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.
Simple Harmonic Motion.
F  ma ; Fx  ma x ;
2
Fx   kx ; a x  d x ;
dt 2
2
2
2
d x
d x
d x k
 kx  m 2  m 2  kx  0  2  x  0
m
dt
dt
dt
2
d x
k
2
2
x  A cos(  0 t   0 )


x

0
 0 ;
0
;
2
m
dt
Displacement, velocity, and acceleration in simple
harmonic motion
§2 Energy in Simple Harmonic motion
kx
Ep 
2
2
2
mV
; Ek 
2
2
2
2
2
kx mV kA mVm
E  E p  Ek 



 const
2
2
2
2
§3 Damped Oscillations.
Oscillations with little damping
F  F f  ma ; F x  F f
F f x   rV x
x
 ma x
dx
d 2x
d 2x
r dx k
 kx  r
m


x0

2
2
dt
m dt m
dt
dt
r
d 2x
dx
2
 2 ;

2



0 x  0
2
m
dt
dt
If the damping force is relatively small, so that
 2  0 2   2 >0
  02   2
the solution of differential equation is
x  Ae t cos(t   0 )
Damped Oscillations
A(t )
  ln
 ln e T  T
A(t  T )
§4 Forced Oscillations. Resonance
Fd  Fm cos d t ;
Fx  F f x  Fd x  max
d 2x
dx
Fm cos d t  m 2  r  kx
dt
dt
Fm
d 2x
dx
2
 2  0 x  f m cosd t
 fm ;
2
dt
dt
m
Forced Oscillations. Resonance
x  Ad cos(d t  0 )
Forced Oscillations. Resonance
Ad 
f0
( 0   d ) 2  4  2 d
2
2
2
 d res   0 2  2  2
§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.
Types of Mechanical 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
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.
Waves Characteristics.
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.
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.
Velocity of wave propagation is
V 
S


 f
t
T
This is a general wave relationship which applies to sound and light
waves, other electromagnetic waves, and waves in mechanical media.
§7 Wave function fof a sinusoidal plane wave. Wave equation.
x  A cos(  (t 
x  A cos t
s
0
s
))
V
S
x  A cos t
s
x  A cos(  (t  ))
V
t s
x  A cos 2 (  )
T 
2x 1 2x
 2 2
2
s
V t
Wave function fof a sinusoidal plane wave.
k
2

Wave equation
x  s cos(t  ks)
§8 Energy of wave motion.
Waves transport energy, but not matter, from one region to
another.
The average power transfering any cross-section is
dE
called an energy flux.

dt

The average power per unit cross-section is called an intensity.
I

σ
  1m2
t  1s
S=vt