Resonance vibration of an optical fiber micro-cantilever using electro-thermal actuation

2.1. Actuation part model

Actuators in this system are performing as longitudinal stiffness elements placed underneath the beam. So, stiffness of each actuator wire, $K_{wire}$, is:

where $A_{wire}$, $L_{wire}$ and $E_{wire}$ are the cross sectional area, length and Young’s moduli of each individual wire, respectively. These elements are acting as parallel springs (with an angle) in conjunction with the section of beam from the clamp point to the connection point (i.e, $x=l$). Stiffness of this section of the beam, $K_{beam}$, is provided from basic beam deflection equations and tables [26]:

where $E_{beam}$ and $I$ are the Young moduli of beam material and the second moment of inertia of beam cross section. Similarly, the slope at the end point of the actuation section, $\theta$, can be found via:

where $F_b$ shows the total reaction, force applied on the beam inside the actuator part with length $l$, from actuator wires and the beam part. Taking into account the combined effects of force and thermal expansion and using the superposition method, the displacement at the top of actuator can be written as:

Fig. 2a) Thermal elongation in the actuator wires if not connected to the beam; b) effective thermal elongation in the actuator connected to the beam

In Eq. (4), $u_A^F$ represents the component of displacement from the applied forces between actuator part and the beam part, and $u_A^T$ is the displacement due to temperature change and elongation in the actuator. According to superposition, each can be found by ignoring the other and solving the mechanics of materials equations with respect to that particular effect. For instance, the relation for the applied/interaction force between actuator part and the beam part, $F_A$, and the force induce displacement, $u_A^F$, is found via:

where $\varphi$ is the angle made between actuator wires and vertical axis (see Fig. 1).

Details of finding $u_A^T$ can be described as following using Fig. 2. Superposition is used here again to find the total displacement at the top of the actuator. First, without the beam stiffness, $u_{free}^T$ is found as shown in Fig. 2(a). From basic geometry, length of actuator wires after temperature change, $L_{wire}^{\text{'}}$, can be formulated as:

where ${\alpha }_{wire}$ is the coefficient of linear thermal expansion for the actuator material and $\mathrm{\Delta }T$ is the change in the temperature. Expanding the above equation and ignoring the terms with $O\left({\alpha }^2\right)$ and $O\left({\left[u_{free}^T\right]}^2\right)$ results in the linearized equation:

Note that $u_{free}^T$ is a free expansion without any force involved. However, adding the beam stiffness that resists this displacement, results in compression in the actuator elements and upward elastic deflection (tension in the analogous stiffness element) in the beam section of the actuation part. The interaction results in a middle ground displacement between 0 and $u_{free}^T$ found via:

Combing Eqs. (4), (5) and (8), governing equation in the actuator part that relates reaction force ($F_A$), degree of freedom ($u_A$), and thermal actuation ($\mathrm{\Delta }T$) can be defined as:

2.2. Beam part model

Given small deformations, the equation of motion for a long narrow beam is as follows [25]:

where $u(x,t)$ is vertical displacement along the beam; $m$ signifies the weight per unit length of the beam, and distributed load on the beam (from fluid forces along the beam) is shown with $q(x,t)$. The spatial coordinates, $x$ and $t$, are the coordinate along the length of the beam, and time, respectively. Given the initial and boundary conditions, we can solve the above equation. As Fig. 1(b) shows the following boundary conditions can be identified for this beam:

where $V$ and $M$ represent the shear force and bending moment on beam cross section, respectively.

To solve the differential equation presented in Eq. (1), the method of separation of variables [27] can be used via replacing $u\left(x,t\right)=w\left(x\right)g\left(t\right)$. Keeping in mind that this system is vibrating under small deformation and only free vibration and harmonic response are tackled in this study, the time dependent component is substituted by its harmonic form, $g\left(t\right)=e^{i\omega t}$. This results in the following forms of the equation of motion.

2.3. Effect of surrounding fluid

The harmonic form of hydrodynamic forces on the micro-cantilever, $Q\left(x,\omega \right)$, is represented via following general form [28]:

where $\rho$ is the density of the fluid medium, $b$ is the dominant length scale (diameter for a circle and width for a rectangle), and $\mathrm{\Gamma }\left(\omega \right)$ is a dimensionless function. This equation can be stipulated through solving the fluid motion equations around a rigid beam. For a beam with circular cross section an analytical solution for $\mathrm{\Gamma }\left(\omega \right)$ is available in the following form [29]:

where $Re=\rho \omega b^2/\left(4\eta \right)$, and$K_i\left(x\right)$ is the modified Bessel function of second kind, and $\eta$ is the viscosity of the fluid.

Therefore, Eq. (12) can be simplified to the following form:

Which is a fourth order differential equation with the following generic homogenous answer:

where the $C_i$ coefficients are the constants of this equation; they can be determined by the boundary conditions. Using Eq. (16) in the boundary condition in Eq. (16) results in the following algebraic terms:

These five equations are summarized in a matrix form as:

where:

2.4. Free vibration analysis

The free vibration (modal) analysis of the micro-cantilever is done with no external load on the system. This can be achieved by replacing $\left\{b\right\}={\left\{0\right\}}^T$. In this case, the resulting from of Eq. (18) has a non-trivial zero answer only if the determinant of $\mathbf{A}$ is zero ($\left|\mathbf{A}\right|=0$). Therefore, the following characteristic equation is extracted for the free vibration of the beam:

It can be checked that, this characteristic equation reduces to the equation of a simple cantilever beam (i.e., $1+\cos \left(\beta L\right)\cosh \left(\beta L\right)=0$) if $D_1\to \infty$. Also, note that $l\to 0$ has the same effect, as it makes $K_{beam}\to \infty$ and therefore $D_1\to \infty$, after solving the indeterminate form of $0\times \infty$, it again reduces to the characteristic equation for a regular cantilever beam. The complexity of this non-linearity equation makes it unsolvable analytically; therefore, we use numerical methods for solving it. Eq. (20) has multiple complex roots in the form of $r={\omega }_d-iϵ$, where ${\omega }_d$ is damped natural frequency, and $ϵ$ is the attenuation (damping) in the system [30].

2.5. Harmonic response

The following form of harmonic loading is imposed on system in Eq. (19):

It must be mentioned that, this is a simplified form of the thermal excitation. The simplifications assumptions are as following:

(1) The electrical current applied to the actuators, to generate heat, can be sinusoidal/harmonic; however, because of the nature of the thermal conduction equation, the resulting temperature changes are not. Indeed, there is non-homogeneity in the temperature distribution along the length of actuator. The assumption of perfect sinusoidal $\mathrm{\Delta }T\left(t\right)$ is a simplification of a more complex function. Nonetheless, because of the small time-constant of micro-scale systems [18], the temperature changes in the actuator can be adjusted rapidly according to the magnitude of electrical current and steady-state can be achieved rapidly (for a more detailed discussions check [21]); and

(2) Active cooling cannot be practically implemented in this system. However, assuming that, after achieving steady-state, there is a base temperature and $\mathrm{\Delta }T\left(t\right)={\stackrel{-}{\mathrm{\Delta }T}}_0+\stackrel{-}{\mathrm{\Delta }T}{e}^{i\omega t}$, where (${\stackrel{-}{\mathrm{\Delta }T}}_0\ge \stackrel{-}{\mathrm{\Delta }T}$), the above formula can be valid for the harmonic part. The constant static deformation, due to the steady-state temperature increase (${\stackrel{-}{\mathrm{\Delta }T}}_0$), can be calculated easily and added to the dynamic response (superposition of static and dynamic deformation). We only consider the harmonic part of mechanical displacement.

Although Eq. (18) can be solved analytically to find the closed form of $C_i$ coeffiecnts, it was found that the resulting closed forms leads to complex terms that does not reduce the complexity of algebraic equations and numerical computations. On the other hand, using a numerical matrix solver package for direct evaluation of $C_i$ coefficients from its matrix form proved to be computationally and programmatically more efficient. The numerical linear algebraic (linalg) modules of NumPy package in Python programming language are used for this purpose here.

In addition to displacement that is directly calculated from the Eq. (16), the stress inside the beam can be found via [31]:

Hence:

In which $z$ is the distance from the neutral axis of the beam.

3.1. Free vibration

Fig. 3 show the first 3 natural frequencies of the system with the dimensions listed above, in exposure to vacuum, air (1 and 10 atm), and water. The drop in the natural frequency when exposed to a denser medium is obvious in the figure. It can be related to the fact that in conjunction to a denser fluid, more mass needs to be displaced during vibration, which in turn increases the effective mass of system and subsequently decreases the natural frequency. A comparison to the natural frequencies obtained from a finite element ANSYS model shown in Fig. 4, and the natural frequencies obtained from a simple cantilever with length $L$ is shown in this figure to validate the formulations presented above. The ANSYS model uses a 3D representation of the current system with the described boundary conditions. Also, only one half of this model is created and meshed, however proper boundary conditions are imposed for symmetry. As it can be seen in Fig. 3, results of FEM are acceptably close to the analytical solution. Moreover, as comparing the results of simple clamped beam to the other cases shows, this system in terms of natural frequency resembles a clamped beam with length $L$. Further investigations on the effect of actuator wires (its dimensions and stiffness) as well as distance of the actuator to the clamp point ($l$), showed that these parameters have negligible effect on the natural frequency (less than 0.1 %).

Fig. 3First 3 natural frequencies predicted from the analytical model in cavity and different fluid mediums, compared to the FEM model and simple cantilever model in cavity

Fig. 4The finite element model used to verify the analytical model

The eigenvalues of Eq. (18) was used to find the natural frequencies. Using the eigenvectors related to latter natural frequencies results in mode shapes for corresponding natural frequencies. Fig. 5, depicts mode shapes obtained for the first 3 modes calculated analytically and is compared with mode shapes extracted from ANSYS. Note that only flexural modes are taken into account here and torsional and longitudinal modes are ignored.

Fig. 5First 3 mode shapes obtained from a) analytical model; and b) ANSYS

a)

b)

3.2. Harmonic response

Harmonic response of the beam over a frequency range of 0 to 300 kHz is studied with $\stackrel{-}{\mathrm{\Delta }T}=$ 1 °C. Fig. 6 shows the amplitude of displacement at the tip point of the beam and the maximum stress ($z=\pm d/2$) at the mid-section of the beam. A comparison is made with an analogous ANSYS model where effect of thermal expansion is replaced by directly imposing displacements at the point that actuator is attached to the beam. Normalizing results of analytical solution to 1 nm displacement actuation, shows that there is agreeable correspondence between the results of the analytical solution and the FEM method.

Fig. 6Harmonic response of the micro-cantilever using analytical model compared to the analogous ANSYS model: a) tip displacement; b) stress magnitude for 1 nm actuation

a)

b)

Fig. 7 shows the magnitude of tip displacement and maximum stress at the mid-span of the beam in vacuum. The harmonic response with the dimensions mentioned above are compared to the cases with change in the actuator wire length ($L_{wire}$), the actuation distance ($l$) and the actuator angle ($\varphi$). Although it was observed before that these variables do not have noticeable effect on the natural frequency, as it can be seen in this figure, they can substantially influence the tip displacement and mid-span stress values. Jumps in the magnitude of the displacement as well as stress at the frequencies that corresponds to the natural frequency are also observed in this graph. This clearly shows the resonance phenomenon. It should be mentioned that in in this case the peak points are singular point that go to infinite (no damping in resonance).

Similarly, Fig. 8 shows the effect of change in wire length ($L_{wire}$), the actuation distance ($l$) and the actuator angle ($\varphi$), while the micro-cantilever is exposed to air with 10 atm pressure. As it can be seen, the general response of system is relatively the same compared to vacuum, however, there is a noticeable drop in the magnitude of response at the resonance frequency. This is due to the effect of damping which is particularly dominant at the resonance frequency [25].

Fig. 9 illustrates the tip displacement of the micro-cantilever at its first (fundamental) frequency as a function of actuator distance from the clamp point ($l$) for various values of actuator wire length ($L_{wire}$) at 1 atm air pressure. As it can be seen, increasing the length of the actuator wire increases the tip displacement almost proportionally. Keeping in mind that the longer actuator wire means longer displacement at the top of the actuator Eq. (7), this result is expected. On the other hand, the tip displacement falls drastically as a result of increase in the distance from the base. The latter phenomenon can be related to the fact that shorter distance between the actuator and the base increases the slope of the beam as it can be seen in Eq. (3).

Fig. 7Harmonic response of micro-cantilever in vacuum: a) tip displacement; b) maximum stress magnitude at mid-span

a)

b)

Fig. 8Harmonic response of micro-cantilever in air (10 atm): a) tip displacement; b) maximum stress magnitude at mid-span

a)

b)

3.3. Quality factor

Calculating the quality factor for the system under consideration shows that the quality factor is independent of the parameters of the actuation part and only dependent on the beam part (beam length and diameter) as well as ambient pressure and mode number. Fig. 10 shows the quality factor calculated using the integration method (described in Eqs. (24-27)) for the first two modes of the original model, and changes in the length and diameter of the micro-cantilever. It must be noted that the approximated low dissipation Eq. (28) results in values that are up to –2 % higher than the values that are shown here.

Fig. 9Effect of actuator distance from the clamp point and the length of actuator wire on the tip displacement magnitude at first natural frequency

Despite the fact that each line represents a different resonance frequency (depending on mode number, and beam length and diameter), similar trends in quality factor can be noticed as a function of air pressure. The effect of air pressure on the natural frequency in each system demonstrated here is minimal (i.e., <1 %), however, a considerable change in the quality factor is observed. Increasing the air pressure results in a rapid decline in the quality factor initially but this slows down as the magnitude of pressure is increased. Increasing the beam length and/or decreasing the beam diameter result in more displacement in the beam (due to lower flexural rigidity). Higher displacement amplitudes escalate the effect of drag force and eventually decrease the quality factor, as it can be seen from the lines that represent increasing length ($L=$ 1000) and decreasing diameter ($d=$ 12). A remarkable observation in this figure is substantial increase in the quality factor in the second mode. Generally, the quality of factor of vibrating micro-cantilever systems are expected to increase at higher modes [33]. Nonetheless, in practice the predicted quality factor of higher modes can be affected by factors such as larger displacement gradient at higher modes, and effect of fluid compressibility [34] that are not taken into account in the simplified and linearized models.

Fig. 10Effect of actuator distance from the clamp point and the length of actuator wire on the quality factor at first natural frequency