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 $$F_1{e}^{i{\omega }_1^{\prime }t}+F_2{e}^{{\mathrm{ior}}_{2}l}+F_3{e}^{i\mathrm{cog}T}$$ Write the solution by the principle of superposition. Suppose, for given ${\omega }_1^{\prime },{\omega }_2^{\prime },{\omega }_3^{\prime }$, that we adjust the system so that $\omega ={\omega }_1^{\prime }$; 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

When $\omega ={\omega }_1^{\prime }$, the principal term is $C_1{e}^{i\omega t}$, making the system a filter for that frequency.

Step-by-step solution

- Understand the Given Equation

The given equation for damped forced vibrations involves terms of different frequencies. The right-hand side is given as $$F_1{e}^{i{\omega }_1^{\prime }t}+F_2{e}^{i{\omega }_2^{\prime }t}+F_3{e}^{i{\omega }_3^{\prime }t}$$We need to write the solution using the principle of superposition and show the principal term when \omega = \omega_{1}^{\prime}.

- Apply the Principle of Superposition

According to the principle of superposition, the solution to this equation can be considered as the superposition of the solutions corresponding to each force term:$$x(t)=x_1(t)+x_2(t)+x_3(t)$$where each $x_i(t)$ is the response to the individual force term $F_i{e}^{i{\omega }_i^{\prime }t}$.

- Write the General Solution for Each Term

For each force term, the response can be written as:$$x_i(t)=C_i{e}^{i{\omega }_i^{\prime }t}$$where $C_i$ is a constant determined by the specific conditions of the system. Thus,$$x(t)=C_1{e}^{i{\omega }_1^{\prime }t}+C_2{e}^{i{\omega }_2^{\prime }t}+C_3{e}^{i{\omega }_3^{\prime }t}$$

- Adjust the System Frequency

Set the system frequency $\omega$ equal to ${\omega }_1^{\prime }$. When $\omega ={\omega }_1^{\prime }$, the solution simplifies to:$$x(t)=C_1{e}^{i\omega t}+C_2{e}^{i{\omega }_2^{\prime }t}+C_3{e}^{i{\omega }_3^{\prime }t}$$

- Identify the Principal Term

Since the system is tuned to $\omega ={\omega }_1^{\prime }$, the term with frequency ${\omega }_1^{\prime }$ will dominate due to resonance. Therefore, the principal term in the solution is the first term:$$x(t)\approx C_1{e}^{i\omega t}$$Thus, the system acts as a filter selecting the vibration of frequency $\omega ={\omega }_1^{\prime }$, analogous to a radio tuned to a station.

Key concepts

Superposition Principle

The principle of superposition states that the response caused by two or more forces is the sum of the responses that would have been caused by each force individually. This principle is vital in both mechanical and electrical vibrations. In our problem, the equation for damped forced vibrations includes several forces of different frequencies on the right-hand side. To find the solution, we superimpose the individual responses to each force term. If we denote each force term as $F_i{e}^{i{\omega }_i^{\prime }t}$, the resulting response is the sum of the responses for each of these force terms: $$x(t)=x_1(t)+x_2(t)+x_3(t)=C_1{e}^{i{\omega }_1^{\prime }t}+C_2{e}^{i{\omega }_2^{\prime }t}+C_3{e}^{i{\omega }_3^{\prime }t}$$ where each term corresponds to the individual force terms in the original equation. This method simplifies the analysis of complex systems by breaking down the problem into manageable parts.

Resonance

Resonance occurs when the frequency of external forces matches the natural frequency of the system. When resonance happens, the system's response becomes much larger. In our specific equation, if we set the system's frequency $\omega$ to match ${\omega }_1^{\prime }$, the system resonates with the force term $F_1{e}^{i{\omega }_1^{\prime }t}$. As a result, this term dominates the solution: $$x(t)\approx C_1{e}^{i\omega t}$$ In practical terms, this is like a playground swing pushed at its natural frequency, causing it to swing higher each time. Resonance can be both useful and harmful, enhancing desired signals in radio receivers or causing structural failures in buildings and bridges.

Frequency Filtering

Frequency filtering selects certain frequencies and suppresses others. This concept is vital in many applications, from radio tuning to signal processing. In our problem, we see how the system acts as a filter by tuning the system frequency to match one of the force term frequencies. When the system frequency $\omega$ is set to ${\omega }_1^{\prime }$, the response primarily consists of the term ${\omega }_1^{\prime }$. The other terms $C_2{e}^{i{\omega }_2^{\prime }t}$ and $C_3{e}^{i{\omega }_3^{\prime }t}$ are less significant or neglected: $$x(t)\approx C_1{e}^{i\omega t}$$ This is analogous to tuning a radio to a specific frequency, allowing the receiver to isolate one station from many. Frequency filtering is ubiquitous in technology, ensuring systems respond primarily to desired inputs.

Mechanical Vibrations

Mechanical vibrations refer to the oscillatory motions of physical systems. These can include anything from a guitar string to a car engine. In our problem, we're dealing with damped forced mechanical vibrations, meaning the system is influenced by external forces and experiences resistance that dampens its movement over time. The right-hand side of the equation represents these external forces with different frequencies: $$F_1{e}^{i{\omega }_1^{\prime }t}+F_2{e}^{i{\omega }_2^{\prime }t}+F_3{e}^{i{\omega }_3^{\prime }t}$$ By applying the principle of superposition, we determine how the system behaves under these forces. Adjusting the system frequency to one of these forces reveals how the system can be tuned to favor specific vibrations, which is the essence of resonance in mechanical systems.

Electrical Vibrations

Electrical vibrations work similarly to mechanical vibrations but in electrical systems like circuits. In these systems, oscillations can occur in the current or voltage. Our problem can also be applied to electrical vibrations, where forces could represent electromotive forces (emfs) of different frequencies. Suppose the applied emf terms are written as: $$F_1{e}^{i{\omega }_1^{\prime }t}+F_2{e}^{i{\omega }_2^{\prime }t}+F_3{e}^{i{\omega }_3^{\prime }t}$$ The solution would still use the principle of superposition and, by tuning the system's natural frequency, we can filter and enhance certain frequencies. This is similar to tuning a radio circuit to a specific broadcast frequency, isolating one station from all others. Understanding these principles is key to designing and analyzing efficient electronic systems.