Mechanical oscillations

Acceleration of elastic force
a = -kx/m

Elastic force
F = - kx

Equation of motion of mathematical pendulum
a = - gx / l

Equation of free oscillations
a = - ω^2 * x

Equation of motion of spring pendulum
ω^2 = k/m

Equation of motion of mathematical pendulum
ω^2 = g / l

Free oscillations: declination
x = x_m * cos(ω*t)

Frequency and period of oscillations
ν = 1 / T

Cyclic frequency of oscillations
ω = 2π / T

Cyclic frequency of oscillations
ω = 2π ν

Phase of harmonic oscillations
φ = ω * t

Phase of harmonic oscillations
φ =  2π t / T

Phase of harmonic oscillations
φ =  2π ν t

Harmonic oscillations: declination
x = x_m * cos (ω * t + φ)

Oscillation period of spring pendulum
T = 2 π * saknis(m/k)

Oscillation period of mathematical pendulum
T = 2 π * saknis(l/g)

Harmonic oscillations: body speed
v = v_m * cos (ω * t + π/2)

Harmonic oscillations: body speed
v = v_m * sin (ω * t)

Harmonic oscillations: body acceleration
a = a_m * cos (ω * t + π)

Harmonic oscillations: body acceleration
a = -ω^2 * x *cos(ω * t)

Harmonic oscillations: body speed
v = -ω * x *sin(ω * t)

Harmonic oscillations: body maximum speed
v_m = ω * x_m

Harmonic oscillations: body maximum acceleration
a_m = ω * v_m

Harmonic oscillations: body maximum acceleration
a_m = ω^2 * x_m

Harmonic oscillations: body kinetic energy
E_k = m v^2 / 2

Harmonic oscillations: body potential energy
E_p = k x^2 / 2

Harmonic oscillations: body total energy
E = E__k + E__p

Harmonic oscillations: body total energy
E = {m v^2 // 2} + {k x^2 // 2}

Resonance - oscillation amplitude
x = F / (ω*μ)