Question

Consider the following mass-spring system:

Let $\mathit{x}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$be the deviation of the block from the equilibrium position at time $\mathit{t}$. Consider the velocity $\mathit{v}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}$of the block. There are two forces acting on the mass: the spring force $\mathit{F}$, which is assumed to be proportional to the displacement $\mathit{x}$, and the force ${\mathit{F}}_{\mathbf{f}}$of friction, which is assumed to be proportional to the velocity.

${\mathit{F}}_{\mathbf{s}}\mathbf{=}\mathbf{-}\mathit{p}\mathit{x}$and ${\mathit{F}}_{\mathbf{f}}\mathbf{=}\mathbf{-}\mathit{q}\mathit{v}$

Where$\mathit{p}\mathbf{>}\mathbf{0}$and$\mathit{q}\mathbf{\ge }\mathbf{0}$($\mathit{q}$is 0 if the oscillation is frictionless). Therefore, the total force acting on the mass is

$$\begin{gathered} \mathit{F}\mathbf{=}{\mathit{F}}_{\mathbf{s}}\mathbf{+}{\mathit{F}}_{\mathbf{f}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{-}\mathit{p}\mathit{x}\mathbf{-}\mathit{q}\mathit{v} \end{gathered}$$

By Newton’s second law of motion , we have\

$$\begin{gathered} \mathit{F}\mathbf{=}\mathit{m}\mathit{a} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathit{m}\frac{\mathbf{d}\mathbf{v}\mathbf{,}}{\mathbf{d}\mathbf{t}} \end{gathered}$$

Where represents acceleration and $\mathit{m}$the mass of the block. Combining the last two equations, we find that

$\mathit{m}\frac{\mathbf{d}\mathbf{v}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{-}\mathit{p}\mathit{x}\mathbf{-}\mathit{q}\mathit{v}$

Or

$\frac{\mathbf{d}\mathbf{v}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{-}\frac{\mathbf{p}}{\mathbf{m}}\mathbf{\times }\mathbf{-}\frac{\mathbf{q}}{\mathbf{m}}\mathit{v}$

Let $\mathit{b}\mathbf{=}\frac{\mathbf{p}}{\mathbf{m}}$and $\mathit{c}\mathbf{=}\frac{\mathbf{q}}{\mathbf{m}}$for simplicity. Then the dynamics of this mass-spring system are described by the system

Sketch a phase portrait for this system in each of the following cases, and describe briefly the significance of your trajectories in terms of the movement of the block. Comment on the stability in each case.

a. $\mathit{c}\mathbf{=}\mathbf{0}$(frictionless). Find the period.

b. ${\mathit{c}}^{\mathbf{2}}\mathbf{<}\mathbf{4}\mathit{b}$(underdamped)

c. ${\mathit{c}}^{\mathbf{2}}\mathbf{>}\mathbf{4}\mathit{b}$(overdamped).

Short answer

Thus, (a) Sketch of the system $c=0$is in the explanation.

(b) Sketch of the system $c^2<4b$is in the explanation.

(c) Sketch of the system $c^2>4b$is in the explanation

Step-by-step solution

(a) Phase portrait for c = 0 . 

A phase portrait of

The sketch for the system$c=0$is given below:

Phase portrait for c2<4b.

(b)

If $c^2<4b$then $b=1,c=1$and a phase portrait is sketched as follows:

The sketch for the system $c^2<4b$is given below:

Phase portrait for c2>4b.

(c)

If $c^2>4b$then $b=1,c=3$and a phase portrait is sketched as follows:

The sketch for the system $c^2>4b$is given below: