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Nonperiodic Excitation
Published in William J. Bottega, Engineering Vibrations, 2014
When Eqs. (4.2) and (4.10) are compared, it is evident that () ∫-∞tδ^(t)dt=H(t) Similarly, comparison of Eqs. (4.5) and (4.11) gives the relation () ∫-∞tδ^(t-a)dt=H(t-a) Since the unit step and unit impulse functions are related in this way, the derivatives of the Heaviside step function may be defined as the inverse operation. Equations (4.12) and (4.13) therefore suggest the definition of the derivative of the unit step function as () ddtH(t)=δ^(t) () ddtH(t-a)=δ^(t-a) Both the Dirac delta function and the Heaviside step function will prove to be very useful in describing the loading and response for various types of transient excitation, as well as for the development of the response for arbitrary excitation.
Energy-optimal overtaking manoeuvres of Formula-E cars
Published in Vehicle System Dynamics, 2023
Xuze Liu, Abbas Fotouhi, Daniel J. Auger
Different from the majority of collision-avoidance research that uses quadratic distance (i.e. a circle) to describe the collision exclusive zone [35], in this study the zone is rectangular with the front edge starting from the centre of the target car. The length of the rectangle is the same length as the car and the width is twice the car width laterally symmetrical about the target car centre. There are two main reasons to do so. First, it is to ensure a precise description of the collision zone so that no track space is overlooked which might lead to potential overtaking opportunities. Second, starting the front edge from the CG position of the target car aims to echo the second aforementioned main assumption that to overtake, the actor car only needs to pass the abreast position of the target car. The collision constraint is formulated as follows: where the lateral boundaries and are given by where and are the longitudinal and lateral position of a car on the track based on the curvilinear coordinate system. The subscripts and denote the actor car and the target car, respectively. and are the car width and length. is the road width. The function is the Heaviside step function.