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3.8 Forced Periodic Vibrations

Schematic of a mass m suspended from a ceiling by a spring k and a damper gamma, with an upward force F(t).
\[ mu'' + \gamma u' + ku = F(t) \]

let's focus on a periodic \( F(t) \)

\[ F(t) = F_0 \cos(\omega t) \]

Where \( F_0 \) is the magnitude and \( \omega \) is the freq. of applied force.

let's start w/ undamped case: \( \gamma = 0 \)

\[ mu'' + ku = F_0 \cos(\omega t) \]

Characteristic eq: \( mr^2 + k = 0 \)   \( r = \pm \sqrt{\frac{k}{m}} i \)

\[ u(t) = C_1 \cos\left(\sqrt{\frac{k}{m}} t\right) + C_2 \sin\left(\sqrt{\frac{k}{m}} t\right) + U \]

Where \( U \) is the particular solution due to \( F(t) \).

The terms with \( \sqrt{\frac{k}{m}} \) represent the natural freq. of mass-spring system.

\[ \omega_0 = \sqrt{\frac{k}{m}} \]
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particular solution   \( U = A \cos(\omega t) + B \sin(\omega t) \)

\( \omega \neq \omega_0 \)

\( U' = \dots \)
\( U'' = \dots \)

sub into diff. eq.

finding: \( B = 0 \) ,   \( A = \frac{F_0}{k - m\omega^2} = \frac{F_0}{m(\frac{k}{m} - \omega^2)} = \frac{F_0/m}{(\omega_0^2 - \omega^2)} \)

general solution:

\[ u(t) = C_1 \cos(\omega_0 t) + C_2 \sin(\omega_0 t) + \frac{F_0/m}{(\omega_0^2 - \omega^2)} \cos(\omega t) \]

freq. \( \omega_0 \)

period \( \frac{2\pi}{\omega_0} \)

freq. \( \omega \)

period \( \frac{2\pi}{\omega} \)

as an example, if \( m=1, k=9, F_0=80, \omega=5, u(0)=u'(0)=0 \)

\[ u(t) = 5 \cos(3t) - 5 \cos(5t) \]
  • \( 5 \cos(3t) \) : mass-spring
  • \( -5 \cos(5t) \) : from external force

1st term period \( \frac{2\pi}{3} \)

2nd term period \( \frac{2\pi}{5} \)

ratio is rational so overall period is the least common multiple

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Beats Phenomenon Visualization

← period \( 2\pi \) →
A graph showing a beat pattern with a high-frequency oscillation modulated by a lower-frequency envelope.
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Interesting thing happens if \( w \approx w_0 \)

For example, \( m = 0.1 \), \( F_0 = 50 \), \( k = 302.5 \), \( w = 45 \)

\[ w_0 = \sqrt{\frac{k}{m}} = 55 \]

\( u(t) = C_1 \cos(55t) + C_2 \sin(55t) + \frac{1}{2} \cos(45t) \)

If \( u(0) = u'(0) = 0 \)

\[ u(t) = -\frac{1}{2} \cos(55t) + \frac{1}{2} \cos(45t) \]

If use the identity \( 2 \sin A \sin B = \cos(A-B) - \cos(A+B) \)

\( A = \frac{1}{2}(55+45) \)     \( B = \frac{1}{2}(55-45) \)

\[ u(t) = \sin(5t) \sin(50t) \]

\( \underbrace{\sin(5t)}_{\text{slow}} \)

low \( w \)

\( \underbrace{\sin(50t)}_{\text{fast}} \)

high \( w \)

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Beats Phenomenon

The following graph illustrates the concept of a "beat" in wave mechanics, where two frequencies interact to create an envelope effect.

A graph showing a high-frequency blue wave oscillating within a low-frequency red dotted envelope, labeled 'beat'.

Key Components:

  • Fast frequency: \( \omega = 55 \) (represented by the rapid blue oscillations).
  • Slow frequency: \( \omega = 5 \) (represented by the red dotted envelope).
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What if \( \omega_0 = \omega \)?

The standard solution form \( U = A \cos(\omega t) + B \sin(\omega t) \) needs to be adjusted when the driving frequency matches the natural frequency.

\[ U = A t \cos(\omega t) + B t \sin(\omega t) \]

Consider the specific case where:

  • \( m = 1 \)
  • \( F_0 = 100 \)
  • \( \omega_0 = 50 \)
  • \( \omega = 50 \)
  • Initial conditions: \( u(0) = u'(0) = 0 \)

The differential equation:

\[ m u'' + k u = F_0 \cos(\omega t) \]

has the solution:

\[ u(t) = t \sin(50t) \]

The presence of the \( t \) factor makes \( u \to \infty \) as \( t \to \infty \).

Displacement \( \to \infty \).

Resonance

This phenomenon is called resonance.

BAD for structures

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Oscillation with Increasing Amplitude

A graph showing a blue oscillating wave with an amplitude that increases linearly over time.
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Forced Vibrations with Damping

Now let's put the damper back in:

\[ mu'' + \gamma u' + ku = F_0 \cos(\omega t) \]

The characteristic equation roots are:

\[ r = \frac{-\gamma \pm \sqrt{\gamma^2 - 4km}}{2m} \]

General Solution:

\[ u(t) = c_1 u_1 + c_2 u_2 + U \]

depending on \( r \)

Particular Solution Guess:

\[ U = A \cos(\omega t) + B \sin(\omega t) \]
  • \( U' = \dots \)
  • \( U'' = \dots \)

Sub into \( mu'' + \gamma u' + ku = F_0 \cos(\omega t) \)

\[ A = \frac{(k - m\omega^2) F_0}{(k - m\omega^2)^2 + (\gamma \omega)^2} \]
\[ B = \frac{\gamma \omega F_0}{(k - m\omega^2)^2 + (\gamma \omega)^2} \]
As long as \( \gamma \neq 0 \), both \( A \) and \( B \) are bounded so, \( u \) will NOT go to \( \infty \) as \( t \to \infty \).
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Damped Forced Oscillations

System Parameters:

\[ m = 1, \quad k = 26, \quad \gamma = 2, \quad F_0 = 82, \quad w = 4 \]\[ u(0) = 6, \quad u'(0) = 0 \]

Spring Constant (k = 26):

Spring fights back with force 26 N/m

Damping Coefficient (\(\gamma = 2\)):

Damper fights back with force of 2 N/m/s

Differential Equation

\[ u'' + 2u' + 26u = 82 \cos(4t) \]

General Solution

\[ u(t) = \underbrace{e^{-t}(\cos 5t - 3 \sin 5t)}_{\text{complementary / transient solution}} + \underbrace{5 \cos 4t - 4 \sin 4t}_{\text{due to input / steady-state solution}} \]

Transient Solution

  • Goes to 0 as \( t \to \infty \)
  • Due to the damper which supplies \( e \) to a negative power

Steady-State Solution

(or forced response)

  • Never goes away

Long-term Behavior

\[ \lim_{t \to \infty} u(t) \approx F_0 \cos(wt) \text{ but with a phase shift} \]
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Graphical Analysis of Forced Response

The following graph illustrates the relationship between the actual displacement \( u(t) \) and the driving force function \( F_0 \cos(wt) \) over time.

Plot of u(t) (solid blue) and F_0 cos(wt) (dashed red) vs time, showing phase shift and transient decay.

Key Observations:

  • The solid blue line represents the actual displacement \( u(t) \).
  • The dashed red line represents the driving force function \( F_0 \cos(wt) \).
  • Note the initial discrepancy (transient phase) which settles into a steady-state oscillation with a constant phase shift relative to the input.