A low-frequency vibration isolation device with stepped quasi-zero stiffness characteristics

Abstract

This solution belongs to the field of low-frequency vibration isolation technology and discloses a low-frequency vibration isolation device with stepped quasi-zero stiffness characteristics. It includes a vertically arrayed multi-layer vibration isolation unit, with each layer comprising multiple horizontally arrayed unit cells. Each unit cell includes a horizontally arranged first platform and second platform, and a first spring, a second spring, a third spring, and four sets of connecting rods disposed between the first and second platforms. Each layer of vibration isolation unit has different unit cell parameters, giving the low-frequency vibration isolation device stepped quasi-zero stiffness characteristics. The unit cell parameters include the length l1 and stiffness k1 of the first spring, the stiffness k2 of the second spring, and the length l3 and stiffness k3 of the third spring.

Description

Technical Field

[0001] This solution belongs to the field of low-frequency vibration isolation technology, specifically involving a low-frequency vibration isolation device with stepped quasi-zero stiffness characteristics. Background Technology

[0002] Isolating undesirable vibrations is of paramount importance in many engineering structures and precision equipment. In particular, isolating micro-vibrations in orbiting satellites, optical components, and space telescopes has always been a challenge. From a passive vibration isolation perspective, one of the most common methods is to utilize traditional linear passive vibration isolators. However, linear passive vibration isolators only work when the frequency is more than 2 times the natural frequency. 0.5 Vibrations several times stronger than normal are required for effective isolation. To achieve better vibration isolation performance, isolators can reduce their stiffness to lower the resonant frequency, but this also reduces the structure's load-bearing capacity, making it unable to support the load of the isolated object itself. Therefore, directly using a near-linear, low-stiffness system cannot achieve ultra-low frequency vibration isolation.

[0003] To overcome this drawback, researchers have attempted to design isolators with nonlinear characteristics to improve ultra-low frequency vibration isolation performance. For vibration isolation or suppression, ideal stiffness not only needs to withstand large static effective loads but also possesses extremely low dynamic stiffness, i.e., quasi-zero stiffness (QZS), characterized by high static and low dynamic stiffness. Visually, QZS is a quasi-plateau interval on the force-displacement curve, indicating that the force remains almost constant within this displacement range. Quasi-zero stiffness enables good low-frequency vibration isolation without compromising load-bearing capacity, thus resolving the contradiction between load and low-frequency vibration isolation.

[0004] Traditional nonlinear isolators typically achieve optimal isolation performance only with a single isolated mass. When the isolated target may be several different masses, they cannot simultaneously provide good low-frequency vibration isolation performance. Furthermore, their effective displacement range is narrow, and they can only function as a single vibration isolator, limiting their potential applications. For example, many electronic devices on aircraft are stacked to improve space utilization; however, the micro-vibrations generated by stacked electronic devices can affect the accuracy of adjacent devices. Therefore, for multi-layer vibration isolation systems, in many practical engineering projects, it's not just a single isolator; each layer may serve as a working platform, meaning each layer can potentially generate harmful vibrations. Therefore, developing a new vibration isolation device that can effectively reduce low-frequency vibrations and is suitable for various practical applications is extremely urgent. Summary of the Invention

[0005] This solution aims to overcome at least one deficiency in the existing technology and provide a low-frequency vibration isolation device with stepped quasi-zero stiffness characteristics to solve the problem of narrow effective load range and only being able to function as a single vibration isolator.

[0006] To solve the above-mentioned technical problems, the following technical solution is adopted:

[0007] A low-frequency vibration isolation device with stepped quasi-zero stiffness characteristics includes a vertically arrayed multi-layer vibration isolation unit, each layer comprising multiple horizontally arrayed unit cells. Each unit cell includes a horizontally arranged first platform and a second platform, and a first spring, a second spring, a third spring, and four sets of connecting rods disposed between the first and second platforms. The unit cell has a vertical central axis, and the four sets of connecting rods are distributed at equal angles around the central axis of the unit cell. Each set of connecting rods includes a first inclined rod and a second inclined rod, one end of which is hinged to each other to form a joint. The other end of the first inclined rod is hinged to the first platform, and the other end of the second inclined rod is hinged to the second platform. The first spring is horizontally arranged, with its two ends connected to the joints of the two opposing sets of connecting rods. The second spring is vertically arranged, with its two ends connected to the centers of the first and second platforms, respectively. The third spring is coaxially arranged outside the second spring, with one end connected to the second platform and the other end free. Each layer of vibration isolation unit has different unit cell parameters, giving the low-frequency vibration isolation device stepped quasi-zero stiffness characteristics; the unit cell parameters include the length of the first spring. l 1 and stiffness k 1. Stiffness of the second spring k 2 and the length of the third spring l 3 and stiffness k 3.

[0008] This solution proposes and implements a highly adaptable low-frequency vibration isolation device, composed of three mechanical elements: plates, rods, and linear springs. A broad design space for mechanical behavior is achieved through series and parallel mechanical unit cells. The isolation device, composed of layers with different unit cell parameters, exhibits stepped QZS characteristics. Good low-frequency vibration isolation can be achieved under multiple loads with varying isolation masses through rationally designed QZS loads. Multiple QZS loads can be designed by increasing the number of layers. Furthermore, the isolation device can isolate vibrations generated in any layer and prevent their transmission to other layers, thus realizing a multi-layered working platform with isolation effects. This isolates vibrations from the foundation and any layers generated during operation, which is crucial in many engineering practices. The on-site rapid reverse engineering strategy provides a quick solution to meet various demanding isolation requirements.

[0009] The length of the first spring in each vibration isolation unit l Under the same conditions, the stiffness of the first spring of the first to third vibration isolation units arranged sequentially from top to bottom preferably satisfies the following: k 11 <k 12 < k 13 and k 12 <2 k 11 < k 13 More preferably, satisfying: k 11 : k 12 : k 13 =186: 340: 410; The optimal solution satisfies: k 11 =0.186 N / mm, k 12 =0.340 N / mm, k 13 =0.410 N / mm.

[0010] The length of the second spring in each layer of vibration isolation unit l 2. Under the same conditions, the stiffness of the second spring of the first to third layers of vibration isolation units arranged sequentially from top to bottom preferably satisfies: 2 k 21 < k 22 < k 23 and k 22 <4 k 21 < k 23 More preferably, satisfying: k 21 : k 22 : k 23 =26: 67: 116; The optimal solution satisfies: k 21 =0.026 N / mm, k 22 =0.067 N / mm, k 23 =0.116 N / mm.

[0011] The stiffness of the third spring in each layer of vibration isolation unit k Under the same conditions, the length of the third spring in the first to third layers of vibration isolation units arranged sequentially from top to bottom preferably satisfies the following: l 31 > l 32 > l33 And 2 l 32 > l 31 >2 l 33 More preferably, satisfying: l 31 : l 32 : l 33 =26: 22: 12; The optimal solution satisfies: l 31 =26 mm, l 32 =22 mm, l 33 =12 mm.

[0012] Compared with existing technologies, this solution has the following advantages: by connecting and paralleling unit cells with specific structures and parameters, this solution can achieve low-frequency vibration isolation of multiple quasi-zero stiffness effective loads. Attached Figure Description

[0013] The accompanying drawings are for illustrative purposes only and should not be construed as limiting the scope of this solution. To better illustrate the solution, some components in the drawings may be omitted, enlarged, or reduced, and do not represent the actual dimensions of the product. It is understandable to those skilled in the art that some well-known structures and their descriptions may be omitted in the drawings.

[0014] Figure 1 This is a schematic diagram of a low-frequency vibration isolation device with stepped quasi-zero stiffness characteristics.

[0015] Figure 2 This is a schematic diagram of a single cell.

[0016] Figure 3 This is the design of a vibration isolation device with field-adjustable stepped quasi-zero stiffness characteristics. (a) shows the unit cell structure; (b) shows the force-displacement curve corresponding to the unit cell (the highlighted bar represents the quasi-plateau segment, i.e., the quasi-zero stiffness range); (c) shows the structure of the vibration isolation device; (d) shows the force-displacement curve corresponding to the vibration isolation device; (e) shows a schematic diagram of the isolation condition with multiple supporting masses; (f) shows a schematic diagram of an integrated working platform with low-frequency vibration isolation performance; and (g) shows the excitation acceleration response and output acceleration response curves acquired by the dynamic signal acquisition system (DH5922D) (when the isolated mass corresponds to the quasi-zero stiffness range of the vibration isolation device, the vibration will be almost completely isolated by the vibration isolation device).

[0017] Figure 4 It is the quasi-static loading mechanical behavior of the vibration isolation device 4[S1, S2, S3]T as measured experimentally and predicted numerically.

[0018] Figure 5 The vibration transmittance curves of the vibration isolation device 4[S1, S2, S3]T under five typical support masses were obtained experimentally, where three support masses correspond to the QZS effective load.

[0019] Figure 6 The vibration isolation device 4[S1, S2, S3]T is supported by a mass of M 1 = 490 g, M 2 = 483 g, M Vibrational transmittance curves at 3=483 g. Among them, the vibration source in (a) is... M 1, M 2 and M 3. At the bottom, the vibration source of (b) is... M 2.

[0020] Explanation of reference numerals in the attached drawings: vibration isolation unit 100, unit cell 110, first platform 111, second platform 112, first spring 113, second spring 114, third spring 115, first inclined rod 116, second inclined rod 117. Detailed Implementation

[0021] like Figure 1 As shown, this solution proposes a low-frequency vibration isolation device with stepped quasi-zero stiffness characteristics, comprising a vertically arrayed multi-layer vibration isolation unit 100, each layer of the vibration isolation unit 100 comprising a horizontally arrayed plurality of unit cells 110. Figure 2 As shown, each unit cell 110 includes a horizontally arranged first platform 111 and second platform 112, and a first spring 113, a second spring 114, a third spring 115, and four sets of connecting rods disposed between the first platform 111 and the second platform 112. The unit cell 110 has a vertical central axis, and the four sets of connecting rods are distributed at equal angles around the central axis of the unit cell 110. Each set of connecting rods includes a first inclined rod 116 and a second inclined rod 117, one end of which is hinged to each other to form a joint. The other end of the first inclined rod 116 is hinged to the first platform 111, and the other end of the second inclined rod 117 is hinged to the second platform 112. The first spring 113 is horizontally arranged, with its two ends connected to the joints of the two opposing sets of connecting rods. The second spring 114 is vertically arranged, with its two ends connected to the center of the first platform 111 and the center of the second platform 112, respectively. The third spring 115 is coaxially disposed outside the second spring 114, with one end connected to the second platform 112 and the other end free. Each layer of vibration isolation unit 100 has different unit cell 110 parameters, giving the low-frequency vibration isolation device a stepped quasi-zero stiffness characteristic; the unit cell 110 parameters include the length of the first spring 113. l 1 and stiffness k 1. Stiffness of the second spring 114 k 2 and the length of the third spring 115 l3 and stiffness k 3.

[0022] The aforementioned low-frequency vibration isolation device is composed of series and parallel connections of unit cells with specific structures and parameters, which can achieve low-frequency vibration isolation for multiple quasi-zero stiffness effective loads. The intermediate layer vibration isolation unit can use the second platform of the upper layer vibration isolation unit as the first platform, or the first platform of the lower layer vibration isolation unit as the second platform. In two adjacent layers of vibration isolation units, the second platform of the upper layer vibration isolation unit and the first platform of the lower layer vibration isolation unit are combined into one.

[0023] To enable those skilled in the art to better understand this solution, a more detailed description is provided below.

[0024] The core idea of ​​constructing vibration isolation devices with multiple extremely low stiffness characteristics, such as... Figure 3 As shown. Considering the important role of high static and low dynamic stiffness characteristics in low-frequency vibration isolation, this implies a force-displacement curve with a quasi-plateau within a certain loading range. Our primary task is to develop a novel vibration isolation device with deformation stability, whose load response curve fully conforms to the desired "larger range of quasi-plateau" characteristics. To achieve this, a novel polyhedral unit cell is constructed from two side lengths... a The plate 2 is connected by four pairs of links, such as Figure 3 As shown in (a). Each pair of links consists of two rods of length [missing information]. a The diagonal rod 1 is connected via a shaft hole and then attached to the plate in the same manner. Its length is... l A horizontal spring (spring 1) is connected at the middle joint of two pairs of connecting rods. Figure 3 (a) Joints 1 and 2). When the element is loaded to a certain extent, spring 1 will cause the element to exhibit negative stiffness characteristics ( Figure 3 (b) Orange curve). The vertical spring (spring 2) connects the center of the upper and lower plates. Length l Shorter, larger-diameter spring 3 is connected to the lower plate and nested outside spring 2. Springs 2 and 3 will exhibit positive stiffness characteristics under vertical load. Under a certain load, the positive stiffness provided by the vertical spring 2 ( Figure 3 (b) The dark green curve and the negative stiffness provided by the horizontal spring 1 cancel each other out, allowing QZS to operate over a large displacement range (see...). Figure 3 (b) Columnar highlighted area). Spring 3 can provide more positive stiffness in the vertical direction, thus preventing the unit cell from entering negative stiffness during further deformation. Therefore, the spring parameters are one of the key elements that the unit cell needs to be carefully designed. By changing the spring parameters, the effective load of the unit cell (i.e., the load range from the start to the end of QZS) and the displacement range of QZS can be adjusted.

[0025] Different units are assembled by series and parallel connections (see...) Figure 3(c) allows for a wider range of stiffness tuning behavior. The plates and rods of the vibration isolation device are manufactured using additive manufacturing with curable resin. Cells in the same layer are connected together using the same stainless steel rods as hinges to ensure the stability of the vibration isolation device. Rubber sleeves are installed at both ends of the stainless steel rods to maintain the tightness of the hinges. Since the deformation of each layer is discrete and has the ability to deform independently, when assembled in parallel, its effective load increases in multiples of the number of parallel cells. In the series assembly method, the force on all layers is equal to the external force, and the deformation of each layer is the total deformation. The parallel and series relationships in the structural topology and the unit cells with load curves of quasi-platform shape are crucial for realizing rich stiffness behavior concepts and can produce rich stiffness behaviors. The combination of parallel and series assembly methods allows for a wider range of stiffness tuning of the vibration isolation device. The QZS effective load can be adjusted by the number of cells in each layer and the bearing capacity of each unit cell, and the QZS displacement range can be adjusted by the number of layers and the QZS displacement range of each unit cell. For example, when the stiffness difference between interlayer elements is large, a stepped loading curve with multiple platforms can be generated. Figure 3 (d) In the middle), the isolation quality with large differences can correspond to the QZS regions of different layers ( Figure 3 (e) The vibration isolation device exhibits ultra-low dynamic stiffness, effectively isolating harmful low-frequency vibrations. Furthermore, the different isolation masses of different layers allow each layer to simultaneously exist within the QZS region (…). Figure 3 (f) This isolates vibrations generated in any layer and prevents them from being transmitted to other layers. The force-displacement curve of the quasi-platform provides an ideal stiffness characteristic for attenuating low-frequency vibrations without sacrificing loading capacity. When the mass of the isolated object is within the effective load range, the ultra-low dynamic stiffness of the vibration isolation device perfectly isolates harmful vibrations, such as... Figure 3 As shown in (g).

[0026] To fully demonstrate the rich QZS behavior of this vibration isolation device, this paper mainly uses three elements S1, S2 and S3 with different parameters to construct the vibration isolation device, as shown in Table 1. k 1, k 2and k 3 represents the stiffness of springs 1, 2, and 3. The vibration isolation device consists of units S1, S2, and S3 from top to bottom, with each layer containing 4 cells, such as... Figure 3 As shown in (c), the sample is named 4[S1, S2, S3]T. A uniaxial quasi-static compression test was conducted on the vibration isolation device sample 4[S1, S2, S3]T using a testing machine. The experiment was controlled using a displacement loading method, with a constant displacement rate of 5 mm / min. The reaction force and displacement response of the sample were captured, thus obtaining the force-displacement curve, as shown in (c). Figure 4 and 5 As shown.

[0027] Table 1. Geometric parameters of the unit cell constituting the vibration isolation device

[0028]

[0029] The loading curves corresponding to vibration isolation device 4[S1, S2, S3]T are as follows: Figure 4 As shown, the stiffness-displacement curve is derived from the force-displacement curve. A stiffness of v = 0.1 N / mm is quantitatively chosen as the threshold (i.e., stiffness 0~0.1 N / mm) to define the QZS region. The gradient isolation device gradually achieves three quasi-plateaus on the force-displacement curve, almost corresponding to the QZS ranges of elements S1, S2, and S3. Therefore, the effective load and QZS displacement range of the gradient isolation device are determined by the mechanical properties of the elements. Based on the effective stiffness of the elements in each layer, it is entirely conceivable that the gradient isolation device reaches the QZS plateau sequentially from small to large. In general, the force-displacement curve exhibits a clear stepped QZS characteristic.

[0030] To verify the vibration isolation performance of the vibration isolation device under different masses, the vibration transmission rate of the vibration isolation device 4[S1, S2, S3]T was also measured under five typical support masses. The support masses are as follows: M 1 = 223 g, 490 g, 970 g, 1450 g, and 1928 g, corresponding to Figure 4 Points A, B, C, D, and E in the diagram. (The rest of the text appears to be a list of points and Figure 5 It can be seen that the load M 1 = 490 g (B), 970 g (C), and 1450 g (D) correspond to the effective load. M 1 = 223 g (A) and 1928 g (E) exhibit large positive stiffness. Low-frequency vibration isolation is currently a challenge in the engineering field; therefore, the focus is on the vibration isolation performance between 0 Hz and 20 Hz. For the load... M With loads of 1 = 490 g (B), 970 g (C), and 1450 g (D), the superstructure is in a QZS state. The vibration isolation device begins to suppress vibration after the excitation frequency is higher than 2.2 Hz (B), 1.9 Hz (C), and 3.8 Hz (D), respectively. This indicates that when the support mass corresponds to the QZS effective load, the vibration isolation range of the proposed superstructure almost covers the entire low-frequency band. In contrast, under other loads ( M =1 = 223 g (A) and 1928 g (E)), the critical frequencies for effective isolation increase to 7.4 Hz (A) and 8.3 Hz (E), both higher than the isolation frequencies under QZS conditions. Furthermore, within the studied low-frequency range (0-20 Hz), the load... MThe presence of two transmittance resonance peaks and a high transmittance resonance peak (approximately 14 dB) at 1=223 g (A) and 1928 g (E) is detrimental to low-frequency vibration isolation. The proposed vibration isolation device exhibits superior low-frequency vibration isolation performance within the QZS range compared to other regions with higher positive stiffness. Overall, the proposed vibration isolation device achieves excellent low-frequency vibration isolation performance for isolated objects of varying masses.

[0031] Figure 6 This demonstrates 4[S4, S5, S6]T under load. M 1 = 490 g, M 2 = 483 g and M Under a g=483 g, the vibration transmissibility of the vibration source at different layers. When M 1 = 490 g, M 2 = 483 g and M When 3 = 483 g, all three units are simultaneously within the QZS range. For example... Figure 6 As shown in (a), when the vibration source is at the bottom, M 1. M 2 and M The effective isolation critical frequencies for 3 are 2.9 Hz, 3.2 Hz, and 3.0 Hz, respectively, indicating that under these conditions... M 1. M 2 and M 3. It consistently maintains excellent ultra-low frequency vibration isolation performance. Furthermore, M 1. M 2 and M 3. The transmittance at an excitation frequency of 10Hz was -6.4 dB, -8.9 dB, and -23.6 dB, gradually decreasing, indicating that a multi-layer design can significantly improve vibration isolation when each layer is within the QZS range. For example... Figure 6 As shown in (b), when the bottom is fixed, the vibration source is... M At 2 o'clock, M 1 and M The initial isolation frequencies of the three layers are 2.4 Hz and 0.1 Hz, respectively, and the transmittances at 10 Hz are -13.4 dB and -9.5 dB, respectively. Both the upper and lower layers of the vibration source exhibit good low-frequency vibration isolation performance, especially the lower layer, which provides almost full-frequency isolation. In summary, this vibration isolation device can ensure that each layer of different masses can significantly isolate excitation vibrations from different layers without affecting the stability of other layers, which is highly beneficial for many engineering practices.

[0032] In summary, we proposed and experimentally implemented a highly adaptable vibration isolation device based on rod and spring connections for low-frequency vibration isolation. This device allows for a large customization space for force-displacement curves, such as a stepped QZS characteristic, meaning it has multiple QZS operating ranges, thus achieving excellent low-frequency vibration isolation performance for different isolation masses. Furthermore, each layer can support different isolation masses, thereby suppressing vibration propagation within the isolation device without compromising the stability of other layers. Moreover, the required vibration isolation device can be rapidly reverse-engineered on-site according to the isolation object and excitation amplitude requirements. Notably, the unique mechanical behavior customization strategy of the proposed vibration isolation device enables customized mechanical behavior, opening new avenues for various engineering applications such as automotive suspensions, vibration isolation brackets, and operating tables.

[0033] Obviously, the above embodiments of this solution are merely examples for clearly illustrating this solution, and are not intended to limit the implementation of this solution. Those skilled in the art can make other variations or modifications based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations here. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of this solution should be included within the scope of protection of the claims of this solution.

Claims

1. A low-frequency vibration isolation device with stepped quasi-zero stiffness characteristics, characterized in that, The low-frequency vibration isolation device includes a vertically arrayed multi-layer vibration isolation unit, each layer comprising multiple horizontally arrayed unit cells. Each unit cell includes a horizontally arranged first platform and second platform, and a first spring, a second spring, a third spring, and four sets of connecting rods disposed between the first and second platforms. The unit cell has a vertical central axis, and the four sets of connecting rods are distributed at equal angles around the central axis of the unit cell. Each set of connecting rods includes a first inclined rod and a second inclined rod, one end of which is hinged to each other to form a joint. The other end of the first inclined rod is hinged to the first platform, and the other end of the second inclined rod is hinged to the second platform. The first spring is horizontally arranged, with its two ends connected to the joints of the two opposing sets of connecting rods. The second spring is vertically arranged, with its two ends connected to the center of the first platform and the center of the second platform, respectively. The third spring is coaxially arranged outside the second spring, with one end connected to the second platform and the other end free. Vibration isolation units in the same layer have the same unit cell parameters, while vibration isolation units in different layers have different unit cell parameters, giving the low-frequency vibration isolation device a stepped quasi-zero stiffness characteristic; the unit cell parameters include the length of the first spring. l 1n and stiffness k 1n Stiffness of the second spring k 2n and the length of the third spring l 3n and stiffness k 3n ,in n This represents the number of layers of vibration isolation units from top to bottom; The length of the first spring in each vibration isolation unit l 1n Similarly, the stiffness of the first spring in the first to third layers of vibration isolation units, arranged sequentially from top to bottom, satisfies: k 11 < k 12 < k 13 and k 12 <2 k 11 < k 13 The stiffness of the first spring in the first to third layers of vibration isolation units, arranged sequentially from top to bottom, satisfies: k 11 : k 12 : k 13 =186: 340: 410; The length of the second spring in each vibration isolation unit l 2n Similarly, the stiffness of the second spring in the first to third layers of vibration isolation units, arranged sequentially from top to bottom, satisfies: 2 k 21 < k 22 < k 23 and k 22 <4 k 21 < k 23 The stiffness of the second spring in the first to third layers of vibration isolation units, arranged sequentially from top to bottom, satisfies the following: k 21 : k 22 : k 23 =26: 67: 116; Stiffness of the third spring in each vibration isolation unit k 3n Similarly, the lengths of the third springs in the first to third layers of vibration isolation units, arranged sequentially from top to bottom, satisfy the following: l 31 > l 32 > l 33 And 2 l 32 > l 31 >2 l 33 The lengths of the second springs in the first to third layers of vibration isolation units, arranged sequentially from top to bottom, satisfy the following: l 31 : l 32 : l 33 =26: 22:

12.

2. The low-frequency vibration isolation device with stepped quasi-zero stiffness characteristics according to claim 1, characterized in that, The stiffness of the first spring in the first to third layers of vibration isolation units, arranged sequentially from top to bottom, satisfies: k 11 =0.186 N / mm, k 12 =0.340 N / mm, k 13 =0.410 N / mm.

3. The low-frequency vibration isolation device with stepped quasi-zero stiffness characteristics according to claim 1 or 2, characterized in that, The stiffness of the second spring in the first to third layers of vibration isolation units, arranged sequentially from top to bottom, satisfies: k 21 =0.026 N / mm, k 22 =0.067 N / mm, k 23 =0.116 N / mm.

4. The low-frequency vibration isolation device with stepped quasi-zero stiffness characteristics according to claim 1 or 2, characterized in that, The lengths of the second springs in the first to third layers of vibration isolation units, arranged sequentially from top to bottom, satisfy the following: l 31 =26 mm, l 32 =22 mm, l 33 =12 mm.

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