A numerical prediction method for nonlinear response of multi-point time-series strong impact variable cross-section beam

Abstract

This invention relates to the field of structural impact dynamics and numerical calculation technology, specifically disclosing a numerical prediction method for the nonlinear response of a variable cross-section beam under multi-point time-series strong impact. The method includes: S1, determining the parameters of the variable cross-section beam and establishing an equivalent multi-degree-of-freedom model; S2, determining the multiple loading conditions for the multi-point time-series strong impact; S3, calculating the stiffness changes of each element under large deformation effects; S4, calculating the initial plastic limit displacement difference at both ends of each element; S5, calculating the stiffness matrix affected by geometric nonlinearity at the current moment; S6, calculating the stiffness matrix affected by material nonlinearity at the current moment; S7, calculating the total internal force at the current moment; S8, calculating the acceleration at the current moment; and S9, calculating the velocity and displacement at the current moment. This invention achieves numerical prediction of the nonlinear response of a variable cross-section beam under multi-point time-series strong impact by establishing an equivalent multi-degree-of-freedom model, combining geometric and material nonlinearity to correct the stiffness matrix, and solving the equations of motion.

Description

Technical Field

[0001] This invention relates to the field of structural impact dynamics and numerical calculation technology, and more specifically, to a numerical prediction method for the nonlinear response of a multi-point time-series strong impact variable cross-section beam. Background Technology

[0002] Under severe impact conditions such as strong earthquakes and collisions, engineering structures often exhibit significant nonlinear dynamic response characteristics. This is especially true when load spatial distribution is complex, loading time exhibits a clear temporal sequence, and structural geometric parameters vary non-uniformly, making the structural response problem even more intricate. Compared to equivalent single-degree-of-freedom models, equivalent multi-degree-of-freedom models can more realistically reflect the internal dynamic behavior of structures, possessing greater universality and scalability. They can effectively describe complex conditions such as multi-point loading, non-uniformly distributed loads, temporal impacts, and changes in structural parameters along the length direction. Particularly for variable cross-section beam structures under multi-point, temporal-sequential strong impacts, the response process is often accompanied by geometric and material nonlinearities, making single-degree-of-freedom models insufficient for meeting accuracy requirements. Therefore, developing an efficient and reliable equivalent multi-degree-of-freedom numerical calculation method is of great significance for accurately predicting the nonlinear dynamic response of such structures.

[0003] However, existing research methods still have significant shortcomings in addressing the aforementioned complex impact problems. First, under strong impact loading conditions with multiple points, non-uniform distribution, and obvious temporal characteristics, the equivalent single-degree-of-freedom model has fundamental limitations in the selection of shape functions and mode shape description, often failing to provide accurate solutions or even characterizing the basic dynamic laws of such problems. Second, existing efficient rapid calculation methods mostly use idealized typical structures as research objects, such as beams, plates, or columns with uniform cross-sections. They are poorly applicable to variable cross-section structures with continuously varying geometric parameters along the length direction, and cannot reflect the influence of non-uniform distribution of structural parameters on the dynamic response. Furthermore, under multiple strong impact loading conditions, the location, peak value, duration, and impulse characteristics of each loading are often inconsistent. Currently, there is a lack of efficient and rapid calculation methods capable of simultaneously handling multiple, multi-point, asynchronous loading problems, which to some extent restricts the engineering application of structural dynamic analysis under complex impact conditions.

[0004] As mentioned above, existing research has shortcomings in terms of modeling capability, computational efficiency, and applicability for the nonlinear response of variable cross-section structures under multi-point temporal strong impact, making it difficult to meet the practical requirements of complex engineering problems that demand both accuracy and efficiency. Therefore, researching an equivalent multi-degree-of-freedom numerical calculation method that can quickly predict the nonlinear response of variable cross-section beams under multi-point temporal strong impact is of great significance for revealing the structural dynamics under complex impact conditions, improving the ability to quickly assess engineering projects, and guiding the design of protective structures. Summary of the Invention

[0005] The purpose of this invention is to provide a numerical prediction method for the nonlinear response of a variable cross-section beam under multi-point time-series strong impact. By establishing an equivalent multi-degree-of-freedom model and using geometric nonlinearity and material nonlinearity to correct the stiffness matrix for iterative solution, the nonlinear response of the variable cross-section beam under multi-point time-series strong impact can be numerically predicted.

[0006] To achieve the above objectives, the present invention provides the following technical solution: A numerical prediction method for the nonlinear response of a multi-point time-series strong impact variable cross-section beam includes the following steps: S1. Determine the element division parameters and structural parameters of each element for the variable cross-section beam, and establish an equivalent multi-degree-of-freedom model; S2. Determine the multiple loading conditions for multi-point temporal strong shocks to the equivalent multi-degree-of-freedom model; S3. Using the structural parameters of each element at the current moment, calculate the stiffness change of each element under the large deformation effect caused by strong impact; S4. Using the initial stiffness and plastic limit bending moment of each element, calculate the initial plastic limit displacement difference at both ends of each element. S5. Using the stiffness change of the equivalent multi-degree-of-freedom model and the stiffness matrix of the previous moment, calculate the stiffness matrix of the equivalent multi-degree-of-freedom model under the influence of geometric nonlinearity at the current moment. S6. Using the stiffness matrix and material nonlinearity of the equivalent multi-degree-of-freedom model at the current moment, which are affected by geometric nonlinearity, calculate the stiffness matrix of the equivalent multi-degree-of-freedom model at the current moment, considering the material nonlinearity. S7. Calculate the total internal force of the equivalent multi-degree-of-freedom model at the current moment using the difference between the total stiffness matrix and displacement vector of the equivalent multi-degree-of-freedom model at the current moment. S8. Using the total internal forces, mass matrix, and strong impact load of the equivalent multi-degree-of-freedom model at the current moment, calculate the acceleration of the equivalent multi-degree-of-freedom model at the current moment. S9. Using the current acceleration and the motion parameters from the previous moment, calculate the velocity and displacement of the equivalent multi-degree-of-freedom model at the current moment.

[0007] Furthermore, in S3, the formula for calculating the stiffness change of each element under the large deformation effect is: (1); In the formula, Indicates the first i The stiffness change of each element at the current time step Indicates the first i The axial length of a unit at time n. Indicates the first i The elastic modulus of each unit Indicates the firsti The cross-sectional width of each unit, Indicates the first i The cross-sectional height of each unit Indicates the first i The initial length of each unit.

[0008] Furthermore, in S4, the formula for calculating the initial plastic limit displacement difference at both ends of each unit is: (2); In the formula, Indicates the first i The ultimate displacement difference between the two ends of each element Indicates the first i The plastic limit bending moment of each element. Indicates the first i The initial stiffness of each element. Indicates the first i The initial length of each unit.

[0009] Furthermore, in S5, the formula for calculating the stiffness matrix of the equivalent multi-degree-of-freedom model under the influence of geometric nonlinearity at the current moment is: (3); In the formula, This represents the stiffness matrix considering geometric nonlinearity at time n. Indicates the first n The stiffness matrix at time -1 after being affected by geometric and material nonlinearities. Indicates the first n The change in the stiffness matrix due to geometric nonlinearity at each time step.

[0010] Furthermore, in S6, the formula for calculating the stiffness matrix of the equivalent multi-degree-of-freedom model considering the nonlinear effects of materials at the current moment is: (4); In the formula, Indicates the first n The stiffness matrix after being affected by geometric nonlinearity and material nonlinearity at a given time step. This represents the stiffness matrix affected by geometric nonlinearity at time n. Indicates the first n The influence of geometric nonlinearity on the stiffness matrix of the equivalent multi-degree-of-freedom model at each moment, and the influence of material nonlinearity.

[0011] Furthermore, in S7, the formula for calculating the total internal force of the equivalent multi-degree-of-freedom model at the current moment is: (5); In the formula, Indicates the first n The sum of the total internal force at each moment, Indicates the first n The sum of the total internal force at time -1. Indicates the first n The displacement vector at each moment. Indicates the first n The displacement vector at time -1. Indicates the first n The stiffness matrix after being affected by geometric nonlinearity and material nonlinearity at a given time.

[0012] Furthermore, in S8, the formula for calculating the acceleration of the equivalent multi-degree-of-freedom model at the current moment is: (6); In the formula, C is the damping term. Let n be the acceleration at time n. For the strong impact load at time n, Let the velocity be at time n. The sum of the total internal forces at the nth moment. This is the quality matrix.

[0013] Furthermore, in S9, the formulas for calculating the velocity and displacement of the equivalent multi-degree-of-freedom model at the current moment are as follows: (7); In the formula, Indicates the first The speed at that moment Indicates the first The speed at that moment This represents the acceleration at time n. Indicates the time step. Indicates the first Displacement at any given moment This represents the displacement vector at the nth time step.

[0014] According to specific embodiments provided by the present invention, the present invention has the following technical effects compared to the prior art: This invention constructs a discretized model of a variable cross-section beam and introduces the time history of multi-point time-series strong impact loads, enabling accurate simulation of the mechanical behavior of the beam under extreme dynamic conditions. By calculating the element stiffness changes based on large deformation effects in real time, it effectively considers the correction of system stiffness by geometric nonlinearity. Simultaneously, it utilizes the plastic limit displacement to introduce the influence of material nonlinearity, achieving dual coupling calculation of geometric nonlinearity and material nonlinearity. By iteratively updating the total stiffness matrix of the system and solving the equations of motion, it can accurately obtain the dynamic response parameters such as acceleration, velocity, displacement, and internal force of each degree of freedom of the beam during strong impact. This achieves high-precision numerical prediction of the nonlinear response of complex cross-section beams under multiple impacts, solving the problem that traditional linear theory cannot accurately describe the drastic stiffness changes caused by large deformation and plastic damage under strong impacts. It provides a reliable theoretical basis and calculation method for the impact-resistant design and safety assessment of related engineering structures. Attached Figure Description

[0015] 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, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0016] The following description, in conjunction with the accompanying drawings, further illustrates the numerical prediction method for the nonlinear response of a multi-point time-series strong impact variable cross-section beam according to the present invention. Figure 1 This is a schematic diagram of the overall process of the numerical prediction method for the nonlinear response of a multi-point time-series strong impact variable cross-section beam in an embodiment of the present invention; Figure 2 is a schematic diagram of the time history of three time-series loadings at each loading position of the equivalent multi-degree-of-freedom model in Embodiment 1 of the present invention; wherein (a) is a schematic diagram of the time history of three time-series loadings at the first loading position, (b) is a schematic diagram of the time history of three time-series loadings at the second loading position, (c) is a schematic diagram of the time history of three time-series loadings at the third loading position, (d) is a schematic diagram of the time history of three time-series loadings at the fourth loading position, and (e) is a schematic diagram of the time history of three time-series loadings at the fifth loading position. Figure 3 is a schematic diagram of the mode shape, natural frequency and natural period of each degree of freedom in Embodiment 1 of the present invention; wherein (a) is a schematic diagram of the mode shape, natural frequency and natural period of the first degree of freedom, (b) is a schematic diagram of the mode shape, natural frequency and natural period of the second degree of freedom, (c) is a schematic diagram of the mode shape, natural frequency and natural period of the third degree of freedom, (d) is a schematic diagram of the mode shape, natural frequency and natural period of the fourth degree of freedom, and (e) is a schematic diagram of the mode shape, natural frequency and natural period of the fifth degree of freedom. Figure 4 is a diagram showing the internal force variation at each loading position of the equivalent multi-degree-of-freedom model in Embodiment 1 of the present invention; where (a) is the internal force variation at the first loading position, (b) is the internal force variation at the second loading position, (c) is the internal force variation at the third loading position, (d) is the internal force variation at the fourth loading position, and (e) is the internal force variation at the fifth loading position. Figure 5 shows the velocity response diagrams at each loading position of the equivalent multi-degree-of-freedom model in Embodiment 1 of the present invention; where (a) is the velocity response diagram at the first loading position, (b) is the velocity response diagram at the second loading position, (c) is the velocity response diagram at the third loading position, (d) is the velocity response diagram at the fourth loading position, and (e) is the velocity response diagram at the fifth loading position. Figure 6 shows the displacement response diagrams of the equivalent multi-degree-of-freedom model at each loading position in Embodiment 1 of the present invention; where (a) is the displacement response diagram of the first loading position, (b) is the displacement response diagram of the second loading position, (c) is the displacement response diagram of the third loading position, (d) is the displacement response diagram of the fourth loading position, and (e) is the displacement response diagram of the fifth loading position. Detailed Implementation

[0017] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples. The following examples are for illustrative purposes only and are not intended to limit the scope of the invention.

[0018] To better understand the purpose, structure, and function of this invention, the invention will be described in further detail below with reference to the accompanying drawings.

[0019] Example 1 like Figure 1 As shown, this invention provides a numerical prediction method for the nonlinear response of a multi-point time-series strong impact variable cross-section beam, comprising the following steps: S1. Determine the element division parameters and structural parameters of each element for the variable cross-section beam, and establish an equivalent multi-degree-of-freedom model; In this embodiment, the element division parameters are determined, including the total length of the variable cross-section beam, the element length, the number of elements, and the number of concentrated mass points, as shown in Table 1.

[0020] Table 1. Element division of equivalent multi-degree-of-freedom model (SI)

[0021] In this embodiment, the material / structural parameters of each unit of the variable cross-section beam are determined, including: cross-section height, cross-section width, material density, elastic modulus, yield strength, ultimate bending moment, etc., as shown in Table 2.

[0022] Table 2. Geometric and material parameter settings for each unit (SI).

[0023] S2. Determine the multiple loading conditions for multi-point temporal strong shocks to the equivalent multi-degree-of-freedom model.

[0024] In this embodiment, the conditions for multiple strong impact loading are determined, including the location of each loading, the peak pressure, the duration, and the time interval between each loading, as shown in Tables 3 and 4.

[0025] Table 3. Multi-point timing loading settings (SI)

[0026] Table 4. Timing Loading Time Intervals (SI)

[0027] S3. Using the structural parameters of each element at the current moment, calculate the stiffness change of each element under the large deformation effect caused by strong impact; In S3, the formula for calculating the stiffness change of each element under the large deformation effect is: (1); In the formula, Indicates the first i The stiffness change of each element at the current time step Indicates the first i The axial length of a unit at time n. Indicates the first i The elastic modulus of each unit Indicates the first i The cross-sectional width of each unit, Indicates the first i The cross-sectional height of each unit Indicates the first i The initial length of each unit.

[0028] This embodiment is applied to a variable cross-section beam. Material / geometric binonlinear response prediction calculations under three non-uniformly distributed strong impact loading events yield the following element stiffness under large deformation effects: k=[1.3440 0.9720 0.0603 6.5000 1.8400 0.0967].

[0029] S4. Using the initial stiffness and plastic limit bending moment of each element, calculate the initial plastic limit displacement difference at both ends of each element. In S4, the formula for calculating the initial plastic limit displacement difference at both ends of each unit is: (2); In the formula, Indicates the first i The ultimate displacement difference between the two ends of each element Indicates the first i The plastic limit bending moment of each element. Indicates the first i The initial stiffness of each element. Indicates the first i The initial length of each unit.

[0030] This embodiment is applied to a variable cross-section beam. Material / geometric binonlinear response prediction calculations under three non-uniformly distributed strong impact loading conditions yield the following plastic limit displacements for each element: .

[0031] S5. Using the stiffness change of the equivalent multi-degree-of-freedom model and the stiffness matrix of the previous moment, calculate the stiffness matrix of the equivalent multi-degree-of-freedom model under the influence of geometric nonlinearity at the current moment. In S5, the formula for calculating the stiffness matrix of the equivalent multi-degree-of-freedom model under the influence of geometric nonlinearity at the current moment is: (3); In the formula, This represents the stiffness matrix considering geometric nonlinearity at time n. Indicates the first n The stiffness matrix at time -1 after being affected by geometric and material nonlinearities. Indicates the first n The change in the stiffness matrix due to geometric nonlinearity at each time step.

[0032] S6. Using the stiffness matrix and material nonlinearity of the equivalent multi-degree-of-freedom model at the current moment, which are affected by geometric nonlinearity, calculate the stiffness matrix of the equivalent multi-degree-of-freedom model at the current moment, considering the material nonlinearity. In S6, the formula for calculating the stiffness matrix of the equivalent multi-degree-of-freedom model considering the nonlinear effects of materials at the current moment is: (4); In the formula, This represents the stiffness matrix after being affected by geometric and material nonlinearities at time n. This represents the stiffness matrix affected by geometric nonlinearity at time n. This indicates the influence of geometric nonlinearity and material nonlinearity on the stiffness matrix of the equivalent multi-degree-of-freedom model at time n.

[0033] S7. Calculate the total internal force of the equivalent multi-degree-of-freedom model at the current moment using the difference between the total stiffness matrix and displacement vector of the equivalent multi-degree-of-freedom model at the current moment. In S7, the formula for calculating the total internal force of the equivalent multi-degree-of-freedom model at the current moment is: (5); In the formula, Indicates the first n The sum of the total internal force at each moment, Indicates the first n The sum of the total internal force at time -1. Indicates the first n The displacement vector at each moment. Indicates the first n The displacement vector at time -1. Indicates the first n The stiffness matrix after being affected by geometric nonlinearity and material nonlinearity at a given time.

[0034] S8. Using the total internal forces, mass matrix, and strong impact load of the equivalent multi-degree-of-freedom model at the current moment, calculate the acceleration of the equivalent multi-degree-of-freedom model at the current moment. In S8, the formula for calculating the acceleration of the equivalent multi-degree-of-freedom model at the current moment is: (6); In the formula, C is the damping term. Let n be the acceleration at time n. For the strong impact load at time n, Let the velocity be at time n. The sum of the total internal forces at the nth moment. This is the quality matrix.

[0035] S9. Using the current acceleration and the motion parameters from the previous moment, calculate the velocity and displacement of the equivalent multi-degree-of-freedom model at the current moment.

[0036] In S9, the formulas for calculating the velocity and displacement of the equivalent multi-degree-of-freedom model at the current moment are as follows: (7); In the formula, Indicates the first The speed at that moment Indicates the first The speed at that moment This represents the acceleration at time n. Indicates the time step. Indicates the first Displacement at any given moment This represents the displacement vector at the nth time step.

[0037] This embodiment is applied to a variable cross-section beam to predict the material / geometric bi-nonlinear response under three non-uniformly distributed strong impact loadings. The internal force changes at each loading position (blue area represents the loading period) are shown in Figure 4; the velocity response at each loading position (blue area represents the loading period) is shown in Figure 5; and the displacement response at each loading position (blue area represents the loading period) is shown in Figure 6.

[0038] Example 2 The present invention also provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the numerical prediction method for the nonlinear response of a multi-point time-series strong impact variable cross-section beam in Embodiment 1.

[0039] The present invention also provides a computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the numerical prediction method for the nonlinear response of a multi-point time-series strong impact variable cross-section beam in Embodiment 1.

[0040] The above description of the disclosed embodiments enables those skilled in the art to make or use the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the present invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for numerically predicting nonlinear response of a multi-point time history high-impact variable cross-section beam, characterized in that, Includes the following steps: S1. Determine the element division parameters and structural parameters of each element for the variable cross-section beam, and establish an equivalent multi-degree-of-freedom model; S2. Determine the multiple loading conditions for multi-point temporal strong shocks to the equivalent multi-degree-of-freedom model; S3. Using the structural parameters of each element at the current moment, calculate the stiffness change of each element under the large deformation effect caused by strong impact; S4. Using the initial stiffness and plastic limit bending moment of each element, calculate the initial plastic limit displacement difference at both ends of each element. S5. Using the stiffness change of the equivalent multi-degree-of-freedom model and the stiffness matrix of the previous moment, calculate the stiffness matrix of the equivalent multi-degree-of-freedom model under the influence of geometric nonlinearity at the current moment. S6. Using the stiffness matrix and material nonlinearity of the equivalent multi-degree-of-freedom model at the current moment, which are affected by geometric nonlinearity, calculate the stiffness matrix of the equivalent multi-degree-of-freedom model at the current moment, considering the material nonlinearity. S7. Calculate the total internal force of the equivalent multi-degree-of-freedom model at the current moment using the difference between the total stiffness matrix and displacement vector of the equivalent multi-degree-of-freedom model at the current moment. S8. Using the total internal forces, mass matrix, and strong impact load of the equivalent multi-degree-of-freedom model at the current moment, calculate the acceleration of the equivalent multi-degree-of-freedom model at the current moment. S9. Using the current acceleration and the motion parameters from the previous moment, calculate the velocity and displacement of the equivalent multi-degree-of-freedom model at the current moment.

2. The numerical prediction method for the nonlinear response of a multi-point time-series strong impact variable cross-section beam according to claim 1, characterized in that, In S3, the formula for calculating the stiffness change of each element under the large deformation effect is: (1)； In the formula, Indicates the first i The stiffness change of each element at the current time step Indicates the first i The axial length of a unit at time n. Indicates the first i The elastic modulus of each unit Indicates the first i The cross-sectional width of each unit Indicates the first i The cross-sectional height of each unit Indicates the first i The initial length of each unit.

3. The numerical prediction method for the nonlinear response of a multi-point time-series strong impact variable cross-section beam according to claim 1, characterized in that, In S4, the formula for calculating the initial plastic limit displacement difference at both ends of each unit is: (2)； In the formula, Indicates the first i The ultimate displacement difference between the two ends of each element Indicates the first i The plastic limit bending moment of each element. Indicates the first i The initial stiffness of each element. Indicates the first i The initial length of each unit.

4. The numerical prediction method for the nonlinear response of a multi-point time-series strong impact variable cross-section beam according to claim 1, characterized in that, In S5, the formula for calculating the stiffness matrix of the equivalent multi-degree-of-freedom model under the influence of geometric nonlinearity at the current moment is: (3)； In the formula, This represents the stiffness matrix considering geometric nonlinearity at time n. Indicates the first n The stiffness matrix at time -1 after being affected by geometric and material nonlinearities. Indicates the first n The change in the stiffness matrix due to geometric nonlinearity at each time step.

5. The numerical prediction method for the nonlinear response of a multi-point time-series strong impact variable cross-section beam according to claim 1, characterized in that, In S6, the formula for calculating the stiffness matrix of the equivalent multi-degree-of-freedom model considering the nonlinear effects of materials at the current moment is: (4)； In the formula, Indicates the first n The stiffness matrix after being affected by geometric nonlinearity and material nonlinearity at a given time step. This represents the stiffness matrix affected by geometric nonlinearity at time n. Indicates the first n The influence of geometric nonlinearity on the stiffness matrix of the equivalent multi-degree-of-freedom model at each moment, and the influence of material nonlinearity.

6. The numerical prediction method for the nonlinear response of a multi-point time-series strong impact variable cross-section beam according to claim 1, characterized in that, In S7, the formula for calculating the total internal force of the equivalent multi-degree-of-freedom model at the current moment is: (5)； In the formula, Indicates the first n The sum of the total internal force at each moment, Indicates the first n The sum of the total internal force at time -1. Indicates the first n The displacement vector at each moment. Indicates the first n The displacement vector at time -1. Indicates the first n The stiffness matrix after being affected by geometric nonlinearity and material nonlinearity at a given time.

7. The numerical prediction method for the nonlinear response of a multi-point time-series strong impact variable cross-section beam according to claim 1, characterized in that, In S8, the formula for calculating the acceleration of the equivalent multi-degree-of-freedom model at the current moment is: (6)； In the formula, C is the damping term. Let n be the acceleration at time n. For the strong impact load at the nth moment, Let the velocity be at time n. The sum of the total internal forces at the nth moment. This is the quality matrix.

8. The numerical prediction method for the nonlinear response of a multi-point time-series strong impact variable cross-section beam according to claim 1, characterized in that, In S9, the formulas for calculating the velocity and displacement of the equivalent multi-degree-of-freedom model at the current moment are as follows: (7)； In the formula, Indicates the first The speed at that moment Indicates the first The speed at that moment This represents the acceleration at time n. Indicates the time step. Indicates the first Displacement at any given moment This represents the displacement vector at the nth time step.

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