Nonlinear system dynamic characteristic calculation method based on structural dynamic modification

Abstract

The embodiment of the invention discloses a nonlinear system dynamic characteristic calculation method based on a structural dynamic modification method, and the method achieves the high-precision reduced-order modeling of a nonlinear structure through enabling a complex nonlinear effect to be equivalent to an additional stiffness field and combining with a structural dynamic modification algorithm. And based on the decoupled linear substructure modal space, a nonlinear correction operator is introduced, and the global dynamic response of the nonlinear structure is solved through a resolving method after modification. Structural vibration frequency, vibration shape and energy dissipation characteristics under a nonlinear effect can be efficiently obtained, and dynamic characteristics under different nonlinear amplitudes can be obtained. Compared with a traditional iteration method, the method has the advantage that the calculation complexity is remarkably reduced by equivalently linearizing a nonlinear part.

Description

Technical Field

[0001] The present invention belongs to the field of nonlinear dynamics calculation, and relates to a method for calculating the dynamic characteristics of a nonlinear system based on a structural dynamic modification method. The method can be widely used in fields such as structural vibration analysis, mechanical system optimization, and aerospace engineering. Background Art

[0002] Flexible lightweight structures, owing to their excellent mechanical properties, light weight, and high stiffness, have always played an irreplaceable role in aerospace engineering. Due to the unique characteristics of these structures, their experimental testing and analytical calculations present numerous challenges. Flexible lightweight structures are prone to significant deformation under load, resulting in the system's dynamic characteristics being significantly affected by geometric nonlinearity, making traditional linear modal methods difficult to directly apply. Therefore, developing efficient computational methods to quickly and accurately predict the system's dynamic characteristics for structures with nonlinear stiffness characteristics is of great engineering value and practical significance.

[0003] Nonlinear behavior is widely present in various engineering structures and requires specialized theories and methods for modeling and solving. For example, Vakakis et al. proposed the Nonlinear Normal Modes (NNMs) method, which provides a theoretical framework for modal analysis of nonlinear systems. Lai et al. studied the approximate free vibration solution of nonlinear stiffness systems based on the linearized harmonic balance method, expanding the solution ideas for nonlinear systems. Marinone et al. proposed a modal modification response technology that uses the modal superposition method to significantly reduce the calculation time, making it exhibit high computational efficiency in systems with local nonlinearities. Yao Hongliang et al. proposed a dimensionality reduction incremental harmonic balance method, which effectively reduces the computational complexity by ignoring unimportant high-order information in the linear part and retaining only the main degrees of freedom of the nonlinear part.

[0004] Although the aforementioned methods have made some progress in nonlinear system analysis, most rely on iterative solutions, resulting in high computational complexity, a dependence on initial assumptions for convergence, and divergence issues, making them difficult to meet the efficient and stable computational requirements of practical engineering. Therefore, developing a computational method that can efficiently and accurately predict the dynamic characteristics of structures containing nonlinear forces is of great significance for structural design optimization, health monitoring, and dynamic response prediction in high-end manufacturing fields such as aerospace. Summary of the Invention

[0005] To overcome the problems of existing methods, this paper proposes a nonlinear structural dynamic modification (NLSDM) method for flexible structures. This method can be used to calculate the structural dynamic characteristics of systems containing cubic polynomial nonlinear stiffness. This method has simple implementation requirements and only requires the mass-normalized modal shapes and modal frequencies of a linear system without nonlinear terms, eliminating the need for detailed mass and stiffness matrices. This method is easy to implement, offers higher computational efficiency, and avoids the potential divergence issues associated with iteration. The results are consistent with those of existing iterative methods.

[0006] In order to achieve the above object, the present invention is implemented by adopting the following technical solutions:

[0007] A method for calculating the dynamic characteristics of a nonlinear system based on a structural dynamic modification method is provided for a system in which the nonlinear characteristics can be expressed as a cubic spring term. The method is characterized by comprising the following steps:

[0008] Step 1: Normalized modal parameter extraction and orthogonalization processing:

[0009] When the system does not contain nonlinear forces, it is regarded as a benchmark linear system, and its characteristic equation is solved to obtain the system's undamped natural frequency and mass-normalized modal vibration shape, ensuring that the modal vibration shape meets the orthogonality conditions, thereby laying a benchmark modal framework for subsequent nonlinear system analysis.

[0010] Step 2: Linearization modeling of nonlinear systems:

[0011] For structures with cubic polynomial nonlinear stiffness, the average energy method is used to approximate the equivalent stiffness of local nonlinear force terms and construct an energy balance model of nonlinear stiffness.

[0012] Step 3: Dynamic characteristics analysis of nonlinear system:

[0013] Based on nonlinear normal modes (NNMs), the generalized modal equations of nonlinear systems are established in combination with the structural dynamic modification method (SDM). By decomposing the nonlinear force terms into equivalent linear stiffness matrix components, the extended stiffness matrix is ​​introduced to construct the equivalent linear dynamic equations including nonlinear effects.

[0014] Step 4: Solve using the dual-modal space method:

[0015] The dual-modal space method is used to solve the generalized modal equations of the system and calculate the dynamic characteristics of the nonlinear system. This method can efficiently calculate the frequency-energy curve (FEC) of the nonlinear system and accurately obtain the amplitude-related characteristics of the nonlinear mode.

[0016] Furthermore, the specific method of step 1 is as follows:

[0017] Assume that the structural matrices of the baseline linear system are M and K respectively, and the free vibration equation under the undamped state is:

[0018]

[0019] in, x are the acceleration and displacement vector of the structure respectively.

[0020] Assume that the solution of the free vibration equation under the undamped state is a simple harmonic form:

[0021] x=Acos(ωt) (2)

[0022] Where A is the amplitude vector, t is the time, and ω is the natural frequency. After normalizing the mass of A, the modal vibration shape Φ is obtained. I :

[0023]

[0024] Use modal coordinate transformation to convert physical coordinates into modal coordinates:

[0025] x=Φ I q (4)

[0026] Then transform formula (1) into:

[0027]

[0028] in and q are acceleration vector and displacement vector in modal coordinates respectively, M r ,K r They are the modal mass matrix and modal stiffness matrix:

[0029]

[0030] Where I is the N×N identity matrix, Λ 2 is the frequency matrix:

[0031]

[0032] Furthermore, the specific method of step 2 is as follows:

[0033] When there is a large deformation in the flexible structure, geometric nonlinearity will be introduced, resulting in an overall nonlinear additional stiffness, which will directly affect the dynamic characteristics of the structure. Based on the baseline linear system established in step 1, a nonlinear force term is added, and the motion equation of the nonlinear system is set as:

[0034]

[0035] where f nl (x) is the nonlinear restoring force. Based on the average energy method, the nonlinear restoring force f nl (x) is transformed into a linear system with equivalent stiffness:

[0036] <V nl >= <V eff > (9)

[0037] in, <V nl > is the average potential energy of the nonlinear system, <V eff > is the average potential energy of the equivalent linear system:

[0038]

[0039] Among them, the cycle u is the deformation of the nonlinear spring, which is determined by the position of the nonlinear spring. The nonlinear force term is f nl :

[0040] f nl =k nl u 3 (11)

[0041] Among them, k nl is the cubic nonlinear spring stiffness. The equivalent linear stiffness k can be obtained eq .

[0042] Furthermore, the specific method of step 3 is as follows:

[0043] In conservative systems, nonlinear modes (NNMs) are nonlinear extensions of linear normal modes (LNMs). Nonlinear modes can be described as synchronous periodic motions of the system, where all degrees of freedom of the system oscillate with the same period. An important characteristic of nonlinear modes in this definition is frequency-energy dependence, where the system's vibration frequency changes with the total energy of the system.

[0044] In step 2, find the stiffness term k equivalent to the nonlinear force term eq After that, the stiffness term is the additional stiffness of the local degree of freedom, which needs to be expanded into the additional stiffness matrix K e , establish the equivalent linear motion equation corresponding to formula (8), that is,

[0045] Mx+(K+K e )x=0 (12)

[0046] Furthermore, the specific method of step 4 is as follows:

[0047] The additional stiffness matrix K obtained in step 3 is e Considered as additional stiffness ΔK:

[0048] K'=K+ΔK (13)

[0049] Transform formula (5) into:

[0050]

[0051] According to formula (14), the natural frequencies ω′1, ω′2…ω′ are calculated. N And the mode shape Φ II , perform the second modal coordinate transformation and set

[0052] q=Φ II y (15)

[0053] in and y are the acceleration vector and displacement vector after the second modal transformation, respectively. Then, Equation (14) is transformed into

[0054]

[0055] The free vibration frequencies of the nonlinear system are ω'1, ω'2…ω' N , the vibration mode is Φ I Φ II From the above formula, it can be seen that the proposed nonlinear dynamic calculation method only requires the natural frequency ω of the benchmark undamped linear system. I , mass normalized mode shape Φ I , nonlinear system force term k nl The method is easy to implement and does not require iteration. It can obtain the free vibration frequency and vibration mode of the nonlinear system under different amplitudes r. Compared with the existing technology, the above technical solution has the following beneficial effects:

[0056] The present invention provides a method for calculating the nonlinear structural dynamic characteristics based on structural dynamic modification, which avoids the shortcomings of traditional nonlinear dynamic calculations that require iteration and are slow to calculate. The method has simpler implementation conditions and higher calculation efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Figure 1 The flowchart of the calculation method of nonlinear structural dynamic characteristics based on structural dynamic modification is shown in the figure.

[0058] Figure 2 Schematic diagram of the six-degree-of-freedom nonlinear model;

[0059] Figure 3 This is the residual graph of the iterative process of the traditional method;

[0060] Figure 4 This is the frequency-energy curve obtained using this method. DETAILED DESCRIPTION

[0061] This invention provides a method for calculating the dynamic characteristics of nonlinear systems based on a structural dynamic modification method. This method can be used to calculate the dynamic characteristics of systems containing cubic nonlinear stiffness. The method is easy to implement and has high computational efficiency. This method can be used to predict the dynamic characteristics of nonlinear systems, providing theoretical guidance for dynamic problems in such structures and possessing engineering significance.

[0062] The following is a detailed description of the specific implementation of the present invention with reference to the accompanying drawings and examples. This embodiment provides a six-degree-of-freedom spring-mass model with cubic nonlinear stiffness, and uses the present invention to calculate its nonlinear dynamic characteristics. The creation method includes:

[0063] Step 1: Normalized modal parameter extraction and orthogonalization processing:

[0064] When the system does not contain nonlinear forces, it is regarded as a benchmark linear system, and its characteristic equation is solved to obtain the system's undamped natural frequency and mass-normalized modal vibration shape, ensuring that the modal vibration shape meets the orthogonality conditions, thereby laying a benchmark modal framework for subsequent nonlinear system analysis.

[0065] Step 2: Linearization modeling of nonlinear systems:

[0066] For structures with cubic polynomial nonlinear stiffness, the average energy method is used to approximate the equivalent stiffness of local nonlinear force terms and construct an energy balance model of nonlinear stiffness.

[0067] Step 3: Dynamic characteristics analysis of nonlinear system:

[0068] Based on nonlinear normal modes (NNMs), the generalized modal equations of nonlinear systems are established in combination with the structural dynamic modification method (SDM). By decomposing the nonlinear force terms into equivalent linear stiffness matrix components and introducing the extended stiffness matrix, the equivalent linear dynamic equations including nonlinear effects are constructed.

[0069] Step 4: Solve using the dual-modal space method:

[0070] The dual-modal space method is used to solve the generalized modal equations of the system and calculate the dynamic characteristics of the nonlinear system. This method can efficiently calculate the frequency-energy curve (FEC) of the nonlinear system and accurately obtain the amplitude-related characteristics of the nonlinear mode.

[0071] The specific method of step 1 is as follows:

[0072] like Figure 2 As shown in the figure, a six-degree-of-freedom nonlinear system with a cubic nonlinear spring is established. The mass m1=m2=m3=m4=m5=m6=1kg, and the linear stiffness coefficient k1=k2=k3=k4=k5=k6=1×10 5 N / m. K2 also includes cubic stiffness nonlinear unit, nonlinear stiffness coefficient k nl =1×10 8 N / m 3 , which is reflected as a cubic nonlinear restoring force f nl .

[0073] Assuming that the structural matrices of the baseline linear system without nonlinear force terms are M and K, the free vibration equation in the undamped state is:

[0074] Mx+Kx=0 (1)

[0075] x, x are the acceleration and displacement vectors of the structure respectively. M, K are:

[0076]

[0077] The mass matrix is:

[0078] M=diag(m1,m2,…,m6) (3)

[0079] Assume that the solution to the equation is a simple harmonic form:

[0080] x=Acos(ω Ι t) (4)

[0081] ω I is the natural frequency of the equivalent linear system, t is time, and the displacement vector and amplitude vector are x and A respectively:

[0082] x=(x1 x2 x3 x4 x5 x6) T (5)

[0083] A=(A1 A2 A3 A4 A5 A6) T (6)

[0084] The amplitude vector A is normalized by mass to obtain the vibration mode vector ΦI :

[0085]

[0086] Use modal coordinate transformation to convert physical coordinates into modal coordinates:

[0087] x=Φ I q (8)

[0088] Then formula (1) becomes:

[0089]

[0090] in and q are acceleration vector and displacement vector in modal coordinates respectively, M r ,K r They are the modal mass matrix and modal stiffness matrix:

[0091]

[0092] Where I is the N×N identity matrix, Λ 2 is the frequency matrix:

[0093]

[0094] The specific method of step 2 is as follows:

[0095] The equation of motion for the nonlinear system is:

[0096]

[0097] where f nl (x) is the nonlinear force matrix. For the i-th order, the nonlinear restoring force f on m1 is nl,i for

[0098] f nl,i =k nl u i 3 (13)

[0099] where u i is the deformation of the nonlinear spring, Figure 2 The system shown

[0100] u i =(A i,1 -A i,2 )cos(ω i t) (14)

[0101] Using the average energy method:

[0102]

[0103] The cycle k eq,i is the equivalent linear stiffness. Substituting equations (13) and (14) into equation (15), we can obtain

[0104]

[0105] make Indicates the amplitude, substitute equation (7) into equation (16):

[0106]

[0107] Therefore, the nonlinear force f on m1 is nl,1 for

[0108] f nl,1 =k eq,i (x1-x2) (18)

[0109] The specific method of step 3 is as follows:

[0110] The local stiffness k of each order eq,i All are expanded into six-degree-of-freedom stiffness matrix K eq,i

[0111] K eq,i =k eq,i ·B e T LL T B e (19)

[0112] Among them, L, B e is the transformation matrix

[0113]

[0114] Therefore, the equation of motion (12) can be written as

[0115]

[0116] The specific method of step 4 is as follows:

[0117] The solved K eq,i Considered as additional stiffness ΔK, first use Φ I Perform modal coordinate changes:

[0118]

[0119] From the above formula, we can see that after the mass or stiffness is modified, the originally decoupled modal equation will become a coupled equation again. In order to decouple it again, the natural frequencies ω'1, ω'2…ω' can be calculated according to formula (23): N, the eigenvector Φ II , perform the second modal coordinate transformation, let

[0120] q=Φ II y (24)

[0121] Where y is the modal coordinate after the second coordinate transformation, then formula (14) becomes

[0122]

[0123] The free vibration frequencies of the nonlinear system are ω'1, ω'2…ω' N , the vibration mode is Φ I Φ II From the above formula, it can be seen that the proposed nonlinear dynamic calculation method only requires the natural frequency ω of the benchmark undamped linear system. I , mass normalized mode shape Φ I , nonlinear system force term k nl The method is easy to implement and does not require iteration. It can obtain the free vibration frequency and mode shape of the nonlinear system under different amplitudes r.

[0124] The modified natural frequencies are ω'1, ω'2…ω' N , the modal vibration shape is Φ I Φ II Therefore, based on the changes in mass and stiffness, the changes in the structural modes can be predicted.

Claims

1. A method for calculating the dynamic characteristics of a nonlinear system based on a structural dynamic modification method, wherein the method is applied to a system in which the nonlinear characteristics can be expressed as a cubic spring term, and is characterized in that: The calculation method includes the following steps: Step 1: When the system does not have any nonlinear force acting on it, the system is regarded as a reference linear system, and the characteristic equation of the system is solved to obtain the undamped natural frequency and mass normalized modal vibration shape of the reference linear system; Step 2: When the system has a structure with cubic polynomial nonlinear stiffness, it is called a nonlinear system. The average energy method is used to approximate the equivalent stiffness of the local nonlinear force term and construct an energy balance model of the nonlinear stiffness. Step 3: Based on the nonlinear modal NNMs, the generalized modal equations of the nonlinear system are established in combination with the structural dynamic modification method SDM. By decomposing the nonlinear force terms into equivalent linear stiffness matrix components and introducing the extended stiffness matrix, the equivalent linear dynamic equations including nonlinear effects are constructed. Step 4: Solve the generalized modal equations of the nonlinear system using a dual-modal space method to calculate the dynamic characteristics of the nonlinear system.

2. The method for calculating the dynamic characteristics of a nonlinear system based on the structural dynamic modification method according to claim 1, characterized in that: The specific method of step 1 is as follows: Assume that the structural matrices of the baseline linear system are M and K respectively, and the free vibration equation under the undamped state is: in, x are the acceleration and displacement vector of the structure respectively; Assume that the solution of the free vibration equation under the undamped state is a simple harmonic form: x=Acos(ωt) (2) Where A is the amplitude vector, t is the time, and ω is the natural frequency. After normalizing the mass of A, the modal vibration shape Φ is obtained. I : Use modal coordinate transformation to convert physical coordinates into modal coordinates: x=Φ I q (4) Then transform formula (1) into: in and q are acceleration vector and displacement vector in modal coordinates respectively, M r ,K r They are the modal mass matrix and modal stiffness matrix: Where I is the N×N identity matrix, Λ 2 is the frequency matrix:

3. The method for calculating the dynamic characteristics of a nonlinear system based on the structural dynamic modification method according to claim 1, characterized in that: The specific method of step 2 is as follows: Add nonlinear force terms to the baseline linear system established in step 1 and set the motion equation of the nonlinear system to: where f nl (x) is the nonlinear restoring force; Based on the average energy method, the nonlinear restoring force f nl (x) is transformed into a linear system with equivalent stiffness: <V nl >= <V eff > (9) Among them, <V nl > is the average potential energy of the nonlinear system, <V eff > is the average potential energy of the equivalent linear system: in, u is the deformation of the nonlinear spring, which is determined by the position of the nonlinear spring; the nonlinear force term is: f nl =k nl u 3 (11) Among them, k nl is the cubic nonlinear spring stiffness. The equivalent linear stiffness k can be obtained according to formula (11): eq .

4. The method for calculating the dynamic characteristics of a nonlinear system based on the structural dynamic modification method according to claim 2, characterized in that: The specific method of step 3 is as follows: In step 2, the stiffness term k eq As the additional stiffness of the local degree of freedom, it is expanded into the additional stiffness matrix K e , establish the equivalent linear motion equation corresponding to formula (8), that is, 5. The method for calculating the dynamic characteristics of a nonlinear system based on the structural dynamic modification method according to claim 4, characterized in that: The specific method of step 4 is as follows: The additional stiffness matrix K obtained in step 3 is e Considered as additional stiffness ΔK: K'=K+ΔK (13) Transform formula (5) into: According to formula (14), the natural frequencies ω'1, ω'2…ω' are calculated. N And the mode shape Φ II , perform the second modal coordinate transformation and set q=Φ II y (15) in and y are the acceleration vector and displacement vector after the second modal transformation, respectively. Then, Equation (14) is transformed into The free vibration frequencies of the nonlinear system are ω'1, ω'2…ω' N , the vibration mode is Φ I Φ II .

Images