Mechanical analogues

Whilst obtaining this is the ultimate goal for many rheologists, in practice it is not possible to develop such an expression. However, our mechanical analogues do allow us to develop linear constitutive equations which allow us to relate the phenomena of linear viscoelastic measurements. For a spring the relationship is straightforward. When any form of shear strain is placed on the sample the shear stress responds instantly and is proportional to the strain. The constant of proportionality is the shear modulus 

Here the time derivative of the strain is represented by Newton s dot. This is the response of a purely viscous fluid. Now suppose we consider a combination of these models. The two simplest arrangements that we can visualise is the models in series or parallel. When they are placed in series we have a Maxwell model and in parallel we have a Kelvin (or sometimes a Kelvin-Voigt) model. 

The constitutive equations for these models become more complex. When we apply a stress to the parallel elements of a Kelvin model both elements will respond. Thus a linear addition of the stresses describes the constitutive equation  

For a Maxwell model it is the strain rates that linearly add 

Both of these models show contributions from the viscosity and the elasticity, and so both these models show viscoelastic behaviour. You can visualise a more complex combination of models possessing more complex constitutive equations and thus able to describe more complex rheological profiles. 

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The Hamiltonian operator is the quantum-mechanical analogue of the energy, and we say that the allowed values of the energy, the , above, are the eigenvalues... 

The theory behind molecular vibrations is a science of its own, involving highly complex mathematical models and abstract theories and literally fills books. In practice, almost none of that is needed for building or using vibration spectroscopic sensors. The simple, classical mechanical analogue of mass points connected by springs is more than adequate. 

Equation (31) is known as Heisenberg s equation of motion and is the quantum-mechanical analogue of the classical equation (17). The commutator of two quantum-mechanical operators multiplied by 2mfh) is the analogue of the classical Poisson bracket. In quantum mechanics a dynamical quantity whose operator commutes with the Hamiltonian, [A, H] = 0, is a constant of the motion. 

In which Ein has been replaced by its quantum-mechanical analogue... 

In contrast to the uncertainty with respect to monkeys, the situation in respect of great apes (or at least chimpanzees) is more clear cut. Chimpanzees emerged as the most frequent users of tactical deception in Byrne s (1995) analysis. In addition, evidence from experimental studies by Povinelli et al. (1990) and O Connell (1996) provide convincing evidence that these great apes at least do possess formal theory of mind. Children are not born with a theory of mind ability, but acquire it at about the age of 4 years (Astington 1994). Some individuals (whom we label autistic) never develop this ability (Leslie 1987, Happe 1994). O Connell (1996) devised a mechanical analogue of the standard false belief test which she applied to chimpanzees as well as normal children and autistic adults. Her results demonstrate rather clearly that chimps do better than autistic adults and about as well as 4-year-old children on the same test. In other words, chimps perform about as well as children who have just acquired basic theory of mind. 

The way the three-dimensional quadmpole field acts to keep ions within a certain volume, i.e., within a potential well some electron volts in depth, can be illustrated by a mechanical analogue A ball has to be prevented from rolling from a saddle by rotating the saddle just right to bring the ball back to the middle before it can leave the surface via one of the steeply falling sides (Fig. 4.43). Paul demonstrated the dynamic stabilization of up to three steel balls by such a device in his Noble lecture. [103,104]... 

Fig. 4.43. Visualization of ion motion in the ion trap, (a) Mechanical analogue of the QIT. (b) Photograph of ion trajectories of charged aluminum particles in a quadrupole ion trap.

Bessems JGM, Vermeulen NPE. Paracetamol (acetaminophen) induced toxicity molecular and biochemical mechanisms, analogues and protective approaches. Crit Rev Toxicol 2001 31 55-138. 

Of the three eigenvalue equations, the one of interest to us is the vibrational equation. It has a particularly simple form when normal coordinates are employed because the classical kinetic and potential energies then have no cross terms (see eqns (9-2.17) and (9-2.18)) and this fact leads to a simple form for their quantum mechanical analogues (the kinetic energy and potential energy operators). The vibrational equation is thus... 

Naturally many practical implementation issues arise, including the need to solve the dynamical equations, at least in the regions of importance sampled by the data. In this regard there is a classical mechanical analogue of the coupled dynamical and integral equations. Exploitation of classical inversion may be important, at least as a first step to define the potential in polyatomic cases. The key point at this time is that the new formulation provides a rigorous foundation to build upon for achieving direct practical inversions of temporal and spectroscopic data. 

Let us look at the calculation of the dipole moment within the Hartree-Fock approximation. The quantum mechanical analogue of Eq. (5.205) for the electrons in a molecule is... 

The mathematical term functional, which is akin to function, is explained in Section 7.2.3.1. To the chemist, the main advantage of DFT is that in about the same time needed for an HF calculation one can often obtain results of about the same quality as from MP2 calculations (cf. e.g. Sections 5.5.1 and 5.5.2). Chemical applications of DFT are but one aspect of an ambitious project to recast conventional quantum mechanics, i.e. wave mechanics, in a form in which the electron density, and only the electron density, plays the key role [5]. It is noteworthy that the 1998 Nobel Prize in chemistry was awarded to John Pople (Section 5.3.3), largely for his role in developing practical wavefunction-based methods, and Walter Kohn,1 for the development of density functional methods [6]. The wave-function is the quantum mechanical analogue of the analytically intractable multibody problem (n-body problem) in astronomy [7], and indeed electron-electron interaction, electron correlation, is at the heart of the major problems encountered in... 

With the quantum conditions E = hv, p = h/X, the wave-mechanical analogue becomes ... 

Schrodinger s equation is the wave-mechanical analogue of Hamilton s formulation of the classical laws of motion. Hamilton s function ... 

We have already pointed out the analogy which exists between the wave function 9 and the amplitude of light and other vibrational phenomena. There is now also a mechanical analogue of the occurrence of the two wave functions 9 and 9 as described above. 

The priorities the computer gives to various tasks are a mechanical analogue of human values. Without getting into detailed considerations of emotions yet, we can observe that something you value can be identified... 

In practical applications of the linear-response formalism, it is often more convenient to express the response in terms of a displacement V rather than a flux J, and we will therefore focus on this case here. The response of a gel to an applied stress is, for instance, conveniently expressed as a strain, and likewise, the response of a dielectric material to an applied electric field is often expressed as a polarization [which is closely related to the so-called electric displacement field (89)]. As a mechanical analogue would indicate, a flux is generally proportional to a velocity, and the displacement is therefore computed as the time integral of the flux. 

The continuum mechanical analogues of the fundamental thermodynamic relations are written as ... 

The traditional model of atomic structure has been the quantum-mechanical analogue of a solar system, with electrons perturbing each other only a little from their individual states of definite energy and angular momentum. On the other hand, the traditional model for molecular structure has been the quantum-mechanical counterpart of balls held together by springs in a fairly rigid, well-defined structure. These two pictures are so different and lend themselves to such different computational methods that they have remained as separated, almost unrelated fields within chemical physics. 

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