SDOF Flashcards
What is the equivalent stiffness of springs in a) Parallel b) Series
K1, K2, K3
a) Sum all the springs in parallel
K1 + K2 + K3
b) 1 over the sum of one over the stiffnesses
1 / (1/K1 + 1/K2 + 1/K3)
what is the stiffness of this system?
K1 K2
wwww[ ]wwww
Springs are in parallel so the equivalent stiffness is K1 + K2
What are the equations for stiffness for a rigid and pinned connection?
Rigid = 12EI/L^3
Pinned = 3EI/L^3
How Do you convert from w to fn?
w = 2pi * fn
How do masses relate forces to acceleration?
Newtons 2nd law relates it via mass acting as a coefficient of proportionality
Inertia forces ‘resisting’ acceleration is developed and acts in opposite direction.
What are the Units of K, c and m
K - N/m
c - N/(m/s)
m - kg or N/(m/s^2)
What assumptions are made about the mass of a damped spring system?
1) Springs and dampers have no mass
2) Mass are assumed to behave like rigid bodies and do not deform
What is the dynamic force equilibrium equation?
Inertia Force + Damping force + Spring force = External Force
fl(t) + fc(t) + fk(t) = f(t)
What is the equation of motion of free undamped vibration?
and what is the assumed solution in terms of x(t), x.(t) and x..(t)?
mx..(t) + kx(t) = 0
x..(t) = accelleration
x(t) = Displacement
x(t) = Asin(wnt + Ø)
x.(t) = wnAcos(wnt + Ø)
x..(t) = -wn^2Asin(wn*t + Ø)
wn = angular frequency
Ø = Phase angle of vibration
What is the equation for angular frequency (wn) and what are the units?
wn = √(K/m)
K [N/m]
m [kg]
wn [rad/s]
What is the total energy in undamped free vibration?
E total = Potential Energy (U) + Kinetic Energy (T)
E = 1/2 * K * x(t)^2 + 1/2 * m * x.(t)^2
What is the damped equation of motion?
mx..(t) + cx.(t) + kx(t) = 0
m = mass [kg]
c = damping coefficeitn [Ns/m] or [kg/s]
K = N/m
What are the equations for the critical damping coefficient and damping ratio?
Ccr = 2 * m * wn = 2√(k*m)
ζ = c / Ccr = c / (2 * m * wn) = c / (√(k*m))
How do you convert between angular frequency and damped angular frequency?
wd = wn*√(1 - ζ^2) where wn = √K/m
What is the solution to underdamped free motion?
x(t) = A * e^(-§ * wn * t)*sin(wd * t + Ø)
Where A = formula sheet free response
Ø = formula sheet free response
How do you use logarithmic decrement to calculate the damping ratio?
Ln(Un/Un+1) = 2π*ζ
Ln(Un/Um) = (n-m)2πζ
Un = Magnitude of the nth peak
Un+1 = Magnitude of the n+1th peak
ζ = damping ratio
What is the difference between underdamped, critically damped and overdamped systems
underdamped - § < 1 and C^2 < 4mk. Oscillates with the peaks getting smaller and smaller.
Critically damped - § = 1 and C^2 = 4mk do not oscillate and reach equilibrium faster than any overdamped system. Marks the transition between vibration and non vibration
Overdamped - § > 1 and C^2 > 4mk. Does not oscillate it reaches equilibrium once pushed.
What makes a system stable and which types are stable?
What makes a system unstable?
A system is stable if it vibrates so that its amplitude ιs always less than some finite number for all choices of initial conditions. Normal Damped and undamped systems are stable.
For a system to become unstable, c is negative and m and K are both positive so the motion grows without bound and becomes unstable. in the damped case, it may either grow without bound and not oscillate or it may oscillate as well.
Why is harmonic excitation important?
- Periodic forces van be presented as a sum of harmonics
- As responses to individual harmonics are known and superposition applies, the total response can easily be calculated.
When does beating occur?
When the natural frequence and excitation frequency are close, (wn - w os small)
What is the failure envelope for an undamped system at resonance under harmonic loading
Envelope = fo / (2*wn) * t
fo = Fo / m [N/kg]
Fo = external load
m = mass
What is the difference between the transient and steady-state response of an underdamped system?
Transient - Governed by system properties and usually dies away due to damping
Steady-state - Governed by properties and excitation. Doesn’t die away and is usually the most important part.
Was is the solution for resonant excitation for an underdamped system?
X(t) ≈ Fo/k * 1/(2§) * (e^(-§wnt) - 1) * sin(wn*t)
What is the steady state amplitude at resonance for an underdamped system?
What are the change in static (Δstatic) and dynamic (Δdynamic) responses to an underdamped system?
1/(2ζ)
∆static = Fo/k
∆dynamic = Fo/k * 1/(2ζ)