A fast correction estimation method for continuous elastomer impact force waveforms

Abstract

This invention relates to the field of mechanical engineering and discloses a rapid correction and estimation method for the collision force waveform of a continuous elastic body. Addressing the issue that while the closed-form solution of the collision force pulse obtained under the assumption of a half-period sinusoidal waveform for continuous elastic body collisions has accurate peak value and impulse, it suffers from large errors in waveform and duration. Using the impact of an elastic ball on an elastic plate as the basic physical model, a collision force pulse with a Hertzian contact model is generated, similar to the actual waveform. Then, based on the principle that the peak value and impulse of the collision force pulse remain unchanged, the collision force pulse of the basic physical model is scaled proportionally to obtain a corrected collision force pulse for the continuous elastic body. This corrected pulse has a waveform and duration that are closer to the true waveform, which is significant for improving the prediction accuracy of collision vibration response.

Description

Technical Field

[0001] This invention relates to the field of mechanical engineering, and more specifically to a rapid correction and estimation method for the collision force waveform of a continuous elastic body. Background Technology

[0002] In real-world mechanical systems, due to insufficient structural stiffness or assembly issues, localized collisions or impacts often occur in areas with small gaps under dynamic environments or operating conditions. Furthermore, real-world mechanical structures are essentially continuous elastic bodies. Predicting impact noises in continuous elastic bodies hinges on calculating the collision or impact vibration response of the continuous elastic structure. Calculating the vibration response of structures with nonlinear characteristics such as collisions using traditional finite element methods incurs high time costs. A faster approach to predicting collision vibration response is to fully utilize the system's local nonlinear characteristics, isolating the impact kinematic pairs for separate treatment. The impact force is treated as a secondary excitation force of the continuous elastic mechanical structure, allowing the complex mechanical structure to still be treated as a linear system. Linear system solving algorithms can then be used to quickly predict the response of the entire collision system. For example, Chinese patent CN202111614142.3 discloses an equivalent model and modeling method for a continuous elastic body impact kinematic pair. It represents the secondary excitation force with an equivalent collision force model and assumes that the impact force pulse is a half-cycle sine wave. The Hertz contact model and the origin transfer function of the mechanical structure at the impact contact point are used to express the contact mechanics of the impact pair material pair and the impedance properties of the mechanical structure at the impact point, respectively. Thus, the relationship between the output and input of the nonlinear equivalent model is derived. With the initial collision velocity as the input, the peak value and frequency of the half-wave sine wave can be obtained.

[0003] While this assumption can yield relatively accurate peak collision force and impulse, the assumed half-wave sinusoidal force pulse waveform differs significantly from the actual force pulse waveform, and the force pulse duration also differs significantly from the measured value. This will lead to substantial discrepancies in subsequent collision vibration response analysis results. Summary of the Invention

[0004] The present invention aims to provide a rapid correction estimation method for the collision force waveform of a continuous elastic body, in order to solve the problem that there is a large difference between the assumed half-wave sinusoidal force pulse waveform and the actual force pulse waveform in the prior art, and to lay the foundation for improving the analysis accuracy of structural collision vibration response.

[0005] To achieve the above objectives, the basic solution of the present invention is as follows: a fast correction estimation method for the collision force waveform of a continuous elastic body. Based on the closed-form solution of the collision force pulse obtained under the assumption of a half-cycle sine waveform for the collision of a continuous elastic body, the basic physical model is the impact of an elastic ball with a constant initial velocity on an elastic plate. The Hertzian contact force model is used to express the deformation coordination relationship of the collision material pair, thereby obtaining a Hertzian contact force waveform similar to the actual collision force waveform, and replacing the assumed half-cycle sine waveform with it.

[0006] The principle and beneficial effects of this invention are as follows: The Hertzian contact force model is used to express the contact mechanics properties of the striking pair of materials, and a simple ball-plate collision model is used to establish dynamic equations to accurately solve for the Hertzian contact force pulse. Based on the basic physical model of the collision between a small ball and a plate, the established dynamic equations are as follows:

[0007]

[0008] in

[0009]

[0010] In the formula, m is the mass of the small ball, R is the radius of the small ball, δ is the depth of the indentation caused by deformation, and F Hertz For Hertzian contact force, K H α and α represent the contact stiffness and Hertzian contact constant of the striking material pair, respectively; v1 and v2 represent the Poisson's ratios of the small ball and the plate, respectively; and E1 and E2 represent the elastic moduli of the small ball and the plate, respectively.

[0011] To simulate the deformation behavior of colliding material pairs, the contact stiffness K in the corresponding ball-plate collision model is used. H The contact stiffness should be the same as that of the actual continuous elastic body collision point. Therefore, the material pair of the ball colliding with the plate can be the same as that of the actual continuous elastic body collision, and the radius R of the ball can be the equivalent radius at the contact point of the actual collision structure. Thus, the parameters in the Hertz contact force model are determined, and the mass of the ball in equation (1) can be obtained from the material parameters. Using the numerical solution method of nonlinear equations, such as the fourth-order Rung-Kutta numerical algorithm, the Hertz contact force pulse F of the ball-plate collision model at a certain initial collision velocity can be solved. Hertz .

[0012] Furthermore, based on the peak value and impulse of the closed-form solution of the half-cycle sinusoidal collision force pulse, the Hertzian contact force waveform of the obtained basic physical model is scaled to obtain the corrected continuous elastic body collision force pulse.

[0013] The principle and beneficial effects of this invention: The scaled Hertz contact force pulse maintains the same peak value and impulse as the half-wave sinusoidal force pulse, and can reflect the deformation coordination relationship of the colliding or striking material pair. Its waveform is closer to the real waveform, and the pulse force duration is also closer to the real value, thus achieving the purpose of waveform correction.

[0014] Furthermore, the scaling criterion for the Hertzian contact force waveform is to ensure that the pulse peak value and impulse are equal to the peak value and impulse of the closed-form solution of the collision force pulse, respectively, and the specific relationship is as follows:

[0015]

[0016]

[0017] Among them, F sin For a half-cycle sinusoidal impact force pulse, F sin-max Force pulse F sin The peak value, t sin-max Force pulse F sin The duration of F. Hertz For Hertzian contact force pulse, F Hertz-max Force pulse F Hertz The maximum value of F. pre For the corrected continuous elastic body collision force pulse, t pre-max Force pulse F pre The duration.

[0018] The principle and beneficial effects of this invention: Since the half-wave sinusoidal force pulse assumed by the equivalent collision force model can usually obtain a relatively accurate peak value and impulse, the amplitude of the Hertzian contact force can be scaled proportionally using equation (3) so that its maximum value is equal to the peak value of the half-wave sinusoidal force pulse. The width of the Hertzian contact force can be scaled using equation (4) so ​​that its impulse is equal to the impulse of the half-wave sinusoidal force pulse. The scaled Hertzian contact force pulse F pre It retains the peak value and impulse of the half-wave sinusoidal force pulse, has a waveform similar to the shape of the actual force pulse, and at the same time makes the duration of the force pulse closer to the measured value, thus realizing the correction of the half-wave sinusoidal force pulse waveform.

[0019] Furthermore, when assigning material properties to the sphere and the plate, it is only necessary to satisfy that the collision material pair is the same as the collision material pair of the actual continuous elastic body.

[0020] The principle and beneficial effects of this invention: When the colliding material pair of a continuous elastic body consists of two different materials, the mass of the small ball will vary depending on the different material parameters of the small ball. To verify the influence of the small ball's mass on the scaling of the force pulse shape, an initial velocity of v0 = 10 mm / s and a contact stiffness K were selected. H=1.63×105N / mm, exponent α=1.5. Under the condition that other conditions remain unchanged, the mass of the ball is changed, the Hertz contact force is solved, and the Hertz contact forces under different masses are scaled according to a certain peak value and impulse using equations (3) and (4). It can be seen that the force pulses under different masses are completely overlapped after scaling, which shows that the mass of the ball does not affect the shape of the Hertz force pulse. Therefore, when assigning material properties to the ball and the plate, it is sufficient to satisfy that the collision material pair is the same as the actual continuous elastic body material pair, and there is no special order requirement.

[0021] Furthermore, the shape of the half-wave sinusoidal force pulse is rapidly corrected. The specific implementation steps are as follows:

[0022] Step 1: For the continuous elastic body collision kinematic pair, establish the basic physical model of the collision between the elastic ball and the elastic plate;

[0023] Step 2: Establish a Hertzian contact force model for the kinematic pair of the ball and the board during the collision.

[0024] Step 3: Establish the dynamic equations of the ball-plate collision using the Hertz contact force model, and use the numerical solution method of nonlinear equations to accurately solve the collision force waveform at a certain initial collision velocity.

[0025] Step 4: Based on the principle that the peak value and impulse of the closed-form solution of the half-cycle sinusoidal collision force pulse of the continuous elastic body remain unchanged, the collision force pulse waveform of the basic physical model is scaled, that is, its amplitude and duration are scaled respectively, so as to obtain the corrected collision force pulse of the continuous elastic body. Attached Figure Description

[0026] Figure 1 This is a schematic diagram of the physical model of an elastic ball impacting a fixed elastic plate, which is a fast correction estimation method for the collision force waveform of a continuous elastic body in an embodiment of the present invention.

[0027] Figure 2 This is a comparison diagram of Hertzian contact force shape scaling for different ball masses using the fast correction estimation method for the collision force waveform of a continuous elastic body in an embodiment of the present invention.

[0028] Figure 3 This is a photograph of the actual experimental setup for the collision of a cantilever beam and a fixed hammer, which is part of the rapid correction estimation method for the collision force waveform of a continuous elastic body in this embodiment of the invention.

[0029] Figure 4 This is a schematic diagram of a cantilever beam and fixed hammer collision test apparatus for a rapid correction estimation method of the collision force waveform of a continuous elastic body in an embodiment of the present invention.

[0030] Figure 5This is a comparison chart of the predicted and measured data of the equivalent collision force model for a 45# steel-aluminum alloy material pair in the fast correction estimation method for the collision force waveform of a continuous elastic body in this embodiment of the invention.

[0031] Figure 6 This is a comparison chart of the predicted and measured data of the equivalent collision force model for a stainless steel-aluminum alloy material pair in the fast correction estimation method for the collision force waveform of a continuous elastic body in this embodiment of the invention.

[0032] Figure 7 This is a comparison chart of the predicted and measured data of the equivalent collision force model for aluminum alloy-aluminum alloy material pairs in the fast correction estimation method for the collision force waveform of continuous elastic bodies in this embodiment of the invention.

[0033] Figure 8 This is a comparison chart of the predicted force pulse curve, the corrected force pulse curve, and the measured force pulse curve of the equivalent collision force model for the 45# steel-aluminum alloy material pair in the fast correction estimation method for the collision force waveform of the continuous elastic body in the embodiments of the present invention.

[0034] Figure 9 This is a comparison chart of the predicted force pulse curve, the corrected force pulse curve, and the measured force pulse curve of the equivalent collision force model for a stainless steel-aluminum alloy material pair in the fast correction estimation method for the collision force waveform of a continuous elastic body in the embodiments of the present invention.

[0035] Figure 10 This is a comparison chart of the predicted force pulse curve, the corrected force pulse curve, and the measured force pulse curve of the equivalent collision force model for the aluminum alloy-aluminum alloy material pair in the fast correction estimation method for the collision force waveform of the continuous elastic body in the embodiments of the present invention.

[0036] Figure 11 This is a comparison chart of the predicted duration of the equivalent collision force model, the corrected force pulse duration, and the measured force pulse duration under different material pairs for the fast correction estimation method of the collision force waveform of the continuous elastic body in the embodiments of the present invention. Detailed Implementation

[0037] The following detailed description illustrates the specific implementation method:

[0038] The basic implementation examples are as follows: Figure 1 , Figure 2 As shown: A fast correction and estimation method for the collision force waveform of a continuous elastic body is proposed. The specific steps for correcting the closed-form solution waveform of the collision force pulse based on the half-period sine wave assumption are as follows:

[0039] Step 1: For the collision kinematic pair of continuous elastic bodies, establish a basic physical model of the collision between an elastic ball and an elastic plate made of the same material. Based on the basic physical model of the collision between the ball and the plate, the dynamic equations are as follows.

[0040]

[0041] in

[0042]

[0043] In the formula, ρ is the mass of the ball, R is the radius of the ball, ρ is the depth of the indentation caused by deformation, ρ is the Hertzian contact force, ρ and ρ are the contact stiffness and Hertzian contact constant of the striking material pair, v1 and v2 are the Poisson's ratios of the ball and the plate, respectively, and E1 and E2 are the elastic moduli v1 and v2 of the ball and the plate, respectively. From this, the radius R, Poisson's ratio v1, elastic modulus E1, and density ρ1 of the ball and the Poisson's ratio v2 and elastic modulus E2 of the plate can be obtained.

[0044] At this point, the material parameters of both the ball and the plate are derived from the material parameters of the continuous elastic body collision material pair. It is sufficient that the material pair of the ball-plate collision is the same as the material pair of the actual continuous elastic body collision. The radius R of the ball is the equivalent radius at the contact point of the continuous elastic body collision, characterizing the local geometry of the contact area, and can be determined by the following formula:

[0045]

[0046] R1 and R2 represent the surface curvature radii at the collision contact point of the continuous elastic bodies, respectively. The negative sign is used when the two structures are in contact with the same shape, and the positive sign is used when they are in contact with different shapes.

[0047] Step 2: For the kinematic pair of the ball and the board, establish the Hertzian contact force model, that is, obtain the contact stiffness K. H And Hertz contact constant α.

[0048] Step 3: Establish the dynamic equation of the ball-plate collision shown in Equation (1) using the Hertzian contact force model, and solve for the Hertzian contact force pulse, where the initial collision velocity of the ball is the initial relative velocity of the continuous elastic body collision. Using numerical methods for solving nonlinear equations, such as the fourth-order Rung-Kutta numerical algorithm, the collision force waveform at a certain initial collision velocity can be accurately solved.

[0049] Step 4: Based on the principle that the peak value and impulse of the closed-form solution of the half-cycle sinusoidal collision force pulse of a continuous elastic body remain unchanged, the maximum value and width of the Hertzian contact force pulse are scaled. The scaled collision force pulse is the corrected collision force pulse of the continuous elastic body. The scaling formula is as follows.

[0050]

[0051]

[0052] Among them, Fsin For a continuous elastic body, a half-cycle sinusoidal impact force pulse, F sin-max Force pulse F sin The peak value, t sin-max Force pulse F sin The duration of F. Hertz For the Hertzian contact force pulse in the basic physical model of the ball-plate, F Hertz-max Force pulse F Hertz The maximum value of F. pre For the corrected continuous elastic body collision force pulse, t pre-max Force pulse F pre The duration.

[0053] Since the half-wave sinusoidal force pulse assumed by the equivalent collision force model can usually obtain a relatively accurate peak value and impulse, the amplitude of the Hertz contact force pulse can be scaled proportionally using equation (3) so that its maximum value is equal to the peak value of the half-wave sinusoidal force pulse, and the width of the Hertz contact force pulse can be scaled using equation (4) so ​​that its impulse is equal to the impulse of the half-wave sinusoidal force pulse. The scaled Hertz contact force pulse F pre It retains the peak value and impulse of the half-wave sinusoidal force pulse, and has a waveform similar to the shape of the actual force pulse, and the pulse duration is also closer to the actual one. In this way, the waveform of the half-wave sinusoidal force pulse is corrected.

[0054] The experimental method is as follows:

[0055] A fast correction estimation method for the collision force waveform of a continuous elastic body is proposed. A collision test is constructed based on the impact of a continuous elastic body against a fixed force hammer. The test setup is as follows: Figure 3 As shown, it can be used Figure 4 To illustrate simply, this is a cantilever beam, representing a continuous elastic body, which undergoes forced lateral vibration response under external excitation provided by a vibrator. A fixed stop is set at the free end of the cantilever beam, with an adjustable pre-set gap between it and the beam. When the cantilever beam is subjected to forced lateral vibration, it can strike the fixed stop. In practice, a pulsed hammer is used as the fixed stop; the hammerhead material can be easily replaced, and the striking force can be measured by its force sensor. The materials of the free end of the cantilever beam and the striking point of the stop are also replaceable. A laser velocity sensor is used to observe the vibration velocity at the striking point of the cantilever beam, and an external battery is used to measure the duration of the impact.

[0056] Harmonic signals were applied to the vibrator to excite the cantilever beam, and the velocity signal at the free end of the cantilever beam and the impact force signal from the fixed stop (i.e., the hammer) were recorded. Multiple experiments were conducted by adjusting the pre-set interval between the fixed stop and the cantilever beam, adjusting the amplitude and frequency of the harmonic excitation force, and changing the hammer material and the material of the impact part of the cantilever beam. A series of measured data on the impact force pulses and their initial impact velocities were obtained.

[0057] For the aforementioned cantilever beam system, the equivalent collision force model in Chinese Patent 202111614142.3 was used to predict the half-wave sinusoidal collision force pulse. Three collision material pairs were set up in the experiment, and Table 1 lists the specific materials and their Hertzian contact constants for these three collision material pairs.

[0058] Table 1. Materials of the striking pair at the free end of the cantilever beam and their Hertzian contact constants.

[0059]

[0060] Figure 5 , Figure 6 and Figure 7 The relationships between experimental and predicted initial collision velocities and the peak, duration, and impulse of the force pulse are presented for different collision material pairs. It is evident that the equivalent collision force model can predict the peak and impulse of the collision force relatively well. The predicted force pulse duration is smaller than the experimental value, and its error mainly stems from the difference between the assumed waveform and the actual waveform of the collision force pulse.

[0061] The physical model of the ball-plate collision can be determined from the collision material pairs and corresponding Hertzian contact constants in Table 1. The dynamic equation of equation (1) is established and solved to obtain the Hertzian contact force pulse. Then, the maximum value and width of the Hertzian contact force pulse are scaled according to equations (3) and (4), respectively. The corrected Hertzian contact force pulse retains both the accurate peak value and impulse of the half-wave sinusoidal force pulse and a waveform similar in shape to the actual force pulse, thus achieving the correction of the half-wave sinusoidal force pulse waveform.

[0062] Figure 8 , Figure 9 and Figure 10 The curves of the half-wave sinusoidal force pulse predicted by the equivalent collision force model, the corrected force pulse, and the experimental force pulse are compared under different collision material pairs. The left figure shows the case where the corrected force pulse fits the experimental force pulse well, while the right figure shows the case where it fits poorly. As can be seen from the figures, the corrected force pulse waveform fits the experimental waveform better, demonstrating the applicability of the waveform correction method of this invention.

[0063] Figure 11 The comparison of the predicted duration, corrected duration, and experimental duration of the equivalent collision force model for different collision material pairs under all working conditions is presented. It can be seen that the corrected force pulse duration is increased, and its value and trend are closer to the experimental value, which illustrates the rationality of the waveform correction method of the present invention.

[0064] The above descriptions are merely embodiments of the present invention, and common knowledge such as specific structures and / or characteristics in the solutions are not described in detail here. It should be noted that those skilled in the art can make various modifications and improvements without departing from the structure of the present invention, and these should also be considered within the scope of protection of the present invention. These modifications and improvements will not affect the effectiveness of the implementation of the present invention or the practicality of the patent. The scope of protection claimed in this application should be determined by the content of its claims, and the specific embodiments described in the specification can be used to interpret the content of the claims.

Claims

1. A fast correction and estimation method for the collision force waveform of a continuous elastic body, characterized in that: For the closed-form solution of the collision force pulse obtained under the assumption of a half-period sinusoidal waveform in the collision of continuous elastic bodies, the basic physical model is the impact of an elastic ball with a constant initial velocity on an elastic plate. The Hertzian contact force model is used to express the deformation compatibility relationship of the collision material pair, thereby obtaining a Hertzian contact force waveform similar to the actual collision force waveform, and using it as the waveform to correct the collision force pulse of the continuous elastic body. Based on the peak value and impulse of the closed-form solution of the half-cycle sinusoidal collision force pulse, the Hertz contact force waveform of the obtained basic physical model is scaled to obtain the corrected continuous elastic body collision force pulse. The scaling criterion for the Hertzian contact force waveform is to ensure that the pulse peak value and impulse are equal to the peak value and impulse of the closed-form solution of the collision force pulse, respectively. The specific relationship is as follows: ； Among them, F sin F is a half-cycle sinusoidal collision force pulse. sin-max Force pulse F sin The peak value, t sin-max Force pulse F sin The duration; F Hertz For Hertzian contact force pulse, F Hertz-max Force pulse F Hertz The maximum value of F; pre For the corrected continuous elastic body collision force pulse, t pre-max Force pulse F pre The duration.

2. The fast correction and estimation method for the collision force waveform of a continuous elastic body according to claim 1, characterized in that: When assigning material properties to the ball and the plate, it is only necessary to satisfy that the material pair for ball-plate collision is the same as the material pair for actual continuous elastic body collision.

3. The fast correction and estimation method for the collision force waveform of a continuous elastic body according to claim 2, characterized in that: The shape of the half-cycle sinusoidal collision force pulse can be quickly corrected. The specific implementation steps are as follows: Step 1: For continuous elastic body collision pairs, establish a basic physical model of elastic spheres colliding with an elastic plate of the same material. Step 2: Establish a Hertzian contact force model for the kinematic pair of the ball and the board during the collision. Step 3: Establish the dynamic equations of the ball-plate collision using the Hertz contact force model, and use the numerical solution method of nonlinear equations to accurately solve the collision force waveform at a certain initial collision velocity. Step 4: Based on the principle that the peak value and impulse of the closed-form solution of the half-cycle sinusoidal collision force pulse of the continuous elastic body remain unchanged, the collision force pulse waveform of the basic physical model is scaled, that is, its amplitude and duration are scaled respectively, so as to obtain the corrected collision force pulse of the continuous elastic body.

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