A non-excited damped free vibration differential equation verification model
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- ANHUI UNIV OF SCI & TECH
- Filing Date
- 2023-11-06
- Publication Date
- 2026-06-23
Smart Images
Figure CN117523951B_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The present application relates to the technical field of differential equation verification model, and particularly relates to a verification model of a free vibration differential equation without excitation and with damping. BACKGROUND
[0002] The free vibration differential equation without excitation and with damping describes a situation where an object vibrates freely under the influence of internal damping forces without external excitation. This situation is commonly encountered in physics and engineering, with a common example being a spring-mass system. Generally, this situation can be described by a linear second-order ordinary differential equation, which has the following form: This differential equation describes a situation where an object vibrates freely under the influence of damping and elastic forces without external excitation.
[0003] The free vibration differential equation without excitation and with damping is widely used in production and life, for example: mechanical engineering and construction work: in mechanical systems, the free vibration differential equation can be used to analyze the free vibration of spring-mass systems, pendulums, mechanical vibrators and structures. This is very important for designing and optimizing mechanical systems to reduce vibration, noise and stress. In construction engineering, the free vibration differential equation without excitation and with damping can be used to analyze the free vibration of buildings or bridges to ensure the stability and wind resistance of their structures. In the process of scientific calculation, not only the theoretical calculation results are needed, but also the experimental model is needed to verify whether the theoretical structure of the free vibration differential equation without excitation and with damping is correct. However, in the teaching process of the existing free vibration differential equation without excitation and with damping, students cannot intuitively understand the free vibration differential equation without excitation and with damping through the teacher's explanation, and cannot intuitively observe the oscillation curve generated by the vibration of the free vibration differential equation without excitation and with damping, which reduces the learning efficiency of students.
[0004] In view of the above situation, in order to overcome the above technical problems, the present application designs a verification model of a free vibration differential equation without excitation and with damping, which solves the above technical problems. SUMMARY
[0005] The technical purpose to be achieved by the present application is to solve the problem that the existing free vibration differential equation without excitation and with damping in teaching can demonstrate the vibration process of the free vibration differential equation without excitation and with damping very intuitively through the verification model of the free vibration differential equation without excitation and with damping, and draw the oscillation curve graph, which is convenient for students to understand and improves the learning efficiency of students.
[0006] To achieve the above-mentioned technical objectives, the present invention provides the following technical solution: a verification model for the differential equation of unexcited damped free vibration, comprising a support, a base plate, and a motor. The base plate is mounted on the support, and the motor is mounted on the base plate. The invention also includes an experimental mechanism, which is mounted above the base plate and fixedly connected to the motor. The experimental mechanism pulls a U-shaped rod via an operation button, causing a sliding block to move. The moving sliding block vibrates through a telescopic spring and reciprocates back and forth on a fixed box. The reciprocating sliding block causes a conical pen to leave a motion trajectory on flexible drawing paper.
[0007] When verifying the differential equation of unexcited damped free vibration, it is necessary to perform multiple verifications using a verification model and conduct experimental verification on an experimental device. This involves measuring the changes in the position, velocity, and acceleration of the vibrating object over time, and comparing these data with the predictions of the theoretical model to determine its accuracy. However, experimental data often deviates from theoretical data. Therefore, multiple experiments under the same conditions are required, and the final results are compared with the predictions of the theoretical model to verify the differential equation of unexcited damped free vibration. This equation can be used to describe the unexcited damped free vibration system under study.
[0008] The experimental mechanism includes an operating box, a support base, a fixed box, conveying rollers, sliding blocks, and a moving component. The operating box is installed on one side of the support base, the support base is installed on a base plate, the fixed box is installed at the center of the support base, multiple conveying rollers are respectively installed on the top two sides and the bottom one side of the support base, the sliding block is installed on the fixed box, the sliding block is located inside the fixed box and has a gap between it and the inner wall of the fixed box, and they cooperate with each other. The moving component is installed at the bottom of the support base.
[0009] To ensure the accuracy of the experimental verification of the differential equation for unexcited damped free vibration, a certain gap must be maintained between the sliding block and the fixed box during the verification process. This allows the sliding block to slide back and forth on the fixed box, causing the telescopic spring to vibrate back and forth. Simultaneously, the vibration of the telescopic spring is displayed as an oscillation curve by rotating the conveyor roller. It should be noted that during the verification of the differential equation for unexcited damped free vibration, the initial position of the sliding block is adjusted by moving the component, and then the sliding block is released, allowing the sliding block and the telescopic spring to vibrate freely without excitation, thereby obtaining the final experimental data.
[0010] The fixing box has a conical hollow structure. An electromagnet is installed at the bottom of the conical hollow fixing box. A telescopic spring is snapped into one side of the inside of the conical hollow fixing box. The length of the telescopic spring is 1 / 2 of the length of the fixing box. The inner wall of the conical hollow fixing box is an electromagnetic wall.
[0011] To ensure that the differential equation of unexcited damped free vibration is not affected by external friction during verification, the electromagnetic wall of the conical hollow fixed box generates a repulsive force with the sliding block. When gravity equals the repulsive force, the sliding block will suspend in the air, minimizing friction during vibration and preventing other external forces from affecting the free vibration, thus improving the accuracy of experimental verification. Simultaneously, the length of the telescopic spring is half the length of the fixed box. This allows for vibration under different initial displacement conditions, enabling analysis of the differential equation of unexcited damped free vibration. An electromagnet is installed at the bottom of the hollow fixed box. This is because, when adjusting the initial position of the sliding block, the movement of the electromagnet drives the sliding block on the fixed box to move, compressing the telescopic spring and thus determining the initial displacement. Finally, the differential equation of unexcited damped free vibration is verified.
[0012] The sliding block includes a conical block, a rectangular block, a cylinder, and a conical pen; one end of the conical block is engaged with a telescopic spring, a rectangular block is installed on the top of the conical block, a cylinder is installed on one side of the rectangular block, the conical pen is installed on the cylinder, a circular groove is opened at the outer end of the cylinder, a compression spring is engaged inside the circular groove, and the compression spring is engaged with the conical pen, the conical block is a magnetic hollow block, a base plate is provided at the bottom of the conical block, and the outermost end of the conical pen extends 1-2 mm beyond the vertical tangent of the conveyor roller.
[0013] During the vibration of the sliding block, the vibration amplitude of the sliding block and the telescopic spring needs to be represented by an oscillation curve using a conical pen. The conical pen on the sliding block is a thermal pen that can draw the vibration amplitude. The circular groove is fitted with a compression spring, and the outermost end of the conical pen extends 1-2 mm beyond the vertical tangent of the conveyor roller. This is to ensure that the conical pen remains in contact with the flexible drawing paper during the movement of the sliding block. Furthermore, the contact distance between the conical pen and the flexible drawing paper can be adaptively adjusted during the vibration of the sliding block to prevent the conical pen from damaging the flexible drawing paper. At the same time, the conical block is a magnetic hollow block used to reduce the weight of the entire sliding block. The magnetic repulsion between the fixed box and the conical block allows the sliding block to suspend between the fixed boxes, ensuring that the sliding block does not generate friction with the inner wall of the fixed box when vibrating on the fixed box, thereby reducing the verification error of the differential equation of unexcited damped free vibration.
[0014] The moving component includes a U-shaped rod, an operation button, a conductive sheet, a support spring, and a roller; one end of the U-shaped rod is connected to an electromagnet, and the other end of the U-shaped rod is equipped with an operation button. The bottom of the conductive sheet is engaged with a support spring, and the length of the conductive sheet is equal to the displacement of the U-shaped rod. The support spring is engaged with the bottom of the control box, and the roller is installed at the bottom of one end of the U-shaped rod.
[0015] It should be noted that the initial displacement adjustment of the sliding block is achieved through a moving component. A wire is installed inside the U-shaped rod of the moving component and connected to an electromagnet. The contact between the U-shaped rod and the conductive plate allows current to flow through the electromagnet, generating a strong magnetism that attracts the sliding block. Moving the operating button moves the sliding block via the U-shaped rod, thus moving it to its initial position and recording the initial displacement. This allows for adjustment of the initial displacement. Then, by pressing the operating button again, the internal circuit of the electromagnet is cut off, causing the sliding block to vibrate under the action of the extension spring. The length of the conductive plate is equal to the displacement of the U-shaped rod, allowing for different initial position adjustments of the sliding block.
[0016] The operation button includes a circular block, an elastic diaphragm, a circular shell, an insulating post, and a tension spring. An elastic diaphragm is installed on the outer periphery of the circular block and is connected to the top of the circular shell. The insulating post is installed at the bottom of the circular block, and a tension spring is snapped onto the outer side of the insulating post. The distance from the bottom of the circular block to the bottom of the circular shell is the extension distance of the insulating post.
[0017] To ensure that the electromagnet is de-energized after the slider moves to its initial position, the insulating post on the operation button must be moved downwards, pressing down the conductive plate to prevent it from contacting the U-shaped rod, thus breaking the circuit and separating the electromagnet from the slider. This allows the slider to vibrate under the action of the extension spring. The extension spring is used to reset the insulating post after the circuit is broken and the operation button is moved to the far right of the fixed box. The distance from the bottom of the circular block to the bottom of the circular shell is the extension distance of the insulating post, which is to ensure that the conductive plate can completely separate from the U-shaped rod.
[0018] The top of the control box has a rectangular through hole, the width of which is slightly larger than the width of the U-shaped rod, and a scale is provided on the inner side of the rectangular through hole.
[0019] The width of the rectangular through hole is slightly larger than the width of the U-shaped rod. This is to ensure that the U-shaped rod can slide smoothly on the operating box. At the same time, a scale is provided on the inside of the rectangular through hole. The initial displacement of the sliding block can be accurately adjusted according to the scale and the initial displacement can be recorded. Vibration experiments can be conducted on different initial displacements. The initial position of the sliding block can slide to the left or to the right.
[0020] The conveyor belt is provided at one end of the conveyor roller inside the support base, and flexible drawing paper is provided on the conveyor roller at the bottom of the support base. The two ends of the flexible drawing paper are respectively connected to the conveyor rollers on both sides. The flexible drawing paper is semi-transparent.
[0021] To ensure that the flexible drawing paper unfolds and rolls up simultaneously, a conveyor belt connects the two conveyor rollers of the support base. This ensures that the two rollers rotate in the same direction and at the same speed, guaranteeing the unfolding and rolling of the flexible drawing paper while preventing excessive stress and damage to the paper due to different rotation speeds. The flexible drawing paper is semi-transparent so that, during experiments, the operator can directly observe the oscillation curves drawn on it using a conical pen.
[0022] The upper sides of the support base are provided with elongated through holes, the width of which is slightly larger than the thickness of the flexible drawing paper. A strip hole is provided in the center of the support box, and the strip hole is connected to the bottom of the fixed box.
[0023] The width of the elongated through hole is slightly larger than the thickness of the flexible drawing paper. This is to prevent the flexible drawing paper from rubbing against the elongated through hole on the support during the unfolding and rolling process, which would damage the flexible drawing paper. At the same time, a strip-shaped hole is opened in the center of the support box so that the electromagnet can contact the bottom plate of the sliding block, thereby driving the sliding block to move.
[0024] The front side of the support is provided with a paper outlet hole. The top of the paper outlet hole is provided with a serrated protrusion. The height of the paper outlet hole is slightly greater than the thickness of the flexible drawing paper. Two transport rollers are provided inside the paper outlet hole. One end of the two transport rollers is provided with a gear that meshes with each other, and any one of the transport rollers is connected to the conveyor belt.
[0025] It should be noted that during the vibration of the sliding block, the vibration amplitude will be drawn on the flexible drawing paper using a conical pen. As the sliding block continues to vibrate, the motor will drive the conveyor roller to rotate, thereby unfolding the flexible drawing paper. An oscillation curve of the vibration amplitude will appear on the flexible drawing paper. The height of the paper outlet is slightly greater than the thickness of the flexible drawing paper to ensure that the flexible drawing paper can come out of the paper outlet. As the conveyor roller rotates, the flexible drawing paper will pass between the two conveyor rollers and come out of the paper outlet until the slider stops oscillating. Then, the flexible drawing paper needs to be removed by the serrated protrusions to facilitate the observation of the oscillation curve on the flexible drawing paper.
[0026] The beneficial effects of this invention are as follows:
[0027] 1. This invention uses multiple conveying rollers on an experimental mechanism, and a sliding block that interacts with flexible drawing paper on the conveying rollers. When the sliding block vibrates, the flexible drawing paper is rotated by the conveying rollers, thus visually plotting the oscillation curve of the sliding block on the flexible drawing paper. This clearly demonstrates the process of generating the oscillation curve of the differential equation of unexcited damped free vibration, making it easier for students to understand the differential equation of unexcited damped free vibration and greatly improving their learning efficiency.
[0028] 2. This invention features a movable component on the experimental mechanism, which allows for arbitrary adjustment of the initial displacement during the experiment. The theoretical model of the differential equation of unexcited damped free vibration can be verified through multiple experiments under the same conditions. Furthermore, the flexible drawing paper can be removed, allowing students to intuitively observe the oscillation curve on the flexible drawing paper.
[0029] 3. This invention uses a fixed box and a sliding block in the experimental mechanism. The fixed box and the sliding block work together to generate a magnetic repulsive force, which creates a gap between the sliding block and the inner wall of the fixed box. This reduces the friction between the sliding block and the fixed box and improves the accuracy of verifying the differential equation of unexcited damped free vibration. Attached Figure Description
[0030] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.
[0031] The above and other aspects of the invention will now be described by way of example only, with reference to the accompanying drawings, in which:
[0032] Figure 1This is a three-dimensional structural diagram of the entire invention;
[0033] Figure 2 This is a three-dimensional structural diagram of the flexible drawing paper of the present invention;
[0034] Figure 3 This is a schematic diagram of the internal three-dimensional structure of the support base of the present invention;
[0035] Figure 4 This is a three-dimensional structural schematic diagram of the support base of the present invention;
[0036] Figure 5 This is a three-dimensional structural diagram of the sliding block of the present invention;
[0037] Figure 6 This is a cross-sectional view of the sliding block of the present invention;
[0038] Figure 7 This is a three-dimensional structural diagram of the internal structure of the operating box of the present invention;
[0039] Figure 8 This is a cross-sectional view of the operation button of the present invention;
[0040] Figure 9 This is a partial cross-sectional view of the entire invention;
[0041] Figure 10 This is a force analysis diagram of the sliding block and the fixed box of the present invention;
[0042] Figure 11 This is a schematic diagram of the working principle of the present invention;
[0043] Figure 12 This is the oscillation curve diagram of the present invention.
[0044] In the diagram: 1. Support; 2. Base plate; 3. Experimental mechanism; 31. Control box; 311. Rectangular through hole; 312. Scale; 32. Support seat; 321. Long through hole; 322. Strip hole; 323. Paper outlet hole; 324. Protrusion; 325. Transport roller; 326. Gear; 33. Fixing box; 331. Electromagnet; 332. Telescopic spring; 34. Conveyor roller; 341. Conveyor belt; 342. Flexible drawing paper; 35. Sliding block; 51. Conical block; 352. Rectangular block; 353. Cylinder; 354. Conical pen; 355. Circular groove; 356. Compression spring; 357. Base plate; 36. Moving component; 361. U-shaped rod; 362. Operation button; 3621. Circular block; 3622. Elastic film; 3623. Circular shell; 3624. Tension spring; 3625. Insulating post; 363. Conductive sheet; 364. Support spring; 365. Roller; 4. Motor. Detailed Implementation
[0045] To better understand the above technical solutions, the following will provide a detailed explanation of the technical solutions in conjunction with the accompanying drawings and specific implementation methods.
[0046] like Figures 1 to 12 As shown, a verification model for the differential equation of unexcited damped free vibration includes a support 1, a base plate 2, and a motor 4. The base plate 2 is mounted on the support 1, and the motor 4 is mounted on the base plate 2. The model also includes an experimental mechanism 3, which is mounted above the base plate 2 and is fixedly connected to the motor 4. The experimental mechanism 3 pulls the U-shaped rod 361 through the operation button 362, which drives the sliding block 35 to move. The moving sliding block 35 vibrates through the extension spring 332 and moves back and forth on the fixed box 33. The reciprocating sliding block 35 drives the conical pen 354 to leave a motion trajectory on the flexible drawing paper 342.
[0047] When verifying the differential equation of unexcited damped free vibration, the data on the experimental mechanism 3 needs to be adjusted. First, initial displacement data needs to be provided. Then, the reciprocating vibration of the extension spring 332 is used, and the velocity and acceleration of the vibrating object are measured by a photoelectric sensor. Multiple experiments are conducted for verification, and the initial conditions must be kept consistent each time to reduce the error generated by the experimental verification and ensure the accuracy of the verification of the differential equation of unexcited damped free vibration. Finally, the experimental results are used to drive the conveyor roller 34 to rotate through the rotation of the motor 4, and an oscillation curve is plotted. The experimental results can be clearly obtained, which is convenient for comparison with the theoretical model. This reduces the error generated during the verification of the differential equation of unexcited damped free vibration and greatly improves the accuracy of the verification.
[0048] like Figure 2 and Figure 3 As shown, the experimental mechanism 3 includes an operation box 31, a support base 32, a fixed box 33, conveying rollers 34, sliding blocks 35, and a moving component 36. The operation box 31 is installed on one side of the support base 32, the support base 32 is installed on the base plate 2, the fixed box 33 is installed at the center of the support base 32, multiple conveying rollers 34 are respectively installed on the top two sides and the bottom one side of the support base 32, the sliding block 35 is installed on the fixed box 33, the sliding block 35 is located inside the fixed box 33, and there is a gap between the sliding block 35 and the inner wall of the fixed box 33, and they cooperate with each other. The moving component 36 is installed at the bottom of the support base 32.
[0049] When experimental unit 3 verifies the differential equation of unexcited damped free vibration, firstly, the moving component 36 needs to be manually adjusted to give the sliding block 35 an initial position. Then, the sliding block 35 is released, and the free vibration of the telescopic spring 332 causes the sliding block 35 to slide back and forth on the fixed box 33 until the sliding block 35 stops sliding. At this time, the telescopic spring 332 will also stop vibrating. During the vibration process, the initial velocity and acceleration of the sliding block 35 are measured by a photoelectric sensor. At the same time, the slider will be drawn into an oscillation curve by the rotation of the conveyor roller 34 during the vibration process. The oscillation curve and various data are analyzed. Multiple experiments are required. Finally, the results are compared with the theoretical model to determine the result of the differential equation of unexcited damped free vibration.
[0050] like Figure 4 and Figure 10 As shown, the fixing box 33 is a conical hollow structure. An electromagnet 331 is provided at the bottom of the conical hollow fixing box 33. A telescopic spring 332 is snapped into one side of the inside of the conical hollow box. The length of the telescopic spring 332 is 1 / 2 of the length of the fixing box 33. The inner wall of the conical hollow fixing box 33 is an electromagnetic wall.
[0051] During the verification of the differential equation of unexcited damped free vibration, the electromagnet 331 is energized by the moving component 36. After being energized, the electromagnet 331 generates a large magnetic attraction force, which attracts the sliding block 35. The moving component 36 drives the electromagnet 331 to move, thereby moving the sliding block 35 to the initial displacement position. Then, the electromagnet 331 is de-energized, and at the same time, the electromagnetic wall on the inner wall of the fixed box 33 is energized, generating a repulsive force on the sliding block 35, causing the sliding block 35 to suspend upward. Through the elastic potential energy of the extension spring 332, the sliding block 35 vibrates under the action of the extension spring 332. This reduces the error generated during the verification of the differential equation of unexcited damped free vibration and greatly improves the accuracy of the verification.
[0052] like Figure 5 , Figure 6 and Figure 10As shown, the sliding block 35 includes a conical block 351, a rectangular block 352, a cylinder 353, and a conical pen 354. One end of the conical block 351 is engaged with a telescopic spring 332. A rectangular block 352 is installed on the top of the conical block 351. A cylinder 353 is installed on one side of the rectangular block 352. A conical pen 354 is installed on the cylinder 353. A circular groove 355 is provided at the outer end of the cylinder 353. A compression spring 356 is engaged inside the circular groove 355, and the compression spring 356 is engaged with the conical pen 354. The conical block 351 is a magnetic hollow block. A base plate 357 is provided at the bottom of the conical block 351. The outermost end of the conical pen 354 extends 1-2 mm beyond the vertical tangent of the conveyor roller 34.
[0053] During the vibration of the sliding block 35, the sliding block 35 will drive the rectangular block 352, the cylinder 353 and the conical pen 354 to move back and forth. The contact distance between the conical pen 354 and the flexible drawing paper 342 will be adaptively adjusted by the compression spring 356. The vibration amplitude of the sliding block 35 and the extension spring 332 will be drawn on the flexible drawing paper 342 by the conical pen 354, thereby obtaining the vibration amplitude of the differential equation of unexcited damped free vibration. This reduces the error generated during the verification of the differential equation of unexcited damped free vibration and improves the accuracy of the verification.
[0054] like Figure 7 and Figure 9 As shown, the moving component 36 includes a U-shaped rod 361, an operation button 362, a conductive sheet 363, a support spring 364, and a roller 365. The top of one end of the U-shaped rod 361 is connected to an electromagnet 331, and the operation button 362 is installed on the top of the other end of the U-shaped rod 361. The bottom of the conductive sheet 363 is engaged with the support spring 364. The length of the conductive sheet 363 is equal to the displacement of the U-shaped rod 361. The support spring 364 is engaged with the bottom of the operation box 31, and the roller 365 is installed at the bottom of one end of the U-shaped rod 361.
[0055] During initial displacement adjustment, the operation button 362 is manually slidable. The button 362 causes the U-shaped rod 361 to slide left and right within the fixed box 33. Simultaneously, the roller 365 at the bottom of the U-shaped rod 361 rolls at the bottom of the fixed box 33. When the operation button 362 is slidable, current flows through the entire U-shaped rod 361 and the electromagnet 331, causing the electromagnet 331 to contact the sliding block 35, thus moving the electromagnet 331. When it reaches the initial position, the operation button 362 is manually pressed, causing the conductive plate 363 to move downwards. The support spring 364 at the bottom of the conductive sheet 363 will be compressed, thereby breaking the circuit. The electromagnet 331 will no longer generate magnetism, and the sliding block 35 will vibrate on the fixed box 33. At the same time, the operation button 362 is moved to the far right of the fixed box 33, and the electromagnet 331 will move away from the sliding block 35. When the operation button 362 is released, the conductive sheet 363 will move upward due to the elastic potential energy of the support spring 364 and contact the U-shaped rod 361, thus connecting the circuit. Since the electromagnet 331 is away from the sliding block 35, it will not affect the vibration of the sliding block 35.
[0056] like Figure 8 As shown, the operation button 362 includes a circular block 3621, an elastic film 3622, a circular shell 3623, an insulating post 3625, and a tension spring 3624. The elastic film 3622 is installed on the outer periphery of the circular block 3621. The elastic film 3622 is connected to the top of the circular shell 3623. The insulating post 3625 is installed on the bottom of the circular block 3621. The tension spring 3624 is snapped onto the outer side of the insulating post 3625. The distance from the bottom of the circular block 3621 to the bottom of the circular shell 3623 is the extension distance of the insulating post 3625.
[0057] When the sliding block 35 moves to the initial position, manually press the round block 3621. The round block 3621 will drive the insulating column 3625 to move downward. At the same time, the tension spring 3624 will be stretched, and the insulating column 3625 will come into contact with the conductive plate 363, pressing the conductive plate 363 down. This will break the circuit between the conductive plate 363 and the U-shaped rod 361, causing the magnetism of the electromagnet 331 to disappear. As a result, the sliding block 35 will vibrate. Then, move the operation button 362 to the far right of the fixed box 33 and release the round block 3621. Under the action of the tension spring 3624, the insulating column 3625 will move upward, causing the conductive plate 363 to move upward and come into contact with the U-shaped rod 361. The circuit will be connected. At this time, since the electromagnet 331 is far away from the sliding block 35, it will not affect the vibration of the sliding block 35.
[0058] like Figure 2As shown, a rectangular through hole 311 is provided on the top of the operation box 31. The width of the rectangular through hole 311 is slightly larger than the width of the U-shaped rod 361. A scale 312 is provided on the inner side of the rectangular through hole 311.
[0059] When adjusting the initial displacement of the slider 35, the operation button 362 is moved to the designated position using the scale 312, and the initial displacement at this position is recorded, thereby conducting experimental verification of the differential equation of unexcited damped free vibration.
[0060] like Figure 3 , Figure 7 and Figure 9 As shown, a conveyor belt 341 is provided at one end of the conveyor roller 34 inside the bearing seat 32, and a flexible drawing paper 342 is provided on the conveyor roller 34 at the bottom of the bearing seat 32. The two ends of the flexible drawing paper 342 are respectively connected to the conveyor rollers 34 on both sides. The flexible drawing paper 342 is semi-transparent.
[0061] When verifying the differential equation of unexcited damped free vibration, the motor 4 will control the rotation of the conveyor roller 34 inside the bearing seat 32, thereby driving the conveyor belt 341 to rotate and driving another conveyor roller 34 to rotate. While the flexible drawing paper 342 is unfolded, it is rolled up. At the same time, the conical pen 354 on the sliding block 35 will draw an oscillation curve on the flexible drawing paper 342.
[0062] like Figure 4 As shown, elongated through holes 321 are respectively opened on the upper two sides of the support base 32. The width of the elongated through holes 321 is slightly larger than the thickness of the flexible drawing paper 342. A strip hole 322 is opened at the center of the support box, and the strip hole 322 is connected to the bottom of the fixed box 33.
[0063] like Figure 2 and Figure 3 As shown, the front side of the support 32 is provided with a paper outlet hole 323. The top of the paper outlet hole 323 is provided with a serrated protrusion 324. The height of the paper outlet hole 323 is slightly greater than the thickness of the flexible drawing paper 342. Two transport rollers 325 are provided inside the paper outlet hole 323. One end of the two transport rollers 325 is provided with a gear 326 that meshes with each other, and any one of the transport rollers 325 is connected to the conveyor belt 341.
[0064] When the conical pen 354 on the sliding block 35 draws an oscillation curve on the flexible drawing paper 342, the flexible drawing paper 342 will be driven by the rotation of the transport roller 325 to be output through the paper outlet 323. When the sliding block 35 stops moving, the transport roller 325 will stop rotating. Then, the flexible drawing paper 342 is manually lifted from one side and broken by the serrated protrusion 324 on the paper outlet 323, thus completing the removal of the flexible drawing paper 342 for subsequent observation.
[0065] In the experiment, the flexible drawing paper 342 needs to be installed inside the support base 32. First, the operator opens the side door of the support base 32, and then inserts a rolled-up flexible drawing paper 342 horizontally from one end of the conveyor roller 34 to the other end. Then, one end of the flexible drawing paper 342 is pulled and passes through the elongated through hole 321 at the top of the support base 32. After passing through the two conveyor rollers 34 at the top of the support base 32, it enters the interior of the support base 32 through the elongated through hole 321 on the other side. After passing between the two conveyor rollers 325, it exits through the paper outlet hole 323. Then, the side door of the support base 32 is closed, thus completing the installation of the flexible drawing paper 342.
[0066] The working process of this invention: as follows Figure 11 and Figure 12As shown, when verifying the differential equation of unexcited damped free vibration, the data on experimental mechanism 3 needs to be adjusted. First, the operator needs to manually slide the operation button 362. The operation button 362 will drive the U-shaped rod 361 to slide left or right within the fixed box 33. During the sliding process, the roller 365 at the bottom of the U-shaped rod 361 rolls at the bottom of the fixed box 33. At the same time, the U-shaped rod 361 slides on the conductive plate 363, and current will flow between the entire U-shaped rod 361 and the electromagnet 331, making... When electromagnet 331 is energized, it generates an electromagnetic attraction force and comes into contact with the base plate 357 at the bottom of sliding block 35. As U-shaped rod 361 moves, electromagnet 331 drives sliding block 35 to move. The operator moves operation button 362 to a designated position by observing scale 312 and records the initial displacement. Then, the operator manually presses the circular block 3621 on operation button 362. The circular block 3621 drives insulating post 3625 downwards, simultaneously stretching tension spring 3624. When 625 comes into contact with the conductive sheet 363 and presses down on the conductive sheet 363, the supporting spring 364 at the bottom of the conductive sheet 363 will be compressed, thereby breaking the circuit between the conductive sheet 363 and the U-shaped rod 361. The magnetism of the electromagnet 331 will disappear. At the same time, the operator will manually turn on the energizing switch of the electromagnetic wall on the inner wall of the fixed box 33, so that the electromagnetic wall on the inner wall of the fixed box 33 is energized, and a repulsive force is generated on the conical block 351 on the inclined side of the sliding block 35, causing the sliding block 35 to levitate upwards and slide. Block 35 vibrates under the action of the extension spring 332 due to the elastic potential energy of the extension spring 332. Then, after moving the operation button 362 to the rightmost side of the fixed box 33, the round block 3621 is released. Under the action of the tension spring 3624, the insulating column 3625 moves upward, and the conductive sheet 363 will move upward due to the elastic potential energy of the supporting spring 364 and contact the U-shaped rod 361, thus connecting the circuit. Since the electromagnet 331 is far away from the sliding block 35, it will not affect the vibration of the sliding block 35.
[0067] At this time, the motor 4 will control the conveyor roller 34 inside the support 32 to rotate, thereby driving the conveyor belt 341 to rotate and driving another conveyor roller 34 to rotate. The flexible drawing paper 342 unfolds on the conveyor roller 34 on one side. During the vibration of the sliding block 35, the sliding block 35 will drive the rectangular block 352, the cylinder 353 and the conical pen 354 to move back and forth. The contact distance between the conical pen 354 and the flexible drawing paper 342 will be adaptively adjusted by the compression spring 356. The conical pen 354 will draw an oscillation curve on the flexible drawing paper 342.
[0068] When the conical pen 354 on the sliding block 35 draws an oscillation curve on the flexible drawing paper 342, the flexible drawing paper 342 will drive the rotation of the transport roller 325 through the rotation of the conveyor belt. The gears 326 on the two transport rollers 325 mesh with each other and rotate, causing the two transport rollers 325 to rotate towards each other and drive the flexible drawing paper 342 to be output through the paper outlet 323. When the sliding block 35 stops moving, the transport roller 325 will stop rotating. Then, the flexible drawing paper 342 can be manually lifted from one side and cut through the serrated protrusion 324 on the paper outlet 323, thus completing the removal of the flexible drawing paper 342 for subsequent observation.
[0069] Furthermore, during the vibration process, the initial velocity and acceleration of the sliding block 35 were measured using a photoelectric sensor. Multiple experiments were then conducted for verification, ensuring consistent initial conditions for each verification to minimize errors and guarantee the accuracy of the verification of the differential equation for unexcited damped free vibration. Analysis of the oscillation curves and data was performed, and the experimental results were compared with the theoretical model. This process effectively verified the differential equation for unexcited damped free vibration, significantly reducing errors and improving accuracy.
[0070] The technical features disclosed above are not limited to combinations of the disclosed features with other features. Those skilled in the art may also make other combinations of the technical features according to the purpose of the disclosure in order to achieve the purpose of this disclosure.
Claims
1. A verification model for the differential equation of unexcited damped free vibration, comprising a support (1), a base plate (2), and a motor (4), wherein the base plate (2) is mounted on the support (1), and the motor (4) is mounted on the base plate (2), characterized in that, It also includes an experimental mechanism (3), which is installed above the base plate (2) and is fixedly connected to the motor (4). The experimental mechanism (3) pulls the U-shaped rod (361) and drives the sliding block (35) to move by operating the button (362). The moving sliding block (35) vibrates by the extension spring (332) and moves back and forth on the fixed box (33). The reciprocating sliding block (35) drives the conical brush (354) to leave a motion trajectory on the flexible drawing paper (342). The experimental mechanism (3) includes an operation box (31), a support seat (32), a fixed box (33), conveying rollers (34), sliding blocks (35), and a moving component (36); the operation box (31) is installed on one side of the support seat (32), the support seat (32) is installed on the base plate (2), the fixed box (33) is installed at the center of the support seat (32), multiple conveying rollers (34) are respectively installed on the top two sides and the bottom one side of the support seat (32), the sliding block (35) is installed on the fixed box (33), the sliding block (35) is located inside the fixed box (33), and there is a gap between the sliding block (35) and the inner wall of the fixed box (33), and they cooperate with each other; the moving component (36) is installed at the bottom of the support seat (32); The fixed box (33) is a conical hollow structure. An electromagnet (331) is provided at the bottom of the fixed box (33). A telescopic spring (332) is snapped into one side of the inside of the box. The length of the telescopic spring (332) is 1 / 2 of the length of the fixed box (33). The inner wall of the fixed box (33) is an electromagnetic wall. The sliding block (35) includes a conical block (351), a rectangular block (352), a cylinder (353), and a conical pen (354); one end of the conical block (351) is engaged with a telescopic spring (332), a rectangular block (352) is installed on the top of the conical block (351), a cylinder (353) is installed on one side of the rectangular block (352), a conical pen (354) is installed on the cylinder (353), a circular groove (355) is opened at the outer end of the cylinder (353), a compression spring (356) is engaged inside the circular groove (355), and the compression spring (356) is engaged with the conical pen (354). The conical block (351) is a magnetic hollow block, a base plate (357) is provided at the bottom of the conical block (351), and the outermost end of the conical pen (354) extends 1-2 mm beyond the vertical tangent of the conveying roller (34). The moving component (36) includes a U-shaped rod (361), an operation button (362), a conductive sheet (363), a support spring (364), and a roller (365). One end of the U-shaped rod (361) is connected to an electromagnet (331), and the other end of the U-shaped rod (361) is equipped with an operation button (362). The bottom of the conductive sheet (363) is engaged with a support spring (364). The length of the conductive sheet (363) is equal to the displacement of the U-shaped rod (361). The support spring (364) is engaged with the bottom of the operating box (31). The roller (365) is installed at the bottom of one end of the U-shaped rod (361).
2. The verification model for the differential equation of unexcited damped free vibration according to claim 1, characterized in that: The operation button (362) includes a circular block (3621), an elastic membrane (3622), a circular shell (3623), an insulating post (3625), and a tension spring (3624). The elastic membrane (3622) is installed on the outer periphery of the circular block (3621). The elastic membrane (3622) is connected to the top of the circular shell (3623). The insulating post (3625) is installed at the bottom of the circular block (3621). The tension spring (3624) is snapped onto the outer side of the insulating post (3625). The distance from the bottom of the circular block (3621) to the bottom of the circular shell (3623) is the extension distance of the insulating post (3625).
3. The verification model for the differential equation of unexcited damped free vibration according to claim 1, characterized in that: The top of the control box (31) is provided with a rectangular through hole (311), the width of which is slightly larger than the width of the U-shaped rod (361), and a scale (312) is provided on the inner side of the rectangular through hole (311).
4. The verification model for the differential equation of unexcited damped free vibration according to claim 1, characterized in that: The conveyor belt (341) is provided at one end of the conveyor roller (34) inside the bearing seat (32), and a flexible drawing paper (342) is provided on the conveyor roller (34) at the bottom of the bearing seat (32). The two ends of the flexible drawing paper (342) are respectively connected to the conveyor rollers (34) on both sides. The flexible drawing paper (342) is semi-transparent.
5. The verification model for the differential equation of unexcited damped free vibration according to claim 1, characterized in that: The upper sides of the support base (32) are provided with elongated through holes (321), the width of which is slightly larger than the thickness of the flexible drawing paper (342). A strip hole (322) is provided in the center of the support box, and the strip hole (322) is connected to the bottom of the fixed box (33).
6. The verification model for the differential equation of unexcited damped free vibration according to claim 1, characterized in that: The front side of the support (32) is provided with a paper outlet hole (323). The top of the paper outlet hole (323) is provided with a serrated protrusion (324). The height of the paper outlet hole (323) is slightly greater than the thickness of the flexible drawing paper (342). Two transport rollers (325) are provided inside the paper outlet hole (323). One end of the two transport rollers (325) is provided with a gear (326) that meshes with each other. Any one of the transport rollers (325) is connected to the conveyor belt (341).