Systems with rigid constraints

We now consider the case where the beads are subject to rigid constraints. This is necessary to deal with the problems of suspensions of a rigid body, or polymers with rigid constraints (such as the rodlike polymer, or the freely jointed model), but the reader who is interested only in flexible polymers can omit this section. 

Here we shall consider only the holonomic constrainjst which can be expressed as equations connecting the coordinates of the beads  

Examples of systems which have such constraints are  

When the constraints are introduced, the force is no longer a known function of / and must be determined by the equation of motion. This 

From a practical viewpoint, there are two ways of doing this. 

---

To apply the theory to such general systems, we have to consider a system with rigid constraints. However, in this section we shall first consider the case in which there are no rigid constraints, i.e., the force SWdR is finite and well behaved. 

Phenomenologically, the viscous stress is the stress which vanishes instantaneously when the flow is stopped. On the other hand the elastic stress does not vanish until the system is in equilibrium. The elastic stress is dominant in concentrated polymer solutions, while viscous stress often dominates in the suspensions of larger particles for which the Brownian motion is not effective. Whichever stress dominates, the rheological properties can be quite complex since both and are functions of the configuration of the beads and therefore depend on the previous values of the velocity gradient. Note that the viscous stress only appears in the system with rigid constraints.t... 

Totally rigid models A special case of a system of particles with holonomic constraints is the rigid body. A rigid body can be thought of as a system of particles with a distance constraint between every pair of particles. A (nonlinear) rigid model has six degrees of freedom three translational and three rotational. 

Various conditions determine what states of a system are physically possible. If a uniform phase has an equation of state, property values must be consistent with this equation. The system may have certain built-in or externally-imposed conditions or constraints that keep some properties from changing with time. For instance, a closed system has constant mass a system with a rigid boundary has constant volume. We may know about other conditions that affect the properties during the time the system is under observation. 

The concepts introduced are illustrated in Fig. 1 with the use of a simple mechanical model. This model is analogous to the well known illustration of the Lagrange-Dirichlet theorem concerning stability of potential mechanical systems (a heavy ball on a smooth curved surface). In the present case, a heavy body, say, a cylinder, is placed on a rigid, "geared" cylindrical surface. The body is restricted from the left-hand-side motion with a constraint, a kind of racheting mechanism. The state of the system in Case 1 is subequilibrium, consequently, stable. That in Case 2 is in equilibrium and stable. Case 3 corresponds to an equilibrium, neutral state, and Case 4 to an equilibrium, unstable state. State 5 is nonequilibrium and, consequently, unstable. 

In Sect. 3.2, we have seen that a dynamical system with unilateral or bilateral frictional contact can possess a peculiar characteristic, namely the inertia matrix may be asymmetric and nonpositive definite. Painleve was the first to point out the difficulties that may arise in such cases [53, 95]. As we will see in this section through examples, the presence of a kinematic constraint with friction could lead to situations where the equations of motion of the system do not have a bounded solution (inconsistency) or the solution is not unique (indeterminacy). These situations where the existence and uniqueness properties of the solution of the equations of motion are violated are known as the Painleve s paradoxes. There is a vast literature on the general theory of the rigid body dynamics with frictional constraints... 

The third and final instability mechanism in the lead screw drives is the kinematic constraint. In Sect. 4.3, Painleve s paradoxes were introduced and - through simple examples - it was shown that under the conditions of the paradoxes, the rigid body equations of motion of a system with frictional contact do not have a bounded solution or the solution is not unique. We have also discussed the relationship between Painleve s paradoxes and the kinematic constraint instability mechanism. 

---