Appendix A: Performing Numerical
Simulation
Solving vibration and stability problems typically involves both analytical and
computational work (and sometimes also experimental). We here briefly touch upon
a few practical aspects of dealing numerically with nonlinear dynamical systems.
This concerns solving differential equations, computing chaos-related quantities,
programming a helpful user interface, and using relevant Internet resources. The
information given here is intended only to give an overview of the kind of tools and
skills required for doing computational dynamics.
A.1. Solving Differential Equations
Mathematical models of vibrating systems usually come in the form of differential
equations having second-order time derivatives. With continuous systems, space
derivatives occur as well. Space derivatives are typically eliminated by some discretization method, e.g., Galerkin, finite element, or finite difference – so that in any
case a set of second-order ODEs (ordinary differential equations) results. These can
almost always be recast into the general first-order form:
~ _
x ¼ ~ fð~ x; tÞ:
ðA:1Þ
where ~ x 2 R
n is a vector of state variables (typically generalized displacements and
velocities) and ~ f is a vector of generally nonlinear functions. By introducing a new
dependent variable to eliminate the explicit time-dependency, such a system can be
further transformed into first-order autonomous form:
_
x ¼ fðxÞ;
ðA:2Þ
where x 2 R
n+1 and f 2 R
n+1 . For example, the Duffing equation
© The Editor(s) (if applicable) and The Author(s), under exclusive license to
Springer Nature Switzerland AG 2021
J. J. Thomsen Vibrations and Stability,
https://doi.org/10.1007/978-3-030-68045-9
449
Simulation
Solving vibration and stability problems typically involves both analytical and
computational work (and sometimes also experimental). We here briefly touch upon
a few practical aspects of dealing numerically with nonlinear dynamical systems.
This concerns solving differential equations, computing chaos-related quantities,
programming a helpful user interface, and using relevant Internet resources. The
information given here is intended only to give an overview of the kind of tools and
skills required for doing computational dynamics.
A.1. Solving Differential Equations
Mathematical models of vibrating systems usually come in the form of differential
equations having second-order time derivatives. With continuous systems, space
derivatives occur as well. Space derivatives are typically eliminated by some discretization method, e.g., Galerkin, finite element, or finite difference – so that in any
case a set of second-order ODEs (ordinary differential equations) results. These can
almost always be recast into the general first-order form:
~ _
x ¼ ~ fð~ x; tÞ:
ðA:1Þ
where ~ x 2 R
n is a vector of state variables (typically generalized displacements and
velocities) and ~ f is a vector of generally nonlinear functions. By introducing a new
dependent variable to eliminate the explicit time-dependency, such a system can be
further transformed into first-order autonomous form:
_
x ¼ fðxÞ;
ðA:2Þ
where x 2 R
n+1 and f 2 R
n+1 . For example, the Duffing equation
© The Editor(s) (if applicable) and The Author(s), under exclusive license to
Springer Nature Switzerland AG 2021
J. J. Thomsen Vibrations and Stability,
https://doi.org/10.1007/978-3-030-68045-9
449