Springs connected in series

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 model represents a liquid (able to have irreversible deformations) with some additional reversible (elastic) deformations. If put under a constant strain, the stresses gradually relax. When a material is put under a constant stress, the strain has two components as per the Maxwell Model. First, an elastic component occurs instantaneously, corresponding to the spring, and relaxes immediately upon release of the stress. The second is a viscous component that grows with time as long as the stress is applied. The Maxwell model predicts that stress decays exponentially with time, which is accurate for most polymers. It is important to note limitations of such a model, as it is unable to predict creep in materials based on a simple dashpot and spring connected in series. The Maxwell model for creep or constant-stress conditions postulates that strain will increase linearly with time. However, polymers for the most part show the strain rate to be decreasing with time [23-26],... 

Fig. 2.11 Equilibrium between two stretched springs connected in series.

Figure 2.3 Two springs connected in series. One spring has a spring constant ki and is stretched by a distance Axj, and the other has a spring constant k2 and is stretched by a distance Ax2. The same force F acts on both springs [2].

Equation 12.6 resembles the force law of a spring with a spring constant k = 47ta set by the interfacial tension. Therefore, the contact of Figure 12.1, which involves two bubbles, acts like two such springs connected in series. Writing the total overlap 8 = 8q + 8i as the sum of the overlaps in each bubble, the effective force law of the contact can be written as... 

Suppose we consider a spring and dashpot connected in series as shown in Fig. 3. 7a such an arrangement is called a Maxwell element. The spring displays a Hookean elastic response and is characterized by a modulus G. The dashpot displays Newtonian behavior with a viscosity 77. These parameters (superscript ) characterize the model whether they have any relationship to the... 

When a spring and a dash pot are connected in series the resulting structure is the simplest mechanical representation of a viscoelastic fluid or Maxwell fluid, as shown in Fig. 3.10(d). When this fluid is stressed due to a strain rate it will elongate as long as the stress is applied. Combining both the Maxwell fluid and Voigt solid models in series gives a better approximation for a polymeric fluid. This model is often referred to as the four-parameter viscoelastic model and is shown in Fig. 3.10(e). Atypical strain response as a function of time for an applied stress for the four-parameter model is found in Fig. 3.12. 

Maxwell element or model Model in which an ideal spring and dashpot are connected in series used to study the stress relaxation of polymers, modulus Stress per unit strain measure of the stiffness of a polymer, newtonian fluid Fluid whose viscosity is proportional to the applied viscosity gradient. 

The visco-elastic interactions of connected micro-elements of semicrystalline polyolefins are approximated by the Maxwell model, which formally consists of the spring with modulus E and the viscous dashpot characterized by the viscosity rf connected in series,... 

The simplest mechanical model which can describe a viscoelastic solution is called Maxwell element. It consists of a spring and a viscous element (dashpot) connected in series. The spring corresponds to a shear modulus Gq and the dashpot to a viscosity r). The behavior of the Maxwell element under harmonic oscillations can be obtained from the following equations ... 

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). 

Once more, the rheological behavior of many pharmaceutical and biomedical materials is more appropriately described by a number of Voigt units connected in series (18). The model illustrated in Figure 10.4 describes the rheological behavior of a viscoelastic solid as, in this case, the elastic contribution is sufficient to ensure that there is no unlimited, nonrecoverable viscous flow. However, if the spring in one of the units possesses zero elasticity (i.e., G = 0), then nonrecoverable viscous flow will be observed, and the material is better described as a viscoelastic liquid or, alternatively, an elastoviscous system. 

Finally, the rate dependent properties are usually modeled by a three-parameter solid which consists of a spring (m2) and a dashpot (h) in parallel connected to another spring (mf) in series. Viscoelastic properties may also be expressed in terms of the dynamic modulus G. A sinusoidal displacement of the form u = is applied to the specimen... 

Dashpot dash- pat (1861) n. (1) A device used for damping vibration and cushioning shock in hydraulic systems. Typically, a dashpot is a liquid- or gas-filled cylinder with a piston that is attached to a moving machine part. (2) A modeling concept useful in visualizing the mechanical behavior of viscoelastic materials, a purely viscous element that may operate alone or connected in series and/or parallel with springs and sliders. 

Low-Speed Units Low-speed units usually consist of jacketed pipes connected in series (by special return bends), where the heat-transfer medium flows in the annulus between the two pipes. The scraper assembly consists of a series of blades attached to springs (which hold the blades against the inner pipe wall) that in turn are attached to a central shaft driven at 15 to 50 rpm. Figure 11-22 illustrates this type of evaporator. 

The first two reduced models (reduced model 1 and reduced model 2) have almost identical predictions for the tire contact force as it is shown in Fig. 2.11 and have an error of 0.02 and 0.31 %, respectively. The third reduced model (reduced model 3) has some visible differences with an accuracy of 2.41%, while the last reduced model (reduced model 4) has an average error of 62.2% and fails to capture the system behavior. As can be seen in Fig. 2.11, the tire contact force prediction of the fourth reduced model has undamped oscillations and it is very different from the one of the full model. This prediction is expected, since this model has no dissipation elements included and consists of only a spring and mass connected in series. Note that in the top plot of Fig. 2.11 there are five curves plotted, one for the full model and one for each one of the four reduced models. However, only three curves (reduced 2, reduced 3, reduced 4) are visible, since the full and reduced 1 curves are almost identical with reduced 2. Similarly, in the bottom plot of Fig. 2.11 there are five curves plotted but only three are visible. 

Figure 1.31. Gaussian chain of N segments is realized by a bead-spring model in which N springs are connected in series.

A simple way of understanding the behavior of viscoelastic liquids is by analogy to mechanical models. The stress response t of a spring of modulus G connected in series to a dashpot having a damping constant t] (the combination is often called a Maxwell element) is given in one dimension by the equation... 

The principal setup needed to carry out such measurements consists of three elements connected in series (1) the sample itself, (2) a mechanical device connected to one end of the sample (e.g., a motor) capable of moving the sample end at a preset rate, and (3) a rather stiff spring connected to the other side of the sample. 

Fig. 13.29. The Maxwell model for a viscoelastic material considers an elastic spring and a viscous dashpot connected in series.

For springs that are connected in series, as shown in Figure 2.3, the force acting on each of the springs would be the same but because they have... 

The top mirror support is four springs acting in series. Each individual spring is comprised of a set of three fixed-guides beams connected end-to-end and acting in parallel. 

Seborg and Stamm [148] connected five or six simple spring units in series. Pidgeon and Maass [149]. Mulligan etal. [ISO] and Stamm and Woodruff [151] connected as many as sixteen springs to the same apparatus. 

---