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