SHM Flashcards
What is the equation for Hooke’s Law
F = -k x
F = Force
k = Spring constant
x = extension
What is the equation for displacement of an undamped oscillator?
x = A sin(ωt+φ)
x = extension
A = Amplitude
ω = Angular Frequency
t = time
φ = Phase offset
What is ω equal to?
ω = (k/m)^0.5 = 2πf = 2π/T
ω = Angular Frequancy
k = spring constant
m = mass
f = frequency
T = Period
What is the equation of Velocity of an undamped oscillator?
v = Aω cos(ωt+φ)
v = Velocity
A = Amplitude
φ = Phase Offset
ω = Angular frequency
t = time
What is the equation of acceleration of an undamped oscillator?
a = A ω² sin(ωt+φ)
a = acceleration
A = Amplitude
φ = Phase Offset
ω = Angular frequency
t = time
What is the equation for total energy for SHM?
E = 1/2 mv² = 1/2 k x²
E = energy
m = Mass
v = max velocity
k = spring constant
x = max extension
What is the equation for the Damping Force?
F(d) = -b v
F(d) = damping force
b = damping constant
v = velocity
What is the Equation of Motion for Damped SHM?
ma + bv + kx = 0
m = mass
a = acceleration
b = damping constant
v = velocity
k = spring constant
x = extenstion
How to calculate the type of damping?
b²/(4m²) - k/m = number
number > 0 Heavy damping
number = 0 critical damping
number < 0 light damping
What is the equation for logarithmic decrement?
δ = b/(2m) T’
δ = ratio of amplitudes over a period
b = damping constant
m = mass
T’ = Period
What is the equation of motion for light damping?
x = Ce ^(-(bt)/(2m)) e^(± i ω’ t)
x = extension
b = spring constant
t = time
m = mass
i = (-1)^0.5
ω’ = dampened angular frequeancy
What is the equation for dampened angular frequency?
ω’ = (k/b - b²/(4m²))^0.5
ω’ = dampened angular frequency
k = spring constant
b = damping constant
m = mass
What is the definition of Mechanical Impedance?
The force required to produce unit velocity.
What is the equation for mechanical impedance?
Z(m) = F/v = b + i(ωm-k/ω)
Z(m) = mechanical impedance
F = Force
v = velocity
i = (-1)^0.5
ω = angular frequency
m = mass
k = spring constant
What is the Polar representation of mechanical imepance?
Z(m) hat = |Z(m)| e^(iφ)
Z(m) hat = unit of mechanical impedance
Z(m) = mechanical impedance
i = (-1)^0.5
φ = arctan((ωm-k/ω)/b) Difference in angle between real and imaginary
What is the equation of motion for a Forced oscillator?
ma + bv + kx = F(0) e ^(iωt)
m = Mass
a = acceleration
b = damping constant
v = velocity
k = spring constant
x = extension
F(0) = Driving Force
i = (-1)^0.5
ω = angular frequency
t = time
What is the velocity of a Forced Oscillator?
v = -(i F(0) e^i(ωt-φ))/ω|Z(m)|
v = velocity
i = (-1)^0.5
F(0) = Driving Force
ω = Angular frequency
t = time
φ = arctan((ωm-k/ω)/b) Difference in angle between real and imaginary.
Z(m) = Mechanical impedance
What controls force oscillatory motion at low frequencies?
Spring constant
What controls force oscillatory motion at high frequencies?
mass
What is the equation for instananeous power supplied by driving force?
P = F(0)²/|Z(m)| cos(ωt) cos(ωt-φ)
P = Power
F(0) = driving force
Z(m) = mechanical impedance
φ = phase diference of real ind imaginary
What is the equation for instananeous power supplied by driving force?
P = F(0)²/|Z(m)| cos(ωt) cos(ωt-φ)
P = Power
F(0) = driving force
Z(m) = mechanical impedance
φ = phase diference of real ind imaginary
ω = angular frequency
What is the equation for average power supplied by the driving force?
P(av) = (b F(0)²)/(2|Z(m)|²)
P(av) = average power from driving force
F(0) = driving force
Z(m) = mechanical impedance
What is a Q-Value?
It is a indicator of how “sharp” the resonance is.
What is the equation for Q-Value?
Q = ω(0)/( ω(2) - ω(1)) = ω(0) m / b
ω(0) = The resonant frequency
ω(1) = the freqeuncy that is transfers 1/(2)^0.5 as much power as the resonant frquency
ω(2) = the freqeuncy that is transfers 1/(2)^0.5 as much power as the resonant frquency
m = mass
b = damping constant
What is the resonant bandwidth?
ω(2) - ω(1)
The total number of cycles below and above the resonant frequency for which the current is equal to or greater than 70.7% of its resonant value
