This presentation has been delivered at the 15th World Conference on Earthquake Engineering in Lisbon (Portugal) on 28th September 2012, and shows some preliminary results on the dynamic analysis on non-linear viscoelastic structures.

1. The 15th World Conference on Earthquake Engineering
Lisbon (Portugal), 28th September 2012
Toward an Improved Computational
Strategy for Vibration-Proof
Structures Equipped with Nano-
Enhanced Viscoelastic Devices
Evangelos Ntotsios, Alessandro Palmeri
School of Civil and Building Engineering, Loughborough University
<A.Palmeri@LBORO.ac.uk>

2. Loughborough University
2 A.Palmeri@LBORO.ac.uk

3. Outline
• Introduction
• Linear Viscoelastic Structures
• Generalised Maxwell (GM) model
• Laguerre’s Polynomial Approximation (LPA)
• State-space equations of motion
• Non-Linear Viscoelastic Structures
• Numerical Application
• Conclusions
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4. Introduction
• Energy-dissipation devices exploiting viscoelastic rubber
improve the performance of civil engineering structures to
dynamic loadings
Soong, Dargush (1997), Passive Energy Dissipation Systems in Structural Engineering
• Very complicated constitutive laws Vs. Oversimplified design
rules (particularly in presence of nano-reinforcement)
Dall'Asta, Ragni (2008), Earthquake Engineering & Structural Dynamics 37: 1511-1526
Johnson, Kienholz (1982), AIAA J. 20: 1284-1290
• It is desirable to perform the dynamic analysis of
viscoelastically-damped structures on a reduced modal space
Zambrano, Inaudi, Kelly (1996), J. Engineering Mechanics 122: 603–612
Palmeri et al (2004), Wind & Structures 7: 89–106
Palmeri, Muscolino (2011), Structural Control & Health Monitoring 18: 519-539
4 A.Palmeri@LBORO.ac.uk

5. Linear Viscoelastic Structures
• Reaction force in the time domain:
+¥ +¥
r(t) = ò j (t - s) u(s)ds = R u(t) + ò g(t - s) u(s)ds
-¥
 0
-¥

• Depends on equilibrium modulus R0
and the time-dependent part of the relaxation
function g(t):
ì R0 = t®+¥ j (t) = j (¥)
ï lim
í
ï g(t) = j (t) - j (¥)
î
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6. Generalised Maxwell (GM) Model
• The reaction force can be expressed as
the superposition of l+1 terms, each one
associated with a different rigidity
coefficient Ri

r(t) = R0 u(t) + åRi li (t)
i=1
• The evolution in time of the l internal
variables λi(t) depends on the
corresponding relaxation time τi
 l (t)
li (t) = u(t) - i

ti
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7. Laguerre’s Polynomial Approximation (LPA)
• The function g(t) is expressed as a single exponential
function, depending on a single relaxation time τ0,
modulated by the Laguerre’s polynomials
Palmeri et al (2003), J. Engineering Mechanics 129: 715–724
æ t ö  æ t ö
g(t) = exp ç - ÷ å Ri Li ç ÷
è t 0 ø i=1 è t0 ø
• Similarly to the GM model:

r(t) = R0 u(t) + åRi li (t)
i=1
i
1

li (t) = u(t) -

ti
å l (t) j
j=1
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8. State-Space Equations of Motion
• Physical space of the actual DoFs
p t
M × u(t) + C × u(t) + K × u(t) + å b j × ò j j (t - s)b T × u(s)ds = f(t)
  j 
j=1 o
• Modal transformation of coordinates
• Equations of motion in the modal space
Modal relaxation matrix
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9. Non-Linear Viscoelastic Structures
• Non-linear device
t
(
r(t) = R0 u(t) + 1+ a u 2 (t) ) ò g(t - s) u(t)ds =  +  
 R u(t) (1+ a u (t)) R l (t)
0

2
1 1
0
r0 (t) r1 (t)
• Matrix form
• Equations of motion in the modal space
z(t) = F0 × z(t) + F1 ( z(t),t ) × z(t)

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10. Numerical Example (1/2)
m3
u3 (t)
k3 k3
2 2
m2
u2 (t)
k2 j 2 (t) k2
2 2
m1
u1 (t)
k1 j1 (t) k1
2 2
ag (t)

3
2
1
0
-1
-2
-3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
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11. Numerical Example (2/2)
m3
u3 (t)
k3 k3
2 2
m2
u2 (t)
k2 j 2 (t) k2
2 2
m1
u1 (t)
k1 j1 (t) k1
2 2
ag (t)

5
0
-5
0 2 4 6 8 10 12 14 16 18 20
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12. Conclusions
• An novel computational framework has been suggested for the dynamic
analysis of non-classically damped structures equipped with linear and non-
linear viscoelastic devices
• The preliminary results show that the inaccuracy introduced in the numerical
solution by reducing the size of the problem in the modal space is
substantially independent of the level of non-linearity of the viscoelastic
devices
• This study should be considered as a first step toward a general strategy to
effectively incorporate accurate rheological information on nano-reinforced
elastomeric devices in the non-linear time-domain dynamic analysis of
viscoelastically damped structures
• Further investigations are currently being developed to validate
experimentally the proposed procedure for frames made of composite
beams made with different rubber mixes
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13. Future Work
Non-classically non-viscously damped frame
= Assembly of sandwich metal-rubber viscoelastic
beams with different properties
Steel thickness: 0.006”, 0.100”, 0.015”
Rubber recipe: Natural (control), Low- and High- load
carbon black, silica
This work is supported by the EPSRC First Grant
EP/I033924/1 “TREViS: Tailoring Nano-Reinforced
Elastomers to Vibrating Structures”
13 A.Palmeri@LBORO.ac.uk