CN114278699A - Two-dimensional plane negative stiffness device - Google Patents
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Abstract
The invention provides a two-dimensional plane negative stiffness device, belongs to the field of negative stiffness devices, and can provide linear or nonlinear negative stiffness force in any direction of a two-dimensional plane. The device comprises a curved surface, universal wheels, telescopic rods and a fixed frame. The universal wheel passes through the telescopic link and links to each other with the vibration isolation structure, and the curved surface passes through fixed frame and links to each other with the basis. The telescopic rod provides pre-pressure and is applied to the curved surface through the universal wheel; the reaction force of the curved surface to the universal wheel can be adjusted through the radian of the curved surface. The radian of the curved surface is designed according to the characteristics of the pre-pressure, the reaction force of the curved surface on the universal wheel is adjusted, so that the required linear or nonlinear negative stiffness force is generated in the direction vertical to the pre-pressure, and is transmitted to the vibration isolation structure through the universal wheel and the telescopic rod, and the vibration attenuation effect is realized. The universal wheel can move along any direction on a curved surface, so that negative stiffness force can be generated in any direction of a two-dimensional plane. The invention has simple manufacture, can generate negative rigidity force in any direction of a two-dimensional plane by designing a proper camber, and can generate different types of negative rigidity force in the vertical direction of the prepressing force.
Description
Technical Field
The invention relates to a precise vibration damper, in particular to a two-dimensional plane negative stiffness device.
Background introduction
Excessive vibration can cause many hazards, such as affecting the performance of precision instruments, reducing the ride comfort of the vehicle, and even causing structural damage. To alleviate the adverse effects caused by vibration, various vibration control techniques have been developed. Currently, vibration control techniques can be divided into three categories: passive control, semi-active control, and active control. The specific damping device includes: viscous fluid dampers, friction dampers, viscoelastic dampers, tuned mass dampers, variable orifice dampers, magnetorheological fluid variable dampers, electrorheological fluid variable dampers, variable friction dampers, active vibration damping systems, and the like.
Vibration control techniques have been widely used in the mechanical and civil engineering fields. Semi-active and active vibration control techniques often achieve better vibration control than passive vibration control techniques. In active control techniques, its actuators are often required to produce a control force-deflection relationship having a significant negative stiffness characteristic. This discovery has motivated researchers to seek a passive-stiffness device (NSD) that produces similar or identical hysteresis characteristics to the active control system, thereby achieving similar or identical control performance.
Existing passive negative stiffness devices include: (1) the passive negative stiffness of the pre-bending beam is utilized, (2) the negative stiffness of the hinged and pre-stressed rod piece or the spring is utilized, (3) the pendulum negative stiffness is realized in the horizontal direction by utilizing gravity, and (4) the magnetic force type negative stiffness is utilized. The 4 types of passive negative stiffness devices can only generate negative stiffness force in a single direction, namely one-dimensional negative stiffness force. The invention relates to a two-dimensional plane negative stiffness device, which is not limited to one direction and can generate negative stiffness force in any direction of a two-dimensional plane.
Disclosure of Invention
The invention aims to provide a two-dimensional plane negative stiffness device to solve the problems in the background technology.
In order to achieve the purpose, the invention provides the following technical scheme:
the invention provides a two-dimensional plane negative stiffness device which comprises a curved surface, universal wheels, telescopic rods and a fixed frame, wherein the curved surface is provided with a first end and a second end; the universal wheel is connected with the vibration isolation structure through a telescopic rod, and the curved surface is connected with the foundation through a fixed frame, or vice versa; the telescopic rod provides pre-pressure and is applied to the curved surface through the universal wheel; the reaction force of the curved surface to the universal wheel can be adjusted through the radian of the curved surface; the radian of the curved surface is designed according to the characteristics of the pre-pressure, the reaction force of the curved surface on the universal wheel is adjusted, so that the required linear or nonlinear negative stiffness force is generated in the direction vertical to the pre-pressure, and is transmitted to the vibration isolation structure through the universal wheel and the telescopic rod, so that the vibration isolation effect is realized; the universal wheel can move along any direction on a curved surface, so that negative stiffness force can be generated in any direction of a two-dimensional plane.
Preferably, the design method of the device is as follows:
establishing a coordinate system by taking the geometric center of the curved surface as an origin, wherein the direction of a longitudinal axis y is parallel to the direction of the prestress, and the direction of a transverse axis x is vertical to the direction of the prestress; the relationship between the pre-pressure p (y) applied to the curved surface and the displacement in the y-axis direction can be expressed by the following formula (1):
wherein, P0To initial pre-stress, k1Is a linear stiffness coefficient, k2n+1Is a nonlinear stiffness coefficient (n)>0)；
Fn(x) The force required for negative stiffness is the reaction force of the curved surface to the universal wheel in the direction perpendicular to the prestress direction, i.e. the direction of the x axis, Fn(x) The displacement relation in the x-axis direction is as follows:
wherein, F0Initial negative stiffness force, k1' is a linear negative stiffness coefficient (k)1’<0)，k2n+1' is a nonlinear stiffness coefficient (n)>0)；
The included angle alpha is the included angle between the tangent direction (dy/dx) of the radian of the curved surface and the direction of the x axis, and the force Fn(x) The relationship between P (y) and camber is:
according to the formula (3), the pre-pressure P (y) and the required negative stiffness force F can be determinedn(x) Designing a specific curve radian; the formula for calculating the radian of the curved surface is as follows:
∫P(y)dy＝∫Fn(x)dx (4)
obtaining a curved surface shape expression after integration:
wherein C is a constant and can be determined according to the stroke requirement of negative stiffness;
according to the radian of the curved surface calculated by the formula (5), the required two-dimensional curved surface can be obtained by rotating 360 degrees around the symmetry axis of the curved surface, and the universal wheel can move on the curved surface along any direction, so that negative stiffness force including linear negative stiffness force and nonlinear negative stiffness force can be generated in any direction of the two-dimensional plane.
Preferably, the two-dimensional plane negative stiffness device can be applied to various single-degree-of-freedom systems (such as seats, vibration isolation tables and engine supports), and can also be applied to multi-degree-of-freedom structures (such as building vibration isolation supports and stay cables).
The solution is preferably specified in the design for linear negative stiffness forces as follows:
1) such as a constant pre-stress applied to the curved surface, i.e. P (y) P0The linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and zero at zero displacement, i.e. Fn(x)＝k1' x, (negative stiffness coefficient k)1’<0) (ii) a The expression of the curved surface shape obtained according to the formula (5) is:
2) the pre-stress applied to the curved surface being a linearly varying force, i.e. p (y) ═ k1y+P0In the vertical directionThe linear negative stiffness force is required to be obtained in the direction perpendicular to the prestress and is zero at zero displacement, i.e. Fn(x)＝k1' x, (negative stiffness coefficient k)1’<0) (ii) a The expression of the curved surface shape obtained according to the formula (5) is:
3) applying a constant force in the vertical direction of the curved surface, i.e. P (y) P0The linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and is not zero at zero displacement, i.e. Fn(x)＝k1'x+F0(negative stiffness coefficient k1’<0) (ii) a The expression for the curved surface shape obtained according to equation (5) is:
4) the pre-stress applied to the curved surface being a linearly varying force, i.e. p (y) ═ k1y+P0The linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and is not zero at zero displacement, i.e. Fn(x)＝k1'x+F0(negative stiffness coefficient k1’<0) (ii) a The expression for the curved surface shape obtained according to equation (5) is:
the solution is preferably designed for a decreasing type non-linear negative stiffness force, i.e. the rate of change of the negative stiffness force decreases with increasing displacement x, as explained below:
1) such as a constant pre-stress applied to the curved surface, i.e. P (y) P0The weakening non-linear negative stiffness force is required to be obtained in the direction perpendicular to the pre-stress and is zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3(k1'<0,k3'>0) (ii) a According to the formula (5)) The expression for obtaining the curved surface shape is:
2) the pre-stress applied to the curved surface being a linearly varying force, i.e. p (y) ═ k1y+P0The weakening non-linear negative stiffness force is required to be obtained in the direction perpendicular to the pre-stress and is zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3(k1'<0,k3'>0) (ii) a The expression of the curved surface shape obtained according to the formula (5) is:
3) applying a constant force in the vertical direction of the curved surface, i.e. P (y) P0The weakening-type nonlinear negative stiffness force is required to be obtained in the direction perpendicular to the prestress and is not zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3+F0(k1'<0,k3'>0) (ii) a The expression of the curved surface shape obtained according to the formula (5) is:
4) the pre-stress applied to the curved surface being a linearly varying force, i.e. p (y) ═ k1y+P0The weakening-type nonlinear negative stiffness force is required to be obtained in the direction perpendicular to the prestress and is not zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3+F0(k1'<0,k3'>0) (ii) a The expression for the curved surface shape obtained according to equation (5) is:
solution preferably, in the design, the enhanced nonlinear negative stiffness force, i.e. the rate of change of the negative stiffness force, increases with increasing displacement x, as illustrated below:
1) such as a constant pre-stress applied to the curved surface, i.e. P (y) P0The enhanced non-linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and is zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3(k1'<0,k3'<0) (ii) a The expression of the curved surface shape obtained according to the formula (5) is:
2) the pre-stress applied to the curved surface being a linearly varying force, i.e. p (y) ═ k1y+P0The enhanced non-linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and is zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3(k1'<0,k3'<0) (ii) a The expression for the curved surface shape obtained according to equation (5) is:
3) applying a constant force in the vertical direction of the curved surface, i.e. P (y) P0The enhanced non-linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and is not zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3+F0(k1'<0,k3'<0) (ii) a The expression for the curved surface shape obtained according to equation (5) is:
4) the pre-stress applied to the curved surface being a linearly varying force, i.e. p (y) ═ k1y+P0At right angles to the prestressThe direction needs to obtain enhanced nonlinear negative stiffness force, and the negative stiffness force is not zero at zero displacement, namely Fn(x)＝k1'x+k3'x3+F0(k1'<0,k3'<0) (ii) a The expression for the curved surface shape obtained according to equation (5) is:
in summary, according to the characteristics of the pre-pressure and the requirement of the negative stiffness, the corresponding curvature of the curved surface is designed, so that the required linear or non-linear negative stiffness is obtained in the direction perpendicular to the pre-pressure.
Compared with the prior art, the two-dimensional plane negative stiffness device has the following beneficial effects:
the invention has simple manufacture and is easy to popularize in practice. By designing a proper curve radian, negative stiffness force can be generated in any direction of a two-dimensional plane, and different types of negative stiffness force can be generated in the vertical direction of pre-pressure, so that not only can linear negative stiffness force be generated, but also weakening type nonlinear negative stiffness force and enhancement type nonlinear negative stiffness force can be generated, and the application is wider. The negative stiffness force can be transmitted to the vibration isolation structure through the universal wheel and the telescopic rod or through the universal wheel and the curved surface, and the vibration isolation effect is achieved.
Drawings
FIG. 1 is a schematic structural diagram of the present invention;
FIG. 2 is a diagram showing stress conditions of a curved surface and a universal wheel;
FIG. 3 is a three-dimensional schematic of the present invention;
fig. 4 is a schematic front view of a cross-sectional structure according to a first embodiment of the present invention. The universal wheel 5 is connected with the vibration isolation structure 1 through the telescopic rod 2, and the curved surface 4 is connected with the foundation through the fixed frame 3;
FIG. 5 is a three-dimensional schematic diagram of a first embodiment of the present invention;
FIG. 6 is a top view of a first embodiment of the present invention;
FIG. 7 is a schematic sectional view taken along line A-A of the first embodiment of the present invention;
FIG. 8 is a schematic cross-sectional view of the first embodiment of the present invention taken along line B-B;
fig. 9 is a sample calculation diagram according to a first embodiment of the present invention. Wherein, the graph (a) is a relation graph of pre-pressure P (y) and displacement y, and the graph (b) is a linear negative stiffness force Fn(x) A graph relating to the displacement x (the displacement x is the relative displacement of two ends of the negative stiffness element), (c) a graph showing the radian of a curved surface to be designed for obtaining the negative stiffness force in the graph (b);
fig. 10 is a schematic structural view of a second embodiment of the present invention, in which a universal wheel 5 is connected to a base through a telescopic rod 2 and a fixed frame 3, and a curved surface 4 is connected to a vibration isolation structure 1;
FIG. 11 is a schematic structural diagram of a third embodiment of the present invention;
FIG. 12 is a schematic structural diagram of a fourth embodiment of the present invention;
the reference numerals in the figures denote:
1-vibration isolation structure; 2, a telescopic rod; 3-fixing the frame; 4-curved surface; 5-universal wheels; 6-a spring; 7-a pulley; 8-stay cables; 9-a base; 10-a frame structure; 11-vibration isolation bearing
Detailed Description
The following detailed description of the invention will be made in conjunction with the accompanying drawings 1-10. The telescopic rod 2 generates pre-pressure and is applied to the curved surface 4 through the universal wheel 5, and the radian of the corresponding curved surface 4 is designed according to the pre-pressure characteristic and the requirement of negative rigidity force, so that the required linear or nonlinear negative rigidity force is obtained in the direction perpendicular to the pre-pressure. The negative stiffness force can be transmitted to the vibration isolation structure 1 through the universal wheels 5 and the telescopic rod 2 or through the curved surface 4, so that the vibration isolation effect is achieved, and adverse effects caused by vibration are relieved.
The first embodiment is as follows:
based on the structural foundation of the first embodiment, referring to the attached figures 2-8, the two-dimensional plane negative stiffness device can be applied to a single-degree-of-freedom system, such as a seat, a vibration isolation table, an engine support and the like.
In the design, the following description will be made for different types of prestressing and for the camber required to produce the negative stiffness, both linear and non-linear.
Linear negative stiffness force:
a constant pre-stressing force, i.e. P (y) P, applied to the curved surface 40The linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and zero at zero displacement, i.e. Fn(x)＝k1' x, (negative stiffness coefficient k)1’<0). The expression of the shape of the curved surface 4 is obtained according to equation 5 as follows:
second, the prestress applied to the curved surface 4 is a linearly changing force, i.e., p (y) k1y+P0The linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and zero at zero displacement, i.e. Fn(x)＝k1' x, (negative stiffness coefficient k)1’<0). The expression of the shape of the curved surface 4 is obtained according to equation 5 as follows:
thirdly, applying a constant force in the vertical direction to the curved surface 4, namely, P (y) P0The linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and is not zero at zero displacement, i.e. Fn(x)＝k1'x+F0(negative stiffness coefficient k1’<0). The expression for the shape of the curved surface 4 is obtained according to equation 5 as:
the prestress applied to the curved surface 4 is a linearly changing force, i.e., p (y) k1y+P0The linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and is not zero at zero displacement, i.e. Fn(x)＝k1'x+F0(negative stiffness coefficient k1’<0). The expression for obtaining the shape of the curved surface 4 according to equation 5 is：
The weakening type nonlinear negative stiffness force, i.e. the rate of change of the negative stiffness force decreases with increasing displacement x:
a constant pre-stressing force, i.e. P (y) P, applied to the curved surface 40The weakening non-linear negative stiffness force is required to be obtained in the direction perpendicular to the pre-stress and is zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3(k1'<0,k3'>0). The expression of the shape of the curved surface 4 is obtained according to equation 5 as follows:
second, the prestress applied to the curved surface 4 is a linearly changing force, i.e., p (y) k1y+P0The weakening non-linear negative stiffness force is required to be obtained in the direction perpendicular to the pre-stress and is zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3(k1'<0,k3'>0). The expression of the shape of the curved surface 4 is obtained according to equation 5 as follows:
thirdly, applying a constant force in the vertical direction to the curved surface 4, namely, P (y) P0The weakening-type nonlinear negative stiffness force is required to be obtained in the direction perpendicular to the prestress and is not zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3+F0(k1'<0,k3'>0). The expression of the shape of the curved surface 4 is obtained according to equation 5 as follows:
the prestress applied to the curved surface 4 is a linearly changing force, i.e., p (y) k1y+P0The weakening-type nonlinear negative stiffness force is required to be obtained in the direction perpendicular to the prestress and is not zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3+F0(k1'<0,k3'>0). The expression for the shape of the curved surface 4 is obtained according to equation 5 as:
enhanced nonlinear negative stiffness force, i.e. the rate of change of the negative stiffness force increases with increasing displacement x:
a constant pre-stressing force, i.e. P (y) P, applied to the curved surface 40The enhanced non-linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and is zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3(k1'<0,k3'<0). The expression of the shape of the curved surface 4 is obtained according to equation 5 as follows:
second, the prestress applied to the curved surface 4 is a linearly changing force, i.e., p (y) k1y+P0The enhanced non-linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and is zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3(k1'<0,k3'<0). The expression for the shape of the curved surface 4 is obtained according to equation 5 as:
thirdly, applying constant force to the curved surface 4 in the vertical directionI.e. P (y) ═ P0The enhanced non-linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and is not zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3+F0(k1'<0,k3'<0). The expression for the shape of the curved surface 4 is obtained according to equation 5 as:
the prestress applied to the curved surface 4 is a linearly changing force, i.e., p (y) k1y+P0The enhanced non-linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and is not zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3+F0(k1'<0,k3'<0). The expression for the shape of the curved surface 4 is obtained according to equation 5 as:
according to the above design method, a calculation example is provided, such as P with constant pre-pressure0When the required negative stiffness is linear negative stiffness with negative stiffness coefficient of-1 KN/m and the maximum stroke is 40mm, the radian of the designed curved surface is as follows:
y＝-10x2+0.02 (18)
the specific form of curvature of the curved surface 4 is shown in fig. 9. Wherein, the graph (a) is a relation graph of pre-pressure P (y) and displacement y, and the graph (b) is a linear negative stiffness force Fn(x) The maximum travel of the negative stiffness is 40mm in relation to the displacement x, and (c) the graph is 4 radians of the curved surface for which the negative stiffness force in graph (b) should be designed.
In summary, according to the pre-pressure characteristic and the requirement of the negative stiffness force, the corresponding curved surface 4 radians are designed, so that the required linear or non-linear negative stiffness force is obtained in the direction perpendicular to the pre-pressure. The negative stiffness force can be transmitted to the vibration isolation structure 1 through the universal wheels 5 and the telescopic rods 2. In installation, the two-dimensional plane negative stiffness device and the structural positive stiffness spring 6 are required to be connected in parallel for structural vibration isolation, and the stiffness of the system at the balance position can be close to 0 according to the principle of cancellation of positive and negative stiffness, so that the natural frequency of the system is reduced, low-frequency and ultralow-frequency vibration isolation is realized, the vibration isolation range is expanded, and the vibration isolation capability is improved.
Example two:
different from the first embodiment, the second embodiment is that the installation positions of the curved surface 4, the universal wheel 5, the telescopic rod 2 and the fixed frame 3 are changed, the universal wheel 5 is connected with the foundation through the telescopic rod 2 and the fixed frame 3, and the curved surface 4 is connected with the vibration isolation structure 1. But the negative stiffness generation method is the same, and with reference to the attached figure 2, the corresponding curved surface 4 radian is designed according to the pre-pressure characteristic and the requirement of the negative stiffness force. Due to the change of the installation position, the linear or nonlinear negative stiffness force generated in the vertical direction of the pre-pressure is transmitted to the vibration isolation structure 1 through the curved surface 4. Similarly, the two-dimensional plane negative stiffness device of the invention is required to be connected with the positive stiffness spring 6 of the structure in parallel, and is used for structural vibration isolation according to the principle of positive and negative stiffness cancellation.
Example three:
referring to fig. 11, the two-dimensional plane negative stiffness device of the present invention can be applied to a stay cable for vibration reduction of the stay cable. In the device, the direction of the prestress applied to the curved surface 4 by the telescopic rod 2 through the universal wheel 5 is parallel to the stay cable, and the direction of the generated negative rigidity force is vertical to the stay cable. The universal wheels 5 are connected with the fixed frame 3 and the bridge floor base 9 through the telescopic rods 2, and the curved surfaces 4 are connected with the stay cables 8. The pre-pressure P (y) parallel to the stay cable is applied to the telescopic rod 2, the pre-pressure P (y) is applied to the curved surface 4 through the universal wheels 5, the radian of the curved surface is designed by combining the characteristics of the pre-pressure, linear or nonlinear negative stiffness force can be generated in the direction perpendicular to the pre-pressure, and the method for generating the negative stiffness is the same as the first embodiment and the second embodiment, and the attached figure 2 is also referred. The negative stiffness force can be transmitted to the stay cable 8 through the curved surface 4, so that the vibration reduction of the stay cable is realized.
Example four:
referring to fig. 12, the two-dimensional plane negative stiffness apparatus of the present invention can be applied to a multi-degree-of-freedom system, such as a frame structure, as shown in fig. 12, in which universal wheels 5 are connected to a frame structure 10 through telescopic rods 2, and curved surfaces 4 are connected to a foundation through a fixed frame 3. The telescopic rod 2 generates pre-pressure, the pre-pressure is acted on the curved surface 4 through the universal wheel 5, and the corresponding radian of the curved surface 4 is designed according to the pre-pressure characteristic and the negative rigidity requirement, so that the required negative rigidity force is obtained in the direction perpendicular to the pre-pressure force. The negative stiffness generating method is the same as the first, second and third embodiments. The negative stiffness force can be transmitted to the frame structure 10 through the universal wheels 5 and the telescopic rods 2, and when the negative stiffness required by the vibration isolation structure is large, a plurality of two-dimensional plane negative stiffness devices can be connected in parallel, so that the required vibration damping effect is achieved.
Claims (6)
1. A two-dimensional plane negative stiffness device is characterized by comprising a curved surface (4), universal wheels (5), telescopic rods (2) and a fixed frame (3); the universal wheel (5) is connected with the vibration isolation structure (1) through the telescopic rod (2), the curved surface (4) is connected with the foundation through the fixed frame (3), and vice versa; the telescopic rod (2) provides pre-pressure and is applied to the curved surface (4) through the universal wheel (5); the reaction force of the curved surface (4) on the universal wheel (5) can be adjusted through the radian of the curved surface (4); the radian of the curved surface (4) is designed according to the characteristics of the pre-pressure, the reaction force of the curved surface (4) on the universal wheel (5) is adjusted, so that the required linear or nonlinear negative stiffness force is generated in the direction vertical to the pre-pressure, and is transmitted to the vibration isolation structure (1) through the universal wheel (5) and the telescopic rod (2) to play a role in vibration reduction.
2. The two-dimensional plane negative stiffness device according to claim 1, wherein the device design method is as follows:
a coordinate system is established by taking the geometric center of the curved surface (4) as an original point, wherein the direction of a longitudinal axis y is parallel to the direction of the prestress, and the direction of a transverse axis x is vertical to the direction of the prestress; the relationship between the pre-pressure P (y) applied to the curved surface (4) and the displacement in the y-axis direction can be expressed by the following formula (1):
wherein, P0To initial pre-stress, k1Is a linear stiffness coefficient, k2n+1Is a nonlinear stiffness coefficient (n)>0)；
Fn(x) The required negative stiffness force is the reaction force of the curved surface (4) to the universal wheel (5) in the direction perpendicular to the prestress direction, namely the direction of the x axis, Fn(x) The displacement relation in the x-axis direction is as follows:
wherein, F0Initial negative stiffness force, k1' is a linear negative stiffness coefficient (k)1’<0)，k2n+1' is a nonlinear stiffness coefficient (n)>0)；
The included angle alpha is the included angle between the radian tangent direction (dy/dx) of the curved surface (4) and the x-axis direction, and the force Fn(x) The relationship between P (y) and the radian of the curved surface (4) is as follows:
according to the formula (3), the pre-pressure P (y) and the required negative stiffness force F can be determinedn(x) Designing the radian of the specific curved surface (4); the radian of the curved surface (4) is calculated by the following formula:
∫P(y)dy＝∫Fn(x)dx (4)
and obtaining a curved surface (4) shape expression after integration:
wherein C is a constant and can be determined according to the stroke requirement of negative stiffness;
the radian of the curved surface (4) calculated according to the formula (5) is rotated for 360 degrees around the symmetry axis of the curved surface, so that the required two-dimensional curved surface (4) can be obtained, and the universal wheel (5) can move on the curved surface (4) along any direction, so that negative stiffness force including linear negative stiffness force and nonlinear negative stiffness force can be generated in any direction of a two-dimensional plane.
3. The device of claim 1, wherein the device can be applied to a single degree of freedom system or a multiple degree of freedom system.
4. A two-dimensional planar negative stiffness device as claimed in claim 1, wherein the linear negative stiffness force is specified in the design as follows:
1) if the prestressing force applied to the curved surface (4) is constant, i.e. P (y) P0The linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and zero at zero displacement, i.e. Fn(x)＝k1' x, (negative stiffness coefficient k)1’<0) (ii) a The expression of the shape of the curved surface (4) is obtained according to the formula (5) as follows:
2) the prestressing force applied to the curved surface (4) is a linearly varying force, i.e. p (y) k1y+P0The linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and zero at zero displacement, i.e. Fn(x)＝k1' x, (negative stiffness coefficient k)1’<0) (ii) a The expression of the shape of the curved surface (4) is obtained according to the formula (5) as follows:
3) applying a constant force in the vertical direction to the curved surface (4), i.e. P (y) P0The linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and is not zero at zero displacement, i.e. Fn(x)＝k1'x+F0(negative stiffness coefficient k1’<0) (ii) a The expression for the shape of the curved surface (4) is obtained according to equation (5):
4) the prestressing force applied to the curved surface (4) is a linearly varying force, i.e. p (y) k1y+P0The linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and is not zero at zero displacement, i.e. Fn(x)＝k1'x+F0(negative stiffness coefficient k1’<0) (ii) a The expression for the shape of the curved surface (4) is obtained according to equation (5):
5. a two-dimensional planar negative stiffness device according to claim 1, wherein the design is specified as follows for a decreasing type nonlinear negative stiffness force, i.e. the rate of change of the negative stiffness force decreases with increasing displacement x:
1) if the prestressing force applied to the curved surface (4) is constant, i.e. P (y) P0The weakening non-linear negative stiffness force is required to be obtained in the direction perpendicular to the pre-stress and is zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3(k1'<0,k3'>0) (ii) a The expression of the shape of the curved surface (4) is obtained according to the formula (5) as follows:
2) the prestressing force applied to the curved surface (4) is a linearly varying force, i.e. p (y) k1y+P0The weakening non-linear negative stiffness force is required to be obtained in the direction perpendicular to the pre-stress and is zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3(k1'<0,k3'>0) (ii) a The expression of the shape of the curved surface (4) is obtained according to the formula (5) as follows:
3) applying a constant force in the vertical direction to the curved surface (4), i.e. P (y) P0The weakening-type nonlinear negative stiffness force is required to be obtained in the direction perpendicular to the prestress and is not zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3+F0(k1'<0,k3'>0) (ii) a The expression of the shape of the curved surface (4) is obtained according to the formula (5) as follows:
4) the prestressing force applied to the curved surface (4) is a linearly varying force, i.e. p (y) k1y+P0The weakening-type nonlinear negative stiffness force is required to be obtained in the direction perpendicular to the prestress and is not zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3+F0(k1'<0,k3'>0) (ii) a The expression for the shape of the curved surface (4) is obtained according to equation (5):
6. a two-dimensional planar negative stiffness device as claimed in claim 1, wherein the enhanced non-linear negative stiffness force, i.e. the rate of change of the negative stiffness force, increases with increasing displacement x in the design as follows:
1) if the prestressing force applied to the curved surface (4) is constant, i.e. P (y) P0Enhanced nonlinearity in the direction perpendicular to the pre-stressNegative stiffness force and at zero displacement the negative stiffness force is zero, i.e. Fn(x)＝k1'x+k3'x3(k1'<0,k3'<0) (ii) a The expression of the shape of the curved surface (4) is obtained according to the formula (5) as follows:
2) the prestressing force applied to the curved surface (4) is a linearly varying force, i.e. p (y) k1y+P0The enhanced non-linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and is zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3(k1'<0,k3'<0) (ii) a The expression for the shape of the curved surface (4) is obtained according to equation (5):
3) applying a constant force in the vertical direction to the curved surface (4), i.e. P (y) P0The enhanced non-linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and is not zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3+F0(k1'<0,k3'<0) (ii) a The expression for the shape of the curved surface (4) is obtained according to equation (5):
4) the prestressing force applied to the curved surface (4) is a linearly varying force, i.e. p (y) k1y+P0The enhanced non-linear negative stiffness force needs to be obtained in the direction perpendicular to the pre-stress and is not zero at zero displacement, i.e. Fn(x)＝k1'x+k3'x3+F0(k1'<0,k3'<0) (ii) a Obtaining the shape of the curved surface (4) according to equation (5)The expression is as follows:
in conclusion, according to the characteristics of the pre-pressure and the requirement of the negative stiffness, the radian of the corresponding curved surface (4) is designed, so that the required linear or nonlinear negative stiffness in the direction perpendicular to the pre-pressure is obtained.
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