A method for calculating dynamic response of a non-classical damped vibration system

Abstract

The application provides a kind of non-classical damping vibration system dynamic response calculation method, comprising the following steps: non-classical damping vibration system vibration equation decoupling;Decoupling transfer function dimension reduction;Laplace domain vibration equation representation and external load extreme value-residue decomposition;Non-classical damping vibration system dynamic response solution.The method avoids the defect of periodicity assumption, ensures the accuracy of transient response solution, does not depend on time step, improves the time domain dynamic response calculation efficiency;And the method is more accurate, can handle the external load containing non-harmonic component, so it can realize fast and accurate dynamic response calculation.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of structural engineering, in particular to a dynamic response calculation method of a non-classical damping vibration system. BACKGROUND

[0002] In model tests or actual engineering, the actual damping matrix is often different from the classical proportional damping, such as the additional damping caused by the nonlinearity of attached water or soil in marine engineering structures. The dynamic response calculation of such a non-classical damping vibration system cannot be decoupled by the real modal shape, and the dynamic response analysis of a large degree of freedom vibration system can only be carried out by the overall mass matrix, the damping matrix and the stiffness matrix, which will greatly consume computer resources.

[0003] For the dynamic response calculation of a non-classical damping vibration system, when using time domain analysis, the calculation efficiency is low and the calculation accuracy depends on the time step of time domain analysis, which cannot meet the demand of fast and accurate dynamic response prediction of actual structures; when using frequency domain method, the transient response part is ignored and only the steady state response part is left by calculating the mutual relationship between the transfer function and the load spectrum, so that the dynamic response analysis cannot consider the transient response caused by the initial condition and cannot cover all the most unfavorable cases; when using frequency-time domain transformation method, the calculation efficiency and the calculation accuracy caused by the time step are considered to some extent, but due to the periodicity assumption caused by FFT transformation, the transient solution is difficult to obtain and the existence of non-harmonic load in actual marine environment cannot be covered; when using Laplace domain method, on the one hand, there is no numerical solution method for arbitrary load type in mathematics, on the other hand, the expression of Laplace domain transfer function after decoupling of the vibration equation is difficult to obtain, so it cannot be directly used in actual engineering.

[0004] It can be seen that for the dynamic response solution of a non-classical damping vibration system, the time domain method has problems such as low efficiency and accuracy depending on the time step, and the frequency domain method has problems such as lack of transient solution and inability to avoid periodicity assumption. Therefore, it is urgent to develop a new algorithm to solve the dynamic response calculation problem of a non-classical damping vibration system in actual engineering. SUMMARY

[0005] In view of the above problems existing in the prior art, the present application provides a new dynamic response calculation method of a non-classical damping vibration system, which avoids the periodicity assumption defect of the frequency domain method, ensures the accuracy of the transient response solution, does not depend on the time step, improves the time domain dynamic response calculation efficiency, and has higher accuracy and can process external loads containing non-harmonic components, so as to realize fast and accurate dynamic response calculation.

[0006] To achieve the above object, the present application provides the following technical solutions:

[0007] A dynamic response calculation method of a non-classical damping vibration system, comprising the following steps:

[0008] Step S1, based on a first-order state equation, using left / right eigenvectors to decouple the vibration equation of the non-classical damping vibration system;

[0009] Step S2, decomposing the left / right eigenvectors into upper and lower two-part eigenvectors, and simultaneously substituting into the decoupled vibration equation to obtain a decoupled vibration equation after matrix reduction;

[0010] Step S3, performing Laplace transformation on the decoupled vibration equation after matrix reduction, representing the dynamic equation in the Laplace domain, obtaining the dynamic response generated by the external load excitation and the transient response caused by the initial condition; using complex exponential decomposition technology, solving the extreme value and residue of the external load represented in the time domain, and simultaneously performing Laplace transformation on the external load to obtain the extreme value-residue expression of the external load in the Laplace domain;

[0011] Step S4, solving the transient dynamic response caused by the initial condition using the initial condition to obtain the transient dynamic response in the form of extreme value-residue in the Laplace domain; using system speed and displacement response to represent the state variable in the Laplace domain, and combining the extreme value-residue expression of the response in the complex modal coordinate system to obtain the displacement response in the Laplace domain, and then obtaining the time domain response of the non-classical damping vibration system through inverse Laplace transformation.

[0012] The non-classical damping vibration system dynamic response calculation method provided by the application has the following beneficial effects:

[0013] 1) The application is based on a first-order state equation, and uses left / right eigenvectors to decouple the vibration equation of the non-classical damping vibration system, which has stronger universality and applicability. The classical proportional damping is only a special case of the application.

[0014] 2) The application adopts an equivalent processing method of eigenvectors to reduce the dimension of the transfer function, reduces the calculation dimension by half, improves the calculation efficiency, and can better solve the dynamic response of a large-degree-of-freedom vibration system.

[0015] 3) The application converts the decoupled vibration differential equation to the Laplace domain, and uses the extreme value-residue to represent the external load, so that the steady-state response solving in the Laplace domain can cover the non-harmonic load component, avoiding the harmonic assumption of the Fourier transform in the existing method, not only ensuring the calculation efficiency, but also improving the solving accuracy.

[0016] 4) The application constructs the dynamic response of the non-classical damping vibration system in the Laplace domain by the eigenvectors and the extreme value and residue of the decoupled vibration differential equation, and only needs to use the Laplace inverse transform to obtain the time domain response. Compared with the existing method, the frequency-time domain method of the application has higher calculation precision of transient response solving, and is beneficial to cover the most unfavorable situation in design and prediction. BRIEF DESCRIPTION OF DRAWINGS

[0017] The specific embodiments of the application will be further described in detail below with reference to the accompanying drawings, in which:

[0018] Figure 1 A flow chart showing the steps of the dynamic response calculation method of the non-classical damping vibration system of the application is shown;

[0019] Figure 2 A schematic diagram of a four-story building structure in the embodiment of the application is shown;

[0020] Figure 3 A transfer function of the decoupled dynamic equation of the four-story building structure in the embodiment of the application is shown;

[0021] Figure 4 A comparison diagram of the reconstructed external load and the original external load based on the extreme value and residue of the external load of the four-story building structure obtained by the calculation method of the application is shown;

[0022] Figure 5 A time domain response comparison of the four-story building structure in the embodiment of the application under different calculation time steps is shown;

[0023] Figure 6 A frequency domain response comparison of the four-story building structure in the embodiment of the application under different calculation time steps is shown. DETAILED DESCRIPTION DETAILED DESCRIPTION

[0025] The embodiment provides a dynamic response calculation method of a non-classical damping vibration system, and the specific steps are as shown in Figure 1 The steps include the following steps:

[0026] Step S1, based on the first-order state equation, the vibration equation of the non-classical damping vibration system is decoupled by using the left / right eigenvectors;

[0027] Step S2, the left / right eigenvectors are decomposed into upper and lower two-part eigenvectors, and are simultaneously substituted into the decoupled vibration equation to obtain the decoupled vibration equation after matrix reduction;

[0028] Step S3, Laplace transform is made on the decoupled vibration equation after order reduction of the matrix, the dynamic equation is expressed in the Laplace domain, the dynamic response generated by the external load excitation and the transient response caused by the initial condition are obtained; the complex exponential decomposition technique is used to solve the extreme value and residue of the external load expressed in the time domain, and the Laplace transform is made on the external load to obtain the extreme value-residue expression of the external load in the Laplace domain;

[0029] Step S4, the transient dynamic response caused by the initial condition is solved by using the initial condition, the transient dynamic response in the form of extreme value-residue in the Laplace domain is obtained; the state variable in the Laplace domain is characterized by the system speed and displacement response, and the extreme value-residue expression of the response in the complex modal coordinate system is combined to obtain the displacement response in the Laplace domain, and then the time domain response of the non-classical damping vibration system is obtained through the inverse Laplace transform. Specific implementation method two

[0031] The specific calculation process of step S1 of the embodiment is as follows:

[0032] ①The dynamic equation of the non-classical damping vibration system is expressed by a first-order state equation:

[0033]

[0034] In equation (1): M is the system mass matrix, K is the system stiffness matrix, and C is the system damping matrix, x(t) and f(t) correspond to the speed, displacement and external load of the system respectively;

[0035] ②Since A=A T , the characteristic equations of the dynamic equation (1) are expressed by the left / right eigenvectors, i.e. (λ k A+B T )θ k =0 and (λ j A+B)Θ j =0, and the following equation is obtained:

[0036]

[0037] In equation (2): θ k is the left eigenvector; Θ j is the right eigenvector, where j, k=1, 2, …, 2N, and N is the number of degrees of freedom of the system. The eigenvectors are conjugate complex pairs and linearly independent, so the state variable z(t) in equation (1) can be described as:

[0038]

[0039] In equation (3), y j (t) is a complex modal coordinate system coordinate.

[0040] ③Substitute equation (3) of the state variable z(t) into the dynamic equation (1), and multiply the left side by the transpose of the left eigenvector Then the state equation can be written as:

[0041]

[0042] Based on the characteristic equation (2), the decoupled vibration equation is obtained by decoupling the state equation (4):

[0043]

[0044] The other steps are the same as in the first embodiment. Embodiment three

[0046] The specific calculation process of step S2 of the embodiment is as follows:

[0047] ④For the solution of the characteristic equation, the calculation matrix dimension is 2N x 2N, and when the number of degrees of freedom is large, the characteristic value solution will occupy a large amount of computer resources and take a lot of time. To solve this problem, the right eigenvector is decomposed into upper and lower two-part eigenvectors, each of which is N x 1, that is, Substitute it back into the right eigenvalue equation, and the following equation can be obtained:

[0048]

[0049] ⑤Based on equation (7), the following equation can be obtained: Let The eigenvector can be rewritten as And substitute it back into equation (6) to obtain the reduced-order vibration system characteristic equation of the matrix:

[0050] (λ j 2 M+λ j C+K)Φ j =0 (8)

[0051] For a non-classical damping vibration system with N degrees of freedom, the characteristic values are 2N, and they are conjugate complex pairs. When solving, |λ j 2 M+λ j C+K|=0 can be solved λ j , and then substitute it into equation (8) to obtain the eigenvector Φ j , that is, the overall eigenvector Θ j of the system can be obtained.

[0052] The left eigenvector is obtained by the same processing as in steps 4 and 5, and is The left / right eigenvectors are substituted into the decoupled vibration equation (5) to obtain the decoupled vibration equation of the reduced-order non-classical damping vibration system:

[0053]

[0054] In equation (9), g j (t) is a complex modal load,

[0055] The other steps are the same as in the second embodiment. Fourth Embodiment

[0057] The specific calculation process of step S3 in this embodiment is as follows:

[0058] Step 7: To quickly solve the dynamic response of the non-classical damping vibration system, the Laplace transform is performed on the decoupled dynamic equation, and the dynamic equation is expressed in the Laplace domain, that is:

[0059]

[0060] In equation (10), and are the Laplace transforms of y j (t) and g j (t).

[0061] Thus, in the Laplace domain, the displacement response expression is:

[0062]

[0063] In equation (11), the first term on the right side represents the dynamic response generated by the external load excitation, and the second term represents the transient response part caused by the initial condition.

[0064] Step 7: In the time domain, the external load can be represented as a superposition of a series of real numbers and a series of conjugate complex numbers. When the time interval is Δt, the discrete external load f k,m can be represented as:

[0065]

[0066] In equation (12), f k,m = f k (mΔt) (m = 0, 1, 2, …, M-1), M is the number of discrete points; Y k,n is a residue; τ k,n is a maximum value.

[0067] The extreme value and residue in equation (12) are solved by using complex exponential decomposition technique. Meanwhile, in order to match the Laplace-domain motion equation, the external load f k,m Taking Laplace transform, the extreme value-residue expression of the external load in Laplace domain is obtained as follows:

[0068]

[0069] In equation (13), is the kth element of the complex modal vector .

[0070] The other steps are the same as those in the third embodiment. The fifth embodiment

[0072] The specific calculation process of step S4 in the embodiment is as follows:

[0073] (11), the transient part of the dynamic response is caused by the initial condition, which is currently unknown. By substituting the initial condition into equation (3) for solving, the initial condition y j (0) in the complex modal coordinate system is obtained, that is:

[0074]

[0075] Therefore, in the case of considering the initial condition and the external load of the system at the same time, the dynamic response equation can be expressed in the form of complex exponential as follows:

[0076]

[0077] In equation (15), the first pN extreme values are contributed by the external load, that is, u j,l = τ k,n , l = 1, 2, …, pN, k = 1, 2, …, N, n = 1, 2, …, p, and the corresponding residue is is the transfer function, The last extreme value is contributed by the non-zero initial condition of the system, that is, u j,pN+1 = λ j , and the corresponding residue is

[0078] (3), the state variable z(s) in Laplace domain is represented by the system velocity and displacement response, and the characteristic vector Φ j is substituted, and the displacement response in Laplace domain is obtained by combining equation (15) as follows:

[0079]

[0080] ​By performing the inverse Laplace transform on equation (16), the time-domain response of the non-classical damped vibration system can be obtained.

[0081] The other steps are the same as in Specific Implementation Method Four.

[0082] Through the above implementation methods, accurate and rapid prediction of the dynamic response of time-domain nonclassical damped vibration systems can be achieved.

[0083] Example

[0084] The calculation object in this embodiment of the invention is a four-story building structure, such as... Figure 2 As shown, m1, m2, m3, and m4 are the mass matrices of each floor of the four-story building structure, c1, c2, c3, and c4 are the damping matrices of each floor, k1, k2, k3, and k4 are the stiffness matrices of each floor, and f(t) is the external load. When calculating the dynamic response of this four-story building structure, its actual damping matrix is ​​defined as a non-classical damping matrix, meaning that the damping matrix cannot be linearly represented by the mass and stiffness matrices.

[0085] The dynamic response of the four-story building structure is calculated using the calculation method of this invention. First, in the complex modal coordinate system, the first-order state equations of the dynamic equations of the four-story building structure are decoupled using left / right eigenvectors, yielding the decoupled transfer function of the dynamic equations, as shown below. Figure 3 As shown. From Figure 3 As can be seen, the four inherently coupled components are decoupled into four single-frequency components, indicating that the calculation method of the present invention can achieve decoupling of non-classical vibration systems.

[0086] Then, by using the eigenvector equivalence method, the transfer function is reduced in dimension to obtain the vibration equation of the non-classical damped vibration system after matrix optimization.

[0087] Next, the decoupled vibration differential equation is transformed into the Laplace domain, and the external load is characterized using extrema and residues. Then, the complex exponential decomposition method is used to solve for the extrema and residues corresponding to the discrete-time external load, thus providing a method for calculating the extrema and residues of the transfer function in non-classical structural dynamic response analysis. Then, based on the obtained extrema and residues, time-domain reconstruction is performed, and the reconstructed external load is compared with the original external load, such as... Figure 4 As shown. From Figure 4 As can be seen, the two agree well, indicating that the estimated extreme values ​​and residues can represent the dynamic response analysis of non-classical damped vibration systems under the original external load. Figure 4 It can also be seen that the calculation method of the present invention, while ensuring that the load information is not lost, avoids Fourier's periodicity assumption, thereby improving the applicability of the method of the present invention.

[0088] Finally, the initial conditions are used to solve the transient dynamic response caused by the initial conditions to obtain a dynamic response equation in a complex exponential form; the system speed and displacement response are used to characterize the state variables in the Laplace domain, and the displacement response in the Laplace domain is obtained by combining the dynamic response equation in the complex exponential form, and the time-domain response of the non-classical damping vibration system is obtained through the inverse Laplace transform. Figure 5 The time-domain response comparison and analysis of the four-layer building structure in the embodiment of the application under different calculation time steps using the Newmark-β method in the prior art and the calculation method of the application (shown as proposed method) is shown. Figure 5 In the figure, the time-domain response under the conditions of the time interval Δt being 0.01s, 0.001s and 0.0001s using the prior art time-domain analysis method and the time-domain response under the condition of Δt being 0.01s using the calculation method of the application can be clearly seen. It can be found that the calculation accuracy of the calculation method of the application under the condition of Δt being 0.01s is consistent with the calculation accuracy of the time-domain analysis method in the prior art under the condition of Δt being 0.0001s. It can be seen that the calculation method of the application does not depend on the time step, and a larger time interval can be used to obtain a more accurate calculation result; on the other hand, the calculation method of the application can use a larger calculation time step to obtain a more accurate calculation result, and the calculation efficiency can be further improved.

[0089] Figure 6 The frequency-domain response comparison and analysis of the four-layer building structure in the embodiment of the application under different calculation time steps using the Newmark-β method in the prior art and the calculation method of the application (shown as proposed method) is shown. Figure 6 It can be seen that the dynamic response obtained by the calculation method of the application has a very high fitting degree with the dynamic response obtained by the Newmark-β method in the prior art. This shows that the calculation method of the application can solve the transient response and the steady-state response in the Laplace domain, and can overcome the disadvantage that the transient response cannot be solved in the frequency domain while ensuring the calculation efficiency.

[0090] The application is not limited to the above embodiments, and various modifications or variations of the application do not deviate from the spirit and scope of the application, and these modifications and variations belong to the claims and equivalent technical scope of the application, and the application also intends to include these modifications and variations.

Claims

1. A method for calculating the dynamic response of a non-classically damped vibrating system, characterized by, The method comprises the following steps: ； Domain response.

2. The method of claim 1, wherein, The specific calculation process of the step S2 is as follows: Substitute the left / right eigenvectors into the decoupled vibration equation (5) simultaneously, and the vibration equation of the non-classical damping vibration system after the matrix reduction is obtained: In equation (9): .

3. The method of claim 2, wherein, The specific calculation process of the step S3 is as follows: And residue: 。 4. The method of claim 3, wherein, The specific calculation process of the step S4 is as follows: The Laplace inverse transform is performed on the equation (16), and the time domain response of the non-classical damping vibration system is obtained.

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