Single Degree of Freedom Models

Question 1

Harmonic Sinusoidal Motion Equation

Answer 1

x = a*sin(ωt+Φ)

x = acos(ωt) + jsin(ωt)
= a*e^(jωt)

where:
a = x(0)

Question 2

Basic Equation for SDOF models

Answer 2

x = Ce^(st)

S[1,2] = √(k/m) = ωn

Question 3

Natural Frequency of a System

Answer 3

ωn = √(k/m)

Question 4

Free Undamped Vibration of SDOF system

Answer 4

x(t) = x(0)cos(ωnt) + (x(0)/ωn) sin(ωnt)

Question 5

Damping Ratio Equation

Answer 5

C/Cc = C/(2*√(km))

Question 6

Damping Ratio of a system

Answer 6

ζ > 1 = Overdamped
ζ = 1 Critically damped
ζ < 1 = Underdamped (oscillatory)

Question 7

Critical Damping Coefficient

Answer 7

Cc = 2*√(km)

Question 8

S[1,2] of Damped Free Vibrations

Answer 8

S[1,2] = ωn(-ζ+-√(ζ²-1))

Question 9

Damped Time Period

Answer 9

Td = 2π/ωd

Question 10

Damped Natural Frequency

Answer 10

ωd = ωn*√(1-ζ²)

Question 11

Logarithmic Decrement Equation

Answer 11

Xₒ/Xₙ = e^(nδ)

δ = (1/n) * ln(xₒ/xₙ)

Question 12

Logarithmic Decrement for ζ < 0.2

Answer 12

δ = 2πζ

Question 13

Max Velocity from Max Displacement

Answer 13

ẋ = ω * abs(x[max])

Question 14

Max Acceleration from Max Displacement

Answer 14

ẍ = ω² * abs(x[max])