“Computational Civil Engineering - "Intersections" International Journal

234 C. C. Comisu⎡⎢⎢⎢⎢⎢⎢⎣1⎪⎧ ⎛ ⎞⎨1−⎜⎟⎪⎩ ⎝ω0 ⎠ω 22⎤⎥⎥⎥ F ⋅ e⋅⎥ k⎪⎫+ γ ⎥⎬ ⎥⎪⎭ ⎦j⋅ω⋅twhich lags behind the force vector by an angle θ given by:θ = tan−1⎡⎢⎢ γ⎢⎢ ⎛ ω ⎞1−⎢⎜⎟⎣ ⎝ω0 ⎠It can be seen that the quadrature response peaks more sharply that the totalresponse, and is equal to the total response at resonance, since the in-phaseresponse is zero. The total response on either side of resonance is relatively large,because the in-phase response varies more slowly that the quadrature response.Whilst the in-phase response is asymptotic to 1/m above resonance, the quadratureresponse rapidly approaches zero on either side of resonance.The amplitude of the total response is larger that of the quadrature response andthat the peak of the total response occurs above the true resonant frequency.Although the difference in the frequency of the two peaks is small, the relativeamount of the non-resonant mode response at the frequency of the peak totalresponse is increased. Since the response of the resonant mode varies rapidly withfrequency, the contribution of non-resonant modes to the total response is increasedfrom approximately 25% at the actual resonant frequency to approximately 65% atthe apparent resonant frequency indicated by the peak of the total response.If the mode shape was determined from the quadrature response, adequate modeshape measurement would be obtained, since the error in the amplitude of thefourth mode as determined by the quadrature response is negligible for this simplesystem. In practical structures, however, resonant modes are often close to eachother, causing modal interaction in the quadrature response.2. 3. Close ResonancesFigure 6 shows two modes that are closely spaced in the frequency domain. Eachmode responds to a sinusoidal excitation at any frequency. This response is smallunless the excitation frequency is in the immediate vicinity of the mode’s resonantfrequency. Although the off-resonant contribution is relatively small at the2⎤⎥⎥⎥⎥⎥⎦