Double-potential-well spring oscillator system experimental device and method

Abstract

The invention discloses a double-potential-well spring oscillator system experiment device and a double-potential-well spring oscillator system experiment method. An air cushion guide rail and a non-inertial reference system motion module are arranged on a horizontal bottom plate, a sliding block is installed on the air cushion guide rail in a sliding mode, an adjustable spring fixing support is arranged in the direction perpendicular to the air cushion guide rail, and the adjustable spring fixing support is connected with the air cushion guide rail through a horizontal limiting tension-compression spring module. A visual module is arranged on the adjustable spring fixing bracket; a digital display pull and push dynamometer is further arranged above the air cushion guide rail, and the detection end of the digital display pull and push dynamometer is connected with the sliding block; the method comprises a quasi-static mode, a dynamic mode and a chaos mode, movement between the adjustable spring fixing support and the sliding block is adjusted, different friction of the sliding block on the air cushion guide rail is adjusted, different experiment testing modes are generated, and experiment simulation is achieved. Various nonlinear dynamic phenomena of the spring oscillator in a double-potential-well system are comprehensively and accurately displayed, rich experiment contents and powerful tools are provided for college physics experiment teaching, and the method has wide application prospects.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of experimental education equipment, and relates to a double-potential-well spring oscillator experimental device and method, which is particularly suitable for physical experiment teaching and is used for experimentally testing the dynamic behavior of a spring oscillator in a double-potential-well system, including force hysteresis, bifurcation and chaos. BACKGROUND

[0002] In physical experiment teaching, a one-dimensional spring oscillator is an important experimental content of classical mechanics, which is used to help students understand basic physical concepts such as vibration and energy conversion. However, the traditional spring oscillator experimental device has a single function, and can usually only demonstrate and observe simple harmonic vibration, forced damped vibration, etc. In order to more deeply explore the dynamic phenomena of the spring oscillator, such as force hysteresis, bifurcation and chaos, it is particularly important to develop a spring oscillator experimental device capable of realizing a double-potential-well system. Such a device not only can enrich the teaching content, but also can stimulate students' interest and desire to explore complex physical phenomena. SUMMARY

[0003] In order to solve the problems in the background art, the present application proposes a double-potential-well spring oscillator system experimental device and method, which aims to provide an experimental platform capable of demonstrating various nonlinear dynamic phenomena for physical experiment teaching.

[0004] The present application adopts two symmetrical limit tension and compression spring modules to connect the sliding block, which is placed on the horizontal air cushion slide rail, and the spring is fixed on the support, thereby constructing a double-potential-well spring oscillator system. In combination with a single-chip microcomputer controlled linear motor, the support drives the spring to move, realizes programmable control of the periodic external force of the oscillator, and obtains the hysteresis effect, bifurcation phenomenon and chaotic dynamic characteristics of the double-potential-well oscillator through video image shooting collection and picture tracking analysis method.

[0005] The technical scheme adopted by the present application is as follows: I. A double-potential-well spring oscillator system experimental device: The device comprises an air cushion guide rail, a sliding block, a limit tension and compression spring module, an adjustable spring fixing support, a non-inertial reference system motion module, a digital display push-pull force gauge and a microphone air pump. The air cushion guide rail is fixedly arranged on the bottom plate, the sliding block is slidably mounted on the air cushion guide rail, the adjustable spring fixing support is arranged perpendicular to the direction of the air cushion guide rail, the adjustable spring fixing support is connected through the horizontal limit tension and compression spring module and the sliding block on the air cushion guide rail, and a visual module for shooting is further arranged on the adjustable spring fixing support; The non-inertial reference frame motion module is connected to the adjustable spring fixing bracket, which is used to drive the adjustable spring fixing bracket to move along the direction of the air cushion guide rail. A digital push-pull force gauge is also installed above the air cushion guide rail, and the probe end of the digital push-pull force gauge is connected to the slider.

[0006] It also includes a micro-sound air pump, whose air outlet is connected to the air cushion guide rail via a hose, for controlling the inflation and deflation of the air cushion guide rail and adjusting the sliding friction of the slider on the air cushion guide rail.

[0007] The adjustable spring fixing bracket spans across both sides of the air cushion guide rail, and the ends of the adjustable spring fixing bracket on both sides of the air cushion guide rail are connected to a limiting tension / compression spring module and a slider, respectively.

[0008] The non-inertial reference frame motion module includes a microcontroller, a driver, a linear motor, and a DC switching power supply. The microcontroller and the DC switching power supply are electrically connected to the driver, the driver is electrically connected to the linear motor, and the moving output end of the linear motor is fixedly connected to the adjustable spring fixing bracket.

[0009] The adjustable spring fixing bracket includes an aluminum plate, a perforated aluminum angle, and a spring suspension slider. The aluminum plate is strip-shaped and is arranged horizontally below the air cushion guide rail. A perforated aluminum angle is fixed on both ends of the aluminum plate. A spring suspension slider is installed on the perforated aluminum angle, which can move up and down. The spring suspension slider is used to limit one end of the tension and compression spring module.

[0010] The limiting tension / compression spring module includes a spring body, a rigid extension rod, a limiting sleeve, and a hook. One end of the spring body and one end of the rigid extension rod are fixedly connected and together fitted inside the limiting sleeve. The other end of the spring body extends out of the limiting sleeve and is fixedly connected to the limiting sleeve. After extending out of the limiting sleeve, the other end of the spring body is connected to the adjustable spring fixing bracket via the hook. The other end of the rigid extension rod is connected to the slider via the hook.

[0011] II. Experimental Apparatus for a Double-Potential-Well Spring Oscillator System: Experimental Method for a Double-Potential-Well Spring Oscillator System The method includes three modes: quasi-static mode, dynamic mode, and chaotic mode. By controlling the operation or inactivity of the non-inertial reference frame motion module and coordinating the stationary or moving position of the digital push-pull force gauge, the relative movement between the adjustable spring fixing bracket and the slider is adjusted. Furthermore, by controlling the inflation or deflation of the air cushion guide rail, different friction conditions of the slider on the air cushion guide rail are adjusted, thereby generating different experimental test modes to achieve experimental simulation of the double potential well spring oscillator and obtain the experimental simulation image results.

[0012] After obtaining the experimental simulation results of the double-potential-well spring oscillator at the end of the method, the characteristics of its nonlinear targeted energy transfer can be simulated and calculated. It can generate large-amplitude vibrations over a wide driving frequency range, is insensitive to changes in the structure's intrinsic frequency, and exhibits broadband robustness. This characteristic overcomes the limitations of traditional single-frequency linearly tuned mass dampers, responding not only to vibrations of a single frequency. Even under small initial vibrations, this device can activate its large-amplitude inter-well motion under specific conditions. This has a very powerful effect in the application of mechanical vibration suppressors, enabling highly efficient vibration suppression. It provides a fundamental research platform for flutter suppression in aerospace structures, wind and earthquake resistance in high-rise buildings and bridges, and precision manufacturing platforms for ultra-stable wings.

[0013] The experimental testing mode includes the quasi-static mode, specifically: The non-inertial reference frame motion module is not working, the adjustable spring fixing bracket keeps the position fixed, and the digital push-pull force gauge is used to pull the slider to move along the air cushion guide rail. After pulling, the distance y from the slider to the origin is fixed in real time. The origin is the intersection between the adjustable spring fixing bracket and the air cushion guide rail. Read the force reading f of the digital push-pull force gauge under stationary conditions at different distances y. Calculate the reduced force of the push-pull force gauge corresponding to the slider's equilibrium on the air track using the following formula, based on the force reading f obtained at each different distance y. A Reduced coordinates of the longitudinal position of the slider track Y : A = f / ( 2kl 0), Y = y / l 0 in, f This indicates the force reading of the digital push-pull force gauge. y This represents the distance from the slider to the origin. k Indicates the spring constant. l 0 indicates the length of the spring in its natural state within the limit tension / compression spring module; The experimental testing mode includes the dynamic mode, specifically: The digital push-pull force gauge remains stationary, controlling the non-inertial reference frame motion module to move the adjustable spring fixing bracket parallel to the air cushion guide rail. This, in turn, via the limit tension spring module, pulls the slider along the air cushion guide rail. After movement, the distance y between the slider and the origin is fixed in real time, and the distance is adjusted accordingly. y Images are acquired by the vision module in a static state, and the acceleration of the support and slider is obtained through image processing, based on different distances. y The acceleration of the corresponding slider is calculated using the following formula to obtain the equivalent force acting on the slider. A’ The coordinates of the longitudinal position of the slider track are reduced.Y : A' = m ×( a 0 +a ) / ( 2kl 0), Y = y / l 0 in, m For the mass of the slider, a 0 represents the acceleration during the movement from the origin. a This represents the acceleration of the slider relative to the origin. y This represents the distance from the slider to the origin. k Indicates the spring constant. l 0 indicates the length of the spring in its natural state within the limit tension / compression spring module; Finally, all the corresponding force parameters A’ and coordinate parameters Y Plotting the dynamically reduced position-force relationship Y - A' Image, as a force hysteresis feature.

[0014] The experimental testing mode includes the chaotic mode, specifically: The digital push-pull force gauge remains stationary, controlling the non-inertial reference frame motion module to move the adjustable spring fixing bracket parallel to the air cushion guide rail. This, in turn, pulls the slider along the air cushion guide rail via the limit tension spring module. During this movement, the air cushion guide rail is inflated and deflated in real time to adjust the friction force of the slider on the guide rail. After movement, the distance between the slider and the origin is fixed in real time. y At different distances y Images are acquired by the vision module in a static state, and the acceleration of the support and slider is obtained through image processing, based on different distances. y The acceleration of the corresponding slider is calculated using the following formula to obtain the longitudinal reduced velocity relative to the corresponding slider track. V Reduced coordinates of the longitudinal position of the slider track Y : V = δv / (2 kl 0), Y = y / l 0 in, y This represents the distance from the slider to the origin. k Indicates the spring constant. l 0 indicates the length of the spring in its natural state within the limit tension / compression spring module; v This indicates the speed of the slider relative to the origin along the direction of the air cushion guide. δ The damping coefficient; Finally, the longitudinal velocity is reduced by all corresponding slider tracks. V and coordinate parametersY Plotting the dynamically reduced position-velocity relationship Y - V Images are used to study the broadband phase diagram characteristics of bistable oscillators.

[0015] This invention employs symmetrically arranged limiting tension and compression spring modules, air cushion guide rails, microcontroller-controlled linear motors, digital push-pull force gauges, and adjustable brackets, which are respectively applied to experiments such as quantitative measurement of hysteresis effects in double potential well systems, observation of bifurcation phenomena, dynamic response analysis under non-inertial force drive, and Poincaré section calculation of chaotic behavior, thus forming a complete nonlinear dynamics experimental device system and experimental method.

[0016] In this invention, a DC power supply box provides precise control over the position of the linear motor; a micro-sound air pump is connected to an air cushion guide rail, on which a freely movable slider is mounted; an adjustable bracket is fixed to the linear motor, perpendicular to the direction of the air cushion guide rail, and the distance from the spring suspension point to the guide rail can be adjusted by screws, nuts, and sliding grooves; a limit tension spring module contains a rigid extension rod placed in a limit sleeve to limit lateral deformation during movement.

[0017] The method of this invention includes a quasi-static experimental mode and a dynamic experimental mode. By keeping the synchronous belt module stationary and changing the magnitude of the external force under quasi-static conditions, the displacement-external force diagram of the slider is plotted, and hysteresis and bifurcation phenomena can be clearly observed. Precise control of the displacement, velocity, and acceleration of the synchronous belt module can be considered as applying an inertial force to the block in a non-inertial frame. Image analysis software is used to plot and process the data to obtain a motion phase diagram under dynamic conditions.

[0018] The beneficial effects of this invention are: This invention designs a double-potential-well spring oscillator experimental device to demonstrate and obtain common bistable and abrupt behavior in complex nonlinear systems. It is suitable for observation and teaching experiments of nonlinear dynamic characteristics and constitutes a new type of mechanical experimental teaching instrument.

[0019] This invention enables the reproduction of typical behavior of double potential well systems under laboratory conditions. Based on this device, a systematic experimental method is designed, which can quantitatively measure the displacement-external force hysteresis curve, bifurcation threshold phenomenon, evolution process of periodic motion and chaotic state of double potential well systems under quasi-static and dynamic conditions, as well as the influence of various parameters on the nonlinear characteristics of the system. It can comprehensively carry out the study of the dynamic behavior of double potential well systems, and is not only suitable for physical experimental teaching of bistable scenarios of mechanical systems, but also has certain scientific research value for nonlinear physical phenomena. Attached Figure Description

[0020] Figure 1 This is a schematic diagram of the complete experimental setup for a double-potential-well spring oscillator system.

[0021] Figure 2 This is a schematic diagram of a non-inertial reference frame motion module.

[0022] Figure 3 This is a schematic diagram of an adjustable spring fixing bracket.

[0023] Figure 4 This is a schematic diagram of the limit tension / compression spring module.

[0024] Figure 5 This is a partial top view of the experimental setup for a double-potential-well spring oscillator system.

[0025] In the diagram: 1. Micro-inlet air pump; 2. Air cushion guide rail; 3. Slider; 4. Limiting tension / compression spring module; 4a. Spring body; 4b. Rigid extension rod; 4c. Limiting sleeve; 4d. Hook; 5. Adjustable spring fixing bracket; 5a. Aluminum sheet; 5b. Perforated angle aluminum; 5c. Spring suspension slider; 6. Non-inertial reference frame motion module; 6a. Microcontroller; 6b. Driver; 6c. Linear motor; 6d. DC switching power supply; 7. Digital display push-pull force gauge; 8. Camera bracket; 9. Camera image acquisition module. Detailed Implementation

[0026] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0027] like Figure 1 As shown, the complete device includes an air cushion guide rail 2, a slider 3, a limiting tension and compression spring module 4, an adjustable spring fixing bracket 5, a non-inertial reference frame motion module 6, a digital display push-pull force gauge 7, and a micro-sound air pump 1; the air cushion guide rail 2 and the non-inertial reference frame motion module 6 are arranged on a horizontal base plate. The air cushion guide rail 2 is fixedly arranged on the base plate. The slider 3 is slidably mounted on the air cushion guide rail 2. The adjustable spring fixing bracket 5 is arranged perpendicular to the direction of the air cushion guide rail 2. The adjustable spring fixing bracket 5 is connected to the slider 3 on the air cushion guide rail 2 via the horizontal limiting tension and compression spring module 4. A vision module for shooting is also set on the adjustable spring fixing bracket 5. The non-inertial reference frame motion module 6 is connected to the adjustable spring fixing bracket 5, which is used to drive the adjustable spring fixing bracket 5 to move along the air cushion guide rail 2. A digital push-pull force gauge 7 is also installed above the air cushion guide rail 2, and the probe end of the digital push-pull force gauge 7 is connected to the slider 3. The contact and force direction between the probe end of the digital push-pull force gauge 7 and the slider 3 are parallel to the direction of the air cushion guide rail 2.

[0028] The air cushion guide has scale markings on both sides and support brackets at both ends, allowing it to be placed on a stable, level table.

[0029] like Figure 1As shown, it also includes a micro-sound air pump 1. The air outlet of the micro-sound air pump 1 is connected to one end of the air inlet of the air cushion guide rail 2 via a hose. It is used to control the inflation and deflation of the air cushion guide rail 2 and adjust the sliding friction of the slider 3 on the air cushion guide rail 2. After the micro-sound air pump 1 is turned on, a stable air film is formed on the surface of the air cushion guide rail 2, and the slider 3 floats on the air cushion guide rail, with minimal friction when moving.

[0030] An adjustable spring fixing bracket 5 spans across both sides of the air cushion guide rail 2. The ends of the adjustable spring fixing bracket 5 on both sides of the air cushion guide rail 2 are connected to the middle of the slider 3 via a limiting tension / compression spring module 4. One end of the slider 3 is used to connect to the detection end of the digital display push-pull force gauge 7. The adjustable spring fixing bracket 5 is connected to the symmetrically arranged limiting tension / compression spring modules 4 on both sides, forming a horizontal stable structure.

[0031] like Figure 2 As shown, the non-inertial reference frame motion module 6 includes a microcontroller 6a, a driver 6b, a linear motor 6c, and a DC switching power supply 6d. The microcontroller 6a and the DC switching power supply 6d are electrically connected to the driver 6b, and the driver 6b is electrically connected to the linear motor 6c. The linear motor 6c is located below the air cushion guide rail 2, and the moving output end of the linear motor 6c is fixedly connected to the aluminum plate 5a of the adjustable spring fixing bracket 5.

[0032] The limiting direction of the air cushion guide rail 2 is parallel to the movement direction of the moving output end of the linear motor 6c. The air cushion guide rail 3 is directly above the linear motor 6c, and the one-dimensional motion limiting axes of the two can be in the same vertical plane.

[0033] The driver is powered by a DC switching power supply and controlled by a microcontroller, driving a linear motor to achieve precise motion. The microcontroller's output signal is programmed via computer software, allowing for arbitrary motion forms such as simple harmonic motion, triangular waveform motion, and square wave motion by adjusting the number of signal waves. An adjustable spring mounting bracket allows for the fixing and adjustment of the spring's suspension point position.

[0034] Specifically, the microcontroller 6a controls the linear motor 6c through the driver 6b, which drives the adjustable spring fixing bracket 5 to move along a straight line parallel to the air cushion guide rail 2 and perform periodic motions such as simple harmonic waves, triangular waves or square waves, thereby applying a controllable inertial force to the slider in a non-inertial reference frame that is stationary relative to this device.

[0035] like Figure 3As shown, the adjustable spring fixing bracket 5 includes an aluminum plate 5a, a perforated aluminum angle 5b, and a spring suspension slider 5c. The aluminum plate 5a is strip-shaped and is arranged horizontally below the air cushion guide rail 2 perpendicular to it. A perforated aluminum angle 5b is fixed on both ends of the aluminum plate 5a. A spring suspension slider 5c is installed on the perforated aluminum angle 5b and can move up and down. The spring suspension slider 5c is used to limit one end of the tension and compression spring module 4.

[0036] The screw holes of the aluminum sheet are rigidly connected to the linear motor by screws, and the waist holes of the aluminum sheet are connected to the perforated angle aluminum by screws and nuts. The perforated angle aluminum is symmetrically fixed and its position is adjustable.

[0037] In practice, the aluminum plate 5a is rigidly fixed to the slide of the linear motor 6c by M4 screws, and two parallel waist holes with a length of 60 mm are opened, allowing for continuous adjustment of ±30 mm.

[0038] The perforated aluminum angle 5b is L-shaped. One side of the L-shape of the perforated aluminum angle 5b is locked to the waist hole by screws and nuts. The other side of the L-shape has a sliding groove to adjust the height of the spring-suspended slider 3. The height of the spring suspension point is consistent with the height of the connection point on the slider 3 used to connect the limit tension spring module 4, so that the slider 3 is kept in the horizontal plane under force, realizing independent vertical and horizontal adjustment.

[0039] In the spring fixing bracket, the strip direction of the aluminum plate 5a is perpendicular to the direction of the air cushion guide rail 2. One side of the perforated aluminum angle 5b is fixed on the waist hole of the aluminum plate 5a, and the other side is fixed at the same height as the center of mass of the slider 3. The vertical distance between the perforated aluminum angle 5b and the air cushion guide rail 2 is adjusted, and the two perforated aluminum angles 5b are symmetrically fixed.

[0040] In the initial state, the adjusting spring-suspended sliders 5c are symmetrically distributed on both sides of slider 3, so that the system potential energy presents a double potential well. Slider 3 moves in the potential field of the double potential well, and the force on it is measured in real time by a digital display push-pull force gauge 7; its trajectory is captured in real time by a camera image acquisition module 9 fixed to the top of the system by a camera bracket 8.

[0041] like Figure 4 As shown, the limiting tension / compression spring module 4 includes a spring body 4a, a rigid extension rod 4b, a limiting sleeve 4c, and a hook 4d. One end of the spring body 4a and one end of the rigid extension rod 4b are fixedly connected and together fitted inside the limiting sleeve 4c. The other end of the spring body 4a extends out of the limiting sleeve 4c and is fixedly connected to the limiting sleeve 4c. After the other end of the spring body 4a extends out of the limiting sleeve 4c, it is connected to the spring suspension slider 5c of the adjustable spring fixing bracket 5 via the first hook 4d. The other end of the rigid extension rod 4b is connected to the slider 3 via the second hook 4d.

[0042] The connection points of the limit tension and compression spring module 4, which are respectively connected to the slider 3 and the adjustable spring fixing bracket 5, are on the same horizontal plane.

[0043] The spring body 4a is a pre-stretched piano wire spring with a spring constant k of 25 N / m and an original length of 0.12 m. The rigid extension rod 5 is a carbon fiber tube with a mass of less than 2 g. The inner diameter of the limiting sleeve 4c is slightly larger than the outer diameter of the spring, and both ends are sealed with PTFE gaskets to limit lateral deformation and reduce friction. Each end of the limiting tension / compression spring module 4 has a hook 4d, which connects to... Figure 1 The middle slider 3 and the adjustable spring fixing bracket 5.

[0044] In practice, the micro-pump 1 is connected to the air cushion guide rail 2. Vertically, from top to bottom, the camera image acquisition module 9, the slider 3, the air cushion guide rail 2, and the non-inertial reference frame motion module 6 are arranged sequentially. The track direction of the air cushion guide rail 2 is consistent with and aligned with the track direction of the linear motor 6c. An adjustable spring fixing bracket 5 is fixed perpendicularly to the track direction on the slide of the linear motor 6c.

[0045] The device of the present invention has three modes: quasi-static mode, dynamic mode, and chaotic mode.

[0046] In the quasi-static mode, the linear motor 6c does not work and remains stationary. The slider 3 is connected to the digital push-pull force gauge 7. The digital push-pull force gauge 7 drives the slider 3 to move and measures the force it is subjected to.

[0047] In dynamic and chaotic modes, the digital push-pull force gauge 7 remains stationary, and the microcontroller 6a, driver 6b, linear motor 6c, and DC switching power supply 6d are connected in sequence. The linear motor 6c drives the adjustable spring fixing bracket 5 to move in a direction parallel to the air cushion guide rail 2.

[0048] The adjustable spring fixing bracket 5 is driven to generate programmable motion. The two limit tension and compression spring modules 4 apply periodic inertial force to the slider 3, causing it to vibrate in the double potential well field, forming typical hysteresis, bifurcation and chaotic phenomena. The images are collected by the camera image acquisition module and then imported into the video analysis software for position tracking and data analysis. Force-displacement hysteresis curves, phase diagrams and Poincaré sections are plotted to observe the periodic, quasi-periodic and chaotic states of the system.

[0049] Driven by a linear motor 6c, the adjustable spring fixing bracket 5 reciprocates between two ends of the air cushion guide rail 2, near or far from the slider 3. During this reciprocating movement, each end of the slider 3 forms a balance position where the spring body 4a of the limiting tension spring module 4 is in a natural, unforced state. Each balance position serves as a potential well, thus forming a double potential well. For each potential well, closer to the slider 3 than the potential well, the spring body 4a of the limiting tension spring module 4 is compressed and deformed, experiencing compressive force and generating potential energy; further away from the slider 3 than the potential well, the spring body 4a of the limiting tension spring module 4 is stretched and deformed, experiencing tensile force and also generating potential energy.

[0050] The dual potential well spring system consists of two symmetrically arranged limiting tension / compression spring modules and a slider, with relative movement between the modules and the slider. As the slider moves along the one-dimensional air cushion guide, it causes the springs to extend and retract. At two specific positions, the springs are in a naturally extended state, forming a dual potential well in a one-dimensional motion space. Alternatively, the dual potential well can be formed by the movement of the limiting tension / compression spring modules.

[0051] like Figure 1 As shown, the spring body 4a of the limiting tension / compression spring module 4 is in a natural state where it is not under force.

[0052] The vision module includes a camera bracket 8 and a camera image acquisition module 9. The camera image acquisition module 9 is fixedly mounted on the adjustable spring fixing bracket 5 via the camera bracket 8 and faces downwards. It is used to capture motion images / videos between the slider 3, the limit tension and compression spring module 4, the adjustable spring fixing bracket 5 and the digital push-pull force gauge 7.

[0053] It also includes a digital push-pull force gauge and a camera image acquisition module. The digital push-pull force gauge is connected to the slider and is used to apply force, allowing the reading of the external force at different positions during quasi-static motion. The camera image acquisition module is mounted on the camera bracket, positioned directly above the system for overhead shooting, and can capture motion images for analysis.

[0054] In specific implementation, the device method of the present invention includes three modes: quasi-static mode, dynamic mode, and chaotic mode. By controlling the linear motor 6c in the non-inertial reference frame motion module 6 to work or not work, and cooperating with the stationary or moving position of the digital display push-pull force gauge 7, the relative movement between the adjustable spring fixing bracket 5 and the slider 3 is adjusted. In addition, by controlling the inflation or deflation of the air cushion guide rail 2, the different friction factors of the slider 3 on the air cushion guide rail 2 are adjusted, thereby generating different experimental test modes and conditions, realizing the experimental simulation of the double potential well spring oscillator, and obtaining the image results of the experimental simulation.

[0055] The experimental test modes include the quasi-static mode of the double-potential-well spring oscillator system, specifically: The methods include the measurement of force hysteresis effect under quasi-static conditions and experiments on bifurcation phenomena.

[0056] The micro-pump 1 does not inflate or deflate the air cushion guide rail 2. The linear motor 6c in the non-inertial reference frame motion module 6 does not work. The adjustable spring fixing bracket 5 keeps its position fixed. The digital push-pull force gauge 7 is used to pull the slider 3 to move along the air cushion guide rail 2, that is, along the Y-axis of the direction of the air cushion guide rail 2. After pulling, the distance y from the slider 3 to the origin is fixed in real time. The origin is the intersection between the central axis of the adjustable spring fixing bracket 5 and the central axis of the air cushion guide rail 2. Specifically, it should be set as the intersection between the spring suspension points on the two perforated angle aluminum 5b on the adjustable spring fixing bracket 5 and the air cushion guide rail 2. The distance from the spring suspension point to the origin can be changed through the waist hole of the adjustable spring fixing bracket 5. The distance from the origin to the slider 3 can be adjusted by pulling the digital push-pull force gauge 7. The two displacements are perpendicular to each other.

[0057] like Figure 5 As shown, the force readings f of the digital push-pull force gauge 7 are read at different distances y while the device is stationary. Based on the force readings f obtained at each different distance y, the reduced force of the push-pull force gauge that keeps the corresponding slider 3 in balance on the air cushion guide rail 2 is calculated using the following formula. A Reduced coordinates of the longitudinal position of the slider track Y : A = f / ( 2kl 0), Y = y / l 0 in, f This indicates the force reading of the digital push-pull force gauge 7, that is, the external force applied to the slider 3 by the digital push-pull force gauge 7 as it pulls the slider 3. y This represents the distance from slider 3 to the origin. k Indicates the spring constant. l 0 indicates the length of the spring in its natural state in the limit tension / compression spring module 4; A corresponding force parameter A and a coordinate parameter Y As a group, the final result consists of all the corresponding force parameters of each group. A and coordinate parameters Y Plotting statically reduced position and force relationships Y - A Images, as experimental results of force hysteresis characteristics.

[0058] In the quasi-static mode, slider 3 remains stationary. The net external force on slider 3 at this time... F = 0.

[0059] In specific implementation, further, the perforated angle aluminum 5b is moved along the waist hole of the aluminum plate 5a, that is, along... X Axis movement ( X(The shaft is along the direction of air cushion guide rail 2) and fixed with screws and nuts. Use a ruler to measure the distance from the fixing points of the two springs to the central axis of air cushion guide rail 2. x , x This indicates the vertical distance from the suspension point of the spring on the adjustable spring mounting bracket 5 to the central axis of the air cushion guide rail 2, with the two spring mounting points remaining symmetrical on both sides. At different spacings... x The reading of the digital push-pull force gauge is shown below. f Distance from slider to origin y The relationship is determined by the following formula: Spacing x Convert to lateral reduced coordinates of the support X And the horizontal reduced coordinates of the support X and the reduction of push and pull forces A Reduced coordinates of the longitudinal position of the slider track Y The relationship is established as follows: X = x / l 0 A = Y [1-( X 2 + Y 2 ) -1 / 2 ] Where [] represents the reduced ratio of the spring's tension or compression; The final fixed device corresponds to the reduced lateral coordinates of the support. X The reduced lateral coordinate parameters of different supports can be obtained. X Below Y - A Images show the characteristics of force hysteresis curves under different spring spacings. Quasi-static force-displacement relationship curves of a double-potential-well spring oscillator are obtained, and typical force hysteresis effects and bifurcation phenomena are observed.

[0060] The experimental test modes include the dynamic modes of the double-potential-well spring oscillator system, specifically: The micro-sound air pump 1 does not inflate or deflate the air cushion guide rail 2. The digital push-pull force gauge 7 remains fixed and controls the linear motor 6c in the non-inertial reference frame motion module 6 to work. The linear motor 6c drives the adjustable spring fixing bracket 5 to move parallel to the air cushion guide rail 2, and then pulls the slider 3 along the direction of the air cushion guide rail 2 through the limit tension spring module 4, that is, along the Y-axis where the direction of the air cushion guide rail 2 is located. After the movement, the distance y between the slider 3 and the origin is fixed in real time. The origin is the intersection between the central axis of the adjustable spring fixing bracket 5 and the central axis of the air cushion guide rail 2. The specific setting of the origin is consistent with the quasi-static mode.

[0061] like Figure 5 As shown, at different distances y Images are acquired by the vision module in a static state, and the acceleration of the support and slider is obtained through image processing, based on different distances. yThe acceleration of the corresponding slider is calculated using the following formula to obtain the equivalent force acting on the slider. A’ The coordinates of the longitudinal position of the slider track are reduced. Y : A' = m ×( a 0 +a ) / ( 2kl 0), Y = y / l 0 in, m For the mass of slider 3, a 0 represents the acceleration of the origin, which is a known quantity obtained from the image acquired by the vision module through image analysis and processing. a This represents the acceleration of the slider relative to the origin. y This represents the distance from slider 3 to the origin. k Indicates the spring constant. l 0 indicates the length of the spring in its natural state in the limit tension / compression spring module 4; A corresponding force parameter A’ and a coordinate parameter Y As a group, the final result consists of all the corresponding force parameters of each group. A’ and coordinate parameters Y Plotting the dynamically reduced position-force relationship Y - A' Images, as experimental results of force hysteresis characteristics.

[0062] In dynamic mode, the spring fixing bracket is fixed on the linear motor controlled by the microcontroller. Vertically, from top to bottom, the components are the vision module, the air cushion guide rail, the linear motor, and the spring fixing end fixed on it.

[0063] Controlling the motion mode of the linear motor to subject the slider to periodic reciprocating inertial force. f inertial The origin is set at the intersection of the line connecting the air cushion guide rail and the suspension points of the two perforated aluminum springs. The slider reciprocates relative to the origin under the action of inertial force.

[0064] Analyzing the image yields acceleration information for the origin and the slider's position, which is then used to measure dynamic force hysteresis characteristics. The net force F acting on the mass block in a non-inertial frame satisfies: F = f inertial - 2 ky (1- l 0 / (( x 2 + y 2 ) 1 / 2 )) Using different drive modes, the coordinates are reduced laterally on a given support. X The following is equivalent to:A’ The distance Y from the equivalent position of the slider relative to the origin first decreases, then increases, and then decreases again as the distance Y increases; this trend is described by the principle above. The reduced lateral coordinates of the support can also be changed for observation and measurement. X The influence of system hysteresis and bifurcation, and the evolution of the experimental system from a double-well to a wellless state.

[0065] The above-mentioned lateral reduced coordinate parameters of the support X Sum of force parameters A’ and a coordinate parameter Y The relationship is established as follows: X = x / l 0 A' = Y [( X 2 +Y 2 ) -1 / 2 -1] Where [] represents the reduced ratio of spring tension or compression.

[0066] The final fixed device corresponds to the reduced lateral coordinate parameters of the support. X The reduced lateral coordinate parameters of different supports can be obtained. X Below Y - A' Images were used to study the dynamic reduced position-force relationship curves of a double-potential-well spring oscillator, and to observe the force hysteresis effect and bifurcation phenomenon.

[0067] The experimental test mode includes a chaotic mode of the double potential well spring oscillator system. Specifically, the digital push-pull force gauge 7 remains stationary, controlling the linear motor 6c in the non-inertial reference frame motion module 6 to work. The linear motor 6c drives the adjustable spring fixing bracket 5 to move parallel to the air cushion guide rail 2, which in turn pulls the slider 3 along the direction of the air cushion guide rail 2 via the limit tension spring module 4. During the movement, the air cushion guide rail 2 is inflated and deflated in real time using a micro-sound air pump 1, randomly adjusting the friction force of the slider 3 on the air cushion guide rail 2, i.e., along the direction of the air cushion guide rail 2. Y The axis moves, and the distance between slider 3 and the origin is fixed in real time after the movement. y The origin is the intersection of the central axis of the adjustable spring fixing bracket 5 and the central axis of the air cushion guide rail 2; the specific setting of the origin is consistent with the quasi-static mode.

[0068] like Figure 5 As shown, at different distances y Images are acquired by the vision module in a static state, and the acceleration of the support and slider is obtained through image processing, based on different distances. y The acceleration of the corresponding slider is calculated using the following formula to obtain the longitudinal reduced velocity relative to the corresponding slider track. V Reduced coordinates of the longitudinal position of the slider track Y: V = δv / (2 kl 0), Y = y / l 0 in, y This represents the distance from slider 3 to the origin. k Indicates the spring constant. l 0 indicates the length of the spring in its natural state in the limit tension / compression spring module 4; v This indicates the speed of slider 3 relative to the origin along the direction of air track 2. δ is the damping coefficient.

[0069] The corresponding longitudinal reduced speed of a slider track V and a coordinate parameter Y As a group, the longitudinal velocity of all corresponding slider tracks in each group is finally reduced. V and coordinate parameters Y Plotting the dynamically reduced position-velocity relationship Y - V Images, as experimental results for studying the broadband phase diagram characteristics of a bistable oscillator.

[0070] In chaotic mode, a linear damping term is introduced into the system. The origin is set at the intersection of the line connecting the air cushion guide and the suspension points of the two perforated aluminum angle springs. The slider reciprocates relative to the origin under the action of inertial force. The motion mode of the linear motor is controlled to regulate the motion state of the non-inertial frame and generate inertial force. f inertial .

[0071] Analyzing the image yields acceleration information for the origin and the slider's position. The net force F acting on the mass block in the non-inertial frame satisfies: F = f inertial - 2 ky (1- l 0 / (( x 2 + y 2 ) 1 / 2 )) - δv in, v This indicates the speed of slider 3 relative to the origin along the direction of air track 2. δ is the damping coefficient.

[0072] Draw the slider using the data obtained from tracking Y - VA scatter plot is a Poincaré cross-section of the system in that phase space. If the points on the Poincaré cross-section form fixed points or closed curves, it indicates that the system is in a periodic or quasi-periodic state. A TY scatter plot of the slider can also be plotted, i.e., the system's time-displacement plot. Plot the experimental Poincaré cross-section phase diagram and time-displacement plot.

[0073] The above-mentioned lateral reduced coordinate parameters of the support X Force parameters A’ and a coordinate parameter Y The relationship is established as follows: X = x / l 0 V = Y [( X 2 + Y 2 ) -1 / 2 -1]- A’ A’ = m ×( a 0+ a ) / (2 kl 0) in, m For the mass of slider 3, a 0 represents the acceleration of the origin, which is a known quantity obtained from the image acquired by the vision module through image analysis and processing. a The value represents the acceleration of the slider relative to the origin, and [] represents the reduced ratio of the spring's tension or compression. V This represents the longitudinal reduced velocity of the block track.

[0074] The working principle of this invention is as follows: like Figure 5 As shown, the characteristic behavior of the double-potential-well spring oscillator system is determined by the distance between the suspension points of the two springs, half of which is represented by the symbol. x Indicates. The adjustable spring-loaded bracket 5 on the linear motor 6c is moved symmetrically, so that... x From the original length of the tension / compression spring module 4 (greater than the limit) l The distance gradually transitions from 0 to equal to and then to less than its original length. In quasi-static mode, the displacement y of slider 3 and the external force f are recorded simultaneously. In dynamic mode, the velocity of slider 3 is recorded simultaneously. v This yields a set of displacement-external force diagrams and displacement-velocity phase diagrams. When xWhen the potential energy is large, it is a single potential well, and there is no hysteresis line in the displacement-tension curve; when x is less than the critical value, the potential energy becomes a double potential well, and the displacement-tension curve shows a typical hysteresis line. The displacement-velocity phase diagram will enter chaos through period-doubling bifurcation as the driving amplitude and angular frequency change, which intuitively reflects the hysteresis and bifurcation characteristics of the double potential well system.

[0075] Keeping x constant, adjust the spring constant 4a of the spring body. k and the original length of the limit tension and compression spring module 4 l 0, repeat quasi-static and dynamic experiments. Spring constant. k Increasing the length will increase the potential barrier energy, increase the force required to cross the critical state, and also increase the hysteresis loop area; the original length of the limiting tension / compression spring module l Decreasing the distance between the two potential wells to 0 will shrink the bifurcation threshold, making chaos more likely to occur. This allows for the quantification of the impact of various parameters on the chaotic evolution path.

[0076] In the quasi-static mode, a digital push-pull force gauge 7 is used to slowly pull the slider 3 along the guide rail direction, and the equilibrium position is recorded. y external forces f Using the dimensionless formula: A = Y [1-( X 2 + Y 2 ) -1 / 2 ] do Y - A Curve. For a given... x The slider 3 is pulled back and forth to form a closed loop, the area of ​​which is... S Proportional to energy dissipation; gradually increasing x The point where the hysteresis line area changes abruptly is the bifurcation point. This can be compared with the theoretical critical value to verify the quantitative relationship between hysteresis and bifurcation.

[0077] In dynamic mode, the microcontroller 6a outputs specific waveforms (sine, triangular, square wave) to drive the linear motor 6c, causing the non-inertial reference frame motion module 6 to generate controllable acceleration. a 0. The equivalent inertial force acting on the slider. f inertial =- ma 0, by controlling its amplitude A With angular frequency ω Adjustable independently. When fixed A ,scanning ω At this time, the system successively exhibits single-cycle, double-cycle, multi-cycle, and chaotic phenomena; when fixed... ω ,scanning A When this happens, the chaotic critical threshold can be obtained. A CBy comparing with the theoretical Melnikov criterion, the critical condition for a double potential well system to enter chaos is verified.

[0078] The displacement-velocity of the slider is plotted using data obtained from software tracking. y , v A scatter plot, i.e., the Poincaré cross-section of the system in the phase space, indicates that the system is in a periodic or quasi-periodic state if the points on the Poincaré cross-section form a fixed point or a closed curve. The second-order differential equation is solved using the fourth-order Runge-Kutta method. F = f inertial - 2 ky (1- l 0 / (( x 2 + y 2 ) 1 / 2 )) - δv Theoretical time-displacement curves and Poincaré phase diagrams can be plotted and compared with experimentally measured images.

[0079] The specific implementation process of this embodiment is as follows: like Figure 1 The complete schematic diagram is shown. After powering the microcontroller 6a, the micro-pump 1 is started. After a stable air film is formed on the surface of the air cushion guide rail 2, the slider 3 is gently placed in the center of the guide rail. Then, the linear motor 6c is driven to reciprocate by the software. The position of the waist hole of the adjustable spring fixing bracket 5 is adjusted so that the suspension points of the limit tension spring modules 4 on both sides are symmetrically distributed. The position change of the slider 3 under quasi-static tension can be measured, or the periodic oscillation or chaotic oscillation of the slider 3 can be observed in the real-time picture of the camera image acquisition module 9.

[0080] When measuring the "influence of suspension spacing on hysteresis and bifurcation," symmetrical adjustments were made using aluminum sheet 5a to achieve... x The distance was gradually increased from 10 mm to 30 mm. Each time the distance was changed, the slider was slowly pushed and pulled using a digital push-pull force gauge 7, and the displacement-tension was recorded. y - f The data is used to create a displacement-external force hysteresis loop.

[0081] When measuring the "influence of suspension spacing on the chaos threshold", under fixed conditions... x Under these conditions, the non-inertial reference frame motion module 6 is made to perform sinusoidal motion, and the displacement-time (DTM) data collected by the camera is recorded. y - t The data was processed by video analysis software to obtain displacement-velocity (VoV). y - v Phase diagrams allow observation of phase changes. x The system undergoes a complete evolution from single-potential-well periodic motion to double-potential-well periodic motion, and then to period-doubling bifurcation and chaos.

[0082] When measuring the "influence of spring parameters on the chaos threshold", the limit tension and compression spring module 4 with different spring coefficients is replaced, and the above push-pull and drive experiments are repeated. It can be found that the critical amplitude of the system entering chaos is different under the same drive amplitude, which can be compared with the theoretical value.

[0083] When measuring the "control of nonlinear response by driving waveform and damping", the output signal of the microcontroller 6a was changed to a square wave; then, adjustable damping plates of different areas were attached to the slider to change the damping coefficient. δ Real-time phase diagrams show that strange attractors appear in low-damped systems, while high-damped systems revert to single-cycle motion. This allows for a quantitative study of the damping effect on chaos suppression.

[0084] The experimental setup of a double-potential-well spring oscillator allows for the simulation and calculation of its nonlinear targeted energy transfer characteristics. It can generate large-amplitude vibrations over a wide driving frequency range, is insensitive to changes in the structure's intrinsic frequencies, and exhibits broadband robustness. This characteristic overcomes the limitations of traditional single-frequency linearly tuned mass dampers, responding not only to vibrations of a single frequency. Even under small initial vibrations, the device can activate large-amplitude inter-well motion under specific conditions. This has a very powerful effect in the application of mechanical vibration suppressors, enabling highly efficient vibration suppression. It provides a fundamental research platform for flutter suppression in aerospace structures, wind and earthquake resistance in high-rise buildings and bridges, and precision manufacturing platforms for ultra-stable wings.

[0085] Therefore, this invention comprehensively and accurately experimentally demonstrates various nonlinear dynamic phenomena of spring oscillators in double potential well systems, providing rich experimental content and powerful tools for physics experiments, and has broad application prospects.

Claims

1. An experimental setup for a double-potential-well spring oscillator system, characterized in that: The device includes an air cushion guide rail (2), a slider (3), a limiting tension / compression spring module (4), an adjustable spring fixing bracket (5), a non-inertial reference frame motion module (6), a digital display push / pull force gauge (7), and a micro-sound air pump (1). The air cushion guide rail (2) and the non-inertial reference frame motion module (6) are arranged on a horizontal base plate. The air cushion guide rail (2) is fixedly arranged on the base plate, and the slider (3) is slidably mounted on the air cushion guide rail (2). The adjustable spring fixing bracket (5) is arranged perpendicular to the direction of the air cushion guide rail (2) and is adjustable. The spring fixing bracket (5) is connected to the slider (3) on the air cushion guide rail (2) via the horizontal limiting tension spring module (4). A vision module for shooting is also provided on the adjustable spring fixing bracket (5). The non-inertial reference frame motion module (6) is connected to the adjustable spring fixing bracket (5) to drive the adjustable spring fixing bracket (5) to move along the direction of the air cushion guide rail (2). A digital push-pull force gauge (7) is also provided on the air cushion guide rail (2). The detection end of the digital push-pull force gauge (7) is connected to the slider (3).

2. The experimental apparatus for a double-potential-well spring oscillator system according to claim 1, characterized in that: It also includes a micro-sound pump (1), the air outlet of which is connected to the air cushion guide rail (2) via a hose, for controlling the inflation and deflation of the air cushion guide rail (2) and adjusting the sliding friction of the slider (3) on the air cushion guide rail (2).

3. The experimental apparatus for a double-potential-well spring oscillator system according to claim 1, characterized in that: The adjustable spring fixing bracket (5) spans across both sides of the air cushion guide rail (2), and the ends of the adjustable spring fixing bracket (5) on both sides of the air cushion guide rail (2) are connected by a limiting tension spring module (4) and a slider (3).

4. The experimental apparatus for a double-potential-well spring oscillator system according to claim 1, characterized in that: The non-inertial reference frame motion module (6) includes a microcontroller (6a), a driver (6b), a linear motor (6c), and a DC switching power supply (6d). The microcontroller (6a) and the DC switching power supply (6d) are electrically connected to the driver (6b), and the driver (6b) is electrically connected to the linear motor (6c). The moving output end of the linear motor (6c) is fixedly connected to the adjustable spring fixing bracket (5).

5. The experimental apparatus for a double-potential-well spring oscillator system according to claim 1, characterized in that: The adjustable spring fixing bracket (5) includes an aluminum plate (5a), a perforated aluminum angle (5b), and a spring suspension slider (5c). The aluminum plate (5a) is strip-shaped and is arranged across the bottom of the air cushion guide rail (2). A perforated aluminum angle (5b) is fixed on both ends of the aluminum plate (5a). A spring suspension slider (5c) is installed on the perforated aluminum angle (5b) and can move up and down. The spring suspension slider (5c) is used to limit one end of the tension and compression spring module (4).

6. The experimental apparatus for a double-potential-well spring oscillator system according to claim 1, characterized in that: The limiting tension and compression spring module (4) includes a spring body (4a), a rigid extension rod (4b), a limiting sleeve (4c), and a hook (4d). One end of the spring body (4a) and one end of the rigid extension rod (4b) are fixedly connected and together fitted into the limiting sleeve (4c). The other end of the spring body (4a) extends out of the limiting sleeve (4c) and is fixedly connected to the limiting sleeve (4c). After the other end of the spring body (4a) extends out of the limiting sleeve (4c), it is connected to the adjustable spring fixing bracket (5) via the hook (4d). The other end of the rigid extension rod (4b) is connected to the slider (3) via the hook (4d).

7. A method for experimentally testing a double-potential-well spring oscillator system applied to any one of the experimental apparatuses described in claims 1-6, characterized in that: The method includes three modes: quasi-static mode, dynamic mode, and chaotic mode. By controlling the non-inertial reference frame motion module (6) to work or not work and cooperating with the stationary or moving position of the digital push-pull force gauge (7), the relative movement between the adjustable spring fixing bracket (5) and the slider (3) is adjusted. In addition, by controlling the inflation or deflation of the air cushion guide rail (2), the slider (3) is adjusted to different friction conditions on the air cushion guide rail (2), thereby generating different experimental test modes, realizing the experimental simulation of the double potential trap spring oscillator, and obtaining the image results of the experimental simulation.

8. The experimental method for a double-potential-well spring oscillator according to claim 7, characterized in that: The experimental testing mode includes the quasi-static mode, specifically: The non-inertial reference frame motion module (6) is not working, the adjustable spring fixing bracket (5) is kept in a fixed position, and the digital push-pull force gauge (7) is used to pull the slider (3) to move along the air cushion guide rail (2). After pulling, the distance y from the slider (3) to the origin is fixed in real time. The origin is the intersection between the adjustable spring fixing bracket (5) and the air cushion guide rail (2). Read the force reading f of the digital push-pull force gauge (7) under stationary conditions at different distances y. Calculate the reduced force of the push-pull force gauge that keeps the corresponding slider (3) in balance on the air cushion guide rail (2) according to the force reading f obtained at each different distance y using the following formula. A Reduced coordinates of the longitudinal position of the slider track Y : A=f / ( 2kl 0)， Y=y / l 0 in, f This indicates the force reading of the digital push-pull force gauge (7). y This represents the distance from slider (3) to the origin. k Indicates the spring constant. l 0 indicates the length of the spring in its natural state in the limit tension spring module (4).

9. The experimental method for a double-potential-well spring oscillator according to claim 7, characterized in that: The experimental testing mode includes the dynamic mode, specifically: The digital push-pull force gauge (7) remains fixed, controlling the non-inertial reference frame motion module (6) to work, driving the adjustable spring fixing bracket (5) to move parallel to the air cushion guide rail (2), and then pulling the slider (3) along the direction of the air cushion guide rail (2) via the limit tension spring module (4). After the movement, the distance y between the slider (3) and the origin is fixed in real time, and at different distances... y Images are acquired by the vision module in a static state, and the acceleration of the support and slider is obtained through image processing, based on different distances. y The acceleration of the corresponding slider is calculated using the following formula to obtain the equivalent force acting on the slider. A’ The coordinates of the longitudinal position of the slider track are reduced. Y : A'=m ×( a 0 +a ) / ( 2kl 0)， Y=y / l 0 in, m For the mass of slider (3), a 0 represents the acceleration during the movement from the origin. a This represents the acceleration of the slider relative to the origin. y This represents the distance from slider (3) to the origin. k Indicates the spring constant. l 0 indicates the length of the spring in its natural state in the limit tension / compression spring module (4); Finally, all the corresponding force parameters A’ and coordinate parameters Y Plotting the dynamically reduced position-force relationship Y-A' Image, as a force hysteresis feature.

10. The experimental method for a double-potential-well spring oscillator according to claim 7, characterized in that: The experimental testing mode includes the chaotic mode, specifically: The digital push-pull force gauge (7) remains fixed, controlling the non-inertial reference frame motion module (6) to work, driving the adjustable spring fixing bracket (5) to move parallel to the air cushion guide rail (2), and then pulling the slider (3) along the direction of the air cushion guide rail (2) through the limit tension spring module (4). During the movement, the air cushion guide rail (2) is inflated and deflated in real time to adjust the friction force of the slider (3) on the air cushion guide rail (2). After the movement, the distance between the slider (3) and the origin is fixed in real time. y At different distances y Images are acquired by the vision module in a static state, and the acceleration of the support and slider is obtained through image processing, based on different distances. y The acceleration of the corresponding slider is calculated using the following formula to obtain the longitudinal reduced velocity relative to the corresponding slider track. V Reduced coordinates of the longitudinal position of the slider track Y : V=δv / (2 kl 0)， Y = y / l 0 in, y This represents the distance from slider (3) to the origin. k Indicates the spring constant. l 0 indicates the length of the spring in its natural state in the limit tension / compression spring module (4); v This indicates the velocity of the slider (3) relative to the origin along the direction of the air track (2). δ The damping coefficient; Finally, the longitudinal velocity is reduced by all corresponding slider tracks. V and coordinate parameters Y Plotting the dynamically reduced position-velocity relationship YV Images are used to study the broadband phase diagram characteristics of bistable oscillators.

Images