Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Integrational mechanics. Lecture and exercises
.pdf
3.6. Elements of Lagrange’s analytical mechanics 81
D’Alembert’s principle for constrained system
If a constrained system of bodies is given by only “lost motions”, then
the bodies remain at rest since these motions eliminate one another. D’Alembert
distinguishes the motion transmitted to body and the really received motion. The
difference between “given” motion and “received” motion was called a “lost”
motion.
The lost forces applied to the particles are brought to the state of balance
by the reactions of constraints.
P
= −Ri,
i
=Fi+Ri,
ma
i
Φ
= −mai,
i
Φ
+Fi+Ri=0,
i
=Pi=Φi+Fi.
−R
i
D’Alembert’s principle in kinetostatics states that the sum of active forces and
inertia forces, applied to the particles of a constrained system, is balanced by
reactions of constraints,
P
=Fi+Φi. (3.59)
i
Kinetostatics is a part of classical mechanics which deals with using meth-
ods of statics for dynamic of problems o theory of machines.
Common dynamics equation
Let’s consider a constrained system with ideal constraints. Let M
the mass of particles,F
¨
r
— the real acceleration of i − th particle, δri— the virtual displacement
i
— the resultant of all forces acting on i-th particle,
i
denotes
i
of i-th particle. Then the equilibrium condition for the system acted on by lost
forces P
has the form of common statics equation:
i
N
i=1
P
· δri=0,
i
N
(Fi+Φi) · δri=0. (3.60)
i=1
For dynamics of constraint system the latter equation was called the com-
mon dynamics equation.
82 Chapter 3
In case of non-ideal constraints if is sometimes written in the form
N
(Fi+Ri+Φi) · δri=0. (3.61)
i=1
3.7. Compact “Impact phenomena in mechanics”
Classification of impact theories
Impact is one of the most complex phenomena in engineering. Mechanics
has three stages of development. The first one is classical mechanics in which
all the basic phenomena (vibration, stability, strength, impact) are considered
separately. At this stages there are appeared the four independent theories of
impact:
• stereo — mechanical theory;
• Hertz’s contact theory;
• energetic theory;
• wave theory (after de Saint – Venant).
At the first stage in accordance with dialectics laws there are elements of
the second stage, the stage of developing mechanics as a system one. System
mechanics synthesizes different problems. In the theory of impact it was realized
in the creation of the following theories:
• combination of stereo–mechanical and energetic theories (Carnot’s theory);
• combination of contact and wave theories (Sirs’ theory for rods);
• combination of wave and energetic theories (de Saint–Venant).
The third stage of mechanics development is integrating mechanics based
upon universal principles of cognition. Its realization in the theory of impact is
an informational pyramid; its sides being theory of vibration, stability, theory of
strength. The base of the pyramid is a theory of impact.
In integrating mechanics impact is a synthesis of interconnected nonlinear
problems. Using integrating mechanics one can obtain a number of different
applied theories.
Usually we deal with seven theories of impact above mentioned.
3.7. Compact “Impact phenomena in mechanics” 83
Stereo-mechanical theory of impact
Stereo —mechanical theory of impact was developed by I. Newton while
studying the collision of two bodies. Masses and velocities of the bodies before
collision are: m
,m2,v1,v1. Ones after collision: m1,m2,v
1
,v
1
.
1
According to the momentum conservation principle:
m
+ m2v2= m1v
1v1
1
+ m2v
. (3.62)
2
We have one equation with two unknown quantities. To solve it we should
use an additional information from experiment. Newton introduces a coefficient
of restitution “K
which depends on bodies material and does not on velocities,
v
− v
1
2
K =|
v1− v
| . (3.63)
2
Using coefficient of restitution K one determines velocities after impact.
For the couple of materials steel — steel the coefficient equals K =0, 90...0, 95
(bearing’s ball on a clean plate. But the practical way is opposite, specialists
assume value of K (for springs used in tommy-gun K =0, 4) and do not
calculate the collisions repeated many times.
Even in the simplest theory of impact mechanics of a deformable body
should be used since the coefficient of restitution depends on the impact plastic
deformations.
The stresses in contact points can be determined by Hertz’s theory if the
coefficient
where T
imp
— time of impact, 2θ — double time of running a deformation wave
β =
T
imp
> 3...8, (3.64)
2θ
in the most extended body of ones colliding each other.
Hertz’s contact theory
This theory denies a general deformation of the colliding bodies but in-
volves a local deformation at the point of contact. It agrees physically with
stereo mechanical theory of impact including the coefficient of restitution. A solution of impact problem is based upon equations of classical theory of elasticity
and was found by Hertz’s for of a body static compression.
For the static compression a contact force depends on compression degree α
according to the law
P = kα
3/2
, (3.65)
where coefficient K depends on material and chape of bodies.
84 Chapter 3
Hertz’s also suggested a formula for the special case of contact impact
where bodies general deformation and wave processes are about absent. Contact
theory gives for an engineer the value of stress and the time of impact. The time
of impact is determined by:
T
imp
=2, 9432(
5m
4k
)
2/5
−1/5
V
, (3.66)
0
where m = m
· m2(m1+ m2)−1.
1
In general, process of the development of a complex theory involves the
ideas of system mechanics: get the basic
results nevertheless obtain a simple solution.
Linearized contact theory
The linearized of nonlinear compression characteristic for two bodies is illustrated in Fig. 3.18.
The basic positions of a modified solution are:
Figure 3.18
1. Equality of values of P
max,αmax
(maximum displacement of two bod-
ies) obtained by exact and approxi-
mate formulae.
2. Equality of values of a deformation energy obtained by exact and approximate formulae
= kα
3/2
max
2
5
= c(α
kα
5/2
max
max
= c
− α0),
(α
max
− α0)
2
2
.
P
α
m
max
P (α)dα =
0
After simple calculations we get:
α
0
=0, 2 α
max
,c=
1/3
P
max
4
· k
1/3
.
5
To find the time of impact we use a synthesis of two simplest theories: a body
free motion in section α
with velocity v0, and the vibration of an equivalent
0
3.7. Compact “Impact phenomena in mechanics” 85
mass m like a linear oscillator of stiffness c,
τ = π
m
α
0
+2
c
. (3.67)
v
Formula (3.66) gives the results which differ from ones of the exact solution
(3.65) in less than 1%.
Energetic theory of impact
In energetic theory the process of impact is not considered, but information
(about velocities) is determined before and after impact. The weight (m
down on the truck of mass m
and stiffness c from height h (see Fig. 3.19).
0
m
1
h
m
o
C
) falls
1
Figure 3.19
The three problems can be considered. The first one: the weight of mass
begins after collision vibrate together with the truck of mass m1without a
m
0
recoil.
t =0,v= v
, ((m1+ m0)¨x + p2x =0,p2=
0
˙x
0
x = x
cos pt+
0
p
sin pt,
c
m1+ m
and for the initial conditions given,
x =
˙x
0
sin pt=
p
v
0
p
sin pt.
,
0
86 Chapter 3
Maximum amplitude equals
v
x =
p
0
.
Velocity of vibration of the weight ((m
˙x = v
+ m0) is
1
cos pt.
0
This is a harmonic law of the velocity change.
The second problem: the weight of mass m
m
¨x + cx=0,
0
˙x
0
x =
sin pt, p
p
recoils from the truck.
1
c
2
.
=
m
The third problem: the weight falls again after recoil on the truck and begins to
vibrate with it. In this case we obtain a compound motion, a synthesis of the
first and second problems.
Wave theory of impact
Let’s consider the derivation of a wave equation
2
∂2u
2
a
∂x
∂
u
=
2
, (3.68)
2
∂t
where u = f(x, t) - displacement of a cross section,
E
2
=
a
,
ρ
E — Young’s index, ρ — mass density of a substance, a — velocity of an exten-
sion wave.
We use Newton’s method (see Fig. 3.20). The equation of forces is
2
∂
−N + N + dN = ρF dx·
dN = ρF dx·
∂
∂t
u
,
2
∂t
2
u
2
— Newton’s law.
3.7. Compact “Impact phenomena in mechanics” 87
σ
θ
θ
E
The absolute deformation equals:
u + du − u = du.
The relative deformation equals
∂u
du
=
∂x
.
dx
Hook’s law (force — relative displacement linear law, or stress — deformation
linear law) gives (see Fig. 3.21):
∂u
σ = E
dN
.
∂x
du
Figure 3.21
uduu +
NdNN +
dx
Figure 3.20
tg =
∂
u
x
∂
The force-stress relation is
N = σ · F,
where F — area of cross section.
dN =
∂N
∂x
dx;
∂N
∂x
= ρF ·
∂
∂t
2
u
.
2
If F =const,then
EF
∂
∂x
2
u
2
= ρF ·
∂
∂t
2
u
.
2
Cancelling F we obtain expression (1.69). What should be memorized for
the case of a body striking against a rod (see Fig. 3.25)? The condition: the
body hits the rod and begins to vibrate with it.
88 Chapter 3
ρ
EF ,,
m
V l
o
Figure 3.22
At the very beginning of impact when the wave doesn’t come back to the
point of impact, the stresses do not depend on the masses of bodies. Using this
condition one can find the critical velocity of the impact:
v = a · ε,
v
= a·|ε |, for ordinary steels —50 ε =0.001,
cr
a =5km/s for steel - 50.
The critical velocity of impact equals
v
=5m/s.
cr
If someone suggests you to prevent plastic deformation in the condition of
velocity s more than 5m/s,thenyoumaytoleaveittodobyhimself.
Synthesized theories of impact. Carnot’s theory
This theory is the synthesis of stereo — mechanical and energetic theories of
impact. The coefficient of restitution and the notion of lost - in - impact energy
are introduced here.
Carnot’s formula
T
− T2=
1
Where T
T
2
— kinetic energy of the system of two bodies before impact,
1
v
m
T
1
=
1
2
— kinetic energy of the system after impact,
v
m
1
=
T
1
1
2
1 − K
1+K
2
1
+
2
+
m
m
∗
.
T
2
v
2
2
.
2
2
v
2
2
,
2
3.8. Compact “Linear and nonlinear problems in dynamics” 89
T∗=1/2m1(v1− v
)2+1/2m2(v2− v
1
)2.
2
If the notion of the lost velocities is introduced
v
= v1− v
1
and v2= v2− v
1
then the kinetic energy lost in impact equals
1 − K
— part of the kinetic energy
1+K
,
2
corresponding to the lost velocities.
Sirs’ synthesized theory
In this case Hertz’s theory is used at the point of contact, at all other points
wave theory is used (see Fig. 3.23).
Wave theory
Hertz ‘s theory
Figure 3.23
Combination of wave theory and energetic theory
Stresses are determined at the spot of an embedment where stresses due to
the direct and reverse waves interferes (see Fig. 3.24).
The maximum deformation equals
ξ
max
= ξ
wave
+ ξ
energ.
,
then
=
P
max
= P
wave
+ P
energ.
.
3.8. Compact “Linear and nonlinear problems in dynamics”
3.8.1. Classifications of vibration problems
In integrational mechanics (mechanics of unity of mathematics, physics,and
dialectics) all the basic problems are subdivided into system problems (with
90 Chapter 3
ea
-
V
0
3
2
1
0
21
3
Wave theory
56
4
Figure 3.24
Energy theory
7
8
at
-
e
united physics of a phenomenon). According to the type of nonlinearity all
problems are subdivided into four classes: linear, mathematically nonlinear,
physically nonlinear, system nonlinear. The first three classes are included into
the class of system linear problems. All the four classes are based upon three
types of mechanics: classical mechanics (Newton, Lagrange); classical mechanics of deformable bodies (Euler); synthesized mechanics (vibration in Newton’s
mechanics and Euler’s mechanics).
The classification of vibration problems will be agree with the general clas-
sification of the integrational mechanics problems.
Let’s consider vibration of systems with discrete masses and massless
spring(see Fig. 3.25).
Differential equation of free vibration is
m¨x + cx =0.
Square of a natural frequency:
c
2
.
=
p
m
Now let’s consider vibration of systems with distributed parameters (see
Fig. 3.26).
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]