University of Waterloo
CivE 331
The Vibrating Beam Problem
It is known from solid mechanics (CivE 204, 205, & 306) that the free vibration of a uniform
beam is governed by a fourthorder PDE:
u xxxx
EI
utt 0
Where u(x,t) is the beam deflection, is the ma
University of Waterloo
CivE 331
inside
21 oC
Wall (15 cm 488 cm)
Heat Conduction through a Wall (revisited)
In this application, we will revisit the cost associated with heating a home but will solve the
problem using numerical methods. Consider the outs
University of Waterloo
CivE 331
Compute the Fourier Series expansion of the following periodic function. Plot the expansion
for n=3, 5, and 10.
3
2
1
0
20
18
16
14
12
10
8
6
4
2
0
2
4
6
1
2
3
By inspection, the period of the function is T=16
CivE 331 Advanced Mathematics for Civil Engineers
3. Fourier Series
Consider a 2nd order linear PDE: A uxx + 2B uxy + C uyy + D ux + E uy + F u = f(x, y)
Determining u(x,y) for such PDE is often difficult, especially when f(x, y) is a periodic forcing
fun
University of Waterloo
CivE 331
Heat Conduction through a Wall
In this application, we will examine the cost associated with heating a home. Consider the
outside wall of a house, as shown below:
As indicated in the figure, the inside of the house is heate
University of Waterloo
CivE 331
Seepage through a Deteriorated Pier
The figures below show the crosssection and planview of a hydraulic control structure which
consists of a pier and sluice gate system. Due to weathering processes, the base of the
concr
University of Waterloo
CivE 331
Week 8 The Wave Equation
Last week we finished up our discussion of the diffusion/heat equation. This week, the wave
equation is introduced and a common application of the wave equation (i.e. vibration of a
guitar string) i
University of Waterloo
CivE 331
Response of Bridge Supports to an Earthquake
As depicted in the figure below, seismic waves from an earthquake cause the ground to shake
and this can trigger vertical (i.e. up and down) motion of bridge supports. Earthquake
CivE 331 Advanced Mathematics for Civil Engineers
Department of Civil and Environmental Engineering
University of Waterloo
Winter 2014
CivE 331 Advanced Mathematics for Civil Engineers
Wave Equation
Department of Civil and Environmental Engineering
Univer
University of Waterloo
CivE 331
Deflection of a Telephone Cable
The transverse motion of a telephone cable is governed by the wave equation, modified to
account for a gravitational force due to the large weight of the cable:
c 2u xx utt g
Where u(x,t) is
CivE 331 Advanced Mathematics for Civil Engineers
Fourier Series
p. 7
3.8 Range Expansions
Often the forcing function f(x) is defined only over a finite interval or range of x (e.g. 0 < x < L).
In such cases, f(x) is not periodic and F.S. expansion is not
CivE 331 Advanced Mathematics for Civil Engineers
1. Introduction (Text sections 1.1, 1.2, 1.3, and 18.2)
Definition:

A differential equation is an equation containing one or more derivatives of the function
under consideration.

A differential equatio
CivE 331 Advanced Mathematics for Civil Engineers
2. Review of Prerequisites
2.1 The Separation of Variables Method for Solving Ordinary Differential Equations
Recipe
Step 0 Write the governing equation
Step 1 Separate variables
Step 2 Integrate both sid
University of Waterloo
CivE 331
Periodic Loading of a Railway Foundation
Consider a stationary train in which every other car is loaded with cargo:
empty
empty
.
.
empty
Track
Track Bed
The train and its cargo exert a force (per unit length) on the track
University of Waterloo
CivE 331
Chapter 1 Introduction
1.1. Mathematical Models (Chapter 1 and Section 18.2)
Many interesting phenomena are governed by differential equations. Some
examples:
1) Mass/spring system
2) Hanging cable
1
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