CN116738589B

CN116738589B - Vibration energy transfer analysis method suitable for discontinuous load-carrying structure system

Info

Publication number: CN116738589B
Application number: CN202310268098.8A
Authority: CN
Inventors: 王永锋; 马艳红; 王治州; 李超; 陈雪骑; 洪杰
Original assignee: Beihang University
Current assignee: Beihang University
Priority date: 2023-03-20
Filing date: 2023-03-20
Publication date: 2024-07-19
Anticipated expiration: 2043-03-20
Also published as: CN116738589A

Abstract

The invention relates to the field of vibration analysis of aeroengines, and particularly discloses a vibration energy transfer analysis method suitable for a discontinuous load-bearing structure system, which comprises the following steps: the energy transmission mechanism of the plate-shell combination isomorphic abrupt structure and the modeling method of the discontinuous load-carrying structure of the abrupt section; solving energy dissipation of a connection interface and an equivalent technology in energy transfer analysis of a structural system; calculating vibration power flow under broadband excitation; and a visual analysis method of energy transmission and distribution characteristics. The invention has the advantages of considering the influence of complex structural features such as geometric mutation, friction energy consumption of a connecting interface and the like, intuitively predicting and describing the energy transmission and distribution features and the like, and is suitable for vibration transmission analysis of a complex structural system with various geometric configurations and connecting structures. The method provides technical support for vibration transfer characteristic analysis and dynamic design of the aeroengine load bearing structure system in engineering practice.

Description

Vibration energy transfer analysis method suitable for discontinuous load-carrying structure system

Technical Field

The invention belongs to the field of vibration analysis of aeroengines, and particularly relates to a vibration energy transfer analysis method suitable for a discontinuous load-bearing structure system.

Background

The aeroengine load bearing structure system consists of a bearing seat, a load bearing frame, a casing and the like, provides support and constraint for the aeroengine, and is a 'skeleton' of the aeroengine. The load-carrying structural system adopts a light plate-shell combined structure and is integrally connected by a flange and a bolt, and the unique structure and load characteristics are key to influence the vibration energy transmission and dissipation characteristics: firstly, the load directions of the bearing support plate and the casing are different, and when the bearing support plate and the casing are combined, the energy aggregation is easily generated near a coupling interface due to the uncoordinated stress and vibration mode, so that the energy transmission and distribution in a bearing structure are influenced; secondly, the behavior such as viscous-sliding exists locally at the connecting interface, and the characteristics such as nonlinear rigidity loss, hysteresis damping energy consumption and the like are provided, so that the structural system response and the energy transfer characteristic are complicated.

Because of the complex structure, the working environment is extremely bad, and the aeroengine load bearing structure system is easy to break down. The damage deformation or failure fault is a serious fault of the bearing structure system, has great influence on the reliable operation of the engine, and the excessive vibration response is a main reason for causing the bearing structure system to generate the fault. Therefore, the method has important significance in timely and accurately estimating the vibration response and the energy transmission/distribution characteristics of the load-bearing structure system under the working load.

While conventional vibration characteristic analysis methods based on structural modal characteristics, transfer functions or responses have difficulty in visually describing the characteristics of vibration energy transfer and distribution in a structure, vibration power flow methods have received attention in recent years because of their characteristics of being able to measure vibration energy. The power refers to the work done by load on an object in unit time, the dynamic change of the power value in the structure is power flow, vibration transmission is measured from the viewpoint of energy, and compared with parameters such as single force or displacement, speed, acceleration and the like, the expressed vibration information is more abundant. Lee H.P and the like realize simulation analysis of sound intensity characteristics of the porous composite material sheet structure; and Y, wang et al realize visual analysis of vibration and acoustic radiation problems of the cylindrical shell of the simple support based on a power flow method. However, such a power flow analysis method is only calculated for a linear structure system, and does not consider a nonlinear structure such as bolting. Wang et al further analyzed for simple structures with bolted joints, revealing the impact of energy dissipation at the interface of contact, but limited to mechanical studies, only simpler structures could be analyzed, and it was difficult to apply to aeroengine load bearing structural systems of complex structures.

Accordingly, there is a need for a vibration energy transfer analysis method that can be adapted for use in a discontinuous load bearing structure system to address the above-described problems.

Disclosure of Invention

The present invention has been made to solve the above-mentioned problems occurring in the prior art. Therefore, a vibration energy transfer analysis method suitable for a discontinuous load bearing structure system is needed, so that accurate prediction of vibration energy transfer characteristics of the aeroengine load bearing structure system under working load is realized, and technical support is provided for developing dynamic design of the load bearing structure system in engineering practice.

According to a first aspect of the present invention there is provided a vibration energy transfer analysis method for a discontinuous load bearing structure system, the method comprising: based on a plate-shell combination and other configuration abrupt structure energy transfer mechanism, carrying out equivalent modeling on a discontinuous load-carrying structure with an abrupt section; solving the energy dissipation of the connection interface and an equivalent technology in the energy transfer analysis of the structural system to obtain an equivalent analysis model; carrying out response solving and system vibration energy distribution based on the equivalent analysis model to obtain response characteristics, and calculating to obtain power flow vectors of each structural unit/node, wherein the response characteristics comprise vibration displacement, speed and node force of each node of the structure; visual representation of vibrational energy transfer and distribution characteristics.

According to a second aspect of the present invention, there is provided a vibration energy transfer analysis apparatus suitable for use in a discontinuous load bearing structure system, the apparatus comprising: the equivalent modeling module is configured to perform equivalent modeling on the discontinuous load-carrying structure with the abrupt cross section based on an energy transmission mechanism of the plate-shell combination and other configuration abrupt structure; the solving module is configured to connect the interface energy dissipation solving and the equivalent technology in the structural system energy transfer analysis to obtain an equivalent analysis model; the power flow vector calculation module is configured to carry out response solution and system vibration energy distribution based on the equivalent analysis model to obtain response characteristics, and calculate and obtain power flow vectors of each structural unit/node, wherein the response characteristics comprise vibration displacement, speed and node force of each node of the structure; and a visualization module configured to visualize the expression of the vibrational energy transfer and distribution characteristics.

According to a third aspect of the present invention there is provided a non-transitory computer readable storage medium storing instructions which, when executed by a processor, perform the method as described above.

According to various embodiments of the invention, the vibration energy transfer analysis method suitable for the discontinuous load-bearing structure system has at least the following technical effects:

The invention has the advantages of considering the influence of complex structural features such as geometric mutation, friction energy consumption of a connecting interface and the like, intuitively predicting and describing the energy transmission and distribution features and the like, and is suitable for vibration transmission analysis of a complex structural system with various geometric configurations and connecting structures. The method provides technical support for vibration transfer characteristic analysis and dynamic design of the aeroengine load bearing structure system in engineering practice.

Drawings

In the drawings, which are not necessarily drawn to scale, like numerals may describe similar components in different views. The same reference numerals with letter suffixes or different letter suffixes may represent different instances of similar components. The accompanying drawings illustrate various embodiments by way of example in general and not by way of limitation, and together with the description and claims serve to explain the inventive embodiments. Wherever possible, the same reference numbers will be used throughout the drawings to refer to the same or like parts. Such embodiments are illustrative and not intended to be exhaustive or exclusive of the present apparatus or method.

FIG. 1 is a flow chart of a vibration energy transfer analysis method suitable for use in a discontinuous load bearing structure system.

FIG. 2 is a schematic diagram of a typical load bearing structure system structure and mechanical features.

Fig. 3a is a schematic view of typical load bearing frame structural features.

Fig. 3b is a schematic diagram of a thin-walled plate-shell combination model.

FIG. 4 is a schematic diagram of a finite element equivalent model of a load bearing structural system.

Fig. 5a is a schematic diagram of flange-bolt mechanical characteristics.

Fig. 5b is a schematic diagram of flange-bolt equivalent modeling.

FIG. 6 is a schematic diagram of a structural degree of freedom class diagram including interfaces.

FIG. 7 is a flow chart of vibration energy transfer analysis taking into account a friction contact interface.

Fig. 8 shows the normal stress state of the transverse vibration sheet micro-body.

Fig. 9 is an internal force characteristic of the transverse vibration sheet.

Fig. 10a is a schematic view of sheet microbody stress during in-plane longitudinal vibration.

Fig. 10b is a schematic view showing the loaded state of the thin plate micro-element body during in-plane longitudinal vibration.

FIG. 11 is a graph of the response transmissibility at different locations of a load bearing structure.

Fig. 12a is a graph of a characteristic vector of the vibration power flow of the load carrying structural system.

Fig. 12b is a diagram of a load carrying structural system vibration power flow characteristic flow.

Fig. 13 is a block diagram of a vibration energy transfer analysis apparatus suitable for use in a discontinuous load bearing structure system.

Detailed Description

The present invention will be described in detail below with reference to the drawings and detailed description to enable those skilled in the art to better understand the technical scheme of the present invention. Embodiments of the present invention will be described in further detail below with reference to the drawings and specific examples, but not by way of limitation. The order in which the steps are described herein by way of example should not be construed as limiting if there is no necessity for a relationship between each other, and it should be understood by those skilled in the art that the steps may be sequentially modified without disrupting the logic of each other so that the overall process is not realized.

The embodiment of the invention provides a vibration energy transfer analysis method suitable for a discontinuous load-bearing structure system, wherein the research object of the method is an aeroengine load-bearing structure system, and the vibration response transfer analysis process is shown in a flow chart of fig. 1. In the following, the steps in the vibration transmission analysis method of the aeroengine load bearing structure system will be further described by specific embodiments.

Step 1): the energy transmission mechanism of the plate-shell combination isomorphic abrupt structure and the equivalent modeling method for the discontinuous load-bearing structure with the abrupt cross section.

In aeroengines, the main function of the load-bearing structural system is to support the rotor, providing it with constraints, transferring the loads generated during the operation of the rotor system to the fuselage; meanwhile, the bearing structure is also the installation foundation of other stator components such as stator blades, pipeline accessories and the like, and the dynamic characteristics of the structures are mutually influenced. In the modeling process, the mechanical properties of each part of the load-bearing structure on the load transmission path need to be maintained, and meanwhile, the coupling influence among the structures is considered, and other stator parts which have important influence on the mass and rigidity distribution of the load-bearing structure also need to be considered to develop equivalent modeling. And (3) carrying out equivalent treatment on the characteristics (such as chamfer angle) with larger stress influence and smaller vibration energy transfer analysis influence.

Step 1.1): and (5) carrying out mechanical characteristic identification and model unit selection on the force bearing structure system.

As shown in fig. 2, the bearing structure system includes numerous components, has complex geometric features, and is divided into the aero-engine bearing structure system according to the differences of bearing mechanical features: non-important parts, plate and shell combined structures and connecting structures.

Step 1.2): the characteristics of typical configuration abrupt structures such as plate-shell combinations are equivalent.

Fig. 3a shows a typical plate-shell combined bearing frame structure in an aero-engine, which consists of an inner ring casing, an outer ring casing and circumferentially distributed bearing webs: the casing is of a thin-wall shell structure and mainly bears along the circumferential direction and the axial direction, and the radial bearing capacity is weaker; the radial plates extend along the radial direction and the axial direction of the casing to be the main bearing direction, and the bearing capacity is weaker in the direction perpendicular to the plate surface. It can be seen that there is a significant difference in the bearing direction between the webs and the case, the configuration of the junction of the bearing webs and the inner/outer case is abrupt, and the bearing characteristics are also abrupt. To reveal the load transfer mechanism within the composite structure, the detailed structure of the load-bearing frame is ignored, its main features are extracted to build a mechanical model of the thin-walled plate-shell composite structure, as shown in fig. 3 b.

The plate-shell combined structure consists of two sub-structure parts of a flat plate and a cylindrical shell, as shown in fig. 3b, the displacement response of any position of a curved surface of the shell along the axial direction, the circumferential direction (tangential direction) and the radial direction (normal direction) is respectively marked as u _s、v_s、w_s, and the displacement of any point on the flat plate along the in-plane direction and the out-of-plane direction is respectively up, vp and wp. The plate-shell composite structure can be divided into two subsystems: one is of a flat plate structure, one end of the flat plate structure along the y direction of the local coordinate of the flat plate is a combined boundary with the shell, and the other end of the flat plate structure is a free boundary; and the second is a shell structure, only shell sectors on two sides of the support plate are taken for analysis, and the middle part of the shell sector is connected with the flat plate in a combined way.

According to classical Reissner thin-wall shell theory, a column shell structure vibration differential equation expressed by neutral plane displacement components under a local cylindrical coordinate system is given:

Wherein h _s、R_s respectively represents the thickness and the radius of the cylindrical shell, u _s、v_s、w_s respectively represents the axial, circumferential (tangential) and radial (normal) displacement response of any position of the curved surface of the shell, θ _s represents the included angle between the flat plate and the connecting position of the shell, t represents the time, x _s, R _s represents axial, angular and radial components in the cylindrical coordinates, E _s、ρ_s、υ_s represents Young's modulus, density and Poisson's ratio of the cylindrical shell material, q _x,Q _r represents the cylindrical shell along x,The external excitation load is applied in the r direction, the v represents the displacement of a certain point of the neutral plane along the y direction,Indicating the rigidity of the membrane of the housing,Representing the differential operator of the device,

The internal force of the thin-walled cylindrical shell comprises axial and tangential internal film forcesAndMoment of force AndShear forceThe relationship between the internal force of each point and the displacement deformation of the internal force can be expressed as follows:

In the method, in the process of the invention, Represents the axial and tangential internal membranous force to which the thin-walled cylindrical shell is subjected, The moment applied to the thin-walled cylindrical shell is represented,Representing the shearing force applied to the thin-walled cylindrical shell, D _s representing the bending stiffness of the shell;

The coordinated relationship of the plate-shell combination cross sections includes continuity of the deformation in each direction and internal force (or moment) balance. The structure can be divided into three parts along the combined section of the plate and the shell, and comprises a shell 1, a shell 2 and a plate substructure, and each substructure is stressed and acted by moment at the combined boundary; for the shell substructure, the generalized forces and deformations are both represented in cylindrical coordinates, while the flat structure generalized forces and deformations are represented in their local rectangular coordinates.

From structural continuity, the same position of the combined section is displaced in the same direction along an absolute coordinate system and is continuous with the corner, namely:

Where u _s1、u_s2、u_p1 denotes the axial displacement response of the housing 1, the housing 2 and the flat plate, respectively, v _s1、v_s2、v_p denotes the circumferential displacement response of the housing 1, the housing 2 and the flat plate, w _s1、w_s2、w_p denotes the radial displacement response of the housing 1, the housing 2 and the flat plate, respectively, Representing the angular component of the housing 1,Representing the angular component of the housing 2, y _p representing the vertical component of the plate displacement response;

meanwhile, the combined section has the balance of the acting force and moment among the structures, namely:

In the method, in the process of the invention, Represents the tangential internal force of the housing 1,Representing the tangential internal force of the housing 2,Representing the vertical inward force of the plate,Representing the vertical shear force of the plate,Indicating the tangential shear force of the housing 1,Indicating the tangential shear force of the housing 2,Indicating the axially inward force of the housing 1,Indicating the axially inward force of the housing 2,Representing the axial internal force of the plate;

And the dynamics equation of the combined structure can be obtained by combining the dynamics equation of the combined thin plate and the thin-wall cylindrical shell and realizing the combination of the dynamics equation and the thin-wall cylindrical shell through the continuous deformation condition of the connecting section and the force/moment balance condition. When the cross-sectional moment of inertia is equivalent, the characteristics of the inner diameter and the outer diameter of the simplified front and rear of the shell case are ensured to be unchanged, the cross-sectional area is kept basically consistent, and the moment of inertia of the cross-section in all directions is further kept unchanged, so that the same radial and bending rigidity of the simplified front and rear case is ensured; ensuring that the dimensional characteristics of the radial plate structure in all directions are unchanged and the cross sectional areas are the same so as to meet the principle of transverse rigidity equivalence; the quality distribution characteristics of the axial direction, the longitudinal direction and the like of the casing are ensured to be unchanged, and the total quality characteristics of all structures are ensured to be unchanged. Finally, the simplified front and rear structures have similar dynamic characteristics.

Step 1.3): the characteristics of the connection structure containing the friction contact interface are equivalent.

The flange-bolt connection is the most widely used rigid connection form in a bearing structure system, and the existence of a friction interface in the flange-bolt connection form can bring about certain nonlinear loss of structural rigidity, can generate obvious dry friction damping energy consumption and influence the transmission characteristic of vibration in the structure system.

In order to facilitate the analysis of energy transfer, the flange-bolt structure can be approximately considered as a system with complete periodic symmetry, the equivalent rigidity and damping parameters obtained by the sector model are periodically expanded to the whole flange-bolt connection, and the structural system finite element model is introduced through MATRIX27 superunit. Under the same load environment, the bolt connection structure at different positions can be approximately considered to have local rigidity loss and damping characteristics proportional to local load characteristics, and the approximate equivalent modeling of other positions is realized according to the vibration load distribution characteristics of each position, as shown in fig. 4. Wherein, the rigidity and damping parameter of the MATRIX27 superunit are obtained by the step 2).

Step 1.4): the simplified equivalent of the vibration energy transfer path outer attachment structure.

In an aeroengine, an oil tank, an oil pump and various pipelines are generally attached to a bearing casing structure, so that the bearing casing structure has a large influence on the overall quality and rigidity distribution characteristics of the casing, and the bearing casing structure system dynamic characteristic analysis cannot be directly ignored. The accessory structure is condensed into a superunit by utilizing the substructure analysis theory, so that the difficulty in dynamic analysis of the whole bearing structure system is reduced.

Step 2): connection interface energy dissipation solution, and equivalent techniques in structural system energy transfer analysis. And solving equivalent rigidity/damping of the flange-bolt structure according to the simplification of the flange-bolt structure in the step 1). The mechanical characteristics of the flange-bolt structure are shown in fig. 5a, and in order to facilitate more accurate vibration energy analysis of the load bearing structure system, the invention equivalently models the flange-bolt connection structure as shown in fig. 5 b.

The connection interface is linearly reduced in substructure, and for a local nonlinear structural system with a contact interface, it includes a contact node degree of freedom set (X _i) at the interface, an active node degree of freedom set (X _a) for applying a load or extracting a response, and a substructure internal node degree of freedom set (X _s) other than both, as shown in fig. 6. The degrees of freedom in each set are denoted as n _i、n_a、n_s. The interface contact node degrees of freedom (X _i) and the active node degrees of freedom (X _a) may be grouped into boundary node degrees of freedom, denoted as X _b, and the degrees of freedom are n _b＝n_a+n_s.

The structural system dynamics equation is:

Where M _bb、M_ss denotes a mass array, M _bs、M_sb denotes a coupling term of the mass array, Representing an acceleration matrix, C _bb、C_ss representing a damping matrix, C _bs、C_sb representing a coupling term in the damping matrix,Represents a velocity array, K _bb、K_ss represents a stiffness array, K _bs、K_sb represents a coupling term in the stiffness array, X _b、X_s represents a displacement array, and F _b、F_s represents a load. B and s in the subscripts represent the degrees of freedom of the boundary nodes and the degrees of freedom of the nodes inside the substructure, respectively. For the degrees of freedom of the internal nodes of the linear substructure, a fixed interface substructure mode synthesis method (Craig-Bampton method, CB) is adopted.

After parameters such as converged friction force, relative vibration displacement of contact points and the like are calculated under each frequency, equivalent stiffness and equivalent damping parameters between contact nodes are obtained, and a finite element model of a structural system is introduced to realize equivalent modeling.

Wherein, for a nonlinear system considering interface friction load, the dynamic equation can be expressed in terms of complex stiffness as:

Wherein m represents the mass of the polymer, The acceleration is represented by k (1+i delta), the complex stiffness is represented by x (t), the displacement is represented by p (θ (t)), the external simple harmonic excitation load is represented by f _t (t), and the interface friction force is represented by f _t (t);

In conducting the kinetic response analysis, the kinetic equation can be transformed into:

Wherein:

And (3) making:

where k represents the rigidity, Represents displacement, m represents mass, ω represents circular frequency, P represents load,Representing the first harmonic method equivalent load, f _t (θ) representing the differential form of the first harmonic method equivalent load, ke _q representing the equivalent stiffness term, c _eq representing the equivalent damping term;

The friction force is converted into an equivalent complex stiffness term k _eq+ic_eq similar to the linear term k (1+iδ), where the real part k _eq is the equivalent stiffness term and the imaginary part c _eq is the equivalent damping. It can be seen that the interface friction effect is represented by additional equivalent stiffness and additional equivalent damping, and after the vibration response and the interface friction force are calculated, the equivalent additional stiffness and damping characteristics can be obtained:

and combining the one-dimensional Jenkins contact unit model to obtain an interface equivalent stiffness and damping expression:

where k _t denotes the tangential stiffness, Representing a displacement response value, x _cr representing a critical slip displacement along the tangential direction;

obtaining characteristic parameters such as equivalent rigidity and equivalent damping between interface contact points, and introducing the characteristic parameters into a finite element model of a complex structural system with high degree of freedom

Step 3): a power flow-based energy transfer analysis method for a complex discontinuous load-carrying structure system. And (3) according to the actual working condition of the aero-engine, aiming at complex frequency load, carrying out response solution and system vibration energy distribution according to the equivalent analysis model established in the step (1) and the step (2), obtaining response characteristics of vibration displacement, speed, node force and the like of each node of the structure, and calculating to obtain a power flow vector of each structural unit/node, wherein the specific flow is shown in figure 7.

Specifically, step 3) further includes:

Step 3.1): using finite element analysis software, importing a structure geometric model, inputting material parameters, determining unit types and the like, and creating a finite element grid model of the structure to be researched in a preprocessing module;

step 3.2): applying load and boundary constraint conditions to the finite element model established in the step 3.1) through a load condition command and a boundary condition command;

step 3.3): entering a solving module, determining a solving type, setting related computing parameters, and carrying out dynamics solving;

Step 3.4): entering a post-processing module, determining analysis time/frequency, data extraction position, data type and the like, and obtaining parameters such as internal force, speed and the like of a structure to be analyzed through internal force and response analysis and extraction commands;

When the transverse vibration occurs, the stress components sigma _x、σ_y、τ_xy are linearly distributed along the thickness direction of the unit, the upper value of the neutral plane is zero, and the upper stress and the lower stress of the neutral plane are opposite, so that the stress integral along the thickness direction is zero, and only bending moment and torque can be synthesized; the lateral shear stresses τ _xz and τ _yz may combine into a lateral shear force, as shown in FIG. 8.

As shown in fig. 9, in a cross section perpendicular to the x direction, the positive stress σ _x integrates in the thickness direction to synthesize a bending moment M _x, and the shear stress τ _xy synthesizes a torque M _xy; similarly, on a section perpendicular to the y direction, the stress integral can obtain a bending moment M _y and a torque M _yx; wherein M _x,M_y,M_xy,M_yx is defined as the value of the unit width of the section, and the calculation formula is as follows:

Wherein h represents the height of the unit body, w represents the displacement of a certain point of the neutral plane along the z direction, E represents the elastic modulus, and v represents the displacement of a certain point of the neutral plane along the y direction;

The transverse shear perpendicular to the x-axis and y-axis can be integrated from the transverse shear stresses τ _xz and τ _yz:

When the plate-shell structure has in-plane longitudinal vibration, the unit internal force state is simpler, and can be regarded as a plane membrane unit, the unit internal force state is shown in fig. 10a, and the internal force characteristic is shown in fig. 10 b.

Through a force balance equation and corresponding integral operation, an interfacial intima force calculation formula can be obtained when in-plane longitudinal vibration:

Wherein N _x represents an x-direction internal force, N _y represents a y-direction internal force, N _xy represents a shear force synthesized internal force, and v represents a displacement of a certain point of a neutral plane along the x-direction;

Step 3.5): and (3) storing parameters such as internal force, speed and the like of the structure to be analyzed, and storing the unit/node coordinate information of the calculated finite element model.

Step 3.6): reading in unit/node coordinate information, internal force of structure and speed calculation results output by a finite element program by using programming software, and arranging the unit/node coordinate information, the internal force of structure and the speed calculation results into a matrix form;

Step 3.7): performing physical space and calculation space coordinate transformation according to the coordinate system information, and determining a power flow calculation domain;

Step 3.8): solving the power flow vector value of each unit in a calculation domain according to the vibration power flow calculation formula in each unit, writing a result matrix according to the unit-node relation, and storing power flow result data;

based on the treatment such as kirchhoff sheet and elastic mechanics operation, the stress of the plate-shell unit can be completely converted into an internal force state acting on the neutral plane of the shell. Based on the force and velocity response per unit length of the resulting plate and shell unit, the power flow strength per unit size within the plate and shell unit can be expressed as:

Wherein, I _x、I_y -represents the power flow intensity (structural sound intensity) transmitted along the x direction and the y direction;

v _x(t)、v_y(t)、v_z (t) -represents the vibration linear velocity in x, y, z directions;

Omega _x (t) and omega _y (t) -represent angular velocity values about the x and y axes;

superscript x—means conjugation of complex quantities.

For structural steady-state vibration response under simple harmonic excitation, the average power flow intensity can be expressed as an internal force and a frequency domain vibration displacement by combining a time average power flow formula of a structural system:

Wherein: u, v, w and θ _x、θ_y —vibration displacements in x, y, z directions, and rotational displacements about x and y axes, ω representing angular velocity.

The total vibration energy transmitted in the x and y directions of the plate shell unit is the sum of the energy carried by various vibration waves by combining the characteristics of the transmission energy of various vibration waves. The power flow sound intensity component transmitted by the longitudinal wave is as follows:

Where I _x,L denotes the component of the power flow intensity transmitted by the longitudinal wave in the x-direction, N _x denotes the force in the x-direction, im { } denotes the imaginary part, Representing the vibration linear velocity in the x-direction, u ^* representing the x-direction displacement, I _y,L representing the component of the power flow intensity transmitted by the longitudinal wave in the y-direction, N _y representing the force in the y-direction,Representing the vibration linear velocity in the y-direction;

The power flow intensity components of the shear wave transfer are:

Where I _x,S represents the component of the power flow strength of the shear wave transfer along the x-direction, N _xy represents the shear internal force, Representing the vibration linear velocity in the y-direction, v ^* representing the x-direction displacement, I _y,S representing the component of the power flow intensity transmitted by the shear wave in the y-direction, N _yx representing the shear internal force,Representing the vibration linear velocity in the x-direction, u ^* representing the y-direction displacement;

the power flow intensity components transmitted by the torsional wave are as follows:

Where I _x,T denotes the component of the power flow strength in the x-direction of the torsional wave transmission, M _xy denotes the moment of shear stress synthesis, Indicating the angular velocity about the x-axis,Represents angular displacement about the x-axis, ω represents angular velocity, I _y,T represents the component of the power flow intensity transmitted by the torsional wave in the y-direction, M _yx represents the moment of shear stress synthesis,Indicating the angular velocity about the y-axis,Represents angular displacement about the y-axis;

the power flow intensity components of bending wave transfer are:

Where I _x,B denotes the component of the bending wave transmitted power flow intensity in the x-direction, M _x denotes the moment about the x-axis, Indicating the angular velocity about the y-axis,Represents the vibration linear velocity in the z-direction, ω represents the angular velocity, w ^* represents the z-direction displacement, I _y,B represents the component of the power flow intensity transmitted by the bending wave in the y-direction, M _y represents the moment about the y-axis,Indicating the angular velocity about the x-axis,Represents angular displacement about the x-axis;

Step 3.9): repeating the steps 3.6) to 3.8) according to the solving requirement until the analysis of the power flow results of all time/frequency point structures is completed;

further, the power flow is a reflection of the vibration energy in the unit, and according to the thin plate theory, the vibration differential equation of the thin plate and plate shell unit is known as follows:

wherein F (x, y, t) =f ₀(x,y)e^iωt represents the external simple harmonic excitation load to which the structural unit is subjected; d represents the bending stiffness of the unit, its value being d=eh ³/12(1-υ²); w represents the cell transient displacement response;

The transient vibratory power flow at the location unit can be expressed as:

The vibration energy in the structural unit comprises two parts, namely kinetic energy T and potential energy U, and the energy change frequency under simple harmonic excitation is twice the load change frequency, so that the vibration energy can be expressed as:

E＝T+U＝(T₀+U₀)e^2iωt (30)

the vibration energy can be obtained by deriving time:

Calculating the gradient value of power to the x and y directions by transient vibration power flow of the position (x, y) unit, and summing to obtain the following steps:

The differential equation of vibration of the thin plate and shell unit can be known:

It can be known that substitution transformation is performed from the summation result of gradient values in the x and y directions of the power flow:

the relation between the vibration energy in the unit and the vibration power flow can be finally obtained by substituting the vibration energy time derivative formula and the power flow x, y direction gradient summation result into the conversion formula, and the unit without the action of external excitation load can be expressed as:

the dynamic change in vibration energy over time in a thin-walled sheet shell structural unit is the sum of the amounts of power flow in different directions.

Step 4): visual representation of vibrational energy transfer and distribution characteristics. And 3) corresponding visualization processing is carried out on the results of the response structure and the vibration power flow in the step 3).

Specifically, step 4) further includes:

Step 4.1) preprocessing the power flow vector data, wherein different local coordinate system types and the like can be adopted in the power flow calculation, and different sub-structure position data point space distribution characteristics are different, and in addition, inconsistent unit systems and the like can exist, so that the data needs to be preprocessed, including the transformation of a coordinate system and the standardization of the data;

and 4.2) mapping, drawing and displaying the power flow vector data, and visually expressing the power flow data through a visualization program by adopting vector arrows or streamlines and the like according to analysis requirements. The post-processing software is utilized to read the power flow vector result, and the structure power flow cloud image and the vector image are visually drawn and stored; and (5) using post-processing software to obtain a power flow chart through the visualization processing of the power flow vector.

The definition of the vector diagram and the flow diagram is as follows:

Vector diagram: the space coordinate values of each position of the structure are used for representing the starting point of the power flow vector, the size of the power flow of the structure is used as the length of a vector arrow, and the key position is determined according to the direction, so that the expression of the power flow by adopting the vector arrow method is obtained, and the size and the flow direction of the vibration power of different positions of the structure are intuitively reflected.

Flow chart diagram: the definition of the streamline in analog fluid mechanics adopts numerical integration method, double-flow function method and the like to construct the streamline. The flow diagram visually displays the trend of the structural vibration power flow field, so that the transmission path, the flow form, the dissipation characteristics and the like of vibration energy in a complex structural system can be analyzed and understood more accurately and intuitively.

The streamline definition in analog hydrodynamic is defined by taking tangent lines of each point in the structural power flow vector field as vibration power streamline, namely, the streamline is full:

In the method, in the process of the invention, As a position vector for the power flow,Is the power flow intensity vector at the moment of the corresponding position t.

Then for a three-dimensional solid element, the streamline equation can be expressed as:

From the above equation, the differential equation expression for the structural power flow line can be further derived:

dx/I_x+dy/I_y+dz/I_z＝0 (38)

For a two-dimensional flat plate structure, the structural power flow streamline expression can be further simplified to:

dx/I_x+dy/I_y＝0 (39)

the tangential direction of each point of the structural power flow streamline is the power flow direction at the point, and the streamline trend reflects the transmission path of vibration energy in the structure.

In this embodiment, for vibration analysis of a bearing structure system of an aeroengine, the result of using a conventional transfer function analysis method is shown in fig. 11, and the result of using the vibration energy transfer analysis method of the present invention is shown in fig. 12, wherein fig. 12a is a vector diagram, and fig. 12b is a flow diagram. The comparison shows that: the invention introduces a vibration power flow theory based on the traditional vibration characteristic analysis method based on structural modal characteristics, transfer functions or response, and can intuitively describe the transmission and distribution characteristics of vibration energy in the structure. Aiming at the universal bolt connection structure in the aeroengine load bearing structure system, the friction energy consumption behavior of the connection local interface is considered, so that the vibration energy transmission characteristic of the load bearing structure system is predicted more accurately.

The embodiment of the invention also provides a vibration energy transmission analysis device suitable for the discontinuous load bearing structure system, referring to fig. 13, fig. 13 shows a structural diagram of the vibration energy transmission analysis device suitable for the discontinuous load bearing structure system according to the embodiment of the invention. The apparatus 1300 includes:

The equivalent modeling module 1301 is configured to perform equivalent modeling on a discontinuous load-carrying structure with a mutation section based on a plate-shell combination and other configuration mutation structure energy transfer mechanism;

the solving module 1302 is configured to connect the interface energy dissipation solving and the equivalent technology in the structural system energy transfer analysis to obtain an equivalent analysis model;

The power flow vector calculation module 1303 is configured to perform response solution and system vibration energy distribution based on the equivalent analysis model to obtain response characteristics, and calculate to obtain power flow vectors of each structural unit/node, where the response characteristics include vibration displacement, speed and node force of each node of the structure;

a visualization module 1304 configured to visually express the vibrational energy transfer and distribution characteristics.

In some embodiments, the visualization module 1304 is further configured to:

Preprocessing the power flow vector data, wherein the preprocessing comprises conversion of a coordinate system and standardization of the data;

According to analysis requirements, processing the power flow data by adopting vector arrows or streamline to obtain a vector diagram and a streamline diagram;

The vector diagram represents the starting point of a power flow vector by using spatial coordinate values of each position of a structure, takes the size of the power flow of the structure as the length of a vector arrow, determines key positions according to directions, and obtains the expression of the power flow by using a vector arrow method, thereby intuitively reflecting the size and the flow direction of vibration power of different positions of the structure;

the flow chart is obtained by constructing a flow line by adopting a numerical integration method and/or a double-flow function method.

It should be noted that the modules described in the embodiments of the present invention may be implemented by software, or may be implemented by hardware, and the described modules may also be provided in a processor. The names of these modules do not constitute a limitation on the module itself in some cases.

The vibration energy transfer analysis device suitable for the discontinuous load bearing structure system in the embodiment of the invention belongs to the same technical conception as the method explained before, and has basically the same technical effects and is not repeated here.

The embodiment of the invention also provides a vibration energy transfer analysis system suitable for the discontinuous load-bearing structure system, which comprises:

A memory for storing a computer program;

A processor for executing the computer program to implement the method of any of the embodiments of the invention.

Embodiments of the present invention also provide a non-transitory computer readable medium storing instructions which, when executed by a processor, perform a method according to any of the embodiments of the present invention.

Furthermore, although exemplary embodiments have been described herein, the scope thereof includes any and all embodiments having equivalent elements, modifications, omissions, combinations (e.g., of the various embodiments across), adaptations or alterations as pertains to the present application. The elements in the claims are to be construed broadly based on the language employed in the claims and are not limited to examples described in the present specification or during the practice of the application, which examples are to be construed as non-exclusive. It is intended, therefore, that the specification and examples be considered as exemplary only, with a true scope and spirit being indicated by the following claims and their full scope of equivalents.

The above description is intended to be illustrative and not restrictive. For example, the above-described examples (or one or more aspects thereof) may be used in combination with each other. For example, other embodiments may be used by those of ordinary skill in the art upon reading the above description. In addition, in the above detailed description, various features may be grouped together to streamline the invention. This is not to be interpreted as an intention that the features of the claimed invention are essential to any of the claims. Rather, inventive subject matter may lie in less than all features of a particular inventive embodiment. Thus, the following claims are hereby incorporated into the detailed description as examples or embodiments, with each claim standing on its own as a separate embodiment, and it is contemplated that these embodiments may be combined with one another in various combinations or permutations. The scope of the invention should be determined with reference to the appended claims, along with the full scope of equivalents to which such claims are entitled.

Claims

1. A method of vibration energy transfer analysis for a discontinuous load bearing structure system, the method comprising:

based on a plate-shell combination and other configuration abrupt structure energy transfer mechanism, carrying out equivalent modeling on a discontinuous load-carrying structure with an abrupt section;

solving the energy dissipation of the connection interface and an equivalent technology in the energy transfer analysis of the structural system to obtain an equivalent analysis model;

carrying out response solving and system vibration energy distribution based on the equivalent analysis model to obtain response characteristics, and calculating to obtain power flow vectors of each structural unit/node, wherein the response characteristics comprise vibration displacement, speed and node force of each node of the structure;

visual expression of vibration energy transmission and distribution characteristics;

The method for equivalently modeling the discontinuous load-carrying structure with the abrupt cross section based on the plate-shell combination isomorphic abrupt structure energy transfer mechanism specifically comprises the following steps:

mechanical characteristic identification of a load-carrying structure system and selection of a model unit;

characteristic equivalents of typical configuration mutation structures such as plate-shell combination:

column shell structure vibration differential equation expressed by neutral plane displacement component under partial cylindrical coordinate system:

Wherein h _s、R_s respectively represents the thickness and the radius of the cylindrical shell, u _s、v_s、w_s respectively represents the axial, circumferential and radial displacement response of any position of the curved surface of the shell, θ _s represents the included angle between the flat plate and the connecting position of the shell, t represents the time, x _s, R _s represents axial, angular and radial components in the cylindrical coordinates, E _s、ρ_s、υ_s represents Young's modulus, density and Poisson's ratio of the cylindrical shell material, q _x,Q _r represents the cylindrical shell along x,The external excitation load is applied in the r direction, the v represents the displacement of a certain point of the neutral plane along the y direction,Indicating the rigidity of the membrane of the housing,Representing the differential operator of the device,

The internal force borne by the thin-wall cylindrical shell comprises axial and tangential internal film force, moment and shearing force, and the displacement deformation relation between each point internal force and the internal film force is expressed as:

The coordination relation of the plate-shell combined section comprises continuity of deformation in all directions and internal force or moment balance, the structure is divided into three parts along the plate-shell combined section, the structure comprises two shells and a plate substructure, and all the substructures are stressed and acted by moment at the combined boundary; for the shell substructure, the generalized force and the deformation are both represented by cylindrical coordinates, and the generalized force and the deformation of the flat plate structure are represented by local rectangular coordinates;

According to structural continuity, the same position of the combined section is displaced in the same direction along an absolute coordinate system and is continuous with a corner, namely:

the dynamic equation of the combined structure is obtained by combining the dynamic equation of the thin-wall cylindrical shell and realizing the combination of the two by connecting the continuous deformation condition of the section and the force/moment balance condition;

characteristic equivalence of connection structure containing friction contact interface:

The flange-bolt structure is approximately considered as a system with complete periodic symmetry, the equivalent rigidity and damping parameter period obtained by the sector model is expanded to the whole flange-bolt connection, and a finite element model of the structural system is introduced through a superunit; under the same load environment, the bolt connection structure at different positions is approximately considered to have local rigidity loss and damping characteristics which are proportional to local load characteristics, and the approximate equivalent modeling of other positions is realized according to the vibration load distribution characteristics of each position;

simplified equivalent of vibration energy transfer path outer attachment structure: the accessory structure is condensed into a superunit by utilizing the substructure analysis theory.

2. The method according to claim 1, wherein the solving of the energy dissipation of the connection interface and the equivalent technique in the analysis of the energy transfer of the structural system, obtain an equivalent analysis model, specifically comprises:

The linear substructure reduction is carried out on the connection interface, and for a local nonlinear structure system with a contact interface, the local nonlinear structure system comprises a contact node degree of freedom set (X _i) at the interface, an active node degree of freedom set (X _a) for applying load or extracting response and a substructure inner node degree of freedom set (X _s) except for the contact node degree of freedom set and the active node degree of freedom set, wherein the degrees of freedom in each set are respectively marked as n _i、n_a、n_s, the interface contact node degree of freedom (X _i) and the active node degree of freedom (X _a) can be grouped as boundary node degree of freedom and marked as X _b, and the degree of freedom is marked as n _b＝n_a+n_s;

The structural system dynamics equation is:

Where M _bb、M_ss denotes a mass array, M _bs、M_sb denotes a coupling term of the mass array, Representing an acceleration matrix, C _bb、C_ss representing a damping matrix, C _bs、C_sb representing a coupling term in the damping matrix,Representing a velocity array, K _bb、K_ss representing a stiffness array, K _bs、K_sb representing a coupling term in the stiffness array, X _b、X_s representing a displacement array, and F _b、F_s representing a load; b and s in the subscript represent the degrees of freedom of the boundary nodes and the degrees of freedom of the nodes in the substructures respectively;

After parameters such as converged friction force, relative vibration displacement of contact points and the like are calculated under each frequency are determined, equivalent stiffness and equivalent damping parameters between contact nodes are obtained, and a finite element model of a structural system is introduced to realize equivalent modeling;

wherein, for a nonlinear system considering interface friction load, the dynamic equation is expressed as complex stiffness:

In the dynamic response analysis, the dynamic equation is transformed into:

Wherein:

And (3) making:

where k represents the rigidity, Represents displacement, m represents mass, ω represents circular frequency, P represents load,Representing the equivalent load of the harmonic balancing method, f _t (theta) represents the differential form of the equivalent load of the harmonic balancing method, ke _q represents the equivalent stiffness term, and c _eq represents the equivalent damping term;

after the vibration response and the interface friction force are calculated, equivalent additional rigidity and damping characteristics can be obtained:

And combining the one-dimensional contact unit model to obtain an interface equivalent stiffness and damping expression:

and obtaining characteristic parameters between interface contact points, and introducing the characteristic parameters into a finite element model of the complex structural system with high degree of freedom, wherein the characteristic parameters comprise equivalent rigidity and equivalent damping.

3. The method according to claim 2, wherein the performing response solution and system vibration energy distribution based on the equivalent analysis model obtains response characteristics, and calculates power flow vectors of each structural unit/node, specifically including:

using finite element analysis software, importing a structural geometric model, inputting material parameters, determining unit types, and creating a finite element grid model of the structure to be researched in a preprocessing module;

Applying load and boundary constraint conditions to the established finite element model through the load condition command and the boundary condition command;

entering a solving module, determining a solving type, setting related computing parameters, and developing dynamics solving

Entering a post-processing module, determining analysis time/frequency, data extraction position and data type, and obtaining internal force and speed of a structure to be analyzed through internal force and response analysis and extraction commands;

preserving parameters such as internal force, speed and the like of the structure to be analyzed, and preserving unit/node coordinate information of the calculated finite element model;

reading unit/node coordinate information, internal force of structure and speed calculation results output by the finite element program, and arranging the results into a matrix form;

performing physical space and calculation space coordinate transformation according to the coordinate system information, and determining a power flow calculation domain;

According to the vibration power flow calculation formula in each unit, solving the power flow vector value of each unit in the calculation domain, writing the result matrix according to the unit-node relation, and storing the power flow result data.

4. A method according to claim 3, wherein the entering post-processing module determines the analysis time/frequency, the data extraction location, the data type, and obtains the internal force and the velocity of the structure to be analyzed by the internal force and the response analysis extraction command, and specifically comprises:

When in transverse vibration, the stress components sigma _x、σ_y、τ_xy are linearly distributed along the thickness direction of the unit, the upper value of the neutral plane is zero, and the upper and lower stresses of the neutral plane are opposite, so that the stress integral along the thickness direction is zero, and only bending moment and torque can be synthesized; the transverse shear stresses τ _xz and τ _yz combine the transverse shear forces;

On a cross section perpendicular to the x direction, the normal stress sigma _x integrates a synthetic bending moment M _x along the thickness direction, and the shear stress tau _xy synthesizes a torque M _xy; on a section perpendicular to the y direction, the stress is integrated to obtain a bending moment M _y and a torque M _yx; wherein M _x、M_y、M_xy、M_yx is defined as the value of the unit width of the section, and the calculation formula is as follows:

The transverse shear perpendicular to the x-axis and y-axis is integrated by the transverse shear stresses τ _xz and τ _yz:

Wherein Q _x and Q _y represent transverse shear perpendicular to the x-axis and y-axis;

When the plate-shell structure has in-plane longitudinal vibration, the plate-shell structure is regarded as a plane membrane unit;

When in-plane longitudinal vibration is obtained through a force balance equation and corresponding integral operation, the interfacial intima force calculation formula is as follows:

Where N _x denotes an x-direction internal force, N _y denotes a y-direction internal force, N _xy denotes a shear force combined internal force, and v denotes a displacement amount of a neutral plane point in the x-direction.

5. The method of claim 4, wherein the solving the power flow vector values of each cell in the computational domain according to the vibration power flow calculation formula in each cell, writing the result matrix according to the cell-node relationship, and storing the power flow result data, specifically comprises:

Based on the treatment such as kirchhoff sheet and elastic mechanical operation, the stress of the plate-shell unit is completely converted into an internal force state acting on the neutral plane of the shell; based on the force and speed response in the unit length of the plate shell unit, the power flow intensity in the unit size of the plate shell unit is expressed as:

Wherein I _x、I_y represents the power flow intensity transmitted along the x-direction and the y-direction; v _x(t)、v_y(t)、v_z (t) represents the vibration linear velocity in the x, y, z directions; omega _x (t) and omega _y (t) represent angular velocity values around the x and y axes;

Superscript denotes conjugation of complex numbers;

For the structural steady-state vibration response under the simple harmonic excitation, the average power flow intensity is expressed by internal force and frequency domain vibration displacement by combining a time average power flow formula of the structural system:

wherein u, v, w and θ _x、θ_y represent vibration displacements in x, y, z directions, and rotational displacements about x and y axes, respectively, ω represents angular velocity;

the total vibration energy transmitted in the x and y directions of the plate-shell unit is the sum of energy carried by various vibration waves, wherein the sound intensity component of power flow transmitted by longitudinal waves is as follows:

Where I _x,L denotes the component of the power flow intensity transmitted by the longitudinal wave in the x-direction, N _x denotes the force in the x-direction, im { } denotes the imaginary part, Representing the vibration linear velocity in the x-direction, u ^* representing the x-direction displacement, v ^* representing the y-direction displacement, I _y,L representing the component of the power flow intensity transmitted by the longitudinal wave in the y-direction, N _y representing the force in the y-direction,Representing the vibration linear velocity in the y-direction;

The power flow intensity components of the shear wave transfer are:

Where I _x,S represents the component of the power flow strength of the shear wave transfer along the x-direction, N _xy represents the shear internal force, Representing the linear velocity of vibration along the y-direction, I _y,S representing the component of the power flow intensity transmitted by the shear wave along the y-direction, N _yx representing the internal shear force,Representing the vibration linear velocity in the x-direction;

Where I _x,T denotes the component of the power flow strength in the x-direction of the torsional wave transmission, M _xy denotes the moment of shear stress synthesis, Indicating the angular velocity about the x-axis,Indicating angular displacement about the x-axis, I _y,T indicating the component of the power flow strength transmitted by the torsional wave in the y-direction, M _yx indicating the moment of shear stress composition,Indicating the angular velocity about the y-axis,Represents angular displacement about the y-axis;

the power flow intensity components of bending wave transfer are:

Where I _x,B denotes the component of the bending wave transmitted power flow intensity in the x-direction, M _x denotes the moment about the x-axis, Indicating the angular velocity about the y-axis,Represents the vibration linear velocity in the z-direction, ω represents the angular velocity, w ^* represents the z-direction displacement, I _y,B represents the component of the power flow intensity transmitted by the bending wave in the y-direction, M _y represents the moment about the y-axis,Indicating the angular velocity about the x-axis,Indicating angular displacement about the x-axis.

6. The method of claim 5, wherein said performing a response solution and system vibration energy distribution based on said equivalent analytical model to obtain response characteristics and calculating power flow vectors for each structural unit/node further comprises:

the vibration differential equation of the plate-and-shell unit is as follows:

the transient vibratory power flow at the location unit is expressed as:

the vibration energy in the structural unit comprises two parts, namely kinetic energy T and potential energy U, and the energy change frequency under simple harmonic excitation is twice the load change frequency, and is expressed as:

E＝T+U＝(T₀+U₀)e^2iωt (3)

the vibration energy can be obtained by deriving time:

Substituting a vibration energy time derivative formula and a power flow x, y direction gradient summation result into a transformation formula to finally obtain the relation between the vibration energy and the vibration power flow in the unit, and for the unit without external excitation load action, expressing as follows:

7. The method according to claim 1, characterized in that said visual representation of vibration energy transfer and distribution characteristics, in particular comprises:

8. A vibration energy transfer analysis apparatus adapted for use in a discontinuous load bearing structure system, the apparatus comprising:

The equivalent modeling module is configured to perform equivalent modeling on the discontinuous load-carrying structure with the abrupt cross section based on an energy transmission mechanism of the plate-shell combination and other configuration abrupt structure;

The solving module is configured to connect the interface energy dissipation solving and the equivalent technology in the structural system energy transfer analysis to obtain an equivalent analysis model;

The power flow vector calculation module is configured to carry out response solution and system vibration energy distribution based on the equivalent analysis model to obtain response characteristics, and calculate and obtain power flow vectors of each structural unit/node, wherein the response characteristics comprise vibration displacement, speed and node force of each node of the structure;

a visualization module configured for visual representation of vibrational energy transfer and distribution characteristics;

Wherein h _s、R_s respectively represents the thickness and the radius of the cylindrical shell, u _s、v_s、w_s respectively represents the axial, circumferential and radial displacement response of any position of the curved surface of the shell, θ _s represents the included angle between the flat plate and the connecting position of the shell, t represents the time, x _s, R _s represents axial, angular and radial components in the cylindrical coordinates, E _s、ρ_s、v_s represents Young's modulus, density and Poisson's ratio of the cylindrical shell material, q _x,Q _r represents the cylindrical shell along x,The external excitation load is applied in the r direction, the v represents the displacement of a certain point of the neutral plane along the y direction,Indicating the rigidity of the membrane of the housing,Representing the differential operator of the device,

9. A non-transitory computer readable storage medium storing instructions which, when executed by a processor, perform the method of any one of claims 1 to 7.