CN110136830B - Method for establishing dynamic correction torque prediction model of auxiliary arch for depression - Google Patents
Method for establishing dynamic correction torque prediction model of auxiliary arch for depression Download PDFInfo
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Abstract
The invention discloses a method for establishing a dynamic correction torque prediction model of an auxiliary arch for depression, 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 auxiliary bow for lowering; 2) establishing an apparent arc correction torque equation of the side surface of the auxiliary arch for lowering; 3) establishing an apparent arc correction torque equation in front of the auxiliary arch for lowering; 4) establishing a dynamic resistance model in the process of simulating tooth movement by a wax jaw dike; 5) and establishing a dynamic correction torque prediction model of the auxiliary arch for depression. The invention can effectively carry out parametric expression on the dynamic correction torque of the auxiliary arch for lowering applied by a doctor, accurately predict the dynamic correction torque of the auxiliary arch for lowering 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
Technical Field
The invention relates to a method for establishing an auxiliary arch correcting moment prediction model for depression, belonging to the technical field of orthodontic treatment.
Background
The deep covering jaw is a specific form of malocclusion which is clinically common in oral cavity, and generally has great influence on occlusion relation of a patient, the auxiliary arch for depression is a common arch for high-angle deep covering jaw cases clinically, and the shape of the formed orthodontic arch wire is a determining factor influencing orthodontic moment. In the traditional diagnosis process, the sizes of the correction torque and the prediction of the correction effect generated by the orthodontic arch wire used in each correction stage are judged 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 torque lacks the quantitative standard, the treatment result completely depends on the level of the doctor, and the patient is easily injured and the treatment efficiency is reduced.
The base bracket wax has the physical characteristics of soft texture, good toughness, no stickiness to hands caused by heating and the like, a wax jaw dike can be manufactured based on tooth data of a patient in oral clinical application, the wax jaw dike is immersed into a water bath box to greatly shorten the orthodontic correction process, and different rigid tooth models and other rigid tooth models can observe the moving condition of teeth along with an orthodontic arch wire in the process of immersing the wax jaw dike into a thermostatic water bath environment, so that the simulation of the dynamic process of interaction between the teeth and periodontal tissues in the orthodontic correction process is realized, and the correction scheme is corrected.
Although the rationality of the correction scheme can be improved to a certain extent by applying the wax-made 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 an auxiliary arch dynamic correction moment prediction model for depression based on the wax-made jaw dike is established, and the auxiliary arch dynamic correction moment value for depression is parameterized and expressed, so that the method has very important significance for developing oral digital diagnosis and treatment and assisting the doctor to improve 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 dynamic correction torque prediction model of an auxiliary arch for depressing, which is used for carrying out parametric expression on the dynamic correction torque of the auxiliary arch for depressing the teeth.
The above purpose is mainly achieved through the following scheme:
the invention discloses a method for establishing an auxiliary bow correction torque prediction model for depression, 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 auxiliary bow for lowering;
2) establishing an apparent arc correction torque equation of the side surface of the auxiliary arch for lowering;
3) establishing an apparent arc correction torque equation in front of the auxiliary arch for lowering;
4) establishing a dynamic resistance model in the process of simulating tooth movement by a wax jaw dike;
5) and establishing a dynamic correction torque prediction model of the auxiliary arch for depression.
Preferably, said step 1) isAccording to the structural characteristics of the auxiliary arch for lowering, when the auxiliary arch for lowering is used for correcting teeth, correction torque is released from the side view circular arc and the front view circular arc, and the side view circular arcs of the auxiliary arch for lowering are mutually symmetrical on two sides of the arch wire, so that only the side view circular arc on one side is analyzed; the top of the side view arc has a gap of ltThe height of the side vertical arm is hdThe bottom is provided with a gap of lbThe orthodontic force is generated after the auxiliary arch for clinical installation and depression generates elastic deformation.
Preferably, in step 2), the lateral arc conforms to the mechanical characteristics of lateral buckling of the prismatic bar when loaded, in which case, when the lateral arc buckles, a reaction force s along the archwire will be generated0Defining the straight line of the bottom gap as x-axis, defining the left side of viewing arc facing front of the auxiliary bow for lowering, the top gap as upper and lower, the straight line passing through the bottom gap and perpendicular to the bottom gap as y-axis, and a vertical downward counter force F0The differential equation of the side view arc is as follows:
wherein E is the elastic modulus of the bending material used for the auxiliary arch for lowering, I is the inertia moment of the arch wire cross section to the central shaft of the orthodontic arch wire, and I is pi D for the round wire4D is the diameter of the round wire, and I is c for the rectangular wire1c2 3/12，c2Is the length of the side parallel to the z-axis on the cross section of the rectangular filament, c1The length of the vertical side of the rectangular wire section and the z axis is shown, and the general solution of the deflection curve differential equation of the side view arc is as follows:
in the formula, p1For determining the constant C, a calculation factor is introduced for solving the differential equation of the side view arc1And C2And unknown reaction forces s0The end point conditions are as follows:
substituting the y value of equation (2) into the endpoint condition yields:
the transcendental equation for calculating the critical load can be obtained by three expressions in equation (3):
tanp1lt＝p1lt(5)
solving the formula (5) to obtain p1ltMinimum value of p1ltWhen the reaction force s is 4.493, the reaction force s can be obtained0The expression of (a) is:
preferably, in the step 3), the urging means of the auxiliary bow for lowering is mainly a side arc, and the reaction force s generated after the side arc is changed is0The front arc deformation is caused, the incisor is further pushed, the aim of correcting the incisor and leveling the dental arch is further achieved, the length between the starting point and the stopping point of the front arc is L, the front arc is symmetrical on two sides of the auxiliary arch for pressing down, and therefore the front arc is subjected to two compression forces which are equal in size and opposite in direction, and the compression force is the counter force s generated after the side arc deformation0The correcting force acting on the teeth after the foresight circular arc is deformed is Q, and the deflection curve differential equation of the foresight circular arc is as follows:
defining the distance from the right end point of the anterior arc to the left end point of the tooth bracket No. 21 by FDI marking method, and using mark p2，
Two expressions in equation (7) can be rewritten using the expression in equation (8):
the deflection of the two ends of the auxiliary bow for lowering is zero, so that C is obtained31＝0，C33＝-C34tanp2L, obtaining the other two integral constants according to the continuous condition of the action point of the correcting force Q acted on the teeth, and obtaining the same deflection and the same slope by the two equations in the equation (7);
get it solved
Substituted into the formula (9) and differentiated to obtain,
the expression of the corrective force Q acting on the teeth obtained by the finishing is:
will be provided with
In formula (11), the correcting force Q acting on the tooth is expressed as:
preferably, the method is applied to a wax jaw wall for simulating tooth movement.
Preferably, the wax jaw wall for simulating tooth movement, which is applied to the method, is composed of an auxiliary arch for depressing, a resin tooth model, an orthodontic bracket and a base wax jaw wall.
Preferably, in the step 4), 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 moving in a waxed jaw wall, Reynolds number ReLess 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 (13):
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 × D1，D1Is the diameter of a 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;
suppose that the jaw wall fluid made of wax in this study is notCompressible ideal fluid, wax-removing volume of
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
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
Average acceleration of jaw dike fluid made of inner wax
The product of (a) and (b), namely:
for the cylinder under investigation,the fluid acceleration at the center of the cylinder axis can be taken
To indicate that, at this time:
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:
let Mw＝CmM0Then equation (16) can be expressed as:
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 law in the viscous fluid energy equation when moving inside the wax jaw levee, and let e represent the internal energy of the fluid per unit massρ e is the internal energy per unit volume of fluid, ρ vt 2The term,/2 denotes the kinetic energy per unit volume, so that the total energy E ═ pe + pv per unit volume of fluidt 2/2；
By simplifying the arrangement, the principle of conservation of energy can be approximately expressed as:
in the formula, cpIs a dimensionless pressure coefficient, Φ is the mechanical work consumed by the tooth model when moving in the wax jaw dam fluid, k is a calculation coefficient, ▽ Τ is the temperature gradient of a base wax-holding fluid thermal field, q is the heat flow density;
solving ▽ gamma, setting the thickness of the wax jaw levee to be 2 delta and the initial temperature 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:
where a is the thermal diffusivity and the two sides of equation (19) are integrated over x, we can obtain:
the even heating of wax jaw dyke under the water bath environment can simplify to one-dimensional thermal field problem, consequently has:
the temperature gradient ▽ T of the wax jaw levee fluid thermal field is collated in formula (18) to obtain:
the two sides of the equation of equation (22) are integrated and arranged for t to obtain:
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:
wherein λ is a thermal conductivity coefficient;
the expression for waxy jaw density ρ as a function of time t can be obtained by substituting formula (24) into formula (23):
the dynamic resistance model in the process of simulating tooth movement by the wax jaw dike can be expressed by the formula (26):
wherein f is dynamic resistance of the wax jaw dike in the process of simulating tooth movement.
Preferably, in the step 5), the dynamic reduction auxiliary arch correcting moment M under the correcting force Q acting on the teeth is:
in the formula I1The distance between the point of action of the orthodontic force Q applied to the teeth and the center of rotation O.
The invention has the beneficial effects that:
1. by adopting a parameterized modeling method, the influence effect of each influence factor on the correction torque of the auxiliary arch for lowering can be reflected more visually, and a doctor can adjust the bent auxiliary arch for lowering conveniently to obtain a proper correction torque.
2. By analyzing the action process of clinically depressing the incisors by the auxiliary arch, the auxiliary arch for depressing is divided into a front view arc and a side view arc when the correction torque prediction model is established, and the accuracy of establishing the model is improved by a sub-module modeling method.
3. The method is suitable for the jaw dike made of the base wax, 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 the force of the orthodontic arch wire on the correcting force in the real orthodontic process, so that the correcting 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 correcting moment 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 present invention for establishing a dynamic moment for auxiliary arch correction for lowering;
FIG. 2 is a schematic view of a side view arc structure of the auxiliary bow for lowering in accordance with the present invention;
FIG. 3 is a front view of the arc viewing structure of the auxiliary bow for lowering in accordance with the present invention;
FIG. 4 is a schematic top view of the front view of the arc structure of the auxiliary bow for lowering in accordance with the present invention;
FIG. 5 is a schematic view of the dynamic moment of the auxiliary arch for lowering in accordance with the present invention.
In the figure: 1-pressing down with auxiliary bow; 1-1-forward view arc; 1-2-arc of side view; 1-3-opening the top; 1-4-bottom gapping; 2-resin tooth model; 3-orthodontic bracket; 4-base wax-holding jaw wall.
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, and fig. 5, the following technical solutions are adopted in the specific embodiments: the invention discloses a method for establishing a dynamic correction torque prediction model of an auxiliary arch for depression, 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 auxiliary bow for lowering;
2) establishing an apparent arc correction torque equation of the side surface of the auxiliary arch for lowering;
3) establishing an apparent arc correction torque equation in front of the auxiliary arch for lowering;
4) establishing a dynamic resistance model in the process of simulating tooth movement by a wax jaw dike;
5) and establishing a dynamic correction torque prediction model of the auxiliary arch for depression.
Furthermore, in the step 1), it can be known from the structural characteristics of the auxiliary arch 1 for lowering, when the auxiliary arch 1 for lowering corrects the teeth, the correcting moment is released from the side view arc 1-2 and the front view arc 1-1, and the side view arcs 1-2 of the auxiliary arch 1 for lowering are symmetrical to each other on both sides of the arch wire, so that only the side view arc 1-2 on one side is analyzed; the top of the side view arc has a gap of ltThe height of the side vertical arm is hdThe bottom is provided with a gap of lbThe orthodontic force is generated after the auxiliary arch 1 for clinical installation and depression generates elastic deformation.
Further, in the step 2), the side view arc 1-2 conforms to the mechanical property of the prism rod transversely buckling when being loaded, in this case, the side view arc 1-2 generates a counter force s along the arch wire when being buckled0Defining the straight line where the bottom gap 1-4 is located as an x-axis, defining the straight line which passes through the bottom gap 1-4 and is perpendicular to the bottom gap 1-4 as a y-axis when the arc 1-1 facing the front of the auxiliary bow 1 for pressing down is on the left side, the top gap 1-3 is on the upper side, and the bottom gap 1-4 is on the lower side, and a vertical downward counter force F0The differential equation of the side view arc is as follows:
wherein E is the elastic modulus of the bending material used for the auxiliary arch for lowering, I is the inertia moment of the arch wire cross section to the central shaft of the orthodontic arch wire, and I is pi D for the round wire4D is the diameter of the round wire, and I is c for the rectangular wire1c2 3/12，c2Is the length of the side parallel to the z-axis on the cross section of the rectangular filament, c1The length of the vertical side of the rectangular wire section and the z axis is shown, and the general solution of the deflection curve differential equation of the side view arc is as follows:
in the formula, p1For determining the constant C, a calculation factor is introduced for solving the differential equation of the side view arc1And C2And unknown reaction forces s0The end point conditions are as follows:
substituting the y value of equation (2) into the endpoint condition yields:
the transcendental equation for calculating the critical load can be obtained by three expressions in equation (3):
tanp1lt＝p1lt(5)
solving the formula (5) to obtain p1ltMinimum value of p1ltWhen the reaction force s is 4.493, the reaction force s can be obtained0The expression of (a) is:
further, the steps described3) The force applying unit of the auxiliary bow 1 for pressing down is mainly a reaction force s generated after the side view arc 1-2 is deformed0The front arc 1-1 is deformed to push the incisor, thereby achieving the purpose of correcting and leveling the dental arch of the incisor, the length between the starting point and the stopping point of the front arc 1-1 is L, the front arc 1-1 is symmetrical at two sides of the auxiliary arch 1 for depressing, therefore, the front arc 1-1 is subjected to two compression forces with equal magnitude and opposite directions, the compression force is the counter force s generated after the side arc 1-2 is deformed0The correcting force acting on the teeth after the foresight arc 1-1 deforms is Q, and the deflection curve differential equation of the foresight arc 1-1 is as follows:
the definition takes the front view direction as a reference, the right end point to the left end point of the front view circular arc 1-1 is the positive direction of an x axis, the right end point of the front view circular arc 1-1 is defined to be vertical to the upward direction of the front view circular arc 1-1 is defined to be the positive direction of a y axis, the value c is the distance between the right end point of the front view circular arc and the left end point of No. 21 tooth bracket of the FDI marking method, and the mark p is used2，
Two expressions in equation (7) can be rewritten using the expression in equation (8):
since the deflection of the two ends of the auxiliary bow 1 for lowering is zero, C is obtained31＝0，C33＝-C34tanp2L, obtaining the other two integral constants according to the continuous condition of the action point of the correcting force Q acted on the teeth, and obtaining the same deflection and the same slope by the two equations in the equation (7);
get it solved
Substituted in formula (9)And the differential is obtained, so as to obtain,
the expression of the corrective force Q acting on the teeth obtained by the finishing is:
further, in the formula
In formula (11), the force applied to the teeth, i.e., Q, can be expressed by the following formula,
further, the method is applied to a wax jaw dike for simulating tooth movement.
Furthermore, the wax jaw dike for simulating tooth movement, which is suitable for the method, consists of an auxiliary arch 1 for depression, a resin tooth model 2, an orthodontic bracket 3 and a base wax jaw dike 4.
Further, in the step 4), 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 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 caused by the separation of the boundary layer 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 the front part and the rear part of the cylinder generateThe pressure difference, in turn, generates a force in the direction of flow, while in fluid flow the cylinder vortex wake is R with Reynolds numbereOf teeth moving in a waxed jaw wall, Reynolds number ReLess 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 (13):
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 × D1，D1Is the diameter of a 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
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 dewaxingCylinder (A) ofThe fluid acting in the direction of flow with an inertial force FkInertial force FkIs equal to the dewaxing mass M of the cylinder0Volume and volume
Average acceleration of jaw dike fluid made of inner wax
The product of (a) and (b), namely:
for the cylinder under investigation,
the fluid acceleration at the center of the cylinder axis can be taken
To indicate that, at this time:
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:
let Mw＝CmM0Then equation (16) can be expressed as:
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 rho e is the internal energy of unit volume fluid, rho vt 2The term,/2 denotes the kinetic energy per unit volume, so that the total energy E ═ pe + pv per unit volume of fluidt 2/2；
By simplifying the arrangement, the principle of conservation of energy can be approximately expressed as:
in the formula, cpIs a dimensionless pressure coefficient, Φ is the mechanical work consumed by the tooth model when moving in the wax jaw dam fluid, k is a calculation coefficient, ▽ Τ is the temperature gradient of a base wax-holding fluid thermal field, q is the heat flow density;
solving ▽ gamma, setting the thickness of the wax jaw levee to be 2 delta and the initial temperature to be t0(ii) a It is placed at a temperature t at the initial instant∞In the fluid, the surface between the fluid and the wax jaw wallThe heat transfer coefficient h is constant, two sides of the wax jaw dike are symmetrically heated, and the temperature distribution in 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 the x axis is arranged 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:
where a is the thermal diffusivity and the two sides of equation (19) are integrated over x, we can obtain:
the even heating of wax jaw dyke under the water bath environment can simplify to one-dimensional thermal field problem, consequently has:
the temperature gradient ▽ T of the wax jaw levee fluid thermal field is collated in formula (18) to obtain:
the two sides of the equation of equation (22) are integrated and arranged for t to obtain:
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:
wherein λ is a thermal conductivity coefficient;
the expression for waxy jaw density ρ as a function of time t can be obtained by substituting formula (24) into formula (23):
the dynamic resistance model in the process of simulating tooth movement by the wax jaw dike can be expressed by the formula (26):
wherein f is dynamic resistance of the wax jaw dike in the process of simulating tooth movement.
Further, in the step 5), the dynamic moment M for the auxiliary arch for depressing under the action of the correcting force Q acting on the teeth is:
in the formula I1The distance between the point of action of the orthodontic force Q applied to the teeth and the center of rotation O.
Further, when the wax jaw dike for simulating tooth movement is applied to predict the correction effect of the auxiliary arch for depressing, firstly, the orthodontic bracket is adhered to the outer surface of the resin tooth model, the auxiliary arch for depressing is fixed on the orthodontic bracket, at the moment, the wax jaw dike and the auxiliary arch for depressing are immersed into the environment of a constant-temperature water bath at 75 ℃ at the same time, the orthodontic bracket is taken out after 2min, the tooth positions before and after the water bath are observed, the tooth movement condition under the action of the auxiliary arch for depressing can be clearly known, the dynamic correction moment prediction model for depressing auxiliary arch, which is provided by the method, can generate the dynamic correction moment through the calculation of the shape parameters of the auxiliary arch for depressing, according to the tooth movement condition, the optimal treatment effect is obtained through the calculation and adjustment of the shape parameters of the auxiliary arch for depressing by the model, and further, the more reasonable correction scheme is formulated by an orthodontist.
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 method for establishing a dynamic correction torque prediction model of an auxiliary arch for depression 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 auxiliary bow for lowering;
2) establishing an apparent arc correction torque equation of the side surface of the auxiliary arch for lowering;
3) establishing an apparent arc correction torque equation in front of the auxiliary arch for lowering;
4) establishing a dynamic resistance model in the process of simulating tooth movement by a wax jaw dike;
5) establishing a dynamic correction torque prediction model of an auxiliary arch for lowering;
in the step 1), the structural characteristics of the auxiliary arch for lowering (1) show that when the auxiliary arch for lowering (1) corrects teeth, correction torque is released by a side view arc (1-2) and a front view arc (1-1), and the side view arcs (1-2) of the auxiliary arch for lowering (1) are mutually symmetrical at two sides of the arch wire, so that only the side view arc (1-2) at one side is analyzed; the top of the side view arc has a gap of ltThe height of the side vertical arm is hdThe bottom is provided with a gap of lbThe orthodontic force is generated after the auxiliary arch (1) for clinical installation and depression generates elastic deformation;
in the step 2), the side arc (1-2) conforms to the mechanical property of the lateral buckling of the prism rod when being loaded, and in this case, a reaction force s along the arch wire is generated when the side arc (1-2) buckles0Defining a straight line where the bottom gap (1-4) is positioned as an x-axis, defining a straight line which passes through the bottom gap (1-4) and is perpendicular to the bottom gap (1-4) as a y-axis when the arc (1-1) facing the front of the auxiliary bow (1) for pressing down is on the left side, the top gap (1-3) is on the upper side and the bottom gap (1-4) is on the lower side, and a vertical downward counter force F0The differential equation of the side view arc is as follows:
wherein E is the elastic modulus of the bending material used for the auxiliary arch for lowering, I is the inertia moment of the arch wire cross section to the central shaft of the orthodontic arch wire, and I is pi D for the round wire4D is the diameter of the round wire, and I is c for the rectangular wire1c2 3/12，c2Is the length of the side parallel to the z-axis on the cross section of the rectangular filament, c1The length of the vertical side of the rectangular wire section and the z axis is shown, and the general solution of the deflection curve differential equation of the side view arc is as follows:
in the formula, p1For determining the constant C, a calculation factor is introduced for solving the differential equation of the side view arc1And C2And unknown reaction forces s0The end point conditions are as follows:
substituting the y value of equation (2) into the endpoint condition yields:
the transcendental equation for calculating the critical load can be obtained by three expressions in equation (3):
tan p1lt＝p1lt(5)
solving the formula (5) to obtain p1ltMinimum value of p1ltWhen the reaction force s is 4.493, the reaction force s can be obtained0The expression of (a) is:
in the step 3), the force applying unit of the auxiliary bow (1) for pressing down is mainly a side view arc (1-2), and the reaction force s generated after the side view arc (1-2) is deformed0The front observation arc (1-1) is caused to deform to push the incisor, so that the aim of correcting the incisor and leveling the dental arch is achieved, the length between the starting point and the stopping point of the front observation arc (1-1) is L, the front observation arc (1-1) is symmetrical on two sides of the auxiliary arch (1) for depression, so that the front observation arc (1-1) is subjected to two compression forces which are equal in size and opposite in direction, and the compression forces are counter forces s generated after the side observation arc (1-2) deforms0The correcting force acting on the teeth after the foresight arc (1-1) deforms is Q, and the deflection curve differential equation of the foresight arc (1-1) is as follows:
the definition takes the front viewing direction as a reference, the right end point to the left end point of the front viewing arc (1-1) is the positive direction of an x axis, the right end point of the front viewing arc (1-1) is defined to be vertical to the upward direction of the front viewing arc (1-1) is defined to be the positive direction of a y axis, the value c is the distance between the right end point of the front viewing arc and the left end point of the No. 21 tooth bracket of the FDI marking method, and the mark p is used2，
Two expressions in equation (7) can be rewritten using the expression in equation (8):
the deflection of the two ends of the auxiliary bow (1) for lowering is zero, so that C is obtained31＝0，C33＝-C34tan p2L, obtaining the other two integral constants according to the continuous condition of the action point of the correcting force Q acted on the teeth, and obtaining the same deflection and the same slope by the two equations in the equation (7);
get it solved
Substituted into the formula (9) and differentiated to obtain,
the expression of the corrective force Q acting on the teeth obtained by the finishing is:
will be provided withIn formula (11), the correcting force Q acting on the tooth is expressed as:
in the step 4), the tooth to be measured is connected with the measuring original through the 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 moving in a waxed jaw wall, Reynolds number ReLess 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 (13):
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 × D1，D1Is the diameter of a 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
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
The cylinder acting in the flow direction with an inertial force FkInertial force FkIs equal to the cylinderMass M of volume dewaxing0Volume and volume
Average acceleration of jaw dike fluid made of inner waxThe product of (a) and (b), namely:
for the cylinder under investigation,
the fluid acceleration at the center of the cylinder axis can be takenTo indicate that, at this time:
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:
let Mw＝CmM0Then equation (16) can be expressed as:
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 rho e is the internal energy of unit volume fluid, rho vt 2The term,/2 denotes the kinetic energy per unit volume, so that the total energy E ═ pe + pv per unit volume of fluidt 2/2；
By simplifying the arrangement, the principle of conservation of energy can be approximately expressed as:
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,
the temperature gradient of the thermal field of the basic wax-holding fluid is shown, and q is the heat flow density;
to pair
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:
where a is the thermal diffusivity and the two sides of equation (19) are integrated over x, we can obtain:
the even heating of wax jaw dyke under the water bath environment can simplify to one-dimensional thermal field problem, consequently has:
making wax to make the temperature gradient of jaw wall fluid thermal field
Substituted into formula (18) to obtain:
the two sides of the equation of equation (22) are integrated and arranged for t to obtain:
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:
wherein λ is a thermal conductivity coefficient;
the expression for waxy jaw density ρ as a function of time t can be obtained by substituting formula (24) into formula (23):
the dynamic resistance model in the process of simulating tooth movement by the wax jaw dike can be expressed by the formula (26):
wherein f is dynamic resistance of the wax jaw dike in the process of simulating tooth movement;
in the step 5), the dynamic correction moment M of the auxiliary arch for depressing under the action of the correction force Q acting on the teeth is as follows:
in the formula I1The distance between the point of action of the orthodontic force Q applied to the teeth and the center of rotation O.
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| CN106580509A (en) * | 2016-12-02 | 2017-04-26 | 浙江工业大学 | Measurement method and measurement device capable of simulating quasi-dynamic tooth correcting force at any time of correcting |
| CN108066034A (en) * | 2016-11-10 | 2018-05-25 | 无锡时代天使医疗器械科技有限公司 | tooth stress measuring device and method |
| CN108520136A (en) * | 2018-04-02 | 2018-09-11 | 哈尔滨理工大学 | A kind of open vertical loop orthodontic force prediction model method for building up |
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| CN109712703A (en) * | 2018-12-12 | 2019-05-03 | 上海牙典软件科技有限公司 | A kind of correction prediction technique and device based on machine learning |
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| CN106580509A (en) * | 2016-12-02 | 2017-04-26 | 浙江工业大学 | Measurement method and measurement device capable of simulating quasi-dynamic tooth correcting force at any time of correcting |
| CN108520136A (en) * | 2018-04-02 | 2018-09-11 | 哈尔滨理工大学 | A kind of open vertical loop orthodontic force prediction model method for building up |
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