**2.1 Operation principle**

The operating principle of the piezoelectric stick–slip actuator designed by the topology optimization method is shown in **Figure 1**. The asymmetric sawtooth wave is used as the excitation signal. A motion cycle can be divided into three phases:

**Initial phase** (as shown in **Figure 1a**): At time *t*0, the input voltage is zero, so the piezoelectric stack is at its original length, the actuator is at rest, and the maximum static friction force between the indenter and slider is taken as the locking force *F*0, point *P* is the contact point between the indenter and slider.

**Stick phase** (as shown in **Figure 1b**): From time *t*<sup>0</sup> to *t*1, the length of piezoelectric stack extends with the increases of voltage. Under the effect of flexure hinge, the driving mechanism produces oblique deformation. The vertical positive pressure *Fy*<sup>1</sup>

on the slider increases, and the horizontal driving force *Fx*<sup>1</sup> drives the slider to move *d*<sup>1</sup> distance by static friction force, resulting in a large output displacement. When it reaches time *t*1, the elongation of piezoelectric stack reaches the maximum.

**Slip phase** (as shown in **Figure 1c**): Contrary to the stick stage, the piezoelectric stack shrinks rapidly with the sudden drop of the input voltage, the driving mechanism returns to its original shape, and the vertical positive force *Fy*<sup>2</sup> of the indenter against the slider decreases. There is relative sliding between the stator and the slider, resulting in horizontal kinetic friction force *Fx*<sup>2</sup> in the opposite direction. When the action of kinetic friction force is greater than that of inertial force, a small backward displacement *d*<sup>2</sup> occurs.

After one motion cycle, the slider moves a net forward distance *d*, and the actuator realizes a large-stroke output by repeating the above process driven by periodic asymmetric sawtooth wave.

## **2.2 Problem description**

In order to better describe the method, the symbols used in the structural topology optimization process are expressed in **Table 1**.

Bendsøe and Sigmund pointed out that the ratio between output and input displacements is an important objective function for compliant mechanism [25]. The problem is how to find the optimal driving mechanism shape that can achieve the design goal in a certain design domain, and we design the driving mechanism on the basis of the theories of Ref. [25, 40].


#### **Table 1.**

*The symbols of topology optimization method.*

*Topology Optimization Methods for Flexure Hinge Type Piezoelectric Actuators DOI: http://dx.doi.org/10.5772/intechopen.103983*

**Figure 2.** *The design domain of driving mechanism.*

The structural characteristics of the stick–slip actuator are considered, the design domain of the flexure hinge mechanism is revealed in **Figure 2**, and the dimensions of the design domain are L1 = 12.5 mm, L2 = 7 mm, L3 = 24 mm, L4 = 3.5 mm, respectively. *ux out*, *uy out*, and *uin* are the output displacement at point B in *x* direction, *y* direction and the input displacement at point A, respectively. *fin* is the input force, *kin* and *kout* are the stiffness of input spring and output spring, respectively.

The magnification factor *λ* is defined as the objective function, which is considered the relationship of output displacement *ux out* and input displacement *uin*, while the parasitic displacement *uy out* is small, here the evaluation indicator *τ* is defined as a constraint condition to restrict the parasitic displacement *uy out*.

The displacement magnification factor *λ* and evaluation indicator *τ* are given as follows:

$$
\lambda = \frac{u\_{out}^{\star}}{u\_{in}} \tag{1}
$$

$$
\pi = \frac{u\_{out}^y}{u\_{out}^x} \tag{2}
$$

### **2.3 Mathematical formulation**

A basic engineering goal of piezoelectric stick–slip actuators is to maximize the stroke amplification in structures and mechanisms, a serious of different objective functions can be formulated. The basic topology optimization model of piezoelectric actuator can be given as:

$$\begin{aligned} \text{find } & \mathfrak{p} = [\rho\_1, \rho\_2, \dots, \rho\_N]^T \in \mathbb{R}^n\\ \text{min } & \mathfrak{g}(\mathfrak{p})\\ \text{s.t. } & \left\{ \mathbf{g}\_j(\mathfrak{p}) \le \mathbf{0}, j = 1, 2, \dots, m \right\} \\ & \mathbf{g}\_v(\mathfrak{p}) = \int\_{\Omega} \rho dV \le V\_f V\_0 \\ & \rho\_i = \mathbf{0} \text{ or } \mathbf{1}, i = \mathbf{1}, 2, \dots, n. \end{aligned} \tag{3}$$

where **ρ** is the vector of design variables, *g*ð Þ **ρ** is the objective function, *g <sup>j</sup>* ð Þ **ρ** is the constraint functions, *gv*ð Þ **ρ** is the volume constraint function respectively.

The optimization models described in Eq. (3) are implemented using the MATLAB programming language. Usually, the design aim of the piezoelectric actuators is to maximize mechanical advantage, so the objective functions can be chosen as magnification factor and mechanical efficiency [23, 24], which can be used for static response. Although the compliant hinge is only a part of the compliant mechanism, and its motion and configuration are relatively simple, we can regard the compliant hinge as a simple compliant mechanism, so the topology optimization idea of the compliant mechanism is also applicable to the design of the compliant hinge. Sigmund [23, 24] developed an inverting displacement amplifier, and in [25] mechanical efficiency (ME) is defined as an objective function.

The objective function of the piezoelectric stick–slip actuator topology optimization problem is to obtain largest magnification factor *λ*, and the evaluation indicator *τ* is used to restrict the parasitic displacement *uy out*. So the topology optimization model is given as follows:

$$\begin{aligned} \text{find } \boldsymbol{\varrho} &= \left[\rho\_1, \rho\_2, \dots, \rho\_N\right]^T \in \boldsymbol{R}^N\\ \min \quad & g(\boldsymbol{\varrho}) = -\frac{\boldsymbol{u}\_{out}^\times}{\boldsymbol{u}\_{in}}\\ \text{s.t. } \mathbf{Ku} &= \mathbf{f}\_{in} \\ \text{g}\_\boldsymbol{\varepsilon}(\boldsymbol{\varrho}) &= \left(\frac{\boldsymbol{u}\_{out}^\times}{\boldsymbol{u}\_{out}^\times}\right)^2 \leq \boldsymbol{\varepsilon}^\* \\ \text{g}\_\boldsymbol{v}(\boldsymbol{\varrho}) &= \sum\_{\epsilon=1}^N \boldsymbol{v}\_\epsilon \rho\_\epsilon \leq \boldsymbol{V}\_f \boldsymbol{V}\_0 \\ & 0 < \boldsymbol{V}\_f < \mathbf{1} \\ 0 < \rho\_{\min} &\leq \rho\_\epsilon \leq \mathbf{1}, \boldsymbol{e} = \mathbf{1}, 2, \dots, N. \end{aligned} \tag{4}$$

where **K** is the global stiffness matrix, **u** is the global displacement vector and **f***in* is input force vector, respectively. **ρ** is the design variables vector, *ρ*min (usually *ρ*min ¼ 0*:*001) is introduced to avoid singularity case. *ve* is the volume of element *e*, *V*<sup>0</sup> and *V <sup>f</sup>* are the design domain volume and the prescribed volume fraction, *N* is the number of elements used to discretize the design domain, respectively.

The element stiffness matrix **k***<sup>e</sup>* can be written as

$$\mathbf{k}\_{\epsilon} = \rho\_{\epsilon}^{p} \mathbf{k}\_{0} \tag{5}$$

where *p* represents the penalization power (typically *p* =3), and **k**<sup>0</sup> indicates the element stiffness matrix for an element with unit Young's modulus.

*Topology Optimization Methods for Flexure Hinge Type Piezoelectric Actuators DOI: http://dx.doi.org/10.5772/intechopen.103983*

The global stiffness matrix **K** can be expressed as

$$\mathbf{K} = \sum\_{\epsilon=1}^{N} \rho\_{\epsilon}^{p} \mathbf{K}\_{\epsilon}^{0} + \mathbf{K}\_{in} + \mathbf{K}\_{out} \tag{6}$$

where **K**<sup>0</sup> *<sup>e</sup>* is the element stiffness matrix for an element with unit Young's modulus in global sense, **K***in* is the stiffness matrix of the input spring *kin* at the global level, **K***out* is the stiffness matrix of the output spring *kout* at the global level, respectively.

#### **2.4 Sensitivity analysis**

In this part, how to determinate the output displacement *ux out*, parasitic displacement *uy out* and input displacement *uin* and their derivatives of *<sup>∂</sup>u<sup>x</sup> out ∂ρe* , *∂uy out ∂ρe* and *<sup>∂</sup>uin ∂ρe* , is a key problem. In addition, the relationships of *ux out*, *uy out* and *uin* can be considered as.

$$\boldsymbol{\mu}\_{\rm out}^{x} = \mathbf{L}\_{\mathbf{x}}^{T} \mathbf{u}, \ \boldsymbol{\mu}\_{\rm out}^{y} = \mathbf{L}\_{\mathbf{y}}^{T} \mathbf{u}, \ \boldsymbol{\mu}\_{\rm in} = \mathbf{L}\_{\rm in} \boldsymbol{\t}^{T} \mathbf{u} \tag{7}$$

where **L***x*, **L**<sup>y</sup> and **L***in* are the vectors for which the inner product with **u** produces the relevant output displacement *u<sup>x</sup> out*, *<sup>u</sup><sup>y</sup> out* and the relevant input displacement *uin* (**L***x*, **L**<sup>y</sup> and **L***in* are interpreted as a (unit) load vector), respectively.

The derivatives of *ux out*, *<sup>u</sup><sup>y</sup> out* and *uin* can be expressed as

$$\frac{\partial \mathbf{u}\_{out}^{\mathbf{x}}}{\partial \rho\_{\varepsilon}} = \mathbf{L}\_{\mathbf{x}} \, ^{T} \frac{\partial \mathbf{u}}{\partial \rho\_{\varepsilon}} \tag{8}$$

$$\frac{\partial \mathbf{u}\_{\text{out}}^{\mathcal{V}}}{\partial \rho\_{\epsilon}} = \mathbf{L}\_{\mathcal{V}} \, ^T \frac{\partial \mathbf{u}}{\partial \rho\_{\epsilon}} \tag{9}$$

$$\frac{\partial u\_{\rm in}}{\partial \rho\_{\varepsilon}} = \mathbf{L}\_{\rm in}{}^{T} \frac{\partial \mathbf{u}}{\partial \rho\_{\varepsilon}} \tag{10}$$

Differentiating the static equation **Ku** ¼ **f***in*, we have

$$\frac{\partial \mathbf{u}}{\partial \rho\_{\epsilon}} = \mathbf{K}^{-1} \left( \frac{\partial \mathbf{f}\_{in}}{\partial \rho\_{\epsilon}} - \frac{\partial \mathbf{K}}{\partial \rho\_{\epsilon}} \mathbf{u} \right) \tag{11}$$

Due to **f***in* is a permanent load vector, we obtain

$$\frac{\partial \mathbf{f}\_{in}}{\partial \rho\_{\epsilon}} = \mathbf{0} \tag{12}$$

Differentiating the global stiffness matrix, we have

$$\frac{\partial \mathbf{K}}{\partial \rho\_{\epsilon}} = p \rho\_{\epsilon}^{p-1} \mathbf{K}\_{\epsilon}^{0} = p \rho\_{\epsilon}^{p-1} \mathbf{k}\_{0} \tag{13}$$

Next, we need to solve the adjoint equilibrium equations.

$$\mathbf{K}\overline{\mathbf{u}}\_{\mathbf{x}} = \mathbf{L}\_{\mathbf{x}}, \; \mathbf{K}\overline{\mathbf{u}}\_{\mathcal{Y}} = \mathbf{L}\_{\mathcal{Y}}, \; \mathbf{K}\overline{\mathbf{u}}\_{in} = \mathbf{L}\_{in} \tag{14}$$

where **u***x*, **u***<sup>y</sup>* and **u***in* are the adjoint displacement vectors, which are determined by adjoint (unit) load vector **L***x*, **L**<sup>y</sup> and **L***in*, respectively.

Combined with Eq. (8), Eqs. (11)–(14), *<sup>∂</sup>u<sup>x</sup> out ∂ρe* can be written as

$$\begin{split} \frac{\partial \boldsymbol{u}\_{\text{out}}^{\text{x}}}{\partial \rho\_{\epsilon}} &= \mathbf{L}\_{\text{x}}^{\text{T}} \frac{\partial \mathbf{u}}{\partial \rho\_{\epsilon}} = \mathbf{L}\_{\text{x}}^{\text{T}} \mathbf{K}^{-1} \left( \frac{\partial \mathbf{f}\_{in}}{\partial \rho\_{\epsilon}} - \frac{\partial \mathbf{K}}{\partial \rho\_{\epsilon}} \mathbf{u} \right) \\ &= \left( \mathbf{K}^{-1} \mathbf{L}\_{\text{x}} \right)^{T} \left( \frac{\partial \mathbf{f}\_{in}}{\partial \rho\_{\epsilon}} - \frac{\partial \mathbf{K}}{\partial \rho\_{\epsilon}} \mathbf{u} \right) = -\overline{\mathbf{u}}\_{\text{x}}^{\text{T}} \frac{\partial \mathbf{K}}{\partial \rho\_{\epsilon}} \mathbf{u} \\ &= -p \rho\_{\epsilon}^{p-1} \overline{\mathbf{u}}\_{\text{x}}^{\text{T}} \mathbf{K}\_{\epsilon}^{0} \mathbf{u} = -p \rho\_{\epsilon}^{p-1} (\overline{\mathbf{u}}\_{\text{x}})\_{\epsilon}^{T} \mathbf{k}\_{0} \mathbf{u}\_{\epsilon} \end{split} \tag{15}$$

Then we have

$$\frac{\partial u\_{\rm out}^{\mathcal{V}}}{\partial \rho\_{\varepsilon}} = -p\rho\_{\varepsilon}^{p-1} \left(\overline{\mathbf{u}}\_{\mathcal{V}}\right)\_{\varepsilon}^{T} \mathbf{k}\_{0} \mathbf{u}\_{\varepsilon} \tag{16}$$
 
$$\partial u\_{\rm in} \qquad \square\_{-\pi}^{-1/\varpi} \quad \square^{T} \mathbf{l}\_{\perp} \dots$$

$$\begin{split} \frac{\partial \mathbf{u}\_{in}}{\partial \rho\_{\epsilon}} &= -p\rho\_{\epsilon}^{p-1} (\overline{\mathbf{u}}\_{in})\_{\epsilon}^{T} \mathbf{k}\_{0} \mathbf{u}\_{\epsilon} \\ &= -p\rho\_{\epsilon}^{p-1} \frac{\mathbf{u}\_{\epsilon}^{T} \mathbf{k}\_{0} \mathbf{u}\_{\epsilon}}{||\mathbf{f}\_{in}||} \end{split} \tag{17}$$

where ð Þ **u***<sup>x</sup> <sup>e</sup>*, **u***<sup>y</sup>* � � *e* , ð Þ **u***in <sup>e</sup>* and **u***<sup>e</sup>* are the elements of displacement vectors of **u***x*, **u***y*, **u***in* and **u**, respectively.

The sensitivity of objective function *g* is found as

$$\frac{\partial \mathbf{g}}{\partial \rho\_{\epsilon}} = -\frac{\frac{\partial u\_{\rm out}^{\mathbf{x}}}{\partial \rho\_{\epsilon}} u\_{in} - \frac{\partial u\_{\rm in}}{\partial \rho\_{\epsilon}} u\_{out}^{\mathbf{x}}}{u\_{in}^{2}} \tag{18}$$

and the sensitivity of constraint functions *g<sup>τ</sup>* and *gv* can be expressed as

$$\frac{\partial \mathbf{g}\_{\varepsilon}}{\partial \rho\_{\varepsilon}} = 2 \left( \frac{u\_{out}^{\mathrm{y}}}{u\_{out}^{\mathrm{x}}} \right) \frac{\frac{\partial u\_{out}^{\mathrm{y}}}{\partial \rho\_{\varepsilon}} u\_{out}^{\mathrm{x}} - \frac{\partial u\_{out}^{\mathrm{x}}}{\partial \rho\_{\varepsilon}} u\_{out}^{\mathrm{y}}}{u\_{out}^{\mathrm{x}}^{\mathrm{2}}} \tag{19}$$

$$\frac{\partial \mathbf{g}\_v}{\partial \rho\_\varepsilon} = v\_\varepsilon \tag{20}$$

To avoid the checkerboards patterns and mesh dependencies phenomena, some restriction on the design must be imposed. Here a filtering technique is used to modify the sensitivity of *<sup>∂</sup><sup>g</sup> ∂ρe* as follows:

$$\frac{\partial \hat{\mathbf{g}}}{\partial \rho\_{\epsilon}} = \frac{1}{\rho\_{\epsilon} \sum\_{k=1}^{N\_k} H\_k^{\epsilon}} \sum\_{k=1}^{N\_k} H\_k^{\epsilon} \rho\_k \frac{\partial \mathbf{g}}{\partial \rho\_k} \tag{21}$$

where *Nk* is the set of elements *e*, *dist e*ð Þ , *k* implies the center-to-center distance of element *e* and element *k*, which is smaller than the filter radius *r*min, and the convolution operator (weight factor) *H<sup>e</sup> <sup>k</sup>* in Eq. (21) is defined as

*Topology Optimization Methods for Flexure Hinge Type Piezoelectric Actuators DOI: http://dx.doi.org/10.5772/intechopen.103983*

$$H\_k^\epsilon = \max\left(0, r\_{\min} - dist(e, k)\right) \tag{22}$$

From Eq. (7) and Eqs. (15)–(17), the solutions of Eqs. (18), (19) are obtained.

**Figure 3** shows the flow chart of topology optimization procedures. Initially, the design domain, boundary conditions and material are defined, the design domain is discretized into finite element meshes and solve the static problem. Then the sensitivities of the objective function and constraint conditions are obtained, design variables are updated by MMA algorithm. Here the MMA algorithm is employed by Svanberg [41, 42] as the optimizer. Finally, check the convergence of the results, if it converges, the iteration will stop; if not, go to step 3.

**Figure 3.** *Flow chart of topology optimization procedures.*
