Question

Consider an equation for damped forced vibrations (mechanical or electrical) in which the right-hand side is a sum of several forces or emfs of different frequencies. For example, in (6.32) let the right-hand side be ${\mathit{F}}_{\mathbf{1}}{\mathit{e}}^{\mathbf{i}{\mathbf{\omega }}_{\mathbf{1}}\mathbf{t}}\mathbf{+}{\mathit{F}}_{\mathbf{2}}{\mathit{e}}^{\mathbf{i}{\mathbf{\omega }}_{\mathbf{2}}\mathbf{t}}\mathbf{+}{\mathit{F}}_{\mathbf{3}}{\mathit{e}}^{\mathbf{i}{\mathbf{\omega }}_{\mathbf{3}}\mathbf{t}}$,

Write the solution by the principle of superposition. Suppose, for giventhat we adjust the system so that $\mathit{\omega }\mathbf{=}{\mathit{\omega }}_{\mathbf{1}}^{\mathbf{\text{'}}}$; show that the principal term in the solution is then the first one. Thus, the system acts as a "filter" to select vibrations of one frequency from a given set (for example, a radio tuned to one station selects principally the vibrations of the frequency of that station).

Short answer

Answer

The solution of the differential equation $\frac{d^{2}Y}{d{t}^2}+2b\frac{dY}{dt}+{\omega }^{2}Y=F_1{e}^{i{\omega }_{1}t}+F_2{e}^{i{\omega }_{2}t}+F_3{e}^{i{\omega }_{3}t}$by using the principal of superposition is $y_p=\frac{F_1}{2b{\omega }_1}\sin \left({\omega }_1^{\text{'}}t-\varphi \right)+\frac{F_2}{\sqrt{{\left({\omega }^2-{\omega }_2^2\right)}^2+4b^2{\omega }^{\text{'}}{2}^2}}\mathrm{sinsin}\left({\omega }_2^{\text{'}}t-\varphi \right)+\frac{F_3}{\sqrt{{\left({\omega }^2-\omega 3_{}^2\right)}^2+4b^2{\omega }^{\text{'}}{3}^2}}\mathrm{sinsin}\left({\omega }_3^{\text{'}}t-\varphi \right)$

Step-by-step solution

Given information from question

The solution of the differential equationis

The superposition principle

As per the superposition principle, when two or more waves overlap in space, the resultant disturbance is equal to the algebraic sum of the individual disturbances.

The solution of the differential equation

The solution of the differential equation $\frac{d^{2}Y}{d{t}^2}+2b\frac{dY}{dt}+{\omega }^{2}Y=F_1{e}^{i{w}_{1}t}+F_2{e}^{i{w}_{2}t}+F_3{e}^{i{w}_{3}t}$by using the principal of superposition can be written as:

$y_P=\left(\begin{array}{c}\frac{F_1}{\sqrt{{\left({\omega }^2-\omega \text{'}{1}^2\right)}^2+4b^2\omega \text{'}{1}^2}}\sin \left(\omega \text{'}1t-\phi \right)+\frac{F_2}{\sqrt{{\left({\omega }^2-\omega \text{'}{2}^2\right)}^2+4b^2\omega \text{'}{2}^2}}\sin \left({\omega }_2^{\text{'}}t-\phi \right)\\ +\frac{F_3}{\sqrt{{\left({\omega }^2-{{\omega }_3^{\text{'}}}^2\right)}^2+4b^2\omega \text{'}{3}^2}}\sin \left(\omega \text{'}3t-\phi \right)\end{array}\right)$

Now, if $\omega ={\omega }_1^{\text{'}}$, then the above solution becomes:

$$\begin{gathered} y_P=\left(\begin{array}{c}\frac{F_1}{\sqrt{{\left({\omega }^2-\omega \text{'}{1}^2\right)}^2+4b^2\omega \text{'}{1}^2}}\sin \left(\omega \text{'}1t-\phi \right)+\frac{F_2}{\sqrt{{\left({\omega }^2-\omega \text{'}{2}^2\right)}^2+4b^2\omega \text{'}{2}^2}}\sin \left({\omega }_2^{\text{'}}t-\phi \right)\\ +\frac{F_3}{\sqrt{{\left({\omega }^2-{{\omega }_3^{\text{'}}}^2\right)}^2+4b^2\omega \text{'}{3}^2}}\sin \left(\omega \text{'}3t-\phi \right)\end{array}\right) \\ =\frac{F_1}{2b{\omega }_1}\sin \left({\omega }_1^{\text{'}}t-\varphi \right)+\frac{F_2}{\sqrt{{\left({\omega }^2-{\omega }_2^2\right)}^2+4b^2\omega 2^2}}\sin \left({\omega }_2^{\text{'}}t-\varphi \right)+\frac{F_3}{\sqrt{{\left({\omega }^2-\omega 3^2\right)}^2+4b^2\omega 3^2}}\sin \left({\omega }_3^{\text{'}}t-\varphi \right) \end{gathered}$$

Thus, the solution of the differential equation $\frac{d^{2}Y}{d{t}^2}+2b\frac{dY}{dt}+{\omega }^{2}Y=F_1{e}^{i{\omega }_{2}t}+F_2{e}^{i{\omega }_{2}t}+F_3{e}^{i{\omega }_{3}t}$by using the principal of superposition is $y_p=\frac{F_1}{2b{\omega }_1}\sin \left({\omega }_1^{\text{'}}t-\varphi \right)+\frac{F_2}{\sqrt{{\left({\omega }^2-{\omega }_2^2\right)}^2+4b^2\omega \text{'}{2}^2}}\sin \left({\omega }_{2}t-\varphi \right)+\frac{F_3}{\sqrt{{\left({\omega }^2-\omega 3^2\right)}^2+4b^2\omega \text{'}{3}^2}}\sin \left({\omega }_{3}t-\varphi \right)$