Chapter 8. Pressurized Cylinders and Spinning Disks

Transcription

1 Chpte Pessuized Cylindes nd Spinning Disks Govening qutions x, yz,, θ, z x cosθ, y sin θ nd x + y, θ tn y x xisymmety τ z τ zθ nd γ z γ zθ constitutive eltionships z (o ε z with supeposition B ρω nd Bz B θ τ θ G γ γ θ θ τθ G α ε + νεθ T ν ( Δ ν ε νθ + α Δ T θ α εθ + νε T ν ( Δ ν εθ θ ν + α Δ T ε ν + θ z + α Δ T equiliium equtions θ τθ B θ + θ + ρω

2 θ τ τ + θ + θ + Bθ ( utomtic θ stin-displcement eltionships ε u, εθ u u + θ θ u, γ θ u uθ u + θ θ diffeentil field eqution d d ( u ( dt ν d α + ν ρω d d solution y integtion d ( d u ( + T ( + C ν α ν ρω d d u T C ( ( + ( + ν α ν ρω 3 ν 4 u α ( +ν T( d + C + C ρω ( C u α + ν T( d + C + ν ρω 3 without theml stesses C u C + ν ρω 3 ε + νε θ ν ( θ εθ + νε ν (

3 ε u C C 3 ( ν ρω ν εθ u + C C ρω ν ν ν + ν C 3 + [( C ] ρω θ ν ν ν + ν + C C + 3 [( ] ρω Thick-Wll Pessue Cylindes p o p i θ z z ν ν [( + ν C C ] θ ν ν [( + ν C + C ]

4 oundy conditions ( p o nd ( p i p i ( p o ( θ pi ( + po ( + p o p i + -p i p i θ p i p o θ -p o + - po ε ν θ ν + pi po z with xil stess p i po z εz z ν θ ν + pi p o thin-wll pessue vessels unde intenl pessue h p i t

5 θ( + θ( pi + 3 mximum in-plne she stess p i t p i ( p o ( θ pi ( + po ( + τmx θ p i po τmax τmx( ( pi po τ MAX pguge > pguge stess concenttion

6 Shink Fits, Compound Cylindes mximum in-plne she stess in simple pessue cylinde τ MAX Y, whee τ MAX pguge > pguge p guge Y < ( mximum in-plne she stess in compound pessue cylinde p o c p i p guge < Y ( Mximum Guge Pessue/Yield Stength compound cylinde simple cylinde Inside Rdius/Outside Rdius

7 Supeposition of stesses fom shink-fit pessue pc inside shell outside shell p c J I D C c p c c B A c A pc, B c c + I pc, J pc c + c C pc, D pc c fom intenl pessue pi + pi H G c F pi F p i G pi ( + c H pi ( c

8 comined stesses mximum she in inside shell ( + ( + ( inside mx ( A B F MAX τ τ mximum she in outside shell ( outside ( ( mx( C + G c D + τ τ H MAX optimum design (simultneous filue τ τ ( inside ( outside Y MAX MAX ( c c pc + p i c c c pi Y c ( c optimum ptition c pi Y ( p c pi + equied offset Δ u co u ci ( outside ( inside ci u u ( c nd u u ( c co ( outside C νd ( inside u nd I νj co cε c uci cε c θ θ

9 3 C D I J c pc c Δ c ν ν ( c ( c Δ + p c Δ pi equied tempetue diffeence Δ α ΔTc p i ΔT p i α

10 Spinning Disks Flywheel A θ dmrω dθ A θ ω R t thin im (t << R, few spokes: the spokes exet no dil foces on the im of negligile flexul stiffness etween spokes dm Rω ρa RdθRω A dθ θ θ ρr ω εθ u θ ρ ω R R R R ur ρ 3 ω

11 Bicycle wheel A θ Δm Rω A θ T Δθ ω n spokes R t ( spoke ( spoke (, θ ( im, ( im, θ θ ρaδθ R ω A θ Δθ + T Δθ π n oundy condition u ( R im u ( R spoke u ( R im R θ R ( R spoke ur ε ( d ( d ( + d S ( S + ρsdω d d ρ ω

12 ρω d + C ρω + C ( R ρω R + C T S C T + ρω S R T + ρω ( R S ( R spoke R 3 T TR ρω R u [ + ρω ( R ] d + S S 3 TR S 3 ρω R + 3 R θ nd ρar ω A + θ nt π T S ρω R nt + ρr ω 3 π A T ρω R n 3( + S π A θ ρω R [ ns 3 ( + π A + ] 3 Flt disks without theml stesses C u C + ν ρω 3 ε + νε θ ν ( θ εθ + νε ν ( ε u C C 3 ( ν ρω ν εθ u + C C ρω

13 ν ν + ν C 3 + [( C ] ν ρω θ ν ν ν + ν + C C + 3 [( ] ρω Flt solid disk C 3 + ν ( ( + νc ρω ν ν ( + ν C 3 + ν ρω 3 + ν ρω ( θ 3 + ν ρω ν + 3 ( 3 + ν in-plne θ 3 3 mx +ν ( 6 + ν ν τ ρω ρω 3 +ν out-of-plne θ 3 +ν + 3ν mx τ ρω ( 6 3 +ν 3 + ν τ MAX τ ρω 6 out-of-plne mx ( Flt disks with centl hole ( ( 3 + ν ρω ( +

14 θ 3 + ν ρω ν ( 3 + ν ν ( mx ( 3 + ρω ( ν ( mx ρω ( in-plne θ 3 +ν ν mx τ ρω ( + 3 +ν in-plne 3 +ν ν MAX mx τ τ ( ρω ( + 3 +ν out-of-plne θ 3 +ν + 3ν mx τ ρω ( ν out-of-plne 3 +ν ν MAX mx τ τ ( ρω ( + 3 +ν Nomlized stess θ disk with hole solid disk Nomlized Rdius, / /,. ν 3., 3 + ν ρω

15 Spinning disk with vile thickness h+dh ( +d θ θ θ ( +d d B h z ( dθ θ ( θ θ d O equiliium equtions in the -diection: ( + d( + d( h + dh dθ ( h dθ d h dθ + B h d dθ θ d( h h d θ + B d θ dh ρω d h d ε + νε θ ν ( θ εθ + νε ν ( ε du ε θ u d hypeolic pofile d u du u du u dh ν + + d d + ν ρω h d d h H s

16 dh d s H s s+ h d u du u s du u ν + d d + ν ρω d Disk of unifom stength nd θ z h z τ mx du d u h u ν + d u du u du u dh ν + + ν ρω d d h d d u dh ( + ν ν ρω h d ν dh ν ρω h d dh h ρω d ln h ρω ρω + C + ln h

17 h ln h h ρω L N M ρω h exp The disk cnnot hve centl hole! O Q P