Mechanical Vibrations

x(t) = A cos(ωt + φ)
v(t) = -Aω sin(ωt + φ)
a(t) = -Aω² cos(ωt + φ)

Where:
x(t) = displacement (m)
v(t) = velocity (m/s)
a(t) = acceleration (m/s²)
A = amplitude (m)
ω = angular frequency (rad/s)
φ = phase angle (rad)
t = time (s)

f = ω/(2π) = 1/T
ω = 2πf = 2π/T

Where:
f = frequency (Hz or cycles/s)
ω = angular frequency (rad/s)
T = period (s)

x_rms = A/√2 = 0.707A
x_peak = A
x_peak-to-peak = 2A

RMS (Root Mean Square) values are commonly used for vibration measurements and analysis.

Displacement: mm, μm, mils (1 mil = 25.4 μm)
Velocity: mm/s, in/s (1 in/s = 25.4 mm/s)
Acceleration: m/s², g (1 g = 9.81 m/s²)
Frequency: Hz, CPM (1 Hz = 60 CPM)

m(d²x/dt²) + c(dx/dt) + kx = F(t)
mẍ + cẋ + kx = F(t)

Where:
m = mass (kg)
c = damping coefficient (N·s/m)
k = spring stiffness (N/m)
x = displacement (m)
F(t) = external force (N)

ω_n = √(k/m)
f_n = ω_n/(2π) = (1/2π)√(k/m)
T_n = 1/f_n = 2π√(m/k)

Where:
ω_n = natural angular frequency (rad/s)
f_n = natural frequency (Hz)
T_n = natural period (s)

ζ = c/(2√(km)) = c/c_c
c_c = 2√(km) = 2mω_n

Where:
ζ = damping ratio (dimensionless)
c_c = critical damping coefficient (N·s/m)

Underdamped (ζ < 1): x(t) = Ae^(-ζω_n t) cos(ω_d t + φ)
ω_d = ω_n√(1 - ζ²)
Critically damped (ζ = 1): x(t) = (A + Bt)e^(-ω_n t)
Overdamped (ζ >1): x(t) = Ae^(r₁t) + Be^(r₂t)

Where:
ω_d = damped natural frequency (rad/s)
r₁,r₂ = roots of characteristic equation

δ = ln(x_n/x_(n+1)) = 2πζ/√(1 - ζ²)
ζ = δ/√(4π² + δ²)

Where:
δ = logarithmic decrement
x_n, x_(n+1) = successive peak amplitudes

F(t) = F₀ cos(ωt)
x(t) = X cos(ωt - φ)
X = F₀/k × M(r,ζ)

Where:
F₀ = force amplitude (N)
X = displacement amplitude (m)
ω = excitation frequency (rad/s)
φ = phase lag (rad)
M(r,ζ) = magnification factor

M = 1/√[(1-r²)² + (2ζr)²]
r = ω/ω_n
φ = tan⁻¹[2ζr/(1-r²)]

Where:
M = magnification factor
r = frequency ratio

At resonance (r = 1): M_max = 1/(2ζ)
Peak occurs at: r_peak = √(1 - 2ζ²)
M_peak = 1/(2ζ√(1 - ζ²))

Resonance amplification can be extremely large for lightly damped systems (small ζ).

Q = 1/(2ζ) = ω_n/(2Δω)
Δω = ω₂ - ω₁ (half-power bandwidth)

Where:
Q = quality factor
Δω = bandwidth at -3 dB points
ω₁, ω₂ = frequencies at M/√2

TR = F_transmitted/F_applied = √[(1 + (2ζr)²)/((1-r²)² + (2ζr)²)]
For r > √2: TR < 1 (isolation)

Transmissibility describes the ratio of transmitted force to applied force in vibration isolation.

[M]ẍ + [C]ẋ + [K]x = F(t)
For free vibration: [M]ẍ + [K]x = 0

Where:
[M] = mass matrix
[C] = damping matrix
[K] = stiffness matrix
x = displacement vector
F(t) = force vector

([[K]]-ω²[[M]]){φ} = {0}
det([[K]]-ω²[[M]]) = 0

Where:
ω² = eigenvalues (natural frequencies squared)
{φ} = eigenvectors (mode shapes)

{x(t) = Σ φᵢ qᵢ(t)
q̈ᵢ + 2ζᵢωᵢq̇ᵢ + ωᵢ²qᵢ = Qᵢ(t)/Mᵢ

Where:
qᵢ(t) = modal coordinates
φᵢ = mode shape vectors
Qᵢ(t) = modal forces
Mᵢ = modal masses

φᵢᵀ[M]φⱼ = 0 (i ≠ j)
φᵢᵀ[K]φⱼ = 0 (i ≠ j)
φᵢᵀ[M]φᵢ = Mᵢ, φᵢᵀ[K]φᵢ = Kᵢ

Orthogonality conditions allow modal decoupling and independent analysis of each mode.

Γᵢ = φᵢᵀ[M]{r}/Mᵢ
Effective modal mass = Γᵢ²Mᵢ

Where:
Γᵢ = modal participation factor for mode i
{r} = influence vector (direction of excitation)

Displacement: Low frequency (0.1-10 Hz), large amplitude
Velocity: Medium frequency (10-1000 Hz), general purpose
Acceleration: High frequency (>1000 Hz), impact/shock
Relationship: a = ω²x, v = ωx (for sinusoidal motion)

X(f) = ∫ x(t)e^(-j2πft) dt (Fourier Transform)
PSD = |X(f)|²/T (Power Spectral Density)

Where:
X(f) = frequency domain representation
x(t) = time domain signal
T = measurement time

RMS = √(∫ x²(t) dt / T)
Peak = max|x(t)|
Crest Factor = Peak/RMS
Kurtosis = μ₄/σ⁴

Where:
μ₄ = fourth central moment
σ = standard deviation
Normal kurtosis = 3, higher values indicate impulsive content

H(ω) = X(ω)/F(ω)
|H(ω)| = magnitude, ∠H(ω) = phase
Coherence = |G_xy|²/(G_xx × G_yy)

Where:
H(ω) = frequency response function
G_xy = cross-power spectral density
G_xx, G_yy = auto-power spectral densities

Isolation efficiency = 1 - TR
For f/f_n >√2: Isolation occurs
f_n = (1/2π)√(k/m)

Where:
TR = transmissibility
f = excitation frequency
f_n = natural frequency of isolator

ω_a = ω_n (tuned absorber)
μ = m_a/m (mass ratio)
ζ_opt = √(3μ/8(1+μ))

Where:
ω_a = absorber natural frequency
m_a = absorber mass
m = primary system mass
ζ_opt = optimal damping ratio

F_control = -G(s) × x_measured
G(s) = controller transfer function

Active control uses sensors, controllers, and actuators to generate forces that cancel vibrations.

Free layer damping: Viscoelastic material bonded to structure
Constrained layer damping: Viscoelastic layer between two stiff layers
Tuned mass dampers: Secondary mass-spring-damper systems
Friction dampers: Energy dissipation through sliding friction