“Computational Civil Engineering - "Intersections" International Journal

242 C. C. Comisurealistic criterion for modal purity that is often used, is a phase scatter no greater0that ±10 around the quadrature components.4. CHARACTERISTIC PHASE-LAG MODESFor a n degree of freedom system with viscous damping, the equation ofmotion for steady state sinusoidal excitation can be written in its general form as:⎧⎨⎩⎫⎬⎭⎧ ⎫⎨ ⎬⎩ ⎭•••[ m] x + [ c] x + [ k]{} x = { F} sinωtwhere the system inertia [m], damping [c] and stiffness matrices [k] are assumed tobe real symmetric and positive – define.If the damping is hysteretic, the equation of motion would be given by:⎧⎨⎩⎫⎬⎭1ω⎧ ⎫⎨ ⎬⎩ ⎭•••[ m] x + [ d] x + [ k]{} x = { F} sinωtwhere [d] is the hysteretic damping matrix.In the general case damping would be non-proportional and thus thedamping matrix cannot be diagonalized using the normal mode transformation. Foran arbitrary set of forces { F } and excitation frequency ω the solution of equation(1) is rather complicated. Althrough the responses at each coordinate x areharmonic with the excitation frequency, thei are not all in phase with each other orwith the excitation force. If a system with n degrees of freedom is excited by a nnumber of forces which are either 0 0 or 180 0 out of phase (often called monophaseor coherently phased forces), then for a particular ratio of forces, the response ateach of the coordinates will be in phase with each other and lag behind the force bya common angle θ called the characteristic phase lag.Thus we have to determine the conditions which will produce a solution ofthe form:(1)(2)