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:
0 2
2
kx
dt
dx
c
dt
d x
m
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 2
nd order RK Huen method over the
time period
0 t 5
sec with a step size
t 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.
THE UNIVERSITY OF WOLLONGONG 1 University of Wollongong Faculty of Engineering and Information Sciences ENGG952 Engineering Computing Spring Session – 2017 Assignment 2 Rules: 1. The assignment may be completed individually or by a group up to 3 students. The group formation is your own responsibility. Members may be from the same or different tutorial groups. 2. No collaboration between groups permitted. Any case of plagiarism will be penalized and students should make themselves aware of the university policies regarding plagiarism (see subject outline under University and Faculty Policies) 3. The assignment is due in week 13 (Friday 27 October by 4 pm) and submitted to Central of EIS using then barcoded sheet (see subject outline under assignment submission). Late submission will incur penalty as described in the subject outline. 4. If the assignment is completed by group, statement indicating the effort or contribution to the assignment by each member and signed by all members must be included in the beginning of the report, or all students agree that they have contributed equally to the report –add a statement at the front of the report, signed by all members. 5. The assignment must be submitted as a formal report. You need to include the analysis of the problem, the procedure for solution, and discussion of the results. It is NOT sufficient to only provide the program and results. All matlab script files, function files and excel files must be included as attachments or main contents in the hard copy report. 6. In addition, all matlab codes and function files must be burned in a CD or USB and submitted with the report for checking. 2 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: 0 2 2 kx dt dx c dt xd m 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 50 t sec with a step size 01.t . 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. 3 Question 2: Transient Heat Conduction (50%) The non-dimensional form for the transient heat conduction in an insulated rod is t u x u 2 2 x is the nondimensional length, t is the nondimensional time, u is the nondimensional temperature. This makes for the following boundary and initial conditions: Boundary conditions u(0, t ) = 0.5 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: l x x , )/( 2 klc t t , ol o tt tt u , in which l = the rod length, k = thermal conductivity of the rod material, = density, c = specific heat, to = temperature at x = 0, and tl = temperature at x = l. solve this nondimensional equation for the temperature distribution using explicit finite- difference method and implicit crank-nicholson method: a) write the finite-difference equation of the differential equation. the report must clearly provide the detailed derivation of the technique. b) programing to obtain the solution for time duration 10 t . (for explicit finite difference method, you can use excel or matlab. for implicit crank-nicholson method, you must use the matlab). please demenstrate that appropriate x and t are needed to solve u until steady- state solution is reached. c) plot the nondimensional temperature versus nondimensional length for a few typical values of nondimensional times, which can demonstrate the evloution of the tempeature at different time. 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 dimensionless x d im e n s io n le s s t e m p e ra tu re t=1.0 t=0.038 t=0.018 t=0.008 dr. hongtao zhu october 2017 figure="" 2.="" heat="" conduction="" problem="" in="" an="" insulated="" rod.="" note:="" l="" x="" x="" ,="" )/(="" 2="" klc="" t="" t="" ="" ="" ,="" ol="" o="" tt="" tt="" u="" ="" ="" ="" ,="" in="" which="" l="the" rod="" length,="" k="thermal" conductivity="" of="" the="" rod="" material,="" ="density," c="specific" heat,="" to="temperature" at="" x="0," and="" tl="temperature" at="" x="L." solve="" this="" nondimensional="" equation="" for="" the="" temperature="" distribution="" using="" explicit="" finite-="" difference="" method="" and="" implicit="" crank-nicholson="" method:="" a)="" write="" the="" finite-difference="" equation="" of="" the="" differential="" equation.="" the="" report="" must="" clearly="" provide="" the="" detailed="" derivation="" of="" the="" technique.="" b)="" programing="" to="" obtain="" the="" solution="" for="" time="" duration="" 10="" ="" t="" .="" (for="" explicit="" finite="" difference="" method,="" you="" can="" use="" excel="" or="" matlab.="" for="" implicit="" crank-nicholson="" method,="" you="" must="" use="" the="" matlab).="" please="" demenstrate="" that="" appropriate="" x="" and="" t="" are="" needed="" to="" solve="" u="" until="" steady-="" state="" solution="" is="" reached.="" c)="" plot="" the="" nondimensional="" temperature="" versus="" nondimensional="" length="" for="" a="" few="" typical="" values="" of="" nondimensional="" times,="" which="" can="" demonstrate="" the="" evloution="" of="" the="" tempeature="" at="" different="" time.="" 4="" 0="" 0.1="" 0.2="" 0.3="" 0.4="" 0.5="" 0.6="" 0.7="" 0.8="" 0.9="" 1="" 0.5="" 1="" 1.5="" 2="" dimensionless="" x="" d="" im="" e="" n="" s="" io="" n="" le="" s="" s="" t="" e="" m="" p="" e="" ra="" tu="" re="" t="1.0" t="0.038" t="0.018" t="0.008" dr.="" hongtao="" zhu="" october="">