Question

Find the maximum displacement response spectrum (normalized to the static displacement) of a damped linear single degree of freedom oscillator with mass=m, stiffness=k, and damping constant=2ζmw₀. I believe the correct way to the solution is to use mx"+cx'+kx=0, after dividing both sides by m, the equation becomes mx"+2ζw₀x'+w_n²x=0, but now I am a little at a loss of what to do next. If the w₀ and w_n were the same values, then the answer would be easy enough to find. As for the maximum displacement response spectrum, I think that the Duhamel's integral will be used since I already did a problem with it for an undamped case, but any help would be appreciated.

Answer 1

,mx t cx t kx t mx t ++= g    (4.1) Substitute ω0= / k m and 0 2 c m ξ = ω and 2 0 1 ω =ω −ξ d The equation (4.1) can be re-written as 2 0 0 ( ) 2 ( ) ( ) - ( ) g   xt xt xt x t + ξω + ω =  (4.2) Using Duhamel’s integral, the solution of SDOF system initially at rest is given by (Agrawal and Shrikhande, 2006) - ( - ) 0 0 ( ) - ( ) s ( - ) t t g d d e x t x in t d ξω τ = τ ω ττ ω ∫  (4.3) The maximum displacement of the SDOF system having parameters of ξ and ω0 and subjected to specified earthquake motion, ( ) g x t is expressed by - ( - ) 0 max 0 max ( ) ( ) s ( - ) t t g d d e x t x in t d ξω τ = τ ω ττ ω ∫  (4.4) The relative displacement spectrum is defined as, d 0 max S ( , )= ( ) ξ ω x t (4.5) where 0 (, ) d S ξ ω is the relative displacement spectra of the earthquake ground motion for the parameters of ξ and ω0. Similarly, the relative velocity spectrum, Sv and absolute acceleration response spectrum, Sa are expressed as, v 0 max S ( , )= ( ) ξ ω x t  (4.6) a 0 max max S ( , )= ( ) ( ) ( ) a g ξω = +    x t xt x t (4.7) The pseudo velocity response spectrum, Spv for the system is defined as S ( , ) = S ( , ) pv 0 0 d 0 ξω ω ξω (4.8) Similarly, the pseudo acceleration response, Spa is obtained by multiplying the Sd to ω0 2 , thus 2 S ( , ) = S ( , ) pa 0 0 d 0 ξω ω ξω (4.9) Consider a case where 2 0 0 . . ( ) ( ) - ( ) g ξ= +ω = ie x t x t x t   max | ( ) ( )| S xt x t a g = +   2 0 max =−ω | ( )| x t 2 0 max = ω | | x 2 = ω0Sd = Spa (4.10) The above equation implies that for an undamped system, Sa = Spa. The quantity Spv is used to calculate the maximum strain energy stored in the structure expressed as 2 2 2 2 max max 0 11 1 22 2 E kx m S mS = = ω= d pv (4.11) The quantity Spa is related to the maximum value of base shear as 2 max max 0 pa