Riassunto esame Space structures, Prof. Dozio Lorenzo, libro consigliato Fundamentals of Structural Dynamics, R.R. Craig, A.J. Kurdila

DYNAMICS OF SPACE STRUCTURES

Deterministic vs NON-det. Function

ex. sin(ωt) ex. Random f

INTRO EXAMPLE

Hp: RIGID body

1° METHOD

∑ θ̈(t) = M_d(t) (From fund. Law of rot. dyn.)

ODE II ord. with CONST coeff.

θ̈ = dθ̇/dt

=> θ = (1/s) ∫ M_d dt

2° METHOD

Θ(t) → Θ(s) = ∫₀^∞ θ̈(t) e^st dt

θ̈(t) → sΘ(s) - sΘ(0) - Θ(0)

M_d(t) → M_d(s)

∑ θ̈(t) = M_d → s² ∑ Θ(s) = M_d(s)

Frequency description approach

I can demonstrate this property with INT. P.P.

Algebraic Eq.

Θ(s) = 1/s · 1/s² M₀(s)

Bode diagrams

s = jω

H_R(s) = 1/s · 1/s²

Transfer function

H_P:

s = jω

40

dB!!!

Rolloff rate (per decade) = -40 dB

or

-2 (relating to the exponent)

MASS EFFECT

ω ↑

apparent mass ↑ (seen from fr. approach)

From freq. POV

THINK

H(jω)

SORT OF INERTIA MASS EFFECT

ω

WARNING!

M₀ value to be taken into account

be careful

Now

EXAMPLES OF DANGEROUS PHEN.

• Sloshing
• Thermal eff
• ...

GOLDEN RULES

• Short in time => large in freq ; short in freq => large in time
• Separation of frequency
  • Rule of thumb: 1 decade of separation (from ω_res)

EXAMPLE

Let's introduce internal instruments: how they transmit vibrations to the structure? How all behave?

2 OPTIONS

1. integrated model (expensive option)
2. Keep the equipment separated, measure Fj, see how Fj interacts with structure

Equipment

(standalone)

m_e ü_e(t) + k_e u_e(t) = F(t)

F_e(t) = k_e u_e(t)

s² m_e u_e(s) + k_e u_e(s) = F(s)

u_e = \frac{1}{s^2 m_e + k_e} F(s)

F_e(s) = \frac{k_e}{s^2 m_e + k_e} F(s)

H_e(s)

w_e = \sqrt{\frac{k_e}{m_e}}

E∫(x) = E∫

m(x) = x

E∫ w_xxxx + m ü = 0

w(x, t) = W(x) T(t)

Technique: separation of variables (to solve particular PDE)

E∫ W_xxxx T + m W T̈ = 0

f(t) f(x) => f(t) = F(x) = CONST (the only possibility that a fcn of x can be equal to a fcn of t)

Ẍ̈ T = E∫ W_xxxx

E∫ W_xxxx W - ω² = E∫ W_xxxx - mω²W = 0

IN GENERAL, WE'LL ASSUME

w(x,t) = W(x)e^jωt

Continuing with E∫, m const

E∫ W_xxxx - mω²W = 0

W_xxxx - β⁴ W = 0

K [W(x)] - ω² M [W(x)] = 0

ESW_xxxx - mω²W = 0

Now, instead,

Mu(t) + Ku(t) = 0

u(t) = ye^jωt

=> (K - ω²M)y = 0

W_xxxx - β⁴W = 0

β⁴ = mω² / ES

W = A cos βx + B sin βx + C cosh βx + D sinh βx

I still don't have used BCs

I have to specify A, B, C, D (From BCs)

det(...) = 0 (enforcing non trivial solution)

characteristic (or frequency) equation

f(βL) = 0

ω_i, W_i(x)

first eigenpair

det(y) = 0

cos β L cosh β L - 1 = 0

(i = 1, 2, ..., ∞)

W_i(x) = A_i [sin β_i x + sinh β_i x : sin β_i L - sinh β_i L

cos β_i L - cosh β_i L (cos β_i x + cosh β_i x)]

Free-Free Beam

Axial Case

1. ∫₀^l δ_u E A u_x dx = - ∫₀^l δ_u m ü dx
2. by P
3. δ_u E A u_x |₀^l - ∫₀^l δ_u (E A u_x)_x dx = - ∫₀^l δ_u m ü dx =>

{ - (E A u_x)_x + m ü = 0 (EoM)

E A u_x (0) = 0

E A u_x (l) = 0 (BCs)

Hp.

E A (x) = E A

m (x) = m =>

{ - E A u_xx + m ü 0

u_x (0) = 0

Ex.

Exact analysis

Dofs: ∞ + 1

virtual work

relative displ. (of the endpoints of the spring)

EOM - beam

EOM - spring/mass

substituted weak form

BCs

Note: this time we have 2 EOM

Note: SAME ω (of the entire system)

NO DEPENDANCE FROM x

δy₀ = ∫ (IₚN₀ₓN₀₀ dx + ∫ me²N₀ₓN₀₀ dx) ₀^L ẏ₀(t) +

+ ∫ N₀w(t)N₀WSW0(t) ^L₀ ẏ₀(t) + G ∫ N₀N₀ dx ẏ₀(t) = 0

^{M_ww ÿ_w(t) + M_w₀ ÿ_₀(t) + k_ww y_w(t) = 0}

^{M_ow ÿ_w(t) + M_oo ÿ_₀(t) + k_oo y_₀(t) = 0}

y = ( y_w(t) y_₀(t) )

=> M ** ÿ(t) + K y(t) = 0

y_i(t) = y_ie^jωt

(K - ω²M) u = 0

det ( ) = 0 => ω_i² ; y_i

M = ( M_ww M_w₀ )

 ( M_ow M_oo )

K = ( k_ww 0 )

 ( 0 k_oo )

note that M_wo = M^T_ow

If the distance btw the 2 axis -> 0

(>) problems decoupled

∑_j Φ_i(x) ∫[Φ_z(x)q_z(t)dρq_z(t) + Φ_i(x)q₀(x)dF(t)]

= Φ_i(x) p₀(x)dρF(t)

(i = 1, 2, ..., ∞)

∑ μ_iq_i(t) + μ_iω_z²q_i(t) = Φ_i(x)p₀(x)dρF(t)

i = 1, 2, ..., ∞

easier than before

ODE oscillators

simple example

EJ, m,l

ρ₉step(t)

w(x,0)=0

ω̇(x,0)=0

begin at rest at initial time

EJ(1) = EJ

m(x) = m

ω(x,t) = ∑ sin(πx/l) q_s(t)

d_(x,t) = ∑ ϕ_i(x)q₉(t)

q̇_i(t) = -ω_iA_isin(ω_it) + ω_iB_icos(ω_it)

q̇_i(0) : o ⇒ B_i : o

q_i(0) = o ⇒ A_i :

q_i(0) = o ⇒ A_i:

=> q_i(t) = ^4p_o/_ωil²> [1-cos(ω_it)]

t > 0

i = 1, 3, 5,...∞

ω_i² = ^E3/_po(iL)⁴

...

ω(x,t) = ∑_i=1,3,5... sin(iπx/L) q_i(t)

= ^4p_oL/_E3π⁵ ∑_i=1,3,5... ¹/_i³ sin(iπx/L) [1-cos(ω_it)]

...

M(x,t) = E3 ω_xxx(x,t)

...

T(x,t) = - ^4p_oL2/_u² ∑_i=1,2 cos(iπx/l) [1-cosω_it]

...

so, if we want

a) ω(x,t)

= ^4p_oL/_E3π⁵ ∑_i=1,3,5... ¹/_i³ sin(iπx/L) [1-cos(ω_it)]

...

HIGHER MODES (ω_i) => have decreasingly... to ω(x,t)