Dynamics and Control of Space Structures - Exercises

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

Date: 20 November 2014

Instructor: Lorenzo Dozio

1. Let’s consider the unrestrained composite rod in Figure 1 made by connecting two uniform rods together.

x x

1 2

E , A , ρ E , A , ρ

1 1 1 2 2 2

ℓ ℓ

1 2

Figure 1: Problem 1.

(a) Write the exact boundary-value problem governing the free longitudinal vibrations of the system.

(b) Write the corresponding eigenvalue problem and frequency equation.

(c) Discuss the special case when the material of the two parts is the same (E = ) and they have the

E

1 2

same length (ℓ = ).

ℓ

1 2

2. A random vibrating table used to test some flying equipments is capable of providing a zero mean value

2

acceleration level with constant power spectral density of 0.02 g /Hz over the frequency range 10 1000

−

Hz.

(a) Compute the root mean square value of the acceleration level of the table.

(b) Compute the root mean square value of the displacement level of the table.

3. A homogeneous clamped-clamped beam is equipped with a collocated actuator-sensor pair in the middle

in order to reduce the transverse vibration of the structure induced by a random distributed disturbance

having uniform spatial variation and constant power spectral density. Show the equations and discuss

the design of a frequency-shaped LQR control system incorporating an appropriate filter on the control

input. Assume an ideal force actuator, an ideal accelerometer sensor and a design model including only

the fundamental model of the beam. [Describe all the steps and define all the quantities you need to solve

the problem] ID codes: 072652 SPA 2 liv, 081225 SPA Mag, 091932 SPA Mag

Date: 4 February 2015

Instructor: Lorenzo Dozio

1. Let’s 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

distribution p and covariance C (τ ) = W δ(τ ). The plate is equipped at locations (x , y ) with an array

pp k k

0

of N identical inertial actuators and collocated accelerometer sensors. Each actuator is modeled as a rigid

casing m , rigidly attached to the plate, and an internal moving mass m and suspension stiffness k .

c a a

The casing of each accelerometer sensor has mass m . Assume that the control force f applied by the

s c

actuator is given by f (t) = k i(t), where the electric current i(t) is related to the control voltage v(t) by

c m

the following equation

di(t) δ̇(t) = v(t)

+ Ri(t) + k

L m

dt

in which δ is the elongation of the actuator, and L and R are the inductance and resistance of the actuator,

respectively.

(a) Derive a reduced-order state-space model required for the design of a stochastic direct output optimal

control given by

= (1)

u(t) −Gy(t)

where is the vector of actuator voltages and is the vector of velocity signals obtained by appro-

u y

priate integration of the accelerometer sensors.

(b) Show the design equations of the gain matrix in Eq. (1) and explain the solution procedure. The

G

control system should be aimed at minimizing the expected value of a cost function including the

kinetic energy of the plate and the control effort.

(c) Discuss the spillover effects related to the design of the previous control system.

2. Let’s consider a homogeneous cantilever beam of mass per unit length m, flexural stiffness EJ and length

ℓ. The beam carries a tip mass M , which is connected to the ground by the viscous dashpot c. The

jωt

clamped boundary of the beam is subjected to a prescribed harmonic motion y(t) = Y e . According

to a one-term approximation of the beam dynamics based on the static deflection curve of the cantilever

with end-load, write the equations governing the motion v(t) of the tip mass and, assuming a harmonic

jωt

response v(t) = V e , discuss the effect of the variation of c on the transmissibility function T (ω) = V /Y .

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

Date: 1 July 2015

Instructor: Lorenzo Dozio

1. A cantilever Euler-Bernoulli beam of mass per unit length bending stiffness and length

m(x), EJ(x) ℓ,

carries a tip rigid body of mass and moment of inertia about given by . The body has a center of

m C I

c c

mass that is offset from the point of attachment (at = by a distance The body mass center is

x ℓ) e. C

assumed to be on the axis so that transverse vibrations do not excite torsional vibrations and vice versa.

x

Write the boundary-value problem governing the free bending vibrations of the system and determine the

condition to find the natural frequencies when the beam has uniform properties.

z m , I

c c

C x

EJ(x), m(x), ℓ e

Figure 1: A cantilever beam with a rigid body attached to the right end.

2. A high-aspect ratio wing of length is idealized as a cantilever Euler-Bernoulli beam in bending deforma-

ℓ

tion and a slender bar in torsion. The transverse deflection of the elastic axis (e.a.) along the spanwise x

direction is indicated by and the twist angle of the section about the elastic axis is

w(x, t) θ(x, t). EJ(x)

and are the wing’s bending and torsional stiffness, respectively. The mass per unit length of

GJ(x) m(x)

the wing is distributed along the span with the local center of mass (c.m.) located at (x) aft of the

y m

elastic axis and the local moment of inertia about the elastic axis is (x). The structure is subjected to

I

θ −a|τ |

a transverse random load of spatial distribution and covariance ) = (a 0), applied along

p(x) C(τ e >

the axis a.c. located at (x) with respect to the elastic axis.

y a

y w

θ

c.m. a.c.

a.c. y

e.a.

x

c.m. e.a. y y

m a

Figure 2: High-aspect ratio wing idealized as a beam.

Assuming the system undergoes small deflections and rotations, write the approximate dynamic model

of the wing according to the assumed-modes method, and show how to compute the variance of the

root bending and torsional moments from a state-space reduced-order lightly-damped modal model of the

structure.

3. Show and explain how to design a linear quadratic regulator having a prescribed degree of stability α.

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

Date: 15 July 2015

Instructor: Lorenzo Dozio

1. A robotic arm is idealized as a rigid hub of moment of inertia J and radius R freely rotating about

0

point O, with a flexible link of length ℓ modeled as a cantilever Euler-Bernoulli beam of bending stiffness

EJ. The rigid body is subjected to a prescribed torque T (t) about the axis of rotation. The flexible

c

link is represented by a lumped-parameter model obtained by discretizing its distributed mass with N

concentrated masses M positioned at distance ℓ from the beam root.

i i

i 2 M N

.

b .

2 .

.

M 2

M EJ, ℓ

1

T b

c 1

θ

O i 1

J , R

0

Figure 1: Rigid hub with a flexible appendage – lumped-parameter model.

(a) Write the linearized equations of motion of the system in terms of the rotation angle θ(t) and the

transverse displacement w (t) of each point mass. Assume small rotational speed θ̇ and small deflec-

i

tions w .

i

(b) Write a reduced-order model of the system derived in (a) based on an appropriate modal represen-

tation of the flexible link.

(c) Write the transfer function H(s) = θ(s)/T (s) from the model in (b) including only the fundamental

c

flexible mode of the link and sketch the Bode diagram of H(s).

2. A LTI system with input u and output y is governed by the following equations

z̈(t) = u(t)

y(t) = z(t)

Compute the LQR gain vector such that the desired closed-loop state matrix is given by

0 1

A = √

d 3

−1 −

3. Let’s assume that, from the fi