Problem 1.1: Newton’s second law can be expressed as
F = ma (1)
where F is the net force acting on the body,m mass of the body, and a the acceleration of the body in the direction of the net force. Use Eq. (1) to determine the mathematical model, i.e., governing equation of a free-falling body. Consider only the forces due to gravity and the air resistance. Assume that the air resistance is linearly proportional to the velocity of the falling body.
Solution: From the free-body-diagram it follows that
m
dv dt
= Fg −Fd,F g = mg, Fd = cv where v is the downward velocity (m/s) of the body, Fg is the downward force (N or kg m/s2) due to gravity,Fd is the upward drag force, m is the mass (kg) of the body, g the acceleration (m/s2) due to gravity, andc is the proportionality constant (drag coefficient, kg/s). The equation of motion is
dv dt
+ αv = g, α =
c m
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2 AN INTRODUCTION TO THE FINITE ELEMENT METHOD
Problem 1.2: A cylindrical storage tank of diameter D contains a liquid at depth (or head) h(x,t). Liquid is supplied to the tank at a rate of qi (m3/day) and drained at a rate of q0 (m3/day). Use the principle of conservation of mass to arrive at the governing equation of the flow problem.
Solution: The conservation of mass requires
time rate of change in mass = mass inflow - mass outflow
The above equation for the problem at hand becomes
d dt
(ρAh)=ρqi −ρq0 or
d(Ah) dt
= qi −q0 where A is the area of cross section of the tank (A = πD2/4) and ρ is the mass density of the liquid.
Problem 1.3: Consider the simple pendulum of Example 1.3.1. Write a computer program to numerically solve the nonlinear equation (1.2.3) using the Euler method. Tabulate the numerical results for two different time steps ∆t =0 .05 and ∆t =0 .025 along with the exact linear solution.
Solution: In order to use the finite difference scheme of Eq. (1.3.3), we rewrite (1.2.3) as a pair of first-order equations
dθ dt
= v,
dv dt
=−λ2 sinθ
Applying the scheme of Eq. (1.3.3) to the two equations at hand, we obtain
θi+1 = θi + ∆tv i; vi+1 = vi −∆t λ2 sinθi The above equations can be programmed to solve for (θi,v i). Table P1.3 contains representative numerical results.
Problem 1.4: An improvement of Euler’s method is provided by Heun’s method, which uses the average of the derivatives at the two ends of the interval to estimate the slope. Applied to the equation
du dt
= f(t,u) (1)
Heun’s scheme has the form
ui+1 = ui +
∆t 2 hf(ti,u i)+f(ti+1,u0 i+1)i,u 0 i+1 = ui + ∆tf (ti,u i) (2)
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SOLUTIONS MANUAL 3
Table P1.3: Comparison of various approximate solutions of the equation (d2θ/dt2)+λ2 sinθ = 0 with its exact linear solution.
Exact Approx. solution θ Exact Approx. solution v t θ ∆t = .05 ∆t = .025 v ∆t = .05 ∆t = .025
0.00 0.78540 0.78540 0.78540 -0.00000 -0.00000 -0.00000 0.05 0.76965 0.78540 0.77828 -0.62801 -0.56922 -0.56922 0.10 0.72302 0.75694 0.74276 -1.23083 -1.13844 -1.13027 0.15 0.64739 0.70002 0.67944 -1.78428 -1.69123 -1.66622 0.20 0.54578 0.58980 0.56482 -2.26615 -2.20984 -2.15879 0.25 0.42229 0.50496 0.47627 -2.65711 -2.67459 -2.58816 0.30 0.28185 0.37123 0.34225 -2.94148 -3.06403 -2.93371 0.35 0.13011 0.21803 0.19218 -3.10785 -3.35605 -3.17573 0.40 -0.02685 0.05023 0.03148 -3.14955 -3.53018 -3.29791 0.45 -0.18274 -0.12628 -0.13374 -3.06491 -3.57060 -3.29007 0.50 -0.33129 -0.30481 -0.29690 -2.85732 -3.46921 -3.15014 0.60 -0.58310 -0.63965 -0.59131 -2.11119 -2.85712 -2.50787 0.80 -0.78356 -1.05068 -0.91171 0.21536 -0.50399 -0.28356 1.00 -0.50591 -0.94062 -0.74672 2.41051 2.29398 2.19765
In books on numerical analysis, the second equation in (2) is called the predictor equation and the first equation is called the corrector equation. Apply Heun’s method to Eqs. (1.3.4) and obtain the numerical solution for ∆t =0 .05.
Solution: Heun’s method applied to the pair
dθ dt
= v,
dv dt
=−λ2 sinθ
yields the following discrete equations:
θ0 i+1 = θi + ∆tv i vi+1 = vi −λ2∆t 2 ³sinθi + sinθ0 i+1´ θi+1 = θi + ∆t 2 (vi + vi+1)
The numereical results obtained with the Heun’s method and Euler’s method are presented in Table P1.4.
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4 AN INTRODUCTION TO THE FINITE ELEMENT METHOD
Table P1.4: Numerical solutions of the nonlinear equation d2θ/dt2 + λ2 sinθ =0 along with the exact solution of the linear equation d2θ/dt2+λ2θ = 0.