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Modern safety-critical systems often feature multi-core multi-SoC platforms and execute both periodic and sporadic workloads with real-time requirements. Periodic tasks benefit from a time-triggered (TT) approach, while sporadic events are best modeled by event-triggered (ET) tasks and scheduled using classical online mechanisms such as fixed-priority. Integrating TT and ET tasks in a multi-core environment has mainly been studied in fully partitioned solutions. However, for certain workloads, it may be beneficial to have a global scheduling approach for ET tasks. In this paper, we extend the current state-of-the-art in Response-Time Analysis (RTA) and Real-Time Calculus (RTC) to analyze the schedulability of sporadic ET tasks in a homogeneous multi-core TT system. We generalize previous promising results integrating TT and ET tasks using affine envelopes to homogeneous multi-core systems. While our method can be applied to any TT schedule generation mechanism, we also present a concrete schedule synthesis method based on a variant of the Least-Laxity First scheduling approach. We demonstrate the performance of our approach, in terms of both schedulability and runtime, through real-world and synthetic experiments inspired by real workloads. We note that our affine-envelope-based interference bound and the generalized burst limiting constraint (BLC) are also independent of the concrete TT synthesis algorithm and of the underlying timing analysis method (RTA or RTC). This modularity allows the framework to be combined with alternative schedulability analyses or TT schedule generation techniques, while preserving the demonstrated gains in schedulability and scalability for mixed TT and ET workloads on homogeneous multi-core platforms.