Energy exchange in multi-degree- of-freedom systems

where:

B R

=

+ +

(

+

)

[ ]

+

(

+

) (

+

)

+

μ ω ω ω ω

γ μ ω ω

2 1 24

2 14

1 2 1 22 2 12

2 1 22

2 12

2

Δ Δ Δ Δ Δ Δ

Δ Δ RR² 1 2

2 12

2 2 2

1 2 1 22

2 12

1

Δ Δ

Δ Δ Δ Δ

+

( )

(

−^μ

)

_⎡⎣

(

^ω −^ω

)

+( + )

(

^ω + ^ω

)

_⎤⎦ ^(7.44)

If resonator 2 has no direct excitation, S_f2 = 0, then the power dissipated by the resonator must equal the power transferred from 1 to 2.

〈P2,dissipated〉=〈P_1,2〉; Δ2ε2= B(ε1−ε2) or

ε ε

2

1 2

= +

B

Δ B (7.45)

The largest value of ε2 is ε1, which occurs when the coupling (determined by B) is strong compared to the damping Δ2.

When the two resonators are identical and have stiffness coupling only, then Equation 7.44 becomes:

B R R

= 2 =

4 2

2 2 2

2 2

Δ

Δ ω Δω (7.46)

ε ε

ω ω

2 1

2 2 2

2 2 2

2

1 2

= + R

R Δ

Δ (7.47)

Additional points can now be added in regard to the power fl ow from reso- nator 1 to 2:

5. The power fl ow is proportional to the actual vibration energy difference of the systems, the constant of proportionality being B.

6. The parameter B is positive defi nite and symmetric in system parame- ters; the system is reciprocal and power fl ows from the more energetic resonator to the less energetic one.

7. If only one resonator is directly excited, the greatest possible value of energy for the second resonator is that of the fi rst resonator.

7.4 Energy exchange in multi-degree-

with the orthogonal property and normalizations given by

〈ψiαψiβ〉ρi=δα,β (7.50)

The boundary conditions satisfi ed for ψiα include the clamped condition.

Suppose that the spectral densities of the modal excitations F_iα(t) =∫p_iψiαdx_i are fl at over a fi nite range of frequency Δω and that within this band there are Ni= niΔω modes of each subsystem. The modes for these subsys- tems are illustrated in Fig. 7.6(a). Each mode group represents a model of the subsystem.

In the application of SEA, this model has some very particular properties as illustrated below.

1. Each mode α of subsystem i is assumed to have a natural frequency ωiα

that is uniformly probable over a frequency interval Δω. This means that each subsystem is a member of a population of systems that are gener- ally similar physically, but differ enough to have randomly distributed parameters. The assumption is based on the fact that normally identical structures or acoustic spaces will have uncertainties in their modal parameters, particularly at higher frequencies.

y1 y1

y2

y2 = 0 y1 = 0

p1

p2

p1

p2

1 2 1 2 1

2 2

(a) (b)

(b)

y2(b)

‘Clamped’

(c) 7.5 Blocked or decoupled subsystems.

Subsystem 1

1 2 a N₁

1 2 s N₂

Subsystem 2 Subsystem 1 Subsystem 2

Total Power Π12

Π_as

(a) Blocked (b) Connected

1 2 a N₁

1 2 s N₂

7.6 Modes for subsystems: (a) individual modal groups with subsystems blocked; (b) interactions of mode pairs with subsystems connected at the junction.

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2. We assume that every mode in a subsystem is equally energetic and that the modal amplitudes y t y

m x

i

i i i

α( ) =

∫

ρ ψα ^d i are incoherent; that is,

y y_{iα iβ} =δ_{α β}, y_i2_α (7.51)

3. This assumption requires that we correctly select mode groups, which is an important guide for appropriate SEA modelling. This also implies that the modal excitation functions Fi are drawn from random popula- tions of functions that have certain similarities (such as equal frequency and wave number spectra) but are individually incoherent.

4. The damping of each mode in a subsystem is the same.

‘Unlock’ the system and consider the new equations of motion, which are:

y c

y y p x y y x y y k

i r

i

i i i

i ij i j j

j

ij i j j i

+ i + = + ( ) + −( ) ( ) +

ρ

μ γ

Λ , 1 , _{jj j}

i

y i j i j

≠ , , =1 2, , . . . ρ (7.52)

One now expands these two equations in the eigenfunctions ψ1α(x1) and ψ2σ(x₂) to obtain:

m y_{1 α 1}y_{α α α}²y F_{α ασ σ}y _{ασ σ}y R y_{ασ σ}

σ

ω μ γ

+ +

[

^Δ

]

^{= +}

∑

^{[ + + ] (7.53)}

m y₂ σ ₂yσ σ σ²y Fσ σα αy σα αy R yσα α α

ω μ γ

+ +

[

^Δ

]

^{= +}

∑

[ ^{+ +} ] ^(7.54)

where α is for subsystem 1, and σ is for subsystem 2, and Δi = c_ri/ρi; the mass of the subsystem is mi. The coupling parameters are [1]:

μασ =

∫

μ¹²[ ^{1 2}( )¹] ( )ψα ¹ ψσ[ ²( )¹] ¹

junction

d etc.

x x x, x x x x, (7.55)

μσα =

∫

μ²¹[ ¹( )^{2 2}]ψσ[ ¹( )² ] ( )ψα ^{2 2} junction

d etc.

x x ,x x x x x, (7.56)

where the integrations are taken along the junction between the subsystems and, therefore, over the same range of x1, x2 in both integrals. Conservative coupling requirements are met by μασ=μσα=μ, γασ=γσα=γ, R_ασ= R_σα= R, or μ21=μ12=μ, γ12=γ21=γ, R₁₂= R₂₁= R.

The coupled systems may now be represented as shown in Fig. 7.6(b) with the interaction lines showing the energy fl ows that result from the coupling. Suppose we study the energy fl ow between mode α of subsystem 1 and mode σ of subsystem 2. Modes α and σ have energies εα and εσ. The modal energies of the subsystem 1 modes are all equal; that is εα = ε1 = constant and εσ=ε2= constant.

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The inter-modal power fl ow:

P_ασ = B_{ασ ω ωa σ}(ε ε1− 2) (7.57)

B_{ασ ω ωα σ ασ}

ω λ

= π 2

1 2

Δ Δ

Δ (7.58)

where

λ≡

[

μ ω^{2 2}+

(

γ²+ μ

)

+ ² ω²

]

1 2

2 R R

Δ Δ (7.59)

when μ²<< 1, Δ/ω<< 1 and λ<< 1.

The total power fl ow from all N1 modes of subsystem 1 to mode σ of subsystem 2 is:

〈P_1,σ〉=〈B_ασ〉N1(ε1−ε2) (7.60) Finally the total power fl ow from subsystem 1 to subsystem 2 is found by summing over the N₂ modes of subsystem 2:

P B N N

R R

12 1 2 1 2

2 2 2 2 2

1 2

1 2

2

2

= ( − )

= +

(

+

)

+ ( − )

ασ ε ε

ωμ ω γ μ ω

δω δω ε ε

πΔ (7.61)

when δωi=Δω/N_i is the average frequency separation between modes.

If the total energy of subsystems 1 and 2 is defi ned as E_1(total) and E_2(total):

ε1 ε

1 2

2 2

= E^{( )} = ^{( )} N

E N

1 total total

, (7.62)

P B N N E

N E

N

E N

N

12 1 2

1 1

2 2

12 1

1

= ⎡ −

⎣⎢ ⎤

⎦⎥

= −

( ) ( )

( ) ασ

ωη

total total

total 2 2

E2_(total)

⎡⎣⎢ ⎤

⎦⎥ (7.63)

where η12=〈B_ασ〉N₂/ω.

If we defi ne η21= N₁η12/N₂ then:

〈P₁₂〉=ω (η12E_1(total)−η21E_2(total)) (7.64)

η12 and η21 are called the coupling loss factors for subsystems 1 and 2.

ωη12E_1(total) represents the power lost by subsystem 1 due to coupling, just as the quantity ω1η1E1(total) represents the power lost by subsystem 1 due to damping. If subsystem 1 is connected to subsystem 2 with energy E_2(total), subsystem 1 will receive an amount of power ωη21E_2(total) from subsystem 2.

In the basic relationship N₁η12= N₂η21, replacing the modal count N_i with modal density time, the frequency bandwidth niΔω gives

n1η12= n2η21 (7.65)

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The system shown in Fig. 7.6(b) is simply represented in Fig. 7.7. The power fl ow equations for subsystems 1 and 2 are:

〈P_1,in〉=〈P1,dissipated〉+〈P₁₂〉

=ω [η1E_1(total)+η12E_1(total)−η21E_2(total)] (7.66)

〈P2,in〉=〈P2,dissipated〉+〈P21〉

=ω [η2E2(total)+η21E2(total)−η12E1(total)] (7.67) Considering the case where only one of the systems is directly excited by one external source, that is, setting <P2,in>= 0:

E E

N N

2 1

2 1

21

2 21

total total ( ) ( ) =

+ η

η η (7.68)

A solution of Equations 7.66 and 7.67 gives:

E P P

1 D

1 2 21 2 21

total

in in

( )= , (η η+ ) + , η

ω (7.69)

E P P

2 D

2 1 12 1 12

total

in in

( )= , (η η+ ) + , η

ω (7.70)

where

D = (η1+η12)(η2+η21) −η12η21 (7.71) The dissipated power in subsystem 1 can be evaluated by

〈P1,dissipated〉= 2πfη1E₁ (7.72)

where E1 is the total dynamic energy of the subsystem modes at frequency f (Hz) and η1 is the damping loss factor.

As shown in Fig. 7.7, the net transmitted power can be represented in several different forms, for example

〈P12〉= 2π Δf β12(ε1−ε2) (7.73)

N1 N2

E1,tot Π12 E2,tot

Π1,in

Π1,diss

Π2,in

Π2,diss

7.7 An energy transfer and storage model.

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where ε1 and ε2 are the average modal energies (energies per mode) of the two subsystems in a frequency band Δf, and β12 is a coupling factor that depends only on the physical properties of the two coupled subsystems:

ε= E = δ N E f

Δf (7.74)

where N is the mode number and δf is the average frequency spacing between the modes of the subsystem in the frequency bandwidth Δf.

〈P12〉= 2πf(η12E1−η21E2) (7.75)

〈P1,in〉=〈P1,dissipated〉+〈P12〉= 2πf(η1+η12)E1− 2πfη21E2 (7.76)

〈P2,in〉=〈P2,dissipated〉+〈P21〉= 2πf(η2+η21)E2− 2πfη12E1 (7.77) ε

ε η η η

2 1

21

2 21

1 0

= + ( ^P,in = ) (7.78)

If N1= 1:

P₁₂ =2^πβ₁₂

(

E f₁^Δ −E₂^δf₂

)

^(7.79)

P E f E f E

n

E

f n

12 12 1 1 2 2 12

1 1

2 2

Δ =2^πβ

(

^δ − ^δ

)

^{=β ⎛}_{⎝⎜ ( )ω} ⁻ _{( )ω} ^⎞_⎠⎟ ^(7.80)

For a complex system with N subsystems, the SEA equations can be written as

ω

η η η η

η η η

n n n

n n

i i

N N

i i

1 11 1

1

21 2 1

12 1 2 22 2

2

⎛ +

⎝⎜ ⎞

⎠⎟ − −

− ⎛ +

≠

≠

∑

∑

...

⎝⎝⎜ ⎞

⎠⎟

− +

...

. . . .

. .

... ...

. . . ... ...

η1Nn1 nN ηNN ηNNi i N

N N

E n

E

≠ n

⎛

∑

⎝⎜ ⎞

⎠⎟

⎡

⎣

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎤

⎦

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎧

⎨

⎪ ₁¹ . . . . . .

⎪⎪⎪

⎪⎪⎪

⎩

⎪⎪

⎪⎪

⎪⎪

⎫

⎬

⎪⎪

⎪⎪

⎪⎪

⎭

⎪⎪

⎪⎪

⎪⎪

=

⎧

⎨

⎪⎪

⎪⎪

⎩

⎪⎪

⎪⎪

⎫

⎬

⎪⎪

⎪⎪

⎭

⎪ P

P_N

1

. . . . . .

⎪⎪⎪

⎪

(7.81)

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If each subsystem has n degrees of freedom, the total of n × N dynamic equations are reduced to the total of N SEA equations, which speeds up the computation.

In summary, modes of a system become localized to various subsystems where local modes of the subsystems of a system are statistically described;

local modal properties/wave properties are assumed to be maximally dis- ordered in a state of maximum entropy. Simple expressions for ensemble average energy fl ow between coupled subsystems exist, since vibration energy fl ow between coupled subsystems is proportional to difference in modal energies, that is, average energy per mode. The SEA power balance equations which govern response of a system in a given frequency band can be derived from the principle of energy conservation.

The above SEA modal description is good for a qualitative introduction to SEA theory since the SEA equations derived from a modal approach are based on several assumptions, in particular for the extension of the two oscillator results to multimodal systems. The exact SEA equations as per Equations 7.66 and 7.67 can also be derived from a wave approach [8]. This is discussed in the next section.

7.5 Wave approach to statistical energy