separate branches of mechanics and biomechanics i. periodic motion. mechanical waves. acoustics
TRANSCRIPT
Separate branches of Mechanics and Biomechanics
I. Periodic Motion. Mechanical waves. Acoustics.
§1 Simple Harmonic Motion
Periodic motion (or oscillation) is motion, that repeats itsef in a regular cycle. Oscillation can occur only when there is a restoring force tending to return the system being displaced from its eqilibrium to the latter.The examples of such systems are the spring – mass system, the simple pendulum (an idealized model), the real physical pendulum, the balance wheel of a mechanical watch, the vibrations of molecules.
The examples of oscillating systems.
Simple Harmonic Motion.
amF ; xx maF ;
kxF x ; 2
2
dt
xda x ;
2
2
dt
xdmkx 0
2
2
kxdt
xdm 0
2
2
xm
k
dt
xd
20
m
k ; 02
02
2
xdt
xd ; )cos( 00 tAx
Simple Harmonic Motion.
Displacement, velocity, and acceleration in simple harmonic motion
2
2kxE p ;
2
2mVEk
constmVkAmVkx
EEE mkp
2222
2222
§2 Energy in Simple Harmonic motion
§3 Damped Oscillations.Oscillations with little dampingamFF f ; xxfx maFF
xxfrVF
2
2
dt
xdm
dt
dxrkx 0
2
2
xm
k
dt
dx
m
r
dt
xd
2m
r ; 02 2
02
2
xdt
dx
dt
xd
If the damping force is relatively small, so that22
02 >0
)cos(0 tAex t
the solution of differential equation is
220
Damped Oscillations
TeTtA
tA T
ln)(
)(ln
§4 Forced Oscillations. Resonance
tFF dmd cos ; xxdxfx maFFF
kxdt
dxr
dt
xdmtF dm
2
2
cos
mm fm
F ; tfx
dt
dx
dt
xddm cos2 2
02
2
Forced Oscillations. Resonance
)cos( 0 tAx dd
Forced Oscillations. Resonance
222220
0
4)( dd
d
fA
22
0 2 resd
§5 Mechanical Waves
A wave is a disturbance from eqilibrium that propagates from one region of space to another.
This chapter is about mechanical waves, that travels through some material called the medium.
Another broad class is electromagnetic waves, including light, radio waves, x-rays and gamma rays. No medium is needed for electromagnetic waves; they can travel through empty space.
Types of Mechanical Waves
A). Waves can be transverse, longitudional, or a combination.
Transverse Waves
For transverse waves the displacement of the medium is perpendicular to the direction of propagation of the wave. A wave on a string are easily visualized transverse waves.
Transverse Waves
Transverse waves cannot propagate in a gas or a liquid because there is no mechanism for driving motion perpendicular to the propagation of the wave.
Types of Mechanical Waves
Longitudinal Waves
In longitudinal waves the displacement of the medium is parallel to the propagation of the wave. A wave in a tube filled with a fluid or a liquid is a good visualization. Sound waves in air are longitudinal waves.
Types of Mechanical Waves
Longitudinal Waves
Types of Mechanical Waves
Types of Mechanical Waves
B). Wave pulse and periodic wave.
When the hand shakes the end of a stretched string up and down just once, the result is a single wave pulse, that travels along the length of the string.
When we give the free end of the string a repetitive, or periodic motion, the each particle in the string will also undergo periodic motion, and we have a periodic wave. In particular, if this periodic motion is simple harmonic motion, we call such wave sinusoidal wave.
§6 Waves Characteristics.
Period is the time required to complete a full cycle, T in seconds/cycle.
Frequency is the number of cycles per second, f (or ν) in 1/seconds or Hertz (Hz).
Amplitude is the maximum displacement from equilibrium A.
Velocity of propagation V.
Wavelength λ.
Wave Graphs
Waves may be graphed as a function of time or distance. A single frequency wave will appear as a sine wave in either case. From the distance graph the wavelength may be determined. From the time graph, the period and frequency can be obtained. From both together, the wave speed can be determined.
Waves Characteristics.
Waves Characteristics.
A wavelength, denoted by λ., is a distance between two oscillating points with phase difference being equal to 2π at the direction of wave propagation.
fTt
SV
This is a general wave relationship which applies to sound and light waves, other electromagnetic waves, and waves in mechanical media.
Velocity of wave propagation is
tAx cos
))(cos(V
stAx
)(2cos
s
T
tAx
2
k )cos( kstsx
2
2
22
2 1
t
x
Vs
x
§7 Wave function fof a sinusoidal plane wave. Wave equation.
S0 s
tAx cos))(cos(
V
stAx
Wave equation
Wave function fof a sinusoidal plane wave.
§8 Energy of wave motion.
Waves transport energy, but not matter, from one region to another.The average power transfering any cross-section is called an energy flux.
dt
dE
The average power per unit cross-section is called an intensity.
I
σ
S=vt
21mst 1
