## Closed-form smoothers and shapers with distributed delay for damped oscillatory modes

The study deals with parametrisation of input shapers and smoothers with distributed delays, which are a common tool used for pre-compensating oscillatory modes of flexible systems. These filtering structures can be easily parametrised, if the oscillatory mode is undamped, leading to fully analytical formulas. For the damped case, however, the parametrisation needs to be done numerically as a rule. Utilising a straightforward complex domain transformation, as the main results, the structure of the filters is turned to the closed-form that can be parametrised analytically for the damped case too. The adjusted smoothers and shapers accommodate the reference and the system output signals without vibration at the same time lengths like the pre-forms for the undamped case. This methodology is applied to Trapezoidal, *S*-curve and Trigonometric smoothers, Jerk Limited shaper and recently proposed Zero Vibration shaper with a distributed delay. The proposed new types of smoothers and shapers, which are essentially based on exponential distribution of delays, are investigated in time and frequency domains. Subsequently, the basic properties, i.e. response performances, spectrum distribution and robustness analysis, are demonstrated and cross-compared in a case study example.