Double-well spring oscillator system experimental device and method

Abstract

The application discloses a double-potential well spring oscillator system experimental device and method. A gas cushion guide rail and a non-inertial reference system motion module are arranged on a horizontal bottom plate, a sliding block is slidably arranged on the gas cushion guide rail, and an adjustable spring fixing support is arranged perpendicularly to the direction of the gas cushion guide rail. The adjustable spring fixing support is connected to the horizontal limiting tension and compression spring module and the gas cushion guide rail through the gas cushion guide rail, and the adjustable spring fixing support is provided with a visual module. A digital display push-pull force gauge is further arranged above the gas cushion guide rail, and the detection end of the digital display push-pull force gauge is connected to the sliding block. The method comprises a quasi-static mode, a dynamic mode and a chaotic mode, the movement between the adjustable spring fixing support and the sliding block is adjusted, the sliding block is adjusted to have different friction on the gas cushion guide rail, different experimental test modes are generated, and experimental simulation is realized. The application comprehensively and accurately displays various nonlinear dynamic phenomena of the spring oscillator in the double-potential well system, provides rich experimental content and a powerful tool for college physics experimental teaching, and has a wide application prospect.

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 methods.

[0005] The technical scheme adopted by the present application is as follows:

[0006] I. A double-potential-well spring oscillator system experimental device:

[0007] 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.

[0008] 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.

[0009] The non-inertial reference system movement module is connected with the adjustable spring fixing support and used to drive the adjustable spring fixing support to move along the air cushion guide rail.

[0010] The air outlet of the micro-sound air pump is connected with the air cushion guide rail through a hose, which is used to control the inflation and deflation of the air cushion guide rail and adjust the sliding friction of the sliding block on the air cushion guide rail.

[0011] The adjustable spring fixing support is arranged across the air cushion guide rail, and the ends of the adjustable spring fixing support on both sides of the air cushion guide rail are connected with the sliding block through the limiting tension and compression spring module.

[0012] The non-inertial reference system movement module includes a single-chip microcomputer, a driver, a linear motor and a DC switching power supply.

[0013] The adjustable spring fixing support includes an aluminum plate, a hole angle aluminum and a spring suspension sliding block.

[0014] The limiting tension and compression spring module includes a spring body, a rigid extension rod, a limiting sleeve and a hook.

[0015] II. Experimental method of the double-potential well spring oscillator system experimental device:

[0016] The method includes three modes of quasi-static mode, dynamic mode and chaotic mode.

[0017] By controlling the working or non-working of the non-inertial reference system movement module and cooperating with the static or movement of the digital push-pull force gauge, the relative movement between the adjustable spring fixing support and the sliding block is adjusted.

[0018] After the experimental simulation of the double-well spring oscillator is obtained at the end of the method, the characteristics of the nonlinear targeted energy transfer can be simulated and calculated, which can produce larger amplitude vibrations in a wide range of driving frequencies, and is not sensitive to the change of the intrinsic frequency of the structure, and has wide frequency robustness. This characteristic can break through the limitations of traditional single-frequency linear tuned mass damper and respond to not only single-frequency vibration. Even if the device is subjected to a small initial vibration, it can activate its large-amplitude inter-well motion under certain conditions. This has a very strong effect in the application of mechanical vibration suppressors and can achieve efficient vibration suppression. It provides a research platform for the flutter suppression of aerospace structures, wind and earthquake resistance of high-rise buildings and bridges, and precision manufacturing platforms for wing super-stability.

[0019] The experimental test mode includes the quasi-static mode, in particular:

[0020] The non-inertial reference system motion control module is not working, the adjustable spring fixed support keeps the position fixed, the digital push-pull gauge is used to pull the slider to move along the air cushion guide rail, and the distance y from the slider to the original point is fixed in real time after pulling.

[0021] The force reading f of the digital push-pull gauge is read in the static state of different distances y, and the corresponding reduced force of the push-pull gauge of the slider balanced on the air cushion guide rail is obtained according to the force reading f obtained at each different distance y through the following formula A And the longitudinal position reduced coordinate of the slider track Y :

[0022] A = f / ( 2kl 0), Y = y / l 0

[0023] Wherein, f represents the force reading of the digital push-pull gauge, y represents the distance from the slider to the original point, k represents the spring stiffness coefficient, l 0 represents the length of the spring in the natural state of the limiting tension and compression spring module;

[0024] The experimental test mode includes the dynamic mode, in particular:

[0025] The digital push-pull gauge keeps fixed, the non-inertial reference system motion control module works, drives the adjustable spring fixed support to move along the air cushion guide rail, and then drives the slider to move along the air cushion guide rail through the limiting tension and compression spring module. The distance y between the slider and the original point is fixed in real time after moving. yThe image is collected by the vision module in the static state, and the acceleration of the bracket and the slider is obtained through image processing, and the acceleration of the slider corresponding to each distance y The corresponding acceleration of the slider is obtained by the following formula A’ and the longitudinal position of the slider track is reduced to the coordinate Y :

[0026] A' = m ×( a 0 +a ) / ( 2kl 0), Y = y / l 0

[0027] wherein, m is the mass of the slider, a 0 is the acceleration of the origin movement a , indicating the acceleration of the slider relative to the origin, y , indicating the distance of the slider to the origin, k , indicating the spring stiffness coefficient, l 0, indicating the length of the spring in the natural state of the limiting tension and compression spring module;

[0028] Finally, all the corresponding force parameters A’ and coordinate parameters Y are drawn to draw the relationship between the dynamic reduced position and the force Y - A' Image as force hysteresis characteristics.

[0029] The experimental test mode includes the chaotic mode, specifically:

[0030] The digital display push-pull force meter remains stationary, controls the non-inertial reference system movement module to work, drives the adjustable spring fixed bracket to move along the air cushion guide rail, and then pulls the slider along the air cushion guide rail through the limiting tension and compression spring module. In the moving process, the air cushion guide rail is inflated and deflated to adjust the friction of the slider on the air cushion guide rail, and after moving, the distance between the slider and the origin is fixed in real time y , the image is collected by the vision module in the static state at different distances y , and the acceleration of the bracket and the slider is obtained through image processing, and the acceleration of the slider corresponding to each distance y The corresponding acceleration of the slider is obtained by the following formula V and the longitudinal position of the slider track is reduced to the coordinate Y :

[0031] V = δv / (2 kl 0), Y = y / l 0

[0032] wherein, y represents the distance of the slider to the origin, k represents the spring stiffness coefficient, l 0 represents the length of the spring in the natural state of the limit tension and compression spring module; v represents the movement speed of the slider along the air cushion guide rail direction relative to the origin, δ is the damping coefficient;

[0033] Finally, the longitudinal reduced speed of all corresponding slider tracks V and the coordinate parameters Y draw the dynamic reduced position and speed relationship Y - V image as a study of the characteristics of the wide frequency phase diagram of the bistable oscillator.

[0034] The present application adopts the symmetrically arranged limit tension and compression spring module, air cushion guide rail, single-chip microcomputer controlled linear motor, digital display push-pull force meter and adjustable support, which are respectively applied to quantitative measurement of hysteresis effect, observation of bifurcation phenomenon, dynamic response analysis under non-inertial force driving and calculation of Poincare section of chaotic behavior in the double potential well system, and constitute a complete nonlinear dynamics experimental device system and experimental method.

[0035] In the present application, the DC power supply box supplies power to provide accurate control of the position of the linear motor; the microphone air pump is connected with the air cushion guide rail, and the movable slider is arranged above; the adjustable support is fixed with the linear motor and is 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 through the screw nut and the sliding groove; the rigid extension rod in the limit tension and compression spring module is placed in the limit sleeve to limit the transverse deformation in the movement.

[0036] The method of the present application includes a quasi-static experimental mode and a dynamic experimental mode. The synchronous belt module is controlled to be stationary, the size of the external force is changed under quasi-static state, the displacement-external force diagram of the slider is plotted, and the hysteresis effect and bifurcation phenomenon can be obviously observed; the displacement, speed and acceleration of the synchronous belt module are accurately controlled, which can be regarded as exerting an inertial force on the mass block in a non-inertial system, and the image analysis software is used to draw and process data to obtain the motion phase diagram under dynamic conditions.

[0037] The present application has the following beneficial effects:

[0038] The present application designs a double potential well spring oscillator experimental device for displaying and obtaining common double stable state and mutation behaviors in complex nonlinear systems, and is suitable for observation and teaching experiment of nonlinear dynamics characteristics, and constitutes a new type of mechanical experimental teaching instrument.

[0039] 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

[0040] Figure 1 This is a schematic diagram of the complete experimental setup for a double-potential-well spring oscillator system.

[0041] Figure 2 This is a schematic diagram of a non-inertial reference frame motion module.

[0042] Figure 3 This is a schematic diagram of an adjustable spring fixing bracket.

[0043] Figure 4 This is a schematic diagram of the limit tension / compression spring module.

[0044] Figure 5 This is a partial top view of the experimental setup for a double-potential-well spring oscillator system.

[0045] 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

[0046] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0047] 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.

[0048] The non-inertial reference system motion module 6 is connected with the adjustable spring fixing support 5, and is used to drive the adjustable spring fixing support 5 to move along the air cushion guide rail 2. A digital push-pull force gauge 7 is further arranged above the air cushion guide rail 2, and a detection end of the digital push-pull force gauge 7 is connected with the sliding block 3. The contact between the detection end of the digital push-pull force gauge 7 and the sliding block 3 is parallel to the direction of the air cushion guide rail 2.

[0049] The air cushion guide rail is marked with a scale on both sides of the guide rail, and is supported by support frames at both ends and placed on a stable horizontal table top.

[0050] As shown in Figure 1 , the device further comprises a micro-sound air pump 1, an air outlet of the micro-sound air pump 1 is connected to one end of an air inlet of the air cushion guide rail 2 through a hose, and is used to control inflation and deflation of the air cushion guide rail 2, so as to adjust the sliding friction of the sliding block 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 sliding block 3 floats on the air cushion guide rail, and the friction is extremely small when moving.

[0051] The adjustable spring fixing support 5 is arranged across the air cushion guide rail 2, and end portions of the adjustable spring fixing support 5 on both sides of the air cushion guide rail 2 are connected with the middle portions of the sliding block 3 and the limiting tension and compression spring modules 4 on the both sides respectively. One end of the sliding block 3 is used to connect with the detection end of the digital push-pull force gauge 7. The limiting tension and compression spring modules 4 arranged symmetrically on the both sides are connected to the adjustable spring fixing support 5, and constitute a horizontal stable structure.

[0052] As shown in Figure 2 , the non-inertial reference system motion module 6 comprises a single-chip microcomputer 6a, a driver 6b, a linear motor 6c and a direct-current switching power supply 6d. The single-chip microcomputer 6a and the direct-current switching power supply 6d are electrically connected with the driver 6b respectively, the driver 6b is electrically connected with the linear motor 6c, the linear motor 6c is located below the air cushion guide rail 2, and a moving output end of the linear motor 6c is fixedly connected with an aluminum plate 5a of the adjustable spring fixing support 5.

[0053] 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 movement limiting axes of the two can be in a common vertical plane.

[0054] The driver is powered by the direct-current switching power supply box and is controlled by the single-chip microcomputer to drive the linear motor to move accurately. The output signal of the single-chip microcomputer is realized by computer software programming, and any form of motion such as simple harmonic motion, triangular wave motion and square wave motion can be realized by adjusting the number of signal waves. The position of the suspension point of the adjustable spring fixed support and the adjustable spring is adjusted.

[0055] 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.

[0056] like Figure 3 As 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.

[0057] 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.

[0058] 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.

[0059] 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.

[0060] 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.

[0061] 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.

[0062] like Figure 4As shown, the limit pull and pressure spring module 4 includes a spring body 4a, a rigid extension rod 4b, a limit 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 sleeved in the limit sleeve 4c, the other end of the spring body 4a extends out of the limit sleeve 4c and is fixedly connected with the limit sleeve 4c, the other end of the spring body 4a extends out of the limit sleeve 4c and is connected with the spring suspension sliding block 5c of the adjustable spring fixed support 5 through the first hook 4d; the other end of the rigid extension rod 4b is connected with the sliding block 3 through the second hook 4d.

[0063] The connection points of the limit pull and pressure spring module 4 connected with the sliding block 3 and the spring suspension sliding block 5c of the adjustable spring fixed support 5 are on the same horizontal plane.

[0064] The spring body 4a is a piano wire spring subjected to pre-stretching treatment, the stiffness coefficient k is 25 N / m, the original length is 0.12 m, the rigid extension rod 5 is a carbon fiber tube with a mass less than 2 g. The inner diameter of the limit sleeve 4c is slightly larger than the outer diameter of the spring, and the two ends are sealed with polytetrafluoroethylene gaskets, which not only limits the transverse deformation, but also reduces the friction. The limit pull and pressure spring module 4 has one hook 4d at each end, which is connected with the sliding block 3 and the adjustable spring fixed support 5 respectively. Figure 1 The sliding block 3 and the adjustable spring fixed support 5.

[0065] In specific implementation, the micro sound air pump 1 is connected with the air cushion guide rail 2, and the camera image acquisition module 9, the sliding block 3 and the air cushion guide rail 2, and the non-inertial reference system motion module 6 are arranged in the vertical direction from top to bottom in sequence. The track direction of the air cushion guide rail 2 is consistent with and aligned with the track direction of the linear motor 6c. The adjustable spring fixed support 5 is fixedly connected on the slide table of the linear motor 6c perpendicularly to the track direction.

[0066] The device has three modes of quasi-static mode, dynamic mode and chaotic mode.

[0067] In the quasi-static mode, the linear motor 6c does not work and does not move, the sliding block 3 is connected with the digital display push-pull force gauge 7, the digital display push-pull force gauge 7 drives the sliding block 3 to move, and the force is measured.

[0068] In the dynamic mode and the chaotic mode, the digital display push-pull force gauge 7 is fixedly connected with the single-chip microcomputer 6a, the driver 6b, the linear motor 6c and the direct current switching power supply 6d in sequence, and the linear motor 6c drives the adjustable spring fixed support 5 to move along the direction parallel to the air cushion guide rail 2.

[0069] The adjustable spring fixed support 5 is driven to produce programmable movement, and periodic inertial force is applied to the sliding block 3 through the two limiting tension and compression spring modules 4, so that the sliding block 3 vibrates in the double potential well potential field, typical hysteresis, bifurcation and chaos phenomena are formed, and the camera image acquisition module is used for collecting and importing video analysis software for position tracking and data analysis, and force-displacement hysteresis curves, phase diagrams and Poincare sections are drawn, and the periodic, quasi-periodic and chaotic states of the system are observed.

[0070] The adjustable spring fixed support 5 is driven to produce programmable movement, and periodic inertial force is applied to the sliding block 3 through the two limiting tension and compression spring modules 4, so that the sliding block 3 vibrates in the double potential well potential field, typical hysteresis, bifurcation and chaos phenomena are formed, and the camera image acquisition module is used for collecting and importing video analysis software for position tracking and data analysis, and force-displacement hysteresis curves, phase diagrams and Poincare sections are drawn, and the periodic, quasi-periodic and chaotic states of the system are observed.

[0071] The double potential well spring system is realized by two symmetrical limiting tension and compression spring modules and a sliding block, and the relative movement between the limiting tension and compression spring modules and the sliding block. When the sliding block moves on the one-dimensional air cushion guide rail, the spring is stretched and compressed, and the spring is in a natural elongation state at two specific positions, forming a double potential well in a one-dimensional motion space. The double potential well can also be formed by the movement of the limiting tension and compression spring module.

[0072] As shown in Figure 1 , the spring body 4a of the limiting tension and compression spring module 4 is in a natural state without force.

[0073] The visual module includes a camera support 8 and a camera image acquisition module 9, and the camera image acquisition module 9 is fixedly installed on the adjustable spring fixed support 5 through the camera support 8 and faces downward, and is used for shooting the moving image / video between the sliding block 3, the limiting tension and compression spring module 4, the adjustable spring fixed support 5 and the digital display push-pull force meter 7.

[0074] It also includes a digital display push-pull force meter and a camera image acquisition module. The digital display push-pull force meter is connected with the sliding block and is used for force application, and the force application can read the external force condition at different positions in quasi-static motion. The camera image acquisition module is arranged on the camera support and is shot from above, and can shoot moving images for analysis.

[0075] 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.

[0076] The experimental test modes include the quasi-static mode of the double-potential-well spring oscillator system, specifically:

[0077] The methods include the measurement of force hysteresis effect under quasi-static conditions and experiments on bifurcation phenomena.

[0078] 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.

[0079] 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 :

[0080] A = f / ( 2kl 0), Y = y / l 0

[0081] 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;

[0082] a corresponding force parameter A and a coordinate parameter Y as a set, finally by all the corresponding force parameters of each set A and coordinate parameters Y plotting the static reduced position versus force relationship Y - A the image, as a force hysteresis characteristic experimental result.

[0083] In the quasi-static mode, the slider 3 remains stationary, at which time the resultant external force of the slider 3 F = 0.

[0084] In the specific implementation, further, the hole aluminum 5b is moved along the waist hole of the aluminum plate 5a, that is, along the X axis (the axis is in the direction of the air cushion guide rail 2), fixed with a screw nut, and the distance between the two spring fixing points and the central axis of the air cushion guide rail 2 is measured with a scale X , x represents the vertical distance from the suspension point of the spring on the adjustable spring fixing bracket 5 to the central axis of the air cushion guide rail 2, and the two spring fixing points are kept symmetrical on both sides. At different distances x , read the reading of the digital push-pull force gauge as above x and the relationship between the distance of the slider to the origin f , the distance is converted into the bracket transverse reduced coordinate y according to the following formula x , and the relationship between the bracket transverse reduced coordinate X and the push-pull force gauge reduced force X and the slider track longitudinal position reduced coordinate A is established as: Y

[0085] X = x / l 0

[0086] A = Y [1-( X 2 + Y 2 ) -1 / 2 ]

[0087] wherein [] represents the reduced proportion of spring stretching or compression;

[0088] the bracket transverse reduced coordinate corresponding to the strain variable of the final fixing device X , the image under different bracket transverse reduced coordinate parameters X can be obtained Y - A , which embodies the force hysteresis curve characteristics under different spring distances. The quasi-static force-displacement relationship curve of the double-well spring oscillator is obtained, and the typical force hysteresis effect and bifurcation phenomenon are observed. ​

[0089] The experimental test mode includes the dynamic mode of the double-potential-well spring oscillator system, specifically:

[0090] The air cushion guide rail 2 is not inflated and deflated by the micro air pump 1, the digital display push-pull force meter 7 remains stationary, the linear motor 6c in the control non-inertial reference system motion module 6 is operated, the adjustable spring fixed support 5 is driven by the linear motor 6c to move along the air cushion guide rail 2, and then the slider 3 is pulled along the direction of the air cushion guide rail 2 by the limiting tension and compression spring module 4, that is, along the Y axis in the direction of the air cushion guide rail 2, and then the distance y between the slider 3 and the origin is fixed in real time after moving, the origin is the intersection between the axis of the adjustable spring fixed support 5 and the axis of the air cushion guide rail 2; The origin is specifically set to be consistent with the quasi-static mode.

[0091] As shown in Figure 5 , images are collected by the vision module in different distance y static states, and the acceleration of the support and the slider is obtained through image processing, and the acceleration of the slider corresponding to each different distance y is calculated according to the following formula to obtain the corresponding equivalent force A’ and the coordinate of the longitudinal position of the slider track Y :

[0092] A' = m ×( a 0 +a ) / ( 2kl 0), Y = y / l 0

[0093] Wherein, m m is the mass of the slider 3, a a0 is the acceleration of the origin moving, which is a known quantity, obtained by image analysis and processing of the images collected by the vision module. a a represents the acceleration of the slider relative to the origin, y y represents the distance from the slider 3 to the origin, k k represents the spring stiffness coefficient, l l0 represents the length of the spring in the limiting tension and compression spring module 4 in the natural state;

[0094] Each force parameter A’ and a coordinate parameter Y corresponding to each other as a group, finally all the force parameters A’ and coordinate parameters Y corresponding to each other of each group are drawn into a dynamic reduced position-force relationship Y - A' image as the experimental result of force hysteresis characteristics.

[0095] 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.

[0096] 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.

[0097] 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:

[0098] F = f inertial - 2 ky (1- l 0 / (( x 2 + y 2 ) 1 / 2 ))

[0099] 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.

[0100] 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:

[0101] X = x / l 0

[0102] A' = Y [( X 2 +Y 2 ) -1 / 2 -1]

[0103] Where [] represents the reduced ratio of spring tension or compression.

[0104] 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.

[0105] 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.

[0106] 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 :

[0107] V = δv / (2 kl 0), Y = y / l 0

[0108] 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.

[0109] 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.

[0110] In the case of chaotic motion, a linear damping term is introduced into the system. The intersection of the line connecting the suspension points of the two perforated angle aluminum springs and the air track is set as the origin. The slider moves back and forth relative to the origin under the action of the inertial force. The motion mode of the linear motor is controlled to regulate the non-inertial system motion state, thereby generating the inertial force f inertial .

[0111] The acceleration information of the origin and the slider position is obtained by analyzing the image. The resultant force F experienced by the mass in the non-inertial system satisfies:

[0112] F = f inertial - 2 ky (1- l 0 / (( x 2 + y 2 ) 1 / 2 )) - δv

[0113] wherein, v represents the motion speed of the slider 3 along the air track 2 relative to the origin, δ is the damping coefficient.

[0114] The data obtained by tracking is used to draw the scatter plot of the slider, i.e. the Poincare section of the system in the phase space. If the points on the Poincare section form fixed points or closed curves, it indicates that the system is in a periodic or quasi-periodic state. The T-Y scatter plot of the slider, i.e. the time-displacement graph of the system, can also be tracked and drawn. The experimental Poincare section phase diagram and the time-displacement graph are drawn. Y - V The above support transverse reduction coordinate parameters

[0115] , force parameters X and a coordinate parameter A’ are related as follows: Y

[0116] X = x / l 0

[0117] V = Y [( X 2 + Y 2 ) -1 / 2 -1]- A’

[0118] A’ = m ×( a 0+ a ) / (2​kl 0)

[0119] 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.

[0120] The working principle of this invention is as follows:

[0121] 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 x When 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.

[0122] 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.

[0123] 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:

[0124] A = Y [1-( X 2 + Y 2 )-1 / 2 ]

[0125] work Y - A curve. For the determined x , the back and forth pulling slider 3 forms a closed loop, the area S of which is proportional to the energy dissipation; gradually increasing x , the hysteresis loop area appears a mutation point, that is, the bifurcation point, which can be compared with the theoretical critical value to verify the quantitative relationship between hysteresis and bifurcation.

[0126] In the dynamic mode, the control single-chip microcomputer 6a outputs a specific waveform (sine, triangle, square wave) to drive the linear motor 6c, so that the non-inertial reference system motion module 6 generates controllable acceleration a 0. The equivalent inertia force f inertial ma 0 received by the slider can be independently adjusted by controlling its amplitude A and angular frequency ω . When fixed A , scanning ω , the system appears single-period, double-period, multi-period and chaotic phenomena in turn; when fixed ω , scanning A , the chaotic critical threshold A C can be obtained. Compared with the theoretical Melnikov criterion, the critical condition for the double-potential well system to enter chaos is verified.

[0127] The data obtained by software tracking are used to draw the displacement-velocity (x-v) y , v ) scatter plot of the slider, that is, the Poincare section of the system in the phase space. If the points on the Poincare section form fixed points or closed curves, it indicates that the system is in periodic or quasi-periodic state. The second-order differential equation is solved by the fourth-order Runge-Kutta method:

[0128] F = f inertial - 2 ky (1- l 0 / (( x 2 + y 2 ) 1 / 2 )) - δv

[0129] The theoretical time-displacement curve and Poincare section phase diagram can be drawn, and compared with the images measured by experiments.

[0130] The implementation process of the specific embodiment is as follows:

[0131] For example 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.

[0132] 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.

[0133] 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.

[0134] 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.

[0135] 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.

[0136] The double potential well spring oscillator experimental device can simulate and calculate the characteristics of nonlinear targeted energy transfer, can generate vibration with large amplitude in a wide driving frequency range, is not sensitive to the change of the intrinsic frequency of the structure, and has wide frequency robustness. This characteristic can break through the limitation of the traditional single frequency linear tuned mass damper and not only respond to the vibration of a single frequency. The device can activate the large amplitude inter-well motion under certain conditions even if it is subjected to a small initial vibration. This has a very strong effect in the application of mechanical vibration suppressors and can achieve efficient vibration suppression. It provides a research platform for the flutter suppression of aerospace structures, wind and earthquake resistance of high-rise buildings and bridges, and precise manufacturing platform for wing super stability.

[0137] Therefore, the present application comprehensively and accurately experiments the various nonlinear dynamics phenomena of the spring oscillator in the double potential well system, provides rich experimental content and powerful tools for physical experiments, and has a wide application prospect.

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 and compression 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). 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). 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).

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 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).

4. 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).

5. A method for experimentally testing a double-potential-well spring oscillator system applied to any one of the experimental apparatuses described in claims 1-4, 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.

6. The experimental method for a double-potential-well spring oscillator according to claim 5, 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).

7. The experimental method for a double-potential-well spring oscillator according to claim 5, 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’ Coordinates reduced to the longitudinal position of the slider track 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.

8. The experimental method for a double-potential-well spring oscillator according to claim 5, 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