ENGG952 Engineering Computing- Engineering Assignment-University of Wollongong- Faculty of Engineering and Information Sciences

Engineering Assignment Question 1: Motion of a Spring-Mass System (50%) The motion of a damped spring-mass system is descirbed by the following ordinary differnetial equation:

m	d 2 x	+ c	dx	+ kx = 0
	dt2		dt

where x is displacement from equlibrium position (meter), t is time (second), m is the mass and equal 20 kg, c is the damping coefficient (N.sec/meter) . The dampping coefficient, c, takes on two values of 5 (under damped), 40 (critically damped. The spring constant k = 20 N/meter. The initial velocity is zero, and the intial displacement x = 1 meter. Figure 1. Damped spring-mass system (a)Transform the problem into a system of two first order initial value ODEs. The report must clearly provide the detailed derivation of the technique. (b)Solve for motion of a spring-mass system using the 2nd order RK Huen method over the time period 0 £ t £ 5 sec with a step size Dt =1.0 . You can calcualte the results manually or by matlab or by excel. (c)Plot the displacment verus time for two values of the damping coefficient on the same figure, and discuss the results. 2 Question 2: Transient Heat Conduction (50%) The non-dimensional form for the transient heat conduction in an insulated rod is

		¶2u		¶u
		¶x 2		=
				¶t
		is the nondimensional time, u is the nondimensional
x is the nondimensional length, t
temperature. This makes for the following boundary and initial conditions:
Boundary conditions				) = 0.5
		u(0, t					u(1, t ) = 2.0
Initial conditions		u( x , 0) = 0.5					0 £ x <1

Figure 2. Heat conduction problem in an insulated rod.

Note: x=

x

,

t

,

u =

T -To

, in which L = the rod length, k = thermal

t =

L

(rC L2 / k)

T

L

-T

o

conductivity of the rod material, r = density, C = specific heat, To = temperature at x = 0, and TL = temperature at x = L. Solve this no dimensional equation for the temperature distribution using explicit finite- difference method and implicit Crank-Nicholson method:

1. a) Write the finite-difference equation of the differential equation. The report must clearly provide the detailed derivation of the technique.
2. b) Programing to obtain the solution for time duration 0 £t £1 . (For explicit finite difference method, you can use EXCEL or matlab. For implicit Crank-Nicholsonmethod, you must use the matlab).

Please demenstrate that appropriate Dx and Dt are needed to solve u until steady- state solution is reached.

1. c) Plot the non dimensional temperature versus nondimensional length for a few typical values of nondimensional times, which can demonstrate the evloution of the tempeature at different time

 

	2
temperature	1.5
				t=1.0						t=0.008
Dimensionless
								t=0.018
	1						t=0.038
	0.5	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
						Dimensionless x