CN115985429A - Dynamic simulation model considering thermal effect and shear effect FGM beam based on non-grid method, and establishment method and simulation method thereof - Google Patents
Dynamic simulation model considering thermal effect and shear effect FGM beam based on non-grid method, and establishment method and simulation method thereof Download PDFInfo
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Abstract
The scheme relates to a dynamic simulation model, an establishing method and a simulation method thereof, wherein the dynamic simulation model comprises the following steps: establishing an FGM beam system considering the additional concentrated mass under the thermal effect and the shearing effect; describing a displacement field which is deformed by large-range rotary motion of any point on an FGM beam in the FGM beam system with the additional concentrated mass in a floating coordinate system by adopting a mixed coordinate method; dispersing the large-range rotation deformation of the FGM beam; establishing a primary approximate rigid-flexible coupling kinetic equation of the FGM beam system with the additional concentrated mass by using a second Lagrange equation to obtain a kinetic simulation model of the FGM beam based on the consideration of thermal effect and shearing effect by a non-grid method; solving a dynamic simulation model of the FGM beam with the additional concentrated mass under the thermal effect by adopting a Newmark method to obtain a deformation schematic diagram of the FGM beam with the additional concentrated mass in large-range rotary motion. The simulation model and the simulation method have the advantages of high efficiency and high precision.
Description
Technical Field
The invention relates to the field of system dynamics modeling, in particular to a dynamic simulation model considering thermal effect and shear effect FGM beams based on a meshless method, and an establishing method and a simulation method thereof.
Background
Helicopter rotor blades, robotic arms, turbine blades, and the like in engineered structures are moving toward higher operational speeds and accuracies. These systems consist of a rigid base and a flexible attachment attached thereto, typical of rigid-flexible coupling systems. In practical application, a system is often in working conditions of high temperature, high rotating speed and the like, and the traditional homogeneous material is difficult to meet the requirements of the actual service environment due to the limitation of the mechanical property of the traditional homogeneous material. Therefore, on the premise of not influencing the structural strength, it is important to design a novel composite material meeting the requirements of service environment and research the dynamic characteristics of the novel composite material.
Among many new composite materials, the Functional Gradient Material (FGM) has been widely noticed by researchers due to its superiority in heat resistance, high strength, stress concentration improvement, and the like, and is applied to the fields of aerospace, nuclear industry, biology, and the like.
In the prior art, a central rigid body-functional gradient material beam system is subjected to dynamic modeling, only transverse bending deformation, axial deformation and axial shortening caused by transverse deformation are considered during modeling, and the influence of temperature on the system is ignored. In addition, the theory related to the multi-body system dynamics is mostly researched by using an Euler-Bernoulli beam, the theory is only suitable for calculating a slender beam, and for a beam system with a small slenderness ratio, a certain system error can be caused because the Euler-Bernoulli beam theory ignores the shearing effect. When the system is dispersed by adopting the assumed modal method, the assumed modal method can well describe the deformation of the beam when the beam has small deformation, but the method has theoretical defects when the flexible mechanical arm has large deformation.
Disclosure of Invention
The invention aims to provide a new discrete model based on a Timoshenko beam theory considering a shearing effect so as to solve the influence of a thermal effect, the shearing effect and an additional concentrated mass on the dynamic characteristics of a rotary FGM beam.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
a method for establishing a dynamic simulation model based on a gridding-free method considering a thermal effect and a shear effect FGM beam at least comprises the following steps:
setting the geometric parameters and material parameters of the FGM beam, and establishing an FGM beam system considering the thermal effect and the additional concentrated mass under the shearing effect;
describing a displacement field which is deformed by large-range rotary motion of any point on an FGM beam in the FGM beam system with the additional concentrated mass in a floating coordinate system by adopting a mixed coordinate method;
adopting a non-grid point interpolation method and a non-grid radial base point interpolation method to disperse the large-range rotation deformation of the FGM beam;
and establishing a primary approximate rigid-flexible coupling kinetic equation of the FGM beam system with the additional concentrated mass by using a second Lagrange equation to obtain a dynamic simulation model considering the thermal effect and the shearing effect FGM beam based on a non-grid method.
Preferably, the establishing method, wherein the geometric parameters of the FGM beam are: the beam has a length of l, a width of b, and a concentrated mass of m t The distance of the lumped mass from the fixed end is l t (ii) a The material parameters are as follows: the density rho, the elastic modulus E and the heat conduction coefficient of the FGM beam are K, the linear expansion coefficient is alpha, and the expressions are respectively as follows:
wherein, the corner mark c represents a ceramic material, the corner mark m represents a metal material, h is the thickness of the FGM beam, N is a functional gradient index, and y is the position of the FGM beam in the thickness direction.
Preferably, the establishing method, wherein: radius r and concentrated mass m of deformation of any point on FGM beam t Radius of mt The expression in the inertial coordinate system O-XYZ is:
r＝Θ(r A +ρ 0 +u)
r mt ＝Θ(r A +ρ 1 +u 1 )
in the formula, r A The radius rho from the center rigid body centroid O to the base point O of the floating base 0 And rho 1 Radial, u and u, relative to the floating base before deformation 1 The vector is a deformation displacement vector, and theta is a normal cosine matrix of the floating base relative to an inertial coordinate system;
the expression of the overall kinetic energy of the FGM beam as a whole is:
the torsion effect of the FGM beam is not counted, so the overall elastic potential energy expression of the FGM beam is:
in the formula, epsilon 11 And γ is the longitudinal positive and shear strain at any point on the FGM beam.
Preferably, the establishing method, wherein: FGM Liang Lisan is divided into a plurality of nodes by a meshless method, node information in a calculation point x support domain is obtained through calculation, and longitudinal and transverse displacement functions are taken as follows:
in the formula phi 1 (x)、Φ 2 (x) And phi 3 (x) Are respectively w 1 、v 1 And v 2 Corresponding to the line matrix of the shape function of different discrete methods, A (t), BETA (t) and C (t) are respectively w 1 、v 1 And v 2 The corresponding array of (a);
the nonlinear coupling deformation of the axial displacement caused by the transverse displacement is as follows:
in the formula, H (x) is a coupling shape function, and the expression is as follows:
preferably, the establishing method, wherein: establishing a primary approximate rigid-flexible coupling kinetic equation of the FGM beam system considering the additional concentrated mass under the thermal effect and the shearing effect by using a second Lagrange equation, and specifically taking a generalized coordinate column vector q = (theta, A) of the system T ,B T ,C T ) T The virtual work done by the temperature load on the FGM beam is:
wherein θ = T-T 0 Is the temperature difference, T is the actual temperature, T 0 Zero was taken in this study for the initial reference temperature; sigma T Is a thermal stress; q T The generalized force column matrix under temperature load is shown as the following specific expression:
the virtual work done by the external driving torque is expressed as:
in the above formula, Q τ ＝[F τ 000] T Is a generalized force array corresponding to the external driving force; wherein F τ The method is characterized in that the method is an external driving force acting on a central rigid body, and a total kinetic energy expression W of the FGM beam as a whole and a potential energy expression U of the FGM beam as a whole are substituted into a Lagrange equation of the second type:
the first approximation rigid-flexible coupling system kinetic equation of the FGM beam considering the additional concentrated mass under the thermal effect and the shear effect is as follows:
the specific matrix expression in the formula is as follows:
M 22 ＝M 1
M 33 ＝M 2
the single underlined term in the above equation is due to the consideration of the nonlinear coupling distortion w c Additional coupling terms are generated, double-underlined are related terms generated in consideration of the shearing effect, K 5 And K 6 Is a temperature load term; y, S x 、S y 、S z 、M 1 、M 2 、M 3 、M 4 、M 5 、M 6 、D、K 1 、K 2 、K 3 、K 4 、K 5 、K 6 Is a constant coefficient matrix; j. the design is a square ob Is the moment of inertia of the beam.
A dynamic simulation model obtained by adopting the establishing method is based on a non-grid method and considering the thermal effect and the shearing effect FGM beam.
A simulation method of a dynamic simulation model considering a thermal effect and a shear effect FGM beam based on a meshless method is characterized in that a Newmark method is adopted to solve the dynamic simulation model considering the thermal effect and the shear effect FGM beam based on the meshless method, and a deformation schematic diagram of the FGM beam with additional concentrated mass of large-range rotary motion is obtained.
The beneficial effects of the invention are:
1. the invention is based on the Timoshenko beam theory considering the shearing effect to carry out the numerical solution of the kinetic equation, and researches the non-negligible influence of the thermal effect, the shearing effect and the additional concentrated mass on the dynamic characteristic of the rotary FGM beam.
2. The method adopts a gridless method for dispersion, does not depend on small deformation hypothesis, and has higher precision and wider application range compared with a hypothesis modal method; the method has wide application prospect in the process of engineering calculation problems, and can provide certain technical support for related workers.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, and it is obvious that the drawings in the following description are some embodiments of the present invention, and other drawings can be obtained by those skilled in the art without creative efforts.
FIG. 1 is a flow chart of a simulation method of the present invention based on a gridless method considering thermal effect and shear effect FGM beam dynamic response.
Fig. 2 is a schematic diagram of a variant of the additional lumped mass FGM beam system.
FIG. 3 shows the temperature field T =10-10K rotation speed Ω 0 Deformation profile of FGM beam at =0.4 rad/s.
FIG. 4 shows the rotational speed Ω 0 Transverse deformation profile of the beam end at =10 rad/s.
FIG. 5 is a FGM beam end with consideration of shear and no shear, taken at different beam thicknessesTransverse deformation diagram of the end, wherein the temperature field is in the form of T c =10K and T m ＝0K。
Fig. 6 is a diagram of the transverse end deformation of FGM beam at different lengths taking into account the shear effect, where h =0.2m, Ω 0 =0.4rad/s, temperature field form T c =10K and T m ＝0K。
FIG. 7 is a graph of the lateral deformation of the beam end with different additional lumped masses applied to the end of the FGM beam and with additional lumped masses applied at different locations, where the temperature field has the form T c =10K and T m ＝0K。
Detailed Description
The technical solutions of the present invention will be described clearly and completely with reference to the accompanying drawings, and it should be understood that the described embodiments are some, but not all embodiments of the present invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Furthermore, the technical features involved in the different embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
With reference to fig. 1 to 7, a method for establishing a dynamic simulation model based on a gratuitous method considering a thermal effect and a shear effect FGM beam at least includes:
setting the geometric parameters and material parameters of the FGM beam, and establishing an FGM beam system considering the thermal effect and the additional concentrated mass under the shearing effect;
describing a displacement field which is deformed by large-range rotary motion of any point on an FGM beam in an FGM beam system with additional concentrated mass in a floating coordinate system by adopting a mixed coordinate method;
dispersing the deformation field of the FGM beam by adopting a non-grid point interpolation method and a radial base point interpolation method;
and (3) establishing a primary approximate rigid-flexible coupling kinetic equation of the FGM beam system with the additional concentrated mass by using a second Lagrange equation to obtain a dynamic simulation model of the FGM beam with the additional concentrated mass under consideration of the thermal effect and the shearing effect.
The geometric parameters of the FGM beam are: the beam has a length of l, a width of b, and a concentrated mass of m t The distance of the lumped mass from the fixed end is l t (ii) a The material parameters are as follows: the density rho, the elastic modulus E and the heat conduction coefficient of the FGM beam are K, the linear expansion coefficient is alpha, and the expressions are respectively as follows:
describing a displacement field of large-range rotary motion deformation of any point on an FGM beam in a central rigid body-FGM beam system in a floating coordinate system by adopting a mixed coordinate method; concentrated mass m on FGM beam t Radius of mt The expression in the inertial coordinate system O-XYZ is:
r mt ＝Θ(r A +ρ 1 +u 1 )
in the formula, theta is a normal cosine matrix of the floating base relative to the inertial coordinate system, r A The vector of rigid body centroid O to O, deformation vector u 1 In a floating coordinate system, can be expressed as:
ρ 1 ＝(l t ,0) T
u 1 ＝(u x1 ,u y1 ) T
u x1 ＝w 1 +w c -yφ
v y1 ＝v 1 +v 2
the first derivative is calculated for time t to obtain the deformation velocity of the lumped mass, which is expressed as:
the total kinetic energy of the system is:
in the formula, the first two terms are the kinetic energy of the FGM beam, and the latter term is the kinetic energy of the lumped mass.
Longitudinal positive strain epsilon of any point on functionally graded material beam 11 The expression of (a) is:
shear strain becomes
Without considering the torsion effect of the FGM beam, the elastic potential energy expression of the system is:
adopting a non-grid point interpolation method and a radial base point interpolation method to disperse the large-range rotation deformation of the FGM beam, and taking longitudinal and transverse displacement functions as follows:
in the formula phi 1 (x)、Φ 2 (x) And phi 3 (x) Are respectively w 1 、v 1 And v 2 Corresponding to the matrixes of shape functions of different discrete methods, A (t), BETA (t) and C (t) are respectively w 1 、v 1 And v 2 To the corresponding array.
The nonlinear coupling deformation of the axial displacement caused by the transverse displacement is as follows:
in the formula, H (x) is a coupling shape function, and the expression is as follows:
taking a generalized coordinate column vector q = (theta, A) of the system T ,B T ,C T ) T The virtual work done by the temperature load on the FGM beam is:
wherein θ = T-T 0 Is the temperature difference, T is the actual temperature, T 0 Zero was taken in this study for the initial reference temperature; sigma T Is a thermal stress; q T The generalized force column matrix under temperature load is shown as the following specific expression:
the virtual work done by the external driving torque is expressed as:
in the above formula, Q τ ＝[F τ 0 0 0] T Is a generalized force array corresponding to the external driving force; wherein F τ The method is characterized in that the method is an external driving force acting on a central rigid body, and a total kinetic energy expression W of the FGM beam as a whole and a potential energy expression U of the FGM beam as a whole are substituted into a Lagrange equation of the second type:
the first approximation rigid-flexible coupling system kinetic equation of the FGM beam considering the additional concentrated mass under the thermal effect and the shear effect is as follows:
the specific matrix expression in the formula is as follows:
M 22 ＝M 1
M 33 ＝M 2
the single underlined term in the formula is due to the consideration of the nonlinear coupling distortion w c Additional coupling terms are generated, double-underlined is a correlation term generated in consideration of the shearing effect, K 5 And K 6 Is a temperature load term. The matrix of correlation constants is:
J 2 ＝∫ V ρy 2 dV
Y＝∫ V ρy 2 Φ 2 ′(x)dV
example 1
The embodiment of the invention discloses a computing method for considering thermal effect and shearing effect FGM Liang Dongli response based on a meshless method, which comprises the following steps:
Shown in table 1. The motion law is as follows:
in the formula T 0 =2s, the rigid body angular velocity reaches steady state.
TABLE 1
3, dispersing the large-range rotary deformation of the FGM by adopting an assumed modal method (comparative example), a finite element method (comparative example), a non-grid point interpolation method and a radial base point interpolation method;
and 5, solving a primary approximate rigid-flexible coupling kinetic equation by adopting a Newmark method, and outputting a deformation schematic diagram of the FGM beam in large-range rotary motion.
As can be seen from fig. 3, the rotation speed Ω in the environment with the temperature field T =10-10K 0 When the temperature field is not larger than 0.4rad/s, the influence of the temperature field on the nonlinear coupling deformation is not obvious, but the influence on the axial deformation is obvious, and the axial deformation generates high-frequency oscillation under the impact of thermal load. Due to axial deformation w of the beam ends 1 The longitudinal deformation of the material also shows a remarkable oscillation phenomenon. Therefore, the axial deformation w cannot be neglected when there is a temperature load, as compared to not taking the temperature field into account 1 The influence of (c).
As can be seen from FIG. 4, the rotational speedΩ 0 The simulation results of the transverse deformation of the beam end using the hypothetical mode method diverge when =10rad/s, while the simulation results of the other three methods converge and are substantially identical. The hypothesis mode method based on the small deformation hypothesis is only suitable for the small deformation condition and cannot process the large deformation problem, and the gridding-free method has a wider application range.
As can be seen from FIG. 5, the temperature field is taken as T c =10K and T m And =0K, the deviation of the deformation of the end of the beam without shearing action and shearing effect gradually increases as the thickness of the beam becomes larger, and the lateral deformation of the end of the beam becomes smaller as the thickness of the beam becomes larger. The larger the rotation speed, the larger the transverse deformation of the beam end, and the larger the influence of the shearing effect.
As can be seen from FIG. 6, the temperature field is T =10-0K, h =0.2m, Ω 0 In the case of =0.4rad/s, the influence of the shearing effect becomes more significant as the length of the FGM beam decreases, and the transverse displacement of the end of the FGM beam also gradually decreases, so that the shearing effect is not negligible when the thickness and length are large.
It can be seen from fig. 7 that the greater the additional lumped mass acting on the FGM beam end, the greater the lateral deformation of the beam end; the farther the concentrated mass is from the central rigid body, the greater the lateral deformation of the beam ends.
Although the embodiment 1 of the present invention is exemplified by a metal/ceramic composite material, the simulation model and the simulation method of the present invention can be fully applied to non-metal/ceramic composite materials, metal/non-metal composite materials, ceramic/ceramic composite materials, and the like.
While embodiments of the invention have been described above, it is not limited to the applications set forth in the description and the embodiments, which are fully applicable in various fields of endeavor to which the invention pertains, and further modifications may readily be made by those skilled in the art, it being understood that the invention is not limited to the details shown and described herein without departing from the general concept defined by the appended claims and their equivalents.
Claims (7)
1. A method for establishing a dynamic simulation model based on a gridding-free method considering a thermal effect and a shear effect FGM (flue gas desulfurization) beam is characterized by at least comprising the following steps of:
setting the geometric parameters and material parameters of the FGM beam, and establishing an FGM beam system considering the thermal effect and the additional concentrated mass under the shearing effect;
describing a displacement field which is deformed by large-range rotary motion of any point on an FGM beam in the FGM beam system with the additional concentrated mass in a floating coordinate system by adopting a mixed coordinate method;
adopting a non-grid point interpolation method and a radial base point interpolation method to disperse the deformation field of the FGM beam;
and establishing a primary approximate rigid-flexible coupling kinetic equation of the FGM beam system with the additional concentrated mass by using a second Lagrange equation to obtain a kinetic simulation model of the FGM beam with the additional concentrated mass under consideration of the thermal effect and the shearing effect.
2. The method of building according to claim 1, wherein the geometric parameters of the FGM beam are: the beam has a length of l, a width of b, and a concentrated mass of m t The distance of the lumped mass from the fixed end is l t (ii) a The material parameters are as follows: the density rho, the elastic modulus E and the heat conduction coefficient of the FGM beam are K, the linear expansion coefficient is alpha, and the expressions are respectively as follows:
wherein, the corner mark c represents ceramic material, the corner mark m represents metal material, h is the thickness of the FGM beam, N is the functional gradient index, and y is the position of the FGM beam in the thickness direction; the elastic modulus and density of the ceramic and the metal are respectively taken as E c ＝1.51×10 11 Pa、E m ＝7.0×10 10 Pa、ρ c ＝3.0×10 3 kg/m3、ρ m ＝2.707×10 3 kg/m, the thermal conductivity and thermal expansion coefficient of the ceramic and the metal are respectively K c ＝2.09W/mK、K m ＝204W/mK、α c ＝1×10 -5 、α m ＝2.3×10 -5 。
3. The method of establishing according to claim 2, wherein: radius r and concentrated mass m of deformation of any point on FGM beam t Radius of mt The expression in the inertial coordinate system O-XYZ is:
r＝Θ(r A +ρ 0 +u)
r mt ＝Θ(r A +ρ 1 +u 1 )
in the formula, r A The radius rho from the center rigid body centroid O to the base point O of the floating base 0 And rho 1 Radial, u and u, relative to the floating base before deformation 1 The vector is a deformation displacement vector, and theta is a normal cosine matrix of the floating base relative to an inertial coordinate system;
the expression for the total kinetic energy of the FGM beam as a whole is:
the torsion effect of the FGM beam is not counted, so the overall elastic potential energy expression of the FGM beam is:
in the formula, epsilon 11 And γ is the longitudinal positive and shear strain at any point on the FGM beam.
4. The method of building according to claim 3, characterized by: FGM Liang Lisan is divided into a plurality of nodes by a meshless method, node information in a calculation point x support domain is obtained through calculation, and longitudinal and transverse displacement functions are taken as follows:
in the formula phi 1 (x)、Φ 2 (x) And phi 3 (x) Are respectively w 1 、v 1 And v 2 Corresponding to the line matrix of the shape function of different discrete methods, A (t), BETA (t) and C (t) are respectively w 1 、v 1 And v 2 The corresponding array of (a);
the nonlinear coupling deformation of the axial displacement caused by the transverse displacement is as follows:
in the formula, H (x) is a coupling shape function, and the expression is as follows:
5. the method of building according to claim 4, characterized by: establishing a primary approximate rigid-flexible coupling kinetic equation of the FGM beam system considering the additional concentrated mass under the thermal effect and the shearing effect by using a second Lagrange equation, and specifically taking a generalized coordinate column vector q = (theta, A) of the system T ,B T ,C T ) T The virtual work done by the temperature load on the FGM beam is:
wherein θ = T-T 0 Is the temperature difference, T is the actual temperature, T 0 Zero was taken in this study for the initial reference temperature; sigma T Is a thermal stress; q T Is a generalized force column matrix under temperature load, and a specific expression thereofAs follows:
the virtual work done by the external driving torque is expressed as:
in the above formula, Q τ ＝[F τ 0 0 0] T Is a generalized force array corresponding to the external driving force; wherein F τ The method is an external driving force acting on a central rigid body, and substitutes a total kinetic energy expression W of the whole FGM beam and a potential energy expression U of the whole FGM beam into a Lagrange equation of a second type:
the first approximation rigid-flexible coupling system kinetic equation of the FGM beam considering the additional concentrated mass under the thermal effect and the shear effect is as follows:
the specific matrix expression in the formula is as follows:
M 22 ＝M 1
M 33 ＝M 2
the upper typeThe single underlined term is due to consideration of the nonlinear coupling distortion w c Additional coupling terms are generated, double-underlined are related terms generated in consideration of the shearing effect, K 5 And K 6 Is a temperature load term; y, S x 、S y 、S z 、M 1 、M 2 、M 3 、M 4 、M 5 、M 6 、D、K 1 、K 2 、K 3 、K 4 、K 5 、K 6 Is a constant coefficient matrix; j. the design is a square ob Is the moment of inertia of the beam.
6. A dynamic simulation model based on a mesh-free method considering thermal effect and shear effect FGM beams, obtained by the establishing method of any one of claims 1 to 5.
7. The simulation method of the dynamic simulation model considering the thermal effect and the shear effect FGM beam based on the meshless method as claimed in claim 6, characterized in that the dynamic simulation model considering the thermal effect and the shear effect FGM beam based on the meshless method is solved by a Newmark method to obtain the deformation schematic diagram of the FGM beam with the additional concentrated mass of the large-range rotational motion.
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