Series-elastic element

Both RLC phosphorylation and active stiffness increase more rapidly than isometric force during the initiation of the contraction (Kamm and Stull, 1986). These observations suggest that phosphorylation of myosin RLC allows cross-bridge attachment to actin. The delay in force development may result from cooperative effects of phosphorylation on activation whereby force depends on formation of doubly phos-phorylated myosin (Persechini and Hartshorne, 1981 Sellers et al., 1983) however, other contributions, including a delay in the expression of force through series elastic element in the tissue, cannot be excluded (Aksoyetfl/., 1983). 

Figure 16.4 illustrates the mechanical components of the oculomotor plant for horizontal eye movements, the lateral and medial rectus muscle, and the eyeball. The agonist muscle is modeled as a parallel combination of an active state tension generator Fag> viscosity element Bag> and elastic element TlT) connected to a series elastic element Rse- The antagonist muscle is similarly modeled as a parallel combination of an active state tension generator Tant> viscosity element Rant> and elastic element TlT) connected to a series elastic element Rse- The eyeball is modeled as a sphere with moment of inertia /p, connected to viscosity element Bp and elastic element Kp. The passive elasticity of each muscle is included in spring Kp for ease in analysis. Each of the elements defined in the oculomotor plant is ideal and linear. 

FIGURE 24.6 Structure of the musculotendonal model (based on the Hill viscoelastic model). Abbreviations CE, contractile element SE, series elastic element PE, parallel viscoelastic element TE, tendonal elastic element , length M, muscle T, tendon. 

The simplest model that can show the most important aspects of viscoelastic behaviour is the Maxwell fluid. A mechanical model of the Maxwell fluid is a viscous element (a piston sliding in a cylinder of oil) in series with an elastic element (a spring). The total extension of this mechanical model is the sum of the extensions of the two elements and the rate of extension is the sum of the two rates of extension. It is assumed that the same form of combination can be applied to the shearing of the Maxwell fluid. 

Another approach that has physical merit is to model the behavior of viscoelastic materials as a series of springs (elastic elements) and dashpots (viscous elements) either in series or parallel (see Figure 8.1). If the spring and dashpot are in series, which is described as a Maxwell mechanical element, the stress in the element is constant and independent of the time and the strain increases with time. 

Figure 8.1. Diagram showing Maxwell mechanical model of viscoelastic behavior of connective tissues. In this model an elastic element (spring) with a stiffness Em is in series with a viscous element (dashpot) with viscosity T m. This model is used to represent time dependent relaxation of stress in a specimen bold of fixed length.

Butter, and other unctuous materials, may be qualitatively described by a modified Bingham body (Elliott and Ganz, 1971 Elliott and Green, 1972), which consists of viscous, plastic and elastic elements in series. The stress-strain behavior for the model proposed by Elliot and Ganz (1971) is shown in Figure 7.12B. Diener and Heldman (1968) proposed a more complex model to describe how butter behaves when a low level of strain is applied. The model consists of plastic and viscous elements in parallel, coupled in series with a viscous element in parallel with a combination of a viscous and an elastic element. Figure 7.12C shows the stress-strain curve for... 

Polymers are viscoelastic, meaning that they have intermediate properties between Newtonian liquids and Hookean solids. The simplest model of viscoelasticity is the Maxwell model, which combines a perfectly elastic element with a perfectly viscous element in series, as shown in Fig. 7.21. -Since the elements are in series, the total shear strain 7 is the sum of the... 

The mixture of latex and emulsion may be regarded as two elastic elements in series with the appropriate volume fractions. 

In an un-cross-linked amorphous polymer, above its glass temperature, the molecular chains are continuously wriggling from one conformation to another. If a mechanical stress is imposed on such a system of wriggling chains, it can respond in three distinct ways instantaneous elastic response retarded (conformational) elastic response or viscous flow. Actually, in order to fit experimental data adequately, the retarded elastic element must be expanded into a whole series of such elements, some with shorter and some with longer response times. The local "kinkiness" of the chains can be straightened out (by stress) more rapidly than the... 

Here s is the stress and is the strain, respectively. If s = 0,e = 0, we have an elastic element or a spring, if s = 0,e = 0, we have a viscous element or a dashpot. If e = 0, we have an elastic element and a viscous element in series, which is called visco-elastic element. 

The modeling and control of movements in this chapter relates to external control of muscles via so-called functional electrical stimulation. Macroscopic viscoelastic models started from the observation that the process of electrical stimulation transforms the viscoelastic material from a compliant, fluent state into the stiff, viscous state. Levin and Wyman [35] proposed a three-element model— damped and undamped elastic element in series. Hill s work [36] demonstrated that the heat transfer depends upon the type of contraction (isometric, slow contracting, etc). The model includes the force generator, damping and elastic elements. Winters [37] generalized Hill s model in a simple enhancement of the original, which... 

FIGURE 48.3 The Hill model of muscle separates the artive properties of muscle into a contractile element, in series with a purely elastic element. The properties of the passive muscle are represented by the parallel elastic element. 

The early observations of Bagge et al. [1977] led them to suggest that the neutrophil behaves as a simple viscoelastic solid with a Maxwell element (an elastic and viscous element in series) in parallel with an elastic element. This elastic element in the model was thought to pull the unstressed cell into its spherical shape. Subsequently, Evans and Kukan [1984] and Evans and Yeung [1989] showed that the cells flow continuously into a pipette, with no apparent approach to a static limit, when a constant suction pressure was applied. Thus, the cytoplasm of the neutrophil should be treated as a liquid rather than a solid, and its surface has a persistent cortical tension that causes the cell to assume a spherical shape. 

An interesting three-parameter model (the Burger model has four parameters) was proposed by Hsueh [6] and is shown in Fig. 3b. He demonstrated that for a Hookean elastic element (Ei) in series with a Kelvin solid (E2,ry), the stress-strain rate relations for constant strain rate and constant stress creep tests are,... 

Guiu and Pratt [32] have shown how a model consisting of an Eyring dashpot in series with an elastic element leads to a simple equation to describe stress relaxation curves in tension. Suppose that there is a total strain e on the system of... 

Maxwell (1867) first proposed this equation for the viscosity of gases Despite his initial misapplication of a good theory, rhe-ologists have forgiven him and embrace eq. 3.2.18 as the Maxwell model. It is often represented as a series combination of springs, elastic elements, and dashpots, viscous ones as shown in Hgure 3.2.2. 

James Clerk Maxwell realized that neither a linear viscous element (dashpot) nor a linear elastic element (spring) was sufficient to describe his experiments on the deformation of asphalts, so he proposed a simple series combination of the two, the Maxwell element (Figure 15.1c). In a Maxwell element, the spring and dashpot support the same stress so... 

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