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)
What is the equation for steady-state response of a damped system?
Xss = Xcos(wt - θ)
X = dynamic displacement (on formula sheet)
w = exitation frequency
θ = phase angle (formula sheet)
What controls the response for harmonic excitation for low, high frequencies and frequencies near resonance?
Low Frequences - Controlled by stiffness
High Frequences - Controlled by mass
Near Resonance - Controlled by damping
Describe the phase of force and displacement response well before, after and at resonance,
Well befire - they are both in phase
Well after - they are approx 180º out of phase
At resonance - the are 90º out of phase
what is the displacement response at resonance for an underdamped system under harmonic loading?
X = Fo/k * 1/(2ζ) = Xpeak
Describe what is meant by an impulse force
the very short duration of loading and can ignore system behaviour during the impulse. impulse is a product of force and time
If Pa is used hat are the units for force and distance?
Pa
N
m
kg
What is the amplitude envelop for a underdamped system?
Ae^(-Zwn*t)
How do you calculate maximum horizontal force using acceleration?
F = m * a * DMF
m = mass
a = max acceleration
DMF = Dynamic magnification force
What is the Duhamel Integral used for?
The Duhamel integral is used to calculate the response of a single degree of freedom (SDOF) system to a time-varying load, given the system’s response to a unit impulse load. It allows the superposition of the response due to the impulse load and the response due to the time-varying load and is useful for solving problems involving dynamic loading, such as earthquake and wind engineering.
When harmonic load is applied what does the frequency of the transient and steady state response show us?
Transient shows the the natural frequency of the system
Steady state shows the frequency of the load.
What is the dynamic magnification factor for an undamped and underdamped system at resonance?
Undamped:
r = 1
Underdamped:
r = sqrt(1-2Z^2)
By using Duhamel’s Integral, calculate the max dynamic response of a
SDOF system when constant load per unit time (undamped).
- no damping so dmaping terms = 0
- Take p(t) out of the equation as it is constant
-intergrate sin(wn(t - T)) between t and 0
- This equals p/(mwn^2) * ( 1 - cos(wnt) which has a max value of 2p/(m*wn^2).
