Q1 E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

In the following problems, take $\mathrm{g}=32\mathrm{ft}/{\sec }^2$for the U.S. Customary System and $\mathrm{g}=9.8\mathrm{m}/{\sec }^2$ for the MKS system.

Sketch the frequency response curve (13) for the system in which m = 4, k = 1, b = 2.

Step-by-Step Solution

Answer

Therefore, the frequency response curve is $\mathrm{M}\left(\gamma \right)=\frac{1}{\sqrt{{\left(1-4{\gamma }^2\right)}^2+4{\gamma }^2}}$

Its sketch is shown below;

Step 1: General form

The angular frequency:

 

The amplitude of the steady-state solution to equation (1) depends on the angular frequency of the forcing function and it is given by $\mathrm{A}\left(\gamma \right)={\mathrm{F}}_0\mathrm{M}\left(\gamma \right)$, where

$\mathrm{M}\left(\gamma \right):=\frac{1}{\sqrt{{\left(\mathrm{k}-\mathrm{m}{\gamma }^2\right)}^2+{\mathrm{b}}^2{\gamma }^2}} \dots \left(1\right)$

Step 2: Evaluate the equation

Given that the frequency response curve is $\mathrm{M}\left(\gamma \right):=\frac{1}{\sqrt{{\left(\mathrm{k}-\mathrm{m}{\gamma }^2\right)}^2+{\mathrm{b}}^2{\gamma }^2}}$

And m = 4, k = 1, and b = 2.

Substitute the values in equation (1),

$\mathrm{M}\left(\gamma \right)=\frac{1}{\sqrt{{\left(1-4{\gamma }^2\right)}^2+4{\gamma }^2}}$

Now draw the graph of it.