CN110569560A - Method for superposing stresses of mirror topology tangential loaded circular ring - Google Patents
Method for superposing stresses of mirror topology tangential loaded circular ring Download PDFInfo
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Abstract
The invention discloses a method for superposing mirror image topological tangential loaded ring stress, which comprises the following steps: establishing a static model of the circular ring under the action of single tangential concentrated force on the micro-segment of the circular ring; calculating a distribution function of the tangential stress and the radial stress of the ring under the action of single tangential concentrated force through a static model; and obtaining the distribution function of the tangential stress and the radial stress of the mirror topology tangential loaded ring by using an superposition method. Compared with the prior art, the method has the characteristics of innovation, high efficiency, accuracy, universality and the like. According to the method, the relationship between the distribution of the tangential concentration force and the stress distribution can be researched.
Description
Technical Field
the invention relates to the field of stress distribution of material mechanics, in particular to a method for superposing stresses of a mirror topology tangential loaded ring.
Background
as early as ancient China, the ring structure is used by workers, and a waterwheel, also called a crown block, is an ancient water lifting irrigation tool and an operation machine developed by the workers in ancient China by fully utilizing hydraulic resources. A water bucket is arranged around the waterwheel, and the water bucket takes water to irrigate the land in the rotating process of the waterwheel. The waterwheel can be simplified into an annular structure, and the gravity generated by taking water from the bucket can be regarded as tangential force applied to the annular structure. It is necessary to study the stress distribution of the ring by the tangential force applied to the ring, because the stress distribution can be used to select the material meeting the allowable stress condition to manufacture the waterwheel, which determines the water intake amount. Ancient workers mainly select materials by experience. In modern engineering applications, the main research objects are engaged gears, racks and gear rings. During the meshing process of the gears, a force which points to the tooth surface along the meshing line is generated between the teeth, the component of the force along the tangential direction of the pitch circle is a driving force applied to the driven wheel, and the magnitude of the driving force determines the magnitude of normal working load. Therefore, the stress distribution of the tangential loaded ring can be studied and can be guided in the design stage of the gear. The document (R.G. Parker.A. physical expansion for the efficiency of the play to the support of the play gear vibration. J.SoundVib,2000,236(4):561-573) investigated the effectiveness of the planetary phase at a certain harmonic in suppressing the vibration of the planet gears based on the forces acting on the sun and planet gears. Due to certain design objectives and assembly limitations, the space between the planet gears is sometimes uneven, the cyclic symmetry is lost, and the literature (j. lin and r. g. parker. structural vision devices of planar meshes with equal spacing of planar mirrors, j. sound Vib,2000,233(5):921-928) analyzes the free vibration with an unequally spaced planetary wheel system. At present, the stress distribution problem of a ring is not considered or simplified in the research of a plurality of gear dynamics, and due to the limitation of a ring stress distribution analysis technology, an analysis method of the ring stress distribution under the action of tangential concentrated force is particularly needed.
Disclosure of Invention
The invention provides a method for superposing stress of a mirror topology tangential loaded ring, which aims at the stress distribution problem of the tangential loaded ring, establishes a statics model on the micro-section of the ring by using a cross-section method, and calculates the stress distribution of the mirror topology tangential loaded ring by using an superposition method, so that the obtained result better meets the actual requirements of engineering, and the details are described as follows:
A method for stress superposition of a mirror topology tangential loaded ring, the method comprising the steps of:
establishing a static model of the circular ring under the action of single tangential concentrated force on the micro-segment of the circular ring;
Calculating a distribution function of the tangential stress and the radial stress of the ring under the action of single tangential concentrated force through a static model;
And obtaining the distribution function of the tangential stress and the radial stress of the mirror topology tangential loaded ring by using an superposition method.
The static model specifically comprises the following components:
where θ is the angle of a certain mass point on the ring, FefFor tangential concentrated forces, Fffto make a virtual holding force, FsfFor radially internal forces, FtfAs a tangential internal force, Mbmis a bending moment.
Further, the distribution function of the tangential stress and the radial stress of the ring under the action of the single tangential concentrated force is specifically as follows:
In the formula, Ftfθas tangential stress, FsfrFor radial stress, a ═ bh is the cross-sectional area of the ring, h is the radial thickness, and b is the axial thickness.
Further, the distribution function of the tangential stress and the radial stress of the mirror topology tangential loaded ring is specifically as follows:
wherein N is the number of tangential concentrated forces, thetai1,j1Is the ith1J in the group1the position angle of the tangential concentrated force.
the technical scheme provided by the invention has the beneficial effects that:
1. The invention utilizes a section method to establish a statics model on the micro-segment of the ring, and solves the stress distribution of the ring under the action of single tangential concentrated force;
2. the invention adopts an superposition method to solve the stress distribution of the mirror image topology tangential loaded ring;
3. Compared with the prior art, the method has the characteristics of innovation, high efficiency, accuracy, universality and the like. According to the method, the relationship between the distribution of the tangential concentration force and the stress distribution can be researched.
Drawings
FIG. 1 is a schematic view of the distribution of force on the whole ring and micro-segment under a single tangential concentrated force provided by the present invention;
Wherein, (a) is a distribution schematic diagram of force on the ring under the action of single tangential force; (b) is a schematic diagram of the distribution of force on the micro-segment of the circular ring under the action of single tangential force.
FIG. 2 shows the rotation θ provided by the present inventionkthe distribution diagram of the force on the whole ring and the micro-segment under the action of single tangential concentrated force of the angle;
FIG. 3 is a schematic diagram of the magnitude and distribution of tangential and radial stresses in a ring under a single tangential force provided by the present invention;
wherein (a) provides the inventionSchematic diagram of the tangential and radial stress distribution of the ring under a single tangential concentration force; (b) the invention provides a schematic diagram of the magnitude of tangential and radial stresses of a ring under the action of a single tangential concentrated force; (c) rotation theta provided for the inventionkThe tangential and radial stress distribution of the ring under the action of the single tangential concentrated force of the angle is shown schematically.
FIG. 4 is a schematic diagram of the force distribution of the mirror topology tangential loaded ring provided by the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, embodiments of the present invention are described in further detail below.
The embodiment of the invention provides a method which is strong in applicability and specially aims at stress superposition of a mirror topology tangential loaded ring. Firstly, establishing a static model of the ring under the action of a single tangential concentrated force on a micro-segment of the ring to obtain the distribution of each internal force of the loaded ring, solving the stress distribution of the ring under the action of the single tangential concentrated force according to the relation between the internal force and the stress in material mechanics, and then calculating the stress distribution of the mirror topology tangential loaded ring by adopting an overlay method. The method can also be used for solving the stress distribution of typical periodic structures such as a stator and a rotor of a rotating motor, an annular component in a micro device and the like.
the circular ring is acted by the tangential concentration force of the mirror topology; the superposition method of the stress distribution is basically characterized in that: the method adopts a superposition method to realize the stress distribution solution of the ring, and comprises the following specific steps:
(1) By utilizing a cross section method, a static model of the ring under the action of single tangential concentrated force is established on the micro-segment of the ring according to the force and moment balance principle:
where θ is the angle of a certain mass point on the ring, Feffor tangential concentrated forces, FffTo make a virtual holding force, FsfFor radially internal forces, Ftfas a tangential internal force, Mbmis a bending moment.
FIG. 1 is a graph showing the distribution of forces on the rings and micro-segments when a single tangential concentrated force is applied. As shown in fig. 1(a), the radius of the neutral circle of the ring is R, the radial thickness is h, the axial thickness is b, and the ring is subjected to a tangential force concentration F at θ ═ 0efacting in a downward direction. For research, a uniform virtual support is distributed around the circle, and the virtual support generates a virtual force Fffand the direction is upward, and the circular ring maintains balance under the action of a tangential concentrated force, a virtual torque and a uniformly distributed virtual force.
In order to research the stress distribution of the ring under a single tangential concentrated force, a micro-segment of d theta is cut at the position of the ring theta (theta epsilon (0,2 pi)), and the micro-segment is subjected to stress analysis by adopting a cross-section method, as shown in figure 1(b), O and O' are respectively the geometric center of the ring and the midpoint of the micro-segment, Fsf、Ftf、Mbmand TtmRadial internal force, tangential internal force, bending moment and torque are respectively.
(2) Since the study is of a circle in a micro-segment, d θ is a trace amount, and the concept of limit is utilized, a trigonometric function containing a trace amount can be simplified as follows:
(3) The formulas (4) to (7) are substituted into the formulas (1) to (3), and the formula can be simplified to obtain:
dMbm＝FsfRdθ (10)
(4) Solving radial internal force
The radially inner force F is obtained from the formulae (8) and (9)sfWith tangential concentrated force FefThe relationship of (1):
Equation (11) is a second-order non-homogeneous differential equation of the radially inner force, and a characteristic equation can be obtained from equation (11):
λsf 2+1＝0 (12)
in the formula, λsfis the eigenvalue of the characteristic equation, solved to obtain lambdasf1,2I is an imaginary unit.
to solve a general solution of a differential equation, let a particular solution of the differential equation be:
Fsf *＝θ(a1cosθ+b1sinθ)+a0 (13)
In the formula, a0、a1And b1Are all real numbers.
By substituting formula (13) for formula (11), the following can be obtained:
A is obtained by undetermined coefficient method0、a1And b1Respectively as follows:
a1＝0 (16)
Thus, the solution of the differential equation is:
From the eigenvalues obtained from the eigen equation and a particular solution of the differential equation, the general solution for the radial internal force can be obtained as:
In the formula, c1And c2are real numbers.
(5) Solving tangential internal force and bending moment
From formulae (8), (10) and (19):
(6) solving radial deformation
For the small-curvature ring, the relation between the bending moment and the radial deformation can be known from the knowledge of material mechanics:
where v is the radial deformation of the ring and E and I are the elastic modulus and moment of inertia of the ring, respectively.
From equations (10), (11) and (22), the fifth-order heterogeneous differential equation of radial deformation can be obtained:
from equation (23), the characteristic equation can be derived as:
λv 5+2λv 3+λv＝0 (24)
Solving to obtain λv1＝0,λv2,3＝±i,λv4,5i is an imaginary unit.
To solve a general solution of a differential equation, let a particular solution of the differential equation be:
v*＝θ2(a2cosθ+b2sinθ)+a0′θ (25)
In the formula, a0′、a2and b2Are all real numbers.
By substituting formula (25) for formula (23), the following can be obtained:
A is obtained by undetermined coefficient method0′、a2and b2Respectively as follows:
a2＝0 (28)
Thus, the solution of the differential equation is:
From the eigenvalues obtained from the eigen equation and a particular solution of the differential equation, the general solution for radial deformation can be found as:
In the formula, av1～av5Are all real numbers.
(7) Determining coefficients of general solutions using boundary conditions
The material mechanics knowledge shows that for the small-curvature ring, the corner on any sectionComprises the following steps:
Taking the section analysis of the position where the circular ring theta is 0, the tangential internal force is as follows:
from the formulae (20) and (33):
The rotation angle of the circular ring at the position where theta is 0 is zero, and the formula (31) shows that:
From the formulae (21), (22) and (31):
the radial deformation of the ring is 0 at θ ═ 0 and θ ═ pi, as can be seen from equation (31):
the small curvature ring is known from the assumption of no elongation, and the tangential deformation u of the ring at (0, pi) is 0, as is known from equation (31):
As can be seen from equations (34) and (36) - (38):
Therefore, when a single tangential concentrated force acts, the radial internal force, the tangential internal force, the bending moment and the radial deformation of the ring are respectively as follows:
from the relationship between internal force and stress in material mechanics and the equations (41) to (42), the tangential stress and the radial stress of the ring are respectively:
wherein A is the cross-sectional area of the ring.
(8) superposition of stresses
as shown in fig. 2, the ring has a length θ ═ θkIs subjected to a tangential concentrated force Fefthe action of (a) is that a virtual force balanced with the concentrated force is distributed around the circumference of the ring, and the tangential force and the virtual force are equal in magnitude and rotate in the direction of θ, respectively, as compared with fig. 1(a)kand (4) an angle.
FIG. 3(a) is a graph showing the distribution of tangential and radial stresses in a ring under a single tangential concentrated force, the magnitude of which is shown in FIG. 3 (b). FIG. 3(c) is a view showing a rotation θ from FIG. 3(a)kThe resulting distribution of tangential and radial stresses after the corner.
Since the period of the tangential stress and the radial stress of the ring is 2 pi under the action of a single tangential concentrated force, the Fourier series expansion of the equations (45) and (46) is:
In the formula (I), the compound is shown in the specification,andRespectively as follows:
This is obtained by simplifying formulae (49) to (51) and substituting formula (47):
this is obtained by simplifying formulae (52) to (54) and substituting formula (48):
Tangential concentrated force rotation thetakafter the angle, the distribution of tangential and radial stresses of the ring should also be rotated by the same angle, i.e.:
In the formula, thetakthe rotation angle when the first concentrated force is rotated to the k-th concentrated force.
As shown in FIG. 4, one circle of the ring is acted by N tangential concentrated forces, which are divided into 2 groups and arranged in a mirror topology, as shown in the graph Gi1(i11,2), each group is N/2, as shown in figure Li1,j1(i1＝1,2,j11, 2.. N/2). Thetai1,j1Described is the ith1J in the group1angle of position of individual concentrated force, theta2,j1＝θ1,j1+ π. By using the superposition method, when the ring is subjected to N tangential concentrated forces, the first tangential concentrated force is taken as 0, and the tangential stress and the radial stress of the ring are respectively as follows:
in summary, the embodiment of the present invention provides a method for superimposing stresses on a mirror topology tangential loaded ring. According to the method, a static model is established on the micro-section of the ring, the internal force of the loaded ring is obtained by adopting a cross-section method, the stress distribution of the loaded ring is obtained according to the relation between the internal force and the stress, the stress distribution of the mirror image topology loaded ring is obtained by utilizing an superposition method, the accuracy, the efficiency and the universality of the stress calculation of the ring are obviously improved, and the actual requirements of engineering are better met.
those skilled in the art will appreciate that the drawings are only schematic illustrations of preferred embodiments, and the above-described embodiments of the present invention are merely provided for description and do not represent the merits of the embodiments.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.
Claims (4)
1. a method for stress superposition of a mirror topology tangential loaded ring is characterized by comprising the following steps:
establishing a static model of the circular ring under the action of single tangential concentrated force on the micro-segment of the circular ring;
calculating a distribution function of the tangential stress and the radial stress of the ring under the action of single tangential concentrated force through a static model;
and obtaining the distribution function of the tangential stress and the radial stress of the mirror topology tangential loaded ring by using an superposition method.
2. The method for superposition of mirror topology tangential loaded ring stress according to claim 1, wherein the statics model specifically is:
Where θ is the angle of a certain mass point on the ring, FefFor tangential concentrated forces, Fffto make a virtual holding force, FsfFor radially internal forces, FtfAs a tangential internal force, MbmIs a bending moment.
3. the method for superimposing stress on a mirror topology tangential loaded ring according to claim 2, wherein the distribution function of the tangential stress and the radial stress of the ring under the action of the single tangential concentrated force is specifically:
In the formula, FtfθAs tangential stress, FsfrFor radial stress, a ═ bh is the cross-sectional area of the ring, h is the radial thickness, and b is the axial thickness.
4. the method for superimposing stress on the mirror topology tangential loaded ring according to claim 3, wherein the distribution function of the tangential stress and the radial stress of the mirror topology tangential loaded ring is specifically:
Wherein N is the number of tangential concentrated forces, thetai1,j1is the ith1J in the group1The position angle of the tangential concentrated force.
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
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| CN112270065A (en) * | 2020-08-26 | 2021-01-26 | 天津大学 | Dynamic stability prediction method for eccentric rotation annular periodic structure |
| CN112347591A (en) * | 2020-11-30 | 2021-02-09 | 天津大学 | Internal force analysis and free vibration modeling method for eccentric rotating ring structure |
| CN113051718A (en) * | 2021-03-07 | 2021-06-29 | 天津大学 | Static analysis method for packet topology radial loaded ring with extension hypothesis |
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Cited By (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112270065A (en) * | 2020-08-26 | 2021-01-26 | 天津大学 | Dynamic stability prediction method for eccentric rotation annular periodic structure |
| CN112270065B (en) * | 2020-08-26 | 2023-09-08 | 天津大学 | Dynamic stability prediction method for eccentric rotary annular periodic structure |
| CN112347591A (en) * | 2020-11-30 | 2021-02-09 | 天津大学 | Internal force analysis and free vibration modeling method for eccentric rotating ring structure |
| CN113051718A (en) * | 2021-03-07 | 2021-06-29 | 天津大学 | Static analysis method for packet topology radial loaded ring with extension hypothesis |
| CN113051718B (en) * | 2021-03-07 | 2023-01-03 | 天津大学 | Static analysis method for packet topology radial loaded ring with extension hypothesis |
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