The structure is modeled using a skewer model, combining mass and stiffness elements, with a single translational degree of freedom and attached to the ground by means of a vertical rod of negligible mass. This model is called a simple oscillator and is illustrated in Figure 1.

Figure 1. Simple oscillator subjected to a seismic signal u ¨ s ( t ) .

When a seismic signal is applied to the base of the oscillator, it begins to oscillate and generates two forces in the same direction, and opposite to the direction of movement:

An elastic restoring force expressed by F 1 = k ⋅ u ( t ) ;A damping force expressed by F 2 = c ⋅ u ˙ ( t ) .

Applying the fundamental principle of dynamics to mass, we obtain the following equation of motion for damped forced oscillations [6].

with U ¨ ( t ) = u ¨ s ( t ) + u ¨ ( t ) , it gives:

where m , c , k respectively designates the mass, the damping constant, and the stiffness of the rod.

U ¨ ( t ) , u ¨ ( t ) , u ˙ ( t ) , u ( t ) respectively denotes the total acceleration, the acceleration, the velocity, and the relative displacement of the mass of the oscillator.

To solve this equation, divide expression (2.2) by m, and set ξ = c 2 m ω , ω 2 = k m , we obtain the refined expression (2.3):

where ξ is the damping ratio and ω the natural frequency of the oscillator.

In [6] and [8], Anil K. Chopra and Alain Pecker propose expression (2.4) as the ideal method for solving the linear differential Equation (2.3). Its general solution, for the underdamped oscillator, is of the form:

where ω D is the angular frequency of the damped oscillations and is expressed by ω D = ω 1 − ξ 2 .

The right-hand side of Equation (2.4) describes free oscillations with damping, while the right-hand side describes forced oscillations with damping. In the general case, the free oscillations disappear in a fraction of a second, giving way to established forced oscillations. The general solution (2.4), therefore, takes the form of expression (2.5):

Equation (2.5) is called the convolution integral or Duhamel integral, which we solved by a quasi-analytical method by considering two hypotheses on the behavior of the ground acceleration during a time increment [2]:

The ground acceleration u ¨ s ( t ) varies linearly within each time increment Δ t ;The ground acceleration u ¨ s ( t ) is an average constant within each time increment Δ t .

The two assumptions and approximations of the seismic signal are represented in Figure 2andFigure 3.

Figure 2. Approximation of the linear seismic acceleration u ¨ s ( t ) during a time increment Δ t .

Figure 2 represents the assumption of linear variation of the seismic acceleration. Using Thales’ relation, I obtain the approximation of this acceleration by expression (2.6).

Figure 3. Approximation of the seismic acceleration u ¨ s ( t ) average constant during a time increment Δ t .

Figure 3 represents the assumption of a constant mean seismic acceleration. Approximation (2.7) is obtained by averaging the largest and smallest seismic accelerations that bracket the time increment.

2.2.1. Solving the Equation of Motion by Considering That the Ground Acceleration Varies Linearly over a Time Increment ∆t

Solving Equation (2.5) requires changes of variables, then the use of trigonometric relations, and finally several part-wise integrations.

By posing h ( t − τ ) = 1 ω D e − ξ ω ( t − τ ) sin ω D ( t − τ ) , the integral (2.5) becomes

Moreover: sin ω D ( t − τ ) = sin ω D t cos ω D τ − cos ω D t sin ω D τ and e − ξ ω ( t − τ ) = e − ξ ω t ⋅ e ξ ω τ = e ξ ω τ e ξ ω t .

And expression (8) becomes:

Substituting expression (2.9) into (2.8), we obtain:

By posing A ( t ) = ∫ 0 t u ¨ s ( t ) e ξ ω τ cos ω D τ d τ and B ( t ) = ∫ 0 t u ¨ s ( t ) e ξ ω τ sin ω D τ d τ , Expression (10) can also be written in the form.

Let’s study the functions A ( t ) and B ( t ) in the interval [ 0 , t ] . To do this, subdivide this interval into small time intervals [ t i , t i + 1 ] .

At time t = t i ; A ( t i ) = ∫ 0 t i u ¨ s ( t ) e ξ ω τ cos ω D τ d τ and B ( t i ) = ∫ 0 t i u ¨ s ( t ) e ξ ω τ sin ω D τ d τ .

At time t = t i + 1 ; A ( t i + 1 ) = ∫ 0 t i + 1 u ¨ s ( t ) e ξ ω τ cos ω D τ d τ and B ( t i + 1 ) = ∫ 0 t i + 1 u ¨ s ( t ) e ξ ω τ sin ω D τ d τ .

The relationship between A ( t i ) and A ( t i + 1 ) and is given by expression (2.12).

Proceeding in the same way as before, the relationship between B ( t i ) and B ( t i + 1 ) is given by expression (2.13).

Introducing the analytical expression for ground acceleration (2.6) into the established expressions (2.12) and (2.13), and after several integrations by parts, we obtain the discretized partial solutions given by the expressions (2.14) and (2.15).

Introducing expressions (2.14) and (2.15) into expression (2.11), we obtain the general discretized solution given by expression (2.16).

2.2.2. Solving the Equation of Motion by Considering That the Ground Acceleration Is Constant on Average over a Time Increment ∆t

For this second hypothesis, the solution of Equation (2.5) follows the same procedure as before, with the sole difference that the ground acceleration no longer varies linearly over a time increment but rather remains constant over the same time increment. By introducing the analytical expression for ground acceleration (2.7) into the established expressions (2.12) and (2.13), and after several part-integrations of Equation (2.5), we obtain the discretized partial solutions given by expressions (2.17) and (2.18).

Introducing expressions (2.17) and (2.18) into expression (2.11), we obtain the general discretized solution given by expression (2.19).

To implement functions (2.16) and (2.19), we consider the data of the North-South component from the accelerogram of the EL CENTRO earthquake of May 18, 1940 [9]. Its duration is 31.18 sec, and the ground accelerations are recorded every 0.02 sec. These accelerations have for unit the acceleration of gravity and are worth 9.81 m/s² for this case. The seismic accelerations were used in their raw state without correction or filtering. The displacements obtained in meters were converted to centimeters. Oscillators have periods ranging from 0.2 sec to 20 secs with a pitch of 0.2 sec. For the damping ratio, they vary from 0% to 10% with a pitch of 1%.

The initial conditions are: When the oscillators are in equilibrium, that is to say, with no displacement, velocity, and acceleration are zero. At time t = 0 , u ( 0 ) = u ˙ ( 0 ) = u ¨ ( 0 ) = 0 . This also implies that A ( 0 ) = B ( 0 ) = 0 .

For each plotted curve, we find the maximum amplitude of the response and the period T of that oscillator. This pair constitutes a coordinate of the linear elastic response spectrum for the fixed relative damping ratio ξ. The next step involves finding the coordinate pairs of the series of oscillators with natural periods varying from 0 to 20 seconds, for the same damping ratio. The curve connecting all the points of the pairs constitutes the linear elastic response spectrum. The other spectral curves are thus obtained by repeating the same process, for the relative damping ratio varying from 0% to 10%.

Figure 4 and Figure 5 show the different linear elastic response spectra for damping rates ranging from 0% to 10% in 1% increments. Figure 4 is obtained under the assumption that the ground acceleration varies linearly over a time increment, and Figure 5 is obtained under the assumption that the ground acceleration is constant over the same time increment.

Figure 4. Elastic response spectrum assuming linear variation of ground acceleration over a time increment.

Figure 5. Elastic response spectrum assuming constant average ground acceleration over a time increment.

Analysis of the obtained response spectra allowed us to determine the oscillation periods corresponding to the maximum responses as a function of the damping rate. This analysis is summarized in Table 1 and Table 2. The first column presents the damping rate, the second column the periods corresponding to the maximum response values, and the third column the values of these maximum responses. The maximum response values (63.86 cm and 63.84 cm) were obtained for a damping rate of 0%.

Table 1. Periods of simple oscillators giving maximum responses according to the assumption of linear variation of ground acceleration during a time increment.

Table 2. Periods of simple oscillators giving maximum responses according to the assumption of a constant average of ground acceleration over a time increment.

The aim of this analysis is not to conduct a detailed study, but to compare the maximum responses obtained by the different approaches for each damping ratio and spectrum. Analysis of these results allowed us to determine the differences between the spectral responses measured in the same configurations, for the same period, and the same damping ratio. This comparison is presented in Table 3.

Table 3. Analysis of the responses of simple oscillators based on the two hypotheses put forward regarding ground acceleration.