**4. Dynamic behavior of a novel MEMS amplifier**

### **4.1. Inertial and geometric parameters:**

It is assumed that micro mechanism is made up of silicon having a density of 2.33 g/cm3. For short length of beams, lengths are 100 micron, widths and heights are 25 micron. The mass of short beams is;

$$\text{M} = 2.33^\*100^\*25^\*25^\*10^{\cdot15} \text{m} 145625^\*10^{\cdot18} \text{ [kg]} \tag{45}$$

For long length of beams, lengths are 800 micron, widths and heights are 25 micron. The mass of short beams is;

$$\text{M} = 2.33^\* 800^\* 25^\* 25^\* 10^{\cdot 15} \text{m} 1165000^\* 10^{\cdot 18} \text{ [kg]} \tag{46}$$

The mass of the slider is accepted as 145625\*10-18 in kilograms.

The mass moments of inertia of the beams are calculated as follows;

For short beams;

100 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

mechanism goes to infinity at zero degree crank angles.

**Figure 10.** Plot of displacement ratio according to first stage crank angle, Θ2

**Figure 11.** Plot of force amplifying according to first stage crank angle, Θ2

is claimed that the toggle position of the micro mechanism is a very crucial issue meaning that if the initial conditions such as crank angles are adjusted properly to enable both crank angle pass 0° at the same time, the ratio of the output to the input force applied to the

$$\mathbf{L} = \mathbf{M}\_{\ast} \, \text{\*} (\mathbf{L}^2 + \mathbf{a}^2) / 12 \mathbf{=} 145625 \, \mathbf{^\*} 10^{18} \text{(} 100^2 \text{+} 25^2 \text{)} / 12 \mathbf{=} 128938802.1 \text{\*} 10^{18} \text{ [} \mathbf{kg} \, \text{\*} \mu \text{m}^2\text{]} \tag{47}$$

For long beams;

$$\text{I} = \text{M}s^\*(\text{L}^2 + \text{a}^2)/12 \equiv 1165000^\*10^{\circ} 10^{\circ} \text{s}^\* (800^{\circ} + 25^{\circ})/12 \equiv 6.219401042^\*10^{\circ} \text{[kg}^\* \text{\mu m}^2\text{]} \tag{48}$$

#### **4.2. Kinematic behavior**

#### *4.2.1. Velocity analysis*

Kinematic simulation is used to calculate and to plot the velocities and acceleration of the beam of the MEMS amplifier.

To understand kinematic behavior of the mechanism, first of all, derivatives of vector loop equations derived in position analysis are taken with respect to time and the velocity equations are arranged as follows;

$$-r\_2 \, ^\ast \sin \theta\_2 \, ^\ast w\_2 - r\_3 \, ^\ast \sin \theta\_3 \, ^\ast w\_3 = \dot{r}\_1 \tag{49}$$

$$r\_2 \, ^\ast \cos \theta\_2 \, ^\ast w\_2 + r\_3 \, ^\ast \cos \theta\_3 \, ^\ast w\_3 = 0 \tag{50}$$

$$-r\_2 \, ^\ast \sin \theta\_2 \, ^\ast w\_2 - r\_5 \, ^\ast \sin \theta\_5 \, ^\ast w\_5 - r\_6 \, ^\ast \sin \theta\_6 \, ^\ast w\_6 = 0 \tag{51}$$

$$r\_2 \, ^\ast \cos \theta\_2 \, ^\ast w\_2 + r\_5 \, ^\ast \cos \theta\_5 \, ^\ast w\_5 + r\_6 \, ^\ast \cos \theta\_6 \, ^\ast w\_6 = 0 \tag{52}$$

The beam 6 are rotated at a constant speed, 0.01 rad/s, in clockwise direction and the initial conditions of w2, w3, w5, r� � are -0.059378175917485 [rad/s], 0.059378175917485 [rad/s], 0.011371580426033 [rad/s ], 2.062182408251533 [µm/s], respectively.

**Figure 12.** Plot of angular velocity of beam 2 and beam 3 versus time for simulation of force amplifier

The angular velocities of beams 2 and 3 in 1 stage are equal to each other in magnitude. As w3 rotate counter clockwise direction, w2 rotate clockwise direction and the absolute values of the changes in w3 and w2 equal to each other according to time as shown in Fig. 12.

Slider slows down until the first stage crank angle, Θ2 pass from 0°. When first stage beams are fully open, as having horizontal position, slider velocity is equal to zero. Then the slider moves to along -x direction and angular velocity of beam 5 decreases according to time as in Fig. 13.

**Figure 13.** Plot of angular velocity of beam 5 and slider velocity versus time for simulation of force amplifier

#### *4.2.2. Acceleration analysis*

102 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

**Figure 12.** Plot of angular velocity of beam 2 and beam 3 versus time for simulation of force amplifier

The angular velocities of beams 2 and 3 in 1 stage are equal to each other in magnitude. As w3 rotate counter clockwise direction, w2 rotate clockwise direction and the absolute values of the changes in w3 and w2 equal to each other according to time as shown in Fig. 12.

Slider slows down until the first stage crank angle, Θ2 pass from 0°. When first stage beams are fully open, as having horizontal position, slider velocity is equal to zero. Then the slider moves to along -x direction and angular velocity of beam 5 decreases according to time as in Fig. 13.

**Figure 13.** Plot of angular velocity of beam 5 and slider velocity versus time for simulation of force

amplifier

To analyze the acceleration of the beams, second derivatives of the terms must be handled. The second derivatives of the vector loop equations for the micro mechanism are as follows;

$$-r\_2 \, ^\ast \cos \theta\_2 \, ^\ast w\_2^2 - r\_2 \, ^\ast \sin \theta\_2 \, ^\ast a\_2 - r\_3 \, ^\ast \cos \theta\_3 \, ^\ast w\_3^2 - r\_3 \, ^\ast \sin \theta\_3 \, ^\ast a\_3 - \overset{\cdots}{r\_1} = 0 \tag{53}$$

$$-r\_2 \, ^\ast \sin \theta\_2 \, ^\ast w\_2^2 + r\_2 \, ^\ast \cos \theta\_2 \, ^\ast a\_2 - r\_3 \, ^\ast \sin \theta\_3 \, ^\ast w\_3^2 + r\_3 \, ^\ast \cos \theta\_3 \, ^\ast a\_3 = 0 \tag{54}$$

$$\begin{aligned} -r\_2 \, \prescript{\*}{}{\cos}\theta\_2 \, \prescript{2}{}{w\_2^2} - r\_2 \, \prescript{\*}{}{\sin}\theta\_2 \, \prescript{\*}{}{a\_2} - r\_5 \, \prescript{\*}{}{\cos}\theta\_5 \, \prescript{2}{}{w\_5^2} \\ -r\_5 \, \prescript{\*}{}{\sin}\theta\_5 \, \prescript{2}{}{a\_5} - r\_6 \, \prescript{\*}{}{\cos}\theta\_6 \, \prescript{2}{}{w\_6^2} - r\_6 \, \prescript{\*}{}{\sin}\theta\_6 \, \prescript{2}{}{a\_6} = 0 \end{aligned} \tag{55}$$

$$\begin{aligned} &-r\_2 \, ^\ast \sin \theta\_2 \, ^\ast w\_2^2 + r\_2 \, ^\ast \cos \theta\_2 \, ^\ast a\_2 - r\_5 \, ^\ast \sin \theta\_5 \, ^\ast w\_5^2 \\ &+ r\_5 \, ^\ast \cos \theta\_5 \, ^\ast a\_5 - r\_6 \, ^\ast \sin \theta\_6 \, ^\ast w\_6^2 - r\_6 \, ^\ast \cos \theta\_6 \, ^\ast a\_6 = 0 \end{aligned} \tag{56}$$

.

In acceleration simulation by Simulink, the velocities such as w2, w3, w5, *r*1 , w6 are considered as known. The beam 6 rotates at a constant speed meaning that acceleration of beam 6 is zero.

Acceleration of beam 2 and beam 3 are shown in fig. 14. Both acceleration of beams decrease as the micro mechanism operates under constant w6, angular velocity. The magnitude of acceleration of beam 2 and beam 3 are equal to each other during simulation. Also, as seen in Fig. 15, acceleration of beam 5 and slider decrease as function of time.

**Figure 14.** Acceleration of beam 2 and beam 3 under constant angular acceleration, ߙ2

**Figure 15.** Acceleration of beam 5 and slider under constant angular acceleration, ߙ2

#### **4.3. Acceleration vector equations according to center of mass**

The linear acceleration of the center of mass equations are not present in vector loop equations that are previously derived. So, there must be equations relating to the acceleration of the center of mass of beams. Equation derivation is as follows and schematic representation of the center of mass acceleration in first and second loops is shown in Fig. 16 and Fig. 17, respectively.

**Figure 16.** The center of mass acceleration in first loop

The center of mass acceleration of beam 2 along x and y direction;

$$\bar{A}\_{c2} = \bar{R}\_{c2} \tag{57}$$

Dynamic and Quasi-Static Simulation of a Novel Compliant MEMS Force Amplifier by Matlab/Simulink 105

$$A\_{c2x} = -r\_{c2}"\sin\theta\_2"\,^\*a\_2 - r\_{c2}"\cos\theta\_2"\,^\*w\_2^2\tag{58}$$

$$A\_{c2y} = r\_{c2} \, \text{\*} \, \cos \theta\_2 \, \text{\*} \, a\_2 - r\_{c2} \, \text{\*} \, \sin \theta\_2 \, \text{\*} \, w\_2^2 \tag{59}$$

**Figure 17.** The center of mass acceleration in second loop

104 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

**Figure 15.** Acceleration of beam 5 and slider under constant angular acceleration, ߙ2

The linear acceleration of the center of mass equations are not present in vector loop equations that are previously derived. So, there must be equations relating to the acceleration of the center of mass of beams. Equation derivation is as follows and schematic representation of the center of mass acceleration in first and second loops is shown in Fig. 16

**θ3**

..

2 2 *c Ac R* (57)

r3

rc3

**Ac3**

**4.3. Acceleration vector equations according to center of mass**

and Fig. 17, respectively.

**Figure 16.** The center of mass acceleration in first loop

The center of mass acceleration of beam 2 along x and y direction;

**θ2**

**Ac2**

r2

rc2

The center of mass acceleration of beam 3 along x and y direction;

$$A\_{c3} = \stackrel{\cdots}{R}\_{2} + \stackrel{\cdots}{R}\_{c3} \tag{60}$$

$$A\_{c3x} = -r\_2 \, ^\ast \sin \theta\_2 \, ^\ast a\_2 - r\_{c2} \, ^\ast \cos \theta\_2 \, ^\ast w\_2^2 - r\_{c3} \, ^\ast \sin \theta\_3 \, ^\ast a\_3 - r\_{c3} \, ^\ast \cos \theta\_3 \, ^\ast w\_3^2 \tag{61}$$

$$A\_{c3y} = r\_2 \, "\, \cos\theta\_2 \, "\, \alpha\_2 - r\_{c2} \, "\, \sin\theta\_2 \, "\, w\_2^2 + r\_{c3} \, "\, \cos\theta\_3 \, "\, \alpha\_3 - r\_{c3} \, "\, \sin\theta\_3 \, "\, w\_3^2 \tag{62}$$

The center of mass acceleration of beam 6 along x and y direction;

$$A\_{c\circ\delta} = \stackrel{\cdots}{R}\_{c\circ\delta} \tag{63}$$

$$A\_{c6y} = r\_{c6}{}^\* \sin \theta\_6{}^\* a\_6 + r\_{c6}{}^\* \cos \theta\_6{}^\* w\_6^2 \tag{64}$$

$$A\_{c\theta x} = -r\_{c\theta} \, ^\ast \cos \theta\_{\theta} \, ^\ast a\_{\theta} + r\_{c\theta} \, ^\ast \sin \theta\_{\theta} \, ^\ast w\_{\theta}^2 \tag{65}$$

The center of mass acceleration of beam 5 along x and y direction;

$$A\_{c5} = \stackrel{\cdot}{R}\_{2} + \stackrel{\cdot}{R}\_{c5} \tag{66}$$

$$A\_{c5y} = r\_6 \, ^\ast \sin \theta\_6 \, ^\ast a\_6 + r\_6 \, ^\ast \cos \theta\_6 \, ^\ast w\_6^2 + r\_{c5} \, ^\ast \sin \theta\_5 \, ^\ast a\_5 + r\_{c5} \, ^\ast \cos \theta\_5 \, ^\ast w\_5^2 \tag{67}$$

$$A\_{c5x} = -r\_6 \, ^\ast \cos \theta\_6 \, ^\ast a\_6 + r\_6 \, ^\ast \sin \theta\_2 \, ^\ast w\_2^2 - r\_{c5} \, ^\ast \cos \theta\_5 \, ^\ast a\_5 - r\_{c5} \, ^\ast \sin \theta\_5 \, ^\ast w\_5^2 \tag{68}$$

#### **4.4. Force and dynamic analysis of the micro mechanism**

The micro mechanism operates under constant angular velocity, 0.01 [rad/s], the slider crank starts increasing and reaches to its maximum value, 200 micron, meaning that the first stage slider crank is fully opened at 3.20 sec. then the first crank angle pass from 0° and slider begins to get close to its initial position and R1 decreases as shown Fig. 18. According to both crank angles, Θ2 and (90-Θ6), the output force increases or decreases. In the first section of the Foutput vs. time curve, first, both crank angles decrease, and two slider cranks start to open and at small crank angles, Foutput sharply increase and at 3.20 sec Θ2 is equal to 0.0013°

**Figure 18.** Displacement of slider and output force versus time

and at 3.25sec. Θ2 is equal to -0.0012° and at Θ2 these values, Foutput goes to its peak values such as -3.07\*104 µN at 3.20 sec. and 3.32\*104 µN at 3.25 sec. Θ2 decreases until 3.20 sec. and then it increases, whereas 90°-Θ6 decreases and gets close to small values during the simulation. The magnitude of first peak of Foutput at 3.25 sec. is higher than the magnitude of second peak of Foutput at 3.20 sec. due to the fact that (90°-Θ6) at 3.25 sec. is smaller than the value of (90°-Θ6) at 3.20 sec., meaning that small crank angle value of (90°-Θ6) contributes to get much more output force.


**3.2.2 Receptor D4 activity and the attentional focus**

In the next series of experiments, we keep up our focus on the tonic mode of spiking. And, for each imposed SN spike frequency, we examine the effects of changes in the receptor D4 activation. Since the different degrees of D4 activation can be associated to not modeled exogenous or endogenous factors, that are not modeled, this approach makes it possible to speculate plausible outcomes of the dopaminergic agonists (or antagonists) action at the synaptic cleft. The simulations results are summarized in Table 4**.** First, we may note that the results presented in the column relative to *ĝd4* = 1.0 agree with the previous set of experiments. It is more interesting, however, to observe that as the receptor D4 activity diminishes, the thalamic neurons become less active, thus reaching a completely inhibited state. Conversely, as the receptor D4 activity increases, thalamic neurons become more active and tend to spike at the same frequency. Finally, we highlight that, except for the baseline case where the interval between spikes in SN equals 10, when *ĝd4* assumes values lower than 1.0, the differences between *Tx* and *Ty* spiking frequencies disappear. So, the mesothalamic hypoactivity does not impose the attention to focus on the stimulus *Y* anymore.

Mesothalamic Dopaminergic Activity: Implications in Sleep Alterations in Parkinson's Disease 519

100 0.8 Tx and Ty Bursting

50 0.8 all Tonic

40, 30, 20, 10 1.0 all Tonic 5 1.0 TRNx and TRNy Bursting

Fig. 3. Decreases in the receptor D4 activation, under mesothalamic dopaminergic

hypoactivity, turned the tonic state into the burst one in Tx.

Table 5. Distinct dopaminergic activities in presence of similar stimuli.

**3.2.4 Variations in the dopaminergic activity and the oscillatory state**

(*ĝd4*) Neurons Spike Mode

0.9 all Tonic 1.0 all Tonic

0.9 all Tonic 1.0 all Tonic

In our last series of simulations, we deepen our investigations on the changes in the spiking mode. Now, we repeat the strategy undertaken in Subsection 3.2.1, where a stimulus *X* is presented to the asymmetrical network before a stronger one, *Y.* Through the extended mathematical model we propose in Section 2, here we look for situations where changes in the dopaminergic action lead to the burst mode in neurons of the thalamic complex. In this set of simulations, we examine only extreme cases of dopaminergic alterations. Such option was because all previous experiments indicated these extreme situations as the more plausible to initiate the ionic events that underlie the burst mode occurrence. Table 6

Activity of Dopamine Receptor D4\*

Interval between Spikes in SN

summarizes our results.

(ms)

Table 4. Receptor D4 activity and the thalamic tonic state.

#### **3.2.3 Dopaminergic activity and the oscillatory state**

Next, we simulate the mathematical model described in Section 2 through the extended symmetrical network illustrated in the Figure 1. In this set of experiments, two identical, external stimuli, *X* and *Y*, activate the network simultaneously. As in the previous experiments, we impose different SN spiking frequencies. We observe, in Table 5, that extreme and opposing situations lead to the bursting spiking mode in the thalamic complex: the drastic mesothalamic dopamine hypoactivity caused the oscillatory pattern in the *Tx* and *Ty* neurons, whereas the mesothalamic dopamine hyperactivity made the TRN neurons to spike through bursts. With relation to the *Tx* and *Ty* neurons, the appearance of the bursting mode comes from the strong inhibition they suffered through the GABAergic projection from the TRN neurons, which were over activated due to the low dopaminergic activity. Conversely, the oscillatory behavior of the *TRNx* and *TRNy* neurons originated from the inhibition suffered by the TRN due to the high dopaminergic level. The dynamics of the ionic currents involved in such processes are the same described in Subsection 2.2. Figure 3 illustrates the changes in the *Tx* behavior due to the diminishing of the receptor D4 activation, throughout the dopamine hypoactivity case, described in the first line of Table 5**.**

Interval between Spikes in SN (ms)

50

30

20

10

5

Spikes in 100ms

Activity of Dopamine Receptor D4 (*ĝd4*)

0.4 0.6 0.8 1.0 1.2 1.4

Tx - 0 2 2 11 14 Ty - 0 2 14 16 19 TRNx - 46 32 14 9 4 TRNy - 46 32 28 15 6

Tx - 0 4 4 12 15 Ty - 0 4 13 17 18 TRNx - 42 30 25 12 2 TRNy - 42 30 25 12 5

Tx - 0 5 7 12 18 Ty - 0 5 13 15 18 TRNx - 40 30 13 10 5 TRNy - 40 30 21 12 2

Tx 0 0 0 10 16 18 Ty 0 6 10 15 17 19 TRNx 52 30 20 5 5 0 TRNy 50 30 30 10 4 0

Tx 0 5 15 18 18 - Ty 0 5 15 19 19 - TRNx 40 25 10 0 0 - TRNy 40 20 5 0 0 -

Next, we simulate the mathematical model described in Section 2 through the extended symmetrical network illustrated in the Figure 1. In this set of experiments, two identical, external stimuli, *X* and *Y*, activate the network simultaneously. As in the previous experiments, we impose different SN spiking frequencies. We observe, in Table 5, that extreme and opposing situations lead to the bursting spiking mode in the thalamic complex: the drastic mesothalamic dopamine hypoactivity caused the oscillatory pattern in the *Tx* and *Ty* neurons, whereas the mesothalamic dopamine hyperactivity made the TRN neurons to spike through bursts. With relation to the *Tx* and *Ty* neurons, the appearance of the bursting mode comes from the strong inhibition they suffered through the GABAergic projection from the TRN neurons, which were over activated due to the low dopaminergic activity. Conversely, the oscillatory behavior of the *TRNx* and *TRNy* neurons originated from the inhibition suffered by the TRN due to the high dopaminergic level. The dynamics of the ionic currents involved in such processes are the same described in Subsection 2.2. Figure 3 illustrates the changes in the *Tx* behavior due to the diminishing of the receptor D4 activation, throughout the dopamine hypoactivity case, described in the first line of Table 5**.**

Table 4. Receptor D4 activity and the thalamic tonic state.

**3.2.3 Dopaminergic activity and the oscillatory state**

Table 5. Distinct dopaminergic activities in presence of similar stimuli.


**2. The history of liposuction**

Liposuction is defined as a technique in cosmetic surgery for removing excess fat from under the skin by suction. In this section we have focused on the history of evolution of this procedure since its inception, however, we excluded all the techniques, which originated early in the 20th century including excision or curating of adipose tissue [1, 2].

More than 4 decades elapsed when in 1975 a father and son cosmetic surgery team developed the technique of liposuction by introducing a cannula attached to a suction machine. In so doing they produced consistent results with lower risks [3].

Illouz in 1977 innovated modified liposuction blunt end cannulas of smaller diameter. He used cannulas of varied sizes thus extending the technique to the entire body. He injected saline and hyaluronidase into the fat prior to suctioning, allowing for hydro-dissection; hence reducing trauma to other tissues. This was the advent of the wet technique [4, 5].

Although initially the evolution in liposuction was related to the refinement of the blunt end cannulas the surgery was performed under general anaesthetic and in some instance without injecting fluid to hydro-dissect, which lead to significant blood loss in the lipo-aspirate.

The main evolution in the last 4 decades, however, has been with the anaesthesia and the composition of the fluid injected allowing the surgery to be performed under local anaesthetic with sedation and later also excluding sedation. This was popularised by Klein in 1987 using large volumes of very dilute local anaesthetic allowing for larger volume of lipo-aspirate and under local anaesthetic only [1, 2, 6–8].

Klein's infusion mixture included 0.05% Lidocaine, 1:1000,000 epinephrine and 10 mls of 8.4% bicarbonate per litre of saline. He also demonstrated that a large volume of this diluted mixture was safe, however, it also significantly reduced bleeding. Bleeding was a problem when using other techniques [6–8]. The tumescent technique meant less hospitalisation, reduced costs and risks to the patient even for large volumes of lipo-aspirate [9].

Liposuction uses a vacuum pump (a suction machine), however, in 1988 Toledo in Brazil used disposable syringes on which can be fitted blunt end cannulas. This technique allowed for more precision and refinement in liposuction in addition to usability of the fat to transfer. This gave the cosmetic surgeon choice; the vacuum pump assisted liposuction was used for large volumes and for more refined liposuction the syringe system was used [10, 11].

In 2006 laser assisted liposuction using coherent light to deliver energy preferentially targeting fat cells was a new advancement in liposuction surgery. The laser caused rupture of the adipocyte membrane releasing oily content into the extracellular matrix. The laser energy also leads to neocollagen formation and remodelling with reorganisation of the dermis. The fluid produced in this process was then aspirated using blunt end cannulas and a vacuum pump [12, 13].

Ultrasonic liposuction was first introduced in 1992 by Zochhi; it created an alternative to conventional blunt cannulas suction. The aim was to ease the surgeon workload, shorten surgical time, improve results of liposuction, and use smaller cannula, thus reducing tissue trauma and protecting important structures like the neurovascular bundles [14].

The ultrasonic technology required the tumescent technique to combine with the technology for safer outcomes. Titanium probes were used to deliver ultrasonic energy, followed by liposuction.

Refinement of ultrasound energy delivery using VASER technology (vibration amplification of sound energy at resonance), was first reported by Jewell on the clinical application of a third generation ultrasound device that deliver utilised

*The Safe Evolution of Liposuction into Liposculpture DOI: http://dx.doi.org/10.5772/intechopen.97351*

pulsed low power ultrasound with high efficiency using different size small diameter strong titanium probes [15].

The energy applied to the tissues was approximately one-quarter that of previous devices, while the pulsed mode reduced heat generation. Expanded applications of VASER lipolysis and liposculpture include treatment of the male and female breast, face and neck; fibrous body areas (trunk and back), in addition to combined excisional body contouring procedures of all types [16].

Colombian plastic surgeon Alfredo Hoyos presented a vast improvement in technique in 2003 at a national Colombian meeting. Hoyos innovated the high definition liposcultpure HDL [17].

Hoyos elaborated that the nomenclature liposcultpure is not simple fat removal, but an artistic approach designed to sharpen the anatomy of the muscles through the skin. VASER high definition liposcultpure combines technology and technique unlike mechanical liposuction it reduces trauma to blood vessels, achieve better result, and removes superficial and deep fat. This allows for high levels of finesse in sculpting the human-form three-dimensional anatomy thus sharpening the body muscles details at the same time [17].