A method for calculating secondary frequency vibration boundary conditions

Abstract

The application provides a calculation method of a sub-frequency vibration boundary condition, which comprises the following steps: obtaining a parameter equation based on a sub-frequency vibration parameter according to a harmonic balance equation and a sub-frequency vibration response equation; establishing a complementary equation based on the sub-frequency vibration parameter according to an energy theory; calculating a plurality of sets of solutions of the sub-frequency vibration parameter according to the parameter equation and the complementary equation; bringing the plurality of sets of solutions of the sub-frequency vibration parameter into the sub-frequency vibration response equation respectively to obtain a plurality of sets of corresponding vibration response curves; obtaining a plurality of frequency amplitude response curves according to the plurality of sets of vibration response curves, and determining the boundary condition of the sub-frequency vibration according to the plurality of frequency amplitude response curves. The technical scheme of the application is used to solve the technical problem that the frequency boundary of the sub-frequency vibration cannot be determined in the prior art.

Description

Technical Field

[0001] This invention relates to the field of vibration response analysis technology for nonlinear systems, and in particular to a method for calculating the boundary conditions of secondary frequency vibrations. Background Technology

[0002] In engineering, complex structural shapes and components can easily induce nonlinear vibrations in a system. When the external excitation frequency reaches a certain value, the nonlinear system may exhibit 1 / 3-order harmonic oscillations. This phenomenon has been extensively studied by scholars, but the frequency boundary corresponding to the 1 / 3-order harmonic oscillations has not yet been theoretically investigated in related studies.

[0003] Currently, in the field of nonlinear vibration, the harmonic balance method is a method for seeking the steady-state periodic vibration of a system. Its general idea is to substitute the response equation into the equilibrium equation to obtain the amplitude-frequency characteristic equation, and then determine the frequency boundary based on the amplitude-frequency characteristic equation. This method has been successfully applied to determine the solution of the first harmonic, but it has not yet been successfully used to determine the 1 / 3rd order harmonic vibration. The reason is that when using the harmonic balance method to study the 1 / 3rd order harmonic vibration, the number of equations is less than the number of unknowns, making it impossible to accurately solve for the 1 / 3rd order harmonic vibration, and thus impossible to determine its vibration boundary. Summary of the Invention

[0004] This invention provides a method for calculating the boundary conditions of secondary frequency vibration, which can solve the technical problem that the boundary of the secondary frequency vibration frequency cannot be determined in the prior art.

[0005] According to one aspect of the present invention, a method for calculating the boundary conditions of secondary frequency vibration is provided, the method comprising:

[0006] Based on the harmonic balance equation and the second-frequency vibration response equation, the parametric equation based on the second-frequency vibration parameters is obtained.

[0007] Based on energy theory, supplementary equations based on secondary frequency vibration parameters are established;

[0008] Multiple solutions to the secondary frequency vibration parameters were obtained by calculating the parametric equations and supplementary equations.

[0009] Substituting multiple solutions of the secondary frequency vibration parameters into the secondary frequency vibration response equation yields multiple corresponding vibration response curves.

[0010] Multiple frequency response curves are obtained based on the corresponding multiple sets of vibration response curves, and the boundary conditions for secondary frequency vibration are determined based on the multiple frequency response curves.

[0011] Furthermore, the second-frequency vibration is a 1 / 3-order harmonic vibration, and the parameters of the second-frequency vibration include the amplitude of the first-order harmonic component, the amplitude of the 1 / 3-order harmonic component, the phase of the first-order harmonic component, and the phase of the 1 / 3-order harmonic component.

[0012] Furthermore, the harmonic balance equation is:

[0013]

[0014] In the above formula, x represents the system displacement response, ω0 represents the system natural frequency, ξ represents the damping coefficient, ε represents the nonlinear coefficient, B represents the excitation force amplitude coefficient, ω represents the excitation force frequency, and t represents the vibration time.

[0015] Furthermore, the secondary frequency vibration response equation is:

[0016]

[0017] In the above formula, A1 represents the amplitude of the first harmonic component, A2 represents the amplitude of the 1 / 3 order harmonic component, θ1 represents the phase of the first harmonic component, and θ2 represents the phase of the 1 / 3 order harmonic component.

[0018] Furthermore, the supplementary equation based on the secondary frequency vibration parameters, established according to energy theory, is as follows:

[0019]

[0020] Furthermore, the number of solutions to the secondary frequency vibration parameters calculated based on the parametric equations and supplementary equations is three.

[0021] Furthermore, substituting multiple solutions of the secondary frequency vibration parameters into the secondary frequency vibration response equation yields multiple corresponding vibration response curves, including:

[0022] Substitute the three sets of solutions of the secondary frequency vibration parameters into the secondary frequency vibration response equation to obtain the three corresponding secondary frequency vibration response sub-equations;

[0023] For each sub-equation of the secondary frequency vibration response, multiple vibration response curves are obtained by changing the value of the excitation force frequency.

[0024] Furthermore, based on the multiple sets of vibration response curves, several frequency amplitude response curves are obtained, including:

[0025] Determine the peak response of each of the multiple vibration response curves in the first secondary frequency vibration response sub-equation, and connect them sequentially to obtain the first amplitude frequency response curve;

[0026] Determine the peak response of each of the multiple vibration response curves in the second secondary frequency vibration response sub-equation, and connect them sequentially to obtain the second amplitude frequency response curve;

[0027] Determine the peak response of each of the multiple vibration response curves in the third secondary frequency vibration response sub-equation, and connect them sequentially to obtain the second amplitude frequency response curve.

[0028] Furthermore, determining the boundary conditions for secondary frequency vibration based on multiple frequency response curves includes: determining the excitation force frequency corresponding to the intersection of the first, second, and third frequency response curves as the boundary conditions for secondary frequency vibration.

[0029] The present invention provides a method for calculating the boundary conditions of secondary frequency vibrations. This method obtains supplementary equations through the energy method, which, together with the parametric equations obtained through the harmonic balance method, form a complete set of equations. This allows for the solution of the amplitude-frequency response curve of the secondary frequency vibration, and subsequently, the boundary conditions of the secondary frequency vibration can be quickly determined based on the amplitude-frequency response curve. This method avoids repetitive numerical integration calculations, improves design efficiency, and provides theoretical and technical support for vibration reduction design in engineering. Attached Figure Description

[0030] The accompanying drawings, which form part of this specification, are provided to further illustrate embodiments of the invention and, together with the textual description, explain the principles of the invention. It is obvious that the drawings described below are merely some embodiments of the invention, and those skilled in the art can obtain other drawings based on these drawings without any creative effort.

[0031] Figure 1 A schematic diagram of the amplitude-frequency response curve of a 1 / 3 order harmonic vibration provided according to a specific embodiment of the present invention is shown. Detailed Implementation

[0032] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. The following description of at least one exemplary embodiment is merely illustrative and is in no way intended to limit the present invention or its application or use. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0033] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the exemplary embodiments according to this application. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0034] Unless otherwise specifically stated, the relative arrangement, numerical expressions, and values ​​of the components and steps set forth in these embodiments do not limit the scope of the invention. It should also be understood that, for ease of description, the dimensions of the various parts shown in the drawings are not drawn to actual scale. Techniques, methods, and devices known to those skilled in the art may not be discussed in detail, but where appropriate, such techniques, methods, and devices should be considered part of the specification. In all examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not as limitations. Therefore, other examples of exemplary embodiments may have different values. It should be noted that similar reference numerals and letters in the following figures denote similar items; therefore, once an item is defined in one figure, it need not be further discussed in subsequent figures.

[0035] According to a specific embodiment of the present invention, a method for calculating the boundary conditions of secondary frequency vibration is provided. The calculation method includes:

[0036] Based on the harmonic balance equation and the second-frequency vibration response equation, the parametric equation based on the second-frequency vibration parameters is obtained.

[0037] Based on energy theory, supplementary equations based on secondary frequency vibration parameters are established;

[0038] Multiple solutions to the secondary frequency vibration parameters were obtained by calculating the parametric equations and supplementary equations.

[0039] Substituting multiple solutions of the secondary frequency vibration parameters into the secondary frequency vibration response equation yields multiple corresponding vibration response curves.

[0040] Multiple frequency response curves are obtained based on the corresponding multiple sets of vibration response curves, and the boundary conditions for secondary frequency vibration are determined based on the multiple frequency response curves.

[0041] This configuration provides a method for calculating the boundary conditions of secondary frequency vibrations. This method derives supplementary equations through the energy method, which, together with the parametric equations obtained through the harmonic balance method, form a complete set of equations. This allows for the solution of the amplitude-frequency response curve of the secondary frequency vibration, and subsequently, the boundary conditions of the secondary frequency vibration can be quickly determined based on the amplitude-frequency response curve. This method avoids repetitive numerical integration calculations, improves design efficiency, and provides theoretical and technical support for vibration reduction design in engineering. Compared with existing technologies, the technical solution of this invention can solve the technical problem of being unable to determine the frequency boundary of secondary frequency vibrations in existing technologies.

[0042] In this embodiment of the invention, taking the vibration characteristics of a typical single-degree-of-freedom nonlinear system as an example, the corresponding harmonic balance equation is:

[0043]

[0044] In the above formula, x represents the system displacement response, ω0 represents the system natural frequency, ξ represents the damping coefficient, 2ξω0 is the damping coefficient, ε represents the nonlinear coefficient, and B represents the excitation force amplitude coefficient. ω represents the external excitation force, t represents the excitation force frequency, and t represents the vibration time.

[0045] Furthermore, in this embodiment of the invention, the second-frequency vibration is a 1 / 3-order harmonic vibration, and the parameters of the second-frequency vibration include the amplitude of the first-order harmonic component, the amplitude of the 1 / 3-order harmonic component, the phase of the first-order harmonic component, and the phase of the 1 / 3-order harmonic component. Taking the 1 / 3-order harmonic vibration as an example, the response equation of the second-frequency vibration is:

[0046]

[0047] In the above formula, A1 represents the amplitude of the first harmonic component, A2 represents the amplitude of the 1 / 3 order harmonic component, θ1 represents the phase of the first harmonic component, and θ2 represents the phase of the 1 / 3 order harmonic component.

[0048] Substituting the secondary frequency vibration response equation (2) into the harmonic balance equation (1), we get:

[0049]

[0050]

[0051]

[0052]

[0053] Introducing the relation θ1-3θ2=ψ, we can obtain the following parametric equation from the above equation:

[0054]

[0055]

[0056]

[0057] A1, A2 and ψ can be determined by solving equations (3), (4) and (5), but θ1 and θ2 cannot be determined, that is, the 1 / 3 order harmonic response cannot be determined.

[0058] To solve for θ1 and θ2, in this embodiment of the invention, a supplementary equation based on the second-frequency vibration parameters is established according to the energy theory that the work done by the external force during the stable periodic response of the 1 / 3-order harmonic oscillation is equal to the energy dissipated by the damping. The work done by the external force can be expressed as:

[0059]

[0060] The energy dissipated by damping can be expressed as:

[0061]

[0062] Setting the two terms equal yields the following supplementary equation based on the second-frequency vibration parameters:

[0063]

[0064] Solving equation (6) yields θ1, and then θ2 can be obtained from the relation θ1-3θ2=ψ.

[0065] Furthermore, in this embodiment of the invention, the number of solutions to the secondary frequency vibration parameters calculated based on the parametric equation and the supplementary equation is three sets, that is, A1, A2, θ1 and θ2 each correspond to three solutions.

[0066] Based on the above embodiments, in this embodiment of the invention, substituting multiple solutions of the secondary frequency vibration parameters into the secondary frequency vibration response equation to obtain multiple corresponding vibration response curves includes:

[0067] Substitute the three sets of solutions of the secondary frequency vibration parameters into the secondary frequency vibration response equation to obtain the three corresponding secondary frequency vibration response sub-equations;

[0068] For each sub-equation of the secondary frequency vibration response, multiple vibration response curves are obtained by changing the value of the excitation force frequency.

[0069] In this way, three sets of vibration response curves corresponding one-to-one with the three sets of solutions can be obtained.

[0070] Furthermore, in this embodiment of the invention, obtaining multiple frequency amplitude response curves based on multiple sets of vibration response curves includes:

[0071] Determine the peak response of each of the multiple vibration response curves in the first secondary frequency vibration response sub-equation, and connect them sequentially to obtain the first amplitude frequency response curve;

[0072] Determine the peak response of each of the multiple vibration response curves in the second secondary frequency vibration response sub-equation, and connect them sequentially to obtain the second amplitude frequency response curve;

[0073] Determine the peak response of each of the multiple vibration response curves in the third secondary frequency vibration response sub-equation, and connect them sequentially to obtain the second amplitude frequency response curve.

[0074] In other words, each vibration response curve has a response peak value Y. Connecting the response peak values ​​of each curve in the first set of solutions yields an amplitude-frequency response curve. Connecting the response peak values ​​of each curve in the second set of solutions yields a second amplitude-frequency response curve. Connecting the response peak values ​​of each curve in the third set of solutions yields a third amplitude-frequency response curve. Figure 1 As shown.

[0075] Further, in this embodiment of the invention, determining the boundary conditions for secondary frequency vibration based on multiple amplitude response curves includes: determining the excitation force frequency corresponding to the intersection point of the first amplitude response curve, the second amplitude response curve, and the third amplitude response curve as the boundary conditions for secondary frequency vibration. Figure 1 For example, the middle curve has a high degree of agreement with the numerical results, while the responses of the other two curves are unstable. The middle curve intersects with the other two curves at ω of 0.191 and 0.343 respectively. The ω value corresponding to the intersection point is the frequency boundary corresponding to the 1 / 3 order harmonic vibration.

[0076] In summary, this invention provides a method for calculating the boundary conditions of secondary frequency vibrations. This method obtains supplementary equations through the energy method, which, together with the parametric equations obtained through the harmonic balance method, form a complete set of equations. This allows for the solution of the amplitude-frequency response curve of the secondary frequency vibration, and subsequently, the boundary conditions of the secondary frequency vibration can be quickly determined based on the amplitude-frequency response curve. This method avoids repetitive numerical integration calculations, improves design efficiency, and provides theoretical and technical support for engineering vibration reduction design. Compared with existing technologies, the technical solution of this invention can solve the technical problem of being unable to determine the frequency boundary of secondary frequency vibrations in existing technologies.

[0077] For ease of description, spatial relative terms such as "above," "on top of," "on the upper surface of," "above," etc., are used herein to describe the spatial positional relationship of a device or feature as shown in the figures to other devices or features. It should be understood that spatial relative terms are intended to encompass different orientations in use or operation beyond the orientation of the device as described in the figures. For example, if the device in the figures were inverted, a device described as "above" or "on top of" other devices or structures would subsequently be positioned as "below" or "under" other devices or structures. Thus, the exemplary term "above" can include both "above" and "below." The device may also be positioned in other different ways (rotated 90 degrees or in other orientations), and the spatial relative descriptions used herein will be interpreted accordingly.

[0078] Furthermore, it should be noted that the use of terms such as "first" and "second" to define components is merely for the purpose of distinguishing the corresponding components. Unless otherwise stated, the above terms have no special meaning and therefore should not be construed as limiting the scope of protection of this invention.

[0079] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method of calculating a secondary frequency vibration boundary condition, characterized by, The calculation method includes: Based on the harmonic balance equation and the second-frequency vibration response equation, the parametric equation based on the second-frequency vibration parameters is obtained. Based on energy theory, a supplementary equation is established based on the aforementioned secondary frequency vibration parameters; Multiple solutions to the secondary frequency vibration parameters are calculated based on the parametric equations and the supplementary equations. Substituting multiple solutions of the secondary frequency vibration parameters into the secondary frequency vibration response equation yields multiple corresponding vibration response curves. Multiple frequency response curves are obtained based on the multiple sets of vibration response curves, and the boundary conditions for secondary frequency vibration are determined based on the multiple frequency response curves. The secondary frequency vibration is a 1 / 3 order harmonic vibration, and the secondary frequency vibration parameters include the amplitude of the first harmonic component, the amplitude of the 1 / 3 order harmonic component, the phase of the first harmonic component, and the phase of the 1 / 3 order harmonic component. The harmonic balance equation is: ， In the above formula, represents the system displacement response, represents the system natural frequency, represents the damping coefficient, represents the nonlinear coefficient, represents the excitation force amplitude coefficient, represents the excitation force frequency, represents the vibration time; The secondary frequency vibration response equation is: ， In the above formulae, denotes the amplitude of the first order harmonic component, denotes the amplitude of the 1 / 3 order subharmonic component, denotes the phase of the first order harmonic component, denotes the phase of the 1 / 3 order subharmonic component; The supplementary equation based on the second-frequency vibration parameters, established according to energy theory, is as follows: 。 2. The computational method of claim 1, wherein, The number of solutions to the secondary frequency vibration parameters calculated based on the parametric equation and the supplementary equation is three.

3. The computational method of claim 2, wherein, Substituting multiple solutions of the secondary frequency vibration parameters into the secondary frequency vibration response equation yields multiple corresponding vibration response curves, including: Substitute the three sets of solutions of the secondary frequency vibration parameters into the secondary frequency vibration response equation to obtain three corresponding secondary frequency vibration response sub-equations; For each sub-equation of the secondary frequency vibration response, multiple vibration response curves are obtained by changing the value of the excitation force frequency.

4. The computational method of claim 3, wherein, Based on the multiple sets of vibration response curves, several frequency amplitude response curves are obtained, including: Determine the peak response of each of the multiple vibration response curves in the first secondary frequency vibration response sub-equation, and connect them sequentially to obtain the first amplitude frequency response curve; Determine the peak response of each of the multiple vibration response curves in the second secondary frequency vibration response sub-equation, and connect them sequentially to obtain the second amplitude frequency response curve; Determine the peak response of each of the multiple vibration response curves in the third secondary frequency vibration response sub-equation, and connect them sequentially to obtain the second amplitude frequency response curve.

5. The computational method of claim 4, wherein, Determining the boundary conditions for secondary frequency vibration based on multiple amplitude response curves includes: determining the excitation force frequency corresponding to the intersection point of the first amplitude response curve, the second amplitude response curve, and the third amplitude response curve as the boundary conditions for secondary frequency vibration.

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