MDOF Systems with Proportional Damping

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Presentation transcript:

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: Assume that the damping matrix is directly proportional to the Stiffness matrix: Then: Saeed Ziaei Rad

Proportional damping – special case Let’s define {p} as: The undamped modal matrix Then the equation of motion becomes (f=0): From which: This is a SDOF system and therefore: Saeed Ziaei Rad

Proportional damping – special case The receptane matrix can be defined as: or Saeed Ziaei Rad

Proportional damping – general case The general form of proportional damping is: Again assume: And then: In this case, the damped system has eigenvalues and eigenvectors as follow: Saeed Ziaei Rad

Proportional Hysteretic damping Consider the general equation of motion for a MDOF with a hysteretic damping: Assume the hysteretic damping as: And again Exercise: Extract the above relations from the equation of motion. Saeed Ziaei Rad