Chap 16: Basics of Vibration (alot of eqns)

Question 1

parameters for lump mass model

Answer 1

Mass, m: include any component contributing to the total mass of system
spring constant, k: stiffness of components that resists deformation in response to applied force
damping coefficient, c: all energy dissipating mechanisms within the system

Question 2

basic equation for single degree of freedom (x axis only) and its forces

Answer 2

Forces: P, spring force (kx), damping force (cẋ)

Sum of Fx = m*ẍ

basic eqn: mẍ + cẋ + kx = P

Question 3

free vibration equation stuff

Answer 3

when P = 0, free vibration equation is:
mẍ + cẋ + kx = 0

theoretically, total energy remains the same over time, amplitude of vibration does not change so, damping = 0

reality, energy will dissipate over time and amplitude will reduce to zero so, damping ≠ 0

Question 4

angular frequency definition

Answer 4

an angular frequency value at which an object vibrates when subject to initial force and absence of driving force

can be modified by changing mass (m) or stiffness (k)

Question 5

undamped free vibration equation

Answer 5

mẍ + cẋ + kx = 0

c = 0, therefore

mẍ + kx = 0, divide by m and substitube wn

ẍ + wn^2 x = 0

Question 6

forced vibration equation

Answer 6

P =/= 0

mẍ + cẋ + kx = P

P = F0sinωt

P = bsinwt, b = F0 / k