1 Introduction

Compliant mechanisms have been widely used in precision fields such as micro/nano-positioning stage (Liu et al. 2015; Thanh-Phong and Huang 2017; Zhang et al. 2017), microgripper (Hoxhold and Buettgenbach 2010; Sun et al. 2015; Thanh-Phong et al. 2017), optics alignment (Chen et al. 2019), microscope (Yang et al. 2016), and cell microinjector (Wei and Xu 2017, 2019) for the capability of nanometer resolution and fast response.

For large stroke and compact design, amplification mechanisms are developed to extend the output displacement of precision actuators such as piezoelectric actuators (PZT) and voice coil motors (VCM). And a guiding mechanism is assigned to the output port of an amplifying mechanism to improve the positioning accuracy by eliminating the parasitic motion. Amplification mechanisms are combinations of flexure hinges in a serial or compound manner while guiding mechanisms are designed with parallel leaf springs. Various displacement amplification mechanisms such as lever type (Chen et al. 2016; Qin et al. 2018; Qu et al. 2016), bridge-type (Dong et al. 2018; Ling et al. 2019), and Scott–Russell mechanism (Tian et al. 2009c) are developed for nano manipulating systems. The bridge type mechanism processes a large displacement amplification ratio, but a small output stiffness. The lever type mechanism presents high output stiffness while the amplification ratio is small. The performance of a Scott–Russell mechanism is in the middle range of these two mechanisms.

The amplification ratio determines the travel range of nano manipulating systems. Methods for the calculation of amplification ratio are developed to provide an accurate prediction for the output displacement (Li et al. 2019; Liu and Yan 2016). The stiffness of a compliant mechanism has a non-negligible influence on the actual output displacement of PZT for the serial connection (Tian et al. 2009b). However, the contact stiffness makes it difficult to obtain the displacement amplification ratio of PZT, and the deflection of the rigid lever linking to flexure hinges is ignored to simplify the design of the motion stage. The accumulation of modeling errors leads to a large deviation in the prediction of static and dynamic performances of nano manipulating systems (Lin et al. 2018; Tian et al. 2009d; Wu and Xu 2019).

The error accumulation caused by the series design of flexure hinges has an adverse effect on the performance of the amplification mechanism. To this end, this work proposed a micro-motion stage designed with a lever-type amplification mechanism whose stiffness and resonant frequency can be adjusted by utilizing the nonlinearity of leaf springs (Zhao et al. 2017). The accurate model for the calculation of amplification ratio is built by taking the deflection of the rigid lever and adjustment mechanism into account. The deviation of output stiffness and frequency caused by machining error and modeling error can be eliminated through the adjustment mechanism. The static and dynamic adjustment performances of the proposed motion stage are explored for further applications.

The main contribution of this work is proposing an accurate design method for the micro-motion stage by taking the deflection of rigid levers and the static and dynamic performance can be adjusted by utilizing the nonlinearity of leaf springs to eliminate the influence of uncertain factors and maintain stability. The rest of this paper is organized as follows. The mechanical design of the proposed motion stage is illustrated in Sect. 2. The performance design of the micro-motion stage is structured in Sect. 3. Section 4 validates the design formulas using finite element analysis. Experimental investigations are carried out in Sect. 5 to illustrate the efficiency of the proposed mechanism in static and dynamic performance. Conclusions are drawn in the final section.

2 Mechanism design of motion stage

2.1 Principle of motion stage

As depicted in Fig. 1, the schematic diagram of the proposed micro-motion stage consists of an amplification mechanism, guiding mechanism, PZT, and adjustment mechanism. The amplification mechanism is designed based on the lever principle. Right circular flexure hinge (RCFH) is used to generate rotation for its high rotation accuracy and low stress. The two ends of the rigid connector are respectively connected with the guiding mechanism and the fixed frame by RCFH. A PZT is installed on the fixed frame with a bolt. The installation position of PZT can be adjusted in the installation slot, which leads to a change in the amplification coefficient of the motion stage. Corner filleted leaf springs (CFLSs) are symmetrically arranged as a guiding mechanism to eliminate parasitic motion. The central motion stage is connected to the adjustment mechanism by bolts. Leaf springs can be tensioned by screwing the fixed bolts of the central motion stage to adjust the stiffness and frequency of the motion stage for the tension-stiffening effect.

Fig. 1
figure 1

Schematic diagram of motion stage

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The main parameters of the motion stage are clearly illustrated in Fig. 2. The fillet radius and minimum thickness of RCFH are \({r}_{1}\) and \({t}_{0}\). The thickness of the rigid lever is \({t}_{0}\). The total length of the amplification mechanism is \({l}_{1}\). The minimum thickness, fillet radius, and length of CFLS are \({t}_{2}\), \({r}_{2}\) and \({l}_{2}\), respectively. The width of the motion stage is \(w\).

Fig. 2
figure 2

Main design parameters of the motion stage

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2.2 Simplified formulas for compliant elements

The linear deflection of RCFH and CFLS have been extensively researched in previous literature. The rotation stiffness of a right circular flexure hinge was derived in Ref. (Wu and Zhou 2002) by normalized the radius of the arc as \(s={r}_{1}/{t}_{0}\), which can be expressed as

$${K}_{az}=\frac{{M}_{z}}{{a}_{z}}=\frac{Ew{{r}_{1}}^{2}}{12{f}_{1}}$$
(1)

where \({M}_{z}\) is the moment applied on flexure hinges, \({a}_{z}\) is the corresponding angle displacement and

$${f}_{1}=\frac{2{s}^{3}\left(6{s}^{2}+4s+1\right)}{\left(2s+1\right){\left(4s+1\right)}^{2}}+\frac{\left.12{s}^{4}(2s+1\right)}{{\left(4s+1\right)}^{2.5}}\mathrm{arctan}\sqrt{4s+1}.$$

CFLSs constituting the guiding mechanism are in a fixed-guided constraint which can be modeled using Castigliano’s second theorem. For the situation the normalized radius (\(a={r}_{2}/{t}_{2}\)) is less than 1, the deflection model of a constant cross-section beam is adapted to simply the design. However, the small fillets have a limited improvement in the stress concentration characteristics. For the accurate stress design of the micro-motion stage, the complete model of CFLS is adopted in this work. The bending stiffness of a fixed-guided CFLS can be calculated as Eq. (2) by normalizing the length as \(b={l}_{2}/{t}_{2}\):

$${K}_{lf}=Ew/\left(\frac{6\mathrm{arctan}\left(\sqrt{4a+1}\right)G}{{\left(4a+1\right)}^{2.5}}+\frac{3H}{2\left(2a+1\right){\left(4a+1\right)}^{2}}+\frac{3\pi }{2}+{\left(b-2a\right)}^{3}\right)$$
(2)

where \(G=6{a}^{2}{b}^{2}\left(2a+1\right)-24{a}^{3}b\left(2a+1\right)+{\left(2a+1\right)}^{3}\left(6{a}^{2}-4a-1\right),\)

$$H=4a{b}^{2}\left(6{a}^{2}+4a+1\right)+8{a}^{2}b\left(4{a}^{2}-1\right)-4a\left(2a+1\right)\left(20{a}^{3}-6{a}^{2}-6a-1\right).$$

The stiffness formulas for RCFH and CFLS are complicated and not easily used for mechanical design. In consideration that the arctan function can be approximated using the series shown in Eq. (3) for the normalized radius large than 1, the coefficient \({f}_{1}\) of the rotational stiffness of RCFH can be modeled using a more concise expression.

$$\mathrm{arctan}\left(\sqrt{4a+1}\right)\approx \frac{\pi }{2}-\left(\frac{1}{\sqrt{4a+1}}-\frac{1}{3{\left(\sqrt{4a+1}\right)}^{3}}\right)$$
(3)
$${f}_{1}=\frac{\left(14s+1){s}^{3}\right.}{{\left(4s+1\right)}^{4}}+\frac{6{s}^{4}\left(2s+1\right)}{{\left(4s+1\right)}^{2.5}}\pi$$
(4)

Taking \(\eta =a/b\), the stiffness of the CFLS can be simplified as Eq. (5) when the radius of fillets is less than 1/5 of the length of CFLS.

$${K}_{lf}=Ew/({b}^{3}(1+(\eta -4{\eta }^{2}){H}_{1}+(\eta -2{\eta }^{2}{)H}_{2}))$$
(5)

where \({H}_{1}=\frac{18\pi a\left(2a+1\right)}{{\left(4a+1\right)}^{2.5}}\) and \({H}_{2}=\frac{-24a\left({16a}^{2}+12a+3\right)}{{\left(4a+1\right)}^{3}}\).

3 Performance of the motion stage

3.1 Static performance

The dynamic model for the micro-motion stage is depicted in Fig. 3 which is equivalent to a lumped mass system. The lever in flexure-based mechanisms is typically treated as a rigid element to facilitate the modeling process (Tian et al. 2009a). This approximation leads to an accumulation of model error for the multi lever systems due to the serial design. Calculations of the approximate model will be much larger than the actual stiffness and resonant frequency of micro-motion systems, which results in a large deviation of displacement predictions. In addition, contact stiffness has a crucial influence on the output displacement of PZT, which should be taken into account for an accurate model. The stiffness of the adjustment mechanism introduced in this work influences the input stiffness of the micro-motion stage which is also under the consideration.

Fig. 3
figure 3

Dynamic model of the motion stage

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The accurate input stiffness of the motion stage can be calculated as:

$${K}_{\mathrm{in}}=\frac{{2K}_{az}{K}_{\mathrm{lever}}}{{x}^{2}{K}_{\mathrm{lever}}+{K}_{az}}+{\left(\frac{{l}_{1}}{x}\right)}^{2}\frac{2n{K}_{\mathrm{adj}}{K}_{lf}}{{K}_{\mathrm{adj}}+2n{K}_{lf}}$$
(6)

where \(n\) is the group numbers of CFLSs.

The lever linked with the amplification mechanism and rigid frame is in a simply supported constraint as shown in Fig. 4. The stiffness of the lever subjected to a driving force at the \(x\) point is calculated as Eq. (7) according to the deflection model shown in Fig. 5.

Fig. 4
figure 4

Diagram of lever subjected to the driving force

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Fig. 5
figure 5

Deflection of the rigid lever

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$${K}_{\mathrm{lever}}=Ew{t}_{1}^{3}{l}_{1}/\left({x}^{2}{\left({l}_{1}-x\right)}^{2}\right)$$
(7)

The actual output displacement of PZT depends on the stiffness of PZT Kpzt, the contact stiffness \({K}_{\text{con}}\), and the input stiffness of the motion stage \({K}_{\mathrm{in}}\). Taking \(\eta ={K}_{\text{con}}/\left({K}_{\text{pzt}}+{K}_{\text{con}}\right)\), the actual output of PZT is expressed as

$${x}_{\mathrm{in}}=\frac{\eta {K}_{\mathrm{pzt}}}{\eta {K}_{\mathrm{pzt}}+{K}_{\mathrm{in}}}{x}_{\mathrm{pzt}}$$
(8)

The output stiffness of the micro-motion stage can be modeled as

$${K}_{\mathrm{out}}=2n{K}_{lf}+\frac{{K}_{\mathrm{adj}}{K}_{1}}{{K}_{\mathrm{adj}}+{K}_{1}}$$
(9)

where \({K}_{1}=\frac{(\eta {{x}^{2}K}_{\mathrm{pzt}}+{K}_{az}){K}_{\mathrm{lever}}}{\eta {{x}^{2}K}_{\mathrm{pzt}}+{K}_{az}+{l}_{1}^{2}{K}_{\mathrm{lever}}}\).

Therefore, the accurate amplification coefficient of the input displacement can be expressed as:

$$\gamma = \frac{{x_{{{\text{out}}}} }}{{x_{{{\text{in}}}} }} = \frac{{l_{1} }}{x}\frac{{K_{{{\text{adj}}}} }}{{K_{{{\text{adj}}}} + 2nK_{{lf}} }}\left( {1 - \frac{{K_{{{\text{in}}}} }}{{K_{{{\text{lever}}}} }}} \right)$$
(10)

Considering the effect of contact stiffness and stiffness of lever, the actual amplification coefficient of the displacement of PZT can be calculated as:

$$\gamma _{1} = \frac{{x_{{{\text{out}}}} }}{{x_{{{\text{pzt}}}} }} = \frac{{l_{1} }}{x}\frac{{\eta K_{{{\text{pzt}}}} K_{{{\text{adj}}}} }}{{\left( {K_{{{\text{adj}}}} + 2nK_{{lf}} } \right)\left( {\eta K_{{{\text{pzt}}}} + K_{{{\text{in}}}} } \right)}}\left( {1 - \frac{{K_{{{\text{in}}}} }}{{K_{{{\text{lever}}}} }}} \right)$$
(11)

The amplification ratio can be approximated as Eq. (12) by neglecting the contact stiffness, stiffness of lever, and adjustment mechanism.

$$\gamma =\frac{{x}_{\mathrm{out}}}{{x}_{\mathrm{in}}}=\frac{{l}_{1}}{x}$$
(12)

3.2 Adjustment of stiffness and frequency

The compress-softening effect of leaf spring is utilized in Ref. (Zhao et al. 2017) to extend the travel range of a linear motion stage by providing a lower stiffness, and a double parallelogram design is adopted to eliminate the parasitic motion. CFLSs proposed in this work present different physical properties with leaf springs for the introduced corner fillet. The tension-stiffening effect of CFLSs is used to improve the stiffness and frequency of the micro-motion stage, which can be modeled using the nonlinear model proposed in Ref. (Li et al. 2021) as:

$${K}_{t}=\frac{1.2N}{\lambda {l}_{2}}$$
(13)

where \(\lambda\) is a coefficient related to the parameters of CFLS.

Therefore, the equivalent stiffness of a CFLS with tension force can be expressed as

$${K}_{lf}^{\mathrm{^{\prime}}}={K}_{lf}+{K}_{t}$$
(14)

The equivalent mass of the rigid lever can be calculated using the kinetic energy equivalence.

$$\frac{1}{2}m_{1} \left( {\theta \omega l} \right)^{2} = \frac{1}{2}\rho wt\int_{0}^{{l_{1} }} {\rho wt\left( {\theta \omega x} \right)^{2} dx}$$
(15)

where \(\omega\) is the angular frequency of the rigid lever.

Then the equivalent mass of the rigid lever is expressed as

$${m}_{1}=\frac{\rho wt{l}_{1}}{3}$$
(16)

The equivalent mass of a CFLS can be obtained in the same way as.

$$m_{2} = \left( {\frac{{13}}{{35}}t_{2} l_{2} + 2\left( {1 - \frac{\pi }{4}} \right)r_{2}^{2} } \right)\rho w$$
(17)

Therefore, the natural resonant frequency of the motion stage with the tension force in the driving direction is yielded.

$$f=\frac{1}{2\pi }\sqrt{\frac{{K}_{\mathrm{out}}}{{m}_{eq}}}$$
(18)

where \({m}_{eq}={m}_{1}+2n{m}_{2}+{m}_{0}\) and \({m}_{0}\) is the mass containing the mass of the adjustment mechanism, sensor, and the central stage.

3.3 Stress of stage

Flexure hinges are the most vulnerable component of the flexure-based mechanisms. The stress of the flexure hinge has a vital influence on the fatigue life of the motion stage. Rotational displacement occurs on the RCFH when the driving force is applied. Therefore, the moment applied on RCFH can be calculated by the following formula.

$${M}_{z}={K}_{az}\frac{{x}_{\mathrm{out}}}{{l}_{1}}$$
(19)

The maximum stress of RCFH can be calculated as

$${\sigma }_{\mathrm{max}}\text{=}\,\frac{{{x}_{\mathrm{out}}k}_{b}{K}_{az}}{w{t}_{0}^{2}{l}_{1}}$$
(20)

where \({k}_{b}\) is the stress concentration factor of RCFH, which was proposed in Ref. (Chen et al. 2014).

Although the travel range of the micro-motion stage is largely less than the length of CFLS, the nonlinearity caused by tensile force should be taken into account for the stress design. The moment at the guided end of CFLS can be calculated with the following equation.

$${M}_{lf}^{^{\prime}}=\frac{{x}_{\mathrm{out}}\left({l}_{2}-2{r}_{2}\right){K}_{lf}^{^{\prime}}}{2}-N{x}_{\mathrm{out}}$$
(21)

The bending stress of CFLS can be calculated using the following equation.

$$\sigma_{\max }^{\prime b} = \frac{{3k_{b}^{\prime } x_{{{\text{out}}}} \left( {k_{lf}^{\prime } \left( {l_{2} - 2r_{2} } \right) - 2N} \right)}}{{wt_{2}^{2} }}$$
(22)

where \({k}_{b}^{^{\prime}}\) is the bending stress concentration factor of CFLSs.

The tensile stress of CFLS can be calculated using the following equation.

$${\sigma }_{\mathrm{max}}^{t}=\frac{{k}_{t}^{\mathrm{^{\prime}}}N}{w{t}_{2}}$$
(23)

where \({k}_{t}^{\mathrm{^{\prime}}}\) is the tension stress concentration factor of CFLSs.

Then the maximum stress of CFLS is yielded.

$$\sigma_{\max }^{\prime } = \sigma_{\max }^{\prime b} + \sigma_{\max }^{t}$$
(24)

3.4 Dynamic performance

The motion stage can be simplified as a one-dimensional vibration system:

$$\ddot{x}+2\xi {\omega }_{n}\dot{x}+{\omega }_{n}^{2}x=\left(F/m\right)\mathrm{sin}\left(\omega t\right)$$
(25)

where \({\omega }_{n}\) is the natural frequency and \(\xi\) is the damp factor of the system.

$$\left\{\begin{array}{c}{\omega }_{n}=\sqrt{{K}_{\mathrm{out}}/{m}_{\mathrm{eq}}}\\ \xi =\frac{c}{2{m}_{\mathrm{eq}}{\omega }_{n}}=\frac{c}{2\sqrt{{m}_{\mathrm{eq}}{K}_{\mathrm{out}}}}\end{array}\right.$$
(26)

Solving Eq. (25), the amplitude of the vibration system is obtained (Kelly 2012):

$$\varepsilon =\frac{F}{{K}_{\mathrm{out}}}\left(\frac{1}{\sqrt{{\left(1-{\beta }^{2}\right)}^{2}+{\left(2\xi \beta \right)}^{2}}}\right)$$
(27)

where \(\beta =\omega /{\omega }_{n}\).

Equation (27) shows that the output displacement of the motion stage is determined by the force amplitude, output stiffness of the motion stage, frequency ratio, and damp factor. Therefore, the dynamic displacement output of the motion stage is difficult to maintain stability when the external conditions lead to a change in these parameters. As plotted in Fig. 6, the amplitude ratio of the first-order vibration system is determined by the frequency ratio and damp ratio of the system. Therefore, it is a feasible way to stabilize the output displacement by utilizing the adjustment mechanism.

Fig. 6
figure 6

Amplitude ratio of the first-order vibration system

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Then the amplification coefficient of response amplitude can be expressed as:

$$|\left.H(\omega \right)|=\frac{1}{\sqrt{{\left(1-{\beta }^{2}\right)}^{2}+{\left(2\xi \beta \right)}^{2}}}$$
(28)

4 Validation with FEA

The Young’s modulus, Poisson’s ratio, and mass density of the material are \(E=\) 210 GPa, \(\upsilon =\) 0.33, and \(\rho =\) 7850 kg/m3, respectively. The parameters of CFLS, RCFH, and rigid lever are listed in Table 1.

Table 1 Parameters of the motion stage
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The finite element analysis (FEA) model is built and solved using the software COMSOL Multiphysics 5.3. The amplification ratio of the proposed mechanism is investigated by inputting displacement at the location \(x=\) 17 mm. The contact stiffness is treated as infinite stiffness in the simulation process and the corresponding displacement of the micro-motion stage is depicted in Fig. 7 when the input displacement is \({x}_{\mathrm{in}}\)= 19 µm. The comparison results listed in Table 2 indicate that the accurate model proposed in Eq. (11) has a precise prediction for the output displacement with an error of less than 0.98% while the calculations of the simplified models Eqs. (10) and (12) produce a large deviation (the corresponding errors are 16.4% and 36.7%).

Fig. 7
figure 7

Displacement of micro-motion stage

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Table 2 Results of output displacement
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The maximum stress of the micro-motion stage is crucial for the mechanism design which is the larger one of the maximum stress of CFLS and the maximum stress of RCFH. When the maximum input displacement and maximum tensile force are applied in the micro-motion stage, the flexure hinges are subjected to the maximum combined loads. Thus, this situation is adopted for the calculation of maximum stress. The predictions of Eqs. (20) and (24) are 128 MPa and 138 MPa, respectively. Thus, the maximum stress is 138 MPa and its error is 4.16% compared with the FEA results shown in Fig. 8, which is less than the proportional limit of the steel \({\sigma }_{\mathrm{p}}=\) 280 MPa.

Fig. 8
figure 8

Stress of micro-motion stage

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The adjustable performance of the micro-motion stage is investigated by applying tensile force on the CFLSs. The frequency results plotted in Fig. 9 show that Eq. (18) can accurately predict the adjustment process of the proposed mechanism and the error is less than 3.5%. Simplified formulas for CFLSs and RCFHs also present high accuracy. The frequency of the micro-motion stage has an adjustment range from 417. Hz to 480.2 Hz when the tensile force applied on a single CFLS is in the range of 0 N ~ 1500 N.

Fig. 9
figure 9

Adjustment performance of frequency

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5 Experimental results

The proposed motion stage is manufactured monolithically by using the wire electrical discharge machining (WEDM) technique whose parameters are listed in Table 1. The type of PZT chosen for the mechanism is PSt-20vs12, whose stiffness is 60 N/µm and nominal travel is 19 µm when the input voltage is 150 V. The displacement of the micro-motion stage is measured by utilizing Capacitive Displacement Senor (NS-DCS14-200, from Sanying Motion Control Instrument Ltd.) The resonant frequency of the micro-motion stage is tested by using Laser Vibrometer (NLV-2500, from Polytec Inc.). A strain gage was attached on the midpoint of each CFLS and a Strain/Bridge Input Module (PXIe-4330, from National Instruments Inc.) is used to obtain the axial strain. The experiment setup for the displacement test is depicted in Fig. 10.

Fig. 10
figure 10

Experiment setup for displacement test

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The attached mass is 114 g which includes the mass of the central stage and fixed bolts. The output stiffness \({K}_{\mathrm{out}}\), frequency of mechanism uninstalling the central motion stage \({f}_{1}\) and the frequency of the mechanism installing the central motion stage \({f}_{2}\) are measured and the experimental results are listed in Table 3. The comparison shown in Table 3 indicates that the actual performances of the micro-motion stage have a large deviation for the machining error.

Table 3 Performance of motion stage
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The axial force applied on CFLS is adjusted by screwing the bolts and the corresponding strain and frequency are recorded. The adjustment performance of the micro-motion stage is plotted in Fig. 11. Although the test results are less than the design values, the relative error decreases with the increase of tensile force. When the tensile force applied on a single CFLS reaches 1534 N, the relative error is decreased from 21.6% to 8.5%. And the adjustment amplitude of frequency reaches 28.4% (the adjustment amplitude of corresponding stiffness reaches 64.9%). Therefore, the proposed mechanism can make up the gaps between actual performances and design values.

Fig. 11
figure 11

Test results of the adjustable frequency

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The displacements of the micro-motion stage are measured for different input voltages of the PZT actuator, which have been plotted in Fig. 12. When the input voltage is 70 V, the displacement is 18.18 m\(\upmu\), which is close to the prediction of Eq. (11) (the error is 6.3%). However, the prediction error of Eq. (10) is 22.4% and the prediction error of Eq. (12) is 43.7%.

Fig. 12
figure 12

Relationship between the input voltage and output displacement

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The bode diagram of the proposed mechanism for different stiffness is obtained by using the dynamic analysis technique and depicted in Fig. 13. The output displacement has an indivisible relationship with the frequency of driving force provided by the PZT in the open-loop working scenario. As discussed in Sect. 3.4, the actual output displacement is sharply changed when the driving frequency is close to the resonant frequency of the proposed mechanism which limited the driving frequency to a low range. But the adjustment mechanism proposed in this work can extend this range to a certain degree.

Fig. 13
figure 13

Bode diagram of micro-motion stage

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6 Conclusions

A stiffness-adjustable micro-motion stage is proposed in this work to offset the performance gaps caused by machining error and model error. Stiffness formulas for RCFH and CFLS are simplified for the easy design of compliant mechanisms. The accurate model for the calculation of amplification ratio is built by taking the deflection of the link lever and adjustment mechanism. The adjustment formula is illustrated by utilizing the nonlinear deflection of CFLS. The stress design is conducted by considering the influence of axial force applied on CFLS to avoid damage of flexure hinges. The amplification coefficient of response amplitude is limited by the frequency ratio of the micro-motion stage and driving force, which can be improved by utilizing the proposed adjustment mechanism. The accuracy of design formulas is verified by FEA results. Experimental investigations are carried out to illustrate the adjustment performances.