Harmonic Motion Flashcards
What is a waveform?
- Plot of change in amplitude of displacement (x) over time
- Representation of changes in air pressure and density
- Units: pressure (uPa), voltage (V), meters (m)
- Waveform of sine tones repeats over time (periodic)
What is simple harmonic motion?
- EX: spring-mass system
- Characterized as projected in a circular motion where the circle = displacement waveform
- Compare: changes from 45 to 90 degrees vs. 0 to 90 degrees
- Same # of degrees
- Different magnitude of linear displacement above baseline - 0, 180, and 360 degrees correspond to equilibrium
- Starting phase vs. instantaneous phase
Describe the different amplitude measurements. (6)
- Instantaneous amplitude (a)
- Peak amplitude (A)
- Peak-to-peak amplitude
- Root Mean Square (RMS) amplitude
- Full Wave Rectified Average Amplitude
- FWavg = 2 * peak amplitude / pi - Half Wave Rectified Average Amplitude
- HWavg = peak amplitude / pi
What are the effects of friction on vibratory motion?
- Limits velocity
- Amplitude of vibration diminishes over time (damping)
- In SHM, damping varies sinusoidally over time (in phase with velocity)
- As velocity increases, kinetic energy is transferred to thermal energy (system is damped) - Magnitude of displacement depends on force applied
- Duration of vibration depends on magnitude of damping re: force applied
Describe the damping factor.
- Ratio of amplitudes of any 2 consecutive cycles is a constant
- If you have less than 2 cycles (critical damping)
- System returned to equilibrium as quickly as possible when oscillating again
Describe the two types of sound resonators.
-Free Vibration: once energy is imparted to a body with low damping factor, the body vibrates freely
- Forced Vibration: system forced to vibrate by some external object
- Most real-world acoustic situations are forced vibrations
What is Diffraction?
- Bending of sound by objects in a sound field
- Refraction: sound passes into new medium
- Reflection: sound reflected back away from object
What is the Inverse Square Law?
- For every doubling of distance from the sound source in a free field situation, the sound intensity will diminish by 6dB
- Damping of sound with distance
- Pressure = force/area
- ->I = k * p / (4 * pi * r^2)
What is the Principle of Superposition?
-For all linear systems, the net response at a given place and time caused by 2+ stimuli is the sum of response which would have been caused by each stimulus individually
What is Interference?
- Two types: Constructive and Destructive
- Relevant to standing waves
- Shows effects of amplitude, phase, and frequency
What are Standing Waves?
-Special interference pattern of 2 sounds with the same frequency traveling in opposite directions
What are Beats?
-Combination of sound waves of different frequencies
x(t)= A1sin(2 * pi * f1 * t + phi1) + A2 * sin(2 * pi * f2 t +phi2)
- Caused by interaction of destructive and constructive interference
- Beat frequency: the difference between the 2 frequencies
- To get nodes to go down to zero, the 2 frequencies must have equal amplitudes
- Only heard with small frequency differences (<20-30 Hz)
What is Impedance?
- The total opposition (including R and X) in a circuit to passing alternating current
- Forces exist that oppose or impede motion
- System engages in SHM: vibrates freely at its natural frequency –> Fnat = sqrt(K/M)
What is Resistance (R)?
- Friction/frictional resistance
- Kinetic energy is transformed into thermal energy
- Measured in ohms
- Independent of frequency
- Energy is dissipated
What is Reactance (X)?
- Forces that oppose motion in a frequency-selective way
- Energy is stored as potential energy
- Measured in ohms
- Depends on mass (m) and compliance (1/k), both of which oppose motion but in opposite ways
What is Mass Reactance (Xm)?
- Dominates the high frequencies
- Negligible at low frequencies
- Smaller amplitude of vibration
- Inversely proportional to frequency
What is Compliance Reactance (Xc)?
- Dominates low frequencies
- Negligible at high frequencies
- Larger amplitude of vibration
- Directly proportional to frequency
What happens if F > Fnat?
- Z increases
- Amplitude of vibrations decreases
- Mass dominant (Xm = 2 * pi * Fm)
What happens if F < Fnat?
- Z increases
- Amplitude of vibrations decreases
- Compliance dominant [Xc = 1 / (2 * pi * Fc)]
Describe the phase relationship between Xm, Xc, and R.
- Xm leads R by 90 degrees
- Xc lags R by 90 degrees
- Xm leads Xc by 180 degrees
Describe the crucial phase relations. (4)
- Opposition to motion from resistance is in phase with velocity
- Resistance: in phase with C, M, and damping - Opposition to motion from compliance is in phase with elasticity (lags R by 90 degrees)
- Compliance: in phase with E and X - Opposition to motion from mass is in phase with acceleration (leads R by 90 degrees)
- Opposition to motion from mass is 180 degrees out of phase with opposition to motion from compliance
What is Resonant Frequency (Fn)?
- All things have a resonance (characteristic) frequency that they want to resonate at
- Depends on stiffness (k) and mass (m) of the object
- Resonator vibrates most at its resonant frequency, not at the frequency of the driving force
- -> Fn = [sqrt(k/m) / (2 *pi)]
What is Immittance?
-Combined concept of admittance and impedance
What is Admittance?
- Forces that allow motion (opposite of impedance)
- Conductance (G)
- Susceptance (B)
- Mass Susceptance (Bm)
- Compliance Susceptance (Bc)
Describe impedance example of the middle ear.
- Resistance: viscosity of cochlear fluids that are moving because of stapes
- Stiffness/Compliance Reactance: elastic properties of muscles and ligaments connecting the TM to the ossicles and the ossicles to each other
- Mass reactance (inertance): mass of the ossicles
Describe impedance effects of middle ear pathologies.
- Negative pressure pathologies (Eustachian tube dysfunction): increased stiffness –> increased Fn
- Otosclerosis: increased stiffness at first, maybe some increased mass later on
- Cholesteatoma: increased mass –> decreased Fn
- Middle ear effusion: increased mass and increased stiffness
