**2. Description of the THA method**

processing, control and monitoring. It also simulates the vibration testing and vibration signal processing. It is an important technique that makes it easy to detect internal damage to the structure. Therefore, it is shown to be the prevailing instrument in the study of the dynamic behavior of structures which had become a major concern of mechanical, civil and aerospace engineers. To better understand the dynamic behavior, it is essential to know the modal parameters of the structure, i.e. its natural frequencies, mode shapes and damping ratios. The precise identification of these parameters can be made through the use of robust and reliable methods

There are different causes of vibration, such as continuous force, degradation, resonance, etc. The response of it can be understood through various control actions such as manual, automatic, sine wave generation and square wave generation on the structure. The preventive measures on the structure may be taken through analysis and monitoring of vibration signal by two processes. When the variation of force with time is known, the variation of response is formulated in time domain. This is referred to as time-domain analysis and this former signal analysis can be used to evaluate the response of any linear SDOF system to any arbitrary input. Sometimes, the force function is random and it is not possible to determine its frequency. Moreover, it may have a variable frequency over its duration and hence it is then convenient to perform the analysis in frequency domain. The frequency domain approach is also conceptually similar to the Fourier analysis procedure. However, to apply the periodic load technique to arbitrary loading, it is necessary to extend the Fourier series concept to the representation of non-periodic functions. Various researches are made in recent years to apprehend the dynamic behavior of the structure using virtual instrument engineering workbench. Sura et al., [3] analyzed the cantilever beam using the virtual instrument in which free vibrations were induced and measured in the beam. The results in the form of modal frequency were obtained for the cantilever beam which was properly fixed and he concluded that the theoretically calculated natural frequency and the experimentally calculated natural frequency are almost the same. Yao et al. [4] built a virtual earthquake simulation system instrumentation and stated that the design concept of LabVIEW is more user-friendly and efficient than others. Hu [5], describes the development of modal recognition computing tools and long-term dynamic monitoring in the LabVIEW framework. These consist mainly of two independent functional toolkits known as Structural Modal Identification (SMI) and Continuous Monitoring (CSMI) respectively. It involves checking the latest output measurements, identifying the maximum vibration amplitudes and performing statistical time series on acceleration. It generates waveform plots to represent the distribution of the frequency component and modal parameter based on automated Enhanced Frequency Domain Decomposition (EFDD) technique. An attempt is made to expand handson activity-based educational module through the integration of PASCO models, LabVIEW, NI hardware, sensors, and MATLAB software. Despite some existing limitations, the results successfully showed that this structure worked precisely and stably, producing good output data. It was proved as a potential tool for structural dynamics as well as Structural Health Monitoring (SHM) education and also study in which, each case of damaged structure had a distinctive property [6]. Ugo Andreaus [7] studied the experimental dynamic response of a base-isolated SDOF oscillator and formulated numerical model excited by a harmonic base acceleration using LabVIEW. The behavior of the system was well understood as the numerical

simulation in LabVIEW platform efficiently agreed with the experimental

In this context of the study, an attempt is being made to propose program for time integration method in LabVIEW to predict the changes in displacement, velocity and acceleration for SDOF model for earthquake excitations. The versions

investigation.

**130**

that belong to the field of research known as modal analysis [2].

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

There are many numerical integration methods available to evaluate the approximate solution of equation of motions. There are two basic characteristics of these methods firstly, the differential equations of these methods are satisfied only at discrete time intervals Δt and secondly, a variation in displacement, velocity and acceleration is assumed within each time interval Δt [9]. Causevic et al., [10] discussed about non-linear dynamic time-history analysis; non-linear static method (Euro code 8); non-linear static procedure NSP (FEMA 356) and improved capacity spectrum method CSM (FEMA 440). An eight-storey reinforced concrete frame building is analyzed as the research subject. It is evident that neither of static procedures takes into consideration the damage which can be significant for long duration earthquakes. The author thus concluded that the non-linear THA was the most accurate method. Lestuzzi et al., [11] discussed about the selection of real ground motion records by considering the response of single-degree-of-freedom (SDOF) system with bilinear hysteretic model. The findings from this study are very limited, i.e., they are applicable only for building structures that can be modeled as a SDOF system. The response parameters considered are maximum displacement and ductility of the SDOF system. The study csoncludes the following points: 1.While selecting the real records of THA, the spectral acceleration records that matches with the design spectrum has to be chosen. 2. The period has to be kept as T0 or in a range between T0 and the period corresponding to the secant stiffness. It is observed that the mathematical computation of these methods is difficult and is time consuming and hence a requirement for alternate and efficient platform is needed.

Thus, the concept of nonlinear behavior of structures and the importance of Time history analysis (THA) is more important even though it's a century old concept. Although the linear elastic analysis and the design methods are well established, nonlinear inelastic analysis and their application to design are still evolving. The answer for the question, "Why do we need a nonlinear analysis?" lies in the fact that under extreme probable loading like earthquake; it is no longer advisable to keep the structure elastic due to the reason of yielding in structural components. Thus, a nonlinear analysis requires a clear understanding of the stress–strain curves of all the materials used in the structure, its inelastic behavior, failure criteria of the components, the capacity of its in failure modes and also the nonlinear analysis techniques. In case a single degree of freedom (SDOF) system or a multi degree of freedom (MDOF) system is subjected to a random acceleration time history, it is very difficult to solve the differential equation using the basic principle of calculus. The direct integration methods or step-by-step integration methods are used for the solutions of such problems. A very small time step Δt, is chosen and the solution is obtained from one step to the next step leading to the linear interpolation of the forces. The expression at time step (t + h) may be entirely in term of quantities at time step t or both at time step t and (t + h) which gives rise to two types of algorithm: explicit algorithm and implicit algorithm. In the former, the expressions at time step (t + h) are in terms of time step t only, whereas, in the latter, the expressions at time step (t + h) are in terms of t and (t + h). The solutions using the explicit algorithm are as easy as compared to those using the implicit algorithm.

Hence an attempt is being made to make LabVIEW programs for the widely used explicit and implicit algorithm. A brief overview of these approaches is given, followed by programming in LabVIEW platform and their validation through examples. Li [12] stated that Finite difference method optimizes the approximation for the differential operator in the central node of the considered space and provides numerical solutions to differential equations. It is noticed that the results of the central difference method approximation show a significant improvement in the accuracy along the smooth region. He also concluded that it is possible to test the function f (x) at values on the left and right of x, to obtain an optimal two-point approximation which includes abscissas that are symmetrically chosen on both sides of x. The advantage of this approach is that, its convergence speed is higher than some other finite differentiating methods, such as forward and backward differentiation. Similarly another method developed by E L Wilson for unconditionally stable linear acceleration method is Wilson θ method. This method is based on the assumption that acceleration varies linearly over an extended time step δt = θδt [13]. Wilson-θ method is highly stable numerically as it converges rapidly to a meaningful solution. In our study, earthquake-induced ground motions of El Centro (1940) and Loma Prieta (1989) earthquake data are fed as input to the SDOF system. Seismic responses considered were in the form of acceleration, velocity, displacement and force and the application example considered was SDOF system. The accuracy which means the chosen numerical methods should converges the exact solution in terms of amplitude accuracy or amplitude decay or period accuracy or period decay was carried out in the workbench. The ground motions records were obtained from the PEER Strong Motion Database (http://peer.berke ley.edu/smcat /) [14].

Incremental Stiffness for the ith time step,

Where, *a* and *b* are constants and given as

*<sup>a</sup>* <sup>¼</sup> *<sup>m</sup>*

<sup>2</sup>*:<sup>β</sup>* <sup>þ</sup> *<sup>γ</sup>:<sup>c</sup>*

Incremental force for ith time step,

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

Displacement for i + 1th time step,

Velocity for ith time step,

Acceleration for ith time step,

**3.2 Algorithm**

problem at time t = 0.

*ki* <sup>¼</sup> *<sup>m</sup>*

*<sup>β</sup>* , *<sup>b</sup>* <sup>¼</sup> *<sup>m</sup>*

*ui*þ<sup>1</sup> <sup>¼</sup> *Fi ki*

*<sup>u</sup>*\_ *<sup>i</sup>* <sup>¼</sup> *ui*þ<sup>1</sup> � *ui*�<sup>1</sup>

*<sup>u</sup>*€ <sup>¼</sup> *ui*þ<sup>1</sup> � <sup>2</sup>*:ui* � *ui*�<sup>1</sup>

Step 1: Initial displacement and velocity are known as initial conditions of the

Step 2: Damping c and stiffness k are computed from the system properties.

Step 8: Compute velocity and acceleration at time step i from Eq. (2) and (3).

A visual block diagram which describes the data flow within the VI is presented in the form algorithm in LabVIEW. LabVIEW accepted the input, and the algorithm was sampled and programmed through appropriate interfaces in accordance with the specification of VI, and the output data was collected. In our software, data such as damping, mass and time period were provided as an input and displacement, velocity and acceleration plot was obtained for the time history data (i.e. it can function as an analog to digital converter). Owing to the sheer quantity and simplicity of the different built-in functions, the data was thus manipulated in a wide

Step 3: Acceleration at time t = 0 is computed from Eq. (2). Step 4: Compute equivalent stiffness *ki*from Eq. (4).

Step 6: Compute constants a and b from Eq. (6). Step 7: Solve for new displacement *ui*þ<sup>1</sup> from Eq. (7).

Step 9: Repeat Steps 6 to 8 for the next time step.

range of forms as shown in **Figure 1**.

**133**

Step 5: For time step i, compute equivalent force *Fi* from Eq. (5).

**3.3 Programs developed in LabVIEW for central difference method**

ð Þ *<sup>Δ</sup><sup>t</sup>* <sup>2</sup> <sup>þ</sup>

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

*c*

<sup>2</sup>*:<sup>β</sup>* <sup>þ</sup> *<sup>Δ</sup>t: <sup>γ</sup>*

*Fi* ¼ �*m:u*€*<sup>g</sup>* � *a:ui*�<sup>1</sup> � *b:ui* (5)

<sup>2</sup>*:<sup>β</sup>* � <sup>1</sup>

<sup>2</sup>*:Δ<sup>t</sup>* (4)

*<sup>c</sup>* (6)

<sup>2</sup>*Δ<sup>t</sup>* (8)

ð Þ *<sup>Δ</sup><sup>t</sup>* <sup>2</sup> (9)

(7)

### **3. Methodology**

In the explicit method the response at time tn+1 is known in terms of known variables at time tn. Thus the response values displacement, velocity and acceleration can be determined directly. Whereas, in implicit method, the response at time tn+1 is known in terms of the known variables at time tn and unknown variables at time tn+1. These implicit algorithms involve either an iterative scheme or solution of linear simultaneous equations because the unknown quantities appear on both sides of the equations.

#### **3.1 Central difference method**

This method is based on the finite difference approximation of the time derivative of displacement, that is, velocity and acceleration [9].

An equation of motion for an SDOF system is given as:

$$
\dot{x}\,m.\ddot{u} + c.\dot{u} + k.u = F\_t \tag{1}
$$

*m* = Mass, *c* = Damping, *k* = Stiffness, *u*€ = Acceleration, *u*\_ = Velocity, *u* = Displacement, *Ft* = Force. Initial acceleration is given as,

$$
\ddot{u}\_o = \frac{-m\,\Delta\ddot{u}\_g - c\,\dot{u}\_o - k.u\_o}{m} \tag{2}
$$

Initial displacement at i-1th time step

$$
\mu\_{i-1} = \mu\_0 + \Delta t(\dot{u}\_o) + \frac{\left(\Delta t\right)^2}{2}\ddot{u}\_o \tag{3}
$$

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

Incremental Stiffness for the ith time step,

$$k\_i = \frac{m}{\left(\Delta t\right)^2} + \frac{c}{2\Delta t} \tag{4}$$

Incremental force for ith time step,

$$F\_i = -m.\ddot{u}\_{\text{g}} - a.u\_{i-1} - b.u\_i \tag{5}$$

Where, *a* and *b* are constants and given as

$$a = \frac{m}{2\beta} + \frac{\gamma \cdot c}{\beta}, b = \frac{m}{2\beta} + \Delta t. \left(\frac{\gamma}{2\beta} - \mathbf{1}\right)c\tag{6}$$

Displacement for i + 1th time step,

$$u\_{i+1} = \frac{F\_i}{k\_i} \tag{7}$$

Velocity for ith time step,

$$
\dot{u}\_i = \frac{u\_{i+1} - u\_{i-1}}{2\Delta t} \tag{8}
$$

Acceleration for ith time step,

$$
\ddot{u} = \frac{u\_{i+1} - 2.u\_i - u\_{i-1}}{\left(\Delta t\right)^2} \tag{9}
$$

#### **3.2 Algorithm**

followed by programming in LabVIEW platform and their validation through examples. Li [12] stated that Finite difference method optimizes the approximation for the differential operator in the central node of the considered space and provides numerical solutions to differential equations. It is noticed that the results of the central difference method approximation show a significant improvement in the accuracy along the smooth region. He also concluded that it is possible to test the function f (x) at values on the left and right of x, to obtain an optimal two-point approximation which includes abscissas that are symmetrically chosen on both sides of x. The advantage of this approach is that, its convergence speed is higher than some other finite differentiating methods, such as forward and backward differentiation. Similarly another method developed by E L Wilson for unconditionally stable

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

linear acceleration method is Wilson θ method. This method is based on the

PEER Strong Motion Database (http://peer.berke ley.edu/smcat /) [14].

**3. Methodology**

of the equations.

**132**

**3.1 Central difference method**

*u* = Displacement, *Ft* = Force. Initial acceleration is given as,

Initial displacement at i-1th time step

assumption that acceleration varies linearly over an extended time step δt = θδt [13]. Wilson-θ method is highly stable numerically as it converges rapidly to a meaningful solution. In our study, earthquake-induced ground motions of El Centro (1940) and Loma Prieta (1989) earthquake data are fed as input to the SDOF system. Seismic responses considered were in the form of acceleration, velocity, displacement and force and the application example considered was SDOF system. The accuracy which means the chosen numerical methods should converges the exact solution in terms of amplitude accuracy or amplitude decay or period accuracy or period decay was carried out in the workbench. The ground motions records were obtained from the

In the explicit method the response at time tn+1 is known in terms of known variables at time tn. Thus the response values displacement, velocity and acceleration can be determined directly. Whereas, in implicit method, the response at time tn+1 is known in terms of the known variables at time tn and unknown variables at time tn+1. These implicit algorithms involve either an iterative scheme or solution of linear simultaneous equations because the unknown quantities appear on both sides

This method is based on the finite difference approximation of the time deriva-

*m* = Mass, *c* = Damping, *k* = Stiffness, *u*€ = Acceleration, *u*\_ = Velocity,

*ui*�<sup>1</sup> ¼ *u*<sup>0</sup> þ *Δt u*\_ ð Þþ *<sup>o</sup>*

*<sup>u</sup>*€*<sup>o</sup>* <sup>¼</sup> �*m:Δu*€*<sup>g</sup>* � *<sup>c</sup>:u*\_ *<sup>o</sup>* � *<sup>k</sup>:uo m*

*m:u*€ þ *c:u*\_ þ *k:u* ¼ *Ft* (1)

ð Þ *<sup>Δ</sup><sup>t</sup>* <sup>2</sup>

<sup>2</sup> *<sup>u</sup>*€*<sup>o</sup>* (3)

(2)

tive of displacement, that is, velocity and acceleration [9]. An equation of motion for an SDOF system is given as:

Step 1: Initial displacement and velocity are known as initial conditions of the problem at time t = 0.

Step 2: Damping c and stiffness k are computed from the system properties.

Step 3: Acceleration at time t = 0 is computed from Eq. (2).

Step 4: Compute equivalent stiffness *ki*from Eq. (4).

Step 5: For time step i, compute equivalent force *Fi* from Eq. (5).

Step 6: Compute constants a and b from Eq. (6).

Step 7: Solve for new displacement *ui*þ<sup>1</sup> from Eq. (7).

Step 8: Compute velocity and acceleration at time step i from Eq. (2) and (3).

Step 9: Repeat Steps 6 to 8 for the next time step.

#### **3.3 Programs developed in LabVIEW for central difference method**

A visual block diagram which describes the data flow within the VI is presented in the form algorithm in LabVIEW. LabVIEW accepted the input, and the algorithm was sampled and programmed through appropriate interfaces in accordance with the specification of VI, and the output data was collected. In our software, data such as damping, mass and time period were provided as an input and displacement, velocity and acceleration plot was obtained for the time history data (i.e. it can function as an analog to digital converter). Owing to the sheer quantity and simplicity of the different built-in functions, the data was thus manipulated in a wide range of forms as shown in **Figure 1**.
