**Dynamics 5 – Assignment**

The aim of this assignment is investigate the nonlinear characteristics of a single-degree-of-freedom system: the Duffing oscillator. This consists of a lumped-mass, which is grounded by a dashpot, a linear spring, and a nonlinear, cubic spring.

The oscillator has mass m = 1, the dashpot introduces damping of ¢ = 0.01, and the linear and cubic stiffnesses are given by & = 4and a = 1.2, respectively. The mass is subjected to periodic forcing, at a frequency of 1 defined by F = Fo cos(Mt) and its displacement is denoted by x. A schematic for this system is given in Fig. 1.

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As has been seen in lectures, the nonlinear spring leads to hysteretic behaviour, in which the natural frequency is dependent on the amplitude of the response. In this assignment, this behaviour is to be investigated, using two methods to produce the forced frequency response near w,,. The first method is to use the fourth-order Runge-Kutta method (ode45 in Matlab) to recreate time histories at different frequencies. This will comprise the majority of the marks available and would obtain a B grade, if completed perfectly. The second method is the multiple scales technique seen in lectures . The full details of the task are as follows:

1. (a) Derive the second-order differential equation that defines the dynamic behaviour of the mass. Rewrite this as a system of first-order ordinary differential equations.

(b) Applying the ode45 function in Matlab to this first-order system, generate the forced frequency response of the

Duffing oscillator. This should demonstrate the hysteretic nature of the system and it will be necessary to select

a value for Fo (or several) to do so.

2. The behaviour of the Duffing oscillator has been approximated using the multiple scales method in lectures. Apply

this technique to the present structure and compare the results with those from the previous question. Are the

responses the same? IH not, can you give a reason for why this might be the case?

Write a report outlining the methodology used to answer the above questions, providing any insight and discussion that

you feel is relevant. In the discussion, use table, plots and diagrams as you see fit. The final document should not exceed

four A4 pages, with 2.54 cm margins, written with font type Times or equivalent and font size 11 pt.

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