Periodic　structures　are　effective　in　attenuating　waves　in　a　low　frequency　range　at　local　resonance　(LR)　conditions,　but　it　is　still　a　challenge　to　achieve　this　in　a　very　low　frequency　range,　because　the　system　stability　can　be　damaged　due　to　the　excessive　quality　or　the　deduction　of　stiffness.　However,　the　structural　stability　theory　shows　that　the　structural　stability　is　closely　related　to　the　support　surface;　the　larger　the　support　surface,　the　more　stable　the　structure,　and　thereby　the　array　structure　has　the　feature　of　maintaining　the　stability　of　the　system　through　expanding　the　support　surface.　Based　on　the　theoretical　principle,　a　new　type　of　local　resonator　with　stiffness　arrays　is　presented　to　further　lower　the　band　gaps　of　flexural　wave　propagation　in　LR　beams,　in　which　the　traditional　stiffness　is　equally　divided　into　an　array　form.　Due　to　the　stiffness　array　connections,　the　system　stiffness　is　not　only　reduced　to　a　very　low　value,　but　also　the　stability　is　still　better　maintained　through　expanding　the　support　surface.　Meanwhile,　the　band　structures　of　the　LR　beams　with　stiffness　array　connections,　obtained　by　the　finite　element　method,　demonstrate　that　the　lower　bound　of　the　band-gap　can　be　successfully　decreased　more　times　than　that　of　conventional　LR　beams　under　the　premise　of　maintaining　the　stability　of　the　system,　and　an　ultra-low-frequency　broadband　of　25-395　Hz　is　realized.　Clearly,　the　strategy　of　dividing　the　traditional　stiffness　into　the　stiffness　array　can　successfully　realize　the　low　frequency　band　gap　and　overcome　the　shortcomings　of　the　system　instability　in　the　traditional　method.　Therefore,　two　puzzles　of　realizing　the　ultra-low-frequency　broadband　and　simultaneously　maintaining　the　system　stability　may　be　successfully　resolved　through　introducing　the　stiffness　arrays　into　the　local　resonance　system,　and　the　new　structures　with　stiffness　arrays　could　have　potential　applications　for　ultra-low-frequency　vibration　and　noise　attenuation.