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Section 4.3 kinematic analysis

Subsection 4.3.1 kinematic matrix

Let us consider a system with \(n\) degrees of freedom and subjected to \(m\) constraint conditions. The equations expressing the prescribed constraint conditions can be written as

\begin{equation} \sum_{j=1}^n A_{ij} q_j = d_i \qquad (i=1 \dots m)\,.\tag{4.3.1} \end{equation}

\(d_i\) is a generic assigned displacement on the \(i\)-th constraint, \(q_j\) is the \(j\)-th degree of freedom and \(A_{ij}\) is the unitary contribution to displacement give by \(q_j\) on \(i\)-th constraint. Using a matrix notation the constraint conditions can be also epxressed as follows

\begin{equation} \mat{A} \vec{q} = \vec{d},\label{kinematik_system_eq}\tag{4.3.2} \end{equation}

where \(\mat{A}\) is the \(m \times n\) kinematic matrix of the system, \(\vec{q}\text{,}\) with \(n\) components, is the vector of the chosen degrees of freedom and \(\vec{d}\text{,}\) with \(m\) components, is the vector containing the assigned displacemnts.

Note 4.3.1.

The components of the vector \(\vec{d} \) can indifferently assume null or non-null values. The most recurrent case is, however, a zero assignment for all components.

Subsection 4.3.2 kinematic classification

The linear system (4.3.2) is used for the calculation of the Lagrangian parameters (chosen degrees of freedom) collected in vector \(\vec{q}\text{.}\) To this end, it is useful to define, on the basis of the solubility conditions of the system, a kinematic classification of the systems. Let \(p = \text{min}(n, m) \text{,}\) then the following cases may occur.

  • \(\text{rank}\mat{A} == p\)
    • \(n==m\text{:}\) determinate system;
    • \(n>m\text{:}\) the system is a mechanism;
    • \(n<m\text{:}\) impossible system.
  • \(\text{rango}\mat{A} < p\)
    • degenerate system.
Remark 4.3.2.

In the case of a degenerate system, one or more lines of the matrix \(\mat{A} \) are a linear combination of the others. The elimination of the dependent lines leads to a system with a lower number of equations for which the solution can be carried out by falling into one of the three cases seen for the non-degenerate system. From a mechanical point of view, this situation is not necessarily a symptom of an error but can be determined by one or more constraint conditions that can be eliminated without changing the resulting kinematics of the system.