By Raymond W. Clough
A senior point text/reference creation to Structural Dynamics as taught in Chemical Engineering departments. Covers newest details on Earthquake Engineering.
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This formulation has the advantage of dealing only with purely scalar energy quantities, whereas the forces and displacements used to represent corresponding effects in the virtual-work procedure are all vectorial in character, even though the work terms themselves are scalars. It is of interest to note that Hamilton’s principle can also be applied to statics OVERVIEW OF STRUCTURAL DYNAMICS 11 problems. In this case, it reduces to the well-known principle of minimum potential energy so widely used in static analyses.
3-25), cancel out the quantity exp(iωt) common to each term, substitute k/ω 2 for m and β for ω/ω, and solve for G yielding G= po k p 1 = o (1 − β 2 ) + i (2ξβ) k (1 − β 2 ) − i (2ξβ) (1 − β 2 )2 + (2ξβ)2 (3-27) Substituting this complex value of G into the first of Eqs. (3-26) and plotting the resulting two vectors in the complex plane, one obtains the representation shown in Fig. 3-5. Note that these two vectors and their resultant along with phase angle θ Im p (1 − β 2) o exp (iω t) k (1 − β 2) 2 + (2ξβ )2 θ ωt Re ρ exp [i (ω t − θ )] ωt p 2ξβ o [ i exp iω t ] k (1 − β 2) 2 + (2ξβ )2 − FIGURE 3-5 Steady-state response using viscous damping.
2-22c) to Eq. (2-22b), and also noting that a negative imaginary component would be associated with a negative vector angle, leads to Euler’s pair of equations that serve to transform from trigonometric to exponential functions: exp(iθ) = cos θ + i sin θ exp(−iθ) = cos θ − i sin θ (2-23a) Furthermore, Eqs. (2-23a) may be solved simultaneously to obtain the inverse form of Euler’s equations: cos θ = 1 2 exp(iθ) + exp(−iθ) sin θ = − 2i exp(iθ) − exp(−iθ) Im G = G R + i GI or G = G exp(iθ) G G θ GR = Gcosθ FIGURE 2-4 Complex constant representation in complex plane.
Dynamics of Structures by Raymond W. Clough