium and solid domain acts as a boundary load on the solid:

$$n \cdot \sigma = p$$
(8)

The boundaries of the cover inner surface, disk surface and HGA surfaces are assumed to be acoustically rigid for the acoustic field.

Part 3 In structural analysis, the motion of HGA induced by flow pressure fluctuations can be described as linear vibration (Liu et al. 2011). From this, the motion of HGA induced by acoustic pressure oscillations can be considered to be linear vibration (Min et al. 2013a). These vibrations can be divided into a number of harmonic vibrations. Based on the vibration theory, the superposition principle can be applied in linear vibration system. The APIV of up suspension tip is calculated in the static flow in the harmonic response simulation. And then, the total vibration x of up suspension tip after suppressing is obtained by superposing the only APIV and the only FIOPV, which can be written as:

$$x = x_{1} + x_{2}$$
(9)
$$x_{1} = A_{1} \cdot cos\left( {\omega t + \varphi_{1} } \right)$$
(10)
$$x_{2} = A_{2} \cdot cos\left( {\omega t + \varphi_{2} } \right)$$
(11)

where A i is the vibration amplitude, ω is the vibration frequency, φ i is the vibration phase, and subscripts 1 and 2 denote the APIV and FIOPV, respectively. The total vibration can be rewritten as:

$$x = A_{1} \cdot cos\left( {\omega t + \varphi_{1} } \right) + A_{2} \cdot cos\left( {\omega t + \varphi_{2} } \right) = A \cdot cos\left( {\omega t + \varphi } \right)$$
(12)

where

$$A^{2} = A_{1}^{2} + A_{2}^{2} + 2A_{1} A_{2} cos\left( {\varphi_{2} - \varphi_{1} } \right)$$
(13)
$$tan\varphi = \frac{{A_{1} sin\varphi_{1} + A_{2} sin\varphi_{2} }}{{A_{1} cos\varphi_{1} + A_{2} cos\varphi_{2} }}$$
(14)

The PZT exciting voltage amplitude and phase are denoted by A 3 and φ 3, respectively. From Eq. (13), the phase difference between the APIV and FIOPV should be 180°, thus cos(φ 2 − φ 1) = −1, to achieve a good suppression on the vibration of the up suspension tip. That is,

$$\left| {\varphi_{2} - \varphi_{1} } \right| = 180$$
(15)

Since the FIOPV phase φ 2 is already obtained in part 1, the required APIV phase φ 1 can be easily calculated. In turn, the required PZT exciting voltage (A 3, φ 3) can be decided, since the PZT exciting voltage is the source of above APIV.

Considering the extreme complexity of multi-physics filed interaction involving fluid–structure-acoustics-piezoelectric, in this simulation, the suppressed vibration of up suspension tip is calculated by superposing the only FIOPV and the only APIV, since the flow field has no obvious effects on the acoustic field in HDD at disk rotation speed of 10,000 rpm (Min et al. 2013a). In details, firstly, we calculate the flow-induced off-plate vibration of HGA, to find out the principal frequencies where large vibration amplitudes appear. Secondly, it tries to design proper PZT exciting voltage amplitude and phase, to efficiently suppress the peak vibration amplitudes at above principal frequencies. Finally, it calculates the superposition of above obtained APIV and FIOPV.

3 Results and discussion

3.1 Flow-induced vibration of HGA

In the structure vibration analysis of FSI, the timestep is set to 20 μs, which corresponds to a Nyquist frequency of 25 kHz. Figure 2 shows the FIOPV of up suspension tip as a function of frequency. It can be seen that the spectrum appears two principal frequencies named as peak1 (1760 Hz) and peak2 (3245 Hz) in the both air and helium-filled drives. Moreover, the first principal frequency (1760 Hz) which has the maximum vibration amplitude is almost the same as the 1st natural frequency 1759 Hz of arm-suspension. It means that the first mode (1st bending mode) of arm-suspension dominates the FIOPV of the HGA. Figures 3 and 4 show the bode plot of the FIOPV spectrum of up suspension tip at 1760 and 3245 Hz in the drives filling with air and helium, respectively. The corresponding FIOPV amplitudes and phases are listed in Table 3.

Fig. 2
figure 2

Off-plate vibration of up suspension tip induced by flow versus frequency

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Fig. 3
figure 3

Spectra of up suspension tip off-plate vibration only induced by air flow at a, b 1760 Hz and c, d 3245 Hz in the air-filled drive

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Fig. 4
figure 4

Spectra of up suspension off-plate vibration tip only induced by helium flow at a, b 1760 Hz and c, d 3245 Hz in the helium-filled drive

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Table 3 The flow-induced off-plate vibration of up suspension tip
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3.2 Acoustic pressure induced vibration of HGA

To improve the flying head stability, it is necessary to suppress the FIOPV, especially the peak amplitudes at two principal frequencies as shown in Fig. 2. Harmonic response analyses are conducted to find out the proper PZT exciting voltage amplitude A 3 and phase φ 3, which can efficiently suppress the peak vibration amplitudes at above two principal frequencies of FIOPV.

Figure 5 shows the APIV amplitude A 1 and the phase φ 1 of the up suspension tip at 1760 Hz as a function of the PZT exciting voltage amplitude A 3 with the phase φ 3 = 0 degree. It can be obviously seen that the APIV amplitude A 1 of the up suspension tip is directly proportional to the PZT exciting voltage amplitude A 3.

Fig. 5
figure 5

The APIV amplitude A 1 and phase φ 1 of up suspension tip at 1760 Hz versus PZT exciting voltage amplitude A 3 (φ 3 = 0) in the air-filled drive

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Figure 6 also shows a directly proportional relationship between PZT exciting voltage phase φ 3 and resultant APIV phase φ 1 of the up suspension tip at 1760 Hz when the PZT exciting voltage amplitude A 3 chosen as 100 V. And the difference between the PZT exciting voltage phase φ 3 and the APIV phase φ 1 is always 178.33, that is, φ 3 − φ 1 = 178.33 degree.

Fig. 6
figure 6

The APIV phase φ 1 of up suspension tip at 1760 Hz versus PZT exciting voltage phase φ 3 (A 3 = 100 V) in the air-filled drive

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According to the discussion in part 3 of methodology section and Eq. (15), it can get φ 3 − φ 2 = (φ 3 − φ 1) − |φ 2 − φ 1| = 178.33 − 180 = −1.67 degree. Due to the FIOPV phase φ 2 is −92.44 degree as listed in Table 3, the required PZT exciting voltage phase φ 3 is calculated and its value equals to −94.11 degree when the PZT exciting voltage amplitude A 3 chosen as 100 V.

Figure 7 shows that at the second principal frequency of 3245 Hz, the APIV amplitude can achieve 1.19 nm when the PZT exciting voltage A 3 chosen as 10 V. It can be found that the difference between the PZT exciting voltage phase φ 3 and the APIV phase φ 1 is −13.19 at 3245 Hz, that is, φ 3 − φ 1 = −13.19 degree. According to the discussion in part 3 of methodology section and Eq. (15), it can get φ 3 − φ 2 = (φ 3 − φ 1) − |φ 2 − φ 1| = −13.19 − (−180) = 166.81 degree. Due to the FIOPV phase φ 2 is −59.86 degree as listed in Table 3, the required PZT exciting voltage phase φ 3 is obtained as 106.95 degree when the PZT exciting voltage amplitude A 3 chosen as 10 V, as shown in Fig. 8.

Fig. 7
figure 7

The APIV amplitude A 1 and phase φ 1 of up suspension tip at 3245 Hz versus PZT exciting voltage amplitude A 3 (φ 3 = 0) in the air-filled drive

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Fig. 8
figure 8

The calculated PZT exciting voltage for suppressing the peak amplitudes at two principal frequencies of FIOPV in Figs. 3 and 4

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Similarly, the required PZT exciting voltage amplitude A 3 and phase φ 3 in the helium-filled drive are obtained as shown in Fig. 8. V I and V II are used for the air-filled HDD, while V III and V IV are used for the helium-filled HDD. The corresponding APIV spectrum of the up suspension tip induced by above four PZT exciting voltages are shown in Figs. 9 and 10. The corresponding APIV amplitudes and phases are summarized in Table 4.

Fig. 9
figure 9

Spectra of up suspension tip off-plate vibration only induced by PZT generating acoustic pressure at a, b 1760 Hz and c, d 3245 Hz in the air-filled drive

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Fig. 10
figure 10

Spectra of up suspension tip off-plate vibration only induced by PZT generating acoustic pressure at a, b 1760 Hz and c, d 3245 Hz in the helium-filled drive

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Table 4 The acoustic pressure induced off-plate vibration of up suspension tip
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Figure 11 shows the acoustic pressure distribution in the air-filled drive at 1760 and 3245 Hz. It can be seen that the pressure value at 1760 Hz is much bigger than that at 3245 Hz. Figure 12 shows the acoustic pressure distribution in the helium-filled drive at 1760 and 3245 Hz, respectively, and its maximum value at 3245 Hz is much smaller.

Fig. 11
figure 11

Acoustic pressure distribution in the air-filled drive at the frequencies of a 1760 Hz and b 3245 Hz

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Fig. 12
figure 12

Acoustic pressure distribution in the helium-filled drive at the frequencies of a 1760 Hz and b 3245 Hz

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3.3 The superposition of FIOPV and APIV

The superposition of APIV (x 1) and FIOPV (x 2) can be obtained according to the Eqs. (12–14) by following the superposition principle. The superposition results of up suspension tip vibration (x) at two principal frequencies are presented in Fig. 13.

Fig. 13
figure 13

Spectra of up suspension tip off-plate vibration before and after introducing acoustic PZT actuator in the a, b air-filled and c, d helium-filled drives

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From Fig. 13a, it shows that a suppression up to 3.35 nm (15.3 %) of peak amplitude is achieved at 1760 Hz in the air-filled drive. Figure 13b shows that there is a 1.19 nm (96.7 %) reduction of peak amplitude at 3245 Hz in the air-filled drive. And the phases after superposition are obviously changed.

By conducting the superposition in the helium-filled drive (Fig. 13c, d), a suppression up to 4.31 (69.4 %) and 0.10 nm (35.7 %) of peak amplitude is obtained, respectively.

4 Summary

In this paper, a three-dimensional finite element model of hard disk drive (HDD) is built with an acoustic PZT actuator, which is glued on the cover inner surface of the HDD to generate acoustic pressure to suppress the flow-induced off-plate vibration of the HGA. The voltage amplitude 100 and 10 V are used as the PZT exciting voltages for suppressing the peaks amplitude at 1760 and 3245 Hz in the flow-induced off-plate vibration spectrum, respectively.

It indicates that it is effective to suppress the FIOPV by an acoustic PZT actuator. The results show that distinct suppression can be achieved at two principal frequencies. For suppressing the peak amplitude at 1760 Hz in the air-filled drive, a distinct reduction of 3.35 nm (15.3 %) is obtained, while the suppression at 3245 Hz is about 1.19 nm (96.7 %). For the suppression in the helium-filled drive, suppression up to 4.31 (69.4 %) and 0.10 (35.7 %) nm can be obtained at 1760 and 3245 Hz, respectively.