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Large amplitude vibration of a cantilever actuated by a high-frequency pulsed laser

Jin Li, Tao Sun, Ziwei Meng, and Xingyu Liao

Open Access Open Access

Abstract

Laser excitation based on the thermoelastic principle is effective for micro-scale actuation, enabled energy conversion from optical to mechanical. The major advantages lie in non-contact actuation, easy miniaturization, and integration. To avoid surface damage, the laser power per unit is limited, leading to several micrometers of the vibration. In this study, a pure nickel millimeter-sized cantilever is successfully actuated at a low-frequency resonance (around Hz) via a nanosecond pulsed laser. By modal interaction, the energy is transferred from a low-intensity, high-frequency (around kHz) excitation to a low-frequency response with millimeter amplitude. The stable low-frequency resonance of the cantilever was maintained by changing the laser pulse parameters and the illumination locations. We also present a method to control the vibration of the cantilever using a modulated wave (MW: the laser wave modulated by a rectangular wave). The cantilever’s amplitude can be efficiently adjusted by changing the laser power or duty cycle of the MW. The resonance frequency of the cantilever also can be altered by optimizing the geometries to meet various actuation requirements. This study enables large actuation (up to tens of millimeters) by laser excitation, facilitating applications in precision manipulation, microfluidic mixing, lab-on-a-chip device, and other related micro actuation devices.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Microscale actuation and manipulation are widely used in semiconductors, bio-microelectromechanical systems, and high-precision manufacturing. However, during the motion and transportation process, the conventional contact actuation approaches may cause damage to the objects, which is unfavorable. In the conventional contact actuation principles, extraordinary performance in electrical and magnetic conductivity is required for the objects. In addition, movable excitation and actuation are limited by the bulky equipment and complex wire-linking, hindering application in hostile environments. In fact, manipulations on micro and millimeter-sized objects cannot be performed appropriately [1,2]. Laser excitation based on the thermoelastic principle is essentially a type of noncontact micro actuation method that could efficiently solve the aforementioned problem without causing any damage. The energy required for object actuation and manipulation can be provided by converting optical energy into mechanical energy. Hence, non-contact and remote actuation, integration, and miniaturization can be easily realized. Laser excitation with various properties (such as broad bandwidth, strong anti-electromagnetic interference ability, large driving amplitude, and fast response.) and waves also extensive its applications in micro actuation [3,4]. The generation of the thermoelastic wave by pulsed laser irradiation on a metal surface was first demonstrated by White [5]. And then amounts of studies have been conducted on the mechanisms and applications. Gao et al. [6] effectively excited the microcantilever by using an intensity-modulated laser. Specifically, the influence of laser parameters and microcantilever structural parameters were investigated. Fu et al. [7,8] achieved selective mode excitation of microcantilever by poisoning the laser spot on a specific region along with the lever. Milde et al. [9] studied the optomechanical effect of microcantilever self-oscillation excitation and the influence of the tuning of laser power in it. Therefore, laser excitation based on the thermoelastic principle could be an appropriate alternative for contact micro actuation.

According to the incident laser energy density, the mechanisms of laser actuation can be generally divided into two categories: thermoelastic mechanism and ablation mechanism. The thermoelastic mechanism (approximately less than 107W/cm2) does not generally cause damage to the materials [10]. Therefore, in the early years, Scruby et al. focused on the inner thermoelastic mechanism based on a point-source or line-focused source [10,11], while Wu et al. [12]and Suh [13] attempted to provide an analytical solution to the thermoelastic coupling equation, such that more details of thermoelastic deformation could be revealed. In recent years, researchers are paying more attention to the applications of laser actuation owing to its non-contact and non-destructive properties. The laser streaming formed in the nanostructure under-liquid environments can be applied to microfluidic systems. Wang et al [4] realized a single point self-generated laser acoustics transducers in solution, which can be applied for liquid control and microparticle operation by transforming the laser acoustics into a flow of liquid. Multi-position self-formed laser acoustic transducers were optimized by Yue et al. [14]. This conveniently generates multi-laser streaming and significantly extends its applications and efficiency. Through laser actuation, the vibration performance of the microcantilever has been studied as well. Demirkiran et al. [15] excited the 1st order and 2nd order resonance of the microcantilever by nanosecond pulsed laser. The relationship between the laser frequency and natural frequency of the microcantilever was discussed. Li et al. and Wu et al. [16,17] realized a high amplitude (up to 11 µm) at a high repeat frequency (14 kHz) of laser, while Kiracofe et al. [18] improved the photothermal efficiency by changing the shape of the microcantilever and irradiation position. Additionally, the dynamic vibration performance of the cantilever can be more complex under different resonance modes [19].

In some weak nonlinearity dynamic systems, experimental results show that energy can be transferred from high to low-frequency modes through modal interaction [20,21]. Bian et al. [22] proposed a new method to control the nonlinear vibration of the manipulator based on the internal resonance, and Wang et al. [23] studied the nonlinear vibration, energy transmission, and bifurcations of the vertical cantilever through the strict control of internal resonance. The modal interaction of the cantilever usually occurs under the following conditions [24,25]: (1) the cantilever excited at a high frequency becomes chaotic, leading to the transference of the resonance mode from high to low, and (2) the excitation frequencies are almost equal to the sum or difference of the two or more natural frequencies [26,27]. These conditions often lead to an energy transfer from high-frequency to low-frequency modes.

The aforementioned thermoelastic mechanism-based actuations are primarily focused on the micrometer-sized cantilevers, large amplitude cannot realize. In this study, we investigate non-contact actuation via a nanosecond pulsed laser based on thermoelastic mechanism and its application to millimeter-sized cantilevers. We employ a high-frequency but low-pulse-energy laser to achieve millimeter-sized amplitude, in case of surface damage. The energy successfully transfers from low-density high-frequency excitation to high-amplitude (millimeter-sized amplitude) low-frequency oscillation due to modal interaction. The amplitude and frequency responses of the cantilever are also investigated here. Under specific conditions, the different low resonant frequencies can be realized as well.

This paper is structured as follows: the mechanisms of thermoelastic wave propagation on a millimeter-sized cantilever are analyzed in Section 2, the non-contact thermoelastic wave excitation testing system is introduced in Section 3, and the experimental validations of modal interaction as well as the modulations of vibration responses of the cantilever are investigated in Section 4. This paper is concluded with Section 5.

2. Theoretical analysis

2.1 Mechanisms of thermoelastic wave generation and thermal diffusion modeling

The mechanisms of thermoelastic wave generation are based on complicated physical processes. As shown in Fig. 1(a), when the cantilever absorbs the heat generated by a pulsed laser, the differential temperature distribution is formed on the upper surface of the cantilever, leading to various degrees of deformation. However, the material is still in the range of elastic deformation, and thermal elastic waves are periodically generated due to the periodic deformation resulting from the illumination of a periodic nanosecond pulsed laser.

 figure: Fig. 1.

Fig. 1. Model of cantilever and excitation principle.

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Figure 1(b) shows a model of the millimeter-sized cantilever, and L, W, and H represent the length, width, and thickness of the cantilever. The left of the cantilever is fixed on the two-axis translation stage for ensuring a directional change of the x-axis and z-axis. The right end is free. The thermoelastic oscillation is shown in Fig. 1(c). According to the energy conservation law, the temperature distribution on the surface of a cantilever follows

$$\frac{{\partial T(x,y,z,t)}}{{\partial t}} = {a^2}\left( {\frac{{{\partial^2}T({x,y,z,t} )}}{{\partial {x^2}}} + \frac{{{\partial^2}T({x,y,z,t} )}}{{\partial {y^2}}} + \frac{{{\partial^2}T({x,y,z,t} )}}{{\partial {z^2}}}} \right) + bQ(x,z,t),$$
where $\rho $ is the material density, ${c_p}$ is the specific heat capacity at constant pressure, and $k$ is the thermal conductivity. $T(x,y,z,t)$ is the time and space temperature distribution in the surface of the cantilever, $Q(x,y,z,t)$ is the heat, generated by nanosecond pulsed laser, and it can be expressed as
$$Q(x,y,z,t) = {I_0}A(T)f(x - {x_0})g(t),$$
where ${I_0}$ is the peak power density of the Gaussian laser, $A(T)$ is the material absorption coefficient, and $f(x - {x_0})$, and $g(t)$ are the time domain and space domain functions of the Gaussian laser, expressed as
$$f(x - {x_0}) = \exp \left( { - \frac{{{{({x - {x_0}} )}^2}}}{{{r^2}}}} \right),$$
$$g(t) = \frac{t}{{{t_0}^2}}\exp \left( { - \frac{t}{{{t_0}}}} \right),$$
where $x$, y are the coordinates in the Cartesian coordinate system, ${x_0}$ is the coordinate of the illuminated point, $r$ is the radius of the pulsed laser, and ${t_0}$ is the laser pulse rise time (full width at half maximum). The time-domain distribution of the Gaussian beam is shown in Fig. 1(d).

Without consideration of convection heat dissipation and radiation heat transfer, the initial conditions and boundary conditions also can be given as follows:

Initial condition:

$$T = {T_0} {\kern 1cm} t = 0.$$

Boundary conditions:

$$- k\frac{{\partial T}}{{\partial x}} = 0 {\kern 1cm} \left( {\begin{array}{cc} {x = 0,}&{x = L} \end{array}} \right),$$
$$- k\frac{{\partial T}}{{\partial y}} = 0 {\kern 1cm} ( \begin{array}{cc} {y = 0,}&{y = W} \end{array} ),$$
$$- k\frac{{\partial T}}{{\partial z}} = 0 {\kern 1cm} ( \begin{array}{cc} {z = 0,}&{z = H} \end{array}).$$

Equations (6)–(8) indicate that each side of the cantilever is thermally insulated.

2.2 Thermoelastic modeling

The laser acoustic deformation $U$ that is generated by the nanosecond pulsed laser can be expressed as

$$({\lambda + \mu } )\nabla ({\nabla \cdot U} )+ \mu {\nabla ^2}U - \rho \frac{{{\partial ^2}U}}{{\partial {t^2}}} = ({3\lambda + 2\mu } )\alpha \nabla T,$$
where $U$ is the time-domain displacement, $\lambda $ and $\mu $ are the lame constants, $\rho $ is the density, and $\alpha $ is the thermoelastic expansion coefficient of the isotropic materials. The initial conditions and boundary conditions are as follows:

Initial condition:

$$U = \frac{{\partial U}}{{\partial t}} = 0 {\kern 1cm} t = 0.$$

Boundary condition:

$$n \cdot [\sigma - ({3\lambda + 2\mu } )\alpha T({x,y,z,t} )I] = 0,$$
where $n$ is the unit vector normal to the surface, $I$ is the unit tensor, and $\sigma $ is the stress tensor.

The displacement can be theoretically solved by the combining of Eq. (1) and Eq. (6)-(11).

Figure 2 shows the temperature simulation of the cantilever with a pulse width of 50 ns, 500 ns, 5000 ns, and 10000 ns. The distance from the illumination point to the fix-end of the cantilever is around approximately 45 mm. The initial temperature is set as 293.15 K. The laser radius and the pulse energy are set as 50 µm and 1 mJ, respectively. There is an obvious gradient of temperature from 290 K to 370 K when the pulse width is 50 ns, and the temperature could almost return to its initial value before the next laser pulse illumination. Thus, the cantilever experiences a periodical deformation without heat accumulation. This is the key point of thermoelastic wave generation. However, the temperature gradient becomes smaller as the pulse width increases, and also with heat accumulation, that may damage the material. Therefore, a periodical thermoelastic wave cannot be formed.

 figure: Fig. 2.

Fig. 2. Temperature simulation of the cantilever under different pulse width laser illumination.

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This analysis provides a clear understanding of the generation of thermoelastic wave, and the coupling thermoelastic Eq. (1) and (7) also provide theoretical guidance for performing the following experiments.

3. Experimental setup

Figure 3 shows a schematic of the pulsed laser actuation experimental system, that comprises a non-contact nanosecond pulsed laser (YFL-PN-30-GM-L, Guangzhi, Wuhan, China), modulation module, laser sensor (LK_G4000A, Keyence, Osaka, Japan), and data acquisition card (NI, PXIe-6361, Austin, Texas, USA).

 figure: Fig. 3.

Fig. 3. Schematic of pulsed laser actuation and measurement system.

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An emitted laser irradiates on the upper surface of the millimeter-sized cantilever through a reflector and a convex lens. The millimeter-sized cantilever generates periodic thermoelastic waves by periodic temperature gradient by laser illumination. The irradiation location on the surface of the cantilever can be adjusted via a positioning stage. The vibration amplitude of the cantilever is measured by a laser displacement laser at the free end of the cantilever. All the experiments were conducted in a clean room (the temperature is 293.15 K).

The parameters of the pulsed laser are shown in Fig. 4. Excitation of the 1st resonance of the cantilever. (a) Time-domain oscillograph, (b) Frequency spectrum response. Table 1. The laser power, the repetition frequency, and the pulse width can be adjusted. The initial radius of the laser point is 6 mm. Additionally, in the experiments, the laser is focused by a convex lens, and the cantilever was adjusted to the focus point of the laser by the positioning stage, at this time, the radius of the laser point is around 40µm. The laser emission is controlled by an external trigger from the controller. Importantly, the excitation frequency can be modulated to a lower frequency (about 1 Hz) by employing a rectangular wave with a variable duty cycle as an external trigger signal. This could be used to excite high-order resonance and improve the energy efficiency through modal interaction.

 figure: Fig. 4.

Fig. 4. Excitation of the 1st resonance of the cantilever. (a) Time-domain oscillograph, (b) Frequency spectrum response.

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Tables Icon

Table 1. Parameters of nanosecond pulsed laser

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4. Experimental observation and discussion

4.1 High-frequency laser excitation to low-frequency vibration

To investigate the frequency response of the millimeter-sized cantilever by high-frequency pulsed laser, the laser power, excitation frequency, and the pulse width were set as 9 W, 25 kHz, and 200 ns, respectively. The length, width, and thickness of the cantilever were set as 85 mm, 6 mm, and 0.1 mm, respectively. The distance of the irradiation points and measure point to the free end of the cantilever is set as 40 mm and 2 mm, respectively. Figure 4(a) shows a time-domain oscillograph of the cantilever with millimeter amplitude. Figure 4(b) shows the 1st resonance of the cantilever excited by a high excitation frequency (25 kHz: considerably larger than the 1st natural frequency). These observations could be attributed to the modal interaction by nonlinear effects. The additional energy overflows from the high mode to the low mode when the excitation frequency is significantly higher than the critical points for modal stability. Here, we use the nanosecond pulsed laser to realize modal interaction with a large vibration amplitude.

To investigate the robustness of the vibration of the cantilever based on the thermoelastic principle, we choose the excitation frequency, irradiation location, and laser power as the controlled parameters. Figure 5 shows the different irradiation locations. d is set to 34 mm, 44 mm, 54 mm, and 58 mm, respectively. We can conclude from Fig. 6 that the change in the excitation frequency (from 25 kHz to 80 kHz) has a minor impact on the vibration frequency (approximately 10 Hz) of the cantilever. By comparing with theoretical value (10.606 Hz) according to the Euler-Bernoulli beam theory, the deviation is expected to be around 0.1 [28].

 figure: Fig. 5.

Fig. 5. Irradiation and measuring locations of the cantilever.

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 figure: Fig. 6.

Fig. 6. Frequency response corresponding to different excitation locations versus laser repetition frequency.

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Figure 7 shows the amplitude and the frequency response to the change in the irradiation location along the cantilever. For the same excitation frequency, with the change of the excitation location d, the vibration response frequency of the cantilever exhibits a slight decrease, whereas the vibration amplitude increases. The frequency response may be attributed to the change in the thermophysical parameters during the illumination. Such as the elasticity modulus ($E$), material density ($\rho $), specific heat capacity (${C_P}$), and thermal conductivity ($k$). The amplitude response may be the result of the different strain distributions along the length of the cantilever in the 1st modal shape.

 figure: Fig. 7.

Fig. 7. Frequency response and amplitude versus excitation location.

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Figure 8 shows the amplitude and frequency responses of the cantilever by laser excitation with different power. We can see that the frequency response is barely influenced by the laser power, while the amplitude shows an increment (from 5.485 mm to 7.724 mm). For a specific pulse width, a pulsed laser with a high laser power could realize tens of millimeters of amplitude. It is considerably larger than the values attained in the previous studies (micrometer-sized amplitude).

 figure: Fig. 8.

Fig. 8. Frequency response and amplitude versus laser power.

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Based on the analysis above, we have successfully validated the modal interaction by the pulsed laser excitation. The energy can transfer from a high mode to a low mode with millimeter amplitude. The robustness of the vibration of the cantilever is investigated as well, the stability of the cantilever is maintained, despite the change in the laser power, excitation frequency, and irradiation location.

4.2 Vibration control

To control the vibration response, we propose a method of modulation of the illumination duty cycle and thus, the temperature cycle. The nanosecond pulsed laser is modulated by a rectangular wave, and the excitation frequency in the following parts is named modulation laser wave frequency (MLWF). The diagram of the modulated laser wave is shown in Fig. 9. The illumination time period and excitation frequency are determined by duty cycle and MLWF.

 figure: Fig. 9.

Fig. 9. Diagram of the modulated rectangular wave. (a) Rectangular wave, (b) Pulsed laser wave, (c) Modulated laser wave.

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The frequency response of the cantilever to the modulated pulsed laser is investigated. The laser power, excitation frequency, duty cycle, and pulse width are set to 10.5 W, 25 kHz, 80%, and 200 ns, respectively. The structural parameters of the cantilever are set to 85 mm (length), 6 mm (width), 0.1 mm (thickness), respectively. The distance from the irradiation point to the free end of the cantilever is set to 27 mm. Figure 10 shows the frequency spectrum of the 1st and 2nd resonance induced at 9.639 Hz or 63.856 Hz. It demonstrates that the high orders of resonance of the cantilever could be excited by the control of the MLWF.

 figure: Fig. 10.

Fig. 10. Frequency spectrum. (a) MLWF at 9.639 Hz, (b) MLWF at 63.856 Hz.

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When the MLWF is set to 60.256 Hz or 181.936 Hz, as shown in Fig. 11, there are additional peaks at the lower orders and multiple frequencies besides the excitation frequency. There is a sideband around the excitation frequency. The sideband spacing is 3.612 Hz in Fig. 11(a) and 19.77 Hz in Fig. 11(b), that is close to one or two times the frequency of the first peak. Modulated motion is developed, and bifurcation appears; this could be attributed to the nonlinear effects of the cantilever. The low-order contributions to the response are enhanced under the excitation frequency.

 figure: Fig. 11.

Fig. 11. Frequency spectrum of the cantilever. (a) MLWF at 60.256Hz, (b) MLWF at 181.936Hz.

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To investigate whether the duty cycle has an impact on the amplitude, the cantilever is also induced by a modulated pulsed laser. The excitation frequency and the pulse width are set to 50 kHz and 200 ns, respectively, while the laser power and the irradiation location are set to 9 W and 42 mm, respectively. The MLWF is set to 67.494 Hz, that equals the 2nd-order natural frequency of the cantilever. The parameters of the cantilever are maintained as before. Figure 12 shows the sharp increase (from 0.196 mm to 8.608 mm) in amplitude; this can be attributed to the fact that a larger duty cycle has a higher energy. This approach can realize a high range amplitude tunability than the adjustment of the laser power.

 figure: Fig. 12.

Fig. 12. Amplitude response of the cantilever under different duty cycles.

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From the analysis above, the 1st and 2nd resonances are excited by the modulated pulsed laser, and bifurcation occurs when the MLWF decreases. It shows that the response frequency and the amplitude of the cantilever can be controlled by the MLWF and the duty cycles. This finding enables applications in some specific micro-actuation fields.

5. Conclusion

In this study, we proved that the response frequency of the cantilever can be modulated through modal interaction. We efficiently realized energy transfer from low-density, high-frequency excitation to low-frequency response with millimeter amplitude. The robustness of the vibration of the cantilever is investigated as the pulsed laser parameters and irradiation location vary. Simultaneously, we propose a method to control the vibration response of the cantilever. The high orders of resonance can be induced by the modulated laser at a specific MLWF. The amplitude of the cantilever can also be adjusted by changing two parameters: laser power and duty cycles. Moreover, for more precise vibration control of the metal sheets with arbitrary shapes, a new modulation method might be considered, such as the modeling of the point-array laser or line-focused laser. In summary, laser actuation based on the thermoelastic principle facilitates micro actuation applications and improves the energy efficiency by converting high-frequency excitation to low-frequency response. Thereby this enabling potential applications pertaining to non-contact micro-scale actuation.

Funding

National Natural Science Foundation of China (52075172); Natural Science Foundation of Shanghai (19ZR1413300).

Disclosures

The authors declare no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (12)
Fig. 1.
Fig. 1. Model of cantilever and excitation principle.

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Fig. 2.
Fig. 2. Temperature simulation of the cantilever under different pulse width laser illumination.

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Fig. 3.
Fig. 3. Schematic of pulsed laser actuation and measurement system.

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Fig. 4.
Fig. 4. Excitation of the 1st resonance of the cantilever. (a) Time-domain oscillograph, (b) Frequency spectrum response.

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Fig. 5.
Fig. 5. Irradiation and measuring locations of the cantilever.

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Fig. 6.
Fig. 6. Frequency response corresponding to different excitation locations versus laser repetition frequency.

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Fig. 7.
Fig. 7. Frequency response and amplitude versus excitation location.

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Fig. 8.
Fig. 8. Frequency response and amplitude versus laser power.

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Fig. 9.
Fig. 9. Diagram of the modulated rectangular wave. (a) Rectangular wave, (b) Pulsed laser wave, (c) Modulated laser wave.

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Fig. 10.
Fig. 10. Frequency spectrum. (a) MLWF at 9.639 Hz, (b) MLWF at 63.856 Hz.

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Fig. 11.
Fig. 11. Frequency spectrum of the cantilever. (a) MLWF at 60.256Hz, (b) MLWF at 181.936Hz.

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Fig. 12.
Fig. 12. Amplitude response of the cantilever under different duty cycles.

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Tables (1)

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Table 1. Parameters of nanosecond pulsed laser

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Equations (11)

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∂ T ( x , y , z , t ) ∂ t = a 2 ( ∂ 2 T ( x , y , z , t ) ∂ x 2 + ∂ 2 T ( x , y , z , t ) ∂ y 2 + ∂ 2 T ( x , y , z , t ) ∂ z 2 ) + b Q ( x , z , t ) ,
Q ( x , y , z , t ) = I 0 A ( T ) f ( x − x 0 ) g ( t ) ,
f ( x − x 0 ) = exp ⁡ ( − ( x − x 0 ) 2 r 2 ) ,
g ( t ) = t t 0 2 exp ⁡ ( − t t 0 ) ,
T = T 0 t = 0.
− k ∂ T ∂ x = 0 ( x = 0 , x = L ) ,
− k ∂ T ∂ y = 0 ( y = 0 , y = W ) ,
− k ∂ T ∂ z = 0 ( z = 0 , z = H ) .
( λ + μ ) ∇ ( ∇ ⋅ U ) + μ ∇ 2 U − ρ ∂ 2 U ∂ t 2 = ( 3 λ + 2 μ ) α ∇ T ,
U = ∂ U ∂ t = 0 t = 0.
n ⋅ [ σ − ( 3 λ + 2 μ ) α T ( x , y , z , t ) I ] = 0 ,
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