A brachistochrone device with adjustable curvature

Abstract

The present application belongs to the technical field of physics teaching and popular science exploration instruments, and relates to a brachistochrone exploration device with adjustable curvature, comprising an L-shaped support, a straight slide path arranged between the two ends of the L-shaped support, a cycloid slide path, an adjustable slide path and an adjusting mechanism, wherein the adjustable slide path is arranged between the two ends of the L-shaped support. The adjusting mechanism comprises a plurality of adjustable elements and a linear driving element, each adjustable element comprising a hinged seat and a clamping piece connected to the hinged seat. The L-shaped support comprises a vertical frame and a horizontal frame, and each of the vertical frame and the horizontal frame is provided with an adjustable element. Two clamping pieces are clamped to the upper end and the lower end of the adjustable slide path respectively. The linear driving element is arranged at the connection between the vertical frame and the horizontal frame, and the output end of the linear driving element is connected to the middle part of the adjustable slide path. The scheme of the present application can not only verify the brachistochrone property of the cycloid, but also explore the quantitative influence of the curvature change of the slide path on the sliding time.

Description

Technical Field

[0001] This invention belongs to the technical field of physics teaching and popular science exploration instruments, and relates to a device for exploring the brachistochrone with adjustable curvature. Background Technology

[0002] The brachistochrone problem can be traced back to Galileo's explorations in the 1630s. In 1696, Johann Bernoulli again publicly challenged mathematicians across Europe: given two points in a vertical plane, find the path of a particle that slides down a slope under the influence of gravity in the shortest time. Newton, Leibniz, and the Bernoulli brothers, among other scholars, all provided correct solutions, proving that the path curve is a cycloid.

[0003] The functional solution of the brachistochrone problem gave rise to the variational method, whose inherent concept of extrema has had a profound impact on analytical mechanics and engineering optimization. In teaching practice, the brachistochrone problem, due to its intuitive phenomena and profound implications, is often integrated into the curriculum of secondary and higher education institutions as a classic case study: in physics, it is used to demonstrate the laws of dynamics, while in mathematics, it is used to introduce the ideas of the variational method. Furthermore, it is also a popular demonstration project in science exhibitions and technological activities.

[0004] Existing brachistochrone exploration devices typically include a support frame, a straight slide, an arc slide, and a cycloidal slide. The upper ends of each slide are set at the same height on the top of the support frame, and the lower ends are set at the same height on the bottom of the support frame. During the demonstration, a small ball is placed at the same height on each slide and released simultaneously. Using timing methods such as photoelectric sensors, the time it takes for the ball to slide down different slides is compared, thereby verifying that the cycloidal descent time is the shortest.

[0005] However, the existing brachistochrone exploration apparatus has significant limitations in its structural design: the geometry of each slide is completely fixed after fabrication, and key parameters such as curvature and arc length cannot be adjusted. This limits the apparatus's function to verifying the brachistochrone's speed, preventing further investigation into the quantitative impact of slide curvature changes on descent time. This structural rigidity restricts the apparatus's wider application in teaching and research, making it difficult to meet more exploratory experimental needs. Summary of the Invention

[0006] The purpose of this invention is to provide a brachistochrone investigation device with adjustable curvature, in order to solve the technical problem that existing brachistochrone investigation devices have fixed slides, making it impossible to investigate the quantitative impact of slide curvature changes on descent time, and thus failing to meet the needs of more in-depth exploratory experiments.

[0007] To achieve the above objectives, the present invention provides a specific technical solution for a device for investigating the brachistochrone with adjustable curvature, as follows: A device for investigating the brachistochrone with adjustable curvature includes an L-shaped support, an adjustment mechanism, a linear slide, a cycloidal slide, and an adjustable slide disposed between the two ends of the L-shaped support. The adjustment mechanism includes multiple adjustable elements and a linear drive element. Each adjustable element includes a hinge seat and a clamping member connected to the hinge seat. The L-shaped support includes a vertical frame and a horizontal frame, each with an adjustable element. Two clamping members clamp the upper and lower ends of the adjustable slide, respectively. The linear drive element is disposed at the connection between the vertical frame and the horizontal frame, and its output end is connected to the middle of the adjustable slide for pushing and pulling the adjustable slide to adjust its curvature.

[0008] Furthermore, the linear drive element includes a gas spring, a hand-cranked winch, and a steel strand; the gas spring includes a cylinder and a piston rod telescopically disposed within the cylinder, the cylinder being located at the connection between the vertical frame and the horizontal frame, and the piston rod being connected to the adjustable slide rail; the hand-cranked winch is disposed on the L-shaped bracket, one end of the steel strand is fixed and wound around the hand-cranked winch, and the other end is connected to the piston rod.

[0009] Furthermore, the linear drive element also includes a limiting disc and a stop bar; the hand-cranked winch has a hand crank for driving the steel strand to be wound up and down; the limiting disc is fixed on the hand-cranked winch, and the limiting disc is evenly provided with multiple insertion holes along its circumference; the stop bar is inserted into the insertion holes to restrict the rotation of the hand crank.

[0010] Furthermore, the hinge seat includes two fixed seats, a sleeve, a bolt, and a nut; each fixed seat includes a fixed block and two connecting lugs connected to the same side of the fixed block, and the connecting lugs are provided with mounting holes; the bolt passes through the mounting holes of the two fixed seats and the sleeve, and the sleeve is located between the connecting lugs; the nut cooperates with the bolt.

[0011] Furthermore, the clamping member includes two claws and a connecting piece, the two claws are connected to the same side of the connecting piece, and the distance between the two claws is adjustable; the connecting piece is connected to the side of the fixing block opposite to the connecting ear plate.

[0012] Furthermore, the adjustable curvature brachistochrone exploration device also includes a reference curve plate, which is located on one side of the L-shaped support. The reference curve plate is provided with straight lines, cycloids, and arcs with different curvatures, which are used as adjustment reference lines for the straight slide, cycloid slide, and adjustable slide.

[0013] Furthermore, the adjustable curvature brachistochrone exploration device also includes a synchronous release mechanism; the synchronous release mechanism is a rotating release frame, which is rotatably mounted on the upper end of the vertical frame around a horizontal axis. The rotating release frame is U-shaped, and the bottom of the rotating release frame is used to limit the release of the small ball at the same height position of the corresponding slide; or, the synchronous release mechanism includes multiple electromagnets respectively mounted at the same height position at the upper end of the straight slide, the cycloidal slide and the adjustable slide, for adsorbing iron balls respectively.

[0014] Furthermore, the adjustable curvature brachistochrone exploration device also includes a timer and a photoelectric gate sensor. The photoelectric gate sensor is located at a preset position on the straight slide, the cycloidal slide, and the adjustable slide. The timer and the photoelectric gate sensor are used to detect the time interval of the ball passing through the straight slide, the cycloidal slide, or the adjustable slide, or the speed at the preset position.

[0015] Furthermore, the linear slide, the cycloidal slide, the vertical frame, and the horizontal frame are all connected by adjustable elements, and the hinge seat is adjustable along the respective length direction of the vertical frame or the horizontal frame.

[0016] Furthermore, the adjustable slide, the linear slide, and the cycloidal slide are all made of aluminum alloy, PVC, or glass fiber reinforced polypropylene composite material.

[0017] The device for investigating the brachistochrone with adjustable curvature according to the present invention has the following advantages: The adjustable curvature brachistochrone investigation device of the present invention uses the output end of a linear drive element to push or pull the middle of an adjustable slide closer to or further away from the L-shaped support. Under the hinge constraint of the hinge seat, the adjustable slide elastically bends, forming an inwardly convex arc shape, thus changing the curvature of the adjustable slide until the desired curvature is achieved, at which point the linear drive element stops moving. This invention, by adjusting the different curvatures of the adjustable slide, can not only verify the brachistochrone of the cycloid but also explore the quantitative effect of changes in slide curvature on the descent time. This invention expands the application of brachistochrone investigation devices in teaching and research, and can meet the needs of more in-depth exploratory experiments. Attached Figure Description

[0018] Figure 1 This is a schematic diagram of the structure of the adjustable curvature brachial descent exploration device of the present invention. Figure 2 This is a side view of the curvature-adjustable brachistochrone exploration device of the present invention. Figure 3 This is a schematic diagram of the adjustable element of the present invention; Figure 4 This is a schematic diagram of the reference curve plate of the present invention.

[0019] Figure label: 1. L-shaped bracket; 11. Vertical bracket; 12. Horizontal bracket; 2. Linear slide; 3. Cycloidal slide; 4. Adjustable slide; 5. Linear drive element; 51. Gas spring; 52. Hand-cranked winch; 53. Limiting plate; 54. Stop bar; 6. Adjustable element; 61. Fixed base; 62. Claw; 63. Connecting piece; 64. Sleeve; 65. Bolt; 66. Nut; 7. Rotation release frame; 8. Reference curve plate; 81. Straight line; 82. Cycloidal; 83. Parabola; 84. Circular arc. Detailed Implementation

[0020] The technical solutions of this application will now be described clearly and in detail with reference to the accompanying drawings. In the description of the embodiments of this application, unless otherwise stated, "multiple" refers to two or more. The terms "first" and "second" are used for descriptive purposes only and should not be construed as implying or suggesting relative importance or implicitly indicating the number of indicated technical features. Therefore, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature.

[0021] like Figure 1 , Figure 2 As shown, this invention provides a device for investigating the brachistochrone with adjustable curvature, comprising an L-shaped support 1, a linear slide 2, a cycloidal slide 3, an adjustable slide 4, and an adjustment mechanism. The linear slide 2, cycloidal slide 3, and adjustable slide 4 are all located between the two ends of the L-shaped support 1. The adjustment mechanism includes multiple adjustable elements 6 and linear drive elements 5. Each adjustable element 6 includes a hinge seat and a clamping member connected to the hinge seat. The L-shaped support 1 includes a vertical frame 11 and a horizontal frame 12, each with an adjustable element 6. Two clamping members clamp the upper and lower ends of the adjustable slide 4, respectively. The linear drive element 5 is located at the connection between the vertical frame 11 and the horizontal frame 12, and its output end is connected to the middle of the adjustable slide 4 for pushing and pulling the adjustable slide 4 to adjust its curvature.

[0022] In the adjustable curvature brachistochrone exploration device of this invention, when adjusting the curvature of the adjustable slide 4, the hinge locking of the corresponding hinge seat is released. The output end of the linear drive element 5 pulls the middle part of the adjustable slide 4 towards the inside of the L-shaped bracket 1. Under the hinge constraint of the hinge seats at both ends, the adjustable slide 4 undergoes elastic bending, forming an inwardly convex arc shape, increasing the curvature of the adjustable slide 4 until it reaches the required curvature. At this point, the linear drive element 5 stops moving and finally locks the hinge of the hinge seat. The output end of the linear drive element 5 moves in the opposite direction, pushing the middle part of the adjustable slide 4 away from the L-shaped bracket 1. Under the hinge constraint of the hinge seats at both ends, the adjustable slide 4 undergoes elastic bending, decreasing its curvature until it reaches the required curvature. At this point, the linear drive element 5 stops moving. During adjustment, the hinged seats at both ends rotate on the vertical frame 11 and the horizontal frame 12, ensuring that the clamping element is always perpendicular to the tangent direction of the adjustable slide 4 at that point. This guarantees that the clamping force always acts along the normal direction, preventing lateral slippage. Thus, continuous, coordinated, and stable control of the global curvature of the entire slide can be achieved through a single degree of freedom input from a linear drive element 5. After the curvature adjustment of the adjustable slide 4 is completed, small balls are placed at the starting points of the linear slide 2, the cycloidal slide 3, and the adjustable slide 4, and released simultaneously. By observing with the naked eye or recording the sliding time of each ball using a timing device, it is proven that the cycloidal slide 3 has the shortest sliding time.

[0023] The adjustable curvature brachistochrone investigation device of the present invention uses the output end of the linear drive element 5 to push or pull the middle of the adjustable slide 4 closer to or further away from the L-shaped support 1. Under the hinge constraint of the hinge seat, the adjustable slide 4 elastically bends, forming an inwardly convex arc shape, until the adjustable slide 4 reaches the desired curvature, at which point the linear drive element 5 stops moving. This invention, by adjusting the different curvatures of the adjustable slide 4, can not only verify the brachistochrone of the cycloid 82 but also explore the quantitative effect of changes in slide curvature on the descent time. The solution of the present invention expands the application of the brachistochrone investigation device in teaching and scientific research, and can meet the needs of more in-depth exploratory experiments.

[0024] Furthermore, since the adjustable element 6 includes a hinge seat and a clamping component, during adjustment, the clamping component can automatically adjust its posture according to the local curvature of the slide under the drive of the hinge seat, ensuring that the clamping force is always perpendicular to the tangential direction of the slide, reducing the deformation resistance of the slide and preventing clamping damage. Because the output end of the linear drive element 5 is directly connected to the middle of the adjustable slide 4, the point of application of the driving force is located at the position of maximum slide deflection, improving the responsiveness and accuracy of curvature adjustment. Since the adjustable slide 4 can achieve continuous, reversible, and stepless adjustment of curvature, the same experimental device can cover curves with different curvatures, significantly expanding the experimental dimension and theoretical depth of the brachistochrone teaching experiment.

[0025] Experiments show that, at a horizontal distance of 57.95 cm and a vertical distance of 62.1 cm between the start and end points, the ball takes the shortest time to slide down the cycloid 82 (the brachistochrone), followed by the circular arc, and the straight line 81 is the slowest. For the adjustable track 4, when its curvature is greater than that of the straight line 81 but less than that of the cycloid 82, the greater the curvature, the shorter the ball's sliding time; when its curvature is greater than that of the cycloid 82, the greater the curvature, the longer the sliding time.

[0026] In some embodiments of the present invention, such as Figure 1 , Figure 2 As shown, the linear slide 2, the cycloidal slide 3, the vertical frame 11, and the horizontal frame 12 are all connected by adjustable elements 6, and the hinge seats are adjustable along the respective length direction of the vertical frame 11 or the horizontal frame 12.

[0027] In this embodiment, both the linear slide 2 and the cycloidal slide 3 are mounted on the L-shaped bracket 1 via adjustable elements 6. The specific installation process is as follows: Remove the linear slide 2 or the cycloidal slide 3 from the clamping member and adjust it to the ideal curve. Release the locking between the adjustable element 6 and the vertical frame 11 or the horizontal frame 12, slide the adjustable element 6 relative to each other to find the position where the clamping member is tangent to the ideal curve, fix the adjustable element 6, and finally reinstall the slide with the adjusted shape back onto the clamping member of the adjustable element 6. Since the linear slide 2 and the cycloidal slide 3 are installed and fixed via the adjustable element 6, it is convenient to quickly install the linear slide 2 and the cycloidal slide 3 and to easily adjust the curvature of the linear slide 2 and the cycloidal slide 3.

[0028] In some embodiments of the present invention, such as Figure 4 As shown, the adjustable curvature brachistochrone exploration device also includes a reference curve plate 8, which is located on one side of the L-shaped support 1. The reference curve plate 8 has a straight line 81, a cycloid 82, and arcs with different curvatures, used as adjustment reference lines for the straight slide 2, the cycloid slide 3, and the adjustable slide 4. The arcs can be parabolic or circular. Figure 4 As shown, the reference curve plate 8 is provided with a straight line 81, a cycloid 82, a parabola 83 and a circular arc 84.

[0029] In this embodiment, the positions of each slide and adjustable element 6 on the vertical frame 11 and horizontal frame 12 are adjusted using the reference curve plate 8, thereby ensuring the accuracy of the slide curvature and ensuring that the clamping part of the adjustable element 6 is tangent to the corresponding slide. The specific installation process is as follows: Remove the straight slide 2, cycloidal slide 3, and adjustable slide 4 from the clamping part, and adjust them to the ideal curve by referring to the straight line 81, cycloidal line 82, and arc on the reference curve plate 8. Release the locking of the adjustable element 6 to the vertical frame 11 or horizontal frame 12, slide the adjustable element 6 relative to find the position where the clamping part is tangent to the ideal curve, fix the adjustable element 6, and finally reinstall the adjusted slide onto the clamping part of the adjustable element 6. After the adjustment is completed, the curvature of the straight slide 2 and cycloidal slide 3 remains unchanged and is fixed on the L-shaped bracket 1 without further adjustment. The adjustable slide 4 can have its curvature adjusted by the linear drive element 5. When the linear drive element 5 drives the adjustable slide 4 to bend and deform, the curvature of the adjustable slide 4 is adjusted by referring to the curve on the reference curve plate 8. This process does not require the intervention of measuring instruments, but only relies on human visual interpretation and manual fine-tuning, which meets the needs of classroom teaching for rapid deployment and immediate verification.

[0030] In some embodiments of the present invention, the reference curve plate 8 can be a KT board, which is made of polystyrene. KT boards are lightweight, easy to handle and install, relatively rigid, not easily bent or deformed, widely applicable, and can be directly screen-printed or inkjet-painted. Standard curve KT boards such as straight lines 81, cycloids 82, and arcs, matching the actual size of the teaching aid at a 1:1 scale, can be drawn and printed using software such as Python to assist users in quickly and accurately adjusting the adjustable slide 4 to the preset theoretical shape.

[0031] In some embodiments of the present invention, the vertical frame 11 and the horizontal frame 12 are connected by a connector, which includes an L-shaped angle iron, two T-bolts, and two connecting nuts. Both the vertical frame 11 and the horizontal frame 12 have T-slots along their respective lengths, and the L-shaped angle iron has two mounting holes. When connecting the vertical frame 11 and the horizontal frame 12, the T-shaped ends of the two T-bolts are respectively inserted into the T-slots of the vertical frame 11 and the horizontal frame 12, and the two T-bolts pass through the two mounting holes of the L-shaped angle iron. The two connecting nuts are threadedly connected to the two T-bolts, thereby achieving a fixed connection between the vertical frame 11 and the horizontal frame 12. The connection method in this embodiment is relatively robust, and installation and disassembly are both convenient.

[0032] In some embodiments of the present invention, such as Figure 1 , Figure 2As shown, the linear drive element 5 includes a gas spring 51, a hand-cranked winch 52, and steel strand. The gas spring 51 includes a cylinder and a piston rod that extends and retracts within the cylinder. The cylinder is located at the connection between the vertical frame 11 and the horizontal frame 12, and the piston rod is connected to the adjustable slide rail 4. The hand-cranked winch 52 is mounted on an L-shaped bracket 1. One end of the steel strand is fixed and wound around the hand-cranked winch 52, and the other end is connected to the piston rod.

[0033] In this embodiment, when it is necessary to increase the curvature of the adjustable slide 4, the hand-cranked winch 52 is rotated, causing the steel strand to wind up and pulling the piston rod of the gas spring 51 to retract into the cylinder. This pulls the middle part of the adjustable slide 4 towards the L-shaped bracket 1, increasing the bulge in the middle of the adjustable slide 4 and thus increasing the overall curvature. When it is necessary to decrease the curvature of the adjustable slide 4, the hand-cranked winch 52 is rotated in the opposite direction. Under the combined action of the compressed gas and the buffer medium inside the gas spring 51, the piston rod is smoothly pushed out, pushing the middle part of the adjustable slide 4 outward, thus decreasing the curvature of the adjustable slide 4. The elastic restoring force and damping characteristics provided by the gas spring 51 effectively suppress vibration and impact during the adjustment process, preventing the adjustable slide 4 from rebounding excessively or failing to clamp due to sudden release. The steel strand, as a flexible force transmission medium, physically isolates the hand-cranked operating point from the gas spring 51 body, allowing the user to safely and conveniently complete all adjustment actions outside the bracket without touching high temperature, high pressure, or moving parts.

[0034] In some embodiments of the present invention, such as Figure 1 , Figure 2 As shown, the linear drive element 5 also includes a limiting disc 53 and a stop bar 54. The hand-cranked winch 52 has a hand crank for winding and unwinding the steel strand. The limiting disc 53 is fixed to the hand-cranked winch 52, and the limiting disc 53 has multiple insertion holes evenly distributed along its circumference. The stop bar 54 is inserted into the insertion holes to limit the rotation of the hand crank.

[0035] In this embodiment, the operator rotates the hand crank to retract and extend the steel strand, simultaneously driving the piston rod of the gas spring 51 to extend and retract, causing the adjustable slide 4 to undergo elastic deformation. Once the curvature reaches the desired state, the stop rod 54 is inserted into the hole on the limiting plate 53 corresponding to the current angle of the hand crank. At this point, the stop rod 54 abuts against the root of the hand crank or the edge of the limiting plate 53, forming a rigid limit and preventing further rotation of the hand crank. This limiting mechanism does not alter the original transmission structure of the hand winch 52; it only intervenes during the adjustment completion stage and does not affect the free rotation performance during the adjustment process.

[0036] This embodiment achieves reliable locking of the rotational position of the hand-cranked winch 52. Because the limiting plate 53 rotates synchronously with the hand-cranked winch 52 and the insertion holes are evenly distributed, the amount of steel strand winding or unwinding caused by each rotation of the hand crank is constant. This results in a linear and predictable relationship between the displacement of the piston rod of the gas spring 51 and the curvature change of the adjustable slide rail 4. Furthermore, since the mechanical embedded constraint of the stop rod 54 on the insertion holes directly acts on the degree of freedom of the hand crank's movement, it effectively prevents curvature drift caused by external disturbances or operational negligence, ensuring the consistency and comparability of the geometric parameters of each slide rail in multiple comparative experiments. In this embodiment, the limiting plate 53 can be a shrink sleeve with multiple insertion holes in its circumferential direction. Two insertion holes are used to fix it to the hand-cranked winch 52 using bolts 65, and the remaining insertion holes are used for limiting the movement. The stop rod 54 can be an Allen wrench or a screwdriver.

[0037] In some embodiments of the present invention, such as Figure 3 As shown, the hinged joint includes two fixed seats 61, a sleeve 64, a bolt 65, and a nut 66. Each fixed seat 61 includes a fixed block and two connecting lugs on the same side as the fixed block, with mounting holes on the connecting lugs. The bolt 65 passes through the mounting holes of the two fixed seats 61 and the sleeve 64, which is located between the connecting lugs. The nut 66 mates with the bolt 65.

[0038] In this embodiment, the two fixed seats 61 rotate relative to each other via sleeve 64, bolt 65, and nut 66. This allows the hinge seat in the adjustable element 6 to rotate during curvature adjustment, thus making the curvature of the adjustable slide 4 adjustable. Specifically, when bolt 65 and nut 66 are loose, the connecting lugs of the two fixed seats 61 can rotate relative to each other, allowing the clamping member to adapt to different tangential directions as the curvature of the adjustable slide 4 changes. When bolt 65 and nut 66 are tightened, the connecting lugs of the two fixed seats 61 are axially pressed, and the two sets of connecting lugs simultaneously press against sleeve 64, forming a self-locking rigid connection. This completes the angle positioning, locks the rotating pair, and ensures that the clamping member does not deflect unexpectedly. This hinge method does not introduce additional bearings or pin structures, simplifying the assembly process and reducing manufacturing costs. At the same time, the symmetrical arrangement of the double connecting lugs improves torsional stiffness, prevents the clamping member from torsional instability around the bolt 65 axis, and ensures the attitude accuracy and operational stability of the adjustable slide 4 under various curvature states.

[0039] In some embodiments of the present invention, the fixed base 61 can be connected to the vertical frame 11 or the horizontal frame 12 via a T-bolt and a connecting nut. Specifically, the T-shaped end of the T-bolt is inserted into the T-slot of the vertical frame 11 or the horizontal frame 12, and then the T-bolt passes through the mounting hole of the fixed base 61. The connecting nut is threadedly connected to the T-bolt, thereby achieving the connection of the fixed base 61. When it is necessary to move the adjustable element 6, the connecting nut is loosened, and the fixed base 61 and the T-bolt can move along the length direction of the T-slot of the vertical frame 11 or the horizontal frame 12, thereby adjusting the position of the adjustable element 6.

[0040] In some embodiments of the present invention, such as Figure 3 As shown, the clamping component includes two jaws 62 and a connecting piece 63. The two jaws 62 are connected to the same side of the connecting piece 63, and the spacing between the two jaws 62 is adjustable. The connecting piece 63 is connected to the side of the fixing block opposite to the connecting lug plate.

[0041] In this embodiment, when it is necessary to install the adjustable slide rail 4 or replace it with an adjustable slide rail 4 of different cross-sectional dimensions, the operator first loosens the adjusting bolt 65 between the claw 62 and the connecting piece 63, adjusts the relative distance between the two claws 62 according to the width of the adjustable slide rail 4, so that the distance between their inner surfaces corresponds to the outer contour dimension of the adjustable slide rail 4 in the corresponding direction, then inserts the adjustable slide rail 4 between the two claws 62, and finally tightens the adjusting bolt 65 to complete the overall fastening. After adjustment, the claws 62 can fasten the adjustable slide rail 4 to transmit the axial push and pull force applied by the linear drive element 5, while the connecting piece 63 mainly bears the shear and bending loads. The components have clear division of labor and reliable cooperation. Since the linear slide rail 2 and the cycloidal slide rail 3 are also provided with adjustable elements 6, the installation or adjustment of the linear slide rail 2 and the cycloidal slide rail 3 can be achieved through the corresponding adjustable elements 6.

[0042] Because the spacing between the two jaws 62 is adjustable, it can accommodate adjustable slides 4, linear slides 2, and cycloidal slides 3 of different widths or thicknesses, improving the clamping structure's compatibility with changes in slide specifications. Since the jaws 62 are located on the same side of the connecting piece 63 and form a U-shaped clamping structure, the slide's degree of freedom perpendicular to the sliding direction can be effectively constrained, preventing lateral disengagement during adjustment. Because the connecting piece 63 is connected to the side of the fixed block opposite to the connecting ear plate, the local force distribution is optimized, reducing the bending deformation of the connecting structure under dynamic adjustment conditions, and improving clamping stability and adjustment repeatability.

[0043] In some embodiments of the present invention, such as Figure 1 , Figure 2As shown, the brachistochrone exploration device with adjustable curvature also includes a synchronous release mechanism. The synchronous release mechanism is a rotating release frame 7, which is rotatably mounted on the upper end of the vertical frame 11 around a horizontal axis. The rotating release frame 7 is U-shaped, and its bottom is used to limit the release of the small ball at the same height position of the corresponding slide.

[0044] In this embodiment, the rotating release frame 7 can be a U-shaped structure made of PVC or aluminum alloy profiles, with its opening facing upwards, spanning the top of the vertical frame 11 of the L-shaped support 11. The two ends of the rotating release frame 7 are rotatably connected to the upper end of the vertical frame 11 via rotating shafts, which are horizontally positioned to allow the rotating release frame 7 to rotate freely around this axis. A horizontally extending limiting strip is provided on the inner bottom of the rotating release frame 7 to simultaneously contact and support the small balls placed at the starting positions of the linear slide 2, cycloidal slide 3, and adjustable slide 4, ensuring that the initial height of each ball is consistent. The contact surface between the limiting strip and the upper end of each slide is a smooth plane or a structure with shallow grooves to stably constrain the position of the small balls without introducing additional rolling resistance. This structure is driven manually or by a linkage mechanism to quickly flip the rotating release frame 7 upwards, causing all the small balls to release their constraints at the same time and begin to slide down from a rest state under the action of gravity. Because a rotating release frame 7 is set up that can simultaneously release the ball constraint at the same height position, the time difference and initial velocity disturbance caused by manual release are eliminated, thus improving the accuracy of the multi-track sliding time comparison experiment.

[0045] In some other embodiments of the present invention, the synchronous release mechanism includes a plurality of electromagnets respectively disposed at the upper end of the linear slide 2, the cycloidal slide 3 and the adjustable slide 4 at the same height, for adsorbing iron balls respectively.

[0046] In this embodiment, the electromagnet can be a miniature DC electromagnet with a rated operating voltage of 5V or 12V, and its attraction force is sufficient to stably attract iron balls with a diameter of 10mm to 30mm. Each electromagnet is fixed to a support at the upper end of its corresponding slide, with its magnetic pole facing the ball's placement position, ensuring that the center of attraction is aligned with the ball's center of mass. The electromagnet is electrically connected to an external control circuit, which includes a power module, a relay array, and a trigger switch. The relay array is used to synchronously control the on / off sequence of multiple electromagnets, with a response delay of less than 10ms, thereby ensuring that all electromagnets de-energize and release the ball at the same time. The installation position of the electromagnet can be independently adjusted according to the actual spacing at the upper end of each slide, for example, by using a slide rail mounting base or a mounting plate with an elongated hole to achieve fine-tuning in the horizontal and vertical directions. The electromagnet's casing can be made of non-magnetic stainless steel or engineering plastic to avoid magnetic field interference. Its operating temperature range is −10℃ to +50℃, meeting the requirements of conventional teaching environments. This application does not impose any special limitations on the specific model of the electromagnet, the number of coil turns, the magnetic circuit structure, or the power supply method, as long as it can achieve multi-point synchronous adsorption and release functions. Because multiple independently arranged and controlled synchronously de-energized electromagnets are used, the initial state and release time of the balls can still be ensured to be consistent even in scenarios with irregular distribution in the slide space or requiring automated intervention, thus expanding the applicability and robustness of the research device under different experimental conditions.

[0047] In some embodiments of the present invention, the brachistochrone exploration device with adjustable curvature further includes a timer and a photoelectric gate sensor, the photoelectric gate sensor being disposed at preset positions on the linear slide 2, the cycloidal slide 3, and the adjustable slide 4. The timer and the photoelectric gate sensor are used to detect the time interval of the ball passing through the linear slide 2, the cycloidal slide 3, or the adjustable slide 4, or the speed at the preset position.

[0048] In this embodiment, the photoelectric gate sensor can be an infrared through-beam photoelectric switch, including a transmitter and a receiver, which are installed opposite each other on both sides of the slide to form a stable optical path. When the ball slides down the slide and blocks the optical path, the receiver outputs a level transition signal, which is captured by the timer as a timing event trigger point. The installation position of the photoelectric gate sensor can be set at any preset height of the slide according to experimental requirements, such as the starting point, ending point, or midpoint. The timer is a digital multi-channel intelligent timer with at least two independent photoelectric gate input interfaces, millisecond-level time resolution, automatic zeroing, and data storage functions. Its internal circuit starts timing when it receives the blocking signal of the first photoelectric gate and stops timing when it receives the blocking signal of the second photoelectric gate, and displays the time interval value on the LCD screen. In this embodiment, the timer and the photoelectric gate sensor work together to form a basic quantitative measurement unit. Its structure and parameters can be flexibly configured according to the actual teaching accuracy requirements, such as a time resolution of 0.1ms, 1ms, or 10ms. This application does not impose any special limitations on this.

[0049] The timer, used in conjunction with a photoelectric gate sensor, enables both interval timing and speed measurement. For interval timing, two photoelectric gate sensors are needed on the same slide: one at the start and one at the end. When the ball is released, the timer automatically starts counting the instant it passes the light-blocking point at the start and stops counting when it reaches the end point where the photoelectric gate sensor blocks the light. The timer automatically calculates the time difference between the two points. Repeating the experiment and averaging the results yields the time required for the ball to slide down the slide of this shape. For speed measurement, the instantaneous speed of the ball at that point is calculated by dividing the light-blocking distance d by the light-blocking time Δt. The light-blocking time Δt is the total duration (Δt) during which the ball blocks the infrared light between the transmitter and receiver of the photoelectric gate sensor when it passes a point in one go. Since the diameter d of the ball is known, the light-blocking distance d is determined.

[0050] Furthermore, this device can be used to investigate the law of conservation of mechanical energy, as follows: Three photoelectric gate sensors are simultaneously clamped at the same height on three different tracks. Because the tracks employ a double-rail supported overhead structure, the ball only has minimal contact with the support points, making rolling friction negligible and rotational kinetic energy insignificant. The ball then slides down approximately in pure translational motion. Therefore, with negligible air resistance and frictional resistance, the ball's mechanical energy is conserved. Since balls of the same size slide from the same height, the amount of gravitational potential energy converted into kinetic energy is equal, further implying that the instantaneous velocities at the endpoints should be equal. By measuring the instantaneous velocities of the ball at the endpoints of the three tracks, the law of conservation of mechanical energy can be verified.

[0051] This embodiment achieves highly repeatable and consistent quantitative measurement of the time it takes for a small ball to move on slides with different curvatures. Because photoelectric gate sensors are positioned at preset locations on the straight slide 2, the cycloidal slide 3, and the adjustable slide 4, and because the timer can independently record the signal intervals between the photoelectric gate sensors, it solves the problems of strong subjectivity, large errors, and non-reproducibility caused by traditional exploratory devices relying solely on visual observation or manual timing. This upgrades the judgment of whether the cycloidal 82 is the fastest from experiential observation to data-driven scientific verification, supporting the teaching leap from qualitative cognition to quantitative analysis.

[0052] In some embodiments of the present invention, the curvature adjustable brachistochrone exploration device further includes an acceleration sensor and a pressure sensor located at preset positions on the straight slide 2, the cycloidal slide 3, and the adjustable slide 4.

[0053] In this embodiment, the accelerometer is used to measure the instantaneous acceleration components of the ball in the spatial coordinate system during its movement along the slide. It is installed in a pre-set embedding groove on the back or side of the slide body and fixed by a flexible adhesive layer or a micro-clamp. The accelerometer can be a triaxial MEMS accelerometer, such as an integrated sensor module of model ADXL345 or MPU6050, and its output signal is an analog voltage. The function of this accelerometer in this application is to acquire the acceleration-time series during the ball's movement, and then deduce the velocity and displacement trends through numerical integration, thus supplementing the discrete limitations of fixed-point velocity measurement by the photoelectric gate.

[0054] The pressure sensor is used to monitor the normal pressure in the contact area between the ball and the slide. It is installed on the inner side of the bottom surface of the slide body or at the load-bearing point of the supporting structure corresponding to the ball's rolling path. It employs a micro-protrusion embedded package or a thin-film attached arrangement. The pressure sensor can be a piezoresistive thin-film sensor or a micro-strain gauge sensor, with a range covering 0–5N and a response frequency not less than 100Hz. In this application, the pressure sensor reflects the changing law of the supporting force on the ball in different curvature sections. Especially in the curved section of the adjustable slide 4, its output pressure value increases as the local radius of curvature decreases, which can be used to verify the inverse proportional relationship between the normal force and the radius of curvature in the centripetal force formula. It forms a mechanical sensing closed loop with the adjustable slide 4: when the adjustment mechanism changes the curvature of the adjustable slide 4, the position and amplitude of the pressure peak recorded by the pressure sensor change synchronously under the same release height conditions for the same ball, thereby realizing a visual mapping of structural parameters and mechanical response. The signal from the pressure sensor is amplified and filtered by the conditioning circuit and output in the form of analog voltage or digital signal, which is connected to the same timer or an independent data acquisition card for synchronous recording.

[0055] Therefore, by installing accelerometers and pressure sensors at preset positions on the straight slide 2, cycloidal slide 3, and adjustable slide 4, the dynamic changes in acceleration and the contact pressure response of the slide throughout the entire ball's motion can be acquired simultaneously, thus overcoming the dimensional limitations of relying solely on time measurements. Since the accelerometers and photoelectric gate sensors work in tandem, the complete kinematic parameters can be reconstructed through data fusion without requiring manual operation. Because the pressure sensors are positioned at characteristic locations on slides with different curvatures, the influence of curvature changes on the normal support force can be visually presented, providing experimental evidence for centripetal force analysis, mechanical energy loss assessment, and force modeling of non-inertial frames.

[0056] In some embodiments of the present invention, the adjustable slide 4 is made of aluminum alloy, PVC or glass fiber reinforced polypropylene composite material.

[0057] In this embodiment, the adjustable slide 4 can be made of aluminum alloy, such as 6063 aluminum, 6061-T6, or 6063-T5 profile. It has good ductility, moderate yield strength, and excellent surface finish, which allows it to bend stably and maintain a set curvature under the action of the angle adjustment of the claw 62 and the push and pull of the linear drive element 5. At the same time, it maintains a low coefficient of friction and structural deformation stability during the repeated sliding of the ball. The adjustable slide 4 made of aluminum alloy is suitable for teaching and experimental scenarios with high requirements for adjustment accuracy, repeatability, and long-term durability.

[0058] The adjustable slide 4 can also be made of PVC, a rigid polyvinyl chloride profile. This PVC has sufficient rigidity to support the ball's sliding path and good cold-bending plasticity, allowing for repeated bending and shaping at room temperature using manual methods or auxiliary clamps without cracking. This PVC material is inexpensive and easy to process, making it suitable for basic physics experiments in middle school or low-cost, widely available teaching aids.

[0059] The adjustable slide 4 can also be made of glass fiber reinforced polypropylene composite material, in which the mass fraction of glass fiber is 20%–30%. The polypropylene matrix provides toughness and formability, while the glass fiber significantly improves its tensile modulus and creep resistance. This material is lightweight, has high specific stiffness and good wear resistance, so that the adjustable slide 4 can effectively suppress vertical deflection under gravity load while ensuring the sensitivity of curvature adjustment response. It is suitable for portable, modular or frequently disassembled teaching demonstration systems.

[0060] In some embodiments of the present invention, the linear slide 2 and the cycloidal slide 3 are also made of aluminum alloy, PVC, or glass fiber reinforced polypropylene composite material. The effect of using the above-mentioned materials for the linear slide 2 and the cycloidal slide 3 is the same as the effect of using the above-mentioned materials for the adjustable slide 4, and will not be described again.

[0061] In some embodiments of the present invention, the rolling process of the ball can also be analyzed using Tracker software to conduct physical exploration experiments. Tracker software is a free and open-source video analysis and modeling software that can analyze the trajectory, velocity, acceleration, etc., of objects in videos and is commonly used in physics experiments. This application establishes a standard operating procedure to guide users in using smartphones to capture videos of the rolling ball, importing them into Tracker software for coordinate system establishment, trajectory tracking, data acquisition, and fitting analysis. This yields displacement-time and velocity-time images of the ball sliding down different tracks, revealing that when the ball slides down a cycloidal track, the angle between its velocity and the vertical direction increases linearly with time.

[0062] In some embodiments of the present invention, the data can also be compared with actual experimental data through a virtual simulation experimental system. Specifically, interactive software developed based on HTML / JavaScript allows users to input gravitational acceleration and slide parameters (simulated through adjustable curve curvature, etc.), simulates the motion of the ball in real time, outputs kinematic parameters, and compares them with physical experimental data.

[0063] It is understood that the present invention has been described through some embodiments, and those skilled in the art will recognize that various changes or equivalent substitutions can be made to these features and embodiments without departing from the spirit and scope of the invention. Furthermore, under the teachings of the present invention, these features and embodiments can be modified to adapt to specific situations and materials without departing from the spirit and scope of the invention. Therefore, the present invention is not limited to the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application are within the protection scope of the present invention.

Claims

1. A device for investigating the brachistochrone with adjustable curvature, comprising an L-shaped support, a linear slide rail and a cycloidal slide rail disposed between the two ends of the L-shaped support, characterized in that, It also includes an adjustable slide and an adjustment mechanism. The adjustable slide is located between the two ends of the L-shaped bracket. The adjustment mechanism includes multiple adjustable elements and a linear drive element. Each adjustable element includes a hinge seat and a clamping member connected to the hinge seat. The L-shaped bracket includes a vertical frame and a horizontal frame, and each of the vertical frame and the horizontal frame is provided with an adjustable element. The two clamping members are respectively clamped at the upper end and the lower end of the adjustable slide. The linear drive element is located at the connection between the vertical frame and the horizontal frame, and the output end of the linear drive element is connected to the middle of the adjustable slide, for pushing and pulling the adjustable slide to adjust the curvature of the adjustable slide.

2. The apparatus for investigating the brachistochrone with adjustable curvature according to claim 1, characterized in that, The linear drive element includes a gas spring, a hand-cranked winch, and a steel strand; the gas spring includes a cylinder and a piston rod telescopically disposed within the cylinder, the cylinder being located at the connection between the vertical frame and the horizontal frame, and the piston rod being connected to the adjustable slide rail; the hand-cranked winch is disposed on the L-shaped bracket, one end of the steel strand is fixed and wound around the hand-cranked winch, and the other end is connected to the piston rod.

3. The apparatus for investigating the brachistochrone with adjustable curvature according to claim 2, characterized in that, The linear drive element further includes a limiting plate and a stop bar; the hand-cranked winch has a hand crank for driving the steel strand to be wound up and down; the limiting plate is fixed on the hand-cranked winch, and the limiting plate is evenly provided with multiple insertion holes along its circumference; the stop bar is inserted into the insertion holes to restrict the rotation of the hand crank.

4. The apparatus for investigating the brachistochrone with adjustable curvature according to claim 1, characterized in that, The hinged seat includes two fixed seats, a sleeve, a bolt, and a nut; each fixed seat includes a fixed block and two connecting lugs on the same side of the fixed block, and the connecting lugs are provided with mounting holes; the bolt passes through the mounting holes of the two fixed seats and the sleeve, and the sleeve is located between the connecting lugs; the nut cooperates with the bolt.

5. The apparatus for investigating the brachistochrone with adjustable curvature according to claim 4, characterized in that, The clamping component includes two claws and a connecting piece. The two claws are connected to the same side of the connecting piece, and the distance between the two claws is adjustable. The connecting piece is connected to the side of the fixing block opposite to the connecting lug.

6. The apparatus for investigating the brachistochrone with adjustable curvature according to claim 1, characterized in that, It also includes a reference curve plate, which is located on one side of the L-shaped bracket. The reference curve plate has straight lines, cycloids and arcs with different curvatures, which are used as adjustment reference lines for the straight slide, cycloid slide and adjustable slide.

7. The apparatus for investigating the brachistochrone with adjustable curvature according to claim 1, characterized in that, It also includes a synchronous release mechanism; the synchronous release mechanism is a rotating release frame, which is rotatably mounted on the upper end of the vertical frame around a horizontal axis. The rotating release frame is U-shaped, and the bottom of the rotating release frame is used to limit the release of the small ball at the same height position of the corresponding slide. Alternatively, the synchronous release mechanism includes multiple electromagnets respectively mounted at the same height position at the upper end of the straight slide, the cycloidal slide, and the adjustable slide, for adsorbing iron balls respectively.

8. The apparatus for investigating the brachistochrone with adjustable curvature according to claim 1, characterized in that, It also includes a timer and a photoelectric gate sensor, wherein the photoelectric gate sensor is located at a preset position on the linear slide, the cycloidal slide, and the adjustable slide; the timer and the photoelectric gate sensor are used to detect the time interval of the ball passing through the linear slide, the cycloidal slide, or the adjustable slide, or the speed at the preset position.

9. The apparatus for investigating the brachistochrone with adjustable curvature according to claim 1, characterized in that, The linear slide, the cycloidal slide, the vertical frame, and the horizontal frame are all connected by adjustable elements, and the hinge seat is adjustable along the respective length direction of the vertical frame or the horizontal frame.

10. The apparatus for investigating the brachistochrone with adjustable curvature according to claim 1, characterized in that, The adjustable slide, the linear slide, and the cycloidal slide are all made of aluminum alloy, PVC, or glass fiber reinforced polypropylene composite material.

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