Chapter 4
Multi-Degree of Freedom (MDOF)
Systems
We consider systems with finite numbers n > 1 of degrees of freedom. Systems
with infinite numbers of degrees of freedom, referred to as continuous systems, are
discussed in the subsequent chapter. It will be seen that the methods for solving
MDOF and continuous systems are conceptually similar and involve three steps.
First, the displacement vectors/functions of MDOF/continuous systems are
viewed as elements of the linear spaces spanned by the eigenvectors/eigenfunctions
of these systems. The methods of Chap. 3 are employed to construct eigenvector/eigenfunction coordinates. The displacement functions are completely defined
by their projections on these coordinates, which are finite for MDOF systems and
(countable) infinite for continuous systems.
Second, differential equations are developed for the projections of the
displacement vectors/functions of MDOF/continuous systems on their eigenvectors/eigenfunctions. These equations are uncoupled and have the structure of the
equations of motion for SDOF systems. They can be solved one-by-one by using
the methods developed in Chap. 2.
Third, the displacement vectors/functions of MDOF/continuous systems are
assembled from their representations in the eigenvector/eigenfunction coordinates
and their projections of these coordinates delivered by the previous step.
In summary, the analysis of MDOF and continuous systems does not introduce
new concepts. It uses tools for solving equations of motion for SDOF systems and
properties of eigenvectors and eigenvalues, which are discussed in the previous two
chapters.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. D. Grigoriu, Linear Dynamical Systems,
https://doi.org/10.1007/978-3-030-64552-6_4
67
Multi-Degree of Freedom (MDOF)
Systems
We consider systems with finite numbers n > 1 of degrees of freedom. Systems
with infinite numbers of degrees of freedom, referred to as continuous systems, are
discussed in the subsequent chapter. It will be seen that the methods for solving
MDOF and continuous systems are conceptually similar and involve three steps.
First, the displacement vectors/functions of MDOF/continuous systems are
viewed as elements of the linear spaces spanned by the eigenvectors/eigenfunctions
of these systems. The methods of Chap. 3 are employed to construct eigenvector/eigenfunction coordinates. The displacement functions are completely defined
by their projections on these coordinates, which are finite for MDOF systems and
(countable) infinite for continuous systems.
Second, differential equations are developed for the projections of the
displacement vectors/functions of MDOF/continuous systems on their eigenvectors/eigenfunctions. These equations are uncoupled and have the structure of the
equations of motion for SDOF systems. They can be solved one-by-one by using
the methods developed in Chap. 2.
Third, the displacement vectors/functions of MDOF/continuous systems are
assembled from their representations in the eigenvector/eigenfunction coordinates
and their projections of these coordinates delivered by the previous step.
In summary, the analysis of MDOF and continuous systems does not introduce
new concepts. It uses tools for solving equations of motion for SDOF systems and
properties of eigenvectors and eigenvalues, which are discussed in the previous two
chapters.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. D. Grigoriu, Linear Dynamical Systems,
https://doi.org/10.1007/978-3-030-64552-6_4
67