SHM

Question 1

What is the equation for Hooke’s Law

Answer 1

F = -k x
F = Force
k = Spring constant
x = extension

Question 2

What is the equation for displacement of an undamped oscillator?

Answer 2

x = A sin(ωt+φ)
x = extension
A = Amplitude
ω = Angular Frequency
t = time
φ = Phase offset

Question 3

What is ω equal to?

Answer 3

ω = (k/m)^0.5 = 2πf = 2π/T
ω = Angular Frequancy
k = spring constant
m = mass
f = frequency
T = Period

Question 4

What is the equation of Velocity of an undamped oscillator?

Answer 4

v = Aω cos(ωt+φ)
v = Velocity
A = Amplitude
φ = Phase Offset
ω = Angular frequency
t = time

Question 5

What is the equation of acceleration of an undamped oscillator?

Answer 5

a = A ω² sin(ωt+φ)
a = acceleration
A = Amplitude
φ = Phase Offset
ω = Angular frequency
t = time

Question 6

What is the equation for total energy for SHM?

Answer 6

E = 1/2 mv² = 1/2 k x²
E = energy
m = Mass
v = max velocity
k = spring constant
x = max extension

Question 7

What is the equation for the Damping Force?

Answer 7

F(d) = -b v
F(d) = damping force
b = damping constant
v = velocity

Question 8

What is the Equation of Motion for Damped SHM?

Answer 8

ma + bv + kx = 0
m = mass
a = acceleration
b = damping constant
v = velocity
k = spring constant
x = extenstion

Question 9

How to calculate the type of damping?

Answer 9

b²/(4m²) - k/m = number
number > 0 Heavy damping
number = 0 critical damping
number < 0 light damping

Question 10

What is the equation for logarithmic decrement?

Answer 10

δ = b/(2m) T’
δ = ratio of amplitudes over a period
b = damping constant
m = mass
T’ = Period

Question 11

What is the equation of motion for light damping?

Answer 11

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

Question 12

What is the equation for dampened angular frequency?

Answer 12

ω’ = (k/b - b²/(4m²))^0.5
ω’ = dampened angular frequency
k = spring constant
b = damping constant
m = mass

Question 13

What is the definition of Mechanical Impedance?

Answer 13

The force required to produce unit velocity.

Question 14

What is the equation for mechanical impedance?

Answer 14

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

Question 15

What is the Polar representation of mechanical imepance?

Answer 15

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

Question 16

What is the equation of motion for a Forced oscillator?

Answer 16

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

Question 17

What is the velocity of a Forced Oscillator?

Answer 17

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

Question 18

What controls force oscillatory motion at low frequencies?

Answer 18

Spring constant

Question 19

What controls force oscillatory motion at high frequencies?

Answer 19

mass

Question 20

What is the equation for instananeous power supplied by driving force?

Answer 20

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

Question 21

What is the equation for instananeous power supplied by driving force?

Answer 21

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

Question 22

What is the equation for average power supplied by the driving force?

Answer 22

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

Question 23

What is a Q-Value?

Answer 23

It is a indicator of how “sharp” the resonance is.

Question 24

What is the equation for Q-Value?

Answer 24

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

Question 25

What is the resonant bandwidth?

Answer 25

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