Elastic damping

The time-dependence of elastic deformation causes elastic damping, a phenomenon investigated in this exercise. 

Assume that a component, for example a tensile specimen, made of a viscoelastic material is loaded cyclically with angular frequency w. After some initial transient effects, the strain will also oscillate with the same angular frequency lu. Due to the time-dependence of elastic deformation, stress and strain are out of phase because the strain follows the current stress only with some delay. The following time-dependence is assumed for stress and strain  

Sketch the time-dependence of stress and strain and explain the meaning of the parameter J  

Write the stress as a function of the strain Use an addition theorem to split the stress into two components, one in-phase with the strain, the other phase-shifted by 90°  

Draw a stress-strain diagram for a complete cycle  

---

The properties of some polyurethane elastomers, used as such as elastic damping components in engineering applications (e.g., in sheet stamping operations and for coating of press and other rolls of paper-making equipment18) are shown in Table 1.3. 

Figure 3. Dressed state basis for atomic collisions. A - The square of the transfer matrix between the excitation Fock state and the dressed state bases for N = M = 100. Darker areas correspond to larger probability. B - Damping spectrum between the N = M = 5000 manifold and the N = 4999, M = 5000 manifold. Dashed line k = 3.2, dotted line k = 1.6 and solid line k = 0.7, q = k/ /2. Inset energy-conserving surfaces for the two center frequencies of the solid line and for elastic damping from mode k (dashed line). The splitting in the spectrum is due to the nonlinear population oscillations due to three-wave mixing of the modes in the time domain. This behavior is analogous to that of a strongly driven two level atom (Mollow splitting).

In elastically damped loading, a more or less static elastic load is applied to the ceramic or stack actuatoi this affords an initially negative strain of the device and/ or material. The applied electric field then suffices to drive the actuator to almost the same saturation strain value as without any mechanical preload, and thus permits a higher total strain output than its uncompressed counterpart The material fatigue does not differ considerably from the purely electrical unipolar case. 

Smyrou, E., Priestley, M. J., Carr, A. J. (2011). Modelling of elastic damping in nonlinear time-history analyses of cantilever RC walls. Bulletin of Earthquake Engineering, 9(5). doi 10.1007/ s10518-011-9286-y... 

Component modelling The modelling of the stabiliser trim system includes one-dimensional mechanics with elasticity, damping, inertia and friction hydraulics with bulk modulus, fluid density, turbulent and laminar flow and resistive electrics. One of the developed component models is explained below. 

Finally, Fig. 8.3 shows a third form of elastic behaviour found in certain materials. This is called anelasfic behaviour. All solids are anelastic to a small extent even in the regime where they are nominally elastic, the loading curve does not exactly follow the unloading curve, and energy is dissipated (equal to the shaded area) when the solid is cycled. Sometimes this is useful - if you wish to damp out vibrations or noise, for example you... 

The interesting fact is that the two men just mentioned were initially working independently of each other. The initial analysis used infinitely stiff bearings and, while the method improved the results, it did not match the compressor critical speed test results consistently. Later work recognized the existence of a bearing oil film with elastic and damping properties. 

Most of the forcing functions generated by V-belt drives can be attributed to the elastic or mbber band effect of the belt material. This elasticity is needed to provide the traction required transmitting power from the drive sheave (i.e., pulley) to the driven sheave. Elasticity causes belts to act like springs, increasing vibration in the direction of belt wrap, but damping it in the opposite direction. As a result, belt elasticity tends to accelerate wear and the failure rate of both the driver and driven unit. 

Dynamic mechanical tests measure the response or deformation of a material to periodic or varying forces. Generally an applied force and its resulting deformation both vary sinusoidally with time. From such tests it is possible to obtain simultaneously an elastic modulus and mechanical damping, the latter of which gives the amount of energy dissipated as heat during the deformation of the material. 

By calculating the energy to heat it is possible to determine the vibration levels to which the structure can be exposed and still exhibit critical damping. There is one area that must be evaluated. Plastics exhibit a spectrum of response to stress and there are certain straining rates that the material will react to almost elastically. If this characteristic response corresponds to a frequency to which the structure is exposed the damping effect is minimal and the structure may be destroyed. In order to avoid the possibility of this occurring, it is desirable to have a curve of energy absorption vs. frequency for the material that will be used. 

Viscoelastic damping The same approach can be used in designing power transmitting units such as belts. In most applications it is desirable that the belts be elastic and stiff enough to minimize heat buildup and to minimize power loss in the belts. In the case of a driver which might be called noisy in that there are a lot of erratic pulse driven forces present, such as an impulse operated drive, it is desirable to remove this noise by damping out the impulse and get a smooth power curve. 

Damping The loss of energy, as dissipated heat, that results when a material or material system is subjected to an oscillatory load or displacement. Perfectly elastic materials have no mechanical damping. Damping reduces vibrations (mechanical and acoustical) and... 

Kallio, M., The Elastic and Damping Properties of Magnetorheological Elastomers, Espoo 2005, VTT Publications 565, pp. 1-146, 2005. 

Polyurethane-based thermoplastic elastomers are extensively used in applications requiring physical resilience and chemical resistance. In addition to their elasticity, they also exhibit vibration damping, abrasion, tear, and cut resistance. 

After the transient shock load phase has damped out, the subsequent confined hot gas pressure can be considered as a steady state load from a structural dynamics point of view. Therefore the design criteria requires that these loadings do not exceed the elastic limit of the structure. Dynamic increase factors are not applicable since loading rate is no longer a consideration. 

---