Lesson 13
Mechanical Shock Theory
第 13课 机械冲击理论
Introduction
? Throughout the distribution system,packages are
manhandled and mishandled in various ways,
dropped,thrown,kicked and otherwise roughly abused;
fall from conveyors or forklifts and crash to the floor;
subjected to a variety of vehicle impacts; trucks starting,
stopping,hitting chuckholes and railroad crossings,railcar
humping,jolting and other moderately violent actions;
suffers an impact with another object,floor,truck bed,pallet,
bulkhead or another package,
? A mechanical shock occurs when an object's position,velocity
or acceleration suddenly changes,Such a shock may be
characterized by a rapid increase in acceleration followed by a
rapid decrease over a very short period of time,
? Figure 13.1,the acceleration versus time plot for most shocks
Figure 13.1 Representation of mechanical shock
A package shock may typically be 20 milliseconds (0.020
seconds) long and have a magnitude or "height" of 150 g's,
need to know both the magnitude of the acceleration and the
duration of the shock,
The Free Falling Package
? the length of time it takes a package to fall from a
drop height,h
? the downward velocity at which it will be traveling a
moment before impact;,the impact velocity,
? As is shown in Figure 13.2,A package will rebound
a little or a lot depending on the nature of the
package and the surface it hits,
ght 2?
g
ht 2?
ghvI 2?
ghv I 2?
Figure 13.2 A falling package
coefficient of restitution,e,describes the rebound velocity as
a function of the impact velocity
13.1
total velocity change,
13.2
13.3
Because 0 1(typical values falling in the 0.3 to 0.5 range),
13.4
velocity change is also numerically equal to the area beneath
the shock pulse as shown in Figure 13.3,
IR evv ?
RI vvv ???
ghevev I 2)1()1( ?????
ghvgh 222 ???
Figure 13.3 The relationship among shock parameters
Package damage is related to the three factors involved in
mechanical shock,
? Peak Acceleration
? Duration
? Velocity Change
Mechanical Shock Theory
? Shown in Figure 13.4,the product-package system
consists of four basic components,the outer container,
the cushion,the product,and a critical element,
? shown in Figure 13.5,the product-package model,
Figure 13.4 A simple product-
package system
F
igure 13.5 A spring-mass model for
the product-package system
? M2 - the mass of the product
? M1- represents the mass of the critical element or CE
? M3 - represents the mass of the outer container
? kl - the linear spring constant of the sprint-mass system representing the critical element
? k2 - the linear spring constant of the cushion system
Assumptions for simplicity,
a,ignore the mass of the outer container and assume that it provides no spring action;
b,the cushion has no mass or damping and suffers no permanent deflection from a shock;
c,the product-package system impacts a perfectly rigid floor;
d.,the mass of the critical element is negligible compared to the mass of the product,
In Figure 13.6,the impact of a product-package,
Figure 13.6 The impact of a product-package system
At Point A,the product is ready to fall,The potential energy
at this moment is,
13.5
where
At Point C,the kinetic energy of the system is given by
13.6
13.7
which show that the kinetic energy at this moment is equal
to the initial potential energy of the system,
In the moments following the initial contact of the container
with the impact surface,the amount of energy absorbed by
the linear cushion is,
13.8
hWghMPE 22 ??
gMW 22 ?
2
22
1
IvMKE ?
hWghMghMKE 222 )2(21 ???
2
222
1 xkE ?
whereX2is the downward displacement of the product on the
cushion,
At Point D,the cushion has absorbed all of the system's kinetic
energy
13.9
where maximum dynamic compression,dm= maximum value
of x2,
The maximum amount of energy absorbed by the cushion must
equal the system's kinetic energy at impact,
13.10
for the maximum dynamic compression,
13.11
2
2m a x 2
1
mdkE ?
2
2m a x2 2
1
mdkEhWKE ???
2
22 2
1
mdkhW ?
2
22
k
hWd
m ?
The static deflection of the product on the linear cushion is,
so Equation 13.11 may also be written as,
13,12
The maximum force exerted upward by the cushion against
the product occurs when,
13.13
The maximum acceleration (or deceleration) experienced by
the product,Gm,may be found from the relationship,
2
2
k
W
st ??
stm hd ?2?
2
2
2222m a x
2
k
hWkdkxkP
m ???
hWkP 22m a x 2?
13.14
13.15
where Gm is unitless,but is understood to be in units of 1 g,
13.16
Gm is proportional to the square root of the drop height,
This means that if we double the drop height,the magnitude
of the shock is not doubled,but rather increased by a factor
of (about 1.4),
Equation 13.15 may be rearranged to produce an expression
for,
13.17
2
m a x
W
PG
m ?
2
2
2
22 22
W
hk
W
hWkG
m ??
st
m
hG
?
2?
hG m ?
h
GWk m
2
2
2
2 ?
Equation 13.11 may also be rearranged to produce an
equivalent expression,
13.18
13.19
giving us an expression for the maximum dynamic
compression as a function of the maximum acceleration
and the drop height,
Conclusions,
1,is related to the drop height,the springiness of the
cushion and the weight of the product,
2,The stiffer the spring or cushion,the larger the value for,
3,The higher the drop height,the larger the value for(),
4,The heavier the product,the smaller the value of as long
as the spring or cushion is working,
2
2
2
2
md
hWk ?
m
m G
h
W
hk
h
hk
W
hd
2
2
2
2
2
2
22
2 ???
Shock Duration
We may approximate the displacement of the product on
the cushion,,at any time during the impact with the function,
13.20
where
13.21
f2 - the natural frequency of the product (M2) on the cushion
(k2),
)s i n ()( 22 tdtx m ??
22 2 f?? ?
2
2
2 2
1
W
gkf
??
Figure 13.7 Shock duration
and maximum displacement Figure 13.8 Shock duration and the natural period of
vibration
T2 - The period of the free vibration of the product on the
cushion,
13.22
where - the time length or duration of the shock,
For any shock of known period,we may calculate the
equivalent shock frequency,
13.23
the shock duration,
13.24
13.25
?21 2
2
?? Tf
?2
1
2 ?f
2
222
1
W
gkf
?
? ??
gk
W
2
2?? ?
Shock Amplification and the Critical
Element
Define,
Ge - the maximum acceleration experienced by the critical
element,resulting from a shock( Gm and),as shown in
Figure 13.9,
Am - an amplification factor,analogous to vibration
magnification relating the input and output shock levels,
13.26
Ge = Am Gm 13.27
m
e
m G
GA ?
Figure 13.9 Shock transmission
During the impact,the amplification factor is given by the
expression,
13.28
where N - an integer
f2 - the equivalent shock frequency
fl - the natural frequency of the critical element
Just after impact,the amplification factor is,
13.29
)1(
2
s in
)1(
)0(
2
1
2
1
2
1
??
???
f
f
N
f
f
f
f
tA m
?
?
2
2
1
2
1
2
1
)(1
)
2
c o s ()(2
)(
f
f
f
f
f
f
tA m
?
?? ?
The largest value of Am for Equation 13.27 depends on the
relationship of and,
1,If
(Equation 13.29)
2,If
(Equation 13.28)
3.Notice that as the ratio becomes large,the value for Am
approaches 1,reflecting the "direct" transfer of the shock to a
stiff element,
4.For small values of the frequency ratio,
,2121o r,1 2121
2
1 ????? TTff
f
f
)( ??? tAA mm
,2121o r,1 2121
2
1 ????? TTff
f
f
)0( ???? tAA mm
20.0),(2
2
1 ??
2
1
m f
f f o r
f
fA
Mechanical Shock Theory
第 13课 机械冲击理论
Introduction
? Throughout the distribution system,packages are
manhandled and mishandled in various ways,
dropped,thrown,kicked and otherwise roughly abused;
fall from conveyors or forklifts and crash to the floor;
subjected to a variety of vehicle impacts; trucks starting,
stopping,hitting chuckholes and railroad crossings,railcar
humping,jolting and other moderately violent actions;
suffers an impact with another object,floor,truck bed,pallet,
bulkhead or another package,
? A mechanical shock occurs when an object's position,velocity
or acceleration suddenly changes,Such a shock may be
characterized by a rapid increase in acceleration followed by a
rapid decrease over a very short period of time,
? Figure 13.1,the acceleration versus time plot for most shocks
Figure 13.1 Representation of mechanical shock
A package shock may typically be 20 milliseconds (0.020
seconds) long and have a magnitude or "height" of 150 g's,
need to know both the magnitude of the acceleration and the
duration of the shock,
The Free Falling Package
? the length of time it takes a package to fall from a
drop height,h
? the downward velocity at which it will be traveling a
moment before impact;,the impact velocity,
? As is shown in Figure 13.2,A package will rebound
a little or a lot depending on the nature of the
package and the surface it hits,
ght 2?
g
ht 2?
ghvI 2?
ghv I 2?
Figure 13.2 A falling package
coefficient of restitution,e,describes the rebound velocity as
a function of the impact velocity
13.1
total velocity change,
13.2
13.3
Because 0 1(typical values falling in the 0.3 to 0.5 range),
13.4
velocity change is also numerically equal to the area beneath
the shock pulse as shown in Figure 13.3,
IR evv ?
RI vvv ???
ghevev I 2)1()1( ?????
ghvgh 222 ???
Figure 13.3 The relationship among shock parameters
Package damage is related to the three factors involved in
mechanical shock,
? Peak Acceleration
? Duration
? Velocity Change
Mechanical Shock Theory
? Shown in Figure 13.4,the product-package system
consists of four basic components,the outer container,
the cushion,the product,and a critical element,
? shown in Figure 13.5,the product-package model,
Figure 13.4 A simple product-
package system
F
igure 13.5 A spring-mass model for
the product-package system
? M2 - the mass of the product
? M1- represents the mass of the critical element or CE
? M3 - represents the mass of the outer container
? kl - the linear spring constant of the sprint-mass system representing the critical element
? k2 - the linear spring constant of the cushion system
Assumptions for simplicity,
a,ignore the mass of the outer container and assume that it provides no spring action;
b,the cushion has no mass or damping and suffers no permanent deflection from a shock;
c,the product-package system impacts a perfectly rigid floor;
d.,the mass of the critical element is negligible compared to the mass of the product,
In Figure 13.6,the impact of a product-package,
Figure 13.6 The impact of a product-package system
At Point A,the product is ready to fall,The potential energy
at this moment is,
13.5
where
At Point C,the kinetic energy of the system is given by
13.6
13.7
which show that the kinetic energy at this moment is equal
to the initial potential energy of the system,
In the moments following the initial contact of the container
with the impact surface,the amount of energy absorbed by
the linear cushion is,
13.8
hWghMPE 22 ??
gMW 22 ?
2
22
1
IvMKE ?
hWghMghMKE 222 )2(21 ???
2
222
1 xkE ?
whereX2is the downward displacement of the product on the
cushion,
At Point D,the cushion has absorbed all of the system's kinetic
energy
13.9
where maximum dynamic compression,dm= maximum value
of x2,
The maximum amount of energy absorbed by the cushion must
equal the system's kinetic energy at impact,
13.10
for the maximum dynamic compression,
13.11
2
2m a x 2
1
mdkE ?
2
2m a x2 2
1
mdkEhWKE ???
2
22 2
1
mdkhW ?
2
22
k
hWd
m ?
The static deflection of the product on the linear cushion is,
so Equation 13.11 may also be written as,
13,12
The maximum force exerted upward by the cushion against
the product occurs when,
13.13
The maximum acceleration (or deceleration) experienced by
the product,Gm,may be found from the relationship,
2
2
k
W
st ??
stm hd ?2?
2
2
2222m a x
2
k
hWkdkxkP
m ???
hWkP 22m a x 2?
13.14
13.15
where Gm is unitless,but is understood to be in units of 1 g,
13.16
Gm is proportional to the square root of the drop height,
This means that if we double the drop height,the magnitude
of the shock is not doubled,but rather increased by a factor
of (about 1.4),
Equation 13.15 may be rearranged to produce an expression
for,
13.17
2
m a x
W
PG
m ?
2
2
2
22 22
W
hk
W
hWkG
m ??
st
m
hG
?
2?
hG m ?
h
GWk m
2
2
2
2 ?
Equation 13.11 may also be rearranged to produce an
equivalent expression,
13.18
13.19
giving us an expression for the maximum dynamic
compression as a function of the maximum acceleration
and the drop height,
Conclusions,
1,is related to the drop height,the springiness of the
cushion and the weight of the product,
2,The stiffer the spring or cushion,the larger the value for,
3,The higher the drop height,the larger the value for(),
4,The heavier the product,the smaller the value of as long
as the spring or cushion is working,
2
2
2
2
md
hWk ?
m
m G
h
W
hk
h
hk
W
hd
2
2
2
2
2
2
22
2 ???
Shock Duration
We may approximate the displacement of the product on
the cushion,,at any time during the impact with the function,
13.20
where
13.21
f2 - the natural frequency of the product (M2) on the cushion
(k2),
)s i n ()( 22 tdtx m ??
22 2 f?? ?
2
2
2 2
1
W
gkf
??
Figure 13.7 Shock duration
and maximum displacement Figure 13.8 Shock duration and the natural period of
vibration
T2 - The period of the free vibration of the product on the
cushion,
13.22
where - the time length or duration of the shock,
For any shock of known period,we may calculate the
equivalent shock frequency,
13.23
the shock duration,
13.24
13.25
?21 2
2
?? Tf
?2
1
2 ?f
2
222
1
W
gkf
?
? ??
gk
W
2
2?? ?
Shock Amplification and the Critical
Element
Define,
Ge - the maximum acceleration experienced by the critical
element,resulting from a shock( Gm and),as shown in
Figure 13.9,
Am - an amplification factor,analogous to vibration
magnification relating the input and output shock levels,
13.26
Ge = Am Gm 13.27
m
e
m G
GA ?
Figure 13.9 Shock transmission
During the impact,the amplification factor is given by the
expression,
13.28
where N - an integer
f2 - the equivalent shock frequency
fl - the natural frequency of the critical element
Just after impact,the amplification factor is,
13.29
)1(
2
s in
)1(
)0(
2
1
2
1
2
1
??
???
f
f
N
f
f
f
f
tA m
?
?
2
2
1
2
1
2
1
)(1
)
2
c o s ()(2
)(
f
f
f
f
f
f
tA m
?
?? ?
The largest value of Am for Equation 13.27 depends on the
relationship of and,
1,If
(Equation 13.29)
2,If
(Equation 13.28)
3.Notice that as the ratio becomes large,the value for Am
approaches 1,reflecting the "direct" transfer of the shock to a
stiff element,
4.For small values of the frequency ratio,
,2121o r,1 2121
2
1 ????? TTff
f
f
)( ??? tAA mm
,2121o r,1 2121
2
1 ????? TTff
f
f
)0( ???? tAA mm
20.0),(2
2
1 ??
2
1
m f
f f o r
f
fA
