1 Introduction

A compliant joint is a basic element of compliant mechanism which is utilized to transfer force, torque, motion, or energy [1, 2]. Compliant mechanism has emerging properties, e.g., free friction, no lubricant, reduced assembly, and low manufacturing cost. It can be monolithically fabricated by emerging techniques such as wire electrical discharge machining or 3D, 4D printing, or CNC machining. Nowadays, compliant joins have been fast grown for use in ultrahigh-precision instruments [3], tissue cutting [4], and microelectronic devices [5,6,7], and vibration suspension [8]. It is noted that the shape and number of degrees of freedom (DOF) of a compliant joint are two common characteristics that have been received great interest. The reasons are that if the shape or number of DOFs is well designed, performances of a joint can be improved accordingly.

In order to meet various positioning tasks, a lot of different shapes for compliant joint, called as flexure hinge, were proposed, e.g., right circular flexure hinge [9], elliptical-arc flexure hinges [10], damped leaf hinge [11], and parabolic flexure hinge [12]. For biomechanical rehabilitation and assistive technology, rotary compliant joints were created thanks to their simple structure, lightweight, and low manufacturing cost [13, 14]. Moreover, a good compliant behavior can be made by changing the thickness of a compliant joint but this may cause a positioning error and undesired oscillation [15]. In overcome this circumstance, a circular hinge was integrated with a damper so as to harvest outside vibrations [16]. On the other hand, the shape of a compliant joint can be adjusted according to designer’s experience or customer. However, the number of DOFs of a compliant joint has been facing challenges for researchers. A new compliant joint with multiple DOFs is motivated in the present paper.

Regarding the number of DOFs of a join, existing compliant joints have been created for 1-DOF positioning mechanisms and suppressed other DOFs. There have been some efforts on designing a hybrid flexure with bidirectional motion, e.g., a power-function-shaped flexure hinge [17], hybrid bidirectional flexure joint [18], right circular hinge, elliptical-arc, parabolic, and hyperbolic shapes [19]. In order to improve the flexibility of joint, a straight-axis flexure hinge was designed [20] and a flexure hinge with multi-cavity was suggested [21]. Besides, many different configurations of a joint were developed, such as a triple-cross-spring flexure joint [3], a serial flexure [22], a flexure hinge with a variable cross section [23], and flexure arrays [24]. In addition, researchers suggested emerging joints, e.g., cross-axis-flexural pivot [25], large-displacement compliant [26], flexure hinge with topology optimization [27], deep-notch elliptical flexure hinge [28], leaf flexure hinge [29], tape-spring hinge [30], hybrid non-symmetric flexure hinge [31], flexure hinge with large displacement [32], cracked right circular flexure hinge [9], flexure magnifying mechanism [33], and a flexure hinge for sensing device [34]. Recently, a compliant joint with a thin wall was designed for camera device [35, 36] and flexural joint with Q shape was proposed so as to enhance the deflection of joint [37].

Although previous studies have been well developed many different types of compliant joints, they almost focused on designing shapes. Up to now, a compliant joint with multiple DOFs is a lack of interest. In order to fulfill this requirement, the present study proposes a new compliant joint by a combination of a foldable mechanism and a butterfly’s profile. A foldable mechanism is adopted to adjust the stiffness of the joint. The stiffness is proportional to the deflection of the compliant joint; therefore, if the stiffness is varied, the deflection is changed accordingly [38]. Meanwhile, observed from the nature, the butterfly’s profile is very flexible based on its wings. So, the proposed joint can mimic the butterfly’s profile to make a good compliance. In this study, a new joint is named as a new butterfly-inspired compliant joint (BICJ) which is a combination of mechanical structure and nature animal. The main contributions of this article are as follows: (i) The proposed BICJ permits three DOFs in its fabricating plane, including 1-DOF translation in the x-axis, 1-DOF translation in the y-axis, and 1-DOF rotary motion around the z-axis. (ii) The proposed BICJ reaches multiple good performances in terms of bending displacement, compressive displacement, and angular displacement.

The goals of this article are to design and analyze a new compliant joint. The proposed BICJ joint is inspired by a butterfly’s profile and a foldable mechanism. It is capable of allowing multiple flexibilities in three directions. Performances of the proposed BICJ are compared with other joints by using numerical analysis. A closed-form model of the joint is derived so as to calculate the stiffness, displacement, and rotation angle. Simulation and experiments are performed to verify the analytical model.

2 Conceptual Design

As mentioned above, the aim of this article is to create a new compliant joint, BICJ, in terms of 1-DOF motion in the x-axis, 1-DOF motion in the y-axis, and 1-DOF rotary motion around the z-axis. In addition, the proposed BICJ has a large range of load being suitable for different applications, and its stress concentration is smaller than the yield strength of the material to ensure a good enough working strength. Before creating the BICJ, it begins with a common compliant joint where a free rigid end, a flexure body, and a fixed end are composed together. This section gives a conceptual design to form a new joint, and the kinematic diagram of different configurations of a joint is analyzed. Various configurations of joints are classified, as shown in Table 1.

Table 1 Single configurations of a compliant joint
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Table 1 gives eight single configurations of a compliant joint. A fixed rigid link is labeled with gray color, a movable rigid link is labeled with orange color, and a flexure body is the thinnest link. Configuration #1(−) includes two leaf hinges, a movable part and a fixed part. Signal #1(−) shows a load at the bottom part. Configuration #1(+) notes that a load is at the top part. Both these two configurations can subject a very small load. They can make a small rotation angle, small bending displacement, and very small stiffness. Next, both configurations #2(−) and #2(+) are assigned by adopting four flexures, three fixed parts, and a movable element. Both these two configurations give a small capacity of load, small rotational angle, small bending displacement, and small stiffness. Both configurations can only achieve 1-DOF motion but limit multiple DOFs.

Next, configurations #3(−) and #3(+) are designed by using six flexures, three fixed parts, and a movable element. Both these configurations permit a middle load, middle rotational angle, middle displacement, and middle stiffness. Subsequently, configurations #4(−) and #4(+) are assigned via adopting six flexures, three fixed parts, and a movable element. Both these configurations make a relatively large load, large rotational angle, large bending displacement, and large stiffness.

In order to develop a new BICJ with three DOFs, a concept of a compound joint is introduced herein. According to previous studies [15, 38,39,40], if the length of the flexure hinge is enlarged, the displacement is increased. From this point of view, a combination of single configurations in Table 1 is suggested to create a compound configuration. Table 2 gives seven different compound configurations. By combining configurations #1(−) and #1(+), a first compound configuration #11 is made with four-leaf hinges, a movable part, and a fixed part. Configuration #11 has some characteristics such as very small load, very large rotational angle, very large bending displacement, and very small stiffness. Next, configuration #2(−) is coupled with configuration #2(+) to make a configuration #22 consisting of 8 leaf hinges, a movable part, and 5 fixed part. Characteristics of configuration #22 include a small load, small rotational angle, small bending displacement, and low stiffness. Configuration #33 is a combination of #3(−) and #3(+) that is composed of 12 leaf hinges, a movable part, and 5 fixed parts. This configuration allows a small load, small rotational angle, small bending displacement, and small stiffness. Configuration #44 is constructed by combining the configuration #4(−) and #4(+). It consists of 16 leaf hinges, a movable part, and 5 fixed parts. Its characteristics include a small load, small rotational angle, small bending displacement, and small stiffness. Besides, configuration #X is a common configuration consisting of 4 leaf hinges, a movable part, and a fixed part. This configuration allows a very small load, very large rotational angle, very large bending displacement, and very small stiffness.

Table 2 Compound configurations of a compliant joint
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In comparison with existing configurations, this paper proposes two different configurations of the BICJ joint, as shown in Fig. 1a and b. The first configuration is labeled as #half BICJ that has 8 leaf hinges, a movable part, and a fixed part. Its characteristics include a middle load, middle rotational angle, middle bending displacement, and middle stiffness. Another configuration is labeled as #full BICJ, called as a proposed BICJ, with four wings that inspired from a butterfly’s profile. This configuration includes 16 leaf hinges, a movable part, and a fixed part. It permits a very large load, large rotational angle, large bending displacement, and very small stiffness. It is assumed that the performances of proposed BICJ are better than other configurations.

Fig. 1
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Model of the joint: a configuration # BICJ, b butterfly’s profile

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The dimensional diagram of the proposed BICJ is provided in Fig. 2 and Table 3. In this study, a foldable mechanism is coupled with the butterfly’s profile to create the proposed BICJ. As depicted in Fig. 2, the flexibility of the proposed BICJ is strongly dependent on four springs with stiffness k1, k2, k3, and k4. It is noted that a compliant joint is easy to flexible and sensitive to oscillation. Thus, the second and third beams of BICJ are utilized to guarantee a good enough stiffness during the joint’s working operation. Emerging performances of BICJ include axial motion, and bending motion, and rotary motion. The configuration #full BICJ is chosen for analyzing in this study because it can achieve 3-DOFs in three axes. The proposed BICJ is a potential candidate for multiple functions in positioning engineering. Performances and effectiveness of the proposed BICJ are proved in a later section by comparing with other configurations.

Fig. 2
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Dimensions of proposed BICJ joint

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Table 3 Dimension of the proposed BICJ joint
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3 Comparison of Compound Configurations

In this section, the performances and effectiveness of the proposed BICJ are validated through a comparison with both configurations #33 and #X. Three configurations, #33 and #X, and # BICJ have the same dimensions. A finite element analysis (FEA) in ANSYS 18.1 software is implemented for this comparison. 3D models of three configurations are constructed in Solidworks 2018 and then imported into ANSYS. Al T73-7075 material is used because of its high yield strength and light density. The mechanical properties are given in Table 4.

Table 4 Mechanical properties of Al T73-7075
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The boundary conditions and loads for three configurations (#33, #X, and #BICJ) are shown in Fig. 3. The tetrahedron method is utilized for meshing. Ten-node tetrahedral element, SOLID92 element configuration, is adopted in this analysis. The meshes of flexure hinges are refined to achieve accurate results. The meshing models of three configurations are given, as in Fig. 4. A meshing refinement for proposed BICJ is presented, as in Fig. 5. Configuration #33 includes a number of nodes of 61,569 and a number of elements of 27,851. For #X configuration, the number of nodes is 15,736 and the number of elements is 7165. For the proposed BICJ, the number of nodes is 47,176 and the number of elements is 21,040. Meshing quality for the proposed BICJ is evaluated by using skewness criteria and its average value is about 0.35, as shown in Fig. 6. This value guarantees an accurate analysis. Moreover, a meshing convergence is investigated on the proposed BICJ to reach a more accurate solution. To study the convergence of the mesh model, a force P1 of 4 N is exerted on the BICJ, and then a convergent curve of the stress versus the number of elements is plotted, as given in Fig. 7. It found that the meshing model is converged at the elements of 21,040.

Fig. 3
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Three configurations: a #33, b #X, and c #full BICJ

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Fig. 4
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Meshing model of three configurations: a #33, b #X, and c #full BICJ

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Fig. 5
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Meshing refinement of proposed BICJ joint

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Fig. 6
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Distribution of skewness criteria for the proposed BICJ joint

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Fig. 7
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Meshing convergence of the proposed BICJ joint

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3.1 Analysis of Bending Displacement

Bending displacement of the proposed BICJ in the x-axis is the first specification. By using FEA simulation, a load P1 with the value from 2 to 14 N is exerted to a free end, as given in Fig. 3. The results of Table 5 show that the displacement of configuration #X joint is highest, followed by configuration #33, and configuration # BICJ is lowest. Also, the von Mises stress equivalent stress of configuration #X is highest, followed by configuration #33, and configuration # BICJ is smallest.

Table 5 Bending displacement: #33 joint, #X joint, and #full BICJ
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Besides, the resulting stress of #33 joint is larger than that of yield strength of proposed material at the load P1 of 6 N. It notes that the #33 joint has a static failure if the load is over 6 N. Considering the #X joint, the strength limitation of this joint is less than 4 N. Meanwhile, the proposed BICJ can work in a wide range of load from 2 to 10 N without static failures. Therefore, the proposed BICJ is more effective than other configurations and it can be promising candidate for use in positioning applications.

3.2 Analysis of Compressive Displacement

For a vibration isolator or energy harvester, a compressive displacement of the proposed BICJ in the y-axis is the second specification which is used to suppress vibrations. To conduct this analysis, a load P2 from 2 to 14 N is used to act to a free end of three mentioned configurations (#33, #X, and #BICJ). Boundary conditions are assigned, as depicted in Fig. 3. Simulation results for three configurations are recorded in Table 6. The results find that the compressive displacement of proposed BICJ is largest, followed by configuration #33, and configuration #X. Besides, the von Mises equivalent stress of three configurations is still under the yield strength of proposed material. This permits a safety working condition. Regarding a large displacement, the proposed BICJ is expected to be an appropriate candidate for future applications.

Table 6 Compressive displacement: #33 joint, # X joint, and #full BICJ
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3.3 Analysis of Buckling Deformation

In addition to the compressive displacement, buckling phenomenon is considered for a vibration harvester. It determines the critical buckling load or called safe factor before buckling phenomenon is occurred. This phenomenon may be an error of an engineering structure but it benefits of flexible structure, a flexible spring, to harvest vibrations for a vibration harvester. Assume that configurations #33, #X, and proposed #BICJ are considered as cantilever column. The critical buckling load, Pc, is as follows [22]:

$$ P_{\text{c}} = \frac{{EI\pi^{2} }}{{4L^{2} }}, $$
(1)

where E is Young’s modulus, I is second moment of area, and L is length of a joint. A simplified equation for the critical buckling load is described by

$$ P_{\text{c}} = \lambda \times P_{I} , $$
(2)

where λ is eigenvalue or multiplier load.

Loads P1 and P2 from 2 to 14 N are acted to three configurations along the x- and y-axes, respectively. By using FEA simulation, the multiplier load, λ, is retrieved in Table 7. Under load P1, the results reveal that multiplier load for both modes 1 and 2 of configuration X is highest, followed the BICJ, and configuration #33. With load P2, the multiplier load of the BICJ is largest, followed by configuration #33, and configuration #X, as given in Table 8. This specification of BICJ can be promising specification for a vibration harvester.

Table 7 Buckling under load P1: #33 joint, #X joint, and #full BICJ
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Table 8 Buckling under load P2: #33 joint, #X joint, and #full BICJ
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3.4 Analysis of Rotational Angle

Rotational angle of the BICJ is the third specification where the proposed joint may be utilized for a rotary platform. By using load P1 from 2 to 14 N and boundary conditions in Fig. 3, the simulation results are recorded in Table 9. It determines that the rotation angle of proposed BICJ is from 3° to 18°. The proposed BICJ can withstand a maximum load of 10 N without static failure. Meanwhile, configuration #33 can rotate an angle from 9° to 18° but it only be used at maximum load of 4 N without static failure. Besides, configuration #X only rotates an angle of 37° and other cases are failed. Considering a capacity of load with a large bending angle, the proposed BICJ is preferred for positioning systems.

Table 9 Bending angles: #33 joint, #X joint, and #full BICJ
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4 Closed-Form Model of Proposed BICJ Joint

When the proposed BICJ is subjected to a load, elastic energy is stored inside the structure of joint. Therefore, force–displacement curve is critically important to discover the BICJ’s characteristics. The relation of force and displacement can be formulated by adopting the joint is simplified as a spring system, and all individual spring constants are combined into a single equivalent spring constant. In this article, the proposed BICJ consists of four significant springs with their stiffnesses labeled as k1, k2, k3, k4, as shown in Fig. 2. These springs are arranged in a parallel and series system, as illustrated in Fig. 8.

Fig. 8
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Equivalent spring system

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According to the Euler–Bernoulli beam theory, bending stiffness of beam is calculated by

$$ k = \frac{3EI}{{l^{3} }} $$
(3)

where k is stiffness or spring constant, E is Young’s modulus, I is beam’s moment of inertia, and l is the length of beam.

Beam’s moment of inertia is accounted as

$$ I = \frac{{bh^{3} }}{12} $$
(4)

where b and h are width and thickness of beam, respectively.

From the relation of force, stiffness, and deflection, the displacement of bending of beam is computed by

$$ \delta = \frac{P}{k} $$
(5)

where δ, P, and k are the displacement, load, and stiffness, respectively.

According to the relation of springs in Fig. 8, the spring system 1 includes k1, k2, k3, and k4 in parallel and series. So, the general form of equivalent stiffness of spring system 1 is found as

$$ \frac{1}{{k_{{{\text{eq}}1}} }} = \frac{1}{{k_{1} }} + \frac{1}{{k_{2} }} + \frac{1}{{k_{3} }} + \frac{1}{{k_{4} }} = \frac{4}{k} $$
(6)

where keq1 is the equivalent stiffness of spring system 1 and k1, k2, k3, and k4 are stiffness of flexure hinge 1, 2, 3, and 4 respectively.

Equation (6) is simplified by

$$ k_{{{\text{eq}}1}} = \frac{k}{4} $$
(7)

As shown in Fig. 8, the spring system 2 is similar to the spring system 1, and the equivalent stiffness of system 2 is equal to the one of system 1 as follows:

$$ k_{{{\text{eq}}2}} = k_{{{\text{eq}}1}} = \frac{k}{4} $$
(8)

where keq2 is the equivalent stiffness of spring system.

Because both spring systems 1 and 2 are parallel each other, the equivalent stiffness of entire system is calculated by

$$ k_{\text{eq}} = k_{{{\text{eq}}1}} + k_{{{\text{eq}}2}} = \frac{k}{2} $$
(9)

where keq is total stiffness of entire system.

In order to evaluate an improvement of performances of the proposed BICJ, the proposed BICJ’s characteristics are compared with the ones of the rectangular flexure joint.

An increase in bending displacement is determined as

$$ I_{\text{b}} = \left| {\frac{{\delta_{\text{x - BICJ}} - \delta_{\text{x - flexure}} }}{{\delta_{\text{x - flexure}} }}} \right| \times 100 $$
(10)

where Ib represents an increase in bending displacement. δx-BICJ and δx-flexure are the bending displacements in the x-axis of the proposed BICJ and rectangular flexure joint.

An increase in angular displacement is determined as

$$ I_{\text{a}} = \left| {\frac{{\theta_{\text{BICJ}} - \phi_{\text{flexure}} }}{{\phi_{\text{flexure}} }}} \right| \times 100 $$
(11)

where Ia represents an increase in angular displacement or rotation angle. θy-BICJ and θflexure are rotation angle of the proposed BICJ and rectangular flexure joint, respectively.

The relation of moment and bending angle is

$$ \theta = \frac{M}{{k_{\text{eq}} }} $$
(12)

where θ is rotation angle of the joint in degree, M is moment load in N mm.

5 Experimental Investigations

5.1 Performance Evaluations

The experimental investigations are established to realize the BICJ’s performances and validate the analytical models. Al T73-7075 material was used to fabricate the prototype of the proposed joint, as shown in Fig. 9a. The prototype was mounted on a base fixed to the vibration isolation system table to suppress any vibrations. In order to measure the bending displacement of the joint along the x-axis, a force gauge (Lutron, tension and compression, maximum 196 N, DC 9 V adapter, Model: NF-9500, Taiwan) was employed to give the force. A digital indicator (high precision 0.001 mm 1PC ID-C112, Mitutoyo, Japan) was utilized to realize the displacement of the BICJ, as given in Fig. 9b. The axial displacement along the y-axis was measured, as illustrated in Fig. 9c. The rotational angle of the joint was measured, as shown in Fig. 9d by using a force gauge (model AK-2, ALGOL Instrument Co., LTD, Taiwan) to give the load and digital indicator. The experimentations were repeated six times to get the average value.

Fig. 9
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Experimental photos: a prototype, b bending, c compress, d angle

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In order to compare the analytical model with simulation and experiments, the displacement of the proposed BICJ along the x-axis was computed. The load P1 from 2 to 14 N exerted to the free end of prototype, and the corresponding bending deflection was recorded. Besides, the FEA simulation in ANSYS 18.1 software was carried out so as to get the simulated displacement. As shown in Fig. 10, the results revealed that the deviation error between the analytical model and FEA simulation is about 1.88%. The error between the analytical model and experiments is approximately 4.16%. These errors are relatively small, and it validates the accuracy of analytical model.

Fig. 10
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Bending x-displacement versus load P1

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Under the load P1, the angular displacement/rotational angle was calculated; the simulation and experiments were performed. As seen in Fig. 11, the results found that the error between analytical model and simulation is about 4.34%. The error between analytical model and experiment is 8.46%. Similarly, the compressive displacements were recorded and compared with simulation and experiments, as seen in Fig. 12. The results showed that the error between the theory and simulation is about 4.6% and the error between the theory and experiment is approximately 6.38%.

Fig. 11
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Angular displacement versus load P1

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Fig. 12
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Compressive displacement versus load P2

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5.2 Improvement of Performances

The effectiveness of the proposed BICJ was compared with the rectangular cross section flexure. The dimensions of flexure included in-of-plane thickness of 4 mm, length of 88 mm, and out-of-plane width of 10 mm. Under load P1 and the use of Eq. (10), the results showed that an increase in bending displacement of the proposed BICJ is about 14.6%, as shown in Table 10. Using Eq. (11), the results found that an increase in angular displacement of the proposed BICJ is approximately 12.6%, as given in Table 10. Besides, the flexibility and compliance of the proposed joint are better than those of rectangular flexure.

Table 10 Increased performances: bending displacement and angular displacement
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Using Eq. (12) and FEA simulation, the relation of moment load versus rotation angle is described in Fig. 13. Under a moment load from 0 to 100 N mm, it found that the rotation angle of the proposed BICJ is much larger than that of the rectangular flexure.

Fig. 13
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Rotational angle versus moment

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6 Conclusions

This paper has presented design and analysis for a new butterfly-inspired compliant joint. By adopting the foldable mechanism and butterfly’s profile, the BICJ’s equivalent spring stiffness was varied easily. The conceptual design of the proposed BICJ was presented. The performances of the BICJ were validated by comparing with the joints #33 and #X through FEA simulation. The results showed that the performances of BICJ outperform other joints.

The closed-form model of the BICJ was provided, and the equations for calculating the stiffness, displacement, and rotation angle for the proposed joint were derived. Compared with a conventional rectangular flexure joint, the BICJ showed a great improvement. Specifically, the results indicated that an increase in bending displacement of proposed BICJ is about 14.6%, and an increase in angular displacement of proposed BICJ is approximately 12.6% in comparison with the rectangular flexure joint.

The results showed that there is a good agreement between the analytical model, simulation, and experimental results. These results well coincided with each other that confirmed the closed-form model. Finally, the proposed BICJ joint can be a promising candidate for different positioning applications. The results of this article may open possibilities for studying a new class of compliant joints with multiple functions in future work.