CN103473452B - A kind of method of the deformed shape predicting stratosphere balloon and capsule cloth stress - Google Patents

Abstract

A kind of method of deformed shape predicting stratosphere balloon and capsule cloth stress, the method has five big steps: step one, according to the overall stress balance equation of balloon, sets up the relation between the meridional stress of capsule cloth and broadwise stress and deformed shape function；Step 2, foundation describe the geometric model of deformed shape, provide the expression of deformed shape function；Step 3, set up total potential-energy function of balloon system；Step 4, according to minimum potential energy principal, solve the undetermined parameter in deformed shape function expression；Step 5, by it has been determined that undetermined parameter substitute in deformed shape function expression, namely determine the deformed shape of balloon, and then determine meridional stress and the broadwise stress of capsule cloth.The present invention is simple and practical, it is thus only necessary to geometric parameter during by the capsule cloth material parameter of balloon, balloon not by load, internal gas parameter and load are updated in model, the deformed shape being just readily available in stratosphere balloon and capsule cloth stress.

Description

A kind of method of the deformed shape predicting stratosphere balloon and capsule cloth stress

Technical field

The present invention provides a kind of method of deformed shape predicting stratosphere balloon and capsule cloth stress, belongs to mechanical analysis and design field.

Background technology

Balloon is a kind of low cost, high efficiency airborne vehicle, and its prevailing lift comes from the buoyancy of self, and therefore, only need to carry little fuel just can perform aerial mission, transport old cheap, there is boundless application prospect.One of the mechanical property and the inefficacy mechanism focus becoming Recent study of prediction balloon.The accurate deformed shape assessing balloon and capsule cloth stress are for extremely important in the design phase, and at present, engineering staff solves this problem mainly through test, numerical simulation and analytic method.Tested by the balloon flight of scale model or full-size(d), although its deformed shape and capsule cloth stress can be recorded, but flight test cost is of a relatively high and the cycle is long, especially needed being pointed out that is accomplished by deformed shape and capsule these data of cloth stress in its design phase, and flight test below is only comparable to checking and correction that method for designing is carried out；Method for numerical simulation needs to set up complicated FEM (finite element) model, calculates complexity, and computational efficiency is low, it has not been convenient to engineers is applied；And also compare the analytic method lacking deformed shape and capsule cloth stress aspect for assessing balloon at present, therefore, the present invention proposes the new method of a kind of deformed shape predicting stratosphere balloon and capsule cloth stress, the method is very simple and practical, geometric parameter when needing only to the capsule cloth material parameter of balloon, balloon not by load, internal gas parameter and load are updated in model, just can be readily available deformed shape and the capsule cloth stress of balloon, it is seen that the present invention has Important Academic meaning and engineer applied is worth.

Summary of the invention

A kind of method that the invention provides deformed shape predicting stratosphere balloon and capsule cloth stress, the method has calculating simplicity, precision advantages of higher, and its technical scheme is as follows:

The method of a kind of deformed shape predicting stratosphere balloon of the present invention and capsule cloth stress, the method specifically comprises the following steps that

Step one, according to the overall stress balance equation of balloon, set up the relation between the meridional stress of capsule cloth and broadwise stress and deformed shape function.

When balloon bearing load, being shaped like droplet-shaped (as shown in Figure 1), in FIG, circular dashed line represents the original-shape of vertical cross-section during non-bearing load, and water-drop-shaped solid line represents the shape of vertical cross-section during bearing load.Balloon bearing load also results in there is fold (as shown in Figure 2), in fig. 2, outer dashed line represents the original-shape of level cross-sectionn during non-bearing load, internal solid line represents the shape of level cross-sectionn during bearing load, and inner dotted line is reduced to the shape of the level cross-sectionn of circle when representing bearing load.During bearing load, the shape of balloon can describe (as shown in Figure 3) with the rotary body that the curve in plane is formed around axis of symmetry, and perpendicular cross-sectional shape can pass through radius of turn r₂Determine, radius of turn r₂It is the function about x:

r₂=f (x) (1)

r₂Describe the shape function (bus of rotary body) of vertical plane of structure shape when being exactly and describe balloon bearing load, be exactly semi-circumference for describing not bus by the shape of the balloon of load.

Fig. 4 and Fig. 5 sets forth the membrane stress of the both direction that balloon capsule cloth bears and the stressing conditions schematic diagram of lower semisphere.Can be obtained by the stress balance equation set up along gravity direction to the lower semisphere shown in Fig. 5

In formula, r₃For the level cross-sectionn axial symmetry curve without load balloon around the radius of turn of x-axis, N₁For the meridional stress of capsule cloth,For shape function r₂Angle between normal and the x-axis of upper any point, Δ p is the inside and outside differential pressure of balloon, and G is the gravity of balloon institute bearing load.

Can be obtained by formula (2)

Fig. 6 gives balloon axisymmetric vertical cross-section stressing conditions schematic diagram, can set up stress balance equation along the circumferential direction equally:

$2{\∫}_0^H{N}_2\sqrt{1+{r_2}^{\′2}}dx=2{\∫}_0^H{\mathrm{\Δpr}}_{2}dx---\left(4\right)$

In formula, H is balloon height when being subject to load, N₂Broadwise stress for capsule cloth.

Can be obtained by formula (4)

$N_2=\frac{{\mathrm{\Δpr}}_2}{\sqrt{1+{r_2}^{\′2}}}---\left(5\right)$

Owing to the bus length before and after balloon stand under load is constant, can obtain

${\∫}_0^x\sqrt{1+{r_2}^{\′2}}dx=R_0\arcsin \frac{r_3}{R_0}---\left(6\right)$

${\∫}_0^H\sqrt{1+{r_2}^{\′2}}dx={\mathrm{\πR}}_0---\left(7\right)$

In formula, R₀For not by the radius of load seasonal epidemic pathogens ball, N₂Broadwise stress for capsule cloth.

Can be obtained by method of geometry

By formula (3), formula (5) and formula (8) it can be seen that the meridional stress N of balloon capsule cloth₁With broadwise stress N₂Size depends on pressure differential deltap p and radius of turn r₂If, Δ p and r₂Can determine, then N₁And N₂Also can determine that.

Step 2, foundation describe the geometric model of deformed shape, provide the expression of deformed shape function.

Fig. 7 gives out balloon load-bearing load and describes the geometric model schematic diagram of its axisymmetric perpendicular cross-sectional shape during not by load.As shown in Figure 7, time not by load, balloon is circular, and its radius is R₀, barycenter be C₀；When by load, for ball taper, namely tangent spherical and conical composition, therefore, when balloon is by load, the shape function of vertical cross-section can be expressed as

$\left\{\begin{array}{c}{(x-a)}^2+r_2^2=a^2\\ r_2=c\left(x-H\right)\end{array}\right.---\left(9\right)$

In formula, a is the radius of spherical part in shape function, and c is the slope of conical portion in shape function.

Formula (9) is substituted in (7) and can obtain

$\mathrm{\π}a-a\arccos \frac{a}{H-a}+\sqrt{H^2-2aH}={\mathrm{\πR}}_0---\left(10\right)$

Solve (9) can obtain

$x=\frac{a+c^{2}H\±\sqrt{{(a+c^{2}H)}^2-c^2{H}^2(1+c^2)}}{c^2+1}---\left(11\right)$

The solution of formula (11) is the function the describing vertical-type shape of cross section abscissa x at A point_A, namely hemispherical curve and the point of contact of bell-shaped curve in shape function, thus can obtain

$x_A=\frac{a+c^{2}H}{c^2+1}---\left(12\right)$

Can be obtained by by formula (11) and (12)

(a+c²H)²-c²H²(1+c²)=0

Or

$a=cH(\sqrt{1+c^2}-c)---\left(13\right)$

By formula (13) substitution formula (9) can obtain the radius of turn at A point place it is

$r_{2A}=\±\frac{\sqrt{c^{4}H(2a-H)+a^2(1+2c^2)}}{1+c^2}---\left(14\right)$

Formula (13) is substituted in formula (10) and can obtain

$H=\frac{{\mathrm{\πR}}_0}{\mathrm{\π}c(\sqrt{1+c^2}-c)-c(\sqrt{1+c^2}-c)\arccos \frac{c(\sqrt{1+c^2}-c)}{1-c(\sqrt{1+c^2}-c)}+\sqrt{1-2c(\sqrt{1+c^2}-c)}}---\left(15\right)$

Simultaneous formula (12) formula, (13) and formula (15), it is known that, x_A, a and H be the function about unique undetermined parameter c.

System according to balloon, internal gas and load composition can obtain at the equilibrium equation of gravity direction

M=V ρ-(m₀+m_G)(16)

In formula, V is balloon volume when being subject to load, and ρ is atmospheric density, m₀For the gross mass of balloon capsule cloth, m_GFor the quality of load, andG is acceleration of gravity.

The shape function of vertical cross-section when describing balloon by load according to formula (9), can obtain balloon volume now is

$V=\frac{1}{3}\mathrm{\π}({\mathrm{ax}}_A^2-{\mathrm{Hx}}_A^2+2{\mathrm{aHx}}_A)---\left(17\right)$

Step 3, set up total potential-energy function of balloon system.

Center-of-mass coordinate when balloon is by load is

$x_c=\frac{-4x_A^4+(10a+2H)x_A^3-(4a+H){\mathrm{Hx}}_A^2+2{\mathrm{aH}}^2{x}_A}{4({\mathrm{ax}}_A^2-{\mathrm{Hx}}_A^2+2{\mathrm{aHx}}_A)}---\left(18\right)$

As shown in Figure 7, barycenter when balloon is not by load is

x_c0=R₀(19)

Can be obtained the displacement of barycenter before and after balloon loading by formula (18) and formula (19) is

$\mathrm{\Δ}h=x_c-x_{c0}=\frac{-4x_A^4+(10a+2H)x_A^3-(4a+H){\mathrm{Hx}}_A^2+2{\mathrm{aH}}^2{x}_A}{4({\mathrm{ax}}_A^2-{\mathrm{Hx}}_A^2+2{\mathrm{aHx}}_A)}-R_0---\left(20\right)$

The potential energy of balloon system is

E=(m+m₀)g·Δh+m_Gg(H-2R₀)(21)

In formula, g is acceleration of gravity.

Formula (16) is substituted into formula (21) can obtain

E=(V ρ-m_G)g·Δh+m_Gg(H-2R₀)(22)

The pressure of balloon interior gas is represented by

$p_1=\frac{{\mathrm{mR}}_{mix}T}{V}---\left(23\right)$

In formula, R_mixFor the gas constant of internal gas, T is the temperature of internal gas.

Balloon inside and outside differential pressure is represented by respectively

$\mathrm{\Δ}p=p_1-p_2=\frac{{\mathrm{mR}}_{mix}T}{V}-p_2---\left(24\right)$

In formula, p₂Pressure for air.

When balloon is by load, volume is by V₀The potential energy of the internal gas changing to V is

$W=\mathrm{\Δ}p\·V\·ln\frac{V}{V_0}=\mathrm{\Δ}p\·V(\ln V-\ln V_0)---\left(25\right)$

Making balloon not internal gas volume by load is the volume in ground free state, can be obtained by formula (23)

$V_0=\frac{{\mathrm{mR}}_{mix}{T}_0}{p_0}---\left(26\right)$

In formula, T₀And p₀The respectively temperature and pressure of ground air.

Formula (24) and formula (26) are substituted in formula (25) and can obtain

$W=({\mathrm{mR}}_{mix}T-p_{2}V)\[\ln V-ln\left(\frac{{\mathrm{mR}}_{mix}{T}_0}{p_0}\right)\]---\left(27\right)$

By formula (22) to formula (27), the total potential energy that can obtain balloon system is

$\mathrm{\Π}=E-W=(V\mathrm{\ρ}-m_G)g\mathrm{\Δ}h+m_{G}g\·(H-2R_0)-({\mathrm{mR}}_{mix}T-p_{2}V)\[\ln V-ln\left(\frac{{\mathrm{mR}}_{mix}{T}_0}{p_0}\right)\]---\left(28\right)$

By analysis above it can be seen that total potential energy ∏ of balloon system is the function about unique undetermined parameter c.

Step 4, according to minimum potential energy principal, solve the undetermined parameter in deformed shape function expression.

Undetermined parameter c can make total potential energy ∏ of balloon system minimum, can be easy to by numerical method try to achieve the c meeting this condition.When c is certain, formula (13) and formula (15) is utilized namely to can determine that H and a.

Step 5, by it has been determined that undetermined parameter substitute in deformed shape function expression, namely can determine that the deformed shape of balloon, and then may determine that meridional stress and the broadwise stress of capsule cloth.

Formula (3), formula (5) is utilized namely to can determine that the deformed shape of balloon and the stress of capsule cloth to formula (9).Geometric parameter during by stratospheric atmospheric parameter and balloon not by load, internal gas parameter and load are updated in model, i.e. the stress of the deformed shape of measurable stratospheric balloon and capsule cloth.

The method of a kind of deformed shape predicting stratosphere balloon of the present invention and capsule cloth stress, it is characterized in very simple and practical, geometric parameter when needing only to the capsule cloth material parameter of balloon, balloon not by load, internal gas parameter and load are updated in model, it is possible to the deformed shape being readily available in stratosphere balloon and capsule cloth stress.

Accompanying drawing explanation

The geometry schematic diagram of vertical cross-section when Fig. 1 is balloon bearing load.

The geometry schematic diagram of level cross-sectionn with fold when Fig. 2 is balloon bearing load.

Fig. 3 is the function of the geometry describing balloon level cross-sectionn.

Fig. 4 is the stress of balloon capsule cloth both direction.

Fig. 5 is balloon lower semisphere force diagram.

Fig. 6 is the axisymmetric vertical cross-section stressing conditions schematic diagram of balloon.

Fig. 7 is the geometric model schematic diagram of balloon bearing load.

Fig. 8 is be the FB(flow block) of the method for the invention.

In figure, symbol description is as follows:

The inside and outside differential pressure that Δ p is balloon in Fig. 1, G is the gravity of balloon institute bearing load.

R in Fig. 2₂For describing the radius of turn of perpendicular cross-sectional shape during for balloon bearing load, namely for describing the shape function of vertical cross-section, r₃For describing the radius of turn of perpendicular cross-sectional shape during for balloon not by load.

R in Fig. 3₂For describing the radius of turn of perpendicular cross-sectional shape during for balloon bearing load, namely for describing the shape function of vertical cross-section, r₃For describing the radius of turn of perpendicular cross-sectional shape, r during for balloon not by load₁The radius of curvature of any point on shape function during for balloon bearing load, H is balloon height when being subject to load,For shape function r₂Angle between normal and the x-axis of upper any point.

N in Fig. 4₁For the meridional stress of capsule cloth, N₂Broadwise stress for capsule cloth.

A in Fig. 7 is the radius of spherical part, R in shape function₀Radius during for balloon not by load, C₀Barycenter during for balloon not by load, C is barycenter during balloon bearing load, x_cCenter-of-mass coordinate during for balloon by load, Δ h is the displacement of barycenter before and after balloon loads, and α is coning angle.

Detailed description of the invention

Fig. 8 is the FB(flow block) of the method for the invention, and the present invention divides five steps to realize, particularly as follows:

Step one, according to the overall stress balance equation of balloon, set up the relation between the meridional stress of capsule cloth and broadwise stress and deformed shape function.

When balloon bearing load, being shaped like droplet-shaped (as shown in Figure 1), in FIG, circular dashed line represents the original-shape of vertical cross-section during non-bearing load, and water-drop-shaped solid line represents the shape of vertical cross-section during bearing load.Balloon bearing load also results in there is fold (as shown in Figure 2), in fig. 2, outer dashed line represents the original-shape of level cross-sectionn during non-bearing load, internal solid line represents the shape of level cross-sectionn during bearing load, and inner dotted line is reduced to the shape of the level cross-sectionn of circle when representing bearing load.During bearing load, the shape of balloon can describe (as shown in Figure 3) with the rotary body that the curve in plane is formed around axis of symmetry, and perpendicular cross-sectional shape can pass through radius of turn r₂Determine, radius of turn r₂It is the function about x:

r₂=f (x) (1)

r₂Describe the shape function (bus of rotary body) of vertical plane of structure shape when being exactly and describe balloon bearing load, be exactly semi-circumference for describing not bus by the shape of the balloon of load.

Fig. 4 and Fig. 5 sets forth the membrane stress of the both direction that balloon capsule cloth bears and the stressing conditions schematic diagram of lower semisphere.Can be obtained by the stress balance equation set up along gravity direction to the lower semisphere shown in Fig. 5

In formula, r₃For the level cross-sectionn axial symmetry curve without load balloon around the radius of turn of x-axis, N₁For the meridional stress of capsule cloth,For shape function r₂Angle between normal and the x-axis of upper any point, Δ p is the inside and outside differential pressure of balloon, and G is the gravity of balloon institute bearing load.

Can be obtained by formula (2)

Fig. 6 gives balloon axisymmetric vertical cross-section stressing conditions schematic diagram, can set up stress balance equation along the circumferential direction equally:

$2{\∫}_0^H{N}_2\sqrt{1+{r_2}^{\′2}}dx=2{\∫}_0^H{\mathrm{\Δpr}}_{2}dx---\left(4\right)$

In formula, H is balloon height when being subject to load, N₂Broadwise stress for capsule cloth.

Can be obtained by formula (4)

$N_2=\frac{{\mathrm{\Δpr}}_2}{\sqrt{1+{r_2}^{\′2}}}---\left(5\right)$

Owing to the bus length before and after balloon stand under load is constant, can obtain

${\∫}_0^x\sqrt{1+{r_2}^{\′2}}dx=R_0\arcsin \frac{r_3}{R_0}---\left(6\right)$

${\∫}_0^H\sqrt{1+{r_2}^{\′2}}dx={\mathrm{\πR}}_0---\left(7\right)$

In formula, R₀For not by the radius of load seasonal epidemic pathogens ball, N₂Broadwise stress for capsule cloth.

Can be obtained by method of geometry

By formula (3), formula (5) and formula (8) it can be seen that the meridional stress N of balloon capsule cloth₁With broadwise stress N₂Size depends on pressure differential deltap p and radius of turn r₂If, Δ p and r₂Can determine, then N₁And N₂Also can determine that.

Step 2, foundation describe the geometric model of deformed shape, provide the expression of deformed shape function.

Fig. 7 gives out balloon load-bearing load and describes the geometric model schematic diagram of its axisymmetric perpendicular cross-sectional shape during not by load.As shown in Figure 7, time not by load, balloon is circular, and its radius is R₀, barycenter be C₀；When by load, for ball taper, namely tangent spherical and conical composition, therefore, when balloon is by load, the shape function of vertical cross-section can be expressed as

$\left\{\begin{array}{c}{(x-a)}^2+r_2^2=a^2\\ r_2=c\left(x-H\right)\end{array}\right.---\left(9\right)$

In formula, a is the radius of spherical part in shape function, and c is the slope of conical portion in shape function.

Formula (9) is substituted in (7) and can obtain

$\mathrm{\π}a-a\arccos \frac{a}{H-a}+\sqrt{H^2-2aH}={\mathrm{\πR}}_0---\left(10\right)$

Solve (9) can obtain

$x=\frac{a+c^{2}H\±\sqrt{{(a+c^{2}H)}^2-c^2{H}^2(1+c^2)}}{c^2+1}---\left(11\right)$

The solution of formula (11) is the function the describing vertical-type shape of cross section abscissa x at A point_A, namely hemispherical curve and the point of contact of bell-shaped curve in shape function, thus can obtain

$x_A=\frac{a+c^{2}H}{c^2+1}---\left(12\right)$

Can be obtained by by formula (11) and (12)

(a+c²H)²-c²H²(1+c²)=0

Or

$a=cH(\sqrt{1+c^2}-c)---\left(13\right)$

By formula (13) substitution formula (9) can obtain the radius of turn at A point place it is

$r_{2A}=\±\frac{\sqrt{c^{4}H(2a-H)+a^2(1+2c^2)}}{1+c^2}---\left(14\right)$

Formula (13) is substituted in formula (10) and can obtain

$H=\frac{{\mathrm{\πR}}_0}{\mathrm{\π}c(\sqrt{1+c^2}-c)-c(\sqrt{1+c^2}-c)\arccos \frac{c(\sqrt{1+c^2}-c)}{1-c(\sqrt{1+c^2}-c)}+\sqrt{1-2c(\sqrt{1+c^2}-c)}}---\left(15\right)$

Simultaneous formula (12) formula, (13) and formula (15), it is known that, x_A, a and H be the function about unique undetermined parameter c.

System according to balloon, internal gas and load composition can obtain at the equilibrium equation of gravity direction

M=V ρ-(m₀+m_G)(16)

In formula, V is balloon volume when being subject to load, and ρ is atmospheric density, m₀For the gross mass of balloon capsule cloth, m_GFor the quality of load, andG is acceleration of gravity.

The shape function of vertical cross-section when describing balloon by load according to formula (9), can obtain balloon volume now is

$V=\frac{1}{3}\mathrm{\π}({\mathrm{ax}}_A^2-{\mathrm{Hx}}_A^2+2{\mathrm{aHx}}_A)---\left(17\right)$

Step 3, set up total potential-energy function of balloon system.

Center-of-mass coordinate when balloon is by load is

$x_c=\frac{-4x_A^4+(10a+2H)x_A^3-(4a+H){\mathrm{Hx}}_A^2+2{\mathrm{aH}}^2{x}_A}{4({\mathrm{ax}}_A^2-{\mathrm{Hx}}_A^2+2{\mathrm{aHx}}_A)}---\left(18\right)$

As shown in Figure 7, barycenter when balloon is not by load is

x_c0=R₀(19)

Can be obtained the displacement of barycenter before and after balloon loading by formula (18) and formula (19) is

$\mathrm{\Δ}h=x_c-x_{c0}=\frac{-4x_A^4+(10a+2H)x_A^3-(4a+H){\mathrm{Hx}}_A^2+2{\mathrm{aH}}^2{x}_A}{4({\mathrm{ax}}_A^2-{\mathrm{Hx}}_A^2+2{\mathrm{aHx}}_A)}-R_0---\left(20\right)$

The potential energy of balloon system is

E=(m+m₀)g·Δh+m_Gg(H-2R₀)(21)

In formula, g is acceleration of gravity.

Formula (16) is substituted into formula (21) can obtain

E=(V ρ-m_G)g·Δh+m_Gg(H-2R₀)(22)

The pressure of balloon interior gas is represented by

$p_1=\frac{{\mathrm{mR}}_{mix}T}{V}---\left(23\right)$

In formula, R_mixFor the gas constant of internal gas, T is the temperature of internal gas.

Balloon inside and outside differential pressure is represented by respectively

$\mathrm{\Δ}p=p_1-p_2=\frac{{\mathrm{mR}}_{mix}T}{V}-p_2---\left(24\right)$

In formula, p₂Pressure for air.

When balloon is by load, volume is by V₀The potential energy of the internal gas changing to V is

$W=\mathrm{\Δ}p\·V\·ln\frac{V}{V_0}=\mathrm{\Δ}p\·V(\ln V-\ln V_0)---\left(25\right)$

Making balloon not internal gas volume by load is the volume in ground free state, can be obtained by formula (23)

$V_0=\frac{{\mathrm{mR}}_{mix}{T}_0}{p_0}---\left(26\right)$

In formula, T₀And p₀The respectively temperature and pressure of ground air.

Formula (24) and formula (26) are substituted in formula (25) and can obtain

$W=({\mathrm{mR}}_{mix}T-p_{2}V)\[\ln V-ln\left(\frac{{\mathrm{mR}}_{mix}{T}_0}{p_0}\right)\]---\left(27\right)$

By formula (22) to formula (27), the total potential energy that can obtain balloon system is

$\mathrm{\Π}=E-W=(V\mathrm{\ρ}-m_G)g\mathrm{\Δ}h+m_{G}g\·(H-2R_0)-({\mathrm{mR}}_{mix}T-p_{2}V)\[\ln V-ln\left(\frac{{\mathrm{mR}}_{mix}{T}_0}{p_0}\right)\]---\left(28\right)$

By analysis above it can be seen that total potential energy ∏ of balloon system is the function about unique undetermined parameter c.

Step 4, according to minimum potential energy principal, solve the undetermined parameter in deformed shape function expression.

Undetermined parameter c can make total potential energy ∏ of balloon system minimum, can be easy to by numerical method try to achieve the c meeting this condition.When c is certain, formula (13) and formula (15) is utilized namely to can determine that H and a.

Step 5, by it has been determined that undetermined parameter substitute in deformed shape function expression, namely can determine that the deformed shape of balloon, and then may determine that meridional stress and the broadwise stress of capsule cloth.

Formula (3), formula (5) is utilized namely to can determine that the deformed shape of balloon and the stress of capsule cloth to formula (9).Geometric parameter during by stratospheric atmospheric parameter and balloon not by load, internal gas parameter and load are updated in model, i.e. the stress of the deformed shape of measurable stratosphere balloon and capsule cloth.

Claims (1)

1. the method for the deformed shape predicting stratosphere balloon and capsule cloth stress, it is characterised in that: the method specifically comprises the following steps that

Step one, according to the overall stress balance equation of balloon, set up the relation between the meridional stress of capsule cloth and broadwise stress and deformed shape function；

When balloon bearing load, it is shaped as droplet-shaped, circular dashed line represents the original-shape of vertical cross-section during non-bearing load, water-drop-shaped solid line represents the shape of vertical cross-section during bearing load, balloon bearing load also results in there is fold, outer dashed line represents the original-shape of level cross-sectionn during non-bearing load, and internal solid line represents the shape of level cross-sectionn during bearing load, and inner dotted line is reduced to the shape of the level cross-sectionn of circle when representing bearing load；The rotary body that during bearing load, the curve in the shape plane of balloon is formed around axis of symmetry describes, and perpendicular cross-sectional shape passes through radius of turn r₂Determine, radius of turn r₂It is the function about x:

r₂=f (x) (1)

r₂Describe the shape function of vertical plane of structure shape when being exactly and describe balloon bearing load, be exactly semi-circumference for describing not bus by the shape of the balloon of load；X is the pattern curve of balloon coordinate on axis of symmetry；

Obtained by the stress balance equation set up along gravity direction to lower semisphere

In formula, r₃For the level cross-sectionn axial symmetry curve without load balloon around the radius of turn of x-axis, N₁For the meridional stress of capsule cloth,For shape function r₂Angle between normal and the x-axis of upper any point, Δ p is the inside and outside differential pressure of balloon, and G is the gravity of balloon institute bearing load；

Obtained by formula (2)

The same stress balance equation set up along the circumferential direction:

$2{\∫}_0^H{N}_2\sqrt{1+{r_2}^{\′2}}dx=2{\∫}_0^H{\mathrm{\Δpr}}_{2}dx---\left(4\right)$

In formula, H is balloon height when being subject to load, N₂Broadwise stress for capsule cloth；R₂' for r₂Derivative to x；

Obtained by formula (4)

$N_2=\frac{{\mathrm{\Δpr}}_2}{\sqrt{1+{r_2}^{\′2}}}---\left(5\right)$

Owing to the bus length before and after balloon stand under load is constant,

${\∫}_0^x\sqrt{1+{r_2}^{\′2}}dx=R_{0}arcsin\frac{r_3}{R_0}---\left(6\right)$

${\∫}_0^H\sqrt{1+{r_2}^{\′2}}dx={\mathrm{\πR}}_0---\left(7\right)$

In formula, R₀For not by the radius of load seasonal epidemic pathogens ball；

Obtained by method of geometry

Known by formula (3), formula (5) and formula (8), the meridional stress N of balloon capsule cloth₁With broadwise stress N₂Size depends on pressure differential deltap p and radius of turn r₂If, Δ p and r₂Can determine, then N₁And N₂Also can determine that；

Step 2, foundation describe the geometric model of deformed shape, provide the expression of deformed shape function；

Time not by load, balloon is circular, and its radius is R₀, barycenter be C₀；When by load, for ball taper, namely tangent spherical and conical composition, therefore, when balloon is by load, the shape function of vertical cross-section is expressed as

$\left\{\begin{array}{c}{(x-a)}^2=r_2^2=a^2\\ r_2=c(x-H)\end{array}\right.---\left(9\right)$

In formula, a is the radius of spherical part in shape function, and c is the slope of conical portion in shape function；

Formula (9) is substituted in (7) and obtains

$\mathrm{\π}a-aarccos\frac{a}{H-a}+\sqrt{H^2-2aH}={\mathrm{\πR}}_0---\left(10\right)$

Solve (9) to obtain

$x=\frac{a+c^{2}H\±\sqrt{{(a+c^{2}H)}^2-c^2{H}^2(1+c^2)}}{c^2+1}---\left(11\right)$

The solution of formula (11) is the function the describing vertical-type shape of cross section abscissa x at A point_A, namely hemispherical curve and the point of contact of bell-shaped curve in shape function, thus obtain

$x_A=\frac{a+c^{2}H}{c^2+1}---\left(12\right)$

Obtained by by formula (11) and (12)

(a+c²H)²-c²H²(1+c²)=0

Or

$a=cH(\sqrt{1+c^2}-c)---\left(13\right)$

By formula (13) substitution formula (9) obtains the radius of turn at A point place it is

$r_{2A}=\±\frac{\sqrt{c^{4}H(2a-H)+a^2(1+2c^2)}}{1+c^2}---\left(14\right)$

Formula (13) is substituted in formula (10) and obtains

$H=\frac{{\mathrm{\πR}}_0}{\mathrm{\π}c(\sqrt{1+c^2}-c)-c(\sqrt{1+c^2}-c)\arccos \frac{c(\sqrt{1+c^2}-c)}{1-c(\sqrt{1+c^2}-c)}+\sqrt{1-2c(\sqrt{1+c^2}-c)}}---\left(15\right)$

Simultaneous formula (12) formula, (13) and formula (15), it is known that, x_A, a and H be the function about unique undetermined parameter c；

System according to balloon, internal gas and load composition obtains at the equilibrium equation of gravity direction

M=V ρ-(m₀+m_G)(16)

In formula, V is balloon volume when being subject to load, and ρ is atmospheric density, m₀For the gross mass of balloon capsule cloth, m_GFor the quality of load, andG is acceleration of gravity；M is the gross mass of gas in balloon；

The shape function of vertical cross-section when describing balloon by load according to formula (9), obtaining balloon volume now is

$V=\frac{1}{3}\mathrm{\π}({\mathrm{ax}}_A^2-{\mathrm{Hx}}_A^2+2{\mathrm{aHx}}_A)---\left(17\right)$

Step 3, set up total potential-energy function of balloon system；

Center-of-mass coordinate when balloon is by load is

$x_c=\frac{-4x_A^4+(10a+2H)x_A^3-(4a+H){\mathrm{Hx}}_A^2+2{\mathrm{aH}}^2{x}_A}{4({\mathrm{ax}}_A^2-{\mathrm{Hx}}_A^2+2{\mathrm{aHx}}_A)}---\left(18\right)$

Barycenter when balloon is not by load is

x_c0=R₀(19)

Being obtained the displacement of barycenter before and after balloon loading by formula (18) and formula (19) is

$\mathrm{\Δ}h=x_c-x_{c0}=\frac{-4x_A^4+(10a+2H)x_A^3-(4a+H){\mathrm{Hx}}_A^2+2{\mathrm{aH}}^2{x}_A}{4({\mathrm{ax}}_A^2-{\mathrm{Hx}}_A^2+2{\mathrm{aHx}}_A)}-R_0---\left(20\right)$

The potential energy of balloon system is

E=(m+m₀)g·Δh+m_Gg(H-2R₀)(21)

In formula, g is acceleration of gravity；

Formula (16) is substituted into formula (21) obtain

E=(V ρ-m_G)g·Δh+m_Gg(H-2R₀)(22)

The pressure representative of balloon interior gas is

$p_1=\frac{{\mathrm{mR}}_{mix}T}{V}---\left(23\right)$

In formula, R_mixFor the gas constant of internal gas, T is the temperature of internal gas；

Balloon inside and outside differential pressure is expressed as

$\mathrm{\Δ}p=p_1-p_2=\frac{{\mathrm{mR}}_{mix}T}{V}-p_2---\left(24\right)$

In formula, p₂Pressure for air；

When balloon is by load, volume is by V₀The potential energy of the internal gas changing to V is

$W=\mathrm{\Δ}p\·V\·ln\frac{V}{V_0}=\mathrm{\Δ}p\·V(\ln V-{\mathrm{lnV}}_0)---\left(25\right)$

Making balloon not internal gas volume by load is the volume in ground free state, formula (23) obtain

$V_0=\frac{{\mathrm{mR}}_{mix}{T}_0}{p_0}---\left(26\right)$

In formula, T₀And p₀The respectively temperature and pressure of ground air；

Formula (24) and formula (26) are substituted in formula (25) and obtains

$W=({\mathrm{mR}}_{mix}T-p_{2}V)\[\ln V-ln\left(\frac{{\mathrm{mR}}_{mix}{T}_0}{p_0}\right)\]---\left(27\right)$

By formula (22) to formula (27), the total potential energy that can obtain balloon system is

$\Π=E-W=(V\mathrm{\ρ}-m_G)g\mathrm{\Δ}h+m_{G}g\·(H-2R_0)-({\mathrm{mR}}_{mix}T-p_{2}V)\[\ln V-ln\left(\frac{{\mathrm{mR}}_{mix}{T}_0}{p_0}\right)\]---\left(28\right)$ By analysis above it can be seen that total potential energy ∏ of balloon system is the function about unique undetermined parameter c；

Step 4, according to minimum potential energy principal, solve the undetermined parameter in deformed shape function expression；

Undetermined parameter c can make total potential energy ∏ of balloon system minimum, is easy to by numerical method try to achieve the c meeting this condition, when c is certain, utilizes formula (13) and formula (15) namely to determine H and a；

Step 5, by it has been determined that undetermined parameter substitute in deformed shape function expression, namely determine the deformed shape of balloon, and then determine meridional stress and the broadwise stress of capsule cloth；

Formula (3), formula (5) is utilized namely to determine the deformed shape of balloon and the stress of capsule cloth to formula (9)；Geometric parameter during by stratospheric atmospheric parameter and balloon not by load, internal gas parameter and load are updated in model, i.e. the stress of the deformed shape of measurable stratospheric balloon and capsule cloth.