CN103473452A - Novel method for predicting deformation shape and bag cloth stress of balloon in stratosphere - Google Patents

Abstract

A novel method for predicting a deformation shape and bag cloth stress of a balloon in the stratosphere comprises the steps of 1 establishing the relation of warp-wise stress and weft-wise stress of bag cloth and a deformation shape function according to a stress balance equation of the whole balloon; 2 establishing a geometric model describing the deformation shape and giving out a specific expression of the deformation shape function; 3 establishing a total-potential-energy function of a balloon system; 4 solving undetermined parameters in the expression of the deformation shape function according to a principle of minimum potential energy; 5 substituting determined parameters into the expression of the deformation shape function, namely confirming the deformation shape of the balloon and further confirming the warp-wise stress and the weft-wise stress of the bag cloth. The novel method for predicting the deformation shape and the bag cloth stress of the balloon in the stratosphere is simple and practical, and the deformation shape and the bag cloth stress of the balloon in the stratosphere can be easily obtained only by substituting bag cloth material parameters of the balloon, geometric parameters when the balloon is free of load, inner air parameters and load into the model.

Description

A kind of deformed shape of stratosphere balloon and new method of capsule cloth stress predicted

Technical field

The invention provides a kind of deformed shape of stratosphere balloon and new method of capsule cloth stress predicted, belong to mechanical analysis and design field.

Background technology

Balloon is a kind of low cost, high efficiency aircraft, and its main lift comes from the buoyancy of self, therefore, only need carry seldom fuel and just can carry out aerial mission, transport old cheap, there is boundless application prospect.Mechanical property and the inefficacy mechanism of prediction balloon become one of focus of Recent study.The deformed shape of accurate assessment balloon and capsule cloth stress are for extremely important in the design phase, and at present, the engineering staff mainly solves this problem by test, numerical simulation and analytic method.Balloon flight test by scale model or full-size(d), although can record its deformed shape and capsule cloth stress, but the flight test cost is relatively high and the cycle is long, especially it is to be noted and just need deformed shape and these data of capsule cloth stress in its design phase, and the flight test of back just is equivalent to checking and correction that method for designing is carried out; Method for numerical simulation need to be set up complicated finite element model, calculation of complex, and counting yield is low, inconvenient engineers application; And also relatively lack at present the analytic method for assessment of deformed shape and the capsule cloth stress aspect of balloon, therefore, the present invention proposes a kind of deformed shape of stratosphere balloon and new method of capsule cloth stress predicted, the method is very simple and practical, geometric parameter, internal gas parameter and load while only needing capsule cloth material parameter, the balloon of balloon are not subject to load are updated in model, just can be easy to obtain deformed shape and the capsule cloth stress of balloon, visible the present invention has Important Academic meaning and engineering using value.

Summary of the invention

The invention provides a kind of deformed shape of stratosphere balloon and new method of capsule cloth stress predicted, it is easy that the method has calculating, the precision advantages of higher, and its technical scheme is as follows:

A kind of deformed shape of stratosphere balloon and new method of capsule cloth stress predicted of the present invention, the method concrete steps are as follows:

Step 1, according to the stress balance equation of balloon integral body, set up the meridional stress of capsule cloth and the relation between broadwise stress and deformed shape function.

When the balloon bearing load, the similar droplet-shaped of its shape (as shown in Figure 1), in Fig. 1, the original-shape of vertical cross-section when circular dashed line means bearing load not, the shape of vertical cross-section when the water-drop-shaped solid line means bearing load.The balloon bearing load also can cause occurring fold (as shown in Figure 2), in Fig. 2, the original-shape of level cross-sectionn when outside dotted line means bearing load not, the shape of level cross-sectionn when inner solid line means bearing load, be reduced to the shape of circular level cross-sectionn when inner dotted line means bearing load.The rotary body that during bearing load, the shape of balloon can form around axis of symmetry with the curve in plane is described (as shown in Figure 3), and perpendicular cross-sectional shape can be passed through radius of turn r ₂determine radius of turn r ₂the function about x:

r ₂=f(x) (1)

R ₂describing the shape function (bus of rotary body) of vertical plane of structure shape while describing exactly the balloon bearing load, is exactly semi-circumference for the bus of the shape of describing the balloon be not subject to load.

Fig. 4 and Fig. 5 have provided respectively the membrane stress of the both direction that balloon capsule cloth bears and the stressing conditions schematic diagram of lower semisphere.Set up along the stress balance equation of gravity direction and can obtain by the lower semisphere to shown in Fig. 5

In formula, r ₃for the radius of turn of the level cross-sectionn rotational symmetry curve without the load balloon around the x axle, N ₁for the meridional stress of capsule cloth,

for shape function r ₂angle between the normal of upper any point and x axle, the inside and outside differential pressure that △ p is balloon, the gravity that G is balloon institute bearing load.

By formula (2), can be obtained

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

$2{\&Integral;}_0^{\mathrm{<figure-callout\; id="2"\; label="H\; N"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">H</figure-callout>}}{N}_{}{}_2\sqrt{1+{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}^{\&prime;2}}\mathrm{dx}=2{\&Integral;}_0^H\mathrm{\&Delta;p}{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2\mathrm{dx}---\left(4\right)$

In formula, H is the height of balloon while being subject to load, N ₂broadwise stress for capsule cloth.

By formula (4), can be obtained

${\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">N</figure-callout>}}_2=\frac{\mathrm{\&Delta;p}{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}{\sqrt{1+{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}^{\&prime;2}}}---\left(5\right)$

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

${\&Integral;}_0^x\sqrt{1+{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}^{\&prime;2}}\mathrm{dx}=R_0\arcsin \frac{r_3}{R_0}---\left(6\right)$

${\&Integral;}_0^{\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="HDA00003794065400032.png"\; state="\{\{state\}\}">H</figure-callout>}}\sqrt{1+{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}^{\&prime;2}}\mathrm{dx}=\mathrm{\&pi;}{R}_0---\left(7\right)$

In formula, R ₀the radius of balloon when not being subject to load, N ₂broadwise stress for capsule cloth.

By method of geometry, can obtain

From formula (3), formula (5) and formula (8), the meridional stress N of balloon capsule cloth ₁with broadwise stress N ₂size depends on pressure reduction △ p and radius of turn r ₂if, △ p and r ₂can determine, N ₁and N ₂also can determine.

The geometric model of deformed shape is described in step 2, foundation, provides the expression of deformed shape function.

Fig. 7 has given the geometric model schematic diagram that the balloon load-bearing carries and do not describe its axisymmetric perpendicular cross-sectional shape while not being subject to load.As shown in Figure 7, while not being subject to load, balloon is circular, and its radius is R ₀, barycenter is C ₀; When being subject to load, be the ball taper, i.e. tangent spherical and conical composition, therefore, when balloon is subject to load, the shape function of vertical cross-section can be expressed as

$\left\{\begin{array}{c}{(x-a)}^2+{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2=a^2\\ {\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=c(x-H)\end{array}\right.---\left(9\right)$

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

To in formula (9) substitution (7), can obtain

$\mathrm{\&pi;a}-a\arccos \frac{a}{H-a}+\sqrt{H^2-2\mathrm{aH}}=\mathrm{\&pi;}{R}_0---\left(10\right)$

Solving (9) can obtain

$x=\frac{a+c^{2}H\&PlusMinus;\sqrt{{(a+c^{2}H)}^2-c^2{H}^2(1+c^2)}}{{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">c</figure-callout>}}^2+1}---\left(11\right)$

The solution of formula (11) is the horizontal ordinate x that the function of description vertical-type shape of cross section is ordered at A _a, the point of contact of spherical curve and bell-shaped curve in shape function, can obtain thus

$x_A=\frac{a+c^{2}H}{{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">c</figure-callout>}}^2+1}---\left(12\right)$

By formula (11) and (12) can be obtained

(a+c ²H) ²-c ²H ²(1+c ²)=0

Perhaps

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

By the radius of turn that can obtain in formula (13) substitution formula (9) at A point place, be

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

To in formula (13) substitution formula (10), can obtain

$H=\frac{\mathrm{\&pi;}{R}_0}{\mathrm{\&pi;c}(\sqrt{1+c^2}-c)-c(\sqrt{1+c^2}-c)\arccos \frac{c(\sqrt{1+c^2}-c)}{1-c(\sqrt{1+c^2}-c)}+\sqrt{1-2c(\sqrt{1+c^2}-c)}}---\left(15\right)$

Simultaneous formula (12) formula, (13) and formula (15), known, x _a, a and H be the function about unique undetermined parameter c.

The system formed according to balloon, internal gas and load can obtain at the balance equation of gravity direction

m=Vρ-(m ₀+m _G) (16)

In formula, V is the volume of balloon while being subject to load, and ρ is atmospheric density, m ₀for the gross mass of balloon capsule cloth, m _gfor the quality of load, and

g is acceleration of gravity.

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

$V=\frac{1}{3}\mathrm{\&pi;}(a{x}_A^2-H{x}_A^2+2\mathrm{aH}{x}_A)---\left(17\right)$

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

Center-of-mass coordinate when balloon is subject to load is

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

Barycenter when as shown in Figure 7, balloon is not subject to load is

x _c0=R ₀ (19)

The displacement that can be obtained balloon loading front and back barycenter by formula (18) and formula (19) is

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

The potential energy of balloon system is

E=(m+m ₀)g·△h+m _Gg(H-2R ₀) (21)

In formula, g is acceleration of gravity.

Formula (16) substitution formula (21) can be obtained

E=(Vρ-m _G)g·△h+m _Gg(H-2R ₀) (22)

The pressure of balloon interior gas can be expressed as

${\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA00003794065400032.png"\; state="\{\{state\}\}">p</figure-callout>}}_1=\frac{m{R}_{\mathrm{mix}}T}{V}---\left(23\right)$

In formula, R _mixfor the gas law constant of internal gas, the temperature that T is internal gas.

The balloon inside and outside differential pressure can be expressed as respectively

$\mathrm{\&Delta;p}=p_1-{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">p</figure-callout>}}_2=\frac{m{R}_{\mathrm{mix}}T}{V}-p_2---\left(24\right)$

In formula, p ₂pressure for air.

When balloon is subject to load, volume is by V ₀the potential energy that changes to the internal gas of V is

$W=\mathrm{\&Delta;p}\&CenterDot;V\&CenterDot;\ln \frac{V}{V_0}=\mathrm{\&Delta;p}\&CenterDot;V(\ln V-\ln V_0)---\left(25\right)$

The internal gas volume that makes balloon not be subject to load, for the volume in the ground free state, can be obtained by formula (23)

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

In formula, T ₀and p ₀the temperature and pressure of difference tail ground air.

To in formula (24) and formula (26) substitution formula (25), can obtain

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

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

$\mathrm{\&Pi;}=E-W=(\mathrm{V\&rho;}-m_G)\mathrm{g\&Delta;h}+m_{G}g\&CenterDot;(H-2R_0)-(m{R}_{\mathrm{mix}}T-p_{2}V)[\ln V-\ln \left(\frac{m{R}_{\mathrm{mix}}{T}_0}{p_0}\right)]---\left(28\right)$

From the analysis of front, 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 the deformed shape function expression.

Undetermined parameter c can make total potential energy ∏ minimum of balloon system, by numerical method, can be easy to try to achieve the c that meets this condition.Certain as c, utilize formula (13) and formula (15) can determine H and a.

Step 5, by definite undetermined parameter substitution deformed shape function expression, can determine the deformed shape of balloon, and then can determine meridional stress and the broadwise stress of capsule cloth.

Utilize formula (3), formula (5) to formula (9) can determine the deformed shape of balloon and the stress of capsule cloth.Geometric parameter, internal gas parameter and load when stratospheric atmospheric parameter and balloon are not subject to load are updated in model, i.e. the stress of the deformed shape of measurable stratospheric balloon and capsule cloth.

A kind of deformed shape of stratosphere balloon and new method of capsule cloth stress predicted of the present invention, be characterized in very simple and practical, geometric parameter, internal gas parameter and load while only needing capsule cloth material parameter, the balloon of balloon are not subject to load are updated in model, just can be easy to obtain deformed shape and the capsule cloth stress of balloon in stratosphere.

The accompanying drawing explanation

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

When Fig. 2 is the balloon bearing load with the geometric configuration schematic diagram of the level cross-sectionn of fold.

Fig. 3 is the function of the geometric configuration of description balloon level cross-sectionn.

The stress that Fig. 4 is balloon capsule cloth both direction.

Fig. 5 is balloon lower semisphere force diagram.

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

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

Fig. 8 is to 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 △ p in Fig. 1 is balloon, the gravity that G is balloon institute bearing load.

R in Fig. 2 ₂be used for describing the radius of turn of perpendicular cross-sectional shape during for the balloon bearing load, for describing the shape function of vertical cross-section, r ₃while for balloon, not being subject to load for describing the radius of turn of perpendicular cross-sectional shape.

R in Fig. 3 ₂be used for describing the radius of turn of perpendicular cross-sectional shape during for the balloon bearing load, for describing the shape function of vertical cross-section, r ₃while for balloon, not being subject to load for describing the radius of turn of perpendicular cross-sectional shape, r ₁the radius-of-curvature of any point on shape function during for the balloon bearing load, H is the height of balloon while being subject to load,

for shape function r ₂angle between the normal of upper any point and x axle.

N in Fig. 4 ₁for the meridional stress of capsule cloth, N ₂broadwise stress for capsule cloth.

The radius that a in Fig. 7 is spherical part in shape function, R ₀radius while for balloon, not being subject to load, C ₀barycenter while for balloon, not being subject to load, barycenter when C is the balloon bearing load, x _ccenter-of-mass coordinate while for balloon, being subject to load, the displacement that △ h is barycenter before and after balloon loads, α is coning angle.

Embodiment

The FB(flow block) that Fig. 8 is the method for the invention, the present invention divides five steps to realize, is specially:

Step 1, according to the stress balance equation of balloon integral body, set up the meridional stress of capsule cloth and the relation between broadwise stress and deformed shape function.

When the balloon bearing load, the similar droplet-shaped of its shape (as shown in Figure 1), in Fig. 1, the original-shape of vertical cross-section when circular dashed line means bearing load not, the shape of vertical cross-section when the water-drop-shaped solid line means bearing load.The balloon bearing load also can cause occurring fold (as shown in Figure 2), in Fig. 2, the original-shape of level cross-sectionn when outside dotted line means bearing load not, the shape of level cross-sectionn when inner solid line means bearing load, be reduced to the shape of circular level cross-sectionn when inner dotted line means bearing load.The rotary body that during bearing load, the shape of balloon can form around axis of symmetry with the curve in plane is described (as shown in Figure 3), and perpendicular cross-sectional shape can be passed through radius of turn r ₂determine radius of turn r ₂the function about x:

r ₂=f(x) (1)

R ₂describing the shape function (bus of rotary body) of vertical plane of structure shape while describing exactly the balloon bearing load, is exactly semi-circumference for the bus of the shape of describing the balloon be not subject to load.

Fig. 4 and Fig. 5 have provided respectively the membrane stress of the both direction that balloon capsule cloth bears and the stressing conditions schematic diagram of lower semisphere.Set up along the stress balance equation of gravity direction and can obtain by the lower semisphere to shown in Fig. 5

In formula, r ₃for the radius of turn of the level cross-sectionn rotational symmetry curve without the load balloon around the x axle, N ₁for the meridional stress of capsule cloth,

for shape function r ₂angle between the normal of upper any point and x axle, the inside and outside differential pressure that △ p is balloon, the gravity that G is balloon institute bearing load.

By formula (2), can be obtained

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

$2{\&Integral;}_0^{\mathrm{<figure-callout\; id="2"\; label="H\; N"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">H</figure-callout>}}{N}_{}{}_2\sqrt{1+{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}^{\&prime;2}}\mathrm{dx}=2{\&Integral;}_0^H\mathrm{\&Delta;p}{r}_2\mathrm{dx}---\left(4\right)$

In formula, H is the height of balloon while being subject to load, N ₂broadwise stress for capsule cloth.

By formula (4), can be obtained

${\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">N</figure-callout>}}_2=\frac{\mathrm{\&Delta;p}{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}{\sqrt{1+{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}^{\&prime;2}}}---\left(5\right)$

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

${\&Integral;}_0^x\sqrt{1+{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}^{\&prime;2}}\mathrm{dx}=R_0\arcsin \frac{r_3}{R_0}---\left(6\right)$

${\&Integral;}_0^{\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="HDA00003794065400032.png"\; state="\{\{state\}\}">H</figure-callout>}}\sqrt{1+{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}^{\&prime;2}}\mathrm{dx}=\mathrm{\&pi;}{R}_0---\left(7\right)$

In formula, R ₀the radius of balloon when not being subject to load, N ₂broadwise stress for capsule cloth.

By method of geometry, can obtain

From formula (3), formula (5) and formula (8), the meridional stress N of balloon capsule cloth ₁with broadwise stress N ₂size depends on pressure reduction △ p and radius of turn r ₂if, △ p and r ₂can determine, N ₁and N ₂also can determine.

The geometric model of deformed shape is described in step 2, foundation, provides the expression of deformed shape function.

Fig. 7 has given the geometric model schematic diagram that the balloon load-bearing carries and do not describe its axisymmetric perpendicular cross-sectional shape while not being subject to load.As shown in Figure 7, while not being subject to load, balloon is circular, and its radius is R ₀, barycenter is C ₀; When being subject to load, be the ball taper, i.e. tangent spherical and conical composition, therefore, when balloon is subject to load, the shape function of vertical cross-section can be expressed as

$\left\{\begin{array}{c}{(x-a)}^2+{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2^2=a^2\\ {\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=c(x-H)\end{array}\right.---\left(9\right)$

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

To in formula (9) substitution (7), can obtain

$\mathrm{\&pi;a}-a\arccos \frac{a}{H-a}+\sqrt{H^2-2\mathrm{aH}}=\mathrm{\&pi;}{R}_0---\left(10\right)$

Solving (9) can obtain

$x=\frac{a+c^{2}H\&PlusMinus;\sqrt{{(a+c^{2}H)}^2-c^2{H}^2(1+c^2)}}{{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">c</figure-callout>}}^2+1}---\left(11\right)$

The solution of formula (11) is the horizontal ordinate x that the function of description vertical-type shape of cross section is ordered at A _a, the point of contact of spherical curve and bell-shaped curve in shape function, can obtain thus

$x_A=\frac{a+c^2\mathrm{<figure-callout\; id="2"\; label="H\; c"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">H</figure-callout>}}{c^{}}---\left(12\right)$

By formula (11) and (12) can be obtained

(a+c ²H) ²-c ²H ²(1+c ²)=0

Perhaps

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

By the radius of turn that can obtain in formula (13) substitution formula (9) at A point place, be

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

To in formula (13) substitution formula (10), can obtain

$H=\frac{\mathrm{\&pi;}{R}_0}{\mathrm{\&pi;c}(\sqrt{1+c^2}-c)-c(\sqrt{1+c^2}-c)\arccos \frac{c(\sqrt{1+c^2}-c)}{1-c(\sqrt{1+c^2}-c)}+\sqrt{1-2c(\sqrt{1+c^2}-c)}}---\left(15\right)$

Simultaneous formula (12) formula, (13) and formula (15), known, x _a, a and H be the function about unique undetermined parameter c.

The system formed according to balloon, internal gas and load can obtain at the balance equation of gravity direction

m=Vρ-(m ₀+m _G) (16)

In formula, V is the volume of balloon while being subject to load, and ρ is atmospheric density, m ₀for the gross mass of balloon capsule cloth, m _gfor the quality of load, and

g is acceleration of gravity.

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

$V=\frac{1}{3}\mathrm{\&pi;}(a{x}_A^2-\mathrm{<figure-callout\; id="2"\; label="H\; x\; A"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">H</figure-callout>}{x}_A^{}{}_2^{}+2\mathrm{aH}{x}_A)---\left(17\right)$

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

Center-of-mass coordinate when balloon is subject to load is

$x_c=\frac{-4x_A^4+(10a+2H)x_A^3-(4a+H)\mathrm{<figure-callout\; id="2"\; label="H\; x\; A"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">H</figure-callout>}{x}_A^{}{}_2^{}+2a{H}^2{x}_A}{4(a{x}_A^2-\mathrm{<figure-callout\; id="2"\; label="H\; x\; A"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">H</figure-callout>}{x}_A^{}{}_2^{}+2\mathrm{aH}{x}_A)}---\left(18\right)$

Barycenter when as shown in Figure 7, balloon is not subject to load is

x _c0=R ₀ (19)

The displacement that can be obtained balloon loading front and back barycenter by formula (18) and formula (19) is

$\mathrm{\&Delta;h}=x_c-x_{c0}=\frac{-4x_A^4+(10a+2H)x_A^3-(4a+H)\mathrm{<figure-callout\; id="2"\; label="H\; x\; A"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">H</figure-callout>}{x}_A^{}{}_2^{}+2a{H}^2{x}_A}{4({\mathrm{ax}}_A^2-\mathrm{<figure-callout\; id="2"\; label="H\; x\; A"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">H</figure-callout>}{x}_A^{}{}_2^{}+2\mathrm{aH}{x}_A)}-R_0---\left(20\right)$

The potential energy of balloon system is

E=(m+m ₀)g·△h+m _Gg(H-2R ₀) (21)

In formula, g is acceleration of gravity.

Formula (16) substitution formula (21) can be obtained

E=(Vρ-m _G)g·△h+m _Gg(H-2R ₀) (22)

The pressure of balloon interior gas can be expressed as

${\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA00003794065400032.png"\; state="\{\{state\}\}">p</figure-callout>}}_1=\frac{m{R}_{\mathrm{mix}}T}{V}---\left(23\right)$

In formula, R _mixfor the gas law constant of internal gas, the temperature that T is internal gas.

The balloon inside and outside differential pressure can be expressed as respectively

$\mathrm{\&Delta;p}=p_1-{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA00003794065400031.png"\; state="\{\{state\}\}">p</figure-callout>}}_2=\frac{m{R}_{\mathrm{mix}}T}{V}-p_2---\left(24\right)$

In formula, p ₂pressure for air.

When balloon is subject to load, volume is by V ₀the potential energy that changes to the internal gas of V is

$W=\mathrm{\&Delta;p}\&CenterDot;V\&CenterDot;\ln \frac{V}{V_0}=\mathrm{\&Delta;p}\&CenterDot;V(\ln V-\ln V_0)---\left(25\right)$

The internal gas volume that makes balloon not be subject to load, for the volume in the ground free state, can be obtained by formula (23)

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

In formula, T ₀and p ₀the temperature and pressure of difference tail ground air.

To in formula (24) and formula (26) substitution formula (25), can obtain

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

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

$\mathrm{\&Pi;}=E-W=(\mathrm{V\&rho;}-m_G)\mathrm{g\&Delta;h}+m_{G}g\&CenterDot;(H-2R_0)-(m{R}_{\mathrm{mix}}T-p_{2}V)[\ln V-\ln \left(\frac{m{R}_{\mathrm{mix}}{T}_0}{p_0}\right)]---\left(28\right)$

From the analysis of front, 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 the deformed shape function expression.

Undetermined parameter c can make total potential energy ∏ minimum of balloon system, by numerical method, can be easy to try to achieve the c that meets this condition.Certain as c, utilize formula (13) and formula (15) can determine H and a.

Step 5, by definite undetermined parameter substitution deformed shape function expression, can determine the deformed shape of balloon, and then can determine meridional stress and the broadwise stress of capsule cloth.

Utilize formula (3), formula (5) to formula (9) can determine the deformed shape of balloon and the stress of capsule cloth.Geometric parameter, internal gas parameter and load when stratospheric atmospheric parameter and balloon are not subject to load are updated in model, i.e. the stress of the deformed shape of measurable stratosphere balloon and capsule cloth.

Claims (1)

1. predict the deformed shape of stratosphere balloon and the new method of capsule cloth stress for one kind, it is characterized in that: the method concrete steps are as follows:

Step 1, according to the stress balance equation of balloon integral body, set up the meridional stress of capsule cloth and the relation between broadwise stress and deformed shape function;

When the balloon bearing load, the similar droplet-shaped of its shape, the original-shape of vertical cross-section when circular dashed line means bearing load not, the shape of vertical cross-section when the water-drop-shaped solid line means bearing load, the balloon bearing load also can cause occurring fold, the original-shape of level cross-sectionn when outside dotted line means bearing load not, the shape of level cross-sectionn when inner solid line means bearing load, be reduced to the shape of circular level cross-sectionn when inner dotted line means bearing load; The rotary body that during bearing load, the shape of balloon forms around axis of symmetry with the curve in plane is described, and perpendicular cross-sectional shape is by radius of turn r ₂determine radius of turn r ₂the function about x:

r ₂=f(x) (1)

R ₂describing the shape function of vertical plane of structure shape while describing exactly the balloon bearing load, is exactly semi-circumference for the bus of the shape of describing the balloon be not subject to load;

By lower semisphere is set up along the stress balance equation of gravity direction and is obtained

In formula, r ₃for the radius of turn of the level cross-sectionn rotational symmetry curve without the load balloon around the x axle, N ₁for the meridional stress of capsule cloth,

for shape function r ₂angle between the normal of upper any point and x axle, the inside and outside differential pressure that △ p is balloon, the gravity that G is balloon institute bearing load;

By formula (2), obtained

The same stress balance equation of setting up along the circumferential direction:

$2{\&Integral;}_0^H{N}_2\sqrt{1+{r_2}^{\&prime;2}}\mathrm{dx}=2{\&Integral;}_0^H\mathrm{\&Delta;p}{r}_2\mathrm{dx}---\left(4\right)$

In formula, H is the height of balloon while being subject to load, N ₂broadwise stress for capsule cloth;

By formula (4), obtained

$N_2=\frac{\mathrm{\&Delta;p}{r}_2}{\sqrt{1+{r_2}^{\&prime;2}}}---\left(5\right)$

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

${\&Integral;}_0^x\sqrt{1+{r_2}^{\&prime;2}}\mathrm{dx}=R_0\arcsin \frac{r_3}{R_0}---\left(6\right)$

${\&Integral;}_0^H\sqrt{1+{r_2}^{\&prime;2}}\mathrm{dx}=\mathrm{\&pi;}{R}_0---\left(7\right)$

In formula, R ₀the radius of balloon when not being subject to load, N ₂broadwise stress for capsule cloth;

By method of geometry, obtain

Known the meridional stress N of balloon capsule cloth by formula (3), formula (5) and formula (8) ₁with broadwise stress N ₂size depends on pressure reduction △ p and radius of turn r ₂if, △ p and r ₂can determine, N ₁and N ₂also can determine;

The geometric model of deformed shape is described in step 2, foundation, provides the expression of deformed shape function;

While not being subject to load, balloon is circular, and its radius is R ₀, barycenter is C ₀; When being subject to load, be the ball taper, i.e. tangent spherical and conical composition, therefore, when balloon is subject to 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, the radius that a is spherical part in shape function, the slope that c is conical portion in shape function;

To in formula (9) substitution (7), obtain

$\mathrm{\&pi;a}-a\arccos \frac{a}{H-a}+\sqrt{H^2-2\mathrm{aH}}=\mathrm{\&pi;}{R}_0---\left(10\right)$

Solving (9) obtains

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

The solution of formula (11) is the horizontal ordinate x that the function of description vertical-type shape of cross section is ordered at A _a, the point of contact of spherical curve and bell-shaped curve in shape function, obtain thus

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

By formula (11) and (12) are obtained

(a+c ²H) ²-c ²H ²(1+c ²)=0

Perhaps

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

By the radius of turn obtained in formula (13) substitution formula (9) at A point place, be

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

To in formula (13) substitution formula (10), obtain

$H=\frac{\mathrm{\&pi;}{R}_0}{\mathrm{\&pi;c}(\sqrt{1+c^2}-c)-c(\sqrt{1+c^2}-c)\arccos \frac{c(\sqrt{1+c^2}-c)}{1-c(\sqrt{1+c^2}-c)}+\sqrt{1-2c(\sqrt{1+c^2}-c)}}---\left(15\right)$

Simultaneous formula (12) formula, (13) and formula (15), known, x _a, a and H be the function about unique undetermined parameter c;

The system formed according to balloon, internal gas and load obtains at the balance equation of gravity direction

m=Vρ-(m ₀+m _G) (16)

In formula, V is the volume of balloon while being subject to load, and ρ is atmospheric density, m ₀for the gross mass of balloon capsule cloth, m _gfor the quality of load, and

g is acceleration of gravity;

The shape function of vertical cross-section when according to formula (9), describing balloon and be subject to load, obtain balloon volume now and be

$V=\frac{1}{3}\mathrm{\&pi;}(a{x}_A^2-H{x}_A^2+2\mathrm{aH}{x}_A)---\left(17\right)$

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

Center-of-mass coordinate when balloon is subject to load is

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

Barycenter when balloon is not subject to load is

x _c0=R ₀ (19)

The displacement that is obtained balloon loading front and back barycenter by formula (18) and formula (19) is

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

The potential energy of balloon system is

E=(m+m ₀)g·△h+m _Gg(H-2R ₀) (21)

In formula, g is acceleration of gravity;

Formula (16) substitution formula (21) is obtained

E=(Vρ-m _G)g·△h+m _Gg(H-2R ₀) (22)

The pressure representative of balloon interior gas is

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

In formula, R _mixfor the gas law constant of internal gas, the temperature that T is internal gas;

The balloon inside and outside differential pressure is expressed as respectively

$\mathrm{\&Delta;p}=p_1-p_2=\frac{m{R}_{\mathrm{mix}}T}{V}-p_2---\left(24\right)$

In formula, p ₂pressure for air;

When balloon is subject to load, volume is by V ₀the potential energy that changes to the internal gas of V is

$W=\mathrm{\&Delta;p}\&CenterDot;V\&CenterDot;\ln \frac{V}{V_0}=\mathrm{\&Delta;p}\&CenterDot;V(\ln V-\ln V_0)---\left(25\right)$

The internal gas volume that makes balloon not be subject to load, for the volume in the ground free state, is obtained by formula (23)

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

In formula, T ₀and p ₀the temperature and pressure of difference tail ground air;

To in formula (24) and formula (26) substitution formula (25), obtain

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

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

$\mathrm{\&Pi;}=E-W=(\mathrm{V\&rho;}-m_G)\mathrm{g\&Delta;h}+m_{G}g\&CenterDot;(H-2R_0)-(m{R}_{\mathrm{mix}}T-p_{2}V)[\ln V-\ln \left(\frac{m{R}_{\mathrm{mix}}{T}_0}{p_0}\right)]---\left(28\right)$

From the analysis of front, 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 the deformed shape function expression;

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

Step 5, by definite undetermined parameter substitution deformed shape function expression, determine the deformed shape of balloon, and then the meridional stress of definite capsule cloth and broadwise stress;

Utilize formula (3), formula (5) to formula (9) to determine the deformed shape of balloon and the stress of capsule cloth; Geometric parameter, internal gas parameter and load when stratospheric atmospheric parameter and balloon are not subject to load are updated in model, i.e. the stress of the deformed shape of measurable stratospheric balloon and capsule cloth.

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