Further Analytical Methods

Purpose of this assignment

The purpose of this assignment is to allow you the opportunity to display your knowledge and capability of solving Differential Equations and to provide evidence on how you may formulate and solve them.

Scenario

Whilst working on several projects at BCoTyou have come across several instances where various quantities appear linked to the values of their rates of change in a set of electrical and mechanical systems. Upon reading you discover that the most accurate form of modelling for these values is a differential equation therefore youare to set out to solve the following problems in order to complete your assigned projects.

Task 1

Whilst testing the resistance of a copper conductor in your test rig, you observe that the rate of change of its resistance as temperature changes is equal to a constant (the temperature coefficient of resistance, which is 3.9× 10^-3 for copper) multiplied by its current resistance.

Formulate a first-order Differential Equation to describe this situation.

Given the following observational data about your conductor, solve the differential equation to find a formula for its resistance at a given temperature.

At 0°C, the resistance of the component is 50Ω.

When the conductor's resistance exceeds (R=58Ω) it causes problems with the system's operation. Using your equation, find out the temperature at which it reaches the system.

Task 2

The equation of motion of a train is given by mdv/dt = mk(1-e^-t)-mcv, where vis the speed, t is the time and m, k and c are constants. Determine the speed v given v=0 at t=0?

Task 3

A mathematical equation you are trying to solve has produced the differential equation dy/dx=6x+3y, with the conditions y=6 when x=0, you are only interested in it over a small range from x=0 to x=3 and do not want to invest the time necessary to properly solve it as you do not need an exact relationship.

Either manually or electronically apply Euler's method to this equation and draw a graph of the approximate solution over the required range.
Using your workings from the previous part where necessary, apply the Euler-Cauchy method to the same range and provide the resulting graph.

Finally, apply the Runge-Kutta method and compare your three results, analysing their accuracy and time-efficiency.

Task 4

Another part of your system involves an oscillating mass which may be looked at as a simple harmonic oscillator.

Taking the given values Newton's Law (F_total = m(d²x)/(dt²)), Hooke's Law (F_spring = -kx) and the formula for a damping force on an oscillating mass (F_damp = -cdx/dt) formulate a second-order differential equation to describe this behaviour, given that the mounting spring has a spring constant k=201, your damping mechanism produces a damping constant c=500 and the oscillating mass has mass m=0.45kg.

Solve this differential equation to find an expression for the position of your oscillator at any given time during its motion, (i.e. x(t)=[function]), given that when t=0 our mass begins with an offset of x=-0.5m, and is also static (dx/dt=0).

Using your equation, find the position of the mass at time t=0.5.

Task 5

An oscillating LCR circuit is used in one of the systems you have been assigned. Its circuit is comprised of a Resistor of R = 100Ω, an Inductor of L=0.5H and a capacitor of C=11mF.

a) Given that it outputs a voltage of V=10sin(3t), using your knowledge of electronic circuits (sum of voltage drops=total voltage) formulate a differential equation to describe the system's behaviour in terms of current.

b) By solving this equation, find the relationship between Current (I) and time (t) given that I=0 at t=0 and I=0 at t=3s.

c) Using this relationship, find the Current at time t=850ms.

Task 6

Evaluate how different models of engineering systems using first and second order differential equations solve engineering problems for Tasks 1,2,4 and 5 above.

Task 7

Critically evaluate first and second order differential equations when generating the solutions to engineering situations using models of engineering systems for Tasks 1,2,4 and 5 above and justify your conclusions by validation with alternative techniques e.g. use of computer programs or other external tools.

Attachment:- Analytical Methods.rar