CN110135087B - T-shaped curve dynamic correction force prediction model establishing method - Google Patents

T-shaped curve dynamic correction force prediction model establishing method Download PDF

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CN110135087B
CN110135087B CN201910420419.5A CN201910420419A CN110135087B CN 110135087 B CN110135087 B CN 110135087B CN 201910420419 A CN201910420419 A CN 201910420419A CN 110135087 B CN110135087 B CN 110135087B
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CN110135087A (en
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姜金刚
马雪峰
闵兆伟
张永德
陈厚鋆
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Harbin University of Science and Technology
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Abstract

The invention discloses a method for establishing a T-shaped curve dynamic correction force prediction model, which relates to the technical field of orthodontic treatment and comprises the following steps: 1) analyzing the structural characteristics and the loading characteristics of the T-shaped curve; 2) establishing a T-shaped curve vertical arm correcting force prediction model; 3) establishing a T-shaped curve horizontal arm correcting force prediction model; 4) establishing a T-shaped curve static correction force prediction model; 5) establishing a dynamic resistance model in the process of simulating tooth movement by a wax jaw dike; 6) and establishing a T-shaped curve dynamic correction force prediction model. The invention can effectively carry out parameterization expression on the T-shaped curve correction force value applied by a doctor, accurately predict the T-shaped curve correction force value applied by the doctor, assist the doctor to improve the safety and predictability of orthodontic treatment and improve the digitization degree of orthodontic treatment.

Description

T-shaped curve dynamic correction force prediction model establishing method
Technical Field
The invention relates to a method for establishing a T-shaped curve dynamic correction force prediction model, and belongs to the technical field of orthodontic treatment.
Background
The fixed correction technology is the most effective orthodontic treatment method at present, the T-shaped curve is the most commonly used curve for closing the tooth gap at present, in the diagnosis process of the traditional fixed correction technology, the correction force and the correction effect generated by the orthodontic arch wire used in each correction stage are predicted according to the experience of an orthodontic doctor, although the traditional orthodontic treatment means depending on the experience of the orthodontic doctor can play a certain role in the treatment of most patients, the orthodontic force lacks the quantitative standard, the treatment result completely depends on the doctor level, and the patient is easily injured and the treatment efficiency is reduced.
The base holds in the palm wax has the texture softer, toughness is good, physical characteristics such as the not sticky hand of heating softening, can make waxy jaw embankment based on patient's tooth data in oral clinical application, immerse wax system jaw embankment in the water bath case can shorten just abnormal correction process greatly, distinguish and other rigidity tooth models, immerse wax system jaw embankment and can observe the removal condition of tooth along with just abnormal arch wire in the constant temperature water bath environment, realize just abnormal the simulation of this dynamic process of interaction between tooth and the periodontal tissue among the correction process, thereby revise the scheme.
Although the rationality of the correction scheme can be improved to a certain extent by applying the wax jaw dike at present, in the process of applying the base support wax to simulate orthodontic treatment, the relationship between the shape of an orthodontic arch wire and the dynamic correction force applied to teeth is not established, and corresponding quantitative standards are lacked, and an orthodontist still cannot predict the size of the correction force applied to a patient under the influence of the orthodontic arch wire through the simulation process, so that a T-shaped curve dynamic correction force prediction model based on the wax jaw dike is established, the T-shaped curve dynamic correction force value is parameterized and expressed, and the method has very important significance for developing oral digital diagnosis and treatment and assisting the doctor in improving the safety and predictability of the orthodontic treatment.
Disclosure of Invention
Aiming at the problems, the invention aims to provide a method for establishing a T-shaped curve dynamic correction force prediction model, which is used for determining the relation between the T-shaped curve shape and the dynamic correction force borne by teeth in the process of simulating orthodontic treatment by using a wax jaw dike and assisting an orthodontic doctor in designing orthodontic arch wire parameters so as to improve the safety and predictability of orthodontic treatment.
The above purpose is mainly achieved through the following scheme:
the invention discloses a method for establishing a T-shaped curve dynamic correction force prediction model, which is characterized by comprising the following steps of: the method comprises the following concrete implementation processes:
1) analyzing the structural characteristics and the loading characteristics of the T-shaped curve;
2) establishing a T-shaped curve vertical arm correcting force prediction model;
3) establishing a prediction model of the correcting force of the horizontal arm of the T-shaped curved arc part;
4) establishing a T-shaped curve static correction force prediction model;
5) establishing a dynamic resistance model in the process of simulating tooth movement by a wax jaw dike;
6) and establishing a T-shaped curve dynamic correction force prediction model.
Preferably, in the step 1), as can be seen from the structural characteristics of the T-shaped curve, when the T-shaped curve corrects the teeth, the correcting force is released by the horizontal arm of the T-shaped curve, and the vertical arm and the arc part of the T-shaped curve, which are deformed, are symmetrical to each other on both sides of the T-shaped curve, so that when the T-shaped curve correcting force model is established, only a part of the symmetry needs to be analyzed; the bending radius of the T-shaped curved arc part is R, the overall height is h, the closing gap is b, and the closing gap is generated when force is applied by withdrawing the arch wire after clinical;
after the deformation condition of the T-shaped curve in the correcting process is analyzed, the T-shaped curve deforms at the vertical arm and the horizontal arm of the arc part of the T-shaped curve, and the orthodontic force generated by the T-shaped curve is composed of restoring forces generated by two deformation areas, so that the vertical arm of the T-shaped curve and the horizontal arm of the arc part of the T-shaped curve need to be subjected to mechanical analysis respectively, and the vertical arm of the T-shaped curve and the horizontal arm of the arc part of the T-shaped curve are superposed to establish a prediction model of the correcting force of the T-;
preferably, in step 2), the rotation angle equation θ (x) and the deflection equation v (x) of the T-shaped vertical arm can be expressed as:
Figure GDA0002382548190000011
where M (x) is the bending moment on the vertical part at a distance x, E is the modulus of elasticity of the material, IzIs the moment of inertia of the arch wire cross-section to the z-axis, for round wire Iz＝πd4D is the diameter of a round wire for a rectangular wire Iz＝c1c2 3/12，c2Is the length of the side parallel to the z-axis on the cross section of the rectangular filament, c1Is the length of the cross section of the rectangular wire perpendicular to the z-axis, C0And D0Is an integral constant, C0And D0The bending moment equation of the T-shaped vertical arm is determined by boundary conditions as follows:
M(x)＝-P(y-x) (2)
wherein, P is the force required for generating the deformation of the corner theta (x) of the vertical arm, and y is the length of the vertical arm before the deformation;
by substituting equation (2) into equation (1) and integrating, it is possible to obtain:
Figure GDA0002382548190000021
to determine the formula (3)Integral constant C in0、D0The boundary condition of the T-shaped curved vertical arm needs to be determined, based on the supporting condition, the deflection or the corner of the T-shaped curved vertical arm is always zero or known, x is 0, namely the deformation of the joint of the T-shaped curved vertical arm and the horizontal arm of the circular arc part is solved, and because the joint of the T-shaped curved vertical arm and the horizontal arm of the circular arc part is provided with a longitudinal symmetrical plane and external force acts on the symmetrical plane, the axis of the deformed curved beam is still positioned in the longitudinal symmetrical plane, and the deformation belongs to the plane bending deformation problem of the curved beam, the circular arc at the joint can be equivalent to the curved beam with the radian of pi/4, and a section of the radian of the curved beam is d α infinitesimal;
under the condition that the plane of the curved beam is bent, external force is in the longitudinal symmetrical plane of the curved beam, the axis after deformation is still a curve in the symmetrical plane, the curved beam does not have torsional deformation, the plane assumption can be still used at the moment, a differential equation of the bending line after the curved beam at the connection part of the vertical arm and the circular arc part and the horizontal arm is deformed can be obtained, the equation is shown as a formula (4), and the deformation of the curved beam can be determined by integrating the equation under the given boundary condition;
Figure GDA0002382548190000022
in the formula, u is the displacement of the cross section of the curved beam at the joint of the vertical arm and the horizontal arm of the arc part in the x direction, and the torque M borne by the joint0＝M|x＝0＝-Py，IωThe moment of inertia of the cross section of the curved beam at the joint to the omega axis is Iω＝Iz；
As can be known from the arc length formula, ds is Rd α, so the differential equation of the bending line after the bending beam at the joint of the vertical arm and the horizontal arm at the arc part deforms can be changed as follows:
Figure GDA0002382548190000023
solving the non-homogeneous differential equation of the constant coefficient of the bending line after the bending beam at the joint is deformed to obtain:
Figure GDA0002382548190000024
because the bent beams at the joints of the vertical arms and the horizontal arms of the arc parts are symmetrical along the longitudinal symmetrical plane, the boundary conditions of the bent beams of the vertical arms and the horizontal arms of the arc parts are as follows
Figure GDA0002382548190000025
Obtaining by solution:
Figure GDA0002382548190000026
the deflection equation of the curved beam at the joint can be expressed as:
Figure GDA0002382548190000027
the corner equation of a camber beam is:
Figure GDA0002382548190000028
therefore, there is a boundary condition
Figure GDA0002382548190000029
Obtaining by solution:
Figure GDA00023825481900000210
c is to be0，D0The following are available in the alternative (3):
Figure GDA0002382548190000031
because the maximum rotation angle and the maximum deflection are generated at the spinning end, namely x is equal to y, and the maximum deflection is the moving distance m of the T-shaped bent horizontal arm at the spinning end, it can be known that:
Figure GDA0002382548190000032
produced by deformation of T-shaped curved vertical arms according to the principle of reaction forceCorrecting force F1I.e. the counterforce to produce the force required for the deformation:
Figure GDA0002382548190000033
preferably, in the step 3), the horizontal arm of the T-shaped curved arc part and the vertical arm of the T-shaped curved arc part deform in the same way, so the same mechanical modeling method can be used, in the orthodontic treatment process, an orthodontist can pull the filament drawing end of the T-shaped curved arc part in advance to cause the T-shaped curved arc to deform, and install the deformed T-shaped curved arc on the teeth of a patient, the restoring force generated after the T-shaped curved arc part pulls the teeth of the patient to move so as to achieve the purpose of correcting the deformed teeth of the patient, the bending deflection of the horizontal arm of the T-shaped curved arc part, namely the displacement of the horizontal arm of the T-shaped curved arc part along the Z-axis direction, in the T-shaped curved arc part, the position of the symmetry center of the connection part of the horizontal arm of the T-shaped curved arc part and the vertical arm of the T-shaped curved arc part in the space is constantly changed, therefore, in order to calculate the bending deflection s of the horizontal arm of the T-, the symmetrical centers of the joints before deformation are overlapped, and the difference value between the theoretical length of the vertical arm after deformation and the length y of the vertical arm before deformation is calculated, namely the bending deflection s of the horizontal arm of the T-shaped curved circular arc part is as follows:
Figure GDA0002382548190000034
the approximate differential equation of the horizontal arm deflection line of the T-shaped curved arc part can be obtained as follows:
Figure GDA0002382548190000035
wherein M (l) is the bending moment of the horizontal arm of the T-shaped curved arc part;
the rotation angle equation theta (l) and the deflection equation v (l) of the horizontal arm of the T-shaped curved arc part can be obtained by integrating the equation (13):
Figure GDA0002382548190000036
Figure GDA0002382548190000037
in the formula, C1And D1Is an integral constant determined by the boundary conditions, and the bending moment equation of the horizontal arm of the T-shaped curved arc portion can be expressed by equation (16):
M(l)＝-G(w-R-l) (16)
in the formula, G is the correcting force generated by the horizontal arm of the arc part, and w is the length of the horizontal arm of the T-shaped curved arc part;
by substituting equations (13) and (16) into equations (14) and (15) and integrating them, it is possible to obtain:
Figure GDA0002382548190000038
Figure GDA0002382548190000039
the solution of the boundary condition of the horizontal arm of the T-shaped curved arc part is the same as that of the T-shaped curved vertical arm, the radian of the side arc on one side is pi/4, and a section of radian is d β infinitesimal, and the solution of the boundary condition of the vertical arm gives a detailed process, so that a boundary condition formula obtained by solving an uneven equation of the deflection constant coefficient of the arc section is directly given:
Figure GDA00023825481900000310
due to the symmetry of the T-shaped curved side arc, the boundary condition of the side arc is
Figure GDA00023825481900000311
Obtaining by solution:
Figure GDA00023825481900000312
the deflection equation of the bent beam is:
Figure GDA0002382548190000041
the corner equation of a camber beam is:
Figure GDA0002382548190000042
therefore, the boundary condition v $l＝0＝u|β＝0＝0，θ|l＝0＝ε|β＝0When the solution is 0, the solution is obtained: c1＝0，D1＝0；
C is to be1And D1Substitution into (17) and (18) can obtain:
Figure GDA0002382548190000043
Figure GDA0002382548190000044
in the formula, the maximum rotation angle and the maximum deflection of the horizontal arm of the T-shaped curved arc part are generated at a position l equal to w-R, and the maximum deflection is the moving distance s of the T-shaped curved horizontal arm (4-3), so that the following results are obtained:
Figure GDA0002382548190000045
orthodontic force F in the direction of the end of the filament2The counter force of the component force of the force required by the deformation of the horizontal arm corner theta (l) of the circular arc part generated by the deformation of the horizontal arm of the T-shaped curved circular arc part along the direction of the drawing end is that:
Figure GDA0002382548190000046
preferably, in the step 4), the static correcting force generated by the T-shaped curve is a resultant force of restoring forces generated after the vertical arm and the horizontal arm of the circular arc portion of the T-shaped curve are deformed, that is:
Figure GDA0002382548190000047
preferably, the method is applied to a wax jaw wall for simulating tooth movement.
Preferably, the wax jaw dike for simulating tooth movement, which is applied to the method, is composed of a base wax jaw dike, a resin tooth model, an orthodontic bracket and a T-shaped curve.
Preferably, in the step 5), the tooth to be measured is connected with the measuring element by a resin cylinder, and the tooth moving in the wax jaw levee is actually the movement of the cylinder connector in the wax jaw levee, so that the analysis is performed by taking the cylinder as a basic component; when the tooth moves in the wax jaw wall, the speed is vtWhen, vtThe flowing speed of the wax jaw levee at the time t is adopted, and acting force on the cylinder along the moving direction is streaming drag force; the friction drag force and the differential pressure drag force jointly form a streaming drag force; the friction drag force is that a boundary layer is formed on the surface of the cylinder due to the viscosity of the fluid, and in the boundary layer range, the fluid generates a velocity gradient, the friction effect is obvious, and friction shear stress is generated; the pressure difference drag force is that the boundary layer is separated at a certain point on the surface of the cylinder, and strong vortex wake is formed at the downstream of the separation point, namely at the rear part of the cylinder, so that pressure difference is generated between the front part and the rear part of the cylinder, and further a force is generated in the flow direction, and in the fluid flow, the vortex wake of the cylinder is R along with the Reynolds numbereOf teeth in a waxed jaw walleLess than 5, therefore, no vortex wake flow is generated, and no pressure difference drag force is generated;
drag force f on cylinder per unit lengthDCan be calculated using equation (27):
Figure GDA0002382548190000048
in the formula, v0For the tooth movement velocity component perpendicular to the cylinder axis unaffected by the streaming, ρ (t) is the density of the wax jaw wall at the test temperature at time t, a is the projected area of the cylinder per unit length perpendicular to the direction of movement, for a cylinder, a ═ 1 × D, D is the diameter of the cylinder, CDIs a mopThe drag coefficient, which collectively reflects the viscous effect due to the viscosity of the fluid, and the Reynolds number ReAnd cylinder surface roughness Ra(ii) related;
assuming the wax chin underlying fluid of this study is an incompressible ideal fluid, the volume of wax removal is
Figure GDA0002382548190000049
At a moving speed vtA wax chin dam flow field of v (x, y, z, t); the influence of the cylinder on the wax jaw levee flow field is not considered for the moment, namely the pressure distribution in the wax jaw levee flow field is assumed not to be changed due to the existence of the cylinder, the boundary of the cylinder is taken as a part of the accelerating fluid boundary, namely the wax jaw levee fluid in the part of the volume replaced by the cylinder, which is supposed to exist in the wax jaw levee flow field in a static state, but actually due to the existence of the movement of the cylinder, the static wax jaw levee fluid is accelerated to the same state as the moving speed of the boundary of the cylinder; thus the accelerated wax jaw dam fluid will be at a volume of dewaxing
Figure GDA0002382548190000051
The cylinder acting in the flow direction with an inertial force FkInertial force FkIs equal to the dewaxing mass M of the cylinder0Volume and volume
Figure GDA0002382548190000052
Average acceleration of jaw dike fluid made of inner wax
Figure GDA0002382548190000053
The product of (a) and (b), namely:
Figure GDA0002382548190000054
for the cylinder under investigation,
Figure GDA0002382548190000055
the fluid acceleration at the center of the cylinder axis can be taken
Figure GDA0002382548190000056
To indicate that, at this time:
Figure GDA0002382548190000057
however, because the cylinder exists in the wax jaw levee flow field, the fluid particles around the cylinder are disturbed to cause speed change, so that the pressure distribution in the wax jaw levee flow field is changed, and the disturbance of the cylinder is the mass M of the part of the additional fluid which changes the original motion state around the cylinderwAn additional inertial force, i.e. an additional mass force, will also be generated on the cylinder in the direction of fluid flow; the streaming inertial force f of the accelerated fluid actually acting on the cylinder in the flow directionLCan be expressed as:
Figure GDA0002382548190000058
let Mw＝CmM0Then equation (30) can be expressed as:
Figure GDA0002382548190000059
in the formula, CmTo add a mass coefficient, CMThe mass coefficient is also called as an inertia force coefficient, and intensively reflects an additional mass effect caused by the change of the speed of a wax jaw dike flow field around the cylinder due to the inertia of the fluid and the existence of the cylinder;
through the analysis, the resistance condition of the teeth in the moving process of the wax jaw levee can be obtained, the teeth move in the wax jaw levee under the influence of the orthodontic force generated by the deformation of the orthodontic arch wire, and the teeth are subjected to the streaming inertia force f due to the streaming characteristic of the flow field in the moving processLAnd drag force fDThe influence of (a);
influenced by heat exchange, the internal temperature of the wax jaw dike model in the thermal field changes along with time, and the change of the internal temperature causes the wax jaw to be formedThe change of the density of the dike model further influences the resistance of teeth moving in the wax jaw dike; the tooth model follows the rule in the viscous fluid energy equation when moving in the wax jaw levee, let e represent the internal energy of unit mass fluid, then ρ (t) e is the internal energy of unit volume fluid, ρ (t) vt 2The kinetic energy per unit volume is represented by/2, so that the total energy E ═ ρ (t) E + ρ (t) v contained in the fluid per unit volumet 2/2；
By simplifying the arrangement, the principle of conservation of energy can be approximately expressed as:
Figure GDA00023825481900000510
in the formula, cpA dimensionless pressure coefficient is adopted, phi is mechanical work consumed by the tooth model when moving in the wax jaw embankment fluid, k is a calculation coefficient, ▽ T is the temperature gradient of the wax-based fluid thermal field, and q is the heat flow density;
solving ▽ T, and setting the thickness of the wax jaw wall as 2 delta and the initial temperature as T0(ii) a It is placed at a temperature t at the initial instant∞In the fluid, the surface heat transfer coefficient h between the fluid and the wax jaw dike is constant, two sides of the wax jaw dike are symmetrically heated, and the internal temperature distribution of the wax jaw dike must take the central section as a symmetrical plane, so that only a half wax jaw dike with the thickness delta needs to be researched, the original point of an x axis is placed on the central section of the wax jaw dike, and for the half wax jaw dike with the x being more than or equal to 0, the following heat conduction differential equation can be listed:
Figure GDA00023825481900000511
where a is the thermal diffusivity, and the two sides of equation (33) are integrated with x, the following can be obtained:
Figure GDA00023825481900000512
the even heating of wax jaw dyke under the water bath environment can simplify to one-dimensional thermal field problem, consequently has:
Figure GDA0002382548190000061
the temperature gradient ▽ T of the wax jaw embankment fluid thermal field is substituted into a formula (32) to be finished to obtain:
Figure GDA0002382548190000062
the two sides of the equation of equation (36) are integrated and arranged for t to obtain:
Figure GDA0002382548190000063
wherein T is the temperature of the wax jaw wall fluid thermal field;
the fourier law, when expressed in terms of heat flow density q, has the following form:
Figure GDA0002382548190000064
wherein λ is a thermal conductivity coefficient;
substitution of equation (38) for equation (37) gives the expression that waxy jaw density ρ varies with time t:
Figure GDA0002382548190000065
the dynamic resistance model in the process of simulating tooth movement by the wax jaw dike can be expressed by the formula (40):
Figure GDA0002382548190000066
wherein f is dynamic resistance of the wax jaw dike in the process of simulating tooth movement.
Preferably, in step 6), based on the T-shaped curve static correction force prediction model and the dynamic resistance model in the process of simulating tooth movement by the wax jaw levee, the T-shaped curve dynamic correction force prediction model can be expressed as:
Figure GDA0002382548190000067
in the formula, F is the dynamic correction force of the T-shaped curve obtained by applying the dynamic correction force prediction model of the T-shaped curve.
The invention has the beneficial effects that:
1. by adopting a parameterized modeling method, the influence effect of each influence factor on the T-shaped curve can be reflected more intuitively, and a doctor can adjust the bent arch wire conveniently to obtain proper orthodontic force.
2. The method is suitable for the denture wax jaw dike, and compared with a prediction model obtained based on the traditional rigid jaw dike, the method can reflect the influence of the relative movement of teeth and the attenuation phenomenon of orthodontic arch wire force on the correction force in the real orthodontic process, so the correction force calculated by the method can reflect the dynamic characteristic in the real orthodontic process, and the method has higher accuracy in the aspect of correction force prediction.
Drawings
For ease of illustration, the invention is described in detail by the following detailed description and the accompanying drawings.
FIG. 1 is a flow chart of the method for establishing a T-shaped curve dynamic correction force prediction model;
FIG. 2 is a schematic diagram of a T-bend vertical arm mechanics analysis of the present invention;
FIG. 3 is a schematic diagram of a force analysis of the T-bend section of the present invention;
FIG. 4 is a schematic diagram illustrating the deformation analysis of the T-shaped curved arc portion according to the present invention;
FIG. 5 is a schematic view of a wax jaw dam for simulating tooth movement used in the present invention;
FIG. 6 is a flow chart of dynamic resistance model establishment in the process of simulating tooth movement by the wax jaw levee according to the method of the invention;
FIG. 7 is a schematic view of the movement of teeth in the wax jaw wall of the present invention;
FIG. 8 is a schematic view of the jaw wall made of wax according to the present invention being heated;
in the figure: 1-base wax-holding jaw dike; 2-resin tooth model; 3-orthodontic bracket; 4-T-shaped curve; 4-1-vertical arm; 4-2-arc segment; 4-3-arc part horizontal arm; 4-4-T-shaped bent horizontal arm.
Detailed Description
In order that the objects, aspects and advantages of the invention will become more apparent, the invention will be described by way of example only, and in connection with the accompanying drawings. It is to be understood that such description is merely illustrative and not intended to limit the scope of the present invention. Moreover, in the following description, descriptions of well-known structures and techniques are omitted so as to not unnecessarily obscure the concepts of the present invention.
As shown in fig. 1, fig. 2, fig. 3, fig. 4, fig. 5, fig. 6, fig. 7, and fig. 8, the present embodiment adopts the following technical solutions: the invention discloses a method for establishing a prediction model of opening vertical curvature orthodontic force, which is characterized by comprising the following steps of: the method comprises the following concrete implementation processes:
1) analyzing the structural characteristics and the loading characteristics of the T-shaped curve;
2) establishing a T-shaped curve vertical arm correcting force prediction model;
3) establishing a prediction model of the correcting force of the horizontal arm of the T-shaped curved arc part;
4) establishing a T-shaped curve static correction force prediction model;
5) establishing a dynamic resistance model in the process of simulating tooth movement by a wax jaw dike;
6) and establishing a T-shaped curve dynamic correction force prediction model.
Furthermore, in the step 1), it can be known from the structural characteristics of the T-shaped curve that when the T-shaped curve corrects the teeth, the correcting force is released by the horizontal arm 4-4 of the T-shaped curve, and the vertical arm 4-1 and the circular arc part 4-2 of the T-shaped curve, which deform, are symmetrical to each other on both sides of the T-shaped curve, so that when the T-shaped curve correcting force model is built, only a part of the symmetry needs to be analyzed; the bending radius of the T-shaped curved arc part 4-2 is R, the overall height is h, the closing gap is b, and the closing gap is generated when an arch wire is drawn out and applied with force after clinical;
after the deformation condition of the T-shaped curve in the correcting process is analyzed, the T-shaped curve deforms at the vertical arm 4-1 and the horizontal arm 4-3 of the arc part of the T-shaped curve, and the correcting force generated by the T-shaped curve is composed of restoring forces generated by two deformation areas, so that the vertical arm 4-1 of the T-shaped curve and the horizontal arm 4-3 of the arc part of the T-shaped curve need to be subjected to mechanical analysis respectively, and are superposed to establish a prediction model of the correcting force of the T-shaped curve.
Further, in the step 2), the rotation angle equation θ (x) and the deflection equation v (x) of the T-shaped vertical arm 4-1 can be expressed as:
Figure GDA0002382548190000071
where M (x) is the bending moment on the vertical part at a distance x, E is the modulus of elasticity of the material, IzIs the moment of inertia of the arch wire cross-section to the z-axis, for round wire Iz＝πd4D is the diameter of a round wire for a rectangular wire Iz＝c1c2 3/12，c2Is the length of the side parallel to the z-axis on the cross section of the rectangular filament, c1Is the length of the cross section of the rectangular wire perpendicular to the z-axis, C0And D0Is an integral constant, C0And D0The bending moment equation of the T-shaped vertical arm 4-1 is determined by boundary conditions as follows:
M(x)＝-P(y-x) (2)
wherein, P is the force required for generating the deformation of the corner theta (x) of the vertical arm, and y is the length of the vertical arm before the deformation;
by substituting equation (2) into equation (1) and integrating, it is possible to obtain:
Figure GDA0002382548190000072
to determine the integration constant C in equation (3)0、D0The boundary condition of the T-shaped curved vertical arm 4-1 needs to be determined, the deflection or the corner of the T-shaped curved vertical arm 4-1 is always zero or known based on the supporting condition, x is equal to 0, namely the deformation of the joint of the T-shaped curved vertical arm 4-1 and the arc part horizontal arm 4-3 is solved, because the joint of the T-shaped curved vertical arm 4-1 and the arc part horizontal arm 4-3 is provided with a longitudinal symmetrical plane, and external force acts on the symmetrical plane, the axis of the deformed curved beam is still positioned in the longitudinal symmetrical plane, and the deformation belongs to the plane bending deformation problem of the curved beam, so that the arc at the joint can be equivalent to the arc curvatureA section of radian d α infinitesimal is taken for the pi/4 bending beam;
under the condition that the plane of the curved beam is bent, external force is in the longitudinal symmetrical plane of the curved beam, the axis after deformation is still the curve in the symmetrical plane, the curved beam does not have torsional deformation, the plane assumption can be still used at the moment, the differential equation of the bending line after the curved beam at the joint of the vertical arm 4-1 and the circular arc part horizontal arm 4-3 is deformed can be obtained as the formula (4), and the deformation of the curved beam can be determined by integrating the equation under the given boundary condition;
Figure GDA0002382548190000081
in the formula, u is the displacement of the cross section of the curved beam at the joint of the vertical arm 4-1 and the horizontal arm 4-3 of the arc part in the x direction, and the torque M borne by the joint0＝M|x＝0＝-Py，IωThe moment of inertia of the cross section of the curved beam at the joint to the omega axis is I because the T-shaped curved vertical arm 4-1 is consistent with the bending type of the curved beam at the jointω＝Iz；
From the arc length formula, ds is Rd α, so the differential equation of the bending line after the bending beam is deformed at the joint of the vertical arm 4-1 and the horizontal arm 4-3 of the arc part can be changed as follows:
Figure GDA0002382548190000082
solving the non-homogeneous differential equation of the constant coefficient of the bending line after the bending beam at the joint is deformed to obtain:
Figure GDA0002382548190000083
because the curved beams at the joints of the vertical arms 4-1 and the circular arc part horizontal arms 4-3 are symmetrical along the longitudinal symmetry plane, the boundary conditions of the curved beams of the vertical arms 4-1 and the circular arc part horizontal arms 4-3 are as follows
Figure GDA0002382548190000084
Obtaining by solution:
Figure GDA0002382548190000085
the deflection equation of the curved beam at the joint can be expressed as:
Figure GDA0002382548190000086
the corner equation of a camber beam is:
Figure GDA0002382548190000087
therefore, there is a boundary condition
Figure GDA0002382548190000088
Obtaining by solution:
Figure GDA0002382548190000089
c is to be0，D0The following are available in the alternative (3):
Figure GDA00023825481900000810
because the maximum rotation angle and the maximum deflection are generated at the spinning end, namely x is equal to y, and the maximum deflection is the moving distance m of the T-shaped bent horizontal arm 4-4 at the spinning end, it can be known that:
Figure GDA00023825481900000811
the correcting force F generated by the deformation of the T-shaped bent vertical arm 4-1 according to the principle of the reaction force1I.e. the counterforce to produce the force required for the deformation:
Figure GDA00023825481900000812
further, in the step 3), the deformation conditions of the horizontal arm 4-3 of the T-shaped curved arc part and the vertical arm 4-1 of the T-shaped curved arc part are the same, so that the same mechanical modeling method can be used, in the orthodontic treatment process, an orthodontist can pull the filament drawing end of the T-shaped curved arc 4 in advance to cause the deformation of the T-shaped curved arc 4, install the deformed T-shaped curved arc on the teeth of a patient, pull the teeth of the patient to move by the restoring force generated after the deformation of the T-shaped curved arc 4 to achieve the purpose of correcting the deformed teeth of the patient, the bending deflection of the horizontal arm 4-3 of the T-shaped curved arc part, namely the displacement of the horizontal arm 4-3 of the T-shaped curved arc part along the Z-axis direction, in the T-shaped curved arc process, the position of the symmetry center of the connection part of the horizontal arm 4-3 of the T-shaped curved arc part and the vertical arm 4-1 of the T-, therefore, in order to calculate the bending deflection s of the horizontal arm 4-3 of the T-shaped curved arc part, the motion condition of the symmetrical center of the connecting part in the deformation process of the T-shaped curved arc 4 needs to be simplified, the symmetrical centers of the connecting part before deformation are overlapped, and the difference value between the theoretical length of the vertical arm after deformation and the length y of the vertical arm before deformation is calculated, namely the bending deflection s of the horizontal arm of the T-shaped curved arc part is as follows:
Figure GDA00023825481900000813
the approximate differential equation of the horizontal arm 4-3 of the T-shaped curved arc part can be obtained as follows:
Figure GDA0002382548190000091
wherein M (l) is the bending moment of a horizontal arm 4-3 of a T-shaped curved arc part;
the rotation angle equation theta (l) and the deflection equation v (l) of the horizontal arm 4-3 of the T-shaped curved arc part can be obtained by integrating the equation (13):
Figure GDA0002382548190000092
Figure GDA0002382548190000093
in the formula, C1And D1Is an integral constant determined by the boundary conditions, and the bending moment equation of the horizontal arm 4-3 of the T-shaped curved arc portion can be expressed by equation (16):
M(l)＝-G(w-R-l) (16)
in the formula, G is the correcting force generated by the horizontal arm of the arc part, and w is the length of the horizontal arm 4-3 of the T-shaped curved arc part;
by substituting equations (13) and (16) into equations (14) and (15) and integrating them, it is possible to obtain:
Figure GDA0002382548190000094
Figure GDA0002382548190000095
the solution of the boundary condition of the horizontal arm 3 of the T-shaped curved arc part is the same as that of the vertical arm 4-1 of the T-shaped curved arc part, the arc radian of the side surface of one side is pi/4, and a section of radian is taken as d β infinitesimal, because the solution of the boundary condition of the vertical arm 4-1 gives a detailed process, the boundary condition formula obtained by solving the non-homogeneous equation of the deflection constant coefficient of the arc section is directly given here:
Figure GDA0002382548190000096
due to the symmetry of the side arc of the T-shaped curve 4, the boundary condition of the side arc is
Figure GDA0002382548190000097
Obtaining by solution:
Figure GDA0002382548190000098
the deflection equation of the bent beam is:
Figure GDA0002382548190000099
the corner equation of a camber beam is:
Figure GDA00023825481900000910
therefore, the boundary condition v $l＝0＝u|β＝0＝0，θ|l＝0＝ε|β＝0When the solution is 0, the solution is obtained: c1＝0，D1＝0；
C is to be1And D1Substitution into (17) and (18) can obtain:
Figure GDA00023825481900000911
Figure GDA00023825481900000912
in the formula, the maximum rotation angle and the maximum deflection of the horizontal arm 4-3 of the T-shaped curved arc part are generated at a position l-w-R, and the maximum deflection is the moving distance s of the horizontal arm 4-3 of the T-shaped curved arc part, so that:
Figure GDA00023825481900000913
orthodontic force F in the direction of the end of the filament2Namely the counterforce of the component force of the force required by the deformation of the corner theta (l) of the horizontal arm of the arc part generated by the deformation of the horizontal arm 4-3 of the T-shaped curved arc part along the direction of the spinning end, namely:
Figure GDA00023825481900000914
further, in the step 4), the static correcting force generated by the T-shaped curve 4 is a resultant force of restoring forces generated after the vertical arm 4-1 and the horizontal arm 4-3 of the circular arc part of the T-shaped curve are deformed, that is:
Figure GDA0002382548190000101
further, the method is applied to a waxy jaw dike for simulating tooth movement.
Furthermore, the wax jaw dike for simulating tooth movement suitable for the method consists of a base wax jaw dike 1, a resin tooth model 2, an orthodontic bracket 3 and a T-shaped curve 4.
Further, in the step 5), the measured tooth and the measuring element are made of resinThe cylinders are connected, and the movement of the teeth in the wax jaw levee is actually the movement of the cylinder connector in the wax jaw levee, so that the cylinder is used as a basic component for analysis; when the tooth moves in the wax jaw wall, the speed is vtWhen, vtThe flowing speed of the wax jaw levee at the time t is adopted, and acting force on the cylinder along the moving direction is streaming drag force; the friction drag force and the differential pressure drag force jointly form a streaming drag force; the friction drag force is that a boundary layer is formed on the surface of the cylinder due to the viscosity of the fluid, and in the boundary layer range, the fluid generates a velocity gradient, the friction effect is obvious, and friction shear stress is generated; the pressure difference drag force is that the boundary layer is separated at a certain point on the surface of the cylinder, and strong vortex wake is formed at the downstream of the separation point, namely at the rear part of the cylinder, so that pressure difference is generated between the front part and the rear part of the cylinder, and further a force is generated in the flow direction, and in the fluid flow, the vortex wake of the cylinder is R along with the Reynolds numbereOf teeth in a waxed jaw walleLess than 5, therefore, no vortex wake flow is generated, and no pressure difference drag force is generated;
drag force f on cylinder per unit lengthDCan be calculated using equation (27):
Figure GDA0002382548190000102
in the formula, v0For the tooth movement velocity component perpendicular to the cylinder axis unaffected by the streaming, ρ (t) is the density of the wax jaw wall at the test temperature at time t, a is the projected area of the cylinder per unit length perpendicular to the direction of movement, for a cylinder, a ═ 1 × D, D is the diameter of the cylinder, CDThe drag coefficient is a viscous effect intensively reflecting the viscosity of the fluid and the Reynolds number ReAnd cylinder surface roughness Ra(ii) related;
assuming the wax chin underlying fluid of this study is an incompressible ideal fluid, the volume of wax removal is
Figure GDA0002382548190000103
At a moving speed of the cylindervtA wax chin dam flow field of v (x, y, z, t); the influence of the cylinder on the wax jaw levee flow field is not considered for the moment, namely the pressure distribution in the wax jaw levee flow field is assumed not to be changed due to the existence of the cylinder, the boundary of the cylinder is taken as a part of the accelerating fluid boundary, namely the wax jaw levee fluid in the part of the volume replaced by the cylinder, which is supposed to exist in the wax jaw levee flow field in a static state, but actually due to the existence of the movement of the cylinder, the static wax jaw levee fluid is accelerated to the same state as the moving speed of the boundary of the cylinder; thus the accelerated wax jaw dam fluid will be at a volume of dewaxing
Figure GDA0002382548190000104
The cylinder acting in the flow direction with an inertial force FkInertial force FkIs equal to the dewaxing mass M of the cylinder0Volume and volume
Figure GDA0002382548190000105
Average acceleration of jaw dike fluid made of inner wax
Figure GDA0002382548190000106
The product of (a) and (b), namely:
Figure GDA0002382548190000107
for the cylinder under investigation,
Figure GDA0002382548190000108
the fluid acceleration at the center of the cylinder axis can be taken
Figure GDA0002382548190000109
To indicate that, at this time:
Figure GDA00023825481900001010
however, because the cylinder is in the flow field of the wax jaw dike, the fluid around the cylinder is ensuredThe point is disturbed to cause speed change, thereby changing the pressure distribution in the flow field of the wax jaw levee, so that the disturbance of the cylinder is the mass M of the part of the additional fluid which changes the original motion state around the cylinderwAn additional inertial force, i.e. an additional mass force, will also be generated on the cylinder in the direction of fluid flow; the streaming inertial force f of the accelerated fluid actually acting on the cylinder in the flow directionLCan be expressed as:
Figure GDA00023825481900001011
let Mw＝CmM0Then equation (30) can be expressed as:
Figure GDA0002382548190000111
in the formula, CmTo add a mass coefficient, CMThe mass coefficient is also called as an inertia force coefficient, and intensively reflects an additional mass effect caused by the change of the speed of a wax jaw dike flow field around the cylinder due to the inertia of the fluid and the existence of the cylinder;
through the analysis, the resistance condition of the teeth in the moving process of the wax jaw levee can be obtained, the teeth move in the wax jaw levee under the influence of the orthodontic force generated by the deformation of the orthodontic arch wire, and the teeth are subjected to the streaming inertia force f due to the streaming characteristic of the flow field in the moving processLAnd drag force fDThe influence of (a);
under the influence of heat exchange, the internal temperature of the wax jaw levee model in the thermal field changes along with time, and the change of the internal temperature causes the change of the density of the wax jaw levee model, so that the resistance of teeth moving in the wax jaw levee is influenced; the tooth model follows the rule in the viscous fluid energy equation when moving in the wax jaw levee, let e represent the internal energy of unit mass fluid, then ρ (t) e is the internal energy of unit volume fluid, ρ (t) vt 2The kinetic energy per unit volume is represented by/2, so that the total energy E ═ ρ (t) E + ρ (t) v contained in the fluid per unit volumet 2/2；
By simplifying the arrangement, the principle of conservation of energy can be approximately expressed as:
Figure GDA0002382548190000112
in the formula, cpA dimensionless pressure coefficient is adopted, phi is mechanical work consumed by the tooth model when moving in the wax jaw embankment fluid, k is a calculation coefficient, ▽ T is the temperature gradient of the wax-based fluid thermal field, and q is the heat flow density;
solving ▽ T, and setting the thickness of the wax jaw wall as 2 delta and the initial temperature as T0(ii) a It is placed at a temperature t at the initial instant∞In the fluid, the surface heat transfer coefficient h between the fluid and the wax jaw dike is constant, two sides of the wax jaw dike are symmetrically heated, and the internal temperature distribution of the wax jaw dike must take the central section as a symmetrical plane, so that only a half wax jaw dike with the thickness delta needs to be researched, the original point of an x axis is placed on the central section of the wax jaw dike, and for the half wax jaw dike with the x being more than or equal to 0, the following heat conduction differential equation can be listed:
Figure GDA0002382548190000113
where a is the thermal diffusivity, and the two sides of equation (33) are integrated with x, the following can be obtained:
Figure GDA0002382548190000114
the even heating of wax jaw dyke under the water bath environment can simplify to one-dimensional thermal field problem, consequently has:
Figure GDA0002382548190000115
the temperature gradient ▽ T of the wax jaw embankment fluid thermal field is substituted into a formula (32) to be finished to obtain:
Figure GDA0002382548190000116
the two sides of the equation of equation (36) are integrated and arranged for t to obtain:
Figure GDA0002382548190000117
wherein T is the temperature of the wax jaw wall fluid thermal field;
the fourier law, when expressed in terms of heat flow density q, has the following form:
Figure GDA0002382548190000118
wherein λ is a thermal conductivity coefficient;
substitution of equation (38) for equation (37) gives the expression that waxy jaw density ρ varies with time t:
Figure GDA0002382548190000119
the dynamic resistance model in the process of simulating tooth movement by the wax jaw dike can be expressed by the formula (40):
Figure GDA00023825481900001110
wherein f is dynamic resistance of the wax jaw dike in the process of simulating tooth movement.
Further, in the step 6), based on the T-shaped curve static correction force prediction model and the dynamic resistance model in the process of simulating tooth movement by the wax jaw levee, the T-shaped curve dynamic correction force prediction model can be expressed as:
Figure GDA0002382548190000121
in the formula, F is the dynamic correction force of the T-shaped curve obtained by applying the dynamic correction force prediction model of the T-shaped curve.
Further, when the wax jaw dike for simulating tooth movement is used for predicting the T-shaped curve correcting effect, the orthodontic bracket is firstly adhered to the outer surface of the resin tooth model, the T-shaped curve is fixed on the orthodontic bracket, at the moment, the wax jaw dike and the T-shaped curve are immersed into the environment of a constant-temperature water bath at 75 ℃ at the same time, the tooth is taken out after 2min, the tooth movement condition under the T-shaped curve effect can be clearly known by observing the tooth positions before and after the water bath, the dynamic correcting force prediction model of the T-shaped curve can generate the dynamic correcting force through the shape parameter calculation of the T-shaped curve, according to the tooth movement condition, the shape parameter of the T-shaped curve is calculated and adjusted through the model to obtain the optimal treatment effect, and further the orthodontic doctor is assisted to formulate a more reasonable correcting scheme.
The foregoing shows and describes the general principles and broad features of the present invention and advantages thereof. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are described in the specification and illustrated only to illustrate the principle of the present invention, but that various changes and modifications may be made therein without departing from the spirit and scope of the present invention, which fall within the scope of the invention as claimed. The scope of the invention is defined by the appended claims and equivalents thereof.

Claims (1)

1. A T-shaped curve dynamic correction force prediction model establishing method is characterized by comprising the following steps: the specific implementation process of the method comprises the following steps:
1) analyzing the structural characteristics and the loading characteristics of the T-shaped curve;
2) establishing a T-shaped curve vertical arm correcting force prediction model;
3) establishing a prediction model of the correcting force of the horizontal arm of the T-shaped curved arc part;
4) establishing a T-shaped curve static correction force prediction model;
5) establishing a dynamic resistance model in the process of simulating tooth movement by a wax jaw dike;
6) establishing a T-shaped curve dynamic correction force prediction model;
in the step 1), the structural characteristics of the T-shaped curve show that when the T-shaped curve corrects teeth, correcting force is released by a horizontal arm (4-4) of the T-shaped curve, and a vertical arm (4-1) and an arc part (4-2) of the T-shaped curve, which deform, are symmetrical to each other on two sides of the T-shaped curve, so that when a T-shaped curve correcting force model is built, only a symmetrical part needs to be analyzed; the bending radius of the T-shaped curved arc part (4-2) is R, the overall height is h, the closing gap is b, and the closing gap is generated when an arch wire is drawn clinically and applied with force;
after the deformation condition of the T-shaped curve in the correcting process is analyzed, the T-shaped curve deforms at the vertical arm (4-1) and the horizontal arm (4-3) of the arc part of the T-shaped curve, the correcting force generated by the T-shaped curve is composed of restoring forces generated by two deformation regions, and therefore the vertical arm (4-1) of the T-shaped curve and the horizontal arm (4-3) of the arc part of the T-shaped curve need to be subjected to mechanical analysis respectively, and the two are superposed to establish a prediction model of the correcting force of the T-shaped curve;
in the step 2), the rotation angle equation theta (x) and the deflection equation v (x) of the T-shaped vertical arm (4-1) can be expressed as follows:
Figure FDA0002382548180000011
where M (x) is the bending moment on the vertical part at a distance x, E is the modulus of elasticity of the material, IzIs the moment of inertia of the arch wire cross-section to the z-axis, for round wire Iz＝πd4D is the diameter of a round wire for a rectangular wire Iz＝c1c2 3/12，c2Is the length of the side parallel to the z-axis on the cross section of the rectangular filament, c1Is the length of the cross section of the rectangular wire perpendicular to the z-axis, C0And D0Is an integral constant, C0And D0The bending moment equation of the T-shaped vertical arm (4-1) is determined by boundary conditions as follows:
M(x)＝-P(y-x) (2)
wherein, P is the force required for generating the deformation of the corner theta (x) of the vertical arm, and y is the length of the vertical arm before the deformation;
by substituting equation (2) into equation (1) and integrating, it is possible to obtain:
Figure FDA0002382548180000012
to determine the integration constant in equation (3)C0、D0The method comprises the following steps that boundary conditions of a T-shaped curved vertical arm (4-1) need to be determined, deflection or a corner of the T-shaped curved vertical arm is always zero or known based on support conditions, x is 0, namely the deformation of the joint of the T-shaped curved vertical arm (4-1) and an arc part horizontal arm (4-3) is solved, and because the joint of the T-shaped curved vertical arm (4-1) and the arc part horizontal arm (4-3) is provided with a longitudinal symmetry plane and external force acts on the symmetry plane, the axis of a deformed curved beam is still located in the longitudinal symmetry plane and the deformation belongs to the plane bending deformation problem of the curved beam, the arc at the joint can be equivalent to the curved beam with the radian pi/4, and a section of radian d α infinitesimal is taken;
under the condition that the plane of the curved beam is bent, external force is in the longitudinal symmetrical plane of the curved beam, the axis after deformation is still the curve in the symmetrical plane, the curved beam does not have torsional deformation, the plane assumption can be still used at the moment, the differential equation of the bending line after the curved beam at the joint of the vertical arm (4-1) and the circular arc part horizontal arm (4-3) is deformed can be obtained as the formula (4), and the deformation of the curved beam can be determined by integrating the equation under the given boundary condition;
Figure FDA0002382548180000013
in the formula, u is the displacement of the cross section of the curved beam at the joint of the vertical arm (4-1) and the circular arc part horizontal arm (4-3) in the x direction, and the torque M borne by the joint0＝M|x＝0＝-Py，IωThe moment of inertia of the cross section of the curved beam at the joint to the omega axis is represented by I due to the fact that the T-shaped curved vertical arm (4-1) is consistent with the bending type of the curved beam at the jointω＝Iz；
From the arc length formula, ds is Rd α, so that the differential equation of the bending line after the bending beam is deformed at the joint of the vertical arm (4-1) and the circular arc part horizontal arm (4-3) can be changed into:
Figure FDA0002382548180000021
solving the non-homogeneous differential equation of the constant coefficient of the bending line after the bending beam at the joint is deformed to obtain:
Figure FDA0002382548180000022
because the curved beams at the joints of the vertical arms (4-1) and the circular arc part horizontal arms (4-3) are symmetrical along the longitudinal symmetry plane, the boundary conditions of the curved beams of the vertical arms (4-1) and the circular arc part horizontal arms (4-3) are as follows
Figure FDA0002382548180000023
Obtaining by solution:
Figure FDA0002382548180000024
the deflection equation of the curved beam at the joint can be expressed as:
Figure FDA0002382548180000025
the corner equation of a camber beam is:
Figure FDA0002382548180000026
therefore, there is a boundary condition
Figure FDA0002382548180000027
Obtaining by solution:
Figure FDA0002382548180000028
c is to be0，D0The following are available in the alternative (3):
Figure FDA0002382548180000029
because the maximum rotation angle and the maximum deflection are generated at the spinning end, namely x is equal to y, and the maximum deflection is the moving distance m of the T-shaped bent horizontal arm (4-4) at the spinning end, it can be known that:
Figure FDA00023825481800000210
the correcting force F generated by the deformation of the T-shaped bent vertical arm (4-1) according to the principle of the reaction force1I.e. the counterforce to produce the force required for the deformation:
Figure FDA00023825481800000211
in the step 3), the deformation conditions of the T-shaped curved arc part horizontal arm (4-3) and the T-shaped curved vertical arm (4-1) are the same, so that the same mechanical modeling method can be used, in the orthodontic treatment process, an orthodontist can pull the wire drawing end of the T-shaped curved part (4) in advance to cause the T-shaped curved part (4) to deform, the deformed T-shaped curved part is installed on the teeth of a patient, the teeth of the patient are pulled to move by restoring force generated after the T-shaped curved part (4) deforms, the purpose of correcting the deformed teeth of the patient is achieved, the bending deflection of the T-shaped curved arc part horizontal arm (4-3), namely the displacement of the T-shaped curved arc part horizontal arm (4-3) along the Z-axis direction, in the T-shaped curved deformation process, the position of the symmetrical center of the connecting part of the T-shaped curved arc part horizontal arm (4-3) and the T-shaped curved vertical arm (4-1) in space is constantly changed, therefore, in order to calculate the bending deflection s of the horizontal arm (4-3) of the T-shaped curved arc part, the motion condition of the symmetrical center of the connecting part in the deformation process of the T-shaped curve (4) needs to be simplified, the symmetrical centers of the connecting part before deformation are overlapped, and the difference value between the theoretical length of the vertical arm after deformation and the length y of the vertical arm before deformation is calculated, namely the bending deflection s of the horizontal arm of the T-shaped curved arc part is as follows:
Figure FDA00023825481800000212
the approximate differential equation of the bending line of the horizontal arm (4-3) at the part of the T-shaped curved arc can be obtained as follows:
Figure FDA00023825481800000213
wherein M (l) is the bending moment of a horizontal arm (4-3) of the T-shaped curved arc part;
the rotation angle equation theta (l) and the deflection equation v (l) of the horizontal arm (4-3) of the T-shaped curved arc part can be obtained by integrating the equation (13):
Figure FDA0002382548180000031
Figure FDA0002382548180000032
in the formula, C1And D1Is an integral constant determined by the boundary conditions, and the bending moment equation of the horizontal arm (4-3) of the T-shaped curved arc portion can be expressed by equation (16):
M(l)＝-G(w-R-l) (16)
in the formula, G is the correcting force generated by the horizontal arm of the arc part, and w is the length of the horizontal arm (4-3) of the T-shaped curved arc part;
by substituting equations (13) and (16) into equations (14) and (15) and integrating them, it is possible to obtain:
Figure FDA0002382548180000033
Figure FDA0002382548180000034
the solving of the boundary condition of the horizontal arm (3) of the T-shaped curved circular arc part is the same as that of the T-shaped curved vertical arm (4-1), the radian of the side circular arc on one side is pi/4, a section of radian is taken as d β infinitesimal, and a detailed process is given for the solving of the boundary condition of the vertical arm (4-1), so that a boundary condition formula obtained by solving an inhomogeneous equation of the constant coefficient of the deflection of the circular arc section is directly given:
Figure FDA0002382548180000035
due to the symmetry of the side arc of the T-shaped curve (4), the boundary condition of the side arc is
Figure FDA0002382548180000036
Obtaining by solution:
Figure FDA0002382548180000037
the deflection equation of the bent beam is:
Figure FDA0002382548180000038
the corner equation of a camber beam is:
Figure FDA0002382548180000039
therefore, the boundary condition v $l＝0＝u|β＝0＝0，θ|l＝0＝ε|β＝0When the solution is 0, the solution is obtained: c1＝0，D1＝0；
C is to be1And D1Substitution into (17) and (18) can obtain:
Figure FDA00023825481800000310
Figure FDA00023825481800000311
in the formula, the maximum rotation angle and the maximum deflection of the horizontal arm (4-3) of the T-shaped curved arc part are generated at a position l-w-R, the maximum deflection is the moving distance s of the T-shaped curved horizontal arm (4-3), and the following results are obtained:
Figure FDA00023825481800000312
orthodontic force F in the direction of the end of the filament2Namely the counterforce of the component force of the force required by the deformation of the corner theta (l) of the horizontal arm of the arc part generated by the deformation of the horizontal arm (4-3) of the T-shaped curved arc part along the direction of the spinning end, namely:
Figure FDA00023825481800000313
in the step 4), the static correction force generated by the T-shaped curve (4) is the resultant force of restoring forces generated after the vertical arm (4-1) and the horizontal arm (4-3) of the T-shaped curve arc part deform, namely:
Figure FDA0002382548180000041
in the step 5), the tooth to be measured is connected with the measuring original through a resin cylinder, the tooth moving in the wax jaw embankment is actually the movement of the cylinder connector in the wax jaw embankment, and therefore, the cylinder is used as a basic component for analysis;
when the tooth moves in the wax jaw wall, the speed is vtWhen, vtThe flowing speed of the wax jaw levee at the time t is adopted, and acting force on the cylinder along the moving direction is streaming drag force; the friction drag force and the differential pressure drag force jointly form a streaming drag force; the friction drag force is that a boundary layer is formed on the surface of the cylinder due to the viscosity of the fluid, and in the boundary layer range, the fluid generates a velocity gradient, the friction effect is obvious, and friction shear stress is generated; the pressure difference drag force is that the boundary layer is separated at a certain point on the surface of the cylinder, and strong vortex wake is formed at the downstream of the separation point, namely at the rear part of the cylinder, so that pressure difference is generated between the front part and the rear part of the cylinder, and further a force is generated in the flow direction, and in the fluid flow, the vortex wake of the cylinder is R along with the Reynolds numbereOf teeth in a waxed jaw walleLess than 5, therefore, no vortex wake flow is generated, and no pressure difference drag force is generated;
drag force f on cylinder per unit lengthDCan be calculated using equation (27):
Figure FDA0002382548180000042
in the formula, v0For the tooth velocity component perpendicular to the cylinder axis unaffected by the streaming, ρ (t) isthe density of the wax jaw levee at the experimental temperature at the time t, wherein A is the projection area of the cylinder with unit length perpendicular to the moving direction, and for the cylinder, A is 1 multiplied by D, D is the diameter of the cylinder, and C isDThe drag coefficient is a viscous effect intensively reflecting the viscosity of the fluid and the Reynolds number ReAnd cylinder surface roughness Ra(ii) related;
assuming the wax chin underlying fluid of this study is an incompressible ideal fluid, the volume of wax removal is
Figure FDA0002382548180000043
At a moving speed vtA wax chin dam flow field of v (x, y, z, t); the influence of the cylinder on the wax jaw levee flow field is not considered for the moment, namely the pressure distribution in the wax jaw levee flow field is assumed not to be changed due to the existence of the cylinder, the boundary of the cylinder is taken as a part of the accelerating fluid boundary, namely the wax jaw levee fluid in the part of the volume replaced by the cylinder, which is supposed to exist in the wax jaw levee flow field in a static state, but actually due to the existence of the movement of the cylinder, the static wax jaw levee fluid is accelerated to the same state as the moving speed of the boundary of the cylinder; thus the accelerated wax jaw dam fluid will be at a volume of dewaxing
Figure FDA0002382548180000044
The cylinder acting in the flow direction with an inertial force FkInertial force FkIs equal to the dewaxing mass M of the cylinder0Volume and volume
Figure FDA0002382548180000045
Average acceleration of jaw dike fluid made of inner wax
Figure FDA0002382548180000046
The product of (a) and (b), namely:
Figure FDA0002382548180000047
for the cylinder under investigation,
Figure FDA0002382548180000048
the fluid acceleration at the center of the cylinder axis can be taken
Figure FDA0002382548180000049
To indicate that, at this time:
Figure FDA00023825481800000410
however, because the cylinder exists in the wax jaw levee flow field, the fluid particles around the cylinder are disturbed to cause speed change, so that the pressure distribution in the wax jaw levee flow field is changed, and the disturbance of the cylinder is the mass M of the part of the additional fluid which changes the original motion state around the cylinderwAn additional inertial force, i.e. an additional mass force, will also be generated on the cylinder in the direction of fluid flow; the streaming inertial force f of the accelerated fluid actually acting on the cylinder in the flow directionLCan be expressed as:
Figure FDA00023825481800000411
let Mw＝CmM0Then equation (30) can be expressed as:
Figure FDA00023825481800000412
in the formula, CmTo add a mass coefficient, CMThe mass coefficient is also called as an inertia force coefficient, and intensively reflects an additional mass effect caused by the change of the speed of a wax jaw dike flow field around the cylinder due to the inertia of the fluid and the existence of the cylinder;
through the analysis, the resistance condition of the teeth in the moving process of the wax jaw levee can be obtained, and the teeth are under the influence of orthodontic force generated by the deformation of the orthodontic arch wireThe teeth are subject to a streaming inertia force f due to the streaming characteristic of a flow field in the moving process of the jaw dikeLAnd drag force fDThe influence of (a);
under the influence of heat exchange, the internal temperature of the wax jaw levee model in the thermal field changes along with time, and the change of the internal temperature causes the change of the density of the wax jaw levee model, so that the resistance of teeth moving in the wax jaw levee is influenced; the tooth model follows the rule in the viscous fluid energy equation when moving in the wax jaw levee, let e represent the internal energy of unit mass fluid, then ρ (t) e is the internal energy of unit volume fluid, ρ (t) vt 2The kinetic energy per unit volume is represented by/2, so that the total energy E ═ ρ (t) E + ρ (t) v contained in the fluid per unit volumet 2/2；
By simplifying the arrangement, the principle of conservation of energy can be approximately expressed as:
Figure FDA0002382548180000051
in the formula, cpIs a dimensionless pressure coefficient, phi is the mechanical work consumed by the tooth model when moving in the wax jaw levee fluid, k is a calculation coefficient,
Figure FDA0002382548180000052
the temperature gradient of the thermal field of the basic wax-holding fluid is shown, and q is the heat flow density;
to pair
Figure FDA0002382548180000053
Solving, the thickness of the wax jaw wall is set to be 2 delta, and the initial temperature is set to be t0(ii) a It is placed at a temperature t at the initial instant∞In the fluid, the surface heat transfer coefficient h between the fluid and the wax jaw dike is constant, two sides of the wax jaw dike are symmetrically heated, and the internal temperature distribution of the wax jaw dike must take the central section as a symmetrical plane, so that only a half wax jaw dike with the thickness delta needs to be researched, the original point of an x axis is placed on the central section of the wax jaw dike, and for the half wax jaw dike with the x being more than or equal to 0, the following heat conduction differential equation can be listed:
Figure FDA0002382548180000054
where a is the thermal diffusivity, and the two sides of equation (33) are integrated with x, the following can be obtained:
Figure FDA0002382548180000055
the even heating of wax jaw dyke under the water bath environment can simplify to one-dimensional thermal field problem, consequently has:
Figure FDA0002382548180000056
making wax to make the temperature gradient of jaw wall fluid thermal field
Figure FDA0002382548180000057
Substituted into formula (32) to obtain:
Figure FDA0002382548180000058
the two sides of the equation of equation (36) are integrated and arranged for t to obtain:
Figure FDA0002382548180000059
wherein T is the temperature of the wax jaw wall fluid thermal field;
the fourier law, when expressed in terms of heat flow density q, has the following form:
Figure FDA00023825481800000510
wherein λ is a thermal conductivity coefficient;
substitution of equation (38) for equation (37) gives the expression that waxy jaw density ρ varies with time t:
Figure FDA00023825481800000511
the dynamic resistance model in the process of simulating tooth movement by the wax jaw dike can be expressed by the formula (40):
Figure FDA00023825481800000512
wherein f is dynamic resistance of the wax jaw dike in the process of simulating tooth movement;
in the step 6), based on the T-shaped curve static correction force prediction model and the dynamic resistance model in the process of simulating tooth movement by the wax jaw levee, the T-shaped curve dynamic correction force prediction model can be expressed as:
Figure FDA00023825481800000513
in the formula, F is the dynamic correction force of the T-shaped curve obtained by applying the dynamic correction force prediction model of the T-shaped curve.
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