SEISMIC RESPONSE ANALYSIS OF SDOF SYSTEMS BY NUMERICAL INTEGRATION OF THE EQUATION OF MOTION LINEAR ACCELERATION METHOD The equation of motion of a SDOF system, subjected to the ground motion, is given by mx(t ) + cx(t ) + kx(t ) = −ma g (t ) (1) where m is the system mass, c is the damping coefficient, k is the stiffness and ag(t) is the ground acceleration. The integral form of equation (1) is obtained by dividing (1) to the mass m: x(t ) + 2ζωx(t ) + ω 2 x(t ) = −a g (t ) (2) where is the damping ratio and ω is the natural circular frequency of the dynamic system. By solving equation (2), the dynamic response expressed in terms of relative response accelerations, velocities and displacements, at instant-time can be obtained. THE LINEAR ACCELERATION METHOD. A BRIEF OVERVIEW. The Linear Acceleration Method is a numerical integration method of the second order differential equations, for SDOF systems, and systems of equations of motion, for MDOF models as well. The method is based on the assumption that the response acceleration varies linearly from step to step. Therefore, the response velocity results of a quadratic form and the response displacement, of cubic form. A brief overview of the method is given below. Using the expansion in Taylor series, the displacement and the velocity at the step ”i+1” can be written in the following form ∆t 2 ∆t 3 ∆t 2 ∆t 2 (3) x i +1 = x i + x i ∆t + x i + xi = x i + x i ∆t + x i + x i +1 2 6 3 6 x i +1 = x i + x i where xi =
∆t ∆t + x i +1 2 2
x i +1 − x i ∆t
(4)
(5)
and ∆t is the integration time-step. By introducing the equations (3) and (4) into equation (1), which must be verified at each timeinstant, the acceleration at step “i+1”is given by 1 xi +1 = − (a g ,i +1 + Bxi + Cxi + Dxi ) (6) A where the integration constants A, B, C and D are given by 2 2 ∆t (7) A = 1 + ζω∆t + ω 6
1
B = ζω∆t + ω 2
∆t 2 3
(8)
C = 2ζω + ω 2 ∆t
(9)
D =ω2
(10)
Therefore, based on equations (3), (4) and (6), the dynamic response at step “i+1” can be computed, based on the response at “i” step. At the first step, the response is computed taking into account the initial conditions, zero ground displacement and velocity, as follows: 1 x1 = − a g ,1 (11) A ∆t (12) x1 = x1 2 ∆t 2 x1 = x1 (13) 6 • On the stability problem of method The Linear Acceleration Method is a conditionally stable algorithm. Its stability is mainly a function of integration time-step ∆t. For stability purpose, the time-step must fulfill the following condition ∆t ≤ 0.55 (14) T where, T is the natural period of vibration of the dynamic system. Other factors that may contribute to errors in results are: 1. Round off 2. Truncation Errors resulting from any causes may be manifested by either or both on the following effects: 1. Phase shift or apparent change of frequency in cyclic results 2. Artificial damping, in which the numerical procedure removes or adds energy to the dynamic system.
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