The　primary　goal　of　this　work　is　to　analyze　the　energized　motion　of　three-degrees-of-freedom　(3DOF)　dynamic　system　consisting　of　a　coupled　double　pendulum　with　the　damped　mass　under　the　external　harmonic　forces　and　moments.　Lagrangian　equations　are　employed　to　derive　the　differential　governing　equations　of　motion　(GEOM)　based　on　the　system　generalized　coordinates.　The　approximate　solutions　(AS)　of　these　equations　are　generated　through　the　utilization　of　the　multiple　scales　technique　(MST)　at　the　third-order　level　of　approximation.　These　solutions　are　ascertained　by　contrasting　them　with　numerical　solutions　(NS)　that　are　derived　utilizing　the　fourth-order　Runge-Kutta　algorithm　(RKA-4).　The　modulation　equations　are　constructed,　and　the　principal　external　resonance　cases　are　scrutinized　concurrently　based　on　the　solvability　constraints.　The　steady-state　solutions　are　studied.　Based　on　Routh-Hurwitz　criteria　(RHC),　the　stability　and　instability　zones　are　examined　and　assessed　in　the　line　with　the　steady-state　solutions.　The　amplitudes　and　phases　over　a　specific　period　of　time　have　been　graphed　to　illustrate　the　movement　at　any　given　instant.　Furthermore,　in　order　to　evaluate　the　advantageous　effects　of　different　values　pertaining　to　the　physical　parameters　on　the　system　behavior,　the　graphed　representations　of　the　obtained　results,　resonance　reactions　and　areas　of　the　stability　are　provided.　The　significance　of　the　research　model　stems　from　its　numerous　applications　such　as　the　gantry　cranes,　robotics,　pump　compressors,　transportation　devices　and　rotor　dynamics.　It　may　be　applied　to　the　study　of　these　systems＇　vibrational　motion.　©　2024　The　Authors