On the stability of a discrete convolution with measured impulse response functions of mechanical components in numerical time integration

The relationship between one input and one output degree of freedom of a linear structure is completely described by the impulse response function (IRF). By convolution of this IRF with any input, the output can be computed. Several publications show in principle how components can be considered in numerical time integration of an overall system based on their IRF. The convolution integral is approximated as a sum, usually with the trapezoidal rule. However, it has been observed that a measured IRF may lead to an unstable behavior in the simulation of the overall system. This is expressed by an increasing amplitude of the involved quantities with increasing simulation time. The cause is suspected to be noise. In this paper, the real reason for the instability is shown which is somehow more general and noise is just a very relevant example which triggers the instability. After a theoretical derivation and discussion of this error, countermeasures for stabilization are explained. In the concluding chapter of the numerical examples, the findings from the theoretical chapter are confirmed. The stabilization strategies lead to the desired success. In the last example, an impulse response is generated based on measured data. After the impulse response is stabilized, the simulation results are compared with the measurement results and show good agreement.