Spring Dashpot Model

3 Key excerpts on "Spring Dashpot Model"

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  eBook - ePub

  Creep and Relaxation of Nonlinear Viscoelastic Materials

  • William N. Findley, Francis A. Davis, Francis A. Davis(Authors)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  In general there are two alternative forms used to represent the stress-strain-time relations of viscoelastic materials. They are called the differential operator method and the integral representation. The differential operator method has been widely used for analysis since the mathematical processes [5.1, 5.2] required are reasonably simple. On the other hand, the integral representation is able to describe the time dependence more generally, but it sometimes leads to difficult mathematics in stress analysis. The differential operator form and the integral representation are discussed in Sections 5.9 and 5.14, respectively, in this chapter.

  5.2 Viscoelastic Models

  In the following discussion of mechanical models, stress σ (force per unit area) and strain e (deformation per unit length) instead of force and deformation of the model will be used to compare the stress-strain-time relation of viscoelastic materials with the viscoelastic models considered.

  5.3 The Basic Elements: Spring and Dashpot

  All linear viscoelastic models are made up of linear springs and linear viscous dashpots. Inertia effects are neglected in such models. In the linear spring shown in Fig. 5.2 a

  (5.1)

  where R in (5.1) can be interpreted as a linear spring constant or a Young’s modulus. The spring element exhibits instantaneous elasticity and instantaneous recovery as shown in Fig. 5.2 b.

  Fig. 5.2 . Behavior of a Linear Spring and Linear Dashpot.

  A linear viscous dashpot element is shown in Fig. 5.2 c where

  (5.2)

  and the constant η is called the coefficient of viscosity. Equation (5.2) states that the strain rate is proportional to the stress or, in other words, the dashpot will be deformed continuously at a constant rate when it is subjected to a step of constant stress as shown in Fig. 5.2 d. On the other hand, when a step of constant strain is imposed on the dashpot the stress will have an infinite value at the instant when the constant strain is imposed and the stress will then rapidly diminish with time to zero at t = 0+ and will remain zero, as shown in Fig. 5.2 e. This behavior for a step change in strain is indicated mathematically by the Dirac delta function10 δ (t ), where δ (t ) = 0 for t ≠ 0, δ (t ) = ∞ for t = 0. Thus the stress resulting from applying a step change in strain ε 0

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  eBook - ePub

  Passive Vibration Control of Structures

  • Suhasini Madhekar, Vasant Matsagar(Authors)
  • 2022(Publication Date)
  • CRC Press

    (Publisher)

  In viscoelastic fluid dampers, viscoelastic fluid is used instead of the solid viscoelastic material. Several types of viscous dampers employed with fluid applications have been developed. Cylindrical plot fluid damper and VDW are the noteworthy examples. These devices cause dissipation of energy through the deformation of a viscous fluid, residing in an open container. However, in order to increase the efficiency of energy dissipation, fluids with larger viscosities, which exhibit both frequency- and temperature-dependent behaviors, are recommended.

  The cylindrical plot fluid damper shown in Figure 4.17 consists of classical dashpot, wherein mechanical energy gets converted to heat energy, causing energy dissipation. The piston provided in the damper deforms a highly viscous fluid, such as silicone gel (viscosity in the range 60,000–90,000 P).

  FIGURE 4.17

  Cylindrical plot fluid damper.

  Mathematical Model

  Although construction of viscoelastic fluid damper is different from viscoelastic solid damper, the analytical models appropriate for the overall force-displacement response of both dampers are similar. Both solid and fluid devices are frequency- and temperature-dependent. In the past, various constitutive models were proposed for viscoelastic fluid dampers. One of the approaches is the fractional derivative technique applied to classical Maxwell model, to represent the behavior of fluid dampers mathematically (Makris et al. 1993 ).

  Fractional derivative of Maxwell force-displacement model is expressed as:

  F t +

  λ v

  d v

  F t

  d

  t v

  =

  C 0

  du t

  dt

  (4.48)

  where

  F t

  is the force applied to the piston;

  u t

  is the resulting piston displacement; and

  C 0

  ,

  λ ,

  and v represent the damping coefficient, relaxation time, and order of fractional derivative, respectively. The relaxation time for the damper is defined as

  λ =

  C 0

  K 0

  , where

  K 0

  is the storage stiffness at a definite frequency. Considering a complex derivative, Maxwell model is used to characterize a highly viscous fluid over a broad range of frequency and temperature. Assuming a very small incompressible deformation, at some reference temperature

  T 0

  , the above proposed model can be represented in terms of shear stress τ and shear strain γ

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  eBook - ePub

  Semi-Active Suspension Control Design for Vehicles

  • Sergio M. Savaresi, Charles Poussot-Vassal, Cristiano Spelta, Olivier Sename, Luc Dugard(Authors)
  • 2010(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  Looking at the future, the three main technologies available today (EH, MR and ER) are still evolving. Another almost-ready technology is air-damping with electronically controlled valves. An air-damping suspension consists in a single device that can be regarded as a gas spring equipped with a piston characterized with variable orifices. Therefore the air-damping suspension mixes the elastic and damping behavior of the gases. Because of this intrinsic dynamical coupling, both the tuning and the electronic control of air-damping suspension seem to be extremely challenging.

  The real innovation however seems to be the introduction of electric motors used as dampers and capable of energy recuperation. This technology is particularly attractive since it can integrate semi-active and active capabilities.

  2.4.4. On “Linearization” of Damping Characteristics

  The characteristics of EH dampers are fairly approximated by an ideal linear damping. For MR and ER dampers, the force-velocity maps are far from linear. However the three technologies are characterized by comparable bandwidth, and also the controllability range is similar: it differs only in terms of shape of the characteristics. Thus, note that with an appropriate electronic control, it is possible to turn the nonlinear maps that characterize ER and MR dampers into linear, similar to those of the EH damper. Under this perspective, the three technologies are comparable. As a consequence, a general concise force-velocity model may be provided, as follows:

  (2.12)

  where the used symbols are according the notation introduced so far.

  2.5. Dynamical Models for Semi-Active Shock Absorber

  As described in Figure 2.16 , a semi-active shock absorber may be viewed as a system with two inputs and one output. The inputs are the deflection speed and the electronic command for the damping actuation . The output is the damping force delivered by the suspension. The peculiarity of the shock absorber is that only the deflection speed is a “driving” input: in fact in the case

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