Elastic micro-beam resonator with two free end boundary conditions

Abstract

The present application relates to a kind of elastic micro beam resonators with two ends free boundary conditions, including substrate, first fixed support, second fixed support, support beam, free-free beam and fixed electrode;The number of support beam and the position of support point of support beam on free-free beam are determined according to the following rules: when free-free beam is required to work at its own N order natural frequency, N+1 identical support beam is vertically supported on the same side of free-free beam, forming N+1 vertical support point, and N+1 vertical support point is the N+1 node of the N order mode function of free-free beam;The cross section of support beam and free-free beam is rectangular, and the side length of rectangle corresponds to equal respectively.The present application realizes the boundary condition of beam two ends free support, and helps to effectively exert its performance advantage in engineering application.

Description

Technical Field

[0001] This invention relates to the field of microelectromechanical systems (MEMS), and in particular to elastic microbeam resonators with free boundary conditions at both ends. Background Technology

[0002] Referring to reference [1], the elastic microbeam resonator mainly consists of a microbeam, a fixed support, and a substrate. The boundary conditions of the microbeam are generally single-end fixed (i.e., cantilever) or double-end fixed. Figure 1 The diagram shown is a schematic diagram of a commonly used cantilever microbeam device. Figure 1 The length, width, and thickness directions of the microbeam are x, y, and z, respectively, and an AC driving voltage is applied between the microbeam and the substrate. Under the action of alternating electrostatic force, the microbeam undergoes elastic vibration deformation along the thickness direction, as shown by the dotted line in the figure. The microbeam vibrates at its natural frequency, and the vibration direction is perpendicular to the substrate. The fixed support and the substrate are relatively rigid and do not deform. The mechanical support boundary conditions of the cantilever are easily implemented in silicon micromachining, so they are widely used. For example, see references [2, 3], for a cantilever microbeam of length L, its deformation curve is y(x, t) = φ(x)f(t), where φ(x) is the amount of deformation of the beam, and f(t) is a function of time. The boundary condition is: at the support of the beam, i.e., x = 0, the amount of deformation of the beam is zero, and the angular deformation of the beam is also zero (the first derivative of the deformation curve is zero), that is...

[0003]

[0004] In engineering, the mechanical boundary conditions for the support can be achieved simply by designing and manufacturing the stiffness at the cantilever beam's support points (x=0) to be absolutely maximum. For a microbeam with fixed ends, the deformation and angular deformation of the beam are zero at both support points, x=0 and x=L. Therefore, the stiffness at both ends (x=0 and x=L) must also be designed and manufactured to be absolutely maximum. Consequently, microbeams with fixed ends are easily implemented in silicon micromachining and have wide applications.

[0005] In vibration dynamics, there is another type of beam where both ends are free, also called a free-free beam (see references [2, 3]), where the bending moment and shear force in the beam are both zero at x = 0 and x = L.

[0006]

[0007] Beams with free ends have unique advantages. For example, given the same structural dimensions (length, width, and thickness), their natural frequencies are lower than those of cantilever beams and beams with fixed ends, making them more suitable for low-frequency vibration devices. However, because free-to-free boundary conditions are difficult to achieve in design and manufacturing, there is currently no effective technical solution to realize the free-support boundary conditions at both ends of the beam. Therefore, the engineering application of free-to-free microbeams is hindered, and their performance advantages cannot be effectively utilized.

[0008] literature:

[0009] [1]Chang Liu, Foundations of MEMS, Pearson Education Asia Ltd, 2008.

[0010] [2]WTThomson, MD Dahleh, Theory of vibration with applications (FifthEdtion), Prentice Hall, 2005.

[0011] [3] Zhang Yimin, Mechanical Vibration, Tsinghua University Press, 2007. Summary of the Invention

[0012] To address the shortcomings of existing technologies, this invention provides an elastic microbeam resonator with free boundary conditions at both ends.

[0013] The technical solution adopted in this invention is as follows:

[0014] An elastic microbeam resonator with free boundary conditions at both ends includes a substrate, a first fixed support, a second fixed support, a support beam, a free-to-free beam, and a fixed electrode. The first fixed support and the second fixed support are disposed at both ends of the substrate and connected to the substrate. The first fixed support is connected to the free-to-free beam via the support beam. The fixed electrode is disposed on the second fixed support and is used to drive the free-to-free beam and the support beam to undergo elastic bending deformation. The number of support beams and the positions of the support points of the support beams on the free-to-free beam are determined according to the following rules:

[0015] When a free-free beam is required to operate at its Nth natural frequency, N+1 structurally identical support beams are vertically supported on the same side of the free-free beam, forming N+1 vertical support points. These N+1 vertical support points are the N+1 nodes of the Nth mode shape function of the free-free beam.

[0016] The cross-sections of both the supporting beam and the free-free beam are rectangular, and the side lengths of the rectangles are respectively equal.

[0017] The further technical solution is as follows:

[0018] The relationship between the length l of the supporting beam and the length L of the free-to-free beam is as follows: λ N Let N be the Nth characteristic constant of a free-free beam, where N ≥ 1.

[0019] The first natural frequency w of the supporting beam 支撑-1With the Nth natural frequency w of the free-free beam 自自-N The relationship is:

[0020]

[0021] When N=1, the relationship between the length l of the supporting beam and the length L of the free-free beam is l=1.25L.

[0022] The supporting beam is made of the same material as the free-free beam, which is either monocrystalline silicon or polycrystalline silicon.

[0023] The beneficial effects of this invention are as follows:

[0024] This invention utilizes the characteristic that the displacement at the nodes of a free-free beam vibration mode is zero, and sets up a supporting beam to approximately realize the free-free boundary conditions.

[0025] The present invention can accurately set the number and position of support beams according to the inherent frequency range, thereby improving the adaptability and effectiveness of free-free beams at different operating frequencies. Attached Figure Description

[0026] Figure 1 This is a schematic diagram of the structural principle of cantilever microbeam devices in the prior art.

[0027] Figure 2 This is a schematic diagram of the structural dimensions of an elastic microbeam resonator operating at the first natural frequency, according to an embodiment of the present invention.

[0028] Figure 3 The curve of the first mode shape of the free-free beam in an embodiment of the present invention is shown.

[0029] Figure 4 This is a schematic diagram of the geometric dimensions of a free-to-free beam with a rectangular cross-section according to an embodiment of the present invention.

[0030] Figure 5 This is a schematic diagram of the structural dimensions of an elastic microbeam resonator operating at the second natural frequency, according to an embodiment of the present invention.

[0031] Figure 6 The curve of the second-order mode shape of the free-free beam in an embodiment of the present invention is shown.

[0032] Figure 7 The bending mode shape of the free-free beam in this embodiment of the invention is calculated using commercial finite element software.

[0033] In the figure: 1. Substrate; 2. First fixed support; 3. Second fixed support; 4. Support beam; 5. Free-to-free beam; 6. Fixed electrode. Detailed Implementation

[0034] The specific embodiments of the present invention are described below with reference to the accompanying drawings.

[0035] This embodiment describes an elastic microbeam resonator with free boundary conditions at both ends. See [link to relevant documentation]. Figure 2 , Figure 2 (a) and (b) are the front view and side view, respectively. The structure includes a substrate 1, a first fixed support, a second fixed support 3, a support beam 4, a free-to-free beam 5, and a fixed electrode 6. The first fixed support 2 and the second fixed support 3 are located at both ends of the substrate 1 and connected to it. One end of the support beam 4 is fixed to the first fixed support 2, and the other end supports and connects to the free-to-free beam 5. The fixed electrode 6 is located on the second fixed support 3 and is used to drive the free-to-free beam 5 and the support beam 4 to undergo elastic bending deformation. The number of support beams 4 and the positions of the support points of the support beams 4 on the free-to-free beam 5 are determined according to the following rules:

[0036] When the free-free beam 5 is required to operate at its Nth natural frequency, N+1 structurally identical support beams 4 are vertically supported on the same side of the free-free beam 5, forming N+1 vertical support points. The N+1 vertical support points are the N+1 nodes of the Nth mode shape function of the free-free beam 5.

[0037] Both the supporting beam 4 and the free-free beam 5 have rectangular cross sections, and the side lengths of the rectangles are equal.

[0038] Furthermore, the relationship between the length l of the supporting beam 4 and the length L of the free-free beam 5 is as follows: λ N For a free-free beam, the Nth-order characteristic constant is given, where N ≥ 1.

[0039] Furthermore, the first natural frequency w of the supporting beam 4 支撑-1 The Nth natural frequency w of the free-free beam 5 自自-N The relationship is:

[0040]

[0041] Specifically, the supporting beam 4 and the free-free beam 5 are made of the same material, namely monocrystalline silicon or polycrystalline silicon.

[0042] like Figure 2 The diagram shows the structural dimensions of the elastic microbeam resonator when N=1. Specifically, when the free-beam 5 is required to operate at its first natural frequency, two identical support beams 4 are vertically supported on the same side of the free-beam 5, forming two vertical support points. These two vertical support points are the two nodes of the first mode shape function of the free-beam 5.

[0043] The two nodes of the first-order mode function are determined by the following method:

[0044] The vibration deformation of a free-beam is given by y(x,t)=φ1(x)q1(t), where q1(t) is the first-order modal coordinate, and φ1(x) is the first-order mode function (modal function) of the free-beam corresponding to the first-order natural frequency, expressed as:

[0045]

[0046] In the above formula, λ1=4.73 and γ1=0.9825 are both first-order mode constants.

[0047] like Figure 3 The figure shows the curve of the first-order mode shape function φ1(x). Clearly, regardless of the beam length L, the mode shape curve φ(x) intersects the zero line at two points, located at 0.224L and 0.776L respectively. At these two intersection points, the mode shape curve value is 0, meaning the beam deformation is zero. These two intersection (zero) points are the nodes of the mode shape function.

[0048] Two rectangular cross-section beams with identical geometric dimensions are used as support beams 4, which are perpendicularly supported on the same side of the free-free beam. Specifically, the long side of the support beam is perpendicular to the long side of the free-free beam, and the perpendicular support points are located at two mode shape nodes on the long side of the free-free beam 5. The axes of the two support beams 4 intersect the axis of the free-free beam 5 perpendicularly. Since theoretically, the nodes of the mode shape function have no vibration displacement, the support beams 4 will not vibrate with the free-free beam.

[0049] When N=1, the relationship between the length l of the supporting beam and the length L of the free-free beam is determined as follows:

[0050] like Figure 4 The figure shows a schematic diagram of the geometric dimensions of a free-to-free beam with a rectangular cross-section. Figure 4 (a) and (b) are the side view and cross-sectional view, respectively. The length, width, and thickness of the beam are L, B, and H, respectively, and the x, z, and y axes are the length, width, and thickness directions, respectively. The beam undergoes elastic deformation along the thickness direction (y-direction). Most beams vibrate at their first natural frequency. The expression for the first natural frequency of a free-free beam is:

[0051]

[0052] In the above formula, E and ρ are the elastic model and density of the beam material, respectively, and λ1 = 4.73 is the first-order characteristic constant of the free-to-free beam. A is the moment of inertia of the cross section of a free-to-free beam. 自自 =BH is the cross-sectional area of ​​a free-to-free beam;

[0053] The two support beams 4 have the same geometric dimensions. The following explains how to determine the length l, width b, and thickness h of the support beams 4.

[0054] The boundary condition for supporting beam 4 is that of a cantilever beam, and its first natural frequency is ω. 支撑-1 Approximately:

[0055]

[0056] To reduce manufacturing costs and simplify the process, the support beam is made of the same material as the free-free beam—monocrystalline or polycrystalline silicon—and is manufactured using conventional silicon etching processes. Both have the same elastic modulus E and density ρ. The width and thickness of the support beam are also equal to those of the free-free beam, i.e., h = H, b = B. Therefore, the moment of inertia and cross-sectional area of ​​the torsional support beam are as follows:

[0057] A 支撑 =BH=A 自自 .

[0058] The first natural frequency w of the free-beam 5 自自-1 as follows:

[0059]

[0060] In mechanics, the vibrational energy of the free-free beam 5 will only not be transmitted to the supporting beam when the thickness h of the supporting beam 4 is zero. Due to strength and stiffness considerations, the thickness h of the supporting beam cannot be zero, so some vibrational energy from the free-free beam will be transmitted to the supporting beam. In engineering, the beam length L is generally much greater than the thickness h, so the energy transmitted to the supporting beam is very small. Furthermore, the free-free beam vibrates at its first natural frequency, which is also the frequency of the AC driving voltage. According to vibration theory, if the first natural frequency of the supporting beam itself is much smaller than the first natural frequency of the free-free beam, the vibration of the supporting beam will be extremely small.

[0061] Therefore, the first natural frequency of the supporting beam proposed in this application is less than or equal to one-tenth of the first natural frequency of the free-free beam.

[0062] use The relationship between the length l of the support beam and the length L of the free-free beam is obtained: l ≥ 1.2534L. Considering that the longer the support beam l is, the larger the overall size and mass of the device will be, this application preferably uses l = 1.25L.

[0063] Similarly, the same principle applies when a beam is required to operate at higher (Nth) natural frequencies, especially the second and third natural frequencies.

[0064] like Figure 5The diagram shows the structural dimensions of the elastic microbeam resonator when N=2. Specifically, when the free-beam 5 is required to operate at its second natural frequency, three identical support beams 4 are vertically supported on the same side of the free-beam 5, forming three vertical support points. These three vertical support points are the two nodes of the second-order mode shape function of the free-beam 5. Figure 5 (a) and (b) are the front view and side view, respectively.

[0065] The two nodes of the second-order mode function are determined by the following method:

[0066] φ2(x) is the second-order mode function (mode function) of the free-beam corresponding to the second-order natural frequency, and its expression is:

[0067]

[0068] In the above formula, λ2=7.853 and γ2=1.0008 are both the second-order mode constants of a free-free beam.

[0069] like Figure 6 The figure shows the curve of the second-order mode shape function φ2(x). Clearly, regardless of the beam length L, the mode shape curve φ2(x) intersects the zero line at three points, located at 0.132L, 0.5L, and 0.868L. At these three intersection points, the mode shape curve value is 0, meaning the beam deformation is zero. These three intersection (zero) points are the nodes of the mode shape function.

[0070] When N=2, the relationship between the length l of the supporting beam and the length L of the free-free beam is determined using the same method as above. The second natural frequency of the free-free beam is:

[0071]

[0072] The first natural frequency w of the supporting beam 支撑-1 It must also be much smaller than the second natural frequency w of a free-beam. 自自-2 ,use The relationship between the length l of the supporting beam and the length L of the free-free beam is: l ≥ 0.76L.

[0073] To avoid excessive size, this application preferably uses a support beam with l = 0.76L.

[0074] In summary, operating at the Nth natural frequency w 自自-N Free-free beam:

[0075]

[0076] The first natural frequency w of the supporting beam 支撑-1 Much smaller than the Nth natural frequency w of a free-beam自自-N ,use The relationship between the length l of the supporting beam and the length L of the free-to-free beam is as follows:

[0077] λ N These are the Nth order characteristic constants of the free beam, λ1 = 4.730, λ2 = 7.853, and λ3 = 10.996;

[0078] When N≥4, λ N = (N+1 / 2)π.

[0079] This application is preferred. The supporting beam.

[0080] The following simulation calculations are used to verify the technical effect of the proposed solution.

[0081] For a free-free beam operating at its first natural frequency, the length, width, and thickness are L = 700 μm, B = 40 μm, and H = 20 μm, respectively. The supporting beam has a length l = 1.25L = 700 * 1.25 = 875 μm, a width b = B = 40 μm, and a thickness h = H = 20 μm. (The rest of the text appears to be unrelated and likely refers to a different beam.) Figure 7 The figure shows the bending mode shapes of a free-to-free beam calculated using commercial finite element software. Clearly, Figure 7 The bending mode shape and Figure 3 The theoretical shape shown is very close.

[0082] The technical solution of this application can realize the boundary condition of free support at both ends of the beam, which helps the engineering application of free-free microbeams to effectively give full play to their performance advantages.

[0083] It will be understood by those skilled in the art that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. An elastic microbeam resonator with free boundary conditions at both ends, comprising a substrate (1), a first fixed support (2), a second fixed support (3), a support beam (4), a free-to-free beam (5), and a fixed electrode (6), wherein the first fixed support (2) and the second fixed support (3) are disposed at both ends of the substrate (1) and connected to the substrate (1), the first fixed support (2) is connected to the free-to-free beam (5) via the support beam (4), and the fixed electrode (6) is disposed on the second fixed support (3) for driving the free-to-free beam (5) and the support beam (4) to undergo elastic bending deformation, characterized in that, The number of support beams (4) and the location of the support points of the support beams (4) on the free-free beam (5) are determined according to the following rules: When the free-free beam (5) is required to operate at its Nth natural frequency, N+1 structurally identical support beams (4) are vertically supported on the same side of the free-free beam (5), forming N+1 vertical support points, which are the N+1 nodes of the Nth mode shape function of the free-free beam (5); The cross sections of both the supporting beam (4) and the free-free beam (5) are rectangular, and the side lengths of the rectangles are equal. The relationship between the length l of the supporting beam (4) and the length L of the free-free beam (5) is as follows: λ N Let N be the Nth characteristic constant of the free-free beam (5), where N ≥ 1; The first natural frequency w of the supporting beam (4) 支撑-1 With respect to the Nth natural frequency w of the free-free beam (5) 自自-N The relationship is:

2. The elastic microbeam resonator with free boundary conditions at both ends according to claim 1, characterized in that, When N=1, the relationship between the length l of the support beam (4) and the length L of the free-free beam (5) is: l=1.25L.

3. The elastic microbeam resonator with free boundary conditions at both ends according to claim 1, characterized in that... The supporting beam (4) and the free-free beam (5) are made of the same material, namely monocrystalline silicon or polycrystalline silicon.

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