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

Date: 4 September 2013

Instructor: Lorenzo Dozio

1. Consider a slender beam of flexural rigidity EJ and mass per unit length m, resting on three simple

supports as shown in Figure 1. Assume the beam to be uniform with ℓ = ℓ = ℓ.

1 2

EJ, m

ℓ ℓ

1 2

Figure 1: Uniform beam resting on three simple supports.

(a) Determine the frequency equation governing the exact free vibration problem of the beam.

(b) Comment on the cases arising from the solution of the equation obtained before.

2. Consider the dynamic system in Figure 2, consisting of a rigid body and two symmetrical cantilever flexible

beams, each modelled by two discrete masses m and m , positioned at a distance ℓ/2 and ℓ, respectively,

1 2

from the beam root. The moment of inertia of the rigid body is J and each massless beam has constant

0

flexural rigidity EJ. The rigid body is equipped with a finite bandwidth actuator capable of applying a

control torque T (t) about the axis of rotation and a finite bandwidth sensor measuring the rotation angle

θ(t). T, θ m 2

m 1

## EJ

J 0 ℓ/2 ℓ/2

Figure 2: Dynamic system.

(a) Assuming a first-order dynamic model for the actuator and sensor devices, write the state-space form

of the equations governing the dynamics of the system.

(b) Discuss the design of a linear state-feedback control law based on a pole placement technique.

(c) Discuss the design of a linear state observer based on a pole placement technique.

3. Based on experimental data, it has been found that a physical quantity of an ergodic random process is

characterized by the following power spectral density:

2

1 + 3(ωa)

S(ω) = (1)

2

2

[1 + (ωa) ]

(a) Write the transfer function of the linear system that, when excited by white noise, has an output

with spectrum S(ω) given by Eq. (1).

(b) Using the Lyapunov equation, compute the corresponding output variance when the white noise has

intensity W .

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

Date: 18 September 2013

Instructor: Lorenzo Dozio

1. Consider the dynamic system in Figure 1, consisting of a lightly damped flexible beam of length connected

at the tip (x = to a homogeneous rigid bar. The beam is fixed at the root (x = 0) and has bending

ℓ)

stiffness torsional stiffness mass per unit length and mass polar moment of inertia

EJ(x), GJ(x), m(x)

per unit length (x). The rigid bar has length mass per unit length and is elastically suspended

I e, m

p R

at one end by a spring of stiffness K.

z y p K

EJ, GJ, m, I , ℓ

p m R e

x

Figure 1: Dynamic system.

The system is forced as in Figure 1 by a random transverse load per unit length which is assumed

p(t),

to have constant spatial distribution , zero mean value and the following power spectral density

p 0

2

ω

(ω) =

S pp 2 2

(1 + )(1 + 4ω )

ω

Assuming the system undergoes small deflections and rotations, write the model and the equations for

computing the variance of the root bending and torsional moments using the mode displacement method.

Use a global Ritz-Galerkin discretization to approximate the beam dynamics.

2. Consider a linear time-invariant system described by the following dynamics

=

θ̈(t) u(t)

where is the rotational angle and is the control torque.

θ u

(a) Find the optimal input which minimizes the control energy

u(t)

Z 1

1 2

= (t)

J u dt

2 0

with the following constraints

= 0, = 0, = 1, = 0.

θ(0) θ̇(0) θ(1) θ̇(1)

(b) Write the closed-loop response of the system in terms of rotation angle and angular velocity and plot

the corresponding trajectories.

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

Date: 7 July 2014

Instructor: Lorenzo Dozio

1. Let us consider a homogeneous bar of length ℓ, torsional stiffness GJ and mass polar moment of inertia

per unit length I . The bar is fixed at one end (x = 0) and carries at the other end (x = ℓ) a rigid disc of

p

moment of inertia I .

d

(a) Determine the exact eigenfrequencies and eigenmodes of the system.

(b) Determine the orthogonality conditions of the eigenmodes.

(c) Determine the approximate value of the fundamental frequency of the system by the Rayleigh’s

quotient using an appropriate static curve as trial function.

## R

(d) Show that provides an upper bound for the lowest eigenfrequency of the system under study.

## R

2. Let us consider a homogeneous isotropic rectangular plate of side lengths a and b, thickness h, with simply

supported edges. The plate is subjected to a transverse distributed random load p(t) of uniform spatial

−a|τ |

distribution p and covariance C (τ ) = e (a > 0). The plate is equipped at location (x , y ) with

pp

0 0 0

an accelerometer and a band-limited inertial actuator modeled as an elastically suspended mass m and

a

internal dynamics expressed in the Laplace domain as

## F

b (b > 0)

δ (s) +

δ(s) = c

s + b k

a

where δ is the elongation of the actuator, δ is the elongation due to the control, k is the stiffness of the

c a

elastic support, and F is the force transmitted by the actuator to the plate.

Write the state-space model of the system to be adopted for the design of a LQG control system aimed

at minimizing the kinetic energy of the plate. Use a modal representation of the plate dynamics.

3. Let us consider a homogeneous rod of length ℓ, axial stiffness EA and mass per unit length m. The rod is

fixed at one end (x = 0) and carries a point mass of nominal value m at x = 3ℓ/4. A disturbance force

p

acts at the tip of the rod, and a point control force is exerted at x = ℓ/2.

(a) Assuming a stiffness-proportional damping model, write the dynamic equations of the system ac-

cording to a finite element model with four equally spaced two-node elements using a mass lumping

approximation.

(b) Write the cost function J corresponding to the design of an infinite-horizon linear quadratic regulator

which is robust to the uncertainty in the value of m and rejects the effect of the disturbance source

p

on the tip position.

(c) Derive the optimal control solution corresponding to the minimization of J.

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

Date: 09 September 2014

Instructor: Lorenzo Dozio

1. The system in Figure 1 consists of a rigid hub with a cantilever flexible appendage carrying a tip payload.

The hub is modeled as a rigid disk hinged at its center, with radius and moment of inertia . The

## R I

h

appendage is modeled as a homogeneous slender beam of length mass per unit length and flexural

ℓ, m

stiffness The payload is modeled as a mass and inertia . A prescribed torque acts on the

EJ. m I M

t t c

hub. Y x

w m , I

t t

y EJ, m, ℓ

θ X

## M

c

R, I h

Figure 1: Dynamic system of problem 1

(a) Write the linearized differential problem governing the planar dynamics of the system (equations

exact

of motion and boundary conditions) in terms of rotation angle and transverse deflection

θ(t) w(x, t)

of the flexible appendage. Assume small rotation speed and small

θ̇ w.

(b) Write the equations of motion of the system according to an appropriate Ritz-Galerkin

approximate

discretization and explain how to compute the bending moment along the flexible appendage using

the mode acceleration method. L L

1 2

2. The system in Figure 2 consists of a rigid body of mass and

M v(t)

moment of inertia with respect to the center of mass (CM).

## J M, J

The body is supported by two massless rods of length and

damping coefficient proportional to the stiffness. The sys-

β β

## CM

β

tem is forced by a prescribed base displacement Write the

w. ℓ,

w(t) ℓ, 2EA,

time-domain equations to compute the variance of the relative EA,

displacement = of CM when the base acceleration is

z v w

a white noise of intensity .

W Figure 2: Dynamic system of problem 2

3. Let’s consider a LTI system = + where the control variable is actually provided by

ż(t) az(t) bu(t), u

an actuator having an internal dynamics with respect to the desired displacement expressed by the

z c

following relation in the Laplace domain

1 1

= (s) +

z(s) z u(s)

c

1 + τs k

(a) Write the equations of the system in state-space form when the output variable is ż.

(b) Compute the open-loop poles of the system when = = and = = 1.

a b k τ

−1 √

3

(c) Design a state feedback controller such that the closed-loop poles of the system are = .

s −3.5±j

1,2 2

(d) Design a state observer such that the observer poles are = and =

s s

−7 −10.

1 2

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

Date: 23 September 2014

Instructor: Lorenzo Dozio

1. A lightly damped, homogeneous, simply supported beam of length ℓ, bending stiffness EJ and mass per

unit length m, is induced to vibrate at the supports by a prescribed sinusoidal acceleration ẅ at frequency

## B

ω . Assuming that the motion of the beam can be well represented by only its fundamental vibration

0

mode,

(a) compute the maximum value of |w |, where w is the vertical displacement at the center of the beam

## C C

relative to the base;

(b) compute the maximum value of the bending moment along the beam. w

EJ, m, ℓ B

w

## C

Figure 1: Problem 1.

2. Let’s consider the system in Figure 2, consisting of a flexible cantilever beam of length ℓ, bending stiffness

EJ(x ) and mass per unit length m(x ), carrying a tip mass M and elastically connected through the

1 1 t

spring K to a rigid bar of mass per unit length m (x ) and length L. The rigid bar is also supported by

1 2

## R

the spring K and dashpot C.

2 x

2 p 0

x

1 K 1 m , L

## R

EJ, m, ℓ K C

2

M t

Figure 2: Problem 2.

The system is subjected to a distributed load p (t)x acting on the rigid bar as an ergodic random process

0 2

of power spectral density

2

ω

S(ω) = 2 2

(1 + ω )(1 + 4ω )

Assuming that the transverse displacement of the cantilever beam can be approximated as w(x , t) =

1

2

(x /ℓ) u, write the state-space equations to compute the variance of the root bending moment of the

1

cantilever beam.

3. Given the system z̈(t) = u(t), design a LQR controller to achieve a prescribed degree of stability α when

= and R = 1. Compute the closed-loop eigenvalues to check your design.

Q 0 2×2

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### DESCRIZIONE APPUNTO

(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.
La raccolta è un ottimo punto di parte anche per la risoluzioni degli esercizi proposti nel corso "Dynamics and Control of Flexible Structures" (prof. Masarati)

(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.
The collection is a good starting point also for the resolution of the exercises submitted during the course "Dynamics and Control of Flexible Structures" (prof. Masarati)

DETTAGLI
Corso di laurea: Corso di laurea magistrale in ingegneria spaziale
SSD:
A.A.: 2018-2019

I contenuti di questa pagina costituiscono rielaborazioni personali del Publisher rrmg di informazioni apprese con la frequenza delle lezioni di Dynamics and Control of Space Structures e studio autonomo di eventuali libri di riferimento in preparazione dell'esame finale o della tesi. Non devono intendersi come materiale ufficiale dell'università Politecnico di Milano - Polimi o del prof Dozio Lorenzo.

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