Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:A General Course of Physics. Mechanics. Textbook
.pdf
101
a kinetic energy store due to which, it goes through the equilibrium posi-
tFF cos
0
kxF
el
x
t
Fig. 55
tion).
The critical behavior of damped oscillations is used, for example,
in automobile anti-vibration system dampers for reducing vibration during motion, in pointer electrical measuring instruments for decreasing
pointer oscillation amplitude during measurements and other instances.
6.8. Forced Oscillations. Resonance
Oscillations occurring in the system, unacted upon by variable
external forces, which cause its initial displacement from the stable equilibrium, are called characteristic or free oscillations. The continuous and
damped oscillations considered here are free oscillations.
Now we will have a closer look at forced oscillations.
Forced oscillations are those which occur in an oscillating system under the action of an external periodic force (constraining force).
Let us consider a simple case when a constraining force is changing in compliance with harmonic law:
.
The frequency is not equal to the natural oscillation frequency
of the system. Because oscillations occur in the system, there is also an
elastic force (or quasi-elastic)
.
In a general case, there can also
be friction force, which, as we have
already seen, results in damped oscillations. But for simplifying the problem,
we will assume, first of all, that there
are no fiction forces. Later we will
show what can happen when friction
forces are taken into account. So, there
are two forces acting upon the system –
constraining and elastic.
102
According to Newton’s second law:
maFF
el
xmkxtF
cos
0
;cos0tFkxxm
t
m
F
x
m
k
x cos
0
2
0
mk
0
t
m
F
xx cos
0
2
0
tAx cos
tAx cos
2
t
m
F
tAtA coscoscos
0
2
0
2
m
F
A
0
22
0
)(
Let us introduce the notation:
;
;
.
where
is the natural oscillation frequency of the system.
Hence follows a differential equation of oscillations:
. (72)
The solution of equation (72) is composed of a general solution
of the homogeneous equation and a particular solution of the heterogeneous equation. The homogeneous equation solution as we have seen
represents harmonic oscillations. They are important only at the initial
stage of the process, while later, forced oscillations set in, and they are
described by a particular solution of the heterogeneous equation. For this
reason, we will further consider only this solution. We will search for a
solution in the following way:
, (73)
i.e. forced oscillations occur at the impressed frequency:
. (74)
Having inserted (73) and (74) into (72), we will get
;
;
103
)(
22
0
0
m
F
A
. (75)
t
m
F
x
cos
)(
22
0
0
0
k
F
m
F
A
0
2
0
0
K
F
Ax
0
0À
0
À
0res
This equation shows the amplitude of forced oscillations. After
insertion (75) into (73), we will get an equation of forced oscillations:
.
It follows from formula (75) that the forced oscillation amplitude
depends on the correlation between impressed frequencies and natural
oscillations of the system 0.
Now we will examine this dependence through a few particular
cases.
Case One. If
.
In this case, there are no oscillations, and displacement equals the
static deformation under the action of a constant force F0:
.
high frequency the force is so changeable in direction that the system is
not able to significantly shift from the equilibrium position.
tion frequency of the system 0, the forced oscillation amplitude increases
sharply. This phenomenon is called resonance, and the frequency
is resonant frequency (Fig. 56).
forces are considered, resonance amplitude will not tend to reach infinity;
however, it will have a finite value. In addition, the stronger the force (the
attenuation coefficient ), the lower the resonance amplitude, i.e. the resonance effect appears to be weaker.
Case Two.
It follows from (75) that
Case Three:
.
. Here
. This is due to the fact that at a
.
Thus, if the impressed frequency approaches the natural oscilla-
Earlier, we neglected the friction force for simplicity. But if friction
104
Apart from that, if friction
forces are considered, the formula
of the resonance frequency will
have the form:
22
0res
2
.
Hence, when the attenuation coefficient increases, the
resonance frequency
res
decreases.
This means that the position of the
maximum resonance shifts to the
low-amplitude area.
The resonance effect is widely used in radio technology (receiver
tx sin
]2[sin ty
tuning) and acoustics. A number of optical phenomena (for example, abnormal dispersion) are related to resonance.
In various constructions and machines, intermittently exposed to
changing loads, resonance is quite dangerous: it can cause their destruction as a result of a significant rise in oscillation amplitude. It has to be
taken into consideration in machine and construction engineering.
1. What happens to the oscillation amplitude if the external force
frequency approaches the natural oscillation frequency of the
system?
2. What are the values of the oscillation amplitude, period and
frequency in Fig. 57? The scale of the vertical axis is presented
in meters.
3. A point particle is in motion in two mutually perpendicular directions in compliance with the equations
particle?
Test Questions
and
. What is the trajectory of the moving point
105
4. A point particle is oscillating along a horizontal axis x in com-
]2[10sin1,0 tx
txx sin
0
0,01
Fig. 57
x
0,03
t, c
1
pliance with the ratio
(in meters). What
is the velocity of the point particle at the time point t = 0?
5. How will the oscillation
period of a load, hung by
a spring, change if the
spring stiffness decreases
16-fold?
6. How will the oscillation
period of a spring pendulum alter if the load mass
increases 4-fold?
7. The initial oscillation phase of a point equals 3. The oscillation period is Т = 0,06 s. Determine the nearest time points
when the velocity and acceleration are half as much as their
amplitude correlates.
8. Determine the oscillation amplitude occurring after composing
two harmonic oscillations with equal frequencies and with the
same direction but with the phase difference 60 and the amplitudes 2 cm and 4 cm.
9. A body with a mass m = 2 kg is performing oscillations in accordance with the law
(х0 = 2 cm; ω = 2 rad/s).
Determine the force acting upon the body and its maximum kinetic energy.
106
7. THE ELEMENTS OF HYDROSTATICS AND
HYDRODYNAMICS
Properties of real fluids are exceptionally complicated. Indeed, it
is necessary to remember that liquid as a physical object does not have its
own external form but acquires the form of the container into which it is
placed. If we also add the fact that individual liquid layers can move in
different directions at different velocities, simultaneously being exposed
to friction forces, then it becomes clear how difficult it is to study liquids
in all their manifestations.
However, a large category of liquids possess low viscosity and
constant density, so for a general description of such liquid properties the
notion of an ideal fluid should be introduced. An ideal fluid is a fluid
model which has no viscosity (internal friction) and the liquid itself is
incompressible.
Examples of liquids with comparatively low viscosity are water,
ether, among others. Properties of these liquids are quite close to the
properties of an ideal fluid.
If there is no motion in a liquid, its viscosity does not exhibit itself, and in such a case, for any fluid at rest both with low and high viscosity, some general regularities can be observed. Among them, the basic
ones are Pascal’s law, the ratio for hydrostatic pressure and Archimedes'
principle. We will also note that fluid mechanics, a branch of physics,
dealing with the consideration of properties of fluids at rest is called hy-
drostatics, and the branch of physics studying liquids in motion is called
hydrodynamics.
7.1. Principal Laws and Relationships in Hydrostatics
Pascal’s Law. A fluid, or gas, at rest exerts external pressure uniformly throughout the whole liquid volume and in all directions.
Special attention should be given to the fact that the law only involves about a fluid at rest and external pressures which act upon the liquid or gas. Apart from the external pressure, there can also be other pressures caused by different forces, for instance, by gravitational force of the
liquid itself, which acts at a depth in a motionless liquid; and also forces
related to the motion of fluid layers, and a number of other factors.
107
The hydraulic press
21
SS
1
F
1
1
2
22
F
S
S
pSF
2
F
2
S
1
S
12
/ SS
ghp
fl
fl
2/flghp
S2
2
F
S2
2
F
Fig. 58
fect is based on the principle of
Pascal’s law. The simplest
draulic press consists of two
communicating vessels, each
with a piston (Fig. 58). The space
between the pistons is filled with
water. Each piston has a differing
surface area:
.
If the piston with a sur-
face area of S1 is acted upon by a
force
, then the liquid will be
subjected to the pressure
p = F1/S1. Such a pressure, according to Pascal, will act upon the volume
of the liquid, including the second piston with a surface area of S2. Then,
the force acting upon piston S2,
.
Hence, the force
applied to piston
many times over, if the ratio
acting upon piston
can exceed the force
is quite high.
Hydrostatic pressure. If a fluid at rest is in the field of gravity,
the base of the vessel is acted upon by the pressure:
, (76)
where
is the fluid density; g is gravitational acceleration; h is the
height of the liquid column. This pressure is called hydrostatic. The pressure exerted on the vessel walls is determined by the formula:
Pressure exerted on the vessel walls is dependent on the wall
coordinates of a point and does not depend on the vessel form itself.
108
Archimedes'
1
h
SpF11
1fl1
ghp
SghF
fl 11
2
h
SghF
fl 22
S
h
2
h
h
1
2
F
2
F
Fig. 59
principle. A body, immersed in a fluid or gas, is
exerted upon by the upward buoyancy force equal
to the weight of the displaced fluid or gas. The
principle is true for any case
where the fluid or gas is motionless, i.e. for a hydrostatic
medium.
We will illustrate
that the Archimedes' principle is an example of hydrostatic pressure acting in a fluid without any inner motions, and the fluid itself is situated in
the Earth’s gravitational field. For this purpose, we will examine a parallelepiped immersed in a fluid (Fig. 59). The forces acting upon its lateral
faces are mutually balanced. The force acting upon the upper face is determined by the hydrostatic pressure at the corresponding depth
. The
force intensity is
, (77),
specifically
; and S is the surface area of the parallelepiped’s
upper face.
Having inserted expression (76) into equation (77), we will get:
.
The force acting upon the lower face is determined by the hydrostatic pressure at the corresponding depth
. This force intensity is higher than
the force acting upon the upper face:
.
The Archimedes upper buoyancy force can be determined by the
subtraction of the forces acting upon the lower and upper faces of the parallelepiped:
109
flflfl
gVhhgSFFF
1212Arch
,
12fl
hhSV
flflArch
gVF
Fig. 60
1
v
2
v
viz.
is the volume of the fluid displaced by the parallele-
piped. Therefore, the Archimedes force is
.
It is necessary to mention that the Archimedes force is determined
by the weight of the displaced fluid, not by the gravitational force of the
displaced fluid. Consequently, if a vessel of fluid containing an immersed
body moves with acceleration in the Earth’s gravitational field, the weight
of the displaced fluid changes and, as a result, the Archimedes force
changes, too. If a fluid with an immersed body is in a state of weightlessness, the Archimedes force equals zero.
7.2. Hydrodynamics of Ideal Fluids: Principal Laws
The motion state of an ideal fluid can be determined by indicating a velocity vector for each space point. The sum total of velocity vectors, plotted in all space points where the fluid is in motion, comprises the
so-called field of velocities. For a graphic representation of the whole velocity field, we will draw the lines in such a way as that the tangent lines
to them coincide with the velocity vector direction at every point (Fig.
60).
These lines are called the lines of fluid flow. If the graph is drawn
is this way, those areas of space where the flow lines are denser, the fluid
velocity is higher and, vice versa, where the flow lines are thinner, the
fluid velocity is lower.
In a general case, the velocity
vector magnitude and direction at
every point of space can change with
time, the streamline pattern also
changes continuously.
If the velocity vector at every
point of space remains constant, the
fluid flow is called steady, or statio-
110
nary. The streamline pattern for
1
S
2
S
dtSdtS
2211
vv
constvS
1
S
2
S
111
SpF
222
SpF
S2
S1
2
v
1
v
Fig. 61
the stationary motion does not
change, and in this case the flow
line coincides with the particle
trajectories.
A part of the fluid in motion within the flow lines is called
a flow tube (Fig. 61). We will also
point out that the fluid particles in
motion do not cross the flow tube
surface area because their velocities are tangential to the flow tube surface.
The theorem on flow continuity. Let us consider a flow tube
(Fig. 61). We will examine two flow tube sections perpendicular to the
fluid velocity at these sections. Let us suggest that the fluid velocity in
section
is v1, and in section
it is v2. Provided that the fluid is not
exposed to discontinuity and it is incompressible, then the liquid volume
flowing through both sections for a short time interval dt must be the
same. Hence,
.
The considered sections are arbitrary; consequently, for any section of a flow tube the following equation is true:
It can be regarded as the theorem on flow continuity. It follows
from this theorem that if the section varies, the flow velocity changes,
too, i.e. the low particles will move with acceleration. This acceleration is
due to the pressure change along the flow tube axis. We will also notice
that the theorem on flow continuity is true for a non-stationary fluid flow
as well.
Bernoulli's Equation. Let us separate a flow tube of a small
cross section in a steadily flowing ideal liquid (Fig. 62). We will consider
sections
and
perpendicular to the streamlines acted upon by the
forces:
and
, (78)
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]