91
CHAPTER 6
Topic AK-6
6.1 STUDY OF IMPACT DYNAMICS BETWEEN RIGID BODIES
Problem 6.1. e cart 1 with total mass m
1
= 6,000 kg is moving horizontally with velocity v
1
= 2.5 m/s and hits
stationary cart 2 with total mass including the container m
2
= 4,000 kg. After the impact the cart gains the horizontal
velocity u
1
= 2 m/s, but the container of mass m
0
= 500 kg ips about point A supported by the stopper. e container
is assumed as a homogeneous rectangular parallelepiped (a = 0.8 m,h = 1.5 m). e vertical impact planes of the carts
are assumed smooth. Determine the velocity of the cart 1 after the impact with the cart 2, and the impact impulse
on the support stopper at point A.
1
2
h
a
A
Figure 6.1.
92 6. TOPIC AK-6
Problem 6.2. A load with mass m
0
= 500 kg falls from the height h = 1 m to the point D of the rigid bar supported
by the stationary pin support A and the spring B with the coecient of the rigidity c = 20,000 N/cm
. e impact
of the load on the bar is assumed as an inelastic. e mass of the bar m = 6,000 kg and its length l = 4 m. e hori-
zontal position of the bar shown in Figure 6.2 corresponds to the static deformation of the spring support under the
weight of the bar only. Consider the bar as a slender homogeneous cylinder, and the weight as a particle. Determine
the impact impulse on the bar at the point D and maximum deformation of the spring support assuming that the
point B moves linearly.
h
B
A
l/4 3l/4
D
Figure 6.2.
Problem 6.3. e string holding the load with the mass m
0
= 500 kg was broken and the load falls from the height
h = 1 m on the platform rested on two identical and symmetrically positioned spring supports. e load strikes at the
point A which is on the vertical symmetrical plane of the platform and at the distance d = 0.6 m from its center of
gravity C. e load impacts the platform not elastically. e mass of the platform is m
0
= 5,000 kg, and its radius of
inertia horizontal axis of symmetry is i
C
= 0.5 m. Dene the velocity of the center of gravity and the angular velocity
of the platform at the end of the impact. Also, determine an impact impulse at the point A. Consider the platform
as a rigid body and the load as a particle.
93
h
d
A
C
Figure 6.3.
Problem 6.4. e homogeneous cylindrical load of mass m = 200 kg and radius r = 0.2 m is moving via a horizontal
conveyor belt with velocity v = 0.6 m/s without skidding on the discs 1 and 2. At some instant the conveyor suddenly
stops. As the surface of the conveyor belt is not smooth, due to the sudden stop of the conveyor the cylinder will roll
on the belt without skidding. e rolling friction is negligibly small. Determine an impact impulse on the absolute
rough surface of the belt due to the sudden stop of the conveyor. Using Carnot’s theorem verify the obtained velocity
of the center of gravity (or an angular velocity) of the cylinder. e cylinder travels a certain distance and hits the
step of the height h = 0.03 m. Determine an impact impulse on this step.
A
2
C
1
O
2
h
O
1
Figure 6.4.
6.1 STUDY OF IMPACT DYNAMICS BETWEEN RIGID BODIES
94 6. TOPIC AK-6
Problem 6.5. e homogeneous cylinder of mass m = 500 kg and radius r = 0.5 m rolls without initial velocity along
an incline (α = 30°) at a distance s
1
= 3 m and continues to roll on the horizontal surface without skidding. e
coecient of the rolling friction is δ = 0.2 cm. At what distance s
2
should a support step of the height h = 0.1 m,
be placed so the cylinder after hitting the corner F just climbs on it without further traveling and skidding? Also,
determine both horizontal and vertical components of the impact impulse on the cylinder from the step.
h
E
D
A
B
C
F
S
2
S
1
Figure 6.5.
Problem 6.6. e pendulum consists of the weight (homogeneous circular disk of radius r = 0.1 m) suspended from
the pivot of l = 1.2 m long. e mass of the weight m
0
= 5 kg, and the mass of the pivot is negligibly small. e
pendulum is displaced sideway from its resting, equilibrium position and it swings about xed point O with angular
velocity ω = 3 s
–1
. e pendulum hits a homogeneous rectangular parallelepiped D of mass m = 6m
0
(a = 0.8 m, b
= 0.4 m, h = 0.2 m) at point B. e coecient of restitution during an impact is k = 0.5. e both surfaces of the
pendulum and the object D are absolutely rough (no skidding during the impact). Determine the angular velocity
of the object D about point A at the end of the impact and the impact impulse on the rough surface at the point A.
l
A
h
r
D
O
b
a
B
Figure 6.6.
95
Problem 6.7. e lever arm consists of two solid bars AB and AD connected under a right angle. e lever arm has a
xed horizontal axis of rotation A and it is held at point B via spring. AD = a = 1.5 m. e load with mass m
0
= 100
kg falls from the height h = 0.5 m and hits the horizontal bar of the lever arm at point D. e mass of the bar is m
= 1,000 kg, and its radius of inertia about axis of rotation is i
A
= 0.5 m. e position of the center of gravity C of the
lever arm is dened by the coordinates x
C
= 0.4 m and y
C
= 0.3 m. Consider the load as a particle and the impact as
inelastic. Determine an impact reaction on the load at point D, as well as both horizontal and vertical components
of the impact impulse on the support A.
A
h
D
y
x
a
B
x
c
y
c
C
Figure 6.7.
6.1 STUDY OF IMPACT DYNAMICS BETWEEN RIGID BODIES
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.