Luo A.C.J. (Ed.) Dynamical Systems: Discontinuity, Stochasticity and Time-Delay

Подождите немного. Документ загружается.

68 K.-L. Li et al.

Fig. 7.2 The time history of the dynamic system at R

D 0; M

D 6:0317

Fig. 7.3 The time history of the dynamic system at R

D 0; M

D 6:0318

Fig. 7.4 Bifurcation diagram of the dynamic system at R

D 0

7 Analysis of Panel Flutter with Thermal Effects in Supersonic Flow 69

7.3.2 R

D 0:056, Mach Number as the Bifurcation Parameter

Taking thermal effect into account and assuming R

D 0:056, it is can be found

that as the Mach number increases from 1.73 to 1.74, the state of the panel varies

from the static state in initial equilibrium position to buckling. The time histories

are shown in Figs. 7.5 and 7.6. As the parameters increase to the values as follows,

D 0:056; M

D 1:73:75; w

D 0; i D 1;:::;N

one of the eigenvalues of the Jacobian matrix is real and equals to zero approxi-

mately, and the real parts of the other eigenvalues are not equal to zero. According to

Fig. 7.5 The time history of the dynamic system at R

D 0:056; M

D 1:73

Fig. 7.6 The time history of the dynamic system at R

D 0:056; M

D 1:74

70 K.-L. Li et al.

Fig. 7.7 Bifurcation diagram of the dynamic system at R

D 0:056

Fig. 7.8 Time history of the dynamic system at R

D 0:056; M

D 2

the nonlinear theory, as the Mach number increases from 1.73 to 1.74, the Pitchfork

bifurcation occurs in the dynamic system at about M

D 1:73075. The bifurcation

diagram is shown in Fig.7.7.

As Mach number increases from 2 to 2.8, several time histories and phase por-

traits are shown in Figs. 7.8–7.12. From the ﬁgures, it can be found that as Mach

number increasing from 2 to 2.8, the panel changes its state from buckling to pe-

riodic oscillation. Some irregular oscillations and a state like chaos occur as the

parameters are varied. Apparently, the bifurcation behavior at R

D 0:056 is differ-

ent from the bifurcation behavior at R

D 0.

7 Analysis of Panel Flutter with Thermal Effects in Supersonic Flow 71

Fig. 7.9 Phase portrait

of the dynamic system at

D 0:056; M

D 2:1

Fig. 7.10 Phase portrait

of the dynamic system at

D 0:056; M

D 2:5

Fig. 7.11 Phase portrait

of the dynamic system at

D 0:056; M

D 2:7

72 K.-L. Li et al.

Fig. 7.12 Phase portrait of the dynamic system at R

D 0:056; M

D 2:8

Fig. 7.13 Time history of the dynamic system at R

D 0:05427; M

D 2

7.3.3 M

D 2, R

as Bifurcation Parameter

From Figs. 7.13 and 7.14, it can be found that the panel changes its state from static

state to periodic oscillation as R

increases. As the parameters are the values as

follows,

D 2; R

D 0:0542714; w

D 0; i D 1;:::;N;

the real parts of a pair of conjugate complex eigenvalues of the Jacobian matrix

equal to zero approximately and the real parts of the other eigenvalues are not equal

to zero. From the nonlinear theory, as R

increases from 0.05427 to 0.05428, Hopf

bifurcation occurs in the dynamic system at about R

D 0:0542714.

7 Analysis of Panel Flutter with Thermal Effects in Supersonic Flow 73

Fig. 7.14 Time history

of the dynamic system at

D 0:05428; M

D 2

Fig. 7.15 Time history

of the dynamic system at

D 0:05596; M

D 2

Fig. 7.16 Time history

of the dynamic system at

D 0:05597; M

D 2

As R

continues to increase, the panel jumps suddenly from periodic oscillation

to buckling at about R

D 0:055965. The time histories are shown in Figs. 7.15 and

7.16, and the bifurcation diagram is demonstrated in Fig. 7.17.

74 K.-L. Li et al.

Fig. 7.17 Bifurcation diagram of the dynamic system at M

D 2

7.4 Conclusions

From the results presented above, some conclusions can be drawn as follows.

As thermal effect is not taken into account, the state of the panel varies from static

state to periodic oscillation at high Mach number (more than 6), which is veriﬁed as

Hopf Bifurcation. Then, as thermal effect is taken into account, the state of the panel

also varies from static state to periodic oscillation, however, at lower Mach number

(less than 3). The pitchfork bifurcation is found as the Mach number is increasing.

Further, as Mach number is assumed to be a constant, the state of the panel changes

from static state to buckling as R

increasing.

So it is can be found that the thermal effect will make the panel unsteady in ini-

tial equilibrium position at lower Mach number (less than 3), the oscillation occurs

easily than the situation without thermal effect.

Appendix 1

Nomenclature

a Length of the panel

c Speciﬁc heat capacity

D Bending stiffness

E Young’s modulus

h Panel thickness

k Thermal conductivity

M Mach number

p Static pressure

7 Analysis of Panel Flutter with Thermal Effects in Supersonic Flow 75

q Dynamic pressure

Steady temperature recovery factor

Unsteady temperature recovery factor

T Temperature

Temperature due to aerodynamic heating

U Velocity

˛ Coefﬁcient of thermal expansion

 Poisson ration

 Density

Subscripts

1 Free stream

p Panel characteristic

s Support structure characteristic

u Condition at external surface

Appendix 2

Nondimensional Variables

Nw D

Nx D

t D



;D



h=D

ˆ D

 D

‚ D



;DD

12.1  

ƒ D

D R

.  1/M

‚

.1 C /

.  1/M

‚

.1 C /

76 K.-L. Li et al.

The Coefﬁcients of the Governing Equation

ba.s/ D .s/

12

1 C v

 12

C R

.s/

bb.s/ D‚

.s/

bc.s/ D ƒ

‚

bf .s; r/ D2



.r/

 r

Œ1  .1/

sCr



bg.s; r/ D 3.s/

.r/

bh.s; r/ D6‚

.s/

Œ1  .1/



r

bi.s; r/ D 2ƒr

 r

Œ1  .1/

sCr



Acknowledgments This research is supported by Program for New Century Excellent Talents in

University in China, No. NCET-07-0685.

References

1. Dugundji J (1966) Theoretical considerations of panel ﬂutter at high super-sonic Mach number.

AIAA J 4:1257–1266

2. Ashley H, Zartarian G (1956) Piston theory-a new aerodynamic tool for the aeroelastician.

J Aeronaut Sci 23:1109–1118

3. Dowell EH (1966) Nonlinear oscillations of a ﬂuttering plate. AIAA J 4:1267–1275

4. Dowell EH (1967) Nonlinear oscillations of a ﬂuttering plate II. AIAA J 5:1856–1862

5. Sipcic SR (1990) Chaotic response of ﬂuttering panel the inﬂuence of maneuvering. Nonlinear

Dyn 1:243–264

6. Gee DJ, Sipcic SR (1999) Coupled thermal model for nonlinear panel ﬂutter. AIAA J

37:642–650

7. Schlichting H (1979) Boundary-layer theory. McGraw-Hill, New York

8. Hildebrand FB (1976) Advanced calculus for applications. Prentice-Hall, Englewood Cliffs, NJ

Chapter 8

A Parameter Study of a Machine Tool

with Multiple Boundaries

Brandon C. Gegg, Steve C.S. Suh, and Albert C.J. Luo

Abstract The parameter study of a machine-tool with intermittent cutting is

completed for eccentricity frequency and amplitude. The effects with respect to

chip length are also incorporated, such that comparisons of the parameter maps

can be accomplished. Speciﬁc areas within the parameter maps are studied, via

switching components, to explain the complicated motions within. In such a case,

the switching characteristics are shown in relation to the eccentricity frequency.

The complexity of the periodic solution structure, with regard to the vector ﬁelds

and mapping quantities, is discussed. Furthermore, the traditional deﬁnition of a

stability boundary is extended beyond that in literature. The most useful data is the

overlay of the number of mappings and minimum switching force product record.

This aspect illustrates the extent and location of complexity in the machine-tool

model studied herein.

8.1 Introduction

The extent of parameter studies on machining systems are typically conﬁned to

descriptions in the frequency and depth of cut plane [1]. In such a case, the

boundary for stable/unstable motions is deﬁned. Other studies focus on the chip

seizure (stick-slip effect) interaction, where a boundary can be deﬁned in parame-

ter space [2]. However, these motions are conﬁned to limitations of a continuous

system. Typically, a system with multiply interconnected domains is not studied for

chip seizure or other phenomena associated with such a system. There are three

parameters studied herein: eccentricity frequency and amplitude, and chip con-

tact length. What makes this study further unique is the output dimension of the

components. There are typically at least two switching components for a simple

B.C. Gegg (



)

Department of Mechanical Engineering, Texas A&M University, College Station,

TX 77843, USA

e-mail: barracuda@tamu.edu

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay,

DOI 10.1007/978-1-4419-5754-2

 Springer Science+Business Media, LLC 2010