Problem 44 - Advanced Engineering Mathematics

Question

Consider a driven undamped spring/mass system described by the initial-value problem $$ \frac{d^{2} x}{d t^{2}}+\omega^{2} x=F_{0} \sin ^{n} \gamma t, x(0)=0, x^{\prime}(0)=0 $$ (a) For \(n=2\), discuss why there is a single frequency \(\gamma_{1} / 2 \pi\) at which the system is in pure resonance. (b) For \(n=3\), discuss why there are two frequencies \(\gamma_{1} / 2 \pi\) and \(\gamma_{2} / 2 \pi\) at which the system is in pure resonance. (c) Suppose \(\omega=1\) and \(F_{0}=1\). Use a numerical solver to obtain the graph of the solution of the initial-value problem for \(n=2\) and \(\gamma=\gamma_{1}\) in part (a). Obtain the graph of the solution of the initial-value problem for \(n=3\) corresponding, in turn, to \(\gamma=\gamma_{1}\) and \(\gamma=\gamma_{2}\) in part (b).

Step-by-Step Solution

Answer

For n=2, there is one resonance frequency at \(\gamma/(2\pi)=\omega/(4\pi)\). For n=3, there are two resonance frequencies at \(\gamma/(2\pi)=\omega/(2\pi)\) and \(\omega/(6\pi)\).

Step 1: Analyze the system for n=2

For \(n=2\), the forcing function is \(F_0 \sin^2(\gamma t)\). Using the identity \(\sin^2(\gamma t) = \frac{1}{2} - \frac{1}{2} \cos(2\gamma t)\), the force becomes \(\frac{F_0}{2} - \frac{F_0}{2} \cos(2\gamma t)\). The system equation is \(\frac{d^2 x}{dt^2} + \omega^2 x = \frac{F_0}{2} - \frac{F_0}{2} \cos(2\gamma t)\). Resonance occurs when the driving frequency matches the natural frequency, so for pure resonance, \(2\gamma = \omega\). Thus, \(\gamma_1 = \frac{\omega}{2}\).

Step 2: Analyze the system for n=3

For \(n=3\), the forcing function is \(F_0 \sin^3(\gamma t)\). Using the identity \(\sin^3(\gamma t) = \frac{3}{4} \sin(\gamma t) - \frac{1}{4} \sin(3\gamma t)\), the system equation is \(\frac{d^2 x}{dt^2} + \omega^2 x = \frac{3F_0}{4} \sin(\gamma t) - \frac{F_0}{4} \sin(3\gamma t)\). Resonance occurs when either driving frequency equals the natural frequency, so for pure resonance, either \(\gamma = \omega\) or \(3\gamma = \omega\). Thus, the resonance frequencies are \(\gamma_1 = \omega\) and \(\gamma_2 = \frac{\omega}{3}\).

Step 3: Graph the solution for n=2 (numerical solution)

With \(n=2\), \(\omega = 1\), and \(F_0 = 1\), solve the differential equation \(\frac{d^2 x}{dt^2} + x = \frac{1}{2} - \frac{1}{2} \cos(\omega t)\) using a numerical method (e.g., Runge-Kutta). For \(\gamma = \frac{1}{2}\), plot the solution to visualize the resonance effect.

Step 4: Graph the solution for n=3 (numerical solution) with two frequencies

With \(n=3\), \(\omega = 1\), and \(F_0 = 1\), solve the differential equation \(\frac{d^2 x}{dt^2} + x = \frac{3}{4} \sin(\gamma t) - \frac{1}{4} \sin(3\gamma t)\) using a numerical method. First, set \(\gamma = 1\) and plot the solution. Then, set \(\gamma = \frac{1}{3}\) and plot the solution to observe the resonance behavior at each frequency.

Key Concepts

Understanding Resonance

Resonance is a fascinating phenomenon that occurs when a system is driven at a frequency matching its natural frequency. When this happens, the system can absorb energy very efficiently, leading to large oscillations.
For the spring-mass system described, resonance is key to understanding which frequencies cause these dramatic effects in the system.
### Pure Resonance in DetailIn the context of the provided problem, the system involves a spring-mass setup with an external driving force. - When the driving force's frequency matches the natural frequency of the system, resonance occurs.- This leads to a condition where the amplitude of the system's oscillations increases substantially. In part (a) of the exercise, for example, a system with a forcing function of the form \( \sin^2(\gamma t) \) shows resonance at a single frequency because of the identity transformation that produces a cosine term with double frequency.
Resonance here would occur when \( 2\gamma = \omega \), which signifies a match with the natural frequency \( \omega \).
### Complexity with Odd PowersAs complexity increases, such as when \( n = 3 \) in the problem, the system resonates not only at its natural frequency \( \omega \) but also harmonically at \( \omega/3 \). - This illustrates how driving forces shaped like \( \sin^3(\gamma t) \) can stimulate both the fundamental frequency and a first harmonic.In practical terms, understanding and predicting resonance is crucial for designing systems to either exploit or avoid these potentially large response amplitudes.

Spring-Mass System Fundamentals

Spring-mass systems are classic examples used in physics and engineering to describe oscillatory motion. They consist of a mass attached to a spring, which can move back and forth.
Such systems are very useful for analyzing simple harmonic motion and vibrations in mechanical structures.
### Key Features of Spring-Mass Systems- **Mass**: The object that oscillates, providing inertia to the system.- **Spring**: Provides a restoring force, often following Hooke's Law, \( F = -kx \), where \( k \) is the spring constant and \( x \) is the displacement. In the given problem, the spring-mass system is undamped, meaning there's no friction or damping force acting contrary to the motion.
This characteristic leads to perpetual oscillations when disturbed, making resonance a critical feature to consider.
### Natural FrequencyThe natural frequency, \( \omega \), is a vital aspect, dictating the rate at which the system naturally oscillates when not influenced by external forces. - It's determined by the properties of the mass and the spring. - Understanding this parameter helps predict resonance conditions as highlighted in the exercise.Spring-mass systems are models for studying vibrations in many real-world applications, from buildings to car suspensions.

Numerical Methods for Solving Differential Equations

Differential equations model dynamic systems like our spring-mass setup, describing how the system evolves over time. However, they can often be complex, requiring numerical methods for practical solutions.
Numerical methods are computational techniques used to approximate solutions to these equations.
### Why Use Numerical Methods?- **Complexity**: Many real-world problems are too complex for analytical solutions, especially with nonlinearities or non-standard forcing functions (like \( \sin^2(\gamma t) \) and \( \sin^3(\gamma t) \) in our problem).- **Efficiency**: These methods allow for the computation of approximate solutions rapidly and accurately enough for engineering applications.In the original problem, numerical methods like the **Runge-Kutta** algorithm are used to solve the differential equations governing the spring-mass system.
These techniques iterate over small time steps to calculate positions and velocities, creating a solvable model.
### Visualizing SolutionsBy applying numerical methods, students can visualize the dynamics of complex systems:- Solutions for cases \( n=2 \) and \( n=3 \) can demonstrate resonance effects vividly through graphs.- They reveal how solutions behave under specific driving frequencies, providing insights into oscillatory patterns.Numerical methods form the backbone of simulations and are critical for exploring phenomena that are analytically challenging.