CENG25200 Report: Computational Modelling 2 - Numerical Analysis

This report presents solutions for a Computational Modelling 2 assignment (CENG25200) focusing on numerical analysis techniques. It covers the Finite Difference Method (FDM) for discretizing a domain and solving differential equations, specifically applied to a free end-point distributed load beam structure. The report also includes Finite Element Analysis (FEA) to solve engineering problems, demonstrated with a truss structure. Furthermore, the report incorporates MATLAB simulations, including Time History Analysis using a Ricker wavelet, and analyzes displacement and acceleration spectrums for a Single Degree of Freedom (SDOF) system excited by a harmonic force. The results are compared with numerical solutions obtained using the MATLAB built-in functions. The report provides detailed analysis and MATLAB code to support the findings, offering a comprehensive understanding of the computational modeling concepts.

Finite difference method:
Finite difference ideology discretizes the domain into a regular grid elaborated by the certain
nodes numbers which are divided by the different intervals. This FDM technique involves of
derivatives replacement in the differential equations by using their equivalent difference
equation referred by Taylor’s series. The outcomes of the Free end point distributed load xy
beam structure and the construction 3 difference equations is applied at every node for system
yielding of the linear algebraic equation that are solved for the deflection at every node in
mathematically manner.
𝑤′′′′(𝑥)+𝑘𝐸𝐼𝑤(𝑥)=𝑞(𝑥)
𝐸𝐼𝑤′′′′
(𝑥)=𝑞(𝑥)−𝑘𝑤(𝑥)
For the values of:
𝐸𝐼=1∙10 5𝑘𝑁/ ,
𝑘=3∙10 4𝑘𝑁/𝑚2 ,
𝐿=1 ,
𝑃1=1200 ,
𝑃2=2000𝑘𝑁
and 𝑃3=1000𝑘𝑁.
Result:

For the values of:
𝐸𝐼=1∙104𝑘𝑁/𝑚
and 1∙108𝑘𝑁𝑚

Finite element analysis:
Finite element investigation is a numerical methodology for solving the engineering problems
and the calculation in physics. A truss is divided in this solution into smaller components
named as members. Then these are assembled and connected at the finite number of the joints
termed as nodes. Each type of member features is obtained and constructed together and
observed as whole to get the solution. Static observation of space truss is completed by the
stiffness methodology where the calculation is simpler for mostly structural analysis
probems.
For the values of:
𝐿= 5𝑚
𝑃=1000𝑘𝑁
𝐸𝐴=2∙106𝑘𝑁
MATLAB Result:

Time History Analysis:
The Ricker wavelet can be elaborated in the time domain mathematically as:
r( ) = (1-↊ 1
2 wp2 ↊2) exp (-1/4 wp2 ↊2)
where,
= time↊
Wp = energetic frequency
The Fourier transform for Ricker wavelet can be elaborated as:
R(w) = (2w2/ √ πw p2) exp (-w2/wp2)
𝑑 2𝑢/𝑑t 2=𝑢𝑜′′∙[1−12𝜔𝑟2 (𝑡−𝑡𝑜 )2 ]∙exp[−14 𝜔𝑟2 (𝑡−𝑡𝑜)2]

MATLAB Result:
Displacement spectrum
𝑇𝑖=0.02:0.02:2𝑠.
The relative displacement response spectrum and the absolute acceleration response spectrum
can easily elaborate. The relative displacement is essential for the design problems at any
circumstances in which experimental calculations are to be taken as the calculation check.
The MATLAB coding of this part elaborates the SDOF system or single degree of freedom is
excited by the harmonic force is computed. It is compared with the numerical solution which is
being provided by the MATLAB built-in function ode 45, and RK method.
The model parameters are shown below:
Eigen-frequency, W0 = 1
Damping ratio = 0.02

Mass, M = 1
Stiffness, K = w02.M
Force amplitude, F0 = 10
MATLAB Result: