Equivalent Model of Continuous Elastomer Knocking Kinematic Pair and Its Modeling Method

Abstract

The present invention belongs to the field of mechanical engineering, and particularly relates to an equivalent model of a knocking kinematic pair of a continuous elastic body and a modeling method thereof. An equivalent model is established for the knocking kinematic pair in a continuous elastic body mechanical structure or system, that is, the non-linear mapping relationship between the knocking pulse force and the knocking initial velocity. By assuming that the knocking force pulse is a half-cycle sine wave, and simultaneously using the Hertz contact model and the origin transfer function or impulse response function of the mechanical structure at the knocking contact point to express the contact mechanical properties of the knocking pair material pair and the impedance properties of the mechanical structure at the knocking part, the relationship between the output and input of the non-linear equivalent model, that is, the knocking pulse force and the knocking initial velocity, is derived. Through the method of the present invention, the problem of rapid prediction of the knocking force pulse of a continuous elastic body mechanical structure can be solved.

Description

Technical Field

[0001] The present invention belongs to the field of mechanical engineering, and particularly relates to an equivalent model of a continuous elastomer percussion kinematic pair and a modeling method thereof. Background Art

[0002] In an actual mechanical system operating in a dynamic environment or dynamic working conditions, vibrations are inevitably generated in a continuous elastomer mechanical structure, and local collisions or percussions often occur. For example, during the driving of an automobile, parts such as the vehicle body, instrument assembly, and car door are prone to knocking abnormal noises, which seriously affect the quality image of the automobile product and the user experience. Therefore, in the development of automobile products, it is urgent to predict the occurrence probability and severity of these abnormal noises and make improvements. However, for such a locally non-linear system, there is currently no accurate and practical method for predicting its vibration response analysis. A feasible idea is to make full use of the local non-linear characteristics of the system, isolate the percussion kinematic pair for separate treatment, and regard the percussion force as the secondary excitation force of the continuous elastomer mechanical structure, so that the complex mechanical structure can still be treated as a linear system, which provides an effective way for the rapid solution of the locally non-linear system (a patent will be separately applied for). The key to realizing this idea is to be able to establish an equivalent model of the percussion kinematic pair.

[0003] Although the research on collision problems has a long history, the research on percussion problems of continuous elastomers such as abnormal noises is not yet mature. For the collision of lumped elements, only the inertial properties of the lumped elements and their local collision contact mechanical characteristics need to be considered, and the relevant theories are already very mature; while the collision between continuous elastomers is a more complex phenomenon, and many scholars have carried out a lot of research on it. Different model simplification methods have been proposed to quickly predict the collision force, including the mass-spring model, the dynamic substructure method, and the multi-variable method, etc., but none of them can effectively balance both the solution efficiency and the result accuracy. For example, the mass-spring model only contains the single or two-mode information of the system and is difficult to describe the continuous collision of the elastic system, so it is generally used for single collision impact problems. The dynamic substructure method describes the system using modal coordinates, and there are literature results showing that modal coordinates are difficult to accurately reflect the actual collision process, and this method can only be used for simple collision problems without slip and friction. The multi-variable method is a method proposed by combining the finite element method and the modal synthesis method for accurately calculating the collision force, but the solution efficiency of this method is low and it is difficult to be applied in the percussion vibration response analysis of actual complex mechanical structures. Therefore, the present invention proposes a new equivalent model and its modeling method, attempting to solve the problem of rapid prediction of the percussion force pulse of a continuous elastomer mechanical structure. Summary of the Invention

[0004] The purpose of the present invention is to provide an equivalent model of a continuous elastomer percussion kinematic pair and a modeling method thereof, and solve the problem of rapid prediction of the percussion force pulse of a continuous elastomer mechanical structure.

[0005] To achieve the above object, the present invention provides an equivalent model of a continuous elastomer impact kinematic pair and a modeling method thereof, which establishes an equivalent model for the impact kinematic pair in a continuous elastomer mechanical structure or system, that is, a non-linear mapping relationship between the impact pulse force and the impact initial velocity, and by assuming that the impact force pulse is a half-cycle sine wave, and simultaneously using the Hertz contact model and the origin transfer function or impulse response function of the mechanical structure at the impact contact point to express the contact mechanical properties of the impact pair materials and the impedance properties of the mechanical structure at the impact part, so as to derive the relationship between the output and input of the non-linear equivalent model, that is, the impact pulse force and the impact initial velocity.

[0006] Furthermore, for the impact kinematic pair between continuous elastomers, the equivalent model established by the present invention, that is, the implicit relationship between the impact force pulse (determined by the sine wave amplitude F0 and frequency ω b ) and the impact initial velocity (V1 + V2) is as follows:

[0007]

[0008]

[0009] where k H , α are respectively the contact stiffness and Hertz contact constant of the impact pair, n1, n2 are positive integers, ω n1 , ω n2 respectively represent the n1th and n2th natural frequencies of the two continuous elastomers of the impact pair, a n1 , a n2 respectively represent the components of the corresponding order normal vibration mode at the impact point. The sum in the formula should theoretically take an infinite number of orders, but in fact, only the sum of the first finite orders is taken to obtain an analysis result with sufficient accuracy.

[0010] Furthermore, for the impact kinematic pair between a continuous elastomer and a fixed boundary, the equivalent model established by the present invention, that is, the implicit relationship between the impact force pulse (determined by the sine wave amplitude F0 and frequency ω b ) and the impact initial velocity V is as follows:

[0011]

[0012]

[0013] where n is a positive integer, ω n , a n respectively represent the nth natural frequency of the continuous elastomer of the impact pair and the component of the normal vibration mode at the impact point. Obviously, the implicit relationship here is a special case of the impact between continuous elastomers.

[0014] Furthermore, the knocking force pulse of the continuous elastomer knocking pair at a given initial knocking velocity can be obtained quickly, and the specific implementation steps are as follows:

[0015] Step 1: Establish a Hertz contact model for the knocking part or knocking kinematic pair of the continuous elastomer, that is, obtain the contact stiffness k H and the Hertz contact constant α;

[0016] Step 2: Establish an origin transfer function or impulse response function model of the knocking point for the continuous elastomer structure, that is, obtain the first several natural frequencies of the continuous elastomer structure and the components of the corresponding normal vibration modes at the knocking point;

[0017] Step 3: If the initial knocking velocity of the continuous elastomer knocking pair is known, then ω b and F0 are obtained according to the two equations of the equivalent model, so as to obtain the knocking force pulse, that is, the half-cycle sine wave determined by ω b and F0.

[0018] Furthermore, after Steps 1 and 2, the equivalent model of the knocking pair has been determined. Note the two equations of the equivalent model. One is a transcendental equation about ω b and has nothing to do with F0. Therefore, any mature numerical algorithm can be used to first obtain the solution of ω b , and then substitute the value of ω b into the other equation, and it is easy to obtain the solution of F0.

[0019] The beneficial effect of the present invention is that: through the method of the present invention, the problem of rapid prediction of the knocking force pulse of the continuous elastomer mechanical structure can be solved. Description of the Drawings

[0020] Figure 1 is a physical photo of the test device for the knocking of the cantilever beam and the fixed stop block;

[0021] Figure 2 is a schematic diagram of the test device for the knocking of the cantilever beam and the fixed stop block;

[0022] Figure 3 is a comparison chart of the predicted and measured waveforms of the knocking force pulse of a typical knocking pair;

[0023] Figure 4 is a comparison chart of the predicted curve of the equivalent model of the knocking force pulse peak - initial knocking velocity and the measured data of the stainless steel - aluminum material pair;

[0024] Figure 5 is a comparison chart of the predicted curve of the equivalent model of the knocking force pulse peak - initial knocking velocity and the measured data of the aluminum - aluminum material pair;

[0025] Figure 6It is a comparison diagram of the predicted curve of the equivalent model of the peak value of the secondary knocking force pulse - knocking initial velocity and the measured data for 45# steel - ABS material;

[0026] Figure 7 It is a comparison diagram of the predicted curve of the equivalent model of the peak value of the secondary knocking force pulse - knocking initial velocity and the measured data for stainless steel - ABS material. Specific implementation manners

[0027] The following is a further detailed description through specific implementation manners.

[0028] For the collision or knocking motion between continuous elastic bodies, the present invention defines a continuous elastic body knocking kinematic pair, which includes two adjacent continuous elastic bodies. They vibrate or move relatively under external excitation and make contact and knocking at the minimum distance between the two adjacent continuous elastic bodies. The present invention proposes a non - linear equivalent model to describe its knocking motion characteristics. The input of this model is the knocking initial velocity of the two continuous elastic bodies, that is, the normal relative motion velocity of the knocking contact point at the beginning of contact knocking. The output of this model is the knocking pulse force between the two continuous elastic bodies. It is a pair of action and reaction forces, equal in magnitude and opposite in direction, acting on the knocking contact points of the two continuous elastic bodies respectively, and pointing towards the inside of the continuous elastic body structure along the normal direction of the contact surface.

[0029] Obviously, the mapping relationship between the output and input of this equivalent model necessarily depends on the physical properties of the continuous elastic body knocking kinematic pair itself. To construct this mapping relationship, the present invention proposes to use the Hertz contact model to express the contact mechanical properties of the knocking pair materials, and use the origin transfer function or impulse response function to characterize the mechanical impedance properties of the continuous elastic body at the knocking contact point. Thus, the knocking force pulse response of the continuous body knocking kinematic pair at a given knocking initial velocity can be determined. To be able to quickly determine this mapping relationship, based on the measured results of a large number of knocking pulse forces, the present invention makes an assumption about the knocking pulse force waveform, making it a sine wave of half a period. Then, only two parameters are needed to determine the knocking pulse force waveform. One is the sine wave amplitude F0, that is, the peak value F0 of the knocking force pulse, and the other is the sine wave frequency ω b That is, the duration ΔT of the knocking force pulse is correspondingly determined as ΔT = π / ω b .

[0030] According to the above idea of the present invention, the specific equation of the equivalent model of the continuous elastic body knocking kinematic pair can be derived.

[0031] According to the waveform assumption, the knocking pulse force can be expressed as

[0032] f(t)=F0sinω b t 0≤t≤π / ω b (I)

[0033] Note that when a continuous elastomer is struck, the impact pulse force always points towards the interior of the continuous elastomer structure along the normal direction of the contact surface. We define this direction as the positive direction of the impact force. At the same time, a generalized coordinate is defined along this direction to express the normal displacement of the impact contact point of the continuous elastomer. The origin of the coordinate is set at the position of the impact point when the impact just starts. When two continuous elastomers strike each other, the normal displacements of the impact contact points on the two continuous elastomers are denoted as x1 and x2 respectively, both of which are functions of time t. Using the Hertz contact model, the impact force can be expressed as

[0034]

[0035] where k H and α are the contact stiffness of the impact material pair and the Hertz contact constant respectively.

[0036] If the impulse response functions (constituting a Laplace transform pair with the origin transfer function) of the two continuous elastomers at the origin of the impact contact point are denoted as h1(t) and h2(t) respectively, they can be expressed using the modal information of the continuous elastomers as

[0037]

[0038] where ω n1 and ω n2 are the n1th and n2th natural frequencies of the two continuous elastomers respectively, and a n1 and a n2 are the components of the corresponding order normal mode shapes of the two continuous elastomers at the impact contact point respectively.

[0039] Suppose that two continuous elastomers start to strike at time t = 0, that is: x1(0) = 0, x2(0) = 0, and the initial velocities of the impact points on the two continuous elastomers are V1 and V2 respectively. During the subsequent impact process, the displacements of the impact points on the two continuous elastomers can be expressed as

[0040]

[0041]

[0042] where the displacement consists of two parts. The first part represents the displacement of the impact point without the action of the impact force, and the second part represents the displacement of the impact point under the action of the impact force.

[0043] According to the assumption of the impact force pulse waveform in equation (1), at t = π / 2ω b , f(t) reaches the maximum value F0, and the relative velocity of the impact contact points of the two continuous elastomers is zero, that is, there are the following relationships:

[0044]

[0045]

[0046] wherein respectively represent the velocities of the contact points of two consecutive elastomer impacts, and can be obtained by differentiating equations (4) and (5).

[0047] By using equations (3) to (7) and substituting the assumed impact force waveform of (1), the following relationship can be derived:

[0048]

[0049]

[0050] Obviously, by using equations (8) and (9), the amplitude F0 and the duration π / ω of the impact force pulse can be obtained based on the initial velocity of the continuous elastomer impact b , that is, the above equations express the input-output mapping relationship of the equivalent model of the continuous elastomer impact pair.

[0051] Equations (8) and (9) are applicable to any continuous elastomer structure. If the continuous elastomer impacts with a fixed boundary under external excitation, it is equivalent to one of the continuous elastomers becoming a fixed rigid body. Let V2 = 0 and a n2 = 0 in equations (8) and (9), and denote n1 as n and V1 as V, then the equation for determining the output of the impact force pulse can be obtained as:

[0052]

[0053]

[0054] Here, by taking the impact between the continuous elastomer and the boundary as a special case of the impact between two continuous elastomers, the equivalent model equation thereof is derived. In fact, if a mechanical model is established separately for the continuous elastomer and the fixed boundary for derivation, the same results as equations (10) and (11) will be obtained.

[0055] According to the method of the present invention, the specific steps for establishing the equivalent model of any continuous elastomer impact pair and thereby determining the impact force response are as follows:

[0056] Step 1: Establish a Hertz contact model for the impact part or the impact kinematic pair of the continuous elastomer, that is, obtain the contact stiffness k H and the Hertz contact constant α.

[0057] The Hertz contact model of the impact pair is related to factors such as material properties and local geometric shapes. For example, for the Hertz contact model of the point contact type impact pair, its contact stiffness can be determined by the following formula:

[0058]

[0059] Among them E i , ν i , R i respectively represent the Young's modulus, Poisson's ratio of the elastomer material, and the surface curvature radius at the percussion point. The Hertz contact constant α can be determined according to the contact force-deformation test curve of the percussion pair materials.

[0060] Step 2: For the continuous elastic body structure, establish the origin transfer function or impulse response function model of the percussion point, that is, obtain the first few finite-order natural frequencies of the continuous elastic body structure and the components of the corresponding normal vibration modes at the percussion point.

[0061] If finite element is used to analyze the continuous elastic body structure, note that when dividing elements, the percussion contact point should be selected as the element node for convenient processing. When using modal superposition to express the origin transfer function or impulse response function, modal truncation should be performed, that is, only the first several modes are selected for superposition, but note that the modal order should not be selected too few. Generally speaking, the shorter the duration of the percussion force pulse, the wider the covered frequency range, and the more structural modes will be excited. Therefore, the modal truncation order needs to be set higher. Here, any mature method can be used to obtain the modes or natural frequencies and vibration modes of the continuous elastic body.

[0062] In fact, so far the equivalent model of the continuous elastic body percussion pair has been determined, that is, all the parameters in equations (8), (9) or (10), (11) have been obtained. As long as the input of the equivalent model, that is, the percussion initial velocity, is given, the percussion force output of the equivalent model can be obtained.

[0063] Step 3: If the percussion initial velocity of the continuous elastic body percussion pair is known, then ω b , F0 are obtained according to the two equations of the equivalent model, so as to obtain the percussion force pulse, that is, the half-cycle sine wave determined by ω b , F0.

[0064] Since equation (9) or (11) only contains one unknown ω b , so this equation should be solved first. This is a transcendental equation about ω b , and only numerical algorithms can be used to obtain the numerical solution. Then substituting the solution of ω b into the other equation can directly obtain the value of F0.

[0065] Example:

[0066] In order to obtain the measured percussion force signal to verify the percussion force predicted by the present invention, we constructed the Figure 1 test device shown, which can be used with Figure 2For a concise illustration. This is a cantilever beam, representing a continuous elastic body, which undergoes a transverse forced vibration response under the external excitation provided by an exciter; a fixed stop is set at the free end of the cantilever beam, with an adjustable reserved gap between it and the cantilever beam. When the cantilever beam undergoes forced transverse vibration, it can strike the fixed stop. In fact, a pulse force hammer is used as the fixed stop, and the hammer head material can be conveniently replaced, and the impact force can be measured through its force sensor; the materials at the free end of the cantilever beam and the striking part of the stop can be replaced; an acceleration sensor is also set on the cantilever beam to observe the vibration acceleration at the striking point of the cantilever beam.

[0067] In the experiment, a harmonic signal is applied to the exciter to excite the cantilever beam, and the acceleration signal at the free end of the cantilever beam and its impact force signal with the fixed stop, i.e., the force hammer, are recorded. By adjusting the reserved gap between the fixed stop and the cantilever beam, adjusting the amplitude and frequency of the harmonic excitation force, replacing the hammer head material and the material of the striking part of the cantilever beam, multiple experiments are carried out, and a series of measured data of the impact force pulse and its initial impact velocity are obtained.

[0068] For the above-mentioned cantilever beam system, its finite element model is established and modal analysis is carried out, and its natural frequencies and normal mode shapes can be obtained. Table 1 lists the first 4 natural frequencies and the components of their normal mode shapes at the striking point.

[0069] Table 1 Measured values of the cantilever beam structure modal and calculated values of the finite element model

[0070]

[0071] For the impact pair composed of the free end of the cantilever beam and the fixed stop (the hammer head of the force hammer), a Hertz contact model is established. Here, the hammer head of the force hammer and the free end of the cantilever beam are both point contacts. Table 2 lists the Hertz contact constants when different material pairs are selected.

[0072] Table 2 Materials of the impact pair at the free end of the cantilever beam and their Hertz contact constants

[0073]

[0074] So far, in fact, the equivalent model of the impact pair at the free end of the cantilever beam has been determined, that is, the parameters required in equations (10) and (11). If the initial impact velocity input is given, the impact force output can be given by this equivalent model.

[0075] According to the measured acceleration signal at the free end of the cantilever beam and the time history of the impact force signal, after necessary processing, the initial impact velocity at the time of each impact can be determined quite accurately. Based on these initial velocities, using the established equivalent model, the impact force pulse of each impact is obtained; we have self-written a numerical calculation program for equations (10) and (11), and first solve equation (11) based on the 0.618 method to obtain ωb , and then substitute it into Equation (10) to calculate F0.

[0076] Figure 3 The comparison between the calculated values and the measured values of the typical impact force pulse waveforms for different material pairs of the impact pair at the free end of the cantilever beam is given. It can be seen that the predicted waveform is basically consistent with the measured waveform.

[0077] Figures 4 to 7 The comparison between the calculated values and the measured values of the impact force pulse amplitudes for different material pairs of the impact pair at the free end of the cantilever beam is given. First, it can be seen from the figure that the impact force pulse amplitude calculated according to the equivalent model increases monotonically with the initial impact velocity, showing a non-linear relationship, but close to linear growth; in fact, it can be seen from the equation of the equivalent model of the impact pair that when the Hertz contact constant α = 1, ω b (corresponding to the impact force pulse duration) is independent of the initial impact velocity and is only related to the material contact stiffness and the nature of the structural mechanical impedance, while F0 is proportional to the initial impact velocity. Secondly, it can be seen from the figure that the measured impact force pulse amplitudes of many pairs, regardless of the material pair, are always near the predicted curve of the equivalent model, indicating that the equivalent model of the present invention truly reflects the mechanical characteristics of the impact pair.

[0078] It should be noted in advance that in the present invention, unless otherwise clearly defined and limited, terms such as "installation", "connection", "connection", "fixation" and other terms should be understood in a broad sense. For example, it can be a fixed connection, a detachable connection, or an integral connection; it can be directly connected, or indirectly connected through an intermediate medium, and it can be the communication inside two components. For those of ordinary skill in the art, the specific meanings of the above terms in the present invention can be understood according to specific circumstances.

[0079] The above are only the embodiments of the present invention, and common general knowledge such as the specific structures and characteristics in the solutions are not described in detail here. It should be pointed out that for those skilled in the art, without departing from the structure of the present invention, several deformations and improvements can still be made, and these should also be regarded as the protection scope of the present invention, and these will not affect the implementation effect of the present invention and the practicality of the patent. The protection scope required by this application should be based on the content of its claims, and the specific implementation manners and the like described in the specification can be used to explain the content of the claims.

Claims

1. Modeling method for equivalent model of continuous elastomer knocking kinematic pair, characterized in that An equivalent model is established for the percussion kinematic pair in a continuous elastic body mechanical structure or system, that is, the non-linear mapping relationship between the percussion pulse force and the percussion initial velocity. By assuming that the percussion force pulse is a half-period sine wave, and simultaneously using the Hertz contact model and the origin transfer function or impulse response function of the mechanical structure at the percussion contact point to express the contact mechanical properties of the percussion pair materials and the impedance properties of the mechanical structure at the percussion part, the relationship between the output and input of the equivalent model, that is, the percussion pulse force and the percussion initial velocity, is derived; The origin impulse response function of two continuous elastic bodies at the percussion contact point is: where \(h_1(t)\) and \(h_2(t)\) are the impulse response functions of the origin at the percussion contact points of two consecutive elastic bodies respectively, and \(\omega\) n1 and \(\omega\) n2 are the \(n_1\)-th and \(n_2\)-th natural frequencies of two consecutive elastic bodies respectively, and \(a\) n1 and \(a\) n2 are the components of the corresponding normal vibration modes of two consecutive elastic bodies at the percussion contact point respectively; The origin impulse response function and the origin transfer function form a Laplace transform pair; For the percussion kinematic pair between continuous elastic bodies, the established equivalent model, that is, the implicit relationship between the percussion force pulse and the percussion initial velocity (V1 + V2), is as follows: Among them, the percussion force pulse is determined by the sine wave amplitude F0 and the frequency ω b , k H , α are respectively the contact stiffness and the Hertz contact constant of the percussion pair, n1 and n2 are positive integers, ω n1 , ω n2 respectively represent the n1th and n2th natural frequencies of the two consecutive elastic bodies of the percussion pair, a n1 , a n2 respectively represent the components of the corresponding order regular vibration mode at the percussion point; Or For the percussion kinematic pair between a continuous elastic body and a fixed boundary, the established equivalent model, that is, the implicit relationship between the percussion force pulse and the percussion initial velocity V, is as follows: Among them, the percussion force pulse is determined by the sine wave amplitude F0 and the frequency ω b where n is a positive integer, and k H , α are the contact stiffness and Hertz contact constant of the percussion pair respectively, ω n , a n represent the nth natural frequency of the continuous elastic body of the percussion pair and the component of the normal vibration mode at the percussion point respectively.

2. The modeling method of the equivalent model of the continuous elastomer percussion kinematic pair according to claim 1, wherein To quickly obtain the percussion force pulse of a continuous elastic body percussion pair at a given percussion initial velocity, the specific implementation steps are as follows: Step 1: Establish a Hertz contact model for the percussion part or percussion kinematic pair of the continuous elastomer, that is, obtain the contact stiffness k H and the Hertz contact constant α; Step 2: For the continuous elastic body structure, establish an origin transfer function or impulse response function model at the percussion point, that is, obtain the components of the first few finite natural frequencies and the corresponding normal vibration modes of the continuous elastic body structure at the percussion point; Step 3: If the initial knocking velocity of the continuous elastomer knocking pair is known, then ω is obtained according to the two equations of the equivalent model b , F0, thereby obtaining the knocking force pulse, that is, a half-cycle sine wave determined by ω b , F0.

3. The modeling method of the equivalent model of the continuous elastomer percussion kinematic pair according to claim 2, characterized in that After steps one and two, the equivalent model of the percussion sub is determined. There are two equations for the equivalent model. One is a transcendental equation regarding ω b which is independent of F0. Using a mature numerical algorithm, the solution of ω b is first obtained, and then the value of ω b is substituted into the other equation to obtain the solution of F0.

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