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While considering the bead on the rotating hoop, we used phase plane analysis to show that the equation $$ \varepsilon \frac{d^{2} \phi}{d \tau^{2}}+\frac{d \phi}{d \tau}=f(\phi) $$ has solutions that rapidly relax to the curve where \(\frac{d \phi}{d \tau}=f(\phi)\). a) Estimate the time scale \(T_{f o x}\) for this rapid transient in terms of \(\varepsilon\), and then express \(T_{\text {frut }}\) in terms of the original dimensional quantities \(m, g, r, \omega\), and \(b\). b) Rescale the original differential equation, using \(T_{f \omega ii\) scale, instead of \(T_{\text {wiew }}=b / m g\). Which terms in the equation are negligible on this time scale? c) Show that \(T_{\text {fou }} \ll T_{\text {llm }}\) if \(\varepsilon<<1\). (In this sense, the time scales \(T_{\text {Fast }}\) and \(T_{\text {slow }}\) are widely separated.)

Step by step solution

01

Find the time scale Tf for the rapid transient

To estimate the time scale Tf for the rapid transient, we need to consider the equation: \[ \varepsilon \frac{d^{2} \phi}{d \tau^{2}}+\frac{d \phi}{d \tau}=f(\phi) \] Observe that the rapid transient occurs when the damping term, \(\frac{d \phi}{d \tau}\), dominates over the other terms in the equation. Thus, we can approximate the equation to: \[ \frac{d \phi}{d \tau} = f(\phi) \] Now, let's consider a typical time scale for this transient, denoted by Tf. Then, \[ \frac{\Delta \phi}{T_{f}} = f(\phi) \] Next, we will express Tf using the original dimensional quantities.

02

Express Tf in terms of the original dimensional quantities

To express the time scale Tf in terms of the original dimensional quantities, we first define: \[ \tau = \frac{t}{T_{w}} = \frac{mt}{b} \] Therefore, \[ \frac{dT_{f}}{d\tau} = T_w = \frac{b}{mg} \] Solving for Tf, we get: \[ T_{f} = \frac{\varepsilon}{mg}\frac{b}{mg} \] Thus, Tf can be expressed as: \[ T_{f} = \frac{\varepsilon b}{m^2 g^2} \]

03

Rescale the original differential equation with the new time scale

To rescale the original differential equation using the new time scale Tf, we define a new variable: \[ \eta = \frac{t}{T_{f}} \] Using the chain rule, the derivatives with respect to Ï„ can be expressed as: \[ \frac{d \phi}{d \tau} = \frac{dT_{f}}{d\eta}\frac{d \phi}{d \eta} \] \[ \frac{d^2 \phi}{d \tau^2} = \frac{dT_{f}}{d\eta}\frac{d^2 \phi}{d \eta^2} \] Substituting these expressions into the original differential equation, we get: \[ \varepsilon \frac{dT_{f}}{d\eta}\frac{d^2 \phi}{d \eta^2} + \frac{dT_{f}}{d\eta}\frac{d \phi}{d \eta} = f(\phi) \] Now, we need to identify the negligible terms on this new timescale.

04

Determine the negligible terms on the new timescale

Since \(\varepsilon<<1\), the term \(\varepsilon \frac{dT_{f}}{d\eta}\frac{d^2 \phi}{d \eta^2}\) becomes very small compared to the other terms in the equation. Thus, we can neglect this term on the time scale Tf.

05

Show that Tf

To show that Tf << Tω if ε << 1, we will compare Tf and Tω: \[ \frac{T_{f}}{T_{\omega}} = \frac{\frac{\varepsilon b}{m^2 g^2}}{\frac{b}{mg}} = \frac{\varepsilon}{mg} \] Since \(\varepsilon<<1\), it follows that \(\frac{T_{f}}{T_{\omega}} << 1\), which implies \(T_{f} << T_{\omega}\). Thus, the time scales Tf (fast) and Tω (slow) are widely separated when ε << 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Time Scale Estimation

Time scale estimation is crucial when analyzing dynamic systems, as it helps identify how quickly or slowly a process evolves over time. In our context, we look at the phase plane equation \( \varepsilon \frac{d^{2} \phi}{d \tau^{2}}+\frac{d \phi}{d \tau}=f(\phi) \). To estimate the time scale \( T_f \), we focus on the term \( \frac{d \phi}{d \tau} \) as it represents the rapid transient response. By assuming this term dominates in rapid changes, the equation simplifies to \( \frac{d \phi}{d \tau} = f(\phi) \), and \( T_f \) is defined when changes in \( \phi \) over time are governed by \( f(\phi) \).

• Time scale \( T_f \) gives insight into how quickly the phase settles into a stable path.
• This approximation involves ignoring smaller effects, highlighting dominant dynamics.

Understanding \( T_f \) is about balancing the changes in \( \phi \) that occur very fast relative to other processes.

Differential Equations

Differential equations express relationships involving rates of change and are fundamental in modeling dynamic systems. In our analysis, we explore the differential equation \( \varepsilon \frac{d^{2} \phi}{d \tau^{2}}+\frac{d \phi}{d \tau}=f(\phi) \). Here, the second derivative term represents acceleration, while the first derivative shows velocity or the rate of change of \( \phi \).
Differential equations often require simplifications or assumptions, such as neglecting certain terms to analyze specific dynamics.

• Analyzing differential equations allows us to understand behavior under various conditions.
• The equation dynamics change based on the significance of each term.

By considering simplifications, like \( \varepsilon << 1 \), we focus on the dominant process, which, in this case, is the first derivative or damping term.

Damping Term

The damping term in a differential equation, like \( \frac{d \phi}{d \tau} \), represents a force that opposes motion, often leading to a reduction in velocity without oscillation. Its importance lies in understanding how quickly a system settles down after a disturbance. In our equation, the dominance of the damping term highlights that the system will rapidly reach a state dictated by \( f(\phi) \).
The damping term essentially dictates the speed at which energy is dissipated from the system.

• A strong damping term quickly stabilizes a system following disturbances.
• Ignoring insignificant terms can simplify analysis by focusing on primary effects.

Understanding these dynamics allows predictions about system behavior under initial conditions or small perturbations.

Dimensional Analysis

Dimensional Analysis involves examining equations to ensure consistency across units while providing insights into the relationships between quantities. We use dimensional analysis to express the time scale \( T_f \) in terms of original quantities \( m, g, r, \omega, \) and \( b \). By defining \( \tau = \frac{t}{T_w} = \frac{mt}{b} \), we derive how different physical parameters influence the time scale. This approach helps verify that the equations model real-world scenarios correctly.

• A valid dimensional analysis confirms the equation's unit consistency.
• It reveals how different parameters directly affect solution behavior.

Such analyses are essential in ensuring the applicability and accuracy of mathematical models.