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MDOF Systems with Proportional Damping Saeed Ziaei Rad General Concepts Proportional damping has the advantage of being easy to include in the analysis so far. The modes of a structure with proportional damping are almost identical to those of the undamped version of the model. It is possible to derive the modal properties of a proportionally damped system by analyzing the undamped version in full and then making a correction for the damping. Saeed Ziaei Rad Proportional damping – special case Consider the general equation of motion for a MDOF with a viscous damping: [ M ]{ x} [ C ]{ x } [ K ]{ x} { f } Assume that the damping matrix is directly proportional to the Stiffness matrix: [C ] [ K ] Then: [ ] [ C ][ ] [ ] [ K ][ ] [ k r ] [ c r ] T T Saeed Ziaei Rad Proportional damping – Let’s define {p} as: special case The undamped modal matrix { x } [ ]{ p } Then the equation of motion becomes (f=0): [ m r ]{ p} [ c r ]{ p } [ k r ]{ p } { 0} From which: m r p r c r p r k r p r 0 This is a SDOF system and therefore: kr cr 2 2 r , d r 1 r , r 0 . 5 r mr 2 krmr Saeed Ziaei Rad Proportional damping – The receptane matrix can be defined as: [ H ( )] ([ K ] i [ C ] [ M ]) 2 or jr kr N H jk ( ) (k r 1 m r ) i c r 2 r Saeed Ziaei Rad 1 special case Proportional damping – general case The general form of proportional damping is: [C ] [ K ] [ M ] Again assume: { x } [ ]{ p } And then: [ ] [ C ][ ] [ k r ] [ m r ] T In this case, the damped system has eigenvalues and eigenvectors as follow: d r 1 r , 2 r 0 . 5 r 0 . 5 / r Saeed Ziaei Rad Proportional Hysteretic damping Consider the general equation of motion for a MDOF with a hysteretic damping: [ M ]{ x} ([ K ] i[ D ]){ x } { f } Assume the hysteretic damping as: [D ] [K ] [M ] And again 2 r kr mr , r r (1 i r ) , 2 2 r / r 2 Exercise: Extract the above relations from the equation of motion. Saeed Ziaei Rad