∫ ∫

Faculty of Mechanical Engineering
Universiti Teknologi Malaysia
30 April 2014
SKMM 3023 Assignment 5 – Numerical Differentiation and Integration
1. Using the forward and backward Taylor series expansions, equation (1) and (2) respectively,
h
h2
h3
f ′( x) +
f ′′( x) +
f ′′′( x) +
1!
2!
3!
f ( x + h )= f ( x) +
( −h ) f ′′( x) + ( −h ) f ′′′( x) +
−h
f ( x − h )= f ( x) +
f ′( x) +
1!
2!
3!
2
(1)
3
(2)
show that the centered finite difference formula for second derivative is
f ′′( x) =
f ( x + h) − 2 f ( x) + f ( x − h)
(3)
h2
Why is the formula second order accurate?
2. Newton’s law of cooling is given as
dT
=
− k (T − Ta )
dt
(4)
where T = temperature of a body (°C), Ta = temperature of the surrounding medium (°C), and
k = cooling rate. A metal ball heated to T = 80°C is dropped into water that is held constant at
Ta = 20°C, and the following data was recorded
Time, min
T, °C
0
80
5
44.5
10
30.0
15
24.1
For the data above,
a. Use 2nd order accurate numerical differentiation to plot equation (4), and
b. Use linear regression to evaluate the cooling rate k.
3. Using Newton-Cotes integration formula,
=
I
∫ f ( x ) dx ≅ ∫
b
b
a
a
pn ( x ) dx
(5)
Show that when pn = p1 = a0 + a1x, the integration becomes
=
I
(b − a )
f ( a ) + f (b)
(6)
2
which is also known as the trapezoidal rule [hint: find the value of a0 and a1 using (a, f(a)) and
(b, f(b))].
4.
The following relationships can be used to analyze uniform beams subject to distributed loads,
dy
= θ ( x);
dx
dθ M ( x )
=
;
dx
EI
dM
= V ( x);
dx
dV
= −w ( x )
dx
(7)
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SKMM 3023 Assignment 5
April 2014
where x = distance along the beam (m), y = deflection (m), θ(x) = slope (m/m), M(x) = moment
(N·m), E = modulus of elasticity (Pa), I = moment of inertia (m4), V(x) = shear (N),
w(x) = distributed load (N/m). For the case of a linearly increasing load (see Figure 1), the slope
can be computed analytically as
θ=
( x)
w0
( −5 x4 + 6L2 x2 − L4 )
120 EIL
(8)
The variable values are: E = 200 GPa, I = 0.0003 m4, w0 = 2.5 kN/cm and L = 3 m.
a. Find the deflection at x = 1.2 m analytically.
b. Use trapezoidal rule to calculate the deflection at x = 1.2 m using ∆x = 0.6 m.
c. Repeat part (b) using ∆x = 0.3 m.
d. Based on the results of part (b) and (c), calculate the deflection at x = 1.2 m using
Richardson’s extrapolation formula.
e. Use two-point Gauss-Legendre formula to calculate the deflection at x = 1.2 m.
f. Which method gives you the best approximation? Which method is the easiest?
Figure 1: (a) Beam with linearly increasing load, and
(b) deflection of the beam due to linearly increasing load
Due date is 7th May 2014 at the beginning of lecture. If there are any questions or any clarifications
on this assignment, please email me at [email protected]. Late homework will NOT be given
credit.
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