Chapter 5
Continuous Systems
It can be argued correctly that all mechanical/structural systems are continuous since
there are no massless system components. Accordingly, they have to be viewed and
analyzed as systems with infinite numbers of degrees of freedom. This observation
may suggest that our work on SDOF/MDOF dynamical systems cannot be used
to analyze continuous systems. This is not the case. There are two methods for
analyzing continuous systems, and both methods involve approximations.
– Method 1: It approximates the solution of continuous systems by that of MDOF
systems obtained by concentrating their distributed mass at finite numbers
of points. There are no formulas delivering MDOF-based representations of
continuous systems. Intuition and experience guide the construction of MDOFbased approximations. The accuracy of the method depends on the quality of the
MDOF representation. The use of more familiar concepts is an advantage of the
method. The number of degrees of freedom, which can be excessive for some
systems, e.g., industrial pipes, and the difficulty to assess solution accuracy are
its main disadvantages.
– Method 2: It uses equations of motions for continuous systems so that its
implementation involves less simple mathematical tools. However, the method is
conceptually similar to Method 1, i.e., solutions are represented by elements of
linear spaces spanned by modal shapes, which are infinite sets of functions rather
than finite sets of vectors. Theoretically, the method delivers the exact solution.
Practically, we can only obtain approximations since the representation of system
displacement has to be truncated, i.e., it has to be based on finite numbers of
modal shapes for numerical solutions.
This section develops Method 2 for flexural and shear beams, see Figs. 5.1
and 5.3. The displacements v(x, t) of these beams are real-valued functions of
space and time arguments which satisfy partial differential equations. The method
© 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_5
115
Continuous Systems
It can be argued correctly that all mechanical/structural systems are continuous since
there are no massless system components. Accordingly, they have to be viewed and
analyzed as systems with infinite numbers of degrees of freedom. This observation
may suggest that our work on SDOF/MDOF dynamical systems cannot be used
to analyze continuous systems. This is not the case. There are two methods for
analyzing continuous systems, and both methods involve approximations.
– Method 1: It approximates the solution of continuous systems by that of MDOF
systems obtained by concentrating their distributed mass at finite numbers
of points. There are no formulas delivering MDOF-based representations of
continuous systems. Intuition and experience guide the construction of MDOFbased approximations. The accuracy of the method depends on the quality of the
MDOF representation. The use of more familiar concepts is an advantage of the
method. The number of degrees of freedom, which can be excessive for some
systems, e.g., industrial pipes, and the difficulty to assess solution accuracy are
its main disadvantages.
– Method 2: It uses equations of motions for continuous systems so that its
implementation involves less simple mathematical tools. However, the method is
conceptually similar to Method 1, i.e., solutions are represented by elements of
linear spaces spanned by modal shapes, which are infinite sets of functions rather
than finite sets of vectors. Theoretically, the method delivers the exact solution.
Practically, we can only obtain approximations since the representation of system
displacement has to be truncated, i.e., it has to be based on finite numbers of
modal shapes for numerical solutions.
This section develops Method 2 for flexural and shear beams, see Figs. 5.1
and 5.3. The displacements v(x, t) of these beams are real-valued functions of
space and time arguments which satisfy partial differential equations. The method
© 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_5
115