Abstract: The rotational stiffness of the zero-stiffness flexible hinge is approximately zero, which overcomes the defect that ordinary flexible hinges require driving torque, and can be applied to flexible grippers and other fields. Taking the inner and outer ring flexible hinges under the action of pure torque as the positive stiffness subsystem, the research Negative stiffness mechanism and matching positive and negative stiffness can construct zero stiffness flexible hinge. Propose a negative stiffness rotation mechanism——Crank spring mechanism, modeled and analyzed its negative stiffness characteristics; by matching positive and negative stiffness, analyzed the influence of structural parameters of crank spring mechanism on zero stiffness quality; proposed a linear spring with customizable stiffness and size——Diamond-shaped leaf spring string, the stiffness model was established and the finite element simulation verification was carried out; finally, the design, processing and testing of a compact zero-stiffness flexible hinge sample were completed. The test results showed that: under the action of pure torque,±18°In the range of rotation angles, the rotational stiffness of the zero-stiffness flexible hinge is 93% lower than that of the inner and outer ring flexible hinges on average. The constructed zero-stiffness flexible hinge has a compact structure and high-quality zero-stiffness; the proposed negative-stiffness rotation mechanism and the linear The spring has great reference value for the study of flexible mechanism.

0 Preface

Flexible hinge (bearing)
^[1-2]
Relying on the elastic deformation of the flexible unit to transmit or convert motion, force and energy, it has been widely used in precision positioning and other fields. Compared with traditional rigid bearings, there is a restoring moment when the flexible hinge rotates. Therefore, the drive unit needs to provide output torque to drive and Keep the rotation of the flexible hinge. Zero stiffness flexible hinge
^[3]
(Zero stiffness flexural pivot, ZSFP) is a flexible rotary joint whose rotational stiffness is approximately zero. This type of flexible hinge can stay at any position within the stroke range, also known as static balance flexible hinge
^[4]
, are mostly used in fields such as flexible grippers.

Based on the modular design concept of the flexible mechanism, the entire zero-stiffness flexible hinge system can be divided into two subsystems of positive and negative stiffness, and the zero-stiffness system can be realized through the matching of positive and negative stiffness
^[5]
. Among them, the positive stiffness subsystem is usually a large-stroke flexible hinge, such as a cross-reed flexible hinge
^[6-7]
, generalized three-cross reed flexible hinge
^[8-9]
and inner and outer ring flexible hinges
^[10-11]
etc. At present, the research on flexible hinges has achieved a lot of results, therefore, the key to design zero-stiffness flexible hinges is to match suitable negative stiffness modules for flexible hinges[3].

Inner and outer ring flexible hinges (Inner and outer ring flexural pivots, IORFP) have excellent characteristics in terms of stiffness, precision and temperature drift. The matching negative stiffness module provides the construction method of the zero-stiffness flexible hinge, and finally, completes the design, sample processing and testing of the zero-stiffness flexible hinge.

1 crank spring mechanism

1.1 Definition of negative stiffness

The general definition of stiffness K is the rate of change between the load F borne by the elastic element and the corresponding deformation dx


K= dF/dx (1)


When the load increment of the elastic element is opposite to the sign of the corresponding deformation increment, it is negative stiffness. Physically, the negative stiffness corresponds to the static instability of the elastic element
^[12]
.Negative stiffness mechanisms play an important role in the field of flexible static balance. Usually, negative stiffness mechanisms have the following characteristics.

(1) The mechanism reserves a certain amount of energy or undergoes a certain deformation.

(2) The mechanism is in a critical instability state.

(3) When the mechanism is slightly disturbed and leaves the equilibrium position, it can release a larger force, which is in the same direction as the movement.

1.2 Construction principle of zero-stiffness flexible hinge

The zero-stiffness flexible hinge can be constructed by using positive and negative stiffness matching, and the principle is shown in Figure 2.

(1) Under the action of pure torque, the inner and outer ring flexible hinges have an approximately linear torque-rotation angle relationship, as shown in Figure 2a. Especially, when the intersection point is located at 12.73% of the reed length, the torque-rotation angle relationship is linear
^[11]
, at this time, the restoring moment Mpivot (clockwise direction) of the flexible hinge is related to the bearing rotation angleθ(counterclockwise) the relationship is


Mpivot=(8EI/L)θ (2)


In the formula, E is the elastic modulus of the material, L is the length of the reed, and I is the moment of inertia of the section.

(2) According to the rotational stiffness model of the inner and outer ring flexible hinges, the negative stiffness rotating mechanism is matched, and its negative stiffness characteristics are shown in Figure 2b.

(3) In view of the instability of the negative stiffness mechanism
^[12]
, the stiffness of the zero-stiffness flexible hinge should be approximately zero and greater than zero, as shown in Figure 2c.

1.3 Definition of crank spring mechanism

According to literature [4], a zero-stiffness flexible hinge can be constructed by introducing a pre-deformed spring between the moving rigid body and the fixed rigid body of the flexible hinge. For the inner and outer ring flexible hinge shown in FIG. 1, a spring is introduced between the inner ring and the outer ring, I .e., a spring-crank mechanisms (SCM) is introduced. Referring to the crank slider mechanism shown in Figure 3, the related parameters of the crank spring mechanism are shown in Figure 4. The crank-spring mechanism is composed of a crank and a spring (set stiffness as k). the initial angle is the included angle between the crank AB and the base AC when the spring is not deformed. R represents the crank length, l represents the base length, and defines the crank length ratio as the ratio of r to l, I .e. = r/l (0<<1).

The construction of the crank-spring mechanism requires the determination of 4 parameters: the base length l, the crank length ratio , the initial angle and the spring stiffness K.

The deformation of the crank spring mechanism under force is shown in Figure 5a, at the moment M
_γ
Under the action, the crank moves from the initial position AB
_Beta
turn to AB
_γ
, during the rotation process, the included angle of the crank relative to the horizontal position
_γ
called the crank angle.

Qualitative analysis shows that the crank rotates from AB (initial position, M & gamma; Zero) to AB0 (“dead point”location, M
_γ
is zero), the crank-spring mechanism has a deformation with negative stiffness characteristics.

1.4 The relationship between torque and rotation angle of crank spring mechanism

In Fig. 5, the torque M & gamma; clockwise is positive, the crank angle & gamma; counterclockwise is positive, and the moment load M is modeled and analyzed below.
_γ
with crank angle
_γ
The relationship between the modeling process is dimensioned.

As shown in Figure 5b, the torque balance equation for crank AB & gamma is listed.

In the formula, F & gamma; is the spring restoring force, d & gamma; is F & gamma; to point A. Assume that the displacement-load relation of the spring is

In the formula, K is the spring stiffness (not necessarily a constant value),δ
_xγ
is the amount of spring deformation (shortened to positive),δ
_xγ
=|B
_Beta
C| – |B
_γ
C|.

Simultaneous type (3)(5), moment M
_γ
with corner
_γ
The relationship is

1.5 Analysis of the negative stiffness characteristics of the crank-spring mechanism

In order to facilitate the analysis of the negative stiffness characteristics of the crank-spring mechanism (moment M
_γ
with corner
_γ
relationship), it may be assumed that the spring has a linear positive stiffness, then formula (4) can be rewritten as

In the formula, Kconst is a constant greater than zero. After the size of the flexible hinge is determined, the length l of the base is also determined. Therefore, assuming that l is a constant, formula (6) can be rewritten as

where Kconstl2 is a constant greater than zero, and the moment coefficient m & gamma; has a dimension of one. The negative stiffness characteristics of the crank-spring mechanism can be obtained by analyzing the relationship between the torque coefficient m & gamma; and the rotation angle & gamma.

From equation (9), Figure 6 shows the initial angle =π relationship between m & gamma; and crank length ratio and rotation angle & gamma;, & isin;[0.1, 0.9],& gamma;& isin;[0, π]. Figure 7 shows the relationship between m & gamma; and rotation angle & gamma; for = 0.2 and different . Figure 8 shows =π When, under different , the relationship between m & gamma; and angle & gamma.

According to the definition of crank spring mechanism (section 1.3) and formula (9), when k and l are constant, m & gamma; Only related to angle & gamma;, crank length ratio and crank initial angle .

(1) If and only if & gamma; is equal to 0 orπ or ,m & gamma; is equal to zero; & gamma; & isin;[0, ],m & gamma; is greater than zero; & gamma; & isin;[,π],m & gamma; less than zero. & isin;[0, ],m & gamma; is greater than zero; & gamma;& isin;[,π],m & gamma; less than zero.

(2) & gamma; When [0, ], the rotation angle & gamma; increases, m & gamma; increases from zero to the inflection point angle & gamma;0 takes the maximum value m & gamma;max, and then gradually decreases.

(3) The negative stiffness characteristic range of the crank spring mechanism: & gamma;& isin;[0, & gamma;0], at this time & gamma; increases (counterclockwise), and the torque M & gamma; increases (clockwise). The inflection point angle & gamma;0 is the maximum rotation angle of the negative stiffness characteristic of the crank-spring mechanism and & gamma;0 & isin;[0, ];m & gamma;max is the maximum negative moment coefficient. Given and , the derivation of equation (9) yields & gamma;0

(4) the larger the initial angle , & gamma; the larger 0, m
_γmax
bigger.

(5) the larger the length ratio , & gamma; the smaller 0, m
_γmax
bigger.

In particular, =πThe negative stiffness characteristics of the crank spring mechanism are the best (the negative stiffness angle range is large, and the torque that can be provided is large). =πAt the same time, under different conditions, the maximum rotation angle & gamma of the negative stiffness characteristic of the crank spring mechanism; 0 and the maximum negative torque coefficient m & gamma; Max is listed in table 1.

2 Construction of zero-stiffness flexible hinge

The matching of positive and negative stiffness of the 2.1 is shown in Figure 9, n(n 2) groups of parallel crank spring mechanisms are evenly distributed around the circumference, forming a negative stiffness mechanism matched with the inner and outer ring flexible hinges.

Using the inner and outer ring flexible hinges as the positive stiffness subsystem, construct a zero-stiffness flexible hinge. In order to achieve zero stiffness, match the positive and negative stiffness

simultaneous (2), (3), (6), (11), and & gamma;=θ, the load F & gamma of the spring can be obtained; and displacementδThe relationship of x & gamma; is

According to section 1.5, the negative stiffness angle range of the crank spring mechanism: & gamma;& isin;[0, & gamma;0] and & gamma;0 & isin;[0, ], the stroke of the zero stiffness flexible hinge shall be less than & gamma;0, I .e. the spring is always in a deformed state (δxγ≠0). The rotation range of the inner and outer ring flexible hinges is±0.35 rad(±20°), simplify the trigonometric functions sin & gamma; and cos & gamma; as follows

After simplification, the load-displacement relationship of the spring

2.2 Error analysis of positive and negative stiffness matching model

Evaluate the error caused by the simplified treatment of equation (13). According to the actual processing parameters of zero stiffness flexible hinge (Section 4.2):n = 3,l = 40mm, =π, = 0.2,E = 73 GPa; The dimensions of the inner and outer ring flexible hinge reed L = 46mm,T = 0.3mm,W = 9.4mm; The comparison formulas (12) and (14) simplify the load displacement relationship and relative error of the front and rear springs as shown in Figures 10a and 10b respectively.

As shown in Figure 10, & gamma; is less than 0.35 rad (20°), the relative error caused by the simplified treatment to the load-displacement curve does not exceed 2.0%, and the formula

The simplified treatment of (13) can be used to construct zero-stiffness flexible hinges.

2.3 Stiffness characteristics of the spring

Assuming the stiffness of the spring is K, the simultaneous (3), (6), (14)

According to the actual processing parameters of zero stiffness flexible hinge (Section 4.2), the change curve of spring stiffness K with angle & gamma; is shown in Figure 11. In particular, when & gamma;= 0, K takes the minimum value.

For the convenience of design and processing, the spring adopts a linear positive stiffness spring, and the stiffness is Kconst. In the whole stroke, if the total stiffness of the zero stiffness flexible hinge is greater than or equal to zero, Kconst should take the minimum value of K

Equation (16) is the stiffness value of the linear positive stiffness spring when constructing the zero stiffness flexible hinge. 2.4 Analysis of zero-stiffness quality The load-displacement relationship of the constructed zero-stiffness flexible hinge is

Simultaneous formula (2), (8), (16) can be obtained

In order to evaluate the quality of zero stiffness, the reduction range of flexible hinge stiffness before and after adding the negative stiffness module is defined as the zero stiffness quality coefficientη

η The closer to 100%, the higher the quality of zero stiffness. Figure 12 is 1-η Relationship with crank length ratio and initial angle η It is independent of the number n of parallel crank-spring mechanisms and the length l of the base, but only related to the crank length ratio , the rotation angle & gamma; and the initial angle .

(1) The initial angle increases and the zero stiffness quality improves.

(2) The length ratio increases and the zero stiffness quality decreases.

(3) Angle & gamma; increases, zero stiffness quality decreases.

In order to improve the zero stiffness quality of the zero stiffness flexible hinge, the initial angle should take a larger value; the crank length ratio should be as small as possible. At the same time, according to the analysis results in Section 1.5, if is too small, the ability of the crank-spring mechanism to provide negative stiffness will be weak. In order to improve the zero stiffness quality of the zero stiffness flexible hinge, the initial angle =π, crank length ratio = 0.2, that is, the actual processing parameters of section 4.2 zero stiffness flexible hinge.

According to the actual processing parameters of the zero-stiffness flexible hinge (Section 4.2), the torque-angle relationship between the inner and outer ring flexible hinges and the zero-stiffness flexible hinge is shown in Figure 13; the decrease in stiffness is the zero-stiffness quality coefficientηThe relationship with the corner & gamma; is shown in Figure 14. By Figure 14: In 0.35 rad (20°) rotation range, the stiffness of the zero-stiffness flexible hinge is reduced by an average of 97%; 0.26 rad(15°) corners, it is reduced by 95%.

3 Design of linear positive stiffness spring

The construction of zero stiffness flexible hinge is usually after the size and stiffness of the flexible hinge are determined, and then the stiffness of the spring in the crank spring mechanism is reversed, so the stiffness and size requirements of the spring are relatively strict. In addition, the initial angle =π, from Figure 5a, during the rotation of the zero-stiffness flexible hinge, the spring is always in a compressed state, that is“Compression spring”.

The stiffness and size of traditional compression springs are difficult to customize precisely, and a guide mechanism is often required in applications. Therefore, a spring whose stiffness and size can be customized is proposed——Diamond-shaped leaf spring string. The diamond-shaped leaf spring string (Figure 15) is composed of multiple diamond-shaped leaf springs connected in series. It has the characteristics of free structural design and high degree of customization. Its processing technology is consistent with that of flexible hinges, and both are processed by precision wire cutting.

3.1 Load-displacement model of diamond-shaped leaf spring string

Due to the symmetry of the rhombic leaf spring, only one leaf spring needs to be subjected to stress analysis, as shown in Figure 16. α is the angle between the reed and the horizontal, the length, width and thickness of the reed are Ld, Wd, Td respectively, f is the dimensionally unified load on the rhombus leaf spring,δy is the deformation of rhombic leaf spring in the y direction, force fy and moment m are equivalent loads on the end of a single reed, fv and fw are component forces of fy in the wov coordinate system.

According to the beam deformation theory of AWTAR[13], the dimensionally unified load-displacement relation of single reed

Due to the constraint relationship of the rigid body on the reed, the end angle of the reed before and after deformation is zero, that isθ = 0. Simultaneous (20)(22)

Equation (23) is the load-displacement dimensional unification model of rhombic leaf spring. n2 rhombic leaf springs are connected in series, and its load-displacement model is

From formula (24), whenαWhen d is small, the stiffness of the diamond-shaped leaf spring string is approximately linear under typical dimensions and typical loads.

3.2 Finite element simulation verification of the model

The finite element simulation verification of the load-displacement model of the diamond-shaped leaf spring is carried out. Using ANSYS Mechanical APDL 15.0, the simulation parameters are shown in Table 2, and a pressure of 8 N is applied to the diamond-shaped leaf spring.

The comparison between the model results and the simulation results of the rhombus leaf spring load-displacement relationship is shown in Fig. 17 (dimensionalization). For four rhombus leaf springs with different inclination angles, the relative error between the model and the finite element simulation results does not exceed 1.5%. The validity and accuracy of the model (24) has been verified.

4 Design and test of zero-stiffness flexible hinge

4.1 Parameter design of zero-stiffness flexible hinge

To design a zero-stiffness flexible hinge, the design parameters of the flexible hinge should be determined according to the service conditions first, and then the relevant parameters of the crank spring mechanism should be calculated inversely.

4.1.1 Flexible hinge parameters

The intersection point of the inner and outer ring flexible hinges is located at 12.73% of the reed length, and its parameters are shown in Table 3. Substituting into equation (2), the torque-rotation angle relationship of the inner and outer ring flexible hinges is

4.1.2 Negative stiffness mechanism parameters

As shown in fig. 18, taking the number n of crank spring mechanisms in parallel as 3, the length l = 40 mm is determined by the size of the flexible hinge. according to the conclusion of section 2.4, the initial angle =π, crank length ratio = 0.2. According to equation (16), the stiffness of the spring (I .e. diamond leaf spring string) is Kconst = 558.81 N/m (26)

4.1.3 Diamond leaf spring string parameters

by l = 40mm, =π, = 0.2, the original length of the spring is 48mm, and the maximum deformation (& gamma;= 0) is 16mm. Due to structural limitations, it is difficult for a single rhombus leaf spring to produce such a large deformation. Using four rhombus leaf springs in series (n2 = 4), the stiffness of a single rhombus leaf spring is


Kd=4Kconst=2235.2 N/m (27)


According to the size of the negative stiffness mechanism (Figure 18), given the reed length, width and reed inclination angle of the diamond-shaped leaf spring, the reed can be deduced from formula (23) and the stiffness formula (27) of the diamond-shaped leaf spring Thickness. The structural parameters of rhombus leaf springs are listed in Table 4.

surface4

In summary, the parameters of the zero-stiffness flexible hinge based on the crank spring mechanism have all been determined, as shown in Table 3 and Table 4.

4.2 Design and processing of the zero-stiffness flexible hinge sample Refer to literature [8] for the processing and testing method of the flexible hinge. The zero-stiffness flexible hinge is composed of a negative stiffness mechanism and an inner and outer ring flexible hinge in parallel. The structural design is shown in Figure 19.

Both the inner and outer ring flexible hinges and diamond-shaped leaf spring strings are processed by precision wire-cutting machine tools. The inner and outer ring flexible hinges are processed and assembled in layers. Figure 20 is the physical picture of three sets of diamond-shaped leaf spring strings, and Figure 21 is the assembled zero-stiffness The physical picture of the flexible hinge sample.

4.3 The rotational stiffness test platform of the zero-stiffness flexible hinge Referring to the rotational stiffness test method in [8], the rotational stiffness test platform of the zero-stiffness flexible hinge is built, as shown in Figure 22.

4.4 Experimental data processing and error analysis

The rotational stiffness of the inner and outer ring flexible hinges and zero-stiffness flexible hinges was tested on the test platform, and the test results are shown in Figure 23. Calculate and draw the zero-stiffness quality curve of the zero-stiffness flexible hinge according to formula (19), as shown in Fig. 24.

The test results show that the rotational stiffness of the zero-stiffness flexible hinge is close to zero. Compared with the inner and outer ring flexible hinges, the zero-stiffness flexible hinge±0.31 rad(18°) stiffness was reduced by an average of 93%; 0.26 rad (15°), the stiffness is reduced by 90%.

As shown in Figures 23 and 24, there is still a certain gap between the test results of the zero stiffness quality and the theoretical model results (the relative error is less than 15%), and the main reasons for the error are as follows.

(1) The model error caused by the simplification of trigonometric functions.

(2) Friction. There is friction between the diamond leaf spring string and the mounting shaft.

(3) Processing error. There are errors in the actual size of the reed, etc.

(4) Assembly error. The gap between the installation hole of the diamond-shaped leaf spring string and the shaft, the installation gap of the test platform device, etc.

4.5 Performance comparison with a typical zero-stiffness flexible hinge In literature [4], a zero-stiffness flexible hinge ZSFP_CAFP was constructed using a cross-axis flexural pivot (CAFP), as shown in Figure 25.

Comparison of the zero-stiffness flexible hinge ZSFP_IORFP (Fig. 21) and ZSFP_CAFP (Fig. 25) constructed using the inner and outer ring flexible hinges

(1) ZSFP_IORFP, the structure is more compact.

(2) The corner range of ZSFP_IORFP is small. The corner range is limited by the corner range of the flexible hinge itself; the corner range of ZSFP_CAFP80°, ZSFP_IORFP corner range40°.

(3) ±18°In the range of corners, ZSFP_IORFP has higher quality of zero stiffness. The average stiffness of ZSFP_CAFP is reduced by 87%, and the average stiffness of ZSFP_IORFP is reduced by 93%.

5 Conclusion

Taking the flexible hinge of the inner and outer rings under pure torque as the positive stiffness subsystem, the following work has been done in order to construct a zero-stiffness flexible hinge.

(1) Propose a negative stiffness rotation mechanism——For the crank spring mechanism, a model (Formula (6)) was established to analyze the influence of structural parameters on its negative stiffness characteristics, and the range of its negative stiffness characteristics was given (Table 1).

(2) By matching the positive and negative stiffnesses, the stiffness characteristics of the spring in the crank spring mechanism (Equation (16)) are obtained, and the model (Equation (19)) is established to analyze the effect of the structural parameters of the crank spring mechanism on the zero stiffness quality of the zero stiffness flexible hinge Influence, theoretically, within the available stroke of the flexible hinge of the inner and outer rings (±20°), the average reduction in stiffness can reach 97%.

(3) Propose a customizable stiffness“spring”——A diamond-shaped leaf spring string was established to establish its stiffness model (Equation (23)) and verified by finite element method.

(4) Completed the design, processing and testing of a compact zero-stiffness flexible hinge sample. The test results show that: under the action of pure torque, the36°In the range of rotation angles, compared with the inner and outer ring flexible hinges, the stiffness of the zero-stiffness flexible hinge is reduced by 93% on average.

The constructed zero-stiffness flexible hinge is only under the action of pure torque, which can realize“zero stiffness”, without considering the case of bearing complex loading conditions. Therefore, the construction of zero-stiffness flexible hinges under complex load conditions is the focus of further research. In addition, reducing the friction that exists during the movement of zero-stiffness flexible hinges is an important optimization direction for zero-stiffness flexible hinges.

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About the author: Bi Shusheng (corresponding author), male, born in 1966, doctor, professor, doctoral supervisor. His main research direction is fully flexible mechanism and bionic robot.