toward overcoming the concept of effective stiffness and damping in the dynamic analysis of...
DESCRIPTION
In the current state-of-practice, the time-domain dynamic analysis of structures incorporating viscoelastic members is generally carried out through the Modal Strain Energy (MSE) method, or other procedures somehow based on the quite simplistic idea of substituting the actual viscoelastically damped structure with an equivalent system featuring a pure viscous damping.This crude approximation in civil engineering applications is very often encouraged by manufacturers of the viscoelastic devices themselves, whose interest is to simplify as much as possible the design procedures for structures embedding their products. As an example, elastomeric seismic isolators are generally advertised and sold with a table listing the equivalent values of elastic stiffness and viscous damping ratio for different amplitudes of vibration. Unfortunately, many experimental and analytical studies confirm that the real dynamic behaviour of such devices is much more complicated, and cannot be bend to the interests of manufacturers and designers.Despite the advances in the field made in the last two decades, two well-established beliefs continue to underpin use and abuse of the concepts of effective stiffness and damping for viscoelastically damped structures: first, MSE method and similar procedures are unconditionally assumed to provide good approximations, which are acceptable for design purposes; second, the implementation of more refined approaches is thought to be computationally too expensive, and hence suitable just for a few very important constructions.In this presentation, as a further contribution to overcome these popular beliefs, a novel time-domain numerical scheme of dynamic analysis is proposed and numerically validated. After a brief review of the LPA (Laguerre’s Polynomial Approximation) technique for one-dimensional viscoelastic members of known relaxation function, the state-space equations of motion for linear structures with viscoelastic components are derived in the modal space. Aimed at making the proposed approach more general, the distribution of the viscoelastic components is allowed to be non-proportional to mass and elastic stiffness, in so removing the most severe limitation of previous formulations. Then, a cascade scheme is derived by decoupling in each time step traditional state variables (i.e. modal displacements and velocities) and additional internal variables. The joint use of modal analysis and improved cascade scheme permits to reduce the size of the problem and to keep low the computational burden. The illustrative application to the small-amplitude vibration of a cable beam made of different viscoelastic materials demonstrates the versatility of the proposed approach. The numerical results confirm a superior accuracy with respect to the classical MSE method, whose underestimate in the low-frequency range can be as large as 75%.TRANSCRIPT
- 1. Toward overcoming the concept of effective stiffness and
damping in the dynamic analysis of structures with viscoelastic
components
Alessandro Palmeri
School of Engineering, Design and Technology
University of Bradford, West Yorkshire, UK
CC2009, Funchal, 3rd September 2009
2. [email protected]
3. Outline
Motivation of this study
Viscoelastic Elastic + Viscous
Linear viscoelastic solids
Generalized Maxwells model
Laguerre Polynomial Approximation
State-space equations of motion
Numerical scheme of solution
Validation
Elastic beam with VE strips (efficiency)
Cable beam made of VE material (versatility)
Concluding remarks
4. Motivation of this study
1.
5. Viscoelastic damping in... Wind Engineering
The first application of viscoelastic materials in Civil
Engineering was aimed at mitigating the wind-induced vibration of
the Twin Towers in the World Trade Center
6. Earthquake Engineering
Seismic applications of viscoelastic dampers are more recent, e.g.
through the use of elastomeric materials placed in the
beam-to-column joints of semi-rigid steel frames
7. Railway Engineering
In the innovative track of the Milan subway, a single elastomeric
pad is placed under the base-plate, aimed at improving passengers
comfort and extending components fatigue life
8. Motivation of this study
Current state-of-practice, for the time-domain dynamic analysis of
structures incorporating viscoelastic members:
Substituting the actual viscoelastically damped structure with an
equivalent system featuring a pure viscous damping
9. Motivation of this study
Manufacturers of viscoelastic devices encourage the use this crude
approximation in civil engineering applications:
They are interested in simplifying as much as possible the design
procedures for structures embedding their products
10. Motivation of this study
As an example, elastomeric seismic isolators are generally
advertised and sold with a table listing somehow equivalent values
of elastic stiffness and viscous damping ratio
11. Motivation of this study
Unfortunately, many experimental and analytical studies confirm
that the real dynamic behaviour of viscoelastic devices can be very
complicated,and cannot be bend to the interests of manufacturers
and designers
12. Motivation of this study
Two well-established beliefs continue to underpin use and abuse of
the concepts of effective stiffness and damping for
viscoelastically damped structures
This simplification always provides good approximations, which are
acceptable for design purposes (FALSE!)
Palmeri et al (2004), J ENG MECH-ASCE 130, 1052
Palmeri et al (2004), WIND STRUCT 7, 89
Muscolino,Palmeri & Ricciarelli (2005), EARTHQUAKE ENG STRUC
34, 1129
Palmeri & Ricciarelli (2006), J WIND ENG IND AEROD 94,
377
Palmeri (2006), ENG STRUCT 28, 1197
Muscolino & Palmeri (2007), INT J SOLIDS STRUCT 44, 1317
More refined approaches are computationally too expensive, and
hence suitable just for a few very important constructions
(FALSE!)
13. LINEAR viscoelastic SOLIDS
2.
14. Linear viscoelastic solids
The term viscoelastic refers to a whole spectrum of possible
mechanical characteristics
At one extreme we have viscous fluids, e.g. air and water
At the other end we have elastic solids, e.g. metals
Viscoelastic behaviour may combine viscous and elastic properties
in any relative portion
15. Linear viscoelastic solids
Two experimental tests can be used to reveal the viscoelastic
behaviour of solids
CREEP TEST: The specimen is subjected to a constant state of
stress, and the resulting variation in strain e as a function of
time t is determined (the strain variation after the stress is
removed corresponds to the recovery test)
Creep function
16. Linear viscoelastic solids
Two experimental tests can be used to reveal the viscoelastic
behaviour of solids
RELAXATION TEST: The specimen is subjected to a constant state of
strain, and the resulting variation in stress s as a function of
time t is determined
Relaxation function
17. Linear viscoelastic solids
The Kelvin-Voigt model, made of a linear spring in parallel with a
linear dashpot, is widely adopted in Structural Dynamics
Interestingly, the relaxation test is impossible for this
model
18. Linear viscoelastic solids
In the Standard Linear Solid (SLS) model the dashpot is substituted
with a Maxwells element
This model allows describing (at least qualitatively) creep and
relaxation processes of actual linear viscoelastic solids
19. Linear viscoelastic solids
The reaction force r(t) experienced by a one-dimensional
viscoelastic component can be expressed in the time domain through
a convolution integral involving the time derivative of the
associated deformation q(t)
pure elastic part
20. Relaxation function(time domain)
j(t)
temperature
21. Linear viscoelastic solids
The complex-valued dynamic modulus k(w) enables one to represent
the viscoelastic behaviour in the frequency domain
where
the REal part is the storage modulusk(w), which is a measure of the
apparent rigidity at a given circular frequency w
the IMaginary part is the loss modulusk(w), which is proportional
to the energy dissipated in a harmonic cycle
Dynamic modulus and relaxation function are interrelated as
22. DYNAMIC MODULUS(Frequency domain)
k(w)
temperature
temperature
Storage modulus (rigidity)
Loss modulus (dissipation)
23. Linear viscoelastic solids
The frequency-dependent behaviour of viscoelastic materials cannot
be captured by the 2-parameter Kelvin-Voigt model
Kelvin-Voigt
Standard Linear Solid
storage
storage
loss
loss
24. Linear viscoelastic solids
Dilemma
On the one hand, more refined models should be used to represent
the dynamic behaviour of actual viscoelastic systems
On the other hand, convolution integrals in the time domain are
computationally burdensome
Proposed approach:
Implementation of state-space models, in which a set of additional
state variables li(t) takes into account the frequency-dependent
behaviour of these systems
25. Linear viscoelastic solids
- The Generalized Maxwells model is made of an elastic spring in parallel with Maxwells rheological elements
26. The relaxation function is the superposition of exponential
functions having different relaxation times tiThe time variation of
the i-th internal variable is given by
27. Linear viscoelastic solids
As an alternative, the Laguerres Polynomial Approximation can be
used
The relaxation function is given by a single exponential function
modulated by a polynomial of order
The evolution in time of the i-th internal variable is ruled
by
t0 being a characteristicrelaxation time of the system
28. Linear viscoelastic solids
GM and LPA models have relative pros and cons
Palmeri et al (2003), J ENG MECH-ASCE 129, 715
GM model is based on a classical chain of elastic springs and
viscous dashpots
The internal variable li(t) is ideally the strain in the elastic
spring of the i-th Maxwells element
The experimental evaluation of the 2 parameters of the GM model is
generally pursued with a non-linear regression based on the results
of small-amplitude vibration tests, which unfortunately turns out
to be an ill-posed problem
Orbey & Dealy (1991),J RHEOL35 1035
Mustapha & Phillips (2000), J PHYS D APPL PHYS 33, 1219
The LPA techniques just require a relaxation test to obtain the +1
parameters characterizing this model
- Both GM model and LPA technique enable one to accurately represent the constitutive law for linear viscoelastic solids