Question

Consider an mass-spring system with the following IVP for its disagreement y (t) at time t greater than or equal to 0. You may assume it is underdamped.

y" + y' + 5y = 0 , y (0) = -2 , y '(0) = -1

(a) Convert this to a DE system IVP in displacement y and velocity v.

(b) Without using technology or solving the second order DE, make a rough sketch of the system solution on a phase plane. It does not need to be precise, but briefly show and explain the direction you give at the initial value, and the long term behavior as t heads towards infinity

Answer 1

Kindly go through the solution provided below.

Dynamic systems are used to model situations where the state of the system is defined by a set of differential equations.

The systems can involve both linear and non-linear components.

But this is a simpler example of a mass-spring system, which involves linear differential equations.

In order to study the behaviour of the solution space, we can draw a phase plane, and analyse the direction of change at different points of the plane, which provides us with insights on how the system behaves around a certain point (Initial value).

Here the initial value of (-2,-1) is represented by the point P on the phase plane.

We notice a pattern in the phase plane. It looks like a downward spiral. As time passes we move towards the centre of the spiral, i.e. at (0,0).

Therefore when t tends to infinity or after a long time, the solution would have reached the equilibrium point at the centre.