Basic Oscillations

Question 1

What is the equation of natural frequency?

Answer 1

sqrt(keff/meff)

Question 2

What is the general solution for oscillating displacement x(t)?

Answer 2

X0sin(ωnt + Φ)

Question 3

What is the root mean square displacement?

Answer 3

Sqrt(1/T * int(x^2)dt)

Question 4

What is the equation of critical damping?

Answer 4

sqrt(4meffkeff)

Question 5

What is the equation of damping ratio?

Answer 5

Ceff/Ccrit

Question 6

Under what damping condition does oscillation occur?

Answer 6

ξ < 1 only

Question 7

What is the general solution for underdamped oscillation displacement?

Answer 7

e^-ξωnt * X0 sin(ωdt + Φ)

Question 8

How is natural frequency converted to damped natural frequency?

Answer 8

Multiply by sqrt(1-ξ^2)

Question 9

Which part of the damped general solution is excluded when overdamped?

Answer 9

The oscillating (sin or cos) component

Question 10

How is the stiffness of springs in parallel calculated? How about in series?

Answer 10

Parallel: Kt = K1 + K2
Series: 1/Kt = 1/K1 + 1/K2

Question 11

What is the 3 step process of finding the steady state motion caused by a linear base excitation pattern?

Answer 11

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

Question 12

What is the angle ϕ in the generic solution also known as?

Answer 12

The phase of vibration

Question 13

If the natural frequency of a system is ωn, what is the frequency of kinetic energy?

Answer 13

2ωn

Question 14

What is the equation of logarithmic decrement?

Answer 14

δ = 2πξ/sqrt(1−ξ^2)

Question 15

ξ and Ceff both have to do with damping. Which is the damping ratio and which is the damping coefficient?

Answer 15

ξ is damping ratio
Ceff is damping coefficient

Question 16

How is natural frequency of a pendulum calculated?

Answer 16

sqrt(mgl/ml^2) or sqrt(g/l)

Question 17

What is the generic solution for angular displacement of a pendulum?

Answer 17

Ae^(−ξωnt) * sin(ωdt)

Question 18

How can you convert a frequency in Hz to rad/s?

Answer 18

Multiply by 2π

Question 19

What does the statement “Assume the effect of damping
negligible away from resonance” mean?

Answer 19

ξ=0 when not at resonance

Question 20

When a sinusoidal vibration is such that amplitude is half the difference between the min and max values, what is true?

Answer 20

The factor Y = 1

Question 21

Describe the difference between transient and steady state solutions

Answer 21

Transient is the instantaneous response to an input
Steady state is consistent regardless of inputs