A cumulative energy-based p-i equivalent damage prediction method

Abstract

This invention relates to the field of damage prediction technology, specifically to a P-I equivalent damage prediction method based on accumulated energy. The purpose of this invention is to construct an equivalent damage relationship for quantifying impact parameters and structural failure, namely, a P-I equivalent damage prediction method based on accumulated energy. Addressing the damage problem of devices under test under different peak values ​​and pulse widths of impact, starting from the dynamic response of the structure under high overload impact, this invention constructs a P-I equivalent damage relationship characterizing the critical damage of the structure by comparing the peak accumulated energy and critical accumulated energy under different peak values ​​and pulse widths of load. This can guide the impact-resistant optimization design of measurement devices under high overload environments, theoretically providing support for the impact-resistant optimization design of devices under high overload environments, and promoting the engineering process of equivalent damage methods for measurement devices.

Description

Technical Field

[0001] This invention relates to the field of damage prediction technology, specifically to a PI equivalent damage prediction method based on accumulated energy. Background Technology

[0002] Measuring devices (such as accelerometers) demonstrate excellent performance in high-overload impact testing; however, their reliability faces significant challenges in real-world applications. Engineering practice and theoretical analysis indicate that two main factors contribute to structural failure: peak impact value and... and impact pulse width When designing measurement devices, using methods that are too simplistic or overly complex to consider structural damage limits is detrimental to the rapid selection of structural dimensions in the early stages and the accuracy of subsequent testing.

[0003] Under single overload impact conditions, it is far from enough to know that a device can withstand a certain peak instantaneous impact. The cumulative effect of energy transferred during the impact process must be comprehensively considered. Therefore, it is urgent to establish an equivalent damage relationship with strong applicability under the impact dynamics theory to quantify the relationship between impact parameters and structural failure, thereby realizing the prediction of damage under dynamic impact environment. Summary of the Invention

[0004] The purpose of this invention is to construct an equivalent damage relationship for quantifying impact parameters and structural failure, namely, a PI equivalent damage prediction method based on cumulative energy.

[0005] This invention is achieved using the following technical solution:

[0006] A method for predicting PI equivalent damage based on accumulated energy includes the following steps:

[0007] 1) Establish a theoretical analysis model for the measuring device under inertial shock;

[0008] 2) Based on the measured acceleration data from the overload impact test, determine the form of the external inertial load for the theoretical analysis model established in step 1).

[0009] 3) Establish the functional expression of the variational principle for the theoretical analysis model;

[0010] 4) Discretize the functional expression established in step 3) in the spatial domain using rectangular elements (REUNC elements);

[0011] 5) Discretize the time domain using the symplectic time subdomain method;

[0012] 6) Taking the first variation of the discretized functional expression from step 5) and setting it equal to zero, we can obtain the recursive relation for the dynamic response by rearranging:

[0013]

[0014] In the formula, The initial displacement vector of each node of the rectangular element within the initial time subdomain. With the initial momentum vector The state array formed From The extracted submatrix is ​​related to the initial conditions, where the 0 before the "_" in the subscript indicates the initial state of the initial time subdomain. This is a column vector consisting of displacement and momentum parameters at all interpolation points in the time subdomain. for A matrix consisting of a zero matrix;

[0015] 7) Array the initial state matrix Δ of the initial time subdomain. 0_2k×1 (Initial displacement vector within the initial time subdomain) With the initial momentum vector Substitute each element (if it is zero) into step 6 to solve. Extract and calculate this time subdomain terminal state As the initial state of the next time subdomain Repeatedly solve for each time subdomain. The terminal states are passed sequentially until all subdomains have been calculated, and the displacement coefficients of all time subdomains are arrayed. ( After splicing, the dynamic response of the recovered elastic cantilever thin plate throughout the entire time history is expressed as:

[0016]

[0017] 8) Using the dynamic response expression obtained in step 7), take the displacement-related parameters. ( Calculate the average difference between four adjacent time steps of each rectangular element as the displacement change of the rectangular element. The acceleration of each node is calculated using the second-order central difference method, and the average acceleration of the four nodes of the rectangular element is taken as the acceleration of the rectangular element.

[0018] Calculate the strain energy increment of each rectangular element. :

[0019]

[0020] in, The mass of the rectangular unit, Represents the displacement values ​​of the four nodes of the rectangular element in the time subdomain. , , , The difference between the initial state average and the final state average. , Let represent the displacements of the initial state of the previous time subdomain and the final state of this time subdomain, respectively. The strain energy increments of each rectangular element are accumulated to obtain the strain energy increment across the entire structure within this time subdomain. ( The energy increment at the end of each time subdomain can be obtained by summing the strain energy increments of all time subdomains:

[0021]

[0022] No. The cumulative energy at the end of each time subdomain is defined as:

[0023]

[0024] 9) Calculate the maximum cumulative energy that the elastic cantilever plate can withstand:

[0025]

[0026] In the formula, , These are the material's fracture strength and elastic modulus, respectively. , , The width, thickness, and length of the flexible cantilever plate;

[0027] 10) Select a series of different overload acceleration peak values Different pulse widths For loads of this type, a cumulative energy contour map is plotted using steps 2)-8) to extract the critical damage point (the critical damage point is the point with the maximum cumulative energy). The corresponding load parameters, i.e., peak acceleration. Pulse width );

[0028] 11) The cumulative energy input to the structure under load. satisfy:

[0029]

[0030] In the formula The load magnitude and structural velocity response are respectively considered. Under the condition of minimal structural deflection, velocity and acceleration are directly related, yielding the relationship between peak cumulative energy, peak acceleration, and pulse width:

[0031]

[0032] in This is the proportionality coefficient;

[0033] 12) Analyze the dynamic response process of the device structure under test. Assuming the area under load is approximately constant, what is the transient pressure on the surface of the rectangular thin plate and the triangular mass block structure? With impulse It can be obtained as follows:

[0034]

[0035] in , These are the surface areas of the upper surfaces of the rectangular thin plate and the triangular mass block, respectively.

[0036] 13) Based on steps 11) and 12), construct the equivalent relationship between pressure and impulse under the cumulative energy state:

[0037]

[0038] The critical damage point extracted in step 10) is used to obtain the transient pressure in step 12). With impulse The result calculated in step 9) Substitute into step 13) Seeking Thus, the equivalent damage relationship of PI is obtained:

[0039]

[0040] in This is the equivalent damage threshold;

[0041] 14) During damage prediction, the structural parameters of the device under test and the impact load parameters (peak acceleration) are used as the basis for prediction. Pulse width ), obtained through step 9) Then, by substituting this into step 13), the equivalent damage threshold is obtained. The transient pressure on the surface of the device under test is obtained through step 12). With impulse and draw The value will eventually be the actual value calculated. Value and equivalent damage threshold The comparison is performed. If the damage is greater than or equal to the damage threshold, it means that the measuring device is predicted to be damaged under the load. If the damage is less than the damage threshold, it means that the measuring device is predicted not to be damaged under the load.

[0042] The beneficial effects of this invention are as follows: This invention addresses the damage problem of the device under test under different peak values ​​and pulse widths. Starting from the dynamic response of the structure under high overload impact, it constructs the PI equivalent damage relationship characterizing the critical damage of the structure by comparing the peak cumulative energy and critical cumulative energy under different peak values ​​and pulse width loads. This can guide the shock resistance optimization design of the measurement device under high overload environment, theoretically provide support for the shock resistance optimization design of the device under high overload environment, and promote the engineering process of the equivalent damage method of measurement device. Attached Figure Description

[0043] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with the invention and, together with the description, serve to explain the principles of the invention.

[0044] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, for those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0045] Figure 1 This is a flowchart of the prediction method described in this invention; Figure 2 This is a physical model diagram of the measuring device described in a specific embodiment of the present invention;

[0046] Figure 3 This refers to the theoretical analysis model established in a specific embodiment of the present invention;

[0047] Figure 4 These are contour plots of cumulative energy under different peak values ​​and pulse width loads, drawn in specific embodiments of the present invention.

[0048] Figure 5 A schematic diagram of the constructed PI critical damage curve;

[0049] Figure 6 This is a schematic diagram of the experimental verification results using the prediction method described in this invention. Detailed Implementation

[0050] To better understand the above-mentioned objectives, features, and advantages of the present invention, the solutions of the present invention will be further described below. It should be noted that, unless otherwise specified, the embodiments of the present invention and the features thereof can be combined with each other.

[0051] Many specific details are set forth in the following description in order to provide a full understanding of the invention, but the invention may also be practiced in other ways different from those described herein; obviously, the embodiments in the specification are only some embodiments of the invention, and not all embodiments.

[0052] The specific embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0053] like Figure 1 As shown, a PI equivalent damage prediction method based on accumulated energy includes the following steps:

[0054] 1) Establish the measuring device (in specific implementation, the measuring device adopts a beam-island structure accelerometer, and the physical model diagram is as follows). Figure 2 (As shown) The theoretical analysis model under inertial impact is simplified to a rectangular thin plate and a triangular mass block, with the rectangular thin plate subjected to a uniformly distributed load. The length is , width is Thick Quality is Elastic cantilever thin plate, triangular mass block If only vertical translation is performed, the inertial force of the plate itself on the elastic cantilever plate can be equivalent to the linear load intensity generated by the center of mass of the triangular mass block at the free end of the elastic cantilever plate. and line bending moment intensity ,like Figure 3 As shown;

[0055] 2) Based on the measured acceleration data from the overload impact test, the external inertial load of the theoretical analysis model established in step 1) is determined to be in the form of a half-sine wave, and the expression is:

[0056] (1)

[0057] in , The masses of the elastic cantilever plate and the triangular mass block are respectively determined. The peak acceleration under a half-sine load. The duration of the load application. The pulse width of the load. Let be the distance from the center of mass of the triangular mass block to the free end of the elastic cantilever plate. This indicates the distance from the fixed end to the flexible cantilever plate.

[0058] 3) Establish the functional expression of the variational principle for the theoretical analysis model:

[0059] (2)

[0060] In the formula, For Hamiltonian functions, , These are functions of deflection and momentum with respect to time, respectively. Indicates deflection right The derivative of Let be the tangent direction of each free boundary. To provide the bending moment in the direction of the outward normal of the free boundary, For the free boundary on the right side of the theoretical analysis model, For the angle of rotation around the normal direction, The damping coefficient is... This refers to the area on the upper surface of the flexible cantilever thin plate. , Represents the coordinate system established in the theoretical analysis model;

[0061] 4) Discretize the functional expression established in step 3) in the spatial domain using rectangular elements (REUNC elements), dividing the theoretical analysis model into multiple equally sized rectangular elements, where... Direction is divided into Group, Direction is divided into Group, calculate the global quality matrix Stiffness matrix Damping matrix and load array And discretize the space of continuous field variables in step 3):

[0062] (3)

[0063] In the formula, For the inertial coupling of momentum and velocity, For scaling transformation of the canonical momentum term, For the density of the physical model, The thickness of the rectangular unit. and Let be the displacement matrix and momentum matrix of the rectangular element node at the end of the considered time subdomain;

[0064] At each rectangular unit node displacement vector at point and momentum vector It can be represented as:

[0065]

[0066]

[0067] In the formula and ( ) represent the nodes at points The generalized displacement matrix and generalized momentum matrix at the location;

[0068] 5) Discretize the time domain using the symplectic time subdomain method. Segments, used on each time subdomain Lagrange interpolation polynomial of degree The matrix formed approximates the displacement function:

[0069] (4)

[0070] In the formula The matrix is ​​composed of Lagrange interpolation polynomials. , These are vectors of undetermined coefficients;

[0071] Substituting formula (4) into the functional expression of formula (3), we get:

[0072] (5)

[0073] In the formula, Dimensionless local time coordinates The local load array, where Step size;

[0074]

[0075] Among them, symbols Represents the Kronecker product. , ;

[0076] 6) For the functional expression in step 5), Taking the first-order variation and setting it equal to zero, we can obtain the recurrence relation for the dynamic response by rearranging:

[0077] (6)

[0078] in:

[0079]

[0080] In the formula, The initial displacement vector of each node of the rectangular element within the initial time subdomain. With the initial momentum vector The state array formed From The extracted submatrix is ​​related to the initial conditions, where the 0 before the "_" in the subscript indicates the initial state of the initial time subdomain. This is a column vector consisting of displacement and momentum parameters at all interpolation points in the time subdomain. for A matrix consisting of a zero matrix;

[0081] 7) Array the initial state matrix of the initial time subdomain. (Initial displacement vector within the initial time subdomain) With the initial momentum vector Substitute each element (if it is zero) into step 6 to solve. Extract and calculate this time subdomain terminal state As the initial state of the next time subdomain Repeatedly solve for each time subdomain. The terminal states are passed sequentially until all subdomains have been calculated, and the displacement coefficients of all time subdomains are arrayed. ( After splicing, the matrix formed by the Lagrange interpolation polynomials in step (5) is obtained. The dynamic response of a recoverable elastic cantilever plate over the entire time history is expressed as:

[0082] (7)

[0083] 8) Using the dynamic response expression obtained in step 7), take the displacement-related parameters. ( Calculate the average difference between four adjacent time steps of each rectangular element as the displacement change of the rectangular element. The acceleration of each node is calculated using the second-order central difference method, and the average acceleration of the four nodes of the rectangular element is taken as the acceleration of the rectangular element.

[0084] Calculate the strain energy increment of each rectangular element. :

[0085] (8)

[0086] in, The mass of the rectangular unit, Represents the displacement values ​​of the four nodes of the rectangular element in the time subdomain. , , , The difference between the initial state average and the final state average. , Let represent the displacements of the initial state of the previous time subdomain and the final state of this time subdomain, respectively. The strain energy increments of each rectangular element are accumulated to obtain the strain energy increment across the entire structure within this time subdomain. ( The energy increment at the end of each time subdomain can be obtained by summing the strain energy increments of all time subdomains:

[0087] (9)

[0088] No. The cumulative energy at the end of each time subdomain is defined as:

[0089] (10)

[0090] 9) Calculate the maximum cumulative energy that the elastic cantilever plate can withstand:

[0091] (11)

[0092] In the formula, , These are the material's fracture strength and elastic modulus, respectively. , , The width, thickness, and length of the flexible cantilever plate;

[0093] 10) Select 1168 sets of half-sine loads with overload acceleration peak values ​​a = 10000, 12000, ..., 300000g and pulse widths T = 0, 20, 40, ..., 140µs. Under these impacts, draw cumulative energy contour maps using steps 2) to 9), as shown below. Figure 4 As shown, the white line represents the critical damage curve, from which the critical damage point (i.e., the maximum accumulated energy) is extracted. The corresponding load parameter is the peak acceleration. Pulse width );

[0094] 11) The cumulative energy input to the structure under a half-sine load. satisfy:

[0095] (12)

[0096] In the formula The load magnitude and structural velocity response are respectively considered. Under the condition of minimal structural deflection, velocity and acceleration are directly related, yielding the relationship between peak cumulative energy, peak acceleration, and pulse width:

[0097] (13)

[0098] in This is the proportionality coefficient;

[0099] 12) Analyze the dynamic response process of the device structure under test. Assuming the area under load is approximately constant, what is the transient pressure on the surface of the rectangular thin plate and the triangular mass block structure? With impulse It can be obtained as follows:

[0100] (14)

[0101] in , These are the surface areas of the upper surfaces of the rectangular thin plate and the triangular mass block, respectively.

[0102] 13) Based on formulas (13) and (14), construct the equivalent relationship between pressure and impulse under the cumulative energy state:

[0103] (15)

[0104] Substituting the critical damage point extracted in step 10) into formula (15), the transient pressure is obtained. With impulse The result calculated in step 9) Substitute into the above formula (15) Seeking Thus, the equivalent damage relationship of PI is obtained:

[0105] (16)

[0106] in The equivalent damage threshold is illustrated in the schematic diagram of the constructed PI critical damage curve. Figure 5 As shown;

[0107] 14) During damage prediction, the structural parameters of the device under test and the impact load parameters (peak acceleration) are used as the basis for prediction. Pulse width ), obtained through formula (11) Substituting this into formula (16) yields the equivalent damage relationship, and the transient pressure on the surface of the device structure to be measured is obtained through formula (14). With impulse and draw The value will eventually be the actual value calculated. Value and equivalent damage threshold The comparison is performed. If the damage is greater than or equal to the damage threshold, it means that the measuring device is predicted to be damaged under the load. If the damage is less than the damage threshold, it means that the measuring device is predicted not to be damaged under the load.

[0108] To verify the feasibility of the above prediction method, multiple impact tests were conducted using different loads. The verification results are as follows: Figure 6 As shown, by Figure 6 It can be seen that in the impact test, the loads that actually caused damage were all greater than the corresponding damage threshold, and the loads that did not actually cause damage were all less than the corresponding damage threshold, proving that the prediction method described in this invention is feasible and has a high prediction accuracy.

[0109] The above description is merely a specific embodiment of the present invention, enabling those skilled in the art to understand or implement the present invention. Although detailed descriptions have been provided with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments, and they should all be covered within the protection scope of the claims.

Claims

1. A method for predicting PI equivalent damage based on accumulated energy, comprising the following steps: 1) Establish a theoretical analysis model for the measuring device under inertial shock; 2) Based on the measured acceleration data from the overload impact test, determine the form of the external inertial load for the theoretical analysis model established in step 1). 3) Establish the functional expression of the variational principle for the theoretical analysis model; 4) Discretize the functional expression established in step 3) using rectangular elements in the spatial domain; 5) Discretize the time domain using the symplectic time subdomain method; 6) Taking the first variation of the discretized functional expression from step 5) and setting it equal to zero, we can obtain the recursive relation for the dynamic response by rearranging: ； In the formula, The initial displacement vector of each node of the rectangular element within the initial time subdomain. With the initial momentum vector The state matrix formed From The extracted submatrix is ​​related to the initial conditions, where the 0 before the "_" in the subscript indicates the initial state of the initial time subdomain. This is a column vector consisting of displacement and momentum parameters at all interpolation points in the time subdomain. for A matrix consisting of a zero matrix; 7) Array the initial state matrix of the initial time subdomain. Substitute into step 6) to solve. Extract and calculate this time subdomain End state As the initial state of the next time subdomain Repeatedly solve for each time subdomain. The final state is passed sequentially until all time subdomains have been calculated, and the displacement coefficients of all time subdomains are arrayed. ( After splicing, the dynamic response of the recovered elastic cantilever thin plate throughout the entire time history is expressed as: ； 8) Using the dynamic response expression obtained in step 7), take the displacement-related parameters. ( Calculate the average difference between four adjacent time steps of each rectangular element as the displacement change of the rectangular element. The acceleration of each node is calculated using the second-order central difference method, and the average acceleration of the four nodes of the rectangular element is taken as the acceleration of the rectangular element. Calculate the strain energy increment of each rectangular element. : ； in, The mass of the rectangular unit, Represents the displacement values ​​of the four nodes of the rectangular element within the time subdomain. , , , The difference between the initial state average and the final state average. , Let represent the displacements of the initial state of the previous time subdomain and the final state of this time subdomain, respectively. The strain energy increments of each rectangular element are accumulated to obtain the strain energy increment across the entire structure within this time subdomain. ( The energy increment at the end of each time subdomain can be obtained by summing the strain energy increments of all time subdomains: ； No. The cumulative energy at the end of each time subdomain is defined as: ； 9) Calculate the maximum cumulative energy that the elastic cantilever plate can withstand: ； In the formula, , These are the material's fracture strength and elastic modulus, respectively. , , The width, thickness, and length of the flexible cantilever plate; 10) Select a series of different overload acceleration peak values Different pulse widths For loads of the form, draw cumulative energy contour maps through steps 2)-8) to extract critical damage points; 11) The cumulative energy input to the structure under load. satisfy: ； In the formula The load magnitude and structural velocity response are respectively considered. Under the condition of minimal structural deflection, velocity and acceleration are directly related, yielding the relationship between peak cumulative energy, peak acceleration, and pulse width: ； in This is the proportionality coefficient; 12) Analyze the dynamic response process of the device structure under test. Assuming the area under load is approximately constant, what is the transient pressure on the surface of the rectangular thin plate and the triangular mass block structure? With impulse It can be obtained as follows: ； in , These are the surface areas of the upper surfaces of the rectangular thin plate and the triangular mass block, respectively. 13) Based on steps 11) and 12), construct the equivalent relationship between pressure and impulse under the cumulative energy state: ； The critical damage point extracted in step 10) is used to obtain the transient pressure in step 12). With impulse The result calculated in step 9) Substitute into step 13) Seeking Thus, the equivalent damage relationship of PI is obtained: ； in This is the equivalent damage threshold; 14) During damage prediction, based on the structural parameters of the device under test and the impact load parameters, the following steps are performed: (1) Obtain the following parameters from step 9. Then, by substituting this into step 13), the equivalent damage threshold is obtained. The transient pressure on the surface of the device under test is obtained through step 12). With impulse and draw The value will eventually be the actual value calculated. Value and equivalent damage threshold The comparison is performed. If the damage is greater than or equal to the damage threshold, it means that the measuring device is predicted to be damaged under the load. If the damage is less than the damage threshold, it means that the measuring device is predicted not to be damaged under the load.

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