The 15th World Conference on Earthquake Engineering       Lisbon (Portugal), 28th September 2012         Toward an Improve...
Loughborough University2                   A.Palmeri@LBORO.ac.uk
Outline• Introduction• Linear Viscoelastic Structures    • Generalised Maxwell (GM) model    • Laguerre’s Polynomial Appro...
Introduction• Energy-dissipation devices exploiting viscoelastic rubber  improve the performance of civil engineering stru...
Linear Viscoelastic Structures• Reaction force in the time domain:             +¥                           +¥    r(t) =  ...
Generalised Maxwell (GM) Model• The reaction force can be expressed as  the superposition of l+1 terms, each one  associat...
Laguerre’s Polynomial Approximation (LPA)• The function g(t) is expressed as a single exponential  function, depending on ...
State-Space Equations of Motion    • Physical space of the actual DoFs                                        p      t    ...
Non-Linear Viscoelastic Structures• Non-linear device                                       t                       (    r...
Numerical Example (1/2)                                         m3                                                        ...
Numerical Example (2/2)                             m3                                                           u3 (t)   ...
Conclusions•    An novel computational framework has been suggested for the dynamic     analysis of non-classically damped...
Future WorkNon-classically non-viscously damped frame= Assembly of sandwich metal-rubber viscoelastic   beams with differe...
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Toward an Improved Computational Strategy for Vibration-Proof Structures Equipped with Nano-Enhanced Viscoelastic Devices

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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.

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Toward an Improved Computational Strategy for Vibration-Proof Structures Equipped with Nano-Enhanced Viscoelastic Devices

  1. 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 PalmeriSchool of Civil and Building Engineering, Loughborough University <A.Palmeri@LBORO.ac.uk>
  2. 2. Loughborough University2 A.Palmeri@LBORO.ac.uk
  3. 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• Conclusions3 A.Palmeri@LBORO.ac.uk
  4. 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) DallAsta, 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: 519-5394 A.Palmeri@LBORO.ac.uk
  5. 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 (¥) î5 A.Palmeri@LBORO.ac.uk
  6. 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  ti6 A.Palmeri@LBORO.ac.uk
  7. 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=17 A.Palmeri@LBORO.ac.uk
  8. 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 matrix8 A.Palmeri@LBORO.ac.uk
  9. 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) 9 A.Palmeri@LBORO.ac.uk
  10. 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 4.5 510 A.Palmeri@LBORO.ac.uk
  11. 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 2011 A.Palmeri@LBORO.ac.uk
  12. 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 mixes12 A.Palmeri@LBORO.ac.uk
  13. 13. Future WorkNon-classically non-viscously damped frame= Assembly of sandwich metal-rubber viscoelastic beams with different propertiesSteel 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

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