## Tesi etd-05122010-103814 |

Thesis type

Tesi di dottorato di ricerca

Author

RIZZO, EMANUELE

URN

etd-05122010-103814

Title

Optimization methods applied to the preliminary design of innovative, non conventional aircraft configurations

Settore scientifico disciplinare

ING-IND/04

Corso di studi

INGEGNERIA AEROSPAZIALE

Supervisors

**tutor**Prof. Frediani, Aldo

Parole chiave

- SQP
- Optimization
- PrandtlPlane
- smoothing technique
- non conventional configurations
- Mads
- direct search
- induced drag
- aerodynamics

Data inizio appello

07/12/2007;

Consultabilità

Parziale

Data di rilascio

07/12/2047

Riassunto analitico

The present work takes origin from studies carried out at the Department of

Aerospace Engineering of University of Pisa on innovative aircraft configurations.

In 1924 the German scientist and father of modern aerodynamics Prof. Ludwing

Prandtl demonstrated the existence of a wing system producing the minimum

induced drag, and he called it Best Wing System (BWS). This wing system is made

of four wings connected together at their tips so that it looks like a box in a frontal

view. The condition of minimum induced drag found by Prandtl states that the two

horizontal wings must have the same lift distribution whereas the vertical wings, or

bulkheads, must carry a butterfly shaped lift distribution. The induced drag of this

system is the lowest among all possible wings carrying the same total lift with a

given wingspan, and the efficiency increases as the vertical gap to wingspan ratio

( h b ) does. The Prandtl’s calculations were based on an approximated method

predicting an induced drag of the BWS as the non-dimensional gap h b! " that

is 0.16 times the reference monoplane induced drag. This is in contradiction with

the fact that when the two equally loaded horizontal wings are at infinite distance,

they can be treated as isolated, therefore they have half of the induced drag of the

equivalent monoplane.

In 1999 Prof. Aldo Frediani at University of Pisa gave a new demonstration based

on more accurate calculations. He obtained the same results as Prandtl in the

range of low values of h b , but now the asymptote is, according with theory, at 0.5.

At the same time, he started the translation into an engineering product of the

mathematical intuition of Prandtl and he called this aircraft PrandtlPlane in his

honor.

The PrandtlPlane is, hence, a new aircraft concept, based on the general aim to

reduce induced drag. Besides, this configuration has showed potential benefits in

different aircraft categories: from Very Large Aircraft (VLA) to small ULM airplanes.

The unconventional nature of this aircraft gives rise to new design challenges

mainly due to a lack of experience and of statistical data; the space of solutions is,

therefore, unexplored. For this reason a solution satisfying different conflicting

requirements such as wings with the same lift, static stability of flight and trim

condition in every point of the flight envelope may be hard to find.

The problem of finding the PrandtlPlane wing planform can be formulated in terms

of optimization, which is to find the geometric parameters defining the wings such

that the drag is minimized and the constraints are satisfied. Constraints involve the

equilibrium and the static stability during cruise and landing, maximum speed stall,

geometrical constraints, and so on. Boundaries on the variables are defined as

well: minimum/maximum swept angles, minimum/maximum dihedral angles,

minimum/maximum elevator and flaps deflections, and so on.

An overview of the main methods and algorithms for the search of minima of

unconstrained and constrained optimization problems is presented. Moreover, an

algorithm based on a derivative free method (MADS) for the search of minima of

unconstrained problems and its extension to constrained problems are presented.

An algorithm for the search of global minima is presented and it is tested on

benchmarking problems.

Finally, these methods are extensively applied to find solutions of a ULM version of

the PrandtlPlane aircraft and a flying model coming from the optimizer is

manufactured and tested.

Aerospace Engineering of University of Pisa on innovative aircraft configurations.

In 1924 the German scientist and father of modern aerodynamics Prof. Ludwing

Prandtl demonstrated the existence of a wing system producing the minimum

induced drag, and he called it Best Wing System (BWS). This wing system is made

of four wings connected together at their tips so that it looks like a box in a frontal

view. The condition of minimum induced drag found by Prandtl states that the two

horizontal wings must have the same lift distribution whereas the vertical wings, or

bulkheads, must carry a butterfly shaped lift distribution. The induced drag of this

system is the lowest among all possible wings carrying the same total lift with a

given wingspan, and the efficiency increases as the vertical gap to wingspan ratio

( h b ) does. The Prandtl’s calculations were based on an approximated method

predicting an induced drag of the BWS as the non-dimensional gap h b! " that

is 0.16 times the reference monoplane induced drag. This is in contradiction with

the fact that when the two equally loaded horizontal wings are at infinite distance,

they can be treated as isolated, therefore they have half of the induced drag of the

equivalent monoplane.

In 1999 Prof. Aldo Frediani at University of Pisa gave a new demonstration based

on more accurate calculations. He obtained the same results as Prandtl in the

range of low values of h b , but now the asymptote is, according with theory, at 0.5.

At the same time, he started the translation into an engineering product of the

mathematical intuition of Prandtl and he called this aircraft PrandtlPlane in his

honor.

The PrandtlPlane is, hence, a new aircraft concept, based on the general aim to

reduce induced drag. Besides, this configuration has showed potential benefits in

different aircraft categories: from Very Large Aircraft (VLA) to small ULM airplanes.

The unconventional nature of this aircraft gives rise to new design challenges

mainly due to a lack of experience and of statistical data; the space of solutions is,

therefore, unexplored. For this reason a solution satisfying different conflicting

requirements such as wings with the same lift, static stability of flight and trim

condition in every point of the flight envelope may be hard to find.

The problem of finding the PrandtlPlane wing planform can be formulated in terms

of optimization, which is to find the geometric parameters defining the wings such

that the drag is minimized and the constraints are satisfied. Constraints involve the

equilibrium and the static stability during cruise and landing, maximum speed stall,

geometrical constraints, and so on. Boundaries on the variables are defined as

well: minimum/maximum swept angles, minimum/maximum dihedral angles,

minimum/maximum elevator and flaps deflections, and so on.

An overview of the main methods and algorithms for the search of minima of

unconstrained and constrained optimization problems is presented. Moreover, an

algorithm based on a derivative free method (MADS) for the search of minima of

unconstrained problems and its extension to constrained problems are presented.

An algorithm for the search of global minima is presented and it is tested on

benchmarking problems.

Finally, these methods are extensively applied to find solutions of a ULM version of

the PrandtlPlane aircraft and a flying model coming from the optimizer is

manufactured and tested.

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