Mechanical Vibrations

Mechanical Vibrations

$Learning Objectives$

Understand the fundamentals of mechanical vibrations, including period, frequency, and amplitude.
Derive and apply equations of motion for free undamped vibrations.
Analyze torsional vibrations in rotational systems.
Evaluate the effects of damping on free vibrations, including overdamped, critically damped, and underdamped cases.
Apply the logarithmic decrement to measure damping experimentally.
Understand the behavior of systems under forced vibrations and the phenomenon of resonance.

Vibration is the periodic motion of a body or system of connected bodies displaced from a position of equilibrium.

Free Undamped Vibrations

Consider a block of mass $m$ attached to a spring of stiffness $k$ . For free vibration (no external force), the system undergoes simple harmonic motion, balancing the inertial force and restoring spring force.

$Simple Harmonic Motion (SHM)$

A type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.

Free Undamped Equation of Motion

The differential equation governing free, undamped vibration and its simple harmonic solution.

m \ddot{x} + k x = 0

x(t) = A \sin(\omega_n t) + B \cos(\omega_n t)

Variables

Symbol	Description	Unit
$m$	Mass of the body	kg
$k$	Spring stiffness	N/m
$x$	Displacement from equilibrium	m
$\ddot{x}$	Acceleration	$m/s^2$
$A, B$	Constants determined by initial conditions	m
$\omega_n$	Natural circular frequency	rad/s
$t$	Time	s

$Amplitude$

The maximum magnitude of displacement from the equilibrium position. It describes the "size" of the oscillation.

$\omega_n$

The rate of oscillation measured in radians per second. $\omega_n = \sqrt{\frac{k}{m}} \quad (\text{rad/s})$

$f_n$

The number of complete cycles of oscillation per second. $f_n = \frac{\omega_n}{2\pi} \quad (\text{Hz})$

$\tau$

The time required for one complete cycle of vibration. $\tau = \frac{1}{f_n} = \frac{2\pi}{\omega_n} = 2\pi \sqrt{\frac{m}{k}} \quad (\text{s})$

$Analyzing Free Undamped Vibrations$

STEP-BY-STEP

Define the Coordinate: Establish a coordinate system indicating the displacement $x$ from the static equilibrium position.
Draw the Free-Body Diagram: Draw the body in a displaced position and identify all active restoring forces (like spring forces).
Apply Newton's Second Law: Write the equation of motion $\sum F_x = m \ddot{x}$ or $\sum M = I \ddot{\theta}$ .
Identify Natural Frequency: Rearrange the equation into the standard form $\ddot{x} + \omega_n^2 x = 0$ to identify $\omega_n^2$ .
Calculate Parameters: Use $\omega_n$ to calculate the frequency $f_n$ and period $\tau$ .
Apply Initial Conditions: Use initial displacement $x(0)$ and velocity $\dot{x}(0)$ to determine the constants $A$ and $B$ in the general solution.

$Interactive Simulation$

Interact with the simulation below to explore mechanical vibrations, varying mass and stiffness to observe changes in frequency and period.

Mechanical Vibrations — Mass-Spring-Damper SystemUnderdamped (ζ = 0.06)

System Parameters

Mass (

m

)5 kg

Stiffness (

k

)50 N/m

Damping (

c

)2 Ns/m

Initial Disp. (

x_0

)1.0 m

Vibration Equations

Equation of Motion:

m\ddot{x} + c\dot{x} + kx = 0

\omega_n = \sqrt{\frac{k}{m}} = 3.16\,\text{rad/s}

\zeta = \frac{c}{c_c} = \frac{c}{2\sqrt{km}} = 0.063

\omega_d = \omega_n\sqrt{1-\zeta^2} = 3.16\,\text{rad/s}

x(t):1.0000 m

ẋ(t):-0.0250 m/s

ẍ(t):-4.993 m/s²

t:0.00 s

Simple Pendulum

For small angles ( $\sin \theta \approx \theta$ ), a simple pendulum behaves like a spring-mass system. The restoring force is provided by the component of gravity acting perpendicular to the pendulum string.

Pendulum Formulas

The natural frequency and period for a simple pendulum. Notice that they are independent of mass.

\omega_n = \sqrt{\frac{g}{L}}

\tau = 2\pi \sqrt{\frac{L}{g}}

Variables

Symbol	Description	Unit
$\omega_n$	Natural frequency	rad/s
$\tau$	Period	s
$g$	Acceleration due to gravity	$m/s^2$
$L$	Length of the pendulum string	m

$Interactive Simulation$

Interact with the simulation below to explore simple pendulum motion. Observe how the period is independent of mass but dependent on length.

Simple Pendulum — Exact Numerical Integration

Pendulum Setup

Length (

L

)1.5 m

Mass (

m

)2.0 kg

Initial Angle (

\theta_0

)30°

Damping Ratio (

\zeta

)0.10

Dynamics Equations

Equation of Motion:

\ddot{\theta} + \frac{g}{L}\sin\theta = 0

Small-Angle Period:

T = 2\pi\sqrt{\frac{L}{g}} = 2.457\,\text{s}

θ (current):30.0°

\omega

:0.000 rad/s

T

(current):0.00 s

Normal (

N

):16.99 N

Energy Balance

PE3.94 J

KE0.00 J

Total E₀3.94 J

Phase Portrait (θ, ω)

Torsional Vibrations

Vibrations can also occur in rotational systems. The angular equivalent to a spring is a torsional spring (e.g., a twisting shaft), which provides a restoring torque $M_t$ proportional to the angle of twist $\theta$ .

$Torsional Vibration Equation$

For a disk with mass moment of inertia $I_0$ attached to a torsional spring, the equation of motion is derived from $\sum M_O = I_O \ddot{\theta}$ , yielding the standard form of the torsional vibration equation.

Torsional Vibration Formulas

The restoring torque, equation of motion, and natural circular frequency for torsional vibration.

M_t = -k_t \theta

I_0 \ddot{\theta} + k_t \theta = 0

\omega_n = \sqrt{\frac{k_t}{I_0}}

Variables

Symbol	Description	Unit
$M_t$	Restoring torque	$N\cdot m$
$k_t$	Torsional stiffness	$N\cdot m/rad$
$\theta$	Angle of twist	rad
$\ddot{\theta}$	Angular acceleration	$rad/s^2$
$I_0$	Mass moment of inertia	$kg\cdot m^2$
$\omega_n$	Natural circular frequency	rad/s

Damped Free Vibrations

In real systems, energy is dissipated through mechanisms like friction or fluid resistance, called damping. If a viscous damper ( $c$ ) is added to the system, the equation of motion must include a damping force proportional to velocity.

Damped Free Vibration Equation

Equation of motion for a system with mass, stiffness, and viscous damping.

m \ddot{x} + c \dot{x} + k x = 0

Variables

Symbol	Description	Unit
$m$	Mass	kg
$\ddot{x}$	Acceleration	$m/s^2$
$c$	Viscous damping coefficient	$N\cdot s/m$
$\dot{x}$	Velocity	m/s
$k$	Spring stiffness	N/m
$x$	Displacement	m

$c_c$

The minimum amount of damping required to return a displaced system to equilibrium without oscillation. $c_c = 2m\omega_n = 2\sqrt{km}$

$\zeta$

A dimensionless measure of the actual damping in a system compared to the critical damping. $\zeta = \frac{c}{c_c}$

$System Behavior Based on Damping Ratio$

The damping ratio determines the nature of the system's response:

Underdamped ( $\zeta \lt 1$ ): The system oscillates, but the amplitude exponentially decays over time. The frequency of this damped oscillation is $\omega_d = \omega_n \sqrt{1 - \zeta^2}$ .
Critically Damped ( $\zeta = 1$ ): The system returns to equilibrium as quickly as possible without oscillating.
Overdamped ( $\zeta \gt 1$ ): The system returns to equilibrium slowly without oscillating. The greater the damping, the slower the return.

Logarithmic Decrement

For an underdamped system ( $\zeta \lt 1$ ), the amplitude of oscillation decays exponentially. A practical way to measure the damping ratio experimentally is by comparing successive peak amplitudes.

Logarithmic Decrement

Calculates the logarithmic decrement, defined as the natural logarithm of the ratio of two successive peak amplitudes.

\delta = \ln\left(\frac{x_1}{x_2}\right) = \frac{2\pi \zeta}{\sqrt{1 - \zeta^2}}

\delta = \frac{1}{n} \ln\left(\frac{x_0}{x_n}\right)

Variables

Symbol	Description	Unit
$\delta$	Logarithmic decrement	-
$x_1, x_2$	Two successive peak amplitudes separated by one full cycle	m
$\zeta$	Damping ratio	-
$n$	Number of cycles	-
$x_0$	Initial peak amplitude	m
$x_n$	Peak amplitude after n cycles	m

$Small Damping Approximation$

For very small damping ( $\zeta \ll 1$ ), the logarithmic decrement equation simplifies to $\delta \approx 2\pi \zeta$ .

Forced Vibrations

When a vibrating system is subjected to a continuous external periodic force, it undergoes forced vibration. Let the driving force be $F(t) = F_0 \sin(\omega_0 t)$ , where $\omega_0$ is the driving frequency.

Forced Undamped Vibration

Equation of motion and steady-state solution amplitude for forced undamped vibration.

m \ddot{x} + k x = F_0 \sin(\omega_0 t)

x_p(t) = X \sin(\omega_0 t)

X = \frac{F_0}{k - m\omega_0^2} = \frac{F_0/k}{1 - (\omega_0/\omega_n)^2}

Variables

Symbol	Description	Unit
$x_p(t)$	Steady-state solution displacement	m
$m$	Mass	kg
$\ddot{x}$	Acceleration	$m/s^2$
$k$	Spring stiffness	N/m
$x$	Displacement	m
$F_0$	Amplitude of the driving force	N
$\omega_0$	Driving frequency	rad/s
$t$	Time	s
$X$	Amplitude of the steady-state solution	m
$\omega_n$	Natural circular frequency	rad/s

Magnification Factor (MF)

The ratio of the dynamic amplitude to the static deflection.

MF = \frac{X}{F_0/k} = \frac{1}{|1 - (\omega_0/\omega_n)^2|}

Variables

Symbol	Description	Unit
$MF$	Magnification factor	-
$X$	Dynamic amplitude	m
$F_0$	Amplitude of the driving force	N
$k$	Spring stiffness	N/m
$\omega_0$	Driving frequency	rad/s
$\omega_n$	Natural circular frequency	rad/s

$Resonance Caution$

When the driving frequency $\omega_0$ exactly equals the natural frequency $\omega_n$ , Resonance occurs. The amplitude theoretically goes to infinity (if undamped). In physical structures like bridges or buildings, this condition can cause severe, catastrophic structural damage.

Key Takeaways

Free Vibration ( $m\ddot{x} + kx = 0$ ) occurs without external forces and results in simple harmonic motion.
Natural Frequency ( $\omega_n = \sqrt{k / m}$ ) depends only on the system's mass and stiffness.
Torsional Vibration follows the same principles, replacing mass with mass moment of inertia ( $I_0$ ) and linear stiffness with torsional stiffness ( $k_t$ ).
Period: $\tau = 2\pi/\omega_n$
Damping ( $c$ ) dissipates energy, usually via fluid resistance or friction, and causes amplitude decay.
Logarithmic Decrement ( $\delta$ ) is used to experimentally measure damping from the decay of successive amplitude peaks.
Critical Damping ( $c_c$ ) defines the boundary between oscillatory and non-oscillatory motion.
Resonance occurs in forced vibration when driving frequency equals natural frequency, causing mathematically unbounded (and practically destructive) amplitude.

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Learning Objectives

Free Undamped Vibrations

Simple Harmonic Motion (SHM)

Free Undamped Equation of Motion

Amplitude

Natural Circular Frequency (ωn\omega_nωn​)

Natural Frequency (fnf_nfn​)

Period (τ\tauτ)

Analyzing Free Undamped Vibrations

Interactive Simulation

System Parameters

Vibration Equations

Simple Pendulum

Pendulum Formulas

Interactive Simulation

Pendulum Setup

Dynamics Equations

Torsional Vibrations

Torsional Vibration Equation

Torsional Vibration Formulas

Damped Free Vibrations

Damped Free Vibration Equation

Critical Damping Coefficient (ccc_ccc​)

Damping Ratio (ζ\zetaζ)

System Behavior Based on Damping Ratio

Logarithmic Decrement

Logarithmic Decrement

Small Damping Approximation

Forced Vibrations

Forced Undamped Vibration

Magnification Factor (MF)

Resonance Caution

$Learning Objectives$

$Simple Harmonic Motion (SHM)$

$Amplitude$

$\omega_n$

$f_n$

$\tau$

$Analyzing Free Undamped Vibrations$

$Interactive Simulation$

$Interactive Simulation$

$Torsional Vibration Equation$

$c_c$

$\zeta$

$System Behavior Based on Damping Ratio$

$Small Damping Approximation$

$Resonance Caution$