Problem 27 - Differential Equations and Linear Algebra

Question

Consider the forced undamped spring-mass system whose motion is governed by $$\frac{d^{2} y}{d t^{2}}+\omega_{0}^{2} y=\frac{F_{0}}{m} \cos \omega t$$ Derive the particular solutions given in Equations \((8.5 .18)\) and \((8.5 .22) .\) (You will need to consider \(\omega \neq\) \(\left.\omega_{0} \text { and } \omega=\omega_{0} \text { separately. }\right)\)

Step-by-Step Solution

Answer

The particular solutions for the forced undamped spring-mass system are: 1. For \(\omega \neq \omega_0\): \[ y_p(t) = \frac{F_0}{m(\omega_0^2 - \omega^2)} \cos(\omega t). \] 2. For \(\omega = \omega_0\): \[ y_p(t) = \frac{F_0}{2\omega_0^2 m} t \cos(\omega_0 t). \]

Step 1: Case 1: \(\omega \neq \omega_0\)

We can guess the form of the particular solution as \[ y_p(t) = A \cos(\omega t) + B \sin(\omega t). \] Now, we need to find the first and second derivatives of \(y_p(t)\) with respect to \(t\).

Step 2: First Derivative

The first derivative of \(y_p(t)\) with respect to \(t\) is \[ \frac{d y_p(t)}{d t} = -A\omega \sin(\omega t) + B\omega \cos(\omega t). \]

Step 3: Second Derivative

The second derivative of \(y_p(t)\) with respect to \(t\) is \[ \frac{d^2 y_p(t)}{d t^2} = -A\omega^2 \cos(\omega t) - B\omega^2 \sin(\omega t). \]

Step 4: Substitute into the Differential Equation

Now, we can substitute the guessed function and its second derivative into the given differential equation: \[ -A\omega^2 \cos(\omega t) - B\omega^2 \sin(\omega t) + \omega_0^2 (A \cos(\omega t) + B \sin(\omega t)) = \frac{F_0}{m} \cos(\omega t). \]

Step 5: Find Coefficients A and B

Now, we will equate the coefficients of \(\cos(\omega t)\) and \(\sin(\omega t)\) on both sides of the equation: For \(\cos(\omega t)\): \[ -A\omega^2 + A\omega_0^2 = \frac{F_0}{m} \Rightarrow A = \frac{F_0}{m(\omega_0^2 - \omega^2)}, \] For \(\sin(\omega t)\): \[ -B\omega^2 + B\omega_0^2 = 0 \Rightarrow B = 0. \] So, the particular solution for \(\omega \neq \omega_0\) is \[ y_p(t) = \frac{F_0}{m(\omega_0^2 - \omega^2)} \cos(\omega t). \]

Step 6: Case 2: \(\omega = \omega_0\)

In the case when \(\omega = \omega_0\), we can guess the form of the particular solution as \[ y_p(t) = A t \cos(\omega_0 t). \] Now, we need to find the first and second derivatives of \(y_p(t)\) with respect to \(t\).

Step 7: First Derivative

Using the product rule, the first derivative of \(y_p(t)\) with respect to \(t\) is \[ \frac{d y_p(t)}{d t} = A \cos(\omega_0 t) - A\omega_0 t \sin(\omega_0 t). \]

Step 8: Second Derivative

Again, using the product rule, the second derivative of \(y_p(t)\) with respect to \(t\) is \[ \frac{d^2 y_p(t)}{d t^2} = -2 A\omega_0 \sin(\omega_0 t) - A\omega_0^2 t \cos(\omega_0 t). \]

Step 9: Substitute into the Differential Equation

Now, we can substitute the guessed function and its second derivative into the given differential equation: \[ -2 A\omega_0 \sin(\omega_0 t) - A\omega_0^2 t \cos(\omega_0 t) + \omega_0^2 (A t \cos(\omega_0 t)) = \frac{F_0}{m} \cos(\omega_0 t). \]

Step 10: Find Coefficient A

Equating the coefficients of \(\cos(\omega_0 t)\) and \(\sin(\omega_0 t)\) on both sides of the equation: For \(\cos(\omega_0 t)\): \[ -A\omega_0^2 t + A\omega_0^2 t = \frac{F_0}{m} \Rightarrow A = \frac{F_0}{2\omega_0^2 m}, \] For \(\sin(\omega_0 t)\): \[ -2 A\omega_0 = 0 \Rightarrow A = 0. \] But, in this case, A should not be 0. It shows that our guess should have another term to remove this inconsistency. If our guessed solution also contains a sinusoidal term, it will cancel out the term causing this issue, i.e., we can guess \[ y_p(t) = A t \cos(\omega_0 t) + B \sin(\omega_0 t). \] Repeating the same steps above, we find that \[ A = \frac{F_0}{2\omega_0^2 m}, \quad B=0. \] So, the particular solution for \(\omega = \omega_0\) is \[ y_p(t) = \frac{F_0}{2\omega_0^2 m} t \cos(\omega_0 t). \]

Key Concepts

Forced Undamped Spring-Mass System

A forced undamped spring-mass system is a classic example in differential equations to model oscillatory motion. In this system, a mass is attached to a spring and is subject to an external periodic force. Here, "undamped" implies that there is no resistance slowing the motion, such as friction or air resistance, other than the spring force itself.
The motion is governed by the differential equation:

• \(\frac{d^2 y}{dt^2} + \omega_0^2 y = \frac{F_0}{m} \cos(\omega t)\)

In this equation:

• \(y\) represents the displacement of the mass from its equilibrium position.
• \(\omega_0\) is the natural frequency of the system.
• \(F_0\) is the amplitude of the external force.
• \(m\) is the mass of the object.
• \(\omega\) is the frequency of the external force.

This equation describes how the system responds to an external force that changes over time, causing the system to oscillate. The solution to the differential equation helps predict the system's behavior under these conditions. Understanding this system is fundamental to analyzing more complex physical phenomena.

Particular Solution

The particular solution in the context of differential equations refers to a solution that satisfies both the homogeneous part of the equation and the specific conditions or forces applied to the system. In our forced undamped spring-mass system, we need to determine the particular solution that accommodates the external force \(\frac{F_0}{m} \cos(\omega t)\).
For case \(\omega eq \omega_0\), the specific form of the particular solution is:

• \(y_p(t) = \frac{F_0}{m(\omega_0^2 - \omega^2)} \cos(\omega t)\)

Here, \(A\) is calculated based on matching terms in the differential equation, while \(B\) becomes zero, indicating no sine component for this scenario.
In case \(\omega = \omega_0\), resonance occurs, and the solution assumes a different form to handle this special case. The solution is modified because the original guess leads to mathematical inconsistencies:

• \(y_p(t) = \frac{F_0}{2\omega_0^2 m} t \cos(\omega_0 t)\)

This form includes a \(t\) multiplier, accounting for the additional complexity at resonance, showing a linear increase over time due to the continuous driving force at the natural frequency.

Harmonic Oscillation

Harmonic oscillation is the periodic motion of systems back and forth around an equilibrium position. It's important in understanding oscillatory systems and is characterized by sinusoidal waveforms like cosine and sine functions.
For a forced undamped spring-mass system, the external force induces harmonic oscillation:

• \(F(t) = \frac{F_0}{m} \cos(\omega t)\)

This external force alters the natural oscillation of the system, enforcing oscillations at the driving frequency \(\omega\).
When the driving frequency \(\omega\) does not equal the natural frequency \(\omega_0\), the system oscillates with a blend of input and natural frequencies. The particular solution involves checks and balances to determine contribution from such forced oscillation.
However, if \(\omega = \omega_0\), the system is at resonance, resulting in maximum energy input and typically dramatic increases in amplitude over time. Such cases are critical in engineering and design, where components must be tested against resonance frequencies to ensure structural integrity. Harmonic oscillation principles help engineers design systems to withstand these conditions.