Problem 10 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Consider a special case of a spring-mass system in which there is no resistance and with a a harmonic forcing function, \(f(t)=\cos \omega t\). Thus examine $$ m y^{\prime \prime}(t)+k y(t)=\cos \omega t $$ Let \(\omega_{0}=\sqrt{\mathbf{k} / \mathbf{m}}\). It is routine to show that if \(\omega \neq \omega_{0},\) the general solution to \(m y^{\prime \prime}(t)+k y(t)=\cos \omega t\) is $$ y(t)=\frac{1}{m\left(\omega_{0}^{2}-\omega^{2}\right)} \cos \omega t+C_{1} \cos \omega_{0} t+C_{2} \sin \omega_{0} t . $$ a. Suppose that the mass is initially at rest so that \(y(0)=0,\) and \(y^{\prime}(0)=0\) and \(\omega \neq \omega_{0} .\) Show that the motion of the mass is approximated by $$ \begin{aligned} y(t) &=\frac{1}{m\left(\omega_{0}^{2}-\omega^{2}\right)} \cos \omega t-\frac{1}{m\left(\omega_{0}^{2}-\omega^{2}\right)} \cos \omega_{0} t \\ &=\frac{1}{m\left(\omega_{0}^{2}-\omega^{2}\right)}\left(\cos \omega t-\cos \omega_{0} t\right) \end{aligned} $$ b. Sketch the graph of \(y(t)\) in Equation 18.16 for the case \(m=1, \omega=1\) and \(k=0.01\) (weak spring) \(\left(\omega_{0}=0.1\right) .\) The impressed force \(\cos t\) appears as the rapid oscillations, and the inherent system frequency appears as the overall gradual wave due to the term \(\cos \omega_{0} t=\cos 0.1 t\) c. Sketch the graph of \(y(t)\) in Equation 18.16 for the case \(m=1, \omega=1\) and \(k=0.81\) (stiff spring) \(\left(\omega_{0}=0.9\right)\) From the identity, \(\cos x-\cos y=2 \sin \frac{x+y}{2} \sin \frac{x-y}{2},\) $$ y(t)=\frac{1}{m\left(\omega_{0}^{2}-\omega^{2}\right)} 2 \sin \left(\frac{\omega_{0}+\omega}{2} t\right) \sin \left(\frac{\omega_{0}-\omega}{2} t\right) $$ The amplitude of the rapid vibrations, \(\sin \left(\frac{\omega_{0}+\omega}{2} t\right)\) is $$ \frac{1}{m\left(\omega_{0}^{2}-\omega^{2}\right)} 2 \sin \left(\frac{\omega_{0}-\omega}{2} t\right) $$ and results in a beat of frequency \(4 \pi /\left(\omega_{0}-\omega\right)\) (that may be heard in mechanical systems.

Step-by-Step Solution

Answer

The motion of the mass is approximated by \( y(t) = \frac{1}{m(\omega_0^2 - \omega^2)}(\cos \omega t - \cos \omega_0 t) \) under given conditions, which describes a beat frequency due to harmonic forcing.

Step 1: Understanding the Problem

We start by understanding the problem, which involves a spring-mass system without resistance and a harmonic forcing function. We are given the equation \( m y''(t) + k y(t) = \cos \omega t \) and need to demonstrate that under initial conditions \( y(0) = 0 \) and \( y'(0) = 0 \), the solution can be simplified.

Step 2: Identify General Solution

The general solution to the equation when \( \omega eq \omega_0 \) is given by: \[ y(t) = \frac{1}{m(\omega_0^2 - \omega^2)} \cos \omega t + C_1 \cos \omega_0 t + C_2 \sin \omega_0 t. \] Our goal is to find \( C_1 \) and \( C_2 \) using the initial conditions.

Step 3: Apply Initial Conditions

To use the initial conditions \( y(0) = 0 \), we substitute \( t = 0 \) into the general solution: \[ 0 = \frac{1}{m(\omega_0^2 - \omega^2)} \cos 0 + C_1 \cos 0 + C_2 \sin 0, \] simplifying to \[ 0 = \frac{1}{m(\omega_0^2 - \omega^2)} + C_1, \] from which \( C_1 = -\frac{1}{m(\omega_0^2 - \omega^2)}. \)

Step 4: Derivative and Second Initial Condition

Take the derivative \( y'(t) \) to apply the second initial condition: \[ y'(t) = -\frac{\omega}{m(\omega_0^2 - \omega^2)} \sin \omega t - \frac{1}{m(\omega_0^2 - \omega^2)} \omega_0 \sin \omega_0 t + C_2 \omega_0 \cos \omega_0 t. \] Applying \( y'(0) = 0 \) gives:\[ 0 = C_2 \omega_0, \] leading to \( C_2 = 0. \)

Step 5: Simplify Expression

Substitute \( C_1 \) and \( C_2 \) back into the general solution: \[ y(t) = \frac{1}{m(\omega_0^2 - \omega^2)} \cos \omega t - \frac{1}{m(\omega_0^2 - \omega^2)} \cos \omega_0 t. \] This confirms the given simplified form of the motion.

Step 6: Use Trigonometric Identity

Using the trigonometric identity \( \cos x - \cos y = -2 \sin \frac{x+y}{2} \sin \frac{x-y}{2} \), rewrite the expression as: \[ y(t) = \frac{1}{m(\omega_0^2 - \omega^2)} \cdot 2 \sin \left(\frac{\omega_0 + \omega}{2} t\right) \sin \left(\frac{\omega_0 - \omega}{2} t\right). \]

Step 7: Graph Sketch for Weak Spring Case

For \( m = 1, \omega = 1, \omega_0 = 0.1, k = 0.01 \), sketch \( y(t) \). This involves rapidly oscillating \( \cos \omega t \) and overall gradual wave \( \cos \omega_0 t \).

Step 8: Graph Sketch for Stiff Spring Case

For \( m = 1, \omega = 1, \omega_0 = 0.9, k = 0.81 \), sketch \( y(t) \). The rapid oscillation is \( \cos \omega t \) and the overall gradual change is due to \( \cos \omega_0 t \).

Step 9: Discuss Beat Frequency

The beat frequency, caused by the term \( 2 \sin\left(\frac{\omega_0 - \omega}{2} t\right) \), can be calculated with \( 4 \pi/(\omega - \omega_0) \). This is a significant feature in mechanical systems sensing vibrations.

Key Concepts

Harmonic Forcing Function

In the context of a spring-mass system, a harmonic forcing function plays a vital role in influencing the system's behavior. A forcing function is an external function applied to a system, in this case represented as \( f(t) = \cos \omega t \). This introduces a periodic external influence on the mass attached to the spring.

The harmonic nature of this function means its influence is oscillatory, with specific frequency \( \omega \). This is akin to nudging the system in a rhythmic pattern, shaping how the system reacts over time. The response becomes predictable—oscillations dominated by this frequency.

• The force is akin to a repeating wave.
• Each cycle pushes the system with the same regularity.
• Other natural behaviors of the system must interact with this frequency.

The harmonic forcing function thereby sets up a scenario where the spring-mass system oscillates with influences both from intrinsic properties and external forces.

General Solution

When analyzing a spring-mass system under a harmonic forcing function, finding the general solution is critical. This helps describe how the system will behave under prescribed conditions. The standard differential equation is:
[m y''(t) + k y(t) = \cos \omega t].

This denotes a second-order differential equation with constant coefficients, representing the setup for a spring mass system with external forcing.

If the natural frequency \( \omega_0 = \sqrt{k/m} \) does not match the frequency of the forcing function \( \omega \), the general solution is provided by:
\[ y(t) = \frac{1}{m(\omega_0^2 - \omega^2)} \cos \omega t + C_1 \cos \omega_0 t + C_2 \sin \omega_0 t. \]

• The first term captures the system's response to the external force \( \cos \omega t \).
• Terms involving \( C_1 \) and \( C_2 \) handle the inherent oscillations of the system itself.
• The entire expression accounts for both forced and natural oscillatory behaviors.

With initial conditions, we adjust it to meet specific physical scenarios.

Initial Conditions

Initial conditions are crucial for tuning the general solution to reflect a specific physical scenario. By applying initial conditions, we align the theoretical model with real-world circumstances. Here, these are given as \( y(0) = 0 \) and \( y'(0) = 0 \), indicating the mass starts at rest.

• Substituting \( t = 0 \) into the general solution aligns the equation with the given state of the mass: \[ 0 = \frac{1}{m(\omega_0^2 - \omega^2)} + C_1. \]
• This simplifies to \( C_1 = -\frac{1}{m(\omega_0^2 - \omega^2)} \), accounting for the initial displacement.
• Taking the derivative \( y'(t) \) and applying \( y'(0) = 0 \), we find \( C_2 = 0 \), leaving: \[ y(t) = \frac{1}{m(\omega_0^2 - \omega^2)} \cos \omega t - \frac{1}{m(\omega_0^2 - \omega^2)} \cos \omega_0 t. \]

These conditions ensure the function mirrors the system's start from rest.

Beat Frequency

In scenarios where the harmonic forcing function's frequency \( \omega \) is close, yet distinct from the system's natural frequency \( \omega_0 \), the phenomenon of beat frequency emerges. This is a result of the interference between these two close frequencies.

This beat frequency is quantitatively described by the modulation term in the equation:
\[ y(t) = \frac{1}{m(\omega_0^2 - \omega^2)} \cdot 2 \sin \left(\frac{\omega_0 + \omega}{2} t\right) \sin \left(\frac{\omega_0 - \omega}{2} t\right). \]

• The term \( \sin \left(\frac{\omega_0 - \omega}{2} t \right) \) dictates the beat frequency.
• This is where the amplitude of oscillations varies, amplifying and diminishing in regular intervals.
• Resulting beats manifest as periodic variations in amplitude, notable in mechanical systems as modifications in oscillatory nuances.

Beat frequency adds a layer of complexity, easily observable as throbbing or pulsating in oscillatory systems, providing insight into the difference between forcing and natural frequencies.