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456 Solutions Manual
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 457
Problem 3.35
Car bumpers are designed to limit the extent of damage to the car
in the case of low-velocity collisions. Consider a passenger car
impacting a concrete barrier while traveling at a speed of
4:0 mph
.
Model the car as a particle of mass
m
, and assume that the bumper
has a spring element in parallel with a shock absorber so that the
overall force exerted by the bumper is
FBDkı C⌘P
ı
, where
k
,
ı
, and
⌘
denote the spring constant, the spring compression, and
the bumper damping coefficient, respectively.
Derive the equations of motion for the car during the colli-
sion.
Solution
Referring to the FBD at the right, we model the car as a particle subject
458 Solutions Manual
Problem 3.36
Car bumpers are designed to limit the extent of damage to the car
in the case of low-velocity collisions. Consider a passenger car
impacting a concrete barrier while traveling at a speed of
4:0 mph
.
Model the car as a particle of mass
m
, and assume that the bumper
has a spring element in parallel with a shock absorber so that the
overall force exerted by the bumper is
FBDkı C⌘P
ı
, where
k
,
ı
, and
⌘
denote the spring constant, the spring compression, and
the bumper damping coefficient, respectively.
Let the weight of the car be
3300 lb
,
kD6500 lb=ft
, and
⌘D300 lbs=ft
, and let the car be traveling at
4:0 mph
at impact.
Determine the maximum compression of the bumper necessary
to bring the car to a stop. Also determine the time required to
stop the car.
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 459
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
460 Solutions Manual
Problem 3.37
A railcar with an overall mass of
75;000 kg
traveling with a speed
vi
is approaching a barrier equipped with a bumper consisting of
a nonlinear spring whose force vs. compression law is given by
FsDˇx3
, where
ˇD640⇥106N=m3
and
x
is the compression
of the bumper.
Treating the system as a particle and assuming that the contact
between the railcar and rails is frictionless, determine the maximum
value of
vi
so that the compression of the bumper is limited to
20 cm.
Solution
Referring to the FBD on the right, once the railcar is in contact with the bumper, we
model it as a particle subject only to its weight
mg
, the normal reaction from the rails
Dynamics 2e 461
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
462 Solutions Manual
Problem 3.38
A railcar with an overall mass of
75;000 kg
traveling with a speed
vi
is approaching a barrier equipped with a bumper consisting of
a nonlinear spring whose force vs. compression law is given by
FsDˇx3
, where
ˇD640⇥106N=m3
and
x
is the compression
of the bumper.
Treating the system as a particle, assuming that the contact
between railcar and rails is frictionless, and letting
viD6km=h
,
determine the bumper compression necessary to bring the railcar
to a stop.
Solution
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 463
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
464 Solutions Manual
Problem 3.39
A railcar with an overall mass of
75;000 kg
traveling with a speed
vi
is approaching a barrier equipped with a bumper consisting of
a nonlinear spring whose force vs. compression law is given by
FsDˇx3
, where
ˇD640⇥106N=m3
and
x
is the compression
of the bumper.
Treating the system as a particle, assuming that the contact be-
tween the railcar and rails is frictionless, and letting
viD6km=h
,
determine how long it takes for the bumper to bring the railcar to a
stop.
Solution
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 465
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
