CN110569560A - A Stress Superposition Method of Mirror Topology Tangentially Loaded Ring - Google Patents

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

A Stress Superposition Method of Mirror Topology Tangentially Loaded Ring

technical field

The invention relates to the field of mechanical stress distribution of materials, in particular to a method for stress superposition of a mirror image topology tangentially loaded circular ring.

Background technique

As early as in ancient China, the ring structure was used by the working people. The water wheel, also called the crane, is an ancient water-lifting irrigation tool. It is a kind of running machine developed by the ancient Chinese working people by making full use of water resources. . Water buckets are arranged around the waterwheel. During the rotation of the waterwheel, the water buckets draw water to irrigate the land. The water wheel can be simplified as a ring structure, and the gravity generated by the water bucket can be regarded as a tangential force exerted on the ring structure. It is necessary to study the tangential force applied on the ring to the stress distribution of the ring, because the material that satisfies the allowable stress conditions can be selected according to the stress distribution to manufacture the waterwheel, which determines the water intake. The ancient working people mainly relied on experience to select materials. In modern engineering applications, our main research objects are meshing gears, racks and ring gears. During the meshing process of the gears, a force will be generated between the teeth and directed to the tooth surface along the meshing line. The component force of this force along the tangential direction of the pitch circle is the driving force applied to the driven wheel. The magnitude of the driving force determines The size of a normal workload. Therefore, the study of the stress distribution of the tangentially loaded ring can be used as a guide in the design stage of the gear. The literature (R.G.Parker.A physical explanation for the effectiveness of planet phasing to suppress planetary gear vibration.J.SoundVib,2000,236(4):561-573) studies the planetary phase according to the force acting on the sun gear and planetary gear The effectiveness of the meshing frequency at certain harmonics to suppress planetary gear vibration. Due to certain design purposes and assembly constraints, the space between planetary gears is sometimes uneven, losing cyclic symmetry, literature (J.Lin and R.G.Parker.Structured vibration characteristics of planetary gears with unevenly spaced planets.J. Sound Vib, 2000, 233(5): 921-928) analyzed the free vibration of a planetary gear system with unequal distances. At present, many studies on gear dynamics do not consider or simplify the stress distribution of the ring. Due to the limitations of the analysis technology of the stress distribution of the ring, an analysis method for the stress distribution of the ring under the action of the tangential concentrated force is especially needed. .

Contents of the invention

The invention provides a method for stress superposition of a mirror-image topology tangentially loaded circular ring. The present invention aims at the stress distribution problem of a tangentially loaded circular ring. The method is used to calculate the stress distribution of the tangentially loaded ring in the mirror image topology, so that the obtained results can better meet the actual needs of the project. See the description below for details:

A method for superimposing stress of a mirrored topological tangentially loaded ring, said method comprising the following steps:

Establish a static model of the ring under the action of a single tangential concentrated force on the micro-segment of the ring;

Calculate the distribution function of the tangential stress and radial stress of the ring under the action of a single tangential concentrated force through the static model;

Using the superposition method, the distribution functions of the tangential stress and the radial stress of the mirror topology tangentially loaded ring are obtained.

Wherein, the static model is specifically:

In the formula, θ is the angle of a particle on the ring, F _ef is the tangential concentrated force, F _ff is the virtual support force, F _sf is the radial internal force, F _tf is the tangential internal force, and M _bm is the bending moment.

Further, the distribution function of the tangential stress and radial stress of the ring under the action of the single tangential concentrated force is specifically:

In the formula, F _tfθ is the tangential stress, F _sfr is the 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 mirrored topological tangentially loaded ring is specifically:

In the formula, N is the number of tangentially concentrated forces, θ _{i1, j1} is the position angle of the j ₁ tangentially concentrated force in the i _1th group.

The beneficial effects of the technical solution provided by the invention are:

1. The present invention utilizes the section method to set up a statics model on the micro-section of the ring to solve the stress distribution of the ring under the action of a single tangential concentrated force;

2. The present invention adopts the superposition method to solve the stress distribution of the mirror image topology tangentially loaded ring;

3. Compared with the existing methods, the present invention has the characteristics of innovation, high efficiency, accuracy and universality. According to this method, the relationship between the distribution of tangential concentrated force and the stress distribution can be studied.

Description of drawings

Fig. 1 is the distribution schematic diagram of whole ring and micro-section upper force under the action of single tangential concentrated force provided by the present invention;

Among them, (a) is a schematic diagram of the force distribution on the ring under the action of a single tangential force; (b) is a schematic diagram of the distribution of the force on the micro-segment of the ring under the action of a single tangential force.

Fig. 2 is the distribution schematic diagram of the force on the whole ring and the micro-section under the action of the single tangential concentrated force of the rotation θ _k angle provided by the present invention;

Fig. 3 is the schematic diagram of the tangential and radial stress size and distribution thereof of the ring under the action of a single tangential force provided by the present invention;

Wherein, (a) is the schematic diagram of the tangential and radial stress distribution of the ring under the action of a single tangential concentrated force provided by the present invention; and the schematic diagram of the radial stress magnitude; (c) is a schematic diagram of the tangential and radial stress distribution of the ring under the single tangential concentrated force of the rotation θ _k angle provided by the present invention.

Fig. 4 is a schematic diagram of the force distribution of the mirror image topology tangential to the loaded ring provided by the present invention.

Detailed ways

In order to make the purpose, technical solution and advantages of the present invention clearer, the implementation manners of the present invention will be further described in detail below.

The embodiment of the present invention proposes a highly applicable method for stress superposition of tangentially loaded circular rings in mirror topology. First, the static model of the ring under the action of a single tangential concentrated force is established on the micro-segment of the ring, and the distribution of internal forces of the loaded ring is obtained. The stress distribution under the action of concentrated force is calculated, and then the superposition method is used to calculate the stress distribution of the mirror-image topological tangentially loaded ring. This method can also be used to solve the stress distribution of typical periodic structures such as stator and rotor of rotating electrical machines and ring components in micro-devices.

The ring is affected by the tangential concentration force of the mirror image topology; the basic feature of the superposition method of the stress distribution is that the stress distribution of the ring is solved by using the superposition method, and the specific steps are as follows:

(1) Using the section method, the static model of the ring under the action of a single tangential concentrated force is established on the micro-segment of the ring according to the principle of force and moment balance:

In the formula, θ is the angle of a particle on the ring, F _ef is the tangential concentrated force, F _ff is the virtual support force, F _sf is the radial internal force, F _tf is the tangential internal force, and M _bm is the bending moment.

Figure 1 is the distribution diagram of the force on the ring and the micro-segment when a single tangential concentrated force acts. As shown in Figure 1(a), the radius of the neutral circle of the ring is R, the radial thickness is h, and the axial thickness is b. The ring is affected by a tangential force concentration F _ef at θ=0, and the direction down. For the convenience of research, uniform virtual supports are distributed around the circle of the ring, and the virtual supports generate virtual force F _ff , the direction is upward, and the ring is under the action of a tangential concentrated force, virtual torque and uniform virtual force maintain balance.

In order to study the stress distribution of the ring under a single tangential concentrated force, the micro-segment of dθ is intercepted at the ring θ(θ∈(0,2π)), and the force analysis of the micro-segment is carried out by using the section method, as shown in Figure 1 As shown in (b), O and O′ are the geometric center of the ring and the midpoint of the micro-segment, respectively, F _sf , F _tf , M _bm and T _tm are the radial internal force, tangential internal force, bending moment and torque, respectively .

(2) Since the micro-segment of the ring is studied, dθ is a micro-quantity, and the idea of limit is used, therefore, the trigonometric function containing a micro-quantity can be simplified as:

(3) Substituting formulas (4)-(7) into formulas (1)-(3) and simplifying can get:

dM _bm = F _sf Rdθ (10)

(4) Find the general solution of the radial internal force

The relationship between the radial internal force F _sf and the tangential concentrated force F _ef can be obtained from formulas (8) and (9):

Equation (11) is the second-order inhomogeneous differential equation of radial internal force, and the characteristic equation can be obtained from Equation (11):

λ _sf ² +1＝0 (12)

In the formula, λ _sf is the eigenvalue of the characteristic equation, and the solution is λ _sf1,2 =±i, where i is the imaginary unit.

In order to solve the general solution of the differential equation, let a special solution of the differential equation be:

F _sf ^* ＝θ(a ₁ cosθ+b ₁ sinθ)+a ₀ (13)

In the formula, a ₀ , a ₁ and b ₁ are all real numbers.

Substituting formula (13) into formula (11), it can be simplified to get:

According to the undetermined coefficient method, a ₀ , a ₁ and b ₁ can be obtained as:

a ₁ =0 (16)

Therefore, the particular solution of the differential equation is:

From the eigenvalue obtained from the characteristic equation and a special solution of the differential equation, the general solution of the radial internal force can be obtained as:

In the formula, c ₁ and c ₂ are real numbers.

(5) Find the general solution of tangential internal force and bending moment

From formulas (8), (10) and (19), we can get:

(6) Find the general solution of radial deformation

For small curvature rings, the relationship between bending moment and radial deformation is known from the knowledge of material mechanics:

In the formula, 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 inhomogeneous differential equation of radial deformation can be obtained:

From formula (23), the characteristic equation can be obtained as:

λ _v ⁵ +2λ _v ³ +λ _v ＝0 (24)

The solution is λ _v1 =0, λ _v2,3 =±i, λ _v4,5 =±i, i is the imaginary unit.

In order to solve the general solution of the differential equation, let a special solution of the differential equation be:

v ^* ＝θ ² (a ₂ cosθ+b ₂ sinθ)+a _0′ θ (25)

In the formula, a _0′ , a ₂ and b ₂ are all real numbers.

Substituting formula (25) into formula (23), simplification can get:

According to the undetermined coefficient method, a _0′ , a ₂ and b ₂ can be obtained as follows:

a ₂ =0 (28)

Therefore, the particular solution of the differential equation is:

From the eigenvalue obtained from the characteristic equation and a special solution of the differential equation, the general solution of the radial deformation can be obtained as:

In the formula, a _v1 ~a _v5 are all real numbers.

(7) Use the boundary conditions to find the coefficients of each general solution

According to the knowledge of material mechanics, for a ring with small curvature, the rotation angle on any section for:

Taking the section analysis of the ring at θ=0, the tangential internal force is:

From equations (20) and (33), we get:

The rotation angle of the ring at θ=0 is zero, and it can be known from formula (31):

From formulas (21), (22) and (31), it can be seen that:

At θ=0 and θ=π, the radial deformation of the ring is 0, as can be seen from formula (31):

For a small curvature ring, it can be known from the assumption of no extension, the tangential deformation of the ring on (0, π) is u = 0, and it can be known from formula (31):

From equations (34) and (36)-(38), it can be seen that:

Therefore, under the action of a single tangential concentrated force, the radial internal force, tangential internal force, bending moment and radial deformation of the ring are respectively:

From the relationship between internal force and stress in material mechanics and formulas (41)-(42), it can be known that the tangential stress and radial stress of the ring are respectively:

In the formula, A is the cross-sectional area of the ring.

(8) Superposition of stress

As shown in Figure 2, the ring is affected by a tangential concentrated force F _ef at θ=θ _k , and a virtual force that is balanced with the concentrated force is distributed around the ring. Compared with Figure 1(a), The magnitudes of the tangential force and the virtual force are respectively equal, and the directions are rotated by θ _k angle respectively.

Figure 3(a) is the distribution diagram of the tangential stress and radial stress of the ring under the action of a single tangential concentrated force, and its size is shown in Figure 3(b). Figure 3(c) is the distribution diagram of the tangential stress and radial stress obtained after rotating the angle θ _k in Figure 3(a).

Since the period of the tangential stress and radial stress of the ring is 2π under the action of a single tangential concentrated force, the Fourier series expansions of equations (45) and (46) are:

In the formula, and They are:

By simplifying formulas (49)-(51) and substituting them into formula (47), we can get:

By simplifying formulas (52)-(54) and substituting them into formula (48), we can get:

After the tangential concentrated force is rotated by θ _k angle, the distribution of tangential stress and radial stress of the ring should also be rotated by the same angle, namely:

In the formula, θ _k is the rotation angle when the first concentrated force rotates to the kth concentrated force.

As shown in Figure 4, the circle of the ring is affected by N tangentially concentrated forces, and these N tangentially concentrated forces are divided into two groups, and the arrangement form of mirror topology is adopted, as shown in Figure G _i1 (i ₁ =1, As shown in 2), each group has N/2 pieces, as shown in Figure L _i1,j1 (i ₁ =1,2,j ₁ =1,2,...N/2). θ _i1,j1 describes the position angle of the j _1st concentrated force in the i _1th group, θ _2,j1 = θ _1,j1 +π. Using the superposition method, when the ring is subjected to N tangential concentrated forces, assuming that the first tangential concentrated force acts at θ=0, the tangential stress and radial stress of the ring are respectively:

To sum up, the embodiment of the present invention provides a method for stress superposition of mirror topology tangentially loaded rings. This method establishes a static model on the micro-segment of the ring, uses the section method to obtain the internal force of the loaded ring, calculates the stress distribution of the loaded ring according to the relationship between the internal force and the stress, and uses the superposition method to obtain the mirror image topology. The stress distribution of the loaded ring significantly improves the accuracy, efficiency and universality of the stress calculation of the ring, and better meets the actual needs of the project.

Those skilled in the art can understand that the accompanying drawing is only a schematic diagram of a preferred embodiment, and the serial numbers of the above-mentioned embodiments of the present invention are for description only, and do not represent the advantages and disadvantages of the embodiments.

The above descriptions are only preferred embodiments of the present invention, and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention shall be included in the protection of the present invention. within range.

Claims (4)

1. a method for superposition of mirror image topology tangentially loaded circular stress, is characterized in that, described method comprises the following steps: Establish a static model of the ring under the action of a single tangential concentrated force on the micro-segment of the ring; Calculate the distribution function of the tangential stress and radial stress of the ring under the action of a single tangential concentrated force through the static model; Using the superposition method, the distribution functions of the tangential stress and the radial stress of the mirror topology tangentially loaded ring are obtained. 2. the method for a kind of mirror image topology tangential loaded ring stress superposition according to claim 1, is characterized in that, described statics model is specifically: In the formula, θ is the angle of a particle on the ring, F _ef is the tangential concentrated force, F _ff is the virtual support force, F _sf is the radial internal force, F _tf is the tangential internal force, and M _bm is the bending moment. 3. the method for a kind of mirror image topology tangentially loaded circular ring stress superposition according to claim 2, is characterized in that, the distribution function of the tangential stress of circular ring under the action of described single tangential concentrated force and radial stress Specifically: In the formula, F _tfθ is the tangential stress, F _sfr is the 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 the stress superposition of a kind of mirror image topology tangentially loaded circular ring according to claim 3, is characterized in that, the distribution function of the tangential stress of described mirror image topology tangentially loaded circular ring and radial stress is specific for: In the formula, N is the number of tangentially concentrated forces, θ _{i1, j1} is the position angle of the j ₁ tangentially concentrated force in the i _1th group.