Springs connected in parallel

Conversely, for low stress states/longer time periods, the time derivative components are negligible and the dashpot can be effectively removed from the system - an open circuit. As a result, only the spring connected in parallel to the dashpot will contribute to the total strain in the system [23-26],... 

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. 

If the fibres are oriented parallel to the loading direction, their arrangement is called parallel connection, analogous to the connection of springs (figure 9.1(a)). Similar to springs connected in parallel, the deformation in the matrix (subscript m ) and in the fibre (subscript f ) must be the same, but the stress may differ ... 

Figure 2.2 Two springs connected in parallel. One spring has a spring constant ki and the other spring has a spring constant k2- Both are stretched by the same amount Aa [1].

Upon associating a Maxwell element and a spring connected in parallel, one obtains the Zener model, which describes very satisfactorily the behavior of highly cross-linked polymers. Such behavior is characterized by an instantaneous elasticity followed by a phase of retarded elasticity, with the deformation exhibiting a completely reversible character. 

Alternatively, if the spring and dashpot are connected in parallel, the following holds ... 

In this model the spring and dashpot elements are connected in parallel as shown in Fig. 2.36. 

When dash pot and spring elements are connected in parallel they simulate the simplest mechanical representation of a viscoelastic solid. The element is referred to as a Voigt or Kelvin solid, and it is shown in Fig. 3.10(c). The strain as a function of time for an applied force for this element is shown in Fig. 3.11. After a force (or stress) elongates or compresses a Voigt solid, releasing the force causes a delay in the recovery due to the viscous drag represented by the dash pot. Due to this time-dependent response the Voigt model is often used to model recoverable creep in solid polymers. Creep is a constant stress phenomenon where the strain is monitored as a function of time. The function that is usually calculated is the creep compliance/(f) /(f) is the instantaneous time-dependent strain e(t) divided by the initial and constant stress o. ... 

Following crystallization, the solid was separated from the mother liquor by filtration, washed with distilled water, and air dried in an oven at 100 C to remove loosely bound water. Samples of the dried powder were sent routinely to the x-ray laboratory. The fact that we could obtain a strip chart recording of the x-ray powder pattern within 30 minutes was an important factor in the pace of our work. Adsorption evaluations were facilitated by use of multiple, quartz spring, McBain-Bakr balances connected in parallel. As many as 16 adsorbent samples could be evaluated simultaneously. 

In practice the stress relaxation behaviour has to be described expressed with N Maxwell elements connected in parallel, each with its own spring constant E and relaxation time t (the so-called Maxwell-Wiechert model) ... 

It is convenient to describe these properties in terms of the following mechanical models [396] the Hooke body (an elastic spring), the Saint-Venant body modeling dry friction (a bar on a solid surface), and the Newton body (a piston in a vessel filled with a viscous fluid). By using various combinations of these elementary models (connected in parallel and/or in series), one can describe situations which are rather complex from the rheological viewpoint. 

In the Kelvin or Voigt model the spring and dashpot elements are connected in parallel, as shown in Figure 3.13a. This model roughly approximates the behavior of rubber. When the load is applied at zero time, the elastic deformation cannot occur immediately because the rate of flow is limited by the dashpot. Displacements continue until the strain equals the elastic deformation of the spring and it resists further movement. On removal of the load the spring recovers the displacement by reversing the... 

In the Voigt model, a spring and dashpot are connected in parallel. Following similar methods to Question 2.5, determine the time dependence of the strain in a creep experiment and the stress during a stress relaxation experiment. 

For springs that are connected in parallel, as shown in Figure 2.2, the deflection of each spring. Ax, would be the same [1] ... 

The folded springs are comprised of fixed-guided beams that are connected in parallel and in series, as shown in Figure 3.8. 

The Voigt Model. It consists of a spring and a dashpot connected in parallel ... 

An actuator capable of developing a thrust of about 12,000 11> was cycled four tiTiies per minute at a stroke of 12 in. and a pres.sure of 80 psig for 265,000 cycles before developing a small leak in the stem sealing bellows. Two and three bellow s-spring assemblies from these units ha e been attached to a common shaft and connected in parallel to a source of air pressure in preliminary tests of an even more pow erful actuator. 

The case of perfect elasticity (connected or not with a latent recovery), of which we find an example in vulcanized rubber, can then only occur when the molecules of the system form one continuous and everywhere connected network, which is not disturbed on deformation. In the model this means that there are no dashpots to which a spring is not connected in parallel (or according to the Figs. 14B and 17B, that the springs arc always rigidly connected to the bottom of the cylinder). Only under that stipubtion can the system return to its original state before the deformation. 

A polymer can be represented by two Maxwell elements connected in parallel, each with the same spring modulus. In a stress relaxation experiment, the stress decreases to 23.7% of its initial value after 10 min. If the relaxation time of the first elanent is 10 min, what is the relaxation time of the second elanent ... 

Fig. 5 When representing mechanical elements with equivalent circuits, elements which are placed in series to each other, physically, have to be drawn as parallel elements in the circuit representation because currents (speeds) are additive for parallel electrical elements. Conversely, mechanical elements which are physically placed in parallel have to be represented in series because the voltage (force) is additive for electrical elements placed in series. In the literature on polymer rheology, springs and dashpots are depicted as on the right-hand side, but connected to each other as on the left-hand side. This convention differs from the convention chosen here. It amounts to a different set of Kirchhoff rules...

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