**5. Results and discussion**

### **5.1 Linear SDOF system response of time integration methods and its results and discussion**

El Centro and Loma Prieta earthquake ground motions were considered for the analysis. The problem was solved using the time step 0.02 to understand the displacement-time history, velocity-time history and acceleration- time history under El Centro and Loma Prieta earthquake using both the methods on LabVIEW and are shown below.

The complex solution determined by the central difference method in terms of velocity is contrasted with the method of wilson-θ. In the case of El Centro earthquake highest peak velocity of 0.653 (m/s.) and lowest peak velocity of 0.606 (m/s.) is responded by wilson-θ method and in case of Loma Prieta the highest peak velocity of 0.363 (m/s.) and lowest peak velocity of 0.341 (m/s.) is displayed by

*Displacement vs time response under Loma Prieta earthquake (a) central difference method (b) wilson-θ method.*

*Displacement vs time response under El Centro earthquake (a) central difference method (b) wilson-θ method.*

*Advanced Modeling of Single Degree of Freedom System for Earthquake Ground Motion Using…*

*DOI: http://dx.doi.org/10.5772/intechopen.96341*

The acceleration response in central difference and wilson-θ method were

Dynamic acceleration response calculated using central difference was compared with wilson-θ method. In case of El Centro earthquake motion, maximum

was displayed by central difference method. Whereas, in case of Loma Prieta

) and minimum peak acceleration of 7.75 (m/s<sup>2</sup>

)

central difference method.

**Figure 3**

**Figure 4.**

**137**

peak acceleration of 10.32 (m/s2

*5.1.3 Response in terms of acceleration of linear SDOF system*

obtained in LabVIEW and are displayed below (**Figures 7** and **8**).

#### *5.1.1 Response in terms of displacement of linear SDOF system*

The displacement response in central difference and wilson-θ method were obtained in LabVIEW and was displayed below in **Figures 3** and **4**.

In case of El Centro earthquake maximum peak displacement of 0.0531 (m) and minimum peak displacement of 0.0456 (m) was given by Wilson-θ method and in case of Loma Prieta the maximum peak displacement of 0.033 (m) and minimum peak displacement of 0.029 (m) was again given by Wilson θ-method.

#### *5.1.2 Response in terms of velocity of linear SDOF system*

The velocity response in central difference and wilson-θ method are obtained in LabVIEW and is displayed below (**Figures 5** and **6**).

*Advanced Modeling of Single Degree of Freedom System for Earthquake Ground Motion Using… DOI: http://dx.doi.org/10.5772/intechopen.96341*

**Figure 3** *Displacement vs time response under El Centro earthquake (a) central difference method (b) wilson-θ method.*

**Figure 4.** *Displacement vs time response under Loma Prieta earthquake (a) central difference method (b) wilson-θ method.*

The complex solution determined by the central difference method in terms of velocity is contrasted with the method of wilson-θ. In the case of El Centro earthquake highest peak velocity of 0.653 (m/s.) and lowest peak velocity of 0.606 (m/s.) is responded by wilson-θ method and in case of Loma Prieta the highest peak velocity of 0.363 (m/s.) and lowest peak velocity of 0.341 (m/s.) is displayed by central difference method.

## *5.1.3 Response in terms of acceleration of linear SDOF system*

The acceleration response in central difference and wilson-θ method were obtained in LabVIEW and are displayed below (**Figures 7** and **8**).

Dynamic acceleration response calculated using central difference was compared with wilson-θ method. In case of El Centro earthquake motion, maximum peak acceleration of 10.32 (m/s2 ) and minimum peak acceleration of 7.75 (m/s<sup>2</sup> ) was displayed by central difference method. Whereas, in case of Loma Prieta

incorporating block diagram from simulate signal express VI, which is located under the signal analysis palette. The functions, such as mathematical operations, express VIs, built analysis tools and signal simulation, were assessed from the palettes by right clicking in the block diagram or front panel, which brought up the palette menu. The frame work was thus created using the algorithm and the waveform was generated

The peak ground motion recorded, magnitude and it's predominated period at real-time data storage station during the 1940 El Centro earthquake or 1940 Imperial Valley earthquake (Mw = 6.9) was considered for THA as the first analytical case study. The SDOF system that was considered has a mass of 1 kg and a damping value of 0.05. The time step that was considered for it was 0.02 s. The ground motion details (horizontal component) are given in **Table 1**. In order to further prove the efficiency of the program, the 1989 Loma Prieta earthquake was considered for analysis and the percentage variation of the LabVIEW is evaluated. The basic parameters of the SDOF system considered has a mass of 1 kg, time step

**5.1 Linear SDOF system response of time integration methods and its results**

**Earthquake Maximum acceleration (g) Magnitude Predominant period** El Centro 0.296 6.9 0.588 Loma Prieta 0.276 7 0.588

analysis. The problem was solved using the time step 0.02 to understand the displacement-time history, velocity-time history and acceleration- time history under El Centro and Loma Prieta earthquake using both the methods on LabVIEW

El Centro and Loma Prieta earthquake ground motions were considered for the

The displacement response in central difference and wilson-θ method were

In case of El Centro earthquake maximum peak displacement of 0.0531 (m) and minimum peak displacement of 0.0456 (m) was given by Wilson-θ method and in case of Loma Prieta the maximum peak displacement of 0.033 (m) and minimum

The velocity response in central difference and wilson-θ method are obtained in

for displacement, acceleration and velocity as shown in the **Figure 2**.

*LabVIEW - A Flexible Environment for Modeling and Daily Laboratory Use*

0.02 s, damping ratio 0.05 and time period of 0.513 s [16].

*5.1.1 Response in terms of displacement of linear SDOF system*

*5.1.2 Response in terms of velocity of linear SDOF system*

LabVIEW and is displayed below (**Figures 5** and **6**).

obtained in LabVIEW and was displayed below in **Figures 3** and **4**.

peak displacement of 0.029 (m) was again given by Wilson θ-method.

**4. Analytical validations**

**5. Results and discussion**

*Earthquake ground motion details (horizontal component).*

**and discussion**

**Table 1.**

**136**

and are shown below.

**Figure 5.** *Velocity vs time response under El Centro earthquake (a) central difference method (b) wilson-θ method.*

**Figure 6.** *Velocity vs time response under Loma Prieta earthquake (a) central difference method (b) wilson-θ method.*

ground motion the maximum peak acceleration of 4.25 (m/s<sup>2</sup> ) was shown by central difference method and minimum peak acceleration of 4.24 (m/s2 ) was given by wilson-θ method.

In the **Table 2** shown above, response results obtained from central difference method and wilson-θ method for El Centro earthquake was worked out. The difference in response was calculated and it was clearly seen that the variation did not exceed more than 0.08.

**6. Conclusions**

**Figure 8.**

**Table 2.**

**139**

**Figure7.**

This paper summarizes, the modeling of linear SDOF system in LabVIEW software using time integration method. The comparative study with the results of an example chosen for the proposed program in LabVIEW clearly stated that the

*Acceleration vs time response under Loma Prieta ground motion (a) central difference method (b) wilson-θ method.*

**El Centro Peak Displacement(m) Peak Velocity (m/s) Peak acceleration (m/s2**

CDM 0.0462 0.635 10.32 WTM 0.0531 0.653 10.20

*Percentage variation of linear SDOF system under El Centro earthquake.*

**)**

*Acceleration vs time response under El Centro ground motion (a) central difference method (b) wilson-θ method.*

*Advanced Modeling of Single Degree of Freedom System for Earthquake Ground Motion Using…*

*DOI: http://dx.doi.org/10.5772/intechopen.96341*

In the **Table 3** shown above, response results obtained from central difference method and wilson-θ method for Loma Prieta earthquake was worked out. The difference in response was calculated and it was clearly seen that the variation did not exceed more than 0.04.

*Advanced Modeling of Single Degree of Freedom System for Earthquake Ground Motion Using… DOI: http://dx.doi.org/10.5772/intechopen.96341*

**Figure7.** *Acceleration vs time response under El Centro ground motion (a) central difference method (b) wilson-θ method.*

**Figure 8.** *Acceleration vs time response under Loma Prieta ground motion (a) central difference method (b) wilson-θ method.*


#### **Table 2.**

ground motion the maximum peak acceleration of 4.25 (m/s<sup>2</sup>

wilson-θ method.

**Figure 6.**

**138**

**Figure 5.**

exceed more than 0.08.

not exceed more than 0.04.

difference method and minimum peak acceleration of 4.24 (m/s2

In the **Table 2** shown above, response results obtained from central difference method and wilson-θ method for El Centro earthquake was worked out. The difference in response was calculated and it was clearly seen that the variation did not

*Velocity vs time response under Loma Prieta earthquake (a) central difference method (b) wilson-θ method.*

*Velocity vs time response under El Centro earthquake (a) central difference method (b) wilson-θ method.*

*LabVIEW - A Flexible Environment for Modeling and Daily Laboratory Use*

In the **Table 3** shown above, response results obtained from central difference method and wilson-θ method for Loma Prieta earthquake was worked out. The difference in response was calculated and it was clearly seen that the variation did

) was shown by central

) was given by

*Percentage variation of linear SDOF system under El Centro earthquake.*
