Basic Oscillations Flashcards
What is the equation of natural frequency?
sqrt(keff/meff)
What is the general solution for oscillating displacement x(t)?
X0sin(ωnt + Φ)
What is the root mean square displacement?
Sqrt(1/T * int(x^2)dt)
What is the equation of critical damping?
sqrt(4meffkeff)
What is the equation of damping ratio?
Ceff/Ccrit
Under what damping condition does oscillation occur?
ξ < 1 only
What is the general solution for underdamped oscillation displacement?
e^-ξωnt * X0 sin(ωdt + Φ)
How is natural frequency converted to damped natural frequency?
Multiply by sqrt(1-ξ^2)
Which part of the damped general solution is excluded when overdamped?
The oscillating (sin or cos) component
How is the stiffness of springs in parallel calculated? How about in series?
Parallel: Kt = K1 + K2
Series: 1/Kt = 1/K1 + 1/K2
What is the 3 step process of finding the steady state motion caused by a linear base excitation pattern?
1) Write the Fourier series expansion of the base excitation
2) For each term calculate the amplitude and phase of resulting motion
3) Apply superposition to sum
What is the angle ϕ in the generic solution also known as?
The phase of vibration
If the natural frequency of a system is ωn, what is the frequency of kinetic energy?
2ωn
What is the equation of logarithmic decrement?
δ = 2πξ/sqrt(1−ξ^2)
ξ and Ceff both have to do with damping. Which is the damping ratio and which is the damping coefficient?
ξ is damping ratio
Ceff is damping coefficient
How is natural frequency of a pendulum calculated?
sqrt(mgl/ml^2) or sqrt(g/l)
What is the generic solution for angular displacement of a pendulum?
Ae^(−ξωnt) * sin(ωdt)
How can you convert a frequency in Hz to rad/s?
Multiply by 2π
What does the statement “Assume the effect of damping
negligible away from resonance” mean?
ξ=0 when not at resonance
When a sinusoidal vibration is such that amplitude is half the difference between the min and max values, what is true?
The factor Y = 1
Describe the difference between transient and steady state solutions
Transient is the instantaneous response to an input
Steady state is consistent regardless of inputs
