dynamic analysis of a hemispherical dome levitated by an air jet

Dynamic analysis of a hemispherical dome levitated by an air jet

Mihir Sen

Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, USA

The dynamics of a hemispherical dome levitated above a table by a verticaljet of air are analyzed. Three different physical regimes are postulated, in each of which the vertical force on the dome is due to a different dominating factor: a lubrication assumption is made for small separations, an inviscid assumption for intermediate distances, and air jet levitation when the dome is far above the table. Estimations of the aerodynamic forces thus obtained are combined into a mathematical model of the system, the static and dynamic characteristics of which are then studied. Periodic perturbation of this dynamical system can result in a response that is either periodic for small perturbations or aperiodic for large ones.

Keywords: hovercraft, levitation, dynamical systems, chaos

Background

A hovercraft is raised above ground by an air cushion; a simple table-top experiment can be devised to study its principle of operation. During the course of such a study the following experiment was conceived and car- ried out. A hat-shaped body shown in Figure 1 consist- ing of a hemispherical dome with a flat rim was con- structed. The body was levitated above the table by a jet of air coming through the table. Air was supplied from a compressor at a known pressure and flow rate. There was a sliding restraint above the dome to make it free to move in the vertical direction only. On perform- ing this experiment, it was observed that there were not one but two stable positions of the dome, one close to the table and another relatively far from it. In between them there was an unstable position also at which any small perturbation would tend to move the dome either upward or downward. The experimental goals did not include taking any quantitative data that would be useful here.

The analysis described here is an attempt to clarify the basic physics of the levitation process, to present a simple mathematical model for the behavior of the dome that was observed, and to study its dynamical characteristics. It must be pointed out that not only is the physical situation different from normal hovercraft operation, but also that only the lowest equilibrium

Address reprint requests to Prof. Sen, Dept. of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA.

Received 22 June 1990; revised 14 August 1992; accepted 25 Septem- ber 1992

position would, in any case, be of practical interest. Though that was the motivation for the present study, the analysis is, however, more general in scope and relevant to other similar applications. It is not restricted by the particular shape or design selected for study.

Analytical models

We will now develop a mathematical model of the aerodynamic forces on the hemispherical dome. The geometry and physical dimensions of the dome are indicated in Figure 1. The radius R is used to scale the other lengths and distances. Thus, the rim is of width hR and the levitated height above the table is yR, both h and y being nondimensional. The quantity y, being always positive, is a measure of the distance above the table at which the dome levitates. The volume flow rate of air Q provided by the compressor and the gauge pressure generated by it are parameters of the system. They are considered to be known, fixed quantities even though in reality the characteristics of the compressor may provide some interrelation between the two.

Apart from inertia and gravity, the principal force acting on the dome is the aerodynamic force of the air, F aeTO. It is obvious that the force exerted by the air jet on the dome varies withy. Furthermore, this variation is nonmonotonic if there is to be more than one position of stable equilibrium. On closer examination, it appears that there are three different physical factors that predominate for different ranges of y.

1. At extremely small values of y, the flow can be assumed to be viscous dominated; this is a lubrica-

226 Appl. Math. Modelling, 1993, Vol. 17, May 0 1993 Butterworth-Heinemann

Dynamics of a hemispherical dome: M. Sen

with u = 0 at z = 0 and z = yR. p’ is the pressure under the rim, r is the radial coordinate, and u is the air velocity of the radial direction. Integration gives

Figure 1. Schematic of hemispherical dome levitated by an air jet.

tion limit and the associated theory can be used to find the pressure distribution in the area of the rim.

2. At intermediate values of y, the flow is essentially inviscid and Bernoulli’s equation determines the pressure.

3 At much larger values of y, the dome is away from the table and levitated only by the impact of the air jet.

Each one of these flow regimes will be analyzed in turn, and the aerodynamic force on the dome estimated for the corresponding range of y.

Lubrication approximation

Assuming the air to be incompressible, the mean velocity V of flow out under the rim is

VEAL 2rrryR

(1)

with R < r < R( 1 + h), where r is the radial coordinate. The reduced Reynolds number Re* defined by’

p \hl (24

(2b)

at r = R, where p and p are the density and the coefficient of dynamic viscosity of air respectively. Re* gives the ratio of inertia to viscous forces, tending to zero as y + 0. This is the lubrication approximation. Thus, for small y (i.e., y << h) inertia forces in the air going through the space between the table and the rim of the dome can be neglected.

Let z be the vertical coordinate in the gap below the rim measured upward from the table. The only compo- nent of the air velocity u is in the radial direction, where u = u(z). Considering the viscous and pressure forces alone, the governing Navier-Stokes equation reduces to

dp’ d2u ,-. -L=qg dr

u(z) = -Ldp’z(yR - z) 2~ dr

The volume flow rate Q is

(4)

2niyRu dz (54 0

(5b)

From this we can get the radial pressure distribution

p’(r) = p. + 3@_lnR(* + h, ,rrR’y’ r

(6)

wherep, is the atmospheric pressure outside the dome. The total force on the dome under the lubrication assumption F,ub is then

R(l+h)

F,ub = (p’ - p) 2gr dr + Ap ,rrR2 (W

6PQ =- Ry3

-ln(l +h) +Ap,rrR2 1 U’b)

Ap is the difference between the pressure below the dome, which is considered uniform, and the outside. Both the displacement and the force are taken to be positive in the upward direction. The expression for the force is valid only for small y.

Inviscid approximation

For intermediate y, the Reynolds number becomes large, indicating that the inertia forces can be neglected in relation to the viscous forces. Essentially the flow can then be considered inviscid. The pressure p’ under the rim can then be related to the pressure pO + Ap under the dome using the Bernoulli equation

2

p,, + Ap = p’ + ‘+ (8)

We have neglected the velocity of the air within the hemispherical part of the dome by assuming that the dome surface is much larger than the exit area. Using equation (1) for the velocity, we have

p’ = pO + Ap - pQ2

8rr2r2y2R2

Integrating the pressure as in equation (7a), we can get the total force on the dome Fi”, under the inviscid assumption

Fin, = -$ ln(1 + h) + Ap,rrR’(l + h)’ (10)

which is now only valid for an intermediate range of y.

Appl. Math. Modelling, 1993, Vol. 17, May 227

Dynamics of a hemispherical dome: A.4 Sen

Air jet approximation For large y, the flow under the rim of the dome does

not play any significant role in determining the force. Rather, it is the direct impact of the jet on the levitated body that determines the force. We simplify the physical process in the following manner. The velocity of the air jet coming out through the table is

u=, ?!a I (11) v P

\ ,

where Ap, is the pressure head that is converted to kinetic energy. We assume that the jet spreads through an angle 2a, so that yR tan CY is the radius of the jet at a distance yR from the table surface. When this radius is smaller than (R + h), the entire momentum of the air jet sustains the levitated body. Assuming that the air jet turns back by 180” after hitting the dome, the momentum transfer is twice the product of the mass flow rate and the airjet velocity. For a larger distance above the table, only the proportion of the jet area that hits the body is used for levitation.

Under these assumptions the force of the jet of air on

c I

0- _V~PAPJ

the dome is

l+h fory 5 -

tana ,._,

I fory>x tan (Y

This estimate is valid only for large values of y.

Static behavior

Figure 2 shows the magnitude of each one of the forces Flub, Fi”,, and F,j in Newtons as a function of y for the following typical parameter values: R = 15 cm, hR = 0.2 cm, Ap = 50 kPa, Q = 0.25 m3/s, CI = 30”, p = 1 kg/m3, p = 2 x 10e5 Ns/m*. The weight of the body F,,, corresponding to a mass of 1 kg is also included for comparison. Note that the lubrication and inviscid models provide forces of very large magnitudes and are thus plotted on a scale different from the other two. Even though all the forces are plotted for lop4 < y < 10, each one is really valid only within a restricted range of y. Furthermore, in an exact theory, obtained

- Lubrication

2000 - 4

-1-*1-- lnvi+J

\ 1000 -

Figure 2. Force on the levitated dome in the different regimes. Lubrication and inviscid models, left scale; air jet model and weight, right scale.

228 Appl. Math. Modelling, 1993, Vol. 17, May


this is accomplished by a smooth transition of a force function of the type UY-~ - by-*, where a and b are both positive. Thus, a single function that combines the qualitative behavior of the piece-wise expressions for the aerodynamic force F,,,, and the constant weight F, is an expression of the form

F(Y) = MRf(y)

where

for example from a detailed numerical analysis of the hydrodynamic equations, there would be a smooth transition between them.

Static equilibrium positions of the body occur when the upward aerodynamic force F,,,, is balanced by the weight F, = mg of the body. For a mass of 1 kg three such positions are observed. The first, to be referred to as the lowest equilibrium position L,, occurs when the force swings from a positive value corresponding to the lubrication model to a negative value obtained from the inviscid analysis. The second, an intermediate equilibrium position y2, comes from the inviscid analysis as it changes back from a negative to a positive value. The third, the highest equilibrium position L3, appears when the force determined from the airjet model drops slowly to zero. It appears that L3 is several orders of magnitude larger than either L1 or Jr, and the y-axis in Figure 2 is drawn on a logarithmic scale to accommo- date all three positions. As an aside it can be mentioned that the experiment also showed such a disparity between the two positions.

The leading term of F,ub from equation (7b) goes as ye3 for small y, whereas that from F,,, in equation (10) is -y-*. Thus, as y is increased, F,,,, should go from a large positive value to a large negative value. In fact

i

21_2+1 -1 ( y3 y2

) for y 5 y0 f(y)=

2 i--L+, (

2

>( >

(13)

Y? Y: 5 - 1 fory>y, Y

The function f(y) represents all the forces on the body including that due to the air and the weight of the body. Figure 3 shows f(y) for y0 = 5.

The main purpose of the fluid-dynamical analysis in the preceding section is to explain the physical factors that are important in each one of the three regimes and to get the general characteristics of the aerodynamic forces in relation to its variation with y. So, from this uoint of view, and for simnlicitv, we will work with f(y) instead of F,,,,(y) and-F,. .

8

6

f(Y)

0.1 1 10 100

Y

Figure 3. Simplified force-position curve with y’, y,, y2, y3, and y” positions marked.



Dynamic behavior

We will now discuss the motion of the levitated body using the same forces that we have worked out in the static case. The equation governing the dynamics of the dome is

d2y dtz+c$-f(Y)=O (14)

where a linear damping term has also been included.

Linear stability There are three equilibrium positions, L, = 0.539,

L2 = 1.675, and jj3 = 6.812, marked in Figure 3, which are all positive solutions of

f(L) = 0 (15)

A linear stability analysis of equation (14) for these equilibria shows that they are stable or unstable if df ldy at y = yi is negative or positive, respectively. So, from Figure 3 it is seen that y2 is unstable whereas L1 and y3 are stable equilibrium positions.

Nonlinear oscillations for zero damping The solution to equation (14) can be expressed in

integral form. However, here we compute the solu-

4

d y/dt

tions numerically using a fourth-order Runge-Kutta scheme with a time step equal to 0.01. The results mentioned below correspond to those obtained from this procedure.

Figure 4 shows the phase plane behavior of the solutions of equation (14) with c = 0 for different initial conditions. 7, and & are stable centers indicated by open circles (0); & is a saddle point indicated by a closed circle (0). The flow is clockwise around both J, and &. There are two homoclinic orbits that start from and end at &. These orbits intersect the abscissa at y = y’ and y = y”, the values of which can be determined from the form of f(y) and are also marked in Figure 3.

Let us consider a trajectory that starts from (y’, 0) and ends at (&, 0). If we multiply the equation

d2y - - f(Y) = 0 dt2

(16)

by dy and integrate over this trajectory, we get

l2

I f(y)& = 0 y’

(17)

because the other term is zero. Substituting for f(y)

I 1

I I c

1

Y

10 IUO

Figure 4. Zero damping oscillations in phase plane. Stable centers (O), unstable saddle (0); (a), (b), and (c) correspond to the time variation in Figure 5.



considerably over a small distance. If the initial condi- tion lies in the range yZ < y(0) < y”, the orbit shown in Figure 4 is around the highest equilibrium position, j& The orbit (c) in Figure 4 is also shown as a function of time in Figure 5.

The change in the period of oscillation T with initial position y(0) (again assuming zero initial velocity) is shown in Figure 6. For y(0) < y’ or y(0) > y”, the orbit is large and so is the period. As y(0) increases the orbit first gets smaller, as does the period. On nearing y(0) = y’, however, the period becomes unbounded because the orbit now passes through the saddle point, Y2, and it takes an infinite amount of time to approach it. In the range y’ < y(0) < L2, the period is seen to be relatively small. At the equilibrium position, r),, the imaginary part of the eigenvalue can be used to calculate the time period of linearized motion around this equilibrium position. The dot e) corresponds to the value thus obtained. Near y(0) = y2 the period again becomes unbounded as the solution takes an infinite amount of time to return to that point. For L2 < y(0) < y”, the orbit is around the higher equilibrium position, j$. The value of T obtained from the imaginary eigenvalue at JJ~ is also indicated, agreeing auite well with the numerical

from equation (13) we get

[ 1 4’2

_‘+4_y =o Y2 Y -’ Y

(18)

which can be solved to give y’ = 0.356. The abscissa crossing of the other homoclinic orbit

(y”, 0) is determined similarly from

Y”

I f(y) dy = 0 (19)

which gives y” = 12.672. Consider initial positions y(0) with zero initial veloc-

ity dyldt(0). This would correspond to the oscillation of the body on releasing it from rest. If the initial position is small enough, i.e., y(0) < y’, the closed orbit takes it around both centers. An example of this is the curve (a) in Figure 4, which is also shown as a time variation y(t) in Figure 5. For y’ < y(0) < Lz, the orbit in Figure 4 is around only one center, 7,. A typical example is (b) in both Figures 4 and 5. The time period and amplitude of oscillation are now seen to be very small because the forces in the lubrication regime are large and vary

_I _. computation.

20

Y

IC

a

Figure 5. Position-time curves for different initial positions. (a), (b), and (c) correspond to the orbits in Figure 4.



31

T

21

\

1(

0

J .l 1 10

Y(O)

100

Figure 6. Time period of oscillation with zero initial velocity and different initial positions y(O). (0) values calculated from eigen analysis; (0) y’, y,, y2, y3, and y”, respectively, from the left.

Wall effect

The solutions of the type (a) in Figure 5 seem to indicate that as the body moves downwards, y --, 0, the force of repulsion Flub + ~0, and the body is deflected back. Equation (14) with c = 0 can be integrated to give

- G(y) = E (20)

where dG/dy = f(y) and E is a constant representing the total mechanical energy. The first term is the kinetic energy of the body and the -G is its potential energy. If the dome approaches the table with a kinetic energy of E,, it comes to rest at y = 6 where G(6) = -E,, before turning back. At any other position the kinetic energy = G(y) - G(6), indicating that it de- pends only on y. The rebound of the dome at the table is completely elastic and none of its energy is lost. It is moreover almost instantaneous because the force becomes very large as y + 0.

Nonzero damping The effect of damping in the system due to friction

and outside air can be included by taking c > 0 in equation (14). The equilibria that were centers for zero

damping are now spiral sinks to which the solutions tend for t +- ~0.

Dynamic behavior with periodic perturbation

We analyze now the effect on the levitated dome system of a periodic perturbation. Such perturbations may come from sources such as oscillations in the pressure or flow rate of the air being delivered by the compressor, or even from motions of the table top. Though these perturbations can be quite complicated, considerable information can be obtained by modifying equation (14) to give

d*y dy - + cz - f(y) = acoswt dt*

(21)

where a is the amplitude of the perturbation and w is the radian frequency, both in suitable units.

The function f(y) in equation (21) has some similar- ity to the form g(y) = y - y3 for the modified Duffmg equation treated by many authors (see, for example, Holmes and Guckenheimer2). This equation arises in some applications in which a system having two stable and one unstable equilibria is periodically perturbed, one example being that of a beam that is attracted in


Dynamics of a hemispherical dome: Al. Sen

_ _ -

150

t

I. ,____ _______________

200 250 300

Figure 7. Periodic response of system with periodic perturbation; a = 2.5, c = 0.1, w = 1, y(0) = 4, dy/dr(O) = 1.

opposite directions by two magnets.3 Analysis of equation (2 1) would be very similar and will not be repeated here. It will just be pointed out that it has been found that for the Duffing equation different combinations of the system parameters and initial conditions lead either to periodic motions close to either of the two stable equilibria, or to chaotic motion. This behavior also holds for the present case.

Figure 7 shows a typical solution y(t) of equation (21) for parameters for which it is ultimately periodic (a = 2.5, c = 0.1, w = 1, with the initial conditions y(0) = 4, dyldr(0) = 1). The broken lines are the stable equilibria for zero perturbation. After an initial series of aperiodic oscillations in which the body approaches the wall several times from the table top, the solution is seen to settle down to a periodic pattern around the higher equilibrium position. The frequency at this stage is that of the forcing perturbation. On increasing the amplitude of this perturbation beyond a certain point the periodic behavior breaks down. Figure 8 shows the solution for a periodic forcing of twice the amplitude (a = 5), the other parameters and initial conditions remaining the same as before. The first 100 or so periods of the forcing function have been eliminated from the plot. No discernible pattern or periodicity is seen to

the variation. The body is seen to move around both equilibria. Data for the same parameters are also shown in Figure 9 as a Poincare plot, the (y, dyldt) value sampled at every time period being represented as a point on the plot. Once again the initial response of the system is removed and the following 1000 periods are shown. The corresponding diagram for a periodic response would be a single point, but here we see considerable scatter in the points even though there appears to be a certain structure to it. Several extremely small and extremely large positions are noted.

Conclusion

It is common in the mathematical modelling of physical phenomena to obtain equations that are intractable. Approximations can then be made in one of two ways. It may be possible at times to take the general governing equations and to apply appropriate scaling or other conditions that would enable one to neglect the unimportant terms. On the other hand it may be easier in certain instances to produce approximations through arguments that are physical in nature but are less rigorous from a mathematical point of view. The latter course is the one chosen here. The theoretical



120

100 -

Y

80 -

60 -

100 200 250 300

Figure 8. y(t) for aperiodic response of system with periodic perturbation; a = 5, c = 0.1, o = 1, y(0) = 4, dy/&(O) = 1.

models so obtained have been simplified further to permit the qualitative behavior of the system to be correctly determined in a relatively simple manner.

Three different physical approximations are the ba- sis of the mathematical model of the hemispherical dome levitated by an air jet. Expressions are found for the force on the dome for various distances from the table top. There are two stable equilibrium positions, just as found by experimentation. The response of the system to periodic perturbation is found to be periodic for small amplitudes and aperiodic for large ones.

Acknowledgment

The author thanks Pedro Sgnchez Upton, a student at the Universidad National Autdnoma de MCxico in Mexico City, who performed the background experiment that led to this work.

Nomenclature

FWJ damping constant total aerodynamic force on body

Faj force on dome in air jet regime

Finv F lub

FW

i

M P’ PO AP API Q R

L?* t T u u V Y Y0

Y’, Y”

Ll

force on dome in inviscid flow regime force on dome in lubrication regime weight of body acceleration due to gravity nondimensional width of rim mass of levitated body pressure under rim atmospheric pressure outside gauge pressure pressure head volume flow rate provided by compressor radius of dome radial coordinate reduced Reynolds number time period of oscillation velocity of air jet radial velocity under rim mean radial velocity under rim nondimensional distance of dome above table y position for change from inviscid to air jet

regimes initial position of orbits that pass through sad-

dle point & lowest equilibrium position



60

dy/dt

30

-20 0.1 1 10 100 1000

Y

Figure 9. Poincare plot for aperiodic response of system with periodic perturbation; a = 5, c = 0.1, w = 1, y(O) = 4, dy/dt(O) = 1.

Lz intermediate equilibrium position References L3 highest equilibrium position 2 coordinate in vertical direction

1 Schlichting, H. Boundary Layer Theory. McGraw-Hill, New York, 1968

a half angle of jet spread 2 Guckenheimer, J. and Holmes, P. Nonlinear Oscillations,

P dynamic viscosity of air Dynamical Systems, and Bifurcations of Vector Fields.

P density of air Springer-Verlag, New York, 1983 3 Moon, F. C. and Holmes, P. J. A Magnetoelastic Strange

Attractor. J. Sound Vib. 1979. 65(2), 275-296


dynamic analysis of a hemispherical dome levitated by an air jet

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