Metamaterial-based Foundation System endowed with Non-Linear Oscillators

Metamaterial-based Foundation

System endowed with Non-Linear

Oscillators

M

ORITZ

W

ENZEL

, R

OBERTO

A

NDREOTTI

, O

RESTE

S. B

URSI

DEPARTMENT OFCIVIL, ENVIRONMENT ANDMECHANICALENGINEERING, UNIVERSITY OFTRENTO, VIAMESIANO77, 38123, TRENTO, ITALY.

1

5

m

8 m

6 mm

Foundation

P-Δ Relationship

Mechanism in Motion

P-Δ Relationship

P-Δ Relationship

𝐹

(𝑣) =

𝑣 −𝑃 + 𝑘

𝑙 − 𝑙 1 −

𝑣

𝑙

𝑙 1 −

𝑣

𝑙

𝐹

(𝑣) = 𝑣 ∙ 𝑎

+ 𝑣

∙ 𝑏

𝐹

(𝑣) = 𝑣 ∙ 𝑎

𝑎

= −

𝑃

𝑙

𝑏

=

𝑘

²

𝑙(2 ∙ 𝑘

∙ 𝑙 − 𝑃)

True Relationship

Polynomial Approximation

Linearized Path

Band Gaps

Band Gaps

𝐶𝑜𝑠 𝑞 = 2k1k2− k2m1𝜔 )

2_{− 2k}

1m2𝜔2− k2m2𝜔2+ m1m2𝜔4+ 𝑎(2k1+ 2k2− (m1+ m2)𝜔2) + 3𝑏B1ഥB1(2k1+ 2k2− (m1+ m2)𝜔2

ഥ 2

Harmonic Balance Method – Poly. Non-Linearity

B₁ =1

FRF Numerical

FRF Numerical

FRF Numerical

Time History Analysis

Time History Analysis

Acknowledgements:

XP-Resilience

Ref: 721816

Conclusion



Metafoundation feasible



Negative Stiffness element

improves the performance

•

Economic design?

•

Details of the

FRF Numerical

Fourier Transformation of EOMs

Optimization

𝑴 (−𝜔

𝑿(𝜔)) +

C

(𝑖𝜔

X

(𝜔)) +

K

(

X

(𝜔))=

P

(𝜔)

(−𝑴𝜔

+ 𝑖𝜔

C

+

K

)

X

(𝜔) =

P

(𝜔)

𝑯(𝜔) = (−𝑴𝜔

+ 𝑖𝜔

C

+

K

)

Transmission Matrix

𝑺

(𝜔, 𝑘

, 𝑐

) = 𝑯(𝜔, 𝑘

, 𝑐

)

𝑺

𝑯(𝜔, 𝑘

, 𝑐

) = (−𝑴𝜔

+ 𝑖𝜔

C

(𝑐

)+

K

(𝑘

))

𝑯(𝜔) numerical Optimization

Power Spectral Density - Response

𝜎

𝑘

, 𝑐

= ׬

𝑆

𝜔, 𝑘

, 𝑐

𝑑𝜔

Variance of the Response

Optimization

Ground Motion Model

Motivation

Economic Loss

Human Lifes

Stability Issue

1

𝑘

+

1

𝑘

+

1

𝑘

+

1

−𝑘

≥ 0

𝑘

≥

𝑘

2

−

𝑘

4

+ 𝑘

𝑘

𝑘

= −

𝑃

𝑙