**3.4 Wilson-θ method, linear SDOF system**

The incremental equation of equilibrium known as Wilson-θ method is developed by Prof.E.L.Wilson, University of California, and Berkeley [9]. The calculations are carried out over an extended time step θΔt, where θ is an amplifier for the time step. It assumes that the variation of acceleration over the extended time step remains unchanged, that is, it is still the same as that of the original time step Δt, a linear variation.

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

$$
\dot{x}m.\ddot{u} + c.\dot{u} + k.u = F\_t \tag{10}
$$

Solve for *δu*€ for ith time step,

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

be calculated,

**3.5 Algorithm**

**Figure 2.**

**135**

Step 1: Compute kt*:*

Step3: Update c and k. Step 4: Repeat steps 1 to 3.

**3.6 Programs developed in LabVIEW**

*Block diagram of wilson-θ method in LabVIEW.*

Incremental velocity,

Incremental displacement,

Incremental Acceleration is given as

*<sup>δ</sup>u*€*<sup>i</sup>* <sup>¼</sup> <sup>6</sup>*:δui*

ð Þ *<sup>θ</sup>:Δ<sup>t</sup>* <sup>2</sup> � <sup>6</sup>*:u*\_

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

*<sup>u</sup>*€*<sup>i</sup>* <sup>¼</sup> *<sup>δ</sup>u*€*<sup>i</sup> kt*

Knowing incremental acceleration, incremental velocity and displacement can

*Δu*\_ *<sup>i</sup>* ¼ *Δt:u*€*<sup>i</sup>* þ

At time *ti+1* displacement, velocity and acceleration can be calculated as

*ui*þ<sup>1</sup> ¼ *ui* þ *Δui u*\_ *<sup>i</sup>*þ<sup>1</sup> ¼ *u*\_ *<sup>i</sup>* þ *Δu*\_ *<sup>i</sup> u*€*<sup>i</sup>*þ<sup>1</sup> ¼ *u*€*<sup>i</sup>* þ *Δu*€*<sup>i</sup>*

It should be noted that in this method, k and c are assumed to remain constant during the extended time step and are updated at the end of the real-time increment **Δt**.**θ** ¼ **1** leads to the linear acceleration method. It is recommended that **θ** is taken >1.37 [15].

The block diagram given below shows all the features that are expressed in VIs.

*<sup>Δ</sup>ui* <sup>¼</sup> *<sup>Δ</sup>t:u*€*<sup>i</sup>* <sup>þ</sup> ð Þ *<sup>Δ</sup><sup>t</sup>* <sup>2</sup>

Step 2: Calculate uiþ1,u\_ <sup>i</sup>þ1, u€<sup>i</sup>þ<sup>1</sup> using the Eqs. (15), (17) and (18).

The input signal was simulated at the first step. This was accomplished by

*θ:Δt*

*Δt:Δu*€*<sup>i</sup>*

*Δu*€*<sup>i</sup>*

*u*\_ *i* <sup>2</sup> <sup>þ</sup> ð Þ *<sup>Δ</sup><sup>t</sup>* <sup>2</sup>

� 3*u*€*<sup>i</sup>* (15)

<sup>2</sup> (17)

<sup>6</sup> (18)

(16)

(19)

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

$$
\ddot{u}\_o = \frac{-m\,\Delta\ddot{u}\_g - c.\dot{u}\_o - k.u\_o}{m} \tag{11}
$$

Incremental force for ith time step,

$$
\delta F\_i = \theta \left( -m.\Delta \ddot{u}\_{\text{g}} \right) + a.\dot{u}\_i + b.\ddot{u}\_i \tag{12}
$$

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

$$a = \frac{6.m}{q\,\Delta t} + 3c, b = \frac{q\,\Delta t.c}{2} + 3.m$$

Tangent Stiffness for ith time step,

$$k\_t = k\_i + \frac{\mathbf{3}.c}{\theta \,\Delta t} + \frac{\mathbf{6}.m}{\theta (\Delta t)^2} \tag{13}$$

Solve for *δu* for ith time step,

$$
\delta u\_i = \frac{\delta F\_i}{k\_t} \tag{14}
$$

**Figure 1.** *Block diagram of central difference method in LabVIEW.*

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

Solve for *δu*€ for ith time step,

**3.4 Wilson-θ method, linear SDOF system**

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

Incremental force for ith time step,

Tangent Stiffness for ith time step,

Solve for *δu* for ith time step,

*Block diagram of central difference method in LabVIEW.*

**Figure 1.**

**134**

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

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

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

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

*δFi* ¼ *θ* �*m:Δu*€*<sup>g</sup>*

*kt* ¼ *ki* þ

*<sup>a</sup>* <sup>¼</sup> <sup>6</sup>*:<sup>m</sup> q:Δt*

*<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*

<sup>þ</sup> <sup>3</sup>*c*, *<sup>b</sup>* <sup>¼</sup> *<sup>q</sup>:Δt:<sup>c</sup>*

3*:c θ:Δt* þ

*<sup>δ</sup>ui* <sup>¼</sup> *<sup>δ</sup>Fi kt*

linear variation.

The incremental equation of equilibrium known as Wilson-θ method is developed by Prof.E.L.Wilson, University of California, and Berkeley [9]. The calculations are carried out over an extended time step θΔt, where θ is an amplifier for the time step. It assumes that the variation of acceleration over the extended time step remains unchanged, that is, it is still the same as that of the original time step Δt, a

*m:u*€ þ *c:u*\_ þ *k:u* ¼ *Ft* (10)

<sup>þ</sup> *<sup>a</sup>:u*\_ *<sup>i</sup>* <sup>þ</sup> *<sup>b</sup>:u*€*<sup>i</sup>* (12)

*θ Δ*ð Þ*<sup>t</sup>* <sup>2</sup> (13)

<sup>2</sup> <sup>þ</sup> <sup>3</sup>*:<sup>m</sup>*

6*:m*

(11)

(14)

$$\delta\ddot{u}\_{i} = \frac{6.\delta u\_{i}}{\left(\theta.\Delta t\right)^{2}} - \frac{6.\dot{u}}{\theta.\Delta t} - 3\ddot{u}\_{i} \tag{15}$$

Incremental Acceleration is given as

$$
\ddot{\boldsymbol{u}}\_i = \frac{\delta \ddot{\boldsymbol{u}}\_i}{k\_t} \tag{16}
$$

Knowing incremental acceleration, incremental velocity and displacement can be calculated,

Incremental velocity,

$$
\Delta \dot{u}\_i = \Delta t.\ddot{u}\_i + \frac{\Delta t.\Delta \ddot{u}\_i}{2} \tag{17}
$$

Incremental displacement,

$$
\Delta u\_i = \Delta t.\ddot{u}\_i + \frac{(\Delta t)^2 \dot{u}\_i}{2} + \frac{(\Delta t)^2 \Delta \ddot{u}\_i}{6} \tag{18}
$$

At time *ti+1* displacement, velocity and acceleration can be calculated as

$$\begin{aligned} \dot{u}\_{i+1} &= u\_i + \Delta u\_i \\ \dot{u}\_{i+1} &= \dot{u}\_i + \Delta \dot{u}\_i \\ \ddot{u}\_{i+1} &= \ddot{u}\_i + \Delta \ddot{u}\_i \end{aligned} \tag{19}$$

#### **3.5 Algorithm**

Step 1: Compute kt*:* Step 2: Calculate uiþ1,u\_ <sup>i</sup>þ1, u€<sup>i</sup>þ<sup>1</sup> using the Eqs. (15), (17) and (18). Step3: Update c and k. Step 4: Repeat steps 1 to 3.

It should be noted that in this method, k and c are assumed to remain constant during the extended time step and are updated at the end of the real-time increment **Δt**.**θ** ¼ **1** leads to the linear acceleration method. It is recommended that **θ** is taken >1.37 [15].

### **3.6 Programs developed in LabVIEW**

The block diagram given below shows all the features that are expressed in VIs. The input signal was simulated at the first step. This was accomplished by

**Figure 2.** *Block diagram of wilson-θ method in LabVIEW.*

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