SDOF

Question 1

What is the equivalent stiffness of springs in a) Parallel b) Series

K1, K2, K3

Answer 1

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)

Question 2

what is the stiffness of this system?
K1 K2
wwww[ ]wwww

Answer 2

Springs are in parallel so the equivalent stiffness is K1 + K2

Question 3

What are the equations for stiffness for a rigid and pinned connection?

Answer 3

Rigid = 12EI/L^3

Pinned = 3EI/L^3

Question 4

How Do you convert from w to fn?

Answer 4

w = 2pi * fn

Question 5

How do masses relate forces to acceleration?

Answer 5

Newtons 2nd law relates it via mass acting as a coefficient of proportionality

Inertia forces ‘resisting’ acceleration is developed and acts in opposite direction.

Question 6

What are the Units of K, c and m

Answer 6

K - N/m
c - N/(m/s)
m - kg or N/(m/s^2)

Question 7

What assumptions are made about the mass of a damped spring system?

Answer 7

1) Springs and dampers have no mass
2) Mass are assumed to behave like rigid bodies and do not deform

Question 8

What is the dynamic force equilibrium equation?

Answer 8

Inertia Force + Damping force + Spring force = External Force

fl(t) + fc(t) + fk(t) = f(t)

Question 9

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)?

Answer 9

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

Question 10

What is the equation for angular frequency (wn) and what are the units?

Answer 10

wn = √(K/m)

K [N/m]
m [kg]
wn [rad/s]

Question 11

What is the total energy in undamped free vibration?

Answer 11

E total = Potential Energy (U) + Kinetic Energy (T)

E = 1/2 * K * x(t)^2 + 1/2 * m * x.(t)^2

Question 12

What is the damped equation of motion?

Answer 12

mx..(t) + cx.(t) + kx(t) = 0
m = mass [kg]
c = damping coefficeitn [Ns/m] or [kg/s]
K = N/m

Question 13

What are the equations for the critical damping coefficient and damping ratio?

Answer 13

Ccr = 2 * m * wn = 2√(k*m)

ζ = c / Ccr = c / (2 * m * wn) = c / (√(k*m))

Question 14

How do you convert between angular frequency and damped angular frequency?

Answer 14

wd = wn*√(1 - ζ^2) where wn = √K/m

Question 15

What is the solution to underdamped free motion?

Answer 15

x(t) = A * e^(-§ * wn * t)*sin(wd * t + Ø)

Where A = formula sheet free response
Ø = formula sheet free response

Question 16

How do you use logarithmic decrement to calculate the damping ratio?

Answer 16

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

Question 17

What is the difference between underdamped, critically damped and overdamped systems

Answer 17

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.

Question 18

What makes a system stable and which types are stable?

What makes a system unstable?

Answer 18

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.

Question 19

Why is harmonic excitation important?

Answer 19

• 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.

Question 20

When does beating occur?

Answer 20

When the natural frequence and excitation frequency are close, (wn - w os small)

Question 21

What is the failure envelope for an undamped system at resonance under harmonic loading

Answer 21

Envelope = fo / (2*wn) * t

fo = Fo / m [N/kg]
Fo = external load
m = mass

Question 22

What is the difference between the transient and steady-state response of an underdamped system?

Answer 22

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.

Question 23

Was is the solution for resonant excitation for an underdamped system?

Answer 23

X(t) ≈ Fo/k * 1/(2§) * (e^(-§wnt) - 1) * sin(wn*t)

Question 24

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?

Answer 24

1/(2ζ)

∆static = Fo/k
∆dynamic = Fo/k * 1/(2ζ)

Question 25

What is the equation for steady-state response of a damped system?

Answer 25

Xss = Xcos(wt - θ)
X = dynamic displacement (on formula sheet)
w = exitation frequency
θ = phase angle (formula sheet)

Question 26

What controls the response for harmonic excitation for low, high frequencies and frequencies near resonance?

Answer 26

Low Frequences - Controlled by stiffness

High Frequences - Controlled by mass

Near Resonance - Controlled by damping

Question 27

Describe the phase of force and displacement response well before, after and at resonance,

Answer 27

Well befire - they are both in phase

Well after - they are approx 180º out of phase

At resonance - the are 90º out of phase

Question 28

what is the displacement response at resonance for an underdamped system under harmonic loading?

Answer 28

X = Fo/k * 1/(2ζ) = Xpeak

Question 29

Describe what is meant by an impulse force

Answer 29

the very short duration of loading and can ignore system behaviour during the impulse. impulse is a product of force and time

Question 30

If Pa is used hat are the units for force and distance?

Answer 30

Pa
N
m
kg

Question 31

What is the amplitude envelop for a underdamped system?

Answer 31

Ae^(-Zwn*t)

Question 32

How do you calculate maximum horizontal force using acceleration?

Answer 32

F = m * a * DMF
m = mass
a = max acceleration
DMF = Dynamic magnification force

Question 33

What is the Duhamel Integral used for?

Answer 33

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.

Question 34

When harmonic load is applied what does the frequency of the transient and steady state response show us?

Answer 34

Transient shows the the natural frequency of the system

Steady state shows the frequency of the load.

Question 35

What is the dynamic magnification factor for an undamped and underdamped system at resonance?

Answer 35

Undamped:

r = 1

Underdamped:

r = sqrt(1-2Z^2)

Question 36

By using Duhamel’s Integral, calculate the max dynamic response of a
SDOF system when constant load per unit time (undamped).

Answer 36

• 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).