Spring-damper models

Figure 3. Mass-spring-damper model of sUp, from Lu et al. [7].

If we look back at our simple mass/spring/damper model from Chapter 4, we note that we assumed that the spring exerted a restoring force that was linearly related to the displacement. This is true for ideal springs or small displacements of real springs. But actual springs exert more force for larger displacements. 

We noted that the ideal string equation and the ideal acoustic tube equation are essentially identical. Just as there are many refinements possible to the plucked-string model to make it more realistic, there are many possible improvements for the clarinet model. Replacing the simple reed model with a variable mass/spring/damper allows the modeling of a lip reed as is found in... 

Flarmonic Excitation of a SDOF System In this section, we will see the behavior of the spring mass damper model when we add a harmonic force in the form below (Fig. 3). 

An extensive study into glottal modelling is given in Flanagan [164], which describes various mass/spring/damper systems. These models can be somewhat difficult to model in discrete-time systems, so instead we adopt models that simply generate a time-domain function that has the properties described above. One such model [368], [376] is given by... 

Shi, J. and Flannagan, R., A simple spring-damper-slider model for laminate slippage . In Poursartip, A. and Street, K. (eds.). Proceedings of the 10th International Conference on Composite Materials (ICCM - 10). Whistler, B.C., Canada, August, 1995, pp. III-197-III-204. 

The TdealGear element introduces these equations to the spur gear model. The Disconnect and Jam elements are rotational spring-dampers described by ... 

The MBS has been used by Zhao and Maisser (2006) to assess the seismic response of a 65 m high HAWT. SSI has been modeled approximately by a frequency-independent discrete parameter model, as a 3D set of uncoupled spring-damper devices, including translations and rotations (Fig. 5). 

Fig. 10.1. A vibrating system vvith one degree of freedom and its transfer fnnction. (a) The vibrating system. A mass M is connected to the frame through a spring and a viscous damper. Regarding STM, there are two realizations of this model. First, the frame represents the floor, and the mass represents the STM. Second, the frame represents the base plate (with the sample) in STM, and the mass represents the tip assembly, (b) The transfer function, which is the ratio of the vibration amplitude of the mass to that of the frame at different frequencies. (After Park and Quate, 1987.)...

The Maxwell model can be represented by a purely viscous damper and a purely elastic spring connected in series, as shown in the diagram. The model can be represented by the following equation ... 

The Kelvin-Voigt model, also known as the Voigt model, consists of a Newtonian damper and Hookean elastic spring connected in parallel, as shown in the picture. It is used to explain the stress relaxation behaviors of polymers. 

Note that the simple Hooke s law behavior of the stress in a solid is analogous to Newton s law for the stress of a fluid. For a simple Newtonian fluid, the shear stress is proportional to the rate of strain, y (shear rate), whereas in a Hookian solid, it is proportional to the strain, y, itself. For a fluid that shares both viscous and elastic behavior, the equation for the shear stress must incorporate both of these laws— Newton s and Hooke s. A possible constitutive relationship between the stress in a fluid and the strain is described by the Maxwell model (Eq. 6.3), which assumes that a purely viscous damper described by Eq. 6.1 and a pure spring described by Eq. 6.2 are connected in series (i.e., the two y from Eqs. 6.1 and 6.2 are additive). 

In addition, a time-dependent viscoelastic component described by the so-called Voigt-Kelvin configuration, i.e. the combination of spring E and damper Yf comes into action. Further, a viscoplastic flow component may exist, modelled by the damper of viscosity... 

The seismic analysis of the core is performed with the two-dimensional special purpose computer codes CRUNCH-2D and MCOCO, which account for the non-linearities in the structural design. Both CRUNCH-2D and MCOCO are based on the use of lumped masses and inertia concepts. A core element, therefore, is created as a rigid body while the element flexibilities are input as discrete springs and dampers at the corners of the element. CRUNCH-2D models a horizontal layer of the core and the core barrel structures (Figure 3.7-7). The model is one element deep and can represent a section of the core at any elevation, MCOCO models a strip of columns in a vertical plane along a core diameter and includes column support posts and core barrel structures (Figure 3.7-8). The strip has a width equal to the width of a permanent reflector block. Both models extend out to the reactor vessel,... 

The Burger model provides a correct graphic description of the elongation-time behavior of most plastics in a first approximation. The spring 1 results in spontaneous elastic load application and relaxation elongation, 1 + 2 in parallel cause creep during load application and creep recovery (delayed viscoelastic reverse deformation) after relaxation, damper 2 results in residual elongatimi. 

Based on its operation principle, the resonator can be modeled as a simple spring-mass-dashpot system, as shown in Fig. 2, with the shuttle being the proof mass, folder beams being the spring, and the surrounding air being the dashpot damper. The displacement of the proof mass can then be obtained by solving the second-order differential equation... 

Figure 15-5. Modelling of the inertial soil-structure interaction by springs and dampers.

Consider a system modelled by N masses, N springs and N dampers. Its forced oscillations under the action of the N external forces, will be... 

Here some practical data and some formulae relevant to modelling the ground (inertial interaction) by equivalent masses, springs and dampers. 

The damping of the soil is composed of two terms (in the model of springs and equivalent dampers) the first is the internal damping which is connected with the energy loss in the cyclic deformation of the soil and depends on the type of soil and on its deformation level (some values are listed in Table 15-14) the second is called radiation damping... 

Simple Lumped Models. At frequencies up to several hundred hertz, the biodynamic response of the human body can be represented theoretically by point masses, springs, and dampers, which constitute the elements of lumped biodynamic models. The simplest one-dimensional model consists of a mass supported by a spring and damper, as sketched in Fig. 10.6, where the system is excited at its base. The equation of motion of a mass m when a spring with stiffness k and damper with resistance proportional to velocity, c, are base driven with a displacement x it) is ... 

FIGURE 10.6 Smgle.degieeK>f-fi elumped-parameter biodynamic model. The mass m is supported by a spring with stiffness k and viscous damper with resistance c. The transmis-sibility of motion to the mass is shown as a function of die fiequency ratio r (=aj/oib) when the base is subjected to a displacement (After Griffin, 1990.)... 

The paradox of Maxwell s model. A popular representation of models in rheology mimics the equivalent electrical circuits with dipolar components. The elastic component is naturally symbolized by a spring and the viscous component by a damper or dashpot (a piston filled with a viscous fluid able to circulate). The viscoelastic relaxation is thus represented with these two components mounted in series, as shown in Figure 11.12a and is known as Maxwell s model (Oswald 2005). (In this representation, the customary notation is used for facilitating comparison with the literature.)... 

The assumed full model of the vehicle is shown in Fig. 2.5 and consists of the sprung mass (car body, engine, etc.) and the unsprung mass that accounts for the wheel and axle masses supported by the tire. The suspension is modeled as a spring and a damper in parallel, which connects the unsprung to the sprung mass. The tire... 

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