Dynamics and Control of Space Structures - Exercises

POLITECNICO DI MILANO

Facoltà di Ingegneria Aerospaziale

Dynamics and Control of

Space Structures

[Exercises]

prof. L. Dozio

a cura di

Giorgio Montorfano

e

Riccardo Rota

DISCLAIMER

(ITA)

Questa dispensa raccoglie temi d'esame del corsi di Dynamics and Control of Space Structures (prof. Dozio):

creata raccogliendo i temi d'esame svolti in preparazione dell'esame, è da intendersi come uno strumento

utile per ripassare prima dell'esame e chiarire eventuali dubbi sorti durantele spiegazioni. Non è concepita

come un sostituto della frequenza alle lezioni, né potrebbe esserlo, poiché la risoluzione degli esercizi

rappresenta una minima parte del necessario alla comprensione della materia. Molti termini, inoltre,

verranno considerati da subito parte integrante del bagaglio culturale dello studente.

Si osservi che le soluzioni proposte non sono state sottoposte alla valutazione da parte del docente, ma sono

basate sugli esercizi svolti in aula e sulla rielaborazione personale dei concetti spiegati: è pertanto possibile

che esista una strada più veloce per giungere alla soluzione dell'esercizio. I temi d'esame presentati sono

riferiti agli anni accademici passati, ma restano validi in quanto gli esercizi vertono frequentemente su alcuni

argomenti principali. Alcuni esercizi presentano una soluzione dettagliata in cui tuttavia, per rapidità di

svolgimento, è stata omessa la sostituzione dei valori numerici.

Si ricorda infine che è fondamentale - tanto per la comprensione della materia quanto per la riuscita positiva

dell'esame - affrontare gli esercizi solo dopo aver studiato attentamente la teoria, poiché, a differenza di altri

corsi, teoria e pratica sono strettamente correlati. Gli esercizi dovrebbero quindi essere sfruttati come

feedback per la propria preparazione: risolvendo nuovi esercizi si affina la comprensione degli argomenti

teorici, migliorando di conseguenza la capacità di affrontare tipologie originali di esercizi e risolvendo così i

problemi posti in esame.

(ENG)

These lecture notes are aimed at collecting exercises for the courses of Dynamics and Control of Space

Structures (prof. Dozio): they include exercises made by students to prepare to face the exam, and they

should be considered as a tool for reviewing notions before the final examination. They are not meant to be

an alternative to attending lessons - nor could they be - as solving exercises is a small part of the effort needed

to grasp the concepts of this subject. Furthermore, many terms will be considered known to the student.

Please note that proposed solutions have not been evaluated by a professor, and that they are based both

on examples shown during lessons and on the personal reinterpretation of what has been explained: it is

thus possible that a shorter solution exists. Exercises refer to exams of past academic years, but they are still

valid as exercises often focus on some main topics. Some exercises have a detailed solution which does not

consider the substitution of numerical values.

As a final remark, please remember that it is of paramount importance - both for understanding the subject

and for passing the exam - to work on exercises after having studied carefully theoretical notions, as,

contrarily to what happens in some other courses, exercises and theory are strictly related. Exercises should

be intended as a feedback on one's preparation: solving new exercises the understanding of topics may be

refined, thus improving the ability to face new exercises and solving problems presented in exams.

ID codes: 072652 SPA 2 liv, 081225 SPA Mag, 091932 SPA Mag

Date: 26 June 2013

Instructor: Lorenzo Dozio

1. Consider the dynamic system in Figure 1, where the beam A-B of constant mass per unit length is

m

assumed to be rigid and the beam B-C of constant bending stiffness is assumed to be massless. The

EJ

elastic beam is rigidly connected at point B and carries the tip point mass . The rigid beam is elastically

M

supported by two springs of stiffness k.

w δ

EJ

m

A C

B k

k φ M

L L

1 2

Figure 1: Dynamic system.

Write the equations of motion of the system in terms of the displacement of the centre of mass of the

w

rigid beam A-B, the displacement of the end-mass measured with respect to line A-B, and the small

δ M

rotation of the rigid beam φ.

2. Consider the rectangular thin symmetric laminate in Figure 2 of side lengths and thickness bending

a b, h,

stiffness matrix and mass density The plate is simply supported along edges and and is

ρ. AB CD

D 0

subjected to the in-plane stress on the free edges and The plate is forced to vibrate by a

σ BC AD.

xx

point transverse load (t) = sin(Ωt) located at (x ).

F F , y

0 0 0

sin(Ωt)

F

0

D y

z C

A 0

σ

xx

b

a B x

Figure 2: Initially stressed SFSF symmetric laminate excited by a transverse point load.

(a) Write the discretized dynamic model of the system using a Ritz approximation for the transverse

displacement = (explain your choice for

w(x, y, t) y)u(t)

N(x, N).

(b) Determine the response to the sinusoidal load using the mode displacement method (x̄,

w(x̄, ȳ, t) F ȳ

are arbitrary locations over the plate).

(c) Determine the response to the sinusoidal load using the mode acceleration method (x̄,

w(x̄, ȳ, t) F ȳ

are arbitrary locations over the plate).

3. Consider the design of a linear quadratic gaussian (LQG) compensator for the following system with input

and output

u y

= +

θ̈ u d (1)

= +

y θ n

where and are uncorrelated white noise processes.

d n

(a) Compute the controller gain matrix in order to minimize the following cost function

G

1 ∞

! 2 2

" #

+ dt

= θ u

J 2 0

(b) Compute the closed-loop poles.

(c) Compute the observer gain matrix when = 1 and = 0.01.

W W

L dd nn

(d) Compute the observer poles.

(e) Write the state-space realization of the LQG compensator.

(f) Write the transfer function of the LQG compensator.

ID codes: 072652 SPA 2 liv, 081225 SPA Mag, 091932 SPA Mag

Date: 10 July 2013

Instructor: Lorenzo Dozio

1. Consider the dynamic system in Figure 1, where the homogeneous elastic beam B-C is assumed to be

massless. The beam B-C is rigidly connected to the homogeneous beam A-B and carries the tip point

mass M . The point B and the tip mass M are elastically supported by two springs of stiffness k and k ,

1 2

respectively. v(t)

w(x, t) M

A C

B x

EJ, ℓ

EJ, m, ℓ k

1 k

2

Figure 1: Dynamic system.

According to a single-term approximation of the transverse dynamics of the beam A-B as w(x, t) =

2

(x/ℓ) u(t), use the assumed modes method to derive the equations of motion governing the free vibrations

of the system in terms of the generalized displacement u(t) and the absolute displacement v(t) of the mass

M .

2. Consider the lightly damped, rectangular thin symmetrically laminated pl