Many critical engineering systems, ranging from aircraft engines and power turbines to robotic devices and precision instruments, contain components that interact through friction and contact. These interactions are difficult to model, yet they play a vital role in determining the reliability, efficiency, and durability of such systems. Inaccurate vibration predictions can lead to fatigue, wear, or even failure, with significant safety and economic consequences. The research funded by this award aims to provide new mathematical tools and computational methods to more accurately simulate the complex dynamic behavior of structures with frictional contact. By reformulating how these interactions are represented, this research aims to eliminate longstanding simplifications and approximations that limit current analysis techniques. These advances will help engineers better understand and predict the performance of engineering systems, enabling safer and more efficient designs and longer-lasting technologies. The outcomes are expected to benefit multiple industries and support broader national goals related to economic prosperity, energy efficiency, and security. The project also supports the training of graduate and undergraduate students and will be integrated into engineering curricula. Outreach activities will introduce engineering topics to high school students through hands-on demonstrations and mentoring, thus fostering interest in STEM careers. The research objective of