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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 shocksz]SS%Scc:ccc4  \  /Figure 13.1 Representation of mechanical shock 00b/   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. Jngg[g  hThe 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. SDSSdSSSe3aaBaasaaah   Figure 13.2 A falling package$ `b(   V 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.JW9( U   5Figure 13.3 The relationship among shock parameters 665   fPackage 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: N0jCCCCG 3CCf  +Figure 13.4 A simple 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: II71]B   I   3Figure 13.6 The impact of a product-package system443    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` 3   ! 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>M6   * hThe 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: >i@(h   + 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 M5wBM      1Equation 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. j G9   9Shock 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). t)  :5Figure 13.7 Shock duration and maximum displacement 665  ;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 :   C-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 13.27Z!N   ]     DFigure 13.9 Shock transmission  EDuring 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: 13.29E #    F6The largest value of Am for Equation 13.27 depends on the relationship of and : 1. If (Equation 13.29) 2. 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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 shocksz]SS%Scc:ccc4  \  /Figure 13.1 Representation of mechanical shock 00b/   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. Jngg[g  hThe 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. SDSSdSSSe3aaBaasaaah   Figure 13.2 A falling package$ `b(   V 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.ZWcgg9g( U   5Figure 13.3 The relationship among shock parameters 66b5   fPackage 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: N0jccccg 3cc`f  +Figure 13.4 A simple product-package system,,c+     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: IIbccc7cc1cc]cccB   I   3Figure 13.6 The impact of a product-package system44c3    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: 13.8t gggg3g   ! 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: 13.11JMgg6g   * hThe 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: Ji@gg(gh   + 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 Mcgg5ggggwgcBM      1Equation 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. g ggg9gb   9Shock Duration (c(`  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). ccc)ccccc  :5Figure 13.7 Shock duration and maximum displacement 66c5  ;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 Fggc   C-Shock Amplification and the Critical Element ..c -  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 13.27ccZcc!cccccccccN   ]     DFigure 13.9 Shock transmissionc  EDuring 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: 13.29Egg gg#gg g   F6The 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: 7gghgRgggg6    K0  P( ; l  C Z-<   H  0޽h ? ̙33y___PPT10Y+D=' = @B +r s? rIm(    lQ_ Equation.30 Microsoft lQ_ 3.0 lQ_ !-lQ_ Equation.30 Microsoft lQ_ 3.0".lQ_ Equation.30 Microsoft lQ_ 3.0#/lQ_ Equation.30 Microsoft lQ_ 3.0$0lQ_ Equation.30 Microsoft lQ_ 3.0%2lQ_ Equation.30 Microsoft lQ_ 3.0&3lQ_ Equation.30 Microsoft lQ_ 3.0'4lQ_ Equation.30 Microsoft lQ_ 3.0(5lQ_ Equation.30 Microsoft lQ_ 3.0)6lQ_ Equation.30 Microsoft lQ_ 3.0*7lQ_ Equation.30 Microsoft lQ_ 3.0+8lQ_ Equation.30 Microsoft lQ_ 3.0,<lQ_ Equation.30 Microsoft lQ_ 3.0-=lQ_ Equation.30 Microsoft lQ_ 3.0.>lQ_ Equation.30 Microsoft lQ_ 3.0/?lQ_ Equation.30 Microsoft lQ_ 3.00@lQ_ Equation.30 Microsoft lQ_ 3.01AlQ_ Equation.30 Microsoft lQ_ 3.02BlQ_ Equation.30 Microsoft lQ_ 3.03GlQ_ Equation.30 Microsoft lQ_ 3.04HlQ_ Equation.30 Microsoft lQ_ 3.05IlQ_ Equation.30 Microsoft lQ_ 3.0/ 0DTimesa Gt+8 08z[ 00DGaramondGt+8 08z[ 00 DWingdingst+8 08z[ 000DTimes New Roman8 08z[ 00@D[SOes New Roman8 08z[ 00PDFranklin Gothic Medium8z[ 00"`DVerdana Gothic Medium8z[ 00" A .@  @@``   @n?" dd@  @@``   @'         !"$%/b$$L܆kZRg1/ R$v$<)7E'As< 2$惽۱NW32$NIFE^jw@ǣ2$O[ޞpjR$v ʗ9'-J2$Rjb9eE0XN%2$[8d ť^ ds2$T|J]2$" m+Q%f춮R$E5 gZ *Y8?R$S!94#6R$Q+mVa{ QCZ2$cY%htʑ?l2$+MV:ދkc2$Sk >b/xo}gί2$3} Kl!<052$ɗP汰. E²2$[2]ߺt=E`2$x : ޜAU 2$+:ު kn\y2$h~JyrBj2$歸ñodr$/2$%/2$q@|%\mnE2$}?ƅ2$8{Z/932$/Jkʋ2$Iv)ɳ>!2$braK*KΙ -#72$biQL(Xs1TF2$#^ŪJg2$}I4yi62$І.^2$'?""#p72$ifY<.z%5+Md2$:5sHR$F"* Emi12$iVd6ga2$(lجgz2$1?kaF2$ }ݽ1L&뼼b2$w@nggȝtR$&Bbʯ~>[[V+9!2$:S? @XIL2$ <qSr* 2$bpCxp2$rȉiD sr $2$>s-! [.2$7t`8 xr=s AA1? 3̙333@83 9x ʚ;ʚ;g4NdNdPz[ 0ppp@ <4dddd| 0L+80___PPT10 pp? %O =E`Lesson 13 Mechanical Shock Theory ,{13 :ghQQt$1%a `4%       Introduction c    (Throughout the distribution system, packages are manhandled and mishandled in various ways: dropped, thrown, kicked and otherwise roughly abused; fall from conveyors or forklifts and crash to the floor; subjected to a variety of vehicle impacts; trucks starting, stopping, hitting chuckholes and railroad crossings, railcar humping, jolting and other moderately violent actions; suffers an impact with another objectfloor, truck bed, pallet, bulkhead or another package. A mechanical shock occurs when an object's position, velocity or acceleration suddenly changes. Such a shock may be characterized by a rapid increase in acceleration followed by a rapid decrease over a very short period of time. Figure 13.1: the acceleration versus time plot for most shocksz]SS%Scc:ccc4  \  /Figure 13.1 Representation of mechanical shock 00b/   A package shock may typically be 20 milliseconds (0.020 seconds) long and have a magnitude or "height" of 150 g's. need to know both the magnitude of the acceleration and the duration of the shock. Jngg[g  hThe Free Falling Package the length of time it takes a package to fall from a drop height, h the downward velocity at which it will be traveling a moment before impact; , the impact velocity: As is shown in Figure 13.2. A package will rebound a little or a lot depending on the nature of the package and the surface it hits. SDSSdSSSe3aaBaasaaah   Figure 13.2 A falling package$ `b(   V 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.ZWcgg9g( U   5Figure 13.3 The relationship among shock parameters 66b5   fPackage 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: N0jccccg 3cc`f  +Figure 13.4 A simple product-package system,,c+     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: IIbccc7cc1cc]cccB   I   3Figure 13.6 The impact of a product-package system44c3    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: 13.8t gggg3g   ! 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: 13.11JMgg6g   * hThe 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: Ji@gg(gh   + 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 Mcgg5ggggwgcBM      1Equation 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. g ggg9gb   9Shock Duration (c(`  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). ccc)ccccc  :5Figure 13.7 Shock duration and maximum displacement 66c5  ;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 Fggc   C-Shock Amplification and the Critical Element ..c -  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 13.27ccZcc!cccccccccN   ]     DFigure 13.9 Shock transmissionc  EDuring 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: 13.29Egg gg#gg g   F6The 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: 7gghgRgggg6    K0 p(  x  c $qF {  q    0e0e]l'.A  73-2#" `|  q  c 0e0ePb'-A  73-2#" `z   qH  0޽h ? ̙33___PPT10i.t3pgE+D=' = @B +r{P? TIs