8.3.2 Damped or Forced Harmonic Motion

Introduction to Damped and Forced Harmonic Motion

Harmonic motion describes systems where displacement from equilibrium is opposed by a restoring force proportional to that displacement. When resistance (damping) or external driving forces (forcing) are introduced, the motion changes.

Second-order differential equations model these dynamics.

Damped Harmonic Motion

Damped harmonic motion is governed by the equation:

m \frac{d^2x}{dt^2} + c \frac{dx}{dt} + kx = 0

where:

$m$ : mass of the object,
$c$ : damping constant (resistance proportional to velocity),
$k$ : spring constant (restoring force proportional to displacement),
$x(t)$ : displacement at time $t$

The nature of the motion depends on the discriminant of the characteristic equation:

m^2 + \frac{c}{m}m + \frac{k}{m} = 0

Cases of Damping

Light Damping ( $c^2 < 4mk$ )

Roots are complex ( $m = \alpha \pm i\beta$ ):

x(t) = e^{-\alpha t} \big(A\cos(\beta t) + B\sin(\beta t)\big)

The system oscillates with exponentially decaying amplitude.

Critical Damping ( $c^2 = 4mk$ ):

Roots are real and equal ( $m = -\frac{c}{2m}$ ):

x(t) = (A + Bt)e^{-\frac{c}{2m}t}

The system returns to equilibrium without oscillating, as quickly as possible.

Heavy Damping ( $c^2 > 4mk$ ):

Roots are real and distinct ( $m_1, m_2$ ):

x(t) = Ae^{m_1t} + Be^{m_2t}

The system slowly returns to equilibrium without oscillating.

Forced Harmonic Motion

Forced harmonic motion occurs when an external periodic force acts on the system, modelled by:

m \frac{d^2x}{dt^2} + c \frac{dx}{dt} + kx = F(t)

where $F(t)$ is the driving force (e.g., $F(t) = F_0\cos(\omega t)$ ).

The general solution is:

x(t) = x_c + x_p

where:

$x_c$ is the complementary function (solution to the homogeneous equation),
$x_p$ is the particular integral (solution to the non-homogeneous equation).

Worked Examples

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Example 1: Lightly Damped Oscillator

A system with mass $m = 1 \, \mathrm{kg}$ , damping constant $c = 2 \, \mathrm{Ns/m}$ , and spring constant $k = 10 \, \mathrm{N/m}$ satisfies the equation:

\frac{d^2x}{dt^2} + 2\frac{dx}{dt} + 10x = 0

Solve for $x(t)$ with initial conditions $x(0)=1$ , $\frac{dx}{dt}(0) = 0$

Step 1: Find the Characteristic Equation

m^2 + 2m + 10 = 0

The discriminant is:

\Delta = 2^2 - 4(1)(10) = -36

Since $\Delta < 0$ , the roots are complex:

m = -1 \pm 3i

Step 2: Write the General Solution

x(t) = e^{-t}(A\cos(3t) + B\sin(3t))

Step 3: Apply Initial Conditions

At $t = 0, x(0) = 1$

1 = e^{0}(A\cos(0) + B\sin(0)) \quad \Rightarrow \quad A = 1

At $t=0, \frac{dx}{dt} = 0$

Differentiate $x(t)$

\frac{dx}{dt} = e^{-t}\big(-A\cos(3t) - B\sin(3t)\big) + e^{-t}\big(-3A\sin(3t) + 3B\cos(3t)\big)

Substitute $t = 0$

0 = (-A + 3B)

Substituting $A = 1$

0 = -1 + 3B \quad \Rightarrow \quad B = \frac{1}{3}

Final Solution

x(t) = e^{-t}\left(\cos(3t) + \frac{1}{3}\sin(3t)\right)

infoNote

Example 2: Forced Oscillator

Solve the forced equation:

\frac{d^2x}{dt^2} + 3\frac{dx}{dt} + 2x = \cos(2t)

Step 1: Solve the Homogeneous Equation

m^2 + 3m + 2 = 0 \quad \Rightarrow \quad (m + 1)(m + 2) = 0

Roots: $m = -1, m = -2$

Complementary Function:

x_c = Ae^{-t} + Be^{-2t}

Step 2: Solve for the Particular Integral

Guess $x_p = C\cos(2t) + D\sin(2t)$

Substitute into the equation:

\frac{d^2x_p}{dt^2} = -4C\cos(2t) - 4D\sin(2t)

\frac{dx_p}{dt} = -2C\sin(2t) + 2D\cos(2t)

Substitute these into the original equation:

(-4C\cos(2t) - 4D\sin(2t)) + 3(-2C\sin(2t) + 2D\cos(2t)) + 2(C\cos(2t) + D\sin(2t)) = \cos(2t)

Group terms:

(-4C + 6D + 2C)\cos(2t) + (-4D - 6C + 2D)\sin(2t) = \cos(2t)

Equating coefficients:

-2C + 6D = 1, \quad -2D - 6C = 0

Solve simultaneously:

D = \frac{1}{10}, \quad C = -\frac{3}{10}

Particular Integral:

x_p = -\frac{3}{10}\cos(2t) + \frac{1}{10}\sin(2t)

Step 3: General Solution

x(t) = x_c + x_p = Ae^{-t} + Be^{-2t} - \frac{3}{10}\cos(2t) + \frac{1}{10}\sin(2t)

Note Summary

infoNote

Common Mistakes

Forgetting exponential decay in damping: Ensure the $e^{-\alpha t}$ factor is included in solutions for damped motion.
Incorrect handling of roots: Misidentifying real or complex roots leads to incorrect solutions.
Misapplying particular integral guesses: Choose forms that match the forcing function $F(t)$
Errors in initial conditions: Double-check substitutions when finding constants.

infoNote

Key Formulas

Damped Harmonic Motion:

m \frac{d^2x}{dt^2} + c \frac{dx}{dt} + kx = 0

Light Damping: $x(t) = e^{-\alpha t}(A\cos(\beta t) + B\sin(\beta t))$
Critical Damping: $x(t) = (A + Bt)e^{-\alpha t}$
Heavy Damping: $x(t) = Ae^{m_1t} + Be^{m_2t}$

Forced Harmonic Motion:

m \frac{d^2x}{dt^2} + c \frac{dx}{dt} + kx = F(t)

General solution: $(x(t) = x_c + x_p)$

Characteristic Equation:

m^2 + \frac{c}{m}m + \frac{k}{m} = 0

Damped or Forced Harmonic Motion Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

8.3.2 Damped or Forced Harmonic Motion

Introduction to Damped and Forced Harmonic Motion

Damped Harmonic Motion

Cases of Damping

Light Damping ( $c^2 < 4mk$ )

Critical Damping ( $c^2 = 4mk$ ):

Heavy Damping ( $c^2 > 4mk$ ):

Forced Harmonic Motion

Worked Examples

Example 1: Lightly Damped Oscillator

Example 2: Forced Oscillator

Note Summary

Common Mistakes

Key Formulas

500K+ Students Use These Powerful Tools to Master Damped or Forced Harmonic Motion For their A-Level Exams.

Other Revision Notes related to Damped or Forced Harmonic Motion you should explore

Damped or Forced Harmonic Motion Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

8.3.2 Damped or Forced Harmonic Motion

Introduction to Damped and Forced Harmonic Motion

Damped Harmonic Motion

Cases of Damping

Light Damping (c2<4mkc^2 < 4mkc2<4mk)

Critical Damping (c2=4mkc^2 = 4mkc2=4mk):

Heavy Damping (c2>4mkc^2 > 4mkc2>4mk):

Forced Harmonic Motion

Worked Examples

Example 1: Lightly Damped Oscillator

Example 2: Forced Oscillator

Note Summary

Common Mistakes

Key Formulas

500K+ Students Use These Powerful Tools to Master Damped or Forced Harmonic Motion For their A-Level Exams.

Other Revision Notes related to Damped or Forced Harmonic Motion you should explore

Light Damping ( $c^2 < 4mk$ )

Critical Damping ( $c^2 = 4mk$ ):

Heavy Damping ( $c^2 > 4mk$ ):