In actuated slider suspension, the tilt motion of the optical actuator causes the variation of focal spot position on the photo-detector, as shown in Figure 2. The middle figure of light intensity distribution is a well-symmetric focal spot on the PD center without tilting angle. If the focusing motion is achieved by z seesaw-type arm which generates a ±0.1° nutating angle, the returning beam would not be orthogonal to the optical sensor. The reflected focal spot on the PD will shift away from the center and the light distributions will approach one side of the photo-diode. Hence, given the increasing accuracy of head positioning, a drive capable of tilt control is necessary to reduce the tilt margin as the optical aberration is correspondingly increased. A compensation scheme is also developed for optical axis in-line calibration, and evaluated by driving a simply bimorph PZT bender.

A common method of piezoelectric actuated suspension is driven along the opposite data (track) tangential direction in an allowable stroke. The proposed seesaw swivel-drive actuator uses a dual-stage leverage mechanism (Figure 3), through use of a coarse movement via VCM control and a secondary fine positioning via PZT. As PZT compensates the tilt angle of seesaw arm in the opposite way—the laser beam will be kept in orthogonal to the disk (Figure 3, left). Hence, PZT tilt control fixes the tangential displacement of focusing, and calibrates the asymmetry of the S-curve. Complete two-axis scanning is then performed by independently controlling the dual stage actuators.

The primary structure with arbitrary distributions of mass, stiffness and damping is subjected to vibration suppression by an absorber subsystem. Sensing variation capabilities in the optic-axis direction are investigated to ensure the suppression of robust vibration. Consisting of two flexible cantilever beams extended from a balanced mass, a cantilever TVA subsystem is illustrated from the perspective of both improving bandwidth and increasing shock protection.

The vibration suppression of structures is based on independent design of dynamic absorbers, while taking into account the selected tuning mass of the subsystem structure. Effective structural stiffness is thus adjusted by adjusting the beam dimensions after a preliminary primitive design. Figure 4 illustrates the structural configuration of this work, which consists of a three-block structure called “soundmetal” that is pinned by a flexible wire.

Figure 5 displays a “sandwich”-like PZT that consists of a carbon fiber chip with two d₃₁-type PZT ceramics. This figure also considers a small piece element in the upper PZT material, in which the corresponding extension can be evaluated as follows: (1) Δ L PZT L PZT   =   d 31 V PZT t PZT

The elongation force ΔF and momentum ΔM generated by the small element in the Figure 4(a) are: (2) Δ M   =   Δ F   ⋅   z   =   d 31 V PZ T b E PZT t PZT zdz

Therefore, integrating Equation (2), the sum of bending moment in Figure 4 generated by the bimorph piezoelectric material is as follows: (3) M   =   d 31 ⋅ V PZT   ⋅   E PZT   ⋅   b   ⋅   ( t PZT   +   t carbon )

Figure 5(a) shows a composed triple-layer beam that consists of both carbon fiber and piezoelectric material. The flexural rigidity of a multilayered beam is: (4) E eq   I eq   =   E PZT I PZT   +   E carbon   I carbonwhere k_PZT is considered the stiffness in rotation direction, and calculated from the equivalent spring on the tip of PZT bender. (5) k PZT   =   k eq   L PZT 2where k_eq in Figure 5(c) is presented as equivalent spring constant. (6) k eq   =   2 E I L PZT 2

Additionally, the rotary angle of PZT is: (7) ϕ   =   M k PZT

The motion of a cantilever beam can be represented as [16]: (8) c 2   ∂ 4   w ( x ,   t ) ∂ x 4   +   ∂ 2   w ( x ,   t ) ∂ t 2   =   0 ,   c   =   E e q I e q m ′

By using the separation of variables method, the general solution can be expressed as (9) w ( x ,   t )   =   X   ( x )   ⋅   T ( t )

Replacing Equation (8) with Equation (9) yields (10) c 2   X ″ ″   ( x ) X   ( x )   =   − T ″ ( t ) T ( t )   =   ω n 2

The general solution of the mode shapes in x-direction can be expressed as (11) X n   ( x )   =   A 1   sin   ( λ n x )   +   A 2   cos   ( λ n x )   +   A 3   sinh   ( λ n x )   +   A 4   cosh   ( λ n x ) ,   λ n 4   =   m ′ ω n 2 E eq I eq

The boundary condition of Equation (10) is: (12) { x   =   0   ⇒   w   =   0 ,   ∂ w ∂ x   =   0 x   =   L PZT   ⇒   E I ∂ 2 w ∂ x 2   =   M   =   0 ,   E I ∂ 3 w ∂ x 3   =   V   =   0

By substituting Equation (12) into Equation (11), the particular solution is (13) X n ( x )   =   cosh ( λ n x L PZT ) − cos ( λ n x L PZT ) −   α ( sinh ( λ n x L PZT ) − sin ( λ n x L PZT ) )where α   =   sinh   ( λ n L PZT )   −   sin   ( λ n L PZT ) cosh   ( λ n   L PZT )   +   cos   ( λ n L PZT )

The natural frequency is solved from the following equation: (14) cos ( λ n x )   cosh ( λ n x )   =   − 1 ;   n = 1 ,   2 ,   3 , …

For n = 1, the fundamental frequency of the bimorph PZT is: (15) ω 1   =   E e q I e q m ′ ⋅ 1.875 2 2 π L PZT 2   ( H z )where m′ = [ρ_PZTt_PZT + ρ_carbont_carbon]b.

From the previous derivations of the equations, two objective functions of the optimization are selected: lateral displacement error E(t_PZT, L_PZT) and fundamental frequency ω(t_PZT, L_PZT). Figure 6 shows the relationship between PZT arm actuator and disk, while Table 1 lists its symbols.

Figure 7 illustrates the optimization procedure, indicating that the PZT bender is optimized. Of the two coupling optimization processes, optimization 1 determines objective functions and sets the initial search interval and accuracy. Meanwhile, optimization 2 copes with constraint conditions for the process. The optimization of length L_PZT and thickness t_PZT, which minimize objective functions to PZT, are 14.13 mm and 0.11 mm after 64 and 59 iterations, respectively.

Figure 8 plots two parameters, rotary angle ϕ [Equation (7)] and fundamental frequency ω₁ [Equation (15)] as a function of effective length. The crossover point is selected to verify the optimization results. Figure 9 illustrates the hysteresis curve of PZT displacement with maximum point is 4 mm when input a sine wave signal is swept from −125 to 125 V. Finally, the experimental results in Figure 10 demonstrate two modes frequency at 985 Hz and 5.8 kHz. Table 2 summarizes the modal analysis results between the numerical optimization and experiments, in which the errors vary by about 1.86% and 3.14%, respectively.

Several attempts have been made to seek optimum absorber parameters when the main system has significant damping. Den Hartog first tackled the optimum solution of a tuned mass damper that is connected to a primary system with the intention of reducing its dynamic response [20]. An optimal broadband attenuation was then achieved by a dynamic absorber design proposed by Ormondroyd and Den Hartog [21]. Based on their fixed-point theory, Brock found the optimum tuning parameter and defined the optimality for the optimum absorber damping [22]. By adopting a frequency locus approach, Thompson found the optimal damper parameters that minimize the structural response [23].

The equation of the damped model in Figure 11 can be expressed as (16) { m sys x ¨ 1   −   c abs   ( x ˙ 2   −   x ˙ 1 )   −   k abs   ( x 2   −   x 1 )   +   c sys x ˙ 1   +   k sys x 1 = f 0 m abs x ¨ 2   +   c abs   ( x ˙ 2   −   x ˙ 1 )   +   k abs   ( x 2   −   x 1 ) = 0where m_sys, c_sys and k_sys denote the mass, damping and stiffness coefficient of primary actuator system; m_abs, c_abs, and k_abs refer to the mass, damping, and stiffness coefficient of tunable vibration absorber system; f₀ and ω denote the amplitude and frequency of the exciting force, respectively; and x₁ and x₂ represent the displacement of the actuator and the absorber, respectively. The normalized amplitude of the steady-state response of the primary mass can be expressed as: (17) { x 1 = X 1 e j ω t x 2 = X 2 e j ω t f 0 = F o e j ω t

The amplitudes of the system displacement, X₁, are expressed as: (18) X 1   =   ( k abs   −   m abs ω 2 )   +   j ω c abs { ( k sys   −   m sys ω 2 ) ( k abs   −   m abs ω 2 )   −   ( k abs m abs   +   c sys c abs ) ω 2 +   j ω [ c abs   ( k sys   −   m sys   ω 2   −   m abs ω 2 )   +   c sys   ( k abs   −   m abs ω 2 ) ] } ⋅   F 0

Taking the absolute of X₁: (19) | X 1 |   =   F 0 [ ( k a m a − ω 2 ) 2   +   ( C a m a ω ) ] 2   ( k sys m sys ) 2 [ ( k sys m sys − ω 2 ) ( k a m a − ω 2 ) − ( k a m a m a m sys + C a m a C sys m sys ) ω 2 ] 2   +   [ C a m a ( k sys m sys − ω 2 − m a m sys ω 2 )   +   C sys m sys ( k a m a − ω 2 ) ] 2

Substituting the natural frequency, ω abs   =   k abs m abs and ω sys   =   k sys m sys, into the Equation (19), the normalized transfer function of an actuator is (20) G   =   | X 1   ⋅   k sys | | f 0 | = | X 1 | ⋅ k sys F 0   =   A 2   ( α ,   β )   +   4 B 2   ( ζ abs ,   α ,   β ) C 2 ( μ ,   ζ sys ,   ζ abs ,   α ,   β )   +   4 D 2   ( μ ,   ζ sys ,   ζ abs ,   α ,   β )where: A ( α ,   β )   =   1 − α 2 β 2 ,   B ( ζ abs ,   α ,   β )   =   ζ abs α β C ( μ ,   ζ sys ,   ζ abs ,   α ,   β )   =   α 4 β 2 − ( 4 ζ sys ζ abs β   +   1 β 2 + μ + 1 ) α 2   +   1 D ( μ ,   ζ sys ,   ζ abs ,   α ,   β )   =   ζ abs   ( α β − α 3 β   −   μ α 3 β ) α 2   +   ζ sys   ( α   −   α 3 β 2 )and the parameters are: μ   =   m abs m sys ,   α   =   ω ω sys ,   ξ sys   =   C sys 2 m sys k sys ,   β   =   ω abs ω sys ,   ξ abs   =   C 2 m abs k abs

According to Den Hartog’s theorem [17], the ratio of natural frequency between absorber and system is: (21) β *   =   1 1   +   μ

Figure 12(a) displays the relationship between μ and G. Hence, the peak is reduced with enlarged μ. According to Figure 12(b), the resonance peak frequency increases when ζ_abs decreases from 1 down to 0.1. The optimal damping ratio of the absorber, ζ abs *, is estimated as: (22) ζ abs   =   ζ abs *   =   3 μ 8 ( 1   +   μ )

Figure 13 shows the normalized amplitude G as a function of μ. By assigning the optimal damping ratio ζ_abs and mass ratio μ, the vibration magnitudes are decreased effectively. (Therefore, the actual values of mass and stiffness of TVA are determined by selecting μ = 0.8 and ζ abs * = 0.408.

Figure 14 shows a prototype of proposed dual-stage PZT-VCM actuator assembly with small passive TVA actuation. The dynamic response of the dual-stage actuator and the PZT bender is obtained experimentally by LDV and a frequency response analyzer, as shown in Figure 15.

Figure 16 shows the effectiveness of TVA vibration suppression. The performance and robustness of the absorber are further enhanced by using optimal processes for dual stage actuation. According to simulation and experimental results, the passive TVA reduces the system vibration by 80% with respect to the peak amplitudes.

Design of the PZT loop compensator in the dual-stage actuator expands the bandwidth of the overall system. The primary VCM continuously provides coarse positioning, while the secondary actuator gives fine positioning, vibration and disturbances rejection. Figure 17 illustrates the open loop frequency response of the VCM and PZT dual-stage actuator, compared with the original single-stage VCM actuator.

The open loop bandwidth of the dual-stage system is approximately 3.8 kHz, with a phase margin of 30 degrees and a gain margin of 5 dB. This represents a significant enhancement over the single-stage VCM system, which has a 0.46 kHz bandwidth for the same stability margins. The PZT actuator can reject high frequency disturbances. Notably, the resonant peak of the PZT actuator at 0.9 kHz is lower than the open loop bandwidth and does not pose stability problems.

Figure 18(a) illustrates the simulated focusing error signal (S-curve) generated by the actuator tilting from −0.5^○ to 0.5^○, whereas the measured asymmetric S-curve is shown in Figure 18(b) for comparison. The astigmatic returning laser spot deteriorates considerably due to the tilt actuation, causing the asymmetric peak shapes of the S-curve. This unbalanced S-curve signal reduces the accuracy of optical sensing. The optical compensation for dual stage actuator is evaluated by designing the schematic layout of the reflection geometry for tilt measurement [24–26], as shown in Figure 19. The rotary angle measurement is performed by mounting a target 45^° micro-prism on the precision VCM-Driven micro-positioning stage. The 635 nm wavelength laser, which is used to adjust tracking angle of dual-stage actuator, produces horizontal beam (green line) and passes through the collimating lens, the polarized beam splitter (PBS), the quarter-wave plate (QWP) to form circular polarization beam and focusing on the side of actuator by objective lens. The returning beam is reflected by the PBS and projected on the collimating detector. Furthermore, the LDV laser beam (red line, vertical) is focusing on the photo-diode to measure the dynamic tilt error motions of focus operation by varying the driving frequency in the range of 5–200 Hz. The distance of the seesaw actuator over the focus actuation stroke is 400 μm or less.

The proposed astigmatic detection method combined with differential phase detection (DPD) is verified through simulation and experimental results. By using the quadrant photo detector array, the astigmatic signal is generated (based on the phase difference between the sum of the diagonal elements, as shown in Equation (22).

Focus error signal (FES), f: (23) f   =   ( A + C )   −   ( B + D ) ( A   +   B   +   C   +   D ) × 100 %

The DPD signal is defined versus the misalignment of the photo-detector in the plane (v, h) by using the following calculation [27,28].

DPD signal, d_h: (24) d h   =   ( A + B )   −   ( C + D ) ( A   +   B   +   C   +   D ) × 100 %

DPD signal, d_v: (25) d v   =   ( A + D )   −   ( B + C ) ( A   +   B   +   C   +   D ) × 100 %where d_h and d_v represent the misalignments along the horizontal (h) and vertical (v) the direction, respectively.

Feasibility of the proposed secondary stage PZT-actuator to compensate for the tilt actuation induced by the primary VCM actuator is demonstrated by conducting an experiment in which the dual stage actuator is kept in the in-focus condition. Figure 20 summarizes the experimental results by moving the prism with reflection geometry. The vertical (black) and horizontal (red) tilt angle are obtained based on differential phase variation with 5 Hz sine-wave excitation. Although the vertical angular displacement involves roll as well as pitch rotation, the phase variation from roll vibration is sufficiently small to be negligible compared with the actuator nutation.

Both results closely correspond to the numerical calculations. The measurement accuracy of the photo-detector is validated. Figure 21 shows the tilt motions for both single and dual stage control. A sinusoidal signal with a frequency of 5 Hz and voltage amplitude of 6 V is applied to the seesaw actuator. As mentioned earlier, the angular displacement is within the restricted range ±0.19° (peak to peak) for the dual stage actuation than ±0.45° (peak to peak) for single-stage control. The tilt signal variance from a single-stage drive is reduced by 60% with the implementation of this dual stage compensation. Measurement results demonstrate a high stability for the application of adjusting the inclination angle of the optical axis with precision positioning, capable of satisfying the requirement for small tilt angle compensation of a dual stage actuator.