CHAPTER 12 TWODEGREE OFFREEDOMSYSTEMS


 Judith Morris
 2 years ago
 Views:
Transcription
1 CHAPTER TWODEGREE OFFREEDOMSYSTEMS Introdution to two deree of freedo systes: The vibratin systes, whih require two oordinates to desribe its otion, are aed twodereesof freedo systes. These oordinates are aed eneraized oordinates when they are independent of eah other and equa in nuber to the derees of freedo of the syste. Unie sine deree of freedo syste, where ony one oordinate and hene one equation of otion is required to epress the vibration of the syste, in twodof systes iniu two oordinates and hene two equations of otion are required to represent the otion of the syste. For a onservative natura syste, these equations an be written by usin ass and stiffness atries. One ay find a nuber of eneraized oordinate systes to represent the otion of the sae syste. Whie usin these oordinates the ass and stiffness atries ay be ouped or unouped. When the ass atri is ouped, the syste is said to be dynaiay ouped and when the stiffness atri is ouped, the syste is nown to be statiay ouped. The set of oordinates for whih both the ass and stiffness atri are unouped, are nown as prinipa oordinates. In this ase both the syste equations are independent and individuay they an be soved as that of a sinedof syste. A twodof syste differs fro the sine dof syste in that it has two natura frequenies, and for eah of the natura frequenies there orresponds a natura state of vibration with a dispaeent onfiuration nown as the nora ode. Matheatia ters assoiated with these quantities are eienvaues and eienvetors. Nora ode vibrations are free vibrations that depend ony on the ass and stiffness of the syste and how they are distributed. A nora ode osiation is defined as one in whih eah ass of the syste underoes haroni otion of sae frequeny and passes the equiibriu position siutaneousy. The study of twodof systes is iportant beause one ay etend the sae onepts used in these ases to ore than dof systes. Aso in these ases one an easiy obtain an anaytia or osedfor soutions. But for ore derees of 09
2 freedo systes nueria anaysis usin oputer is required to find natura frequenies (eienvaues) and ode shapes (eienvetors). The above points wi be eaborated with the hep of eapes in this eture. Few eapes of twodereeoffreedo systes Fiure shows two asses and with three sprins havin sprin stiffness, and 3 free to ove on the horizonta surfae. Let and be the dispaeent of ass and respetivey. 3 Fiure As desribed in the previous etures one ay easiy derive the equation of otion by usin d Aebert prinipe or the enery prinipe (Larane prinipe or Haiton s prinipe) Usin d Aebert prinipe for ass free body diara shown in fiure (b) + ( + ) = 0, fro the () ( ) and siiary for ass + ( + ) = 0 () 3 Iportant points to reeber Inertia fore ats opposite to the diretion of aeeration, so in both the free body diaras inertia fores are shown towards eft. For sprin, assuin >, ( ) 3 Fiure (b), Free body diara 0
3 The sprin wi pu ass towards riht by ( ) and it is strethed by (towards riht) it wi eert a fore of ( ) towards eft on ass. Siiary assuin of >, the sprin et opressed by an aount and eert tensie fore ( ). One ay note that in both ases, free body diara reain unhaned. Now if one uses Larane prinipe, The Kineti enery = T = + and (3) Potentia enery = U = + ( ) + 3 (4) So, the Laranian L = T U = + + ( ) + 3 The equation of otion for this free vibration ase an be found fro the Larane prinipe (5) d L L = 0, (6) dt q q and notin that the eneraized oordinate q = and q = whih yieds + ( + ) = 0 + ( + ) = 0 (8) 3 Sae as obtained before usin d Aebert prinipe. (7) Now writin the equation of otion in atri for = 0 +. (9) Here it ay be noted that for the present two dereeoffreedo syste, the syste is dynaiay unouped but statiay ouped.
4 Eape. Consider a athe ahine, whih an be odeed as a riid bar with its enter of ass not oinidin with its eoetri enter and supported by two sprins,,. ( ) C J e G ( + e ) ( + ) Fiure Fiure 3: Free body diara of the syste In this eape, it wi be shown, how the use of different oordinate systes ead to stati and or dynai ouped or unouped equations of otion. Ceary this is a twodereeof freedo syste and one ay epress the oordinate syste in any different ways. Fiure 3 shows the free body diara of the syste where point G is the enter of ass. Point C represents a point on the bar at whih we want to define the oordinates of this syste. This point is at a distane fro the eft end and between points C and G is e. Assuin fro riht end. Distane is the inear dispaeent of point C and the rotation about point C, the equation of otion of this syste an be obtained by usin d Aeber s prinipe. Now suation of a the fores, viz. the sprin fores and the inertia fores ust be equa to zero eads to the foowin equation. + e + ( ) + ( + ) = 0 (0) Aain tain oent of a the fores about point C J + ( + e ) e ( ) + ( + ) = 0 () G Notin J J e = G +, the above two equations in atri for an be written as e + 0 e J + = () + 0
5 Now dependin on the position of point C, few ases an are studied beow. Case : Considerin written as e = 0, i.e., point C and G oinides, the equation of otion an be ( ) ( + ) J + G = (3) + 0 So in this ase the syste is statiay ouped and if =, this oupin disappears, and we obtained unouped and vibrations. Case : If, =, the equation of otion beoes e e J + 0 =. (4) + 0 Hene in this ase the syste is dynaiay ouped but statiay unouped. Case 3: If we hoose otion wi beoe = 0, i.e. point C oinide with the eft end, the equation of e + 0. (5) e J + = 0 Here the syste is both statiay and dynaiay ouped. Nora Mode Vibration Aain onsiderin the probe of the sprinass syste in fiure with =, = 3, = = =, the equation of otion (9) an be written as + ( ) + = 0 ( ) + = 0 (6) 3
6 We define a nora ode osiation as one in whih eah ass underoes haroni otion of the sae frequeny, passin siutaneousy throuh the equiibriu position. For suh otion, we et iωt = Ae, = Ae (7) i ω t Hene, ( ω ) A A = 0 A + ( ω ) A = 0 or, in atri for ω A 0 = ω A 0 Hene for nonzero vaues of A and A (i.e., for nontrivia response) (8) (9) ω = 0. (0) ω Now substitutin 3 ω = λ, equation 6.. yieds λ (3 ) λ+ ( ) = 0 () Hene, λ = 3 ( 3) = and 3 λ = ( + 3) =.366 So, the natura frequenies of the syste are ω = λ = and ω =.366 Now fro equation ()., it ay be observed that for these frequenies, as both the equations are not independent, one an not et unique vaue of and A. So one shoud A find a noraized vaue. One ay noraize the response by findin the ratio of A to A. Fro the first equation (9) the noraized vaue an be iven by A = = A λ ω and fro the seond equation of (9), the noraized vaue an be iven by () A ω λ = = (3) A Now, substitutin ω = λ = in equation () and (3) yieds the sae vaues, as both these equations are ineary dependent. Here, 4
7 A A λ λ = = 0.73 (4) and siiary for ω = λ =.366 A = A λ λ =.73 (5) It ay be noted Equation (9) ives ony the ratio of the apitudes and not their absoute vaues, whih are arbitrary. If one of the apitudes is hosen to be or any nuber, we say that apitudes ratio is noraized to that nuber. The noraized apitude ratios are aed the nora odes and desinated by φ ( ). n Fro equation (4) and (5), the two nora odes of this probe are: φ( ) φ( ) = = In the st nora ode, the two asses ove in the sae diretion and are said to be in phase and in the nd ode the two asses ove in the opposite diretion and are said to be out of phase. Aso in the first ode when the seond ass oves unit distane, the first ass oves 0.73 units in the sae diretion and in the seond ode, when the seond ass oves unit distane; the first ass oves.73 units in opposite diretion. Free vibration usin nora odes When the syste is disturbed fro its initia position, the resutin freevibration of the syste wi be a obination of the different nora odes. The partiipation of different odes wi depend on the initia onditions of dispaeents and veoities. So for a syste the free vibration an be iven by = φ Asin( ωt+ ψ ) + φ Bsin( ω t+ ψ ) (7) 5
8 Here A and B are part of partiipation of first and seond odes respetivey in the resutin free vibration and ψ and ψ are the phase differene. They depend on the initia onditions. This is epained with the hep of the foowin eape. Eape: Let us onsider the sae sprinass probe (fiure 4) for whih the natura frequenies and nora odes are deterined. We have to deterine the resutin free vibration when the syste is iven an initia dispaeent (0) = 5, (0) = and initia veoity (0) = (0) = 0. Fiure 4 Soution: Any free vibration an be onsidered to be the superposition of its nora odes. For eah of these odes the tie soution an be epressed as: 0.73 = sin ωt.73 = sin ωt.00 The enera soution for the free vibration an then be written as: = A sin( ωt+ ψ) + B sin( ωt+ψ).00 where A and B aow different aounts of eah ode and ψ and ψ aows the two odes different phases or startin vaues. Substitutin: 6
9 (0) = = A sinψ+ B sinψ (0) (0) = = A os + B os (0) 0 ω ψ ω ψ osψ = osψ = 0 => ψ = ψ = 90 0 Substitutin in st set: = A + B 0.73A.73B= 5 A=.33 A+B = B=.33 Hene the resutin free vibration is =.33 osωt.33 osωt Nora odes fro eienvaues The equation of otion for a twodereeof freedo syste an be written in atri for as M + K = 0 (8) where M and K are the ass and stiffness atri respetivey; is the vetor of eneraized oordinates. Now preutipyin ay et I M K M in both side of equation 6.. one + = 0 (9) or, I + A= 0 (30) Here = A M K is nown as the dynai atri. Now to find the nora odes, iωt = Xe, = Xe, the above equation wi redue to i ω t [ A λi] X 0 = (3) 7
10 where { } T X = and λ= ω. Fro equation (3) it is apparent that the free vibration probe in this ase is redued to that of findin the eienvaues and eienvetors of the atri A. Eape: Deterine the nora odes of a doube penduu. Soution Kineti enery of the syste = T = + ( + + os( )) Potentia enery of the syste = ( os ) { ( os ) ( os )} {( ) ( os ) ( os )} U = + + = + + So Laranian of the syste = L= T U = + ( + + os( )) ( + )( os ) + ( os ) So usin Larane prinipe, and assuin sa ane of rotation, the equation of otion an be written in atri for as ( + ) ( + ) = 0 0 Fiure 5 { } Now onsiderin a speia ase when = = and = =, the above equation beoes = or, + = Now A= = 0 To find eienvaues of A, 8
11 λ A λi = 0 = 0 λ Or, 4 4 λ+ λ = 0 Or, λ 4 λ+ = 0 4 ± 4 8 Or, λ = = ( ± ) Hene natura frequenies are ω = , ω =.8478 The nora odes an be deterined fro the eienvaues. The orrespondin prinipa odes are obtained as = = ( ) = = ( ) λ= λ + λ= λ It ay be noted that whie in the first ode Both the penduu oves in the sae diretion, Fiure 6 In the seond ode they ove in opposite diretion One ay sove the sae probe by tain and Here is the horizonta distane oves by ass ass T as the eneraized oordinates. and. Fiure 7 show the free body diara of both the asses. T y y Fiure 7 T is the distane ove by 9
12 Fro the free body diara of ass, T os = Tsin = Aso fro the free body diara of ass, T os T os = T sin T sin + = 0 Assuin and to be sa, sin = tan = = / and sin = tan = = ( ) / Hene T =,and T = ( + ) ( + ) = 0 + = Hene in atri for 0 ( + ) = 0 Considerin the ase in whih = = and = =, the above equation beoes = 0 0
13 3 A = 3 λ and A λi = 0 = 0 Hene λ Or, λ 4 λ+ = 0 ( ) and λ ( ) or, λ = = + Sae as those obtained by tain and as the eneraized oordinates. Now X X λ= λ X X λ= λ = = = = λ = = = =.44 3 λ Fiure The different odes are as shown in the above fiure. Eape Deterine the equation of otion if the doube penduu is started with initia onditions (0) = (0) = 0.5, (0) = (0) = 0. Soution: The resutin free vibration an be onsidered to be the superposition of the nora odes. For eah of these odes, the tie soution an be written as
14 X X = sinω t = sinω t X X The enera soution for the free vibration an be written as = Asin( ωt+ ψ) + Bsin( ωt+ ψ) where Aand B are the aounts of first and seond ode s partiipation and ψ andψ are the startin vaues or phases of the two odes. Substitutin the initia onditions in the above equation = Asinψ+ Bsinψ 0.5 and = Aω osψ+ Bωosψ 0 0 For the seond set of equations to be satisfied, osψ = osψ = 0, so that ψ = ψ = 90. Hene A = and B = So the equation for free vibration an be iven by = osωt0.036osωt Dapedfree vibration of twodof systes Consider a two derees of freedo syste with dapin as shown in fiure 3 3 Fiure 9 Now the equation of otion of this syste an be iven by = + + (3) 0 3 3
15 As in the previous ase, here aso the soution of the above equations an be written as st = Ae and = Ae (33) st where A, A and s are onstant. Substitutin (33) in (3), one ay write s + ( + ) s+ + s A 0 = s s + ( + 3) s+ A Now for a nontrivia response i.e., for nonzero vaues of their oeffiient atri ust vanish. Hene A and A (34), the deterinant of s + ( + ) s+ + s s s + ( + 3) s+ + 3 = 0 or, ( s + ( + ) s+ + )( s + ( + ) s+ + ) + ( s+ ) = whih is a fourth order equation in s and is nown as the harateristi equation of the syste. This equation is to be soved to et four roots. The enera soution of the syste an be iven by st st st 3 s4t = Ae + Ae + Ae + Ae 3 4 = A e + A e + A e + A e st st st 3 s4t 3 4 Here A,,,3,4 i i = onditions and the oeffiients fro equation (34) as (35) (36) (37) are four arbitrary onstants to be deterined fro the initia A,,,3,4 i i = are reated to A i and an be deterined A s + = A s + + s + + i i i i ( ) i For a physia syste with dapin, the otion wi die out with tie. For a stabe syste, a the four roots ust be either rea neative nubers or ope nuber with neative rea parts. It ay be reaed that, if the roots ontain ope onjuate nubers, the otion wi be osiatory. (38) Eape: Find the response of the syste as shown in fiure 9 onsiderin = = 3 = and = 3 = 0and =. Soution. In this ase the harateristis equation beoes ( s s )( s s ) ( s ) = 0 = =, 3
16 ( s + s + ) ( s + ) = or, s + s + (4 + ) s + (4 ) s + 4 = 0 4 3, = 0 or s s s s, ( ) + ( ) = 0 or s s s s s,( + )( ) = 0 or s s s,( + )( ) = 0 or s s s Hene the roots are s, =± i and s3,4 = ± 3 So the syste has a pair of ope onjuate SEMIDEFINITE SYSTEMS The systes with have one of their natura frequenies equa to zero are nown as seidefinite or deenerate systes. One an show that the foowin two systes are deenerate systes. I I Fiure 0 Fiure Fro fiure 0 the equation of otion of the syste is 0 0 = (39) iωt iωt Assuin the soution = Ae and = Ae (40) ω A 0 = A ω 0 (4) So for nonzero vaues of A, A, 4
17 ω = 0 ω ( ω )( ω ) or, = 0 4 or, ( ) ω ω 0 (4) (43) + + = (44) or, ω ( ω ( + )) = 0 (45) ω = 0, and, ω = ( + ) Hene, the syste is a seidefinite or deenerate syste. Correspondin to the first ode frequeny, i.e., ω = 0, A = A. So the syste wi have a riidbody otion. For the seond ode frequeny A = = = = A ω ( + ) apitude ratio is inversey proportiona to the ass ratio the syste. Siiary one ay show for the tworotor syste, I = (48) I the ratio of ane of rotation inversey proportiona to the oent of inertia of the rotors. (46) (47) Fored haroni vibration, Vibration Absorber Consider a syste eited by a haroni fore F sinωtepressed by the atri equation F sinωt + = 0 (49) Sine the syste is undaped, the soution an be assued as X = sinωt X Substitutin equation (50) in equation (49), one obtains ω ω X F sinωt = sin t X ω ω ω 0 X ω ω F or, = X ω ω 0 (50) (5) 5
18 X ω F ω = X 0 ω ω Hene X ω + F ω 0 + ω ω = ω ω ( ω ), ω ω F = (53) Z( ω) [ Z ω ] where ( ) X = ( ω ) ω ω = ω ω F Z( ω) Eape Consider the syste shown in fiure where the ass fore Fsinω t. Find the response of the syste when = and = = 3. (54) (5) is subjeted to a F sinω t 3 Fiure Soution: The equation of otion of this syste an be written as 0 + F sinωt 0 + = Fsinωt 0 + = 0 So assuin the soution 6
19 X = sinωt and proeedin as epained before X [ Z( ω) ] ω = ω 4 4 Z( ω) = ( ω ) = ω 4ω + 3 = ( ω 4 ω + 3 ) or, Z( ω ) = ( ω )( 3 ) ( )( ) ω = ω ω ω ω where, ω = and ω = 3 are nora ode frequenies of this syste. Hene, X X = ( ω ) F ω ω ω ω = ( )( ) F ( )( ) ω ω ω ω So it ay be observed that the syste wi have aiu vibration when ω = ω or, ω = ω. Aso it ay be observed that X = 0, when ω = /. Tuned Vibration Absorber Consider a vibratin syste of ass, stiffness, subjeted to a fore Fsinω t. As studied in ase of fored vibration of sinederee of freedo syste, the syste wi have a steady state response iven by Fsin t = ω,where ωn / = ( ωn ω ) (55) whih wi be aiu when ω = ω n. Now to absorb this vibration, one ay add a seondary sprin and ass syste as shown in fiure 3. Fsinωt Fiure 3 7
20 The equation of otion for this syste an be iven by 0 + Fsinωt 0 + = 0 Coparin equation (49) and (56), (56) =, = 0, = 0, = = +, =, =, and =., Hene, Z( ω) = = ω ω ω + ω 4 + ω ω = ( λω )( λ ω ) (57) where λ and λ are the roots of the harateristi equation Z( ω ) = 0 of the freevibration of this syste., whih an be iven by λ, = ± Now fro equation (53) and (54) X X ( ω ) ( ω ), (58) F F = = (59) Z( ω) Z( ω) F Z( ω) = (60) Fro equation (59), it is ear that, X = 0, when ω =. Hene, by suitaby hoosin the stiffness and ass of the seondary sprin and ass syste, vibration an be opetey eiinated fro the priary syste. For Z( ω ) = + = + = F F and X = = ω =, (6) (6) 8
21 Centrifua Penduu Vibration Absorber The tuned vibration absorber is ony effetive when the frequeny of eterna eitation equas to the natura frequeny of the seondary sprin and ass syste. But in any ases, for eape in ase of an autoobie enine, the eitin torques are proportiona to the rotationa speed n whih ay vary over a wide rane. For the absorber to be effetive, its natura frequeny ust aso be proportiona to the speed. The harateristis of the entrifua penduu are ideay suited for this purpose. Pain the oordinates throuh point O, parae and nora to r, the ine r rotates with anuar veoity ( + φ ) ĵ î r R O O The aeeration of ass ˆ a os sin ( ) sin os ( ) ˆ = R φ + R φ r + φ i + R φ + R φ + r + φ j (63) Sine the oent about O is zero, MO = R sinφ + R os φ + r( + φ) r = 0 (64) Assuin φ to be sa, osφ =, sinφ = φ, so R R+ r r r (65) φ + φ = If we assue the otion of the whee to be a steady rotation osiation of frequeny ω, one ay write n pus a sa sinusoida nt sinωt 0 (66) nt osωt 0 (67) =ω 0 sinωt (68) Substitutin the above equations in equation (65) yieds, 9
22 R R+ r sin 0 t r r ω (69) φ + n φ = ω Hene the natura frequeny of the penduu is R ω n = n (70) r and its steadystate soution is ( R+ r)/ r = t (7) ω + ( Rn / r) φ ω 0 sin ω It ay be noted that the sae penduu in a ravity fied woud have a natura frequeny of r. So it ay be noted that for the entrifua penduu the ravity fied is repaed by the entrifua fied Rn. Torque eerted by the penduu on the whee With the ĵ oponent of a equa to zero, the penduu fore is a tension aon r, iven by ties the î oponent of a. ( osφ ˆ sinφ ˆ) osφ sin φ ( φ) T = R i + R j R R r iˆ + + =Rφ Rω 0 sinωt sinφrn rn r φ r φ Now assuin sa ane of rotation T = ( R+ r) n Rφ (73) Now substitutin the (73) in (7), R( R + r) n / r T = ( Rn / r) ω ω 0 sinωt R ( + r) = = J eff (74) rω / Rn Hene the effetive inertia an be written as J eff R ( + r) R ( + r) = = rω / Rn / ( ω ω ) n whih an be at its natura frequeny. This possesses soe diffiuties in the desin of the penduu. For eape to suppress a disturbin torque of frequeny equa to four ties the natura speed n, the penduu ust eet the requireent ω = (4 n) = n R/ r. Hene, as the enth of the penduu r = R/6 beoes very (7) (75) 30
23 sa it wi be diffiut to desin it. To avoid this one ay o for Chiton bifiar desin. Eerise probes. In a ertain refrieration pant, a setion of pipe arryin the refrierant vibrated vioenty at a opressor speed of 3 rp. To eiinate this diffiuty, it was proposed to ap a antiever sprin ass syste to the pipe to at as an absorber. For a tria test, for a 905. Absorber tuned to 3 p resuted in two natura frequenies of 98 and 7 p. If the absorber syste is to be desined so that the natura frequenies ie outside the reion 60 to 30 p, what ust be the weiht and sprin stiffness?. Derive the nora odes of vibration of a doube penduu with sae enth and ass of the penduu. 3. Deveop a atab ode for deterination of freevibration of a enera twoderee of freedo syste. 4. Derive the equation of otion for the doube penduu shown in fiure p in ters of and usin Larane prinipe. Deterine the natura frequenies and ode shapes of the systes. If the syste is started with the foowin initia onditions: (0) = (0) = X, v (0) =v (0)=0, (v and v are veoity) deterine the equation of otion. If the ower ass is iven an ipuse F 0 δ (t), deterine the response in ters of nora odes. L L Fiure P 5. A entrifua pup rotatin at 500 rp is driven by an eetri otor at 00 rp throuh a sine stae redution earin. The oents of inertia of the pup ipeer and the otor are 600. and 500. respetivey. The enths of the pup shaft and the otor shaft are 450 and 00, and their diaeters are 00 and 50 respetivey. Neetin the inertia of the ears, find the frequenies of torsiona osiations of the syste. Aso deterine the position of the nodes. 3
4/3 Problem for the Gravitational Field
4/3 Proble for the Gravitational Field Serey G. Fedosin PO bo 6488 Sviazeva str. 79 Per Russia Eail: intelli@list.ru Abstrat The ravitational field potentials outside and inside a unifor assive ball
More informationThe Simple Pendulum. by Dr. James E. Parks
by Dr. James E. Parks Department of Physics and Astronomy 401 Niesen Physics Buidin The University of Tennessee Knoxvie, Tennessee 37996100 Copyriht June, 000 by James Edar Parks* *A rihts are reserved.
More informationEnergy Density / Energy Flux / Total Energy in 3D
Lecture 5 Phys 75 Energy Density / Energy Fux / Tota Energy in D Overview and Motivation: In this ecture we extend the discussion of the energy associated with wave otion to waves described by the D wave
More informationAngles formed by 2 Lines being cut by a Transversal
Chapter 4 Anges fored by 2 Lines being cut by a Transversa Now we are going to nae anges that are fored by two ines being intersected by another ine caed a transversa. 1 2 3 4 t 5 6 7 8 If I asked you
More informationMotorcycle Accident Reconstruction Part I  Physical Models
Motoryle Aident Reonstrution Part I  Physial Models Oren Masory Wade Bartlett Bill Wright Dept. of Oean & Mehanial Engr. Mehanial Forensis Engineering Servies IPTM Florida Atlanti University 179 Cross
More information11  KINETIC THEORY OF GASES Page 1
 KIETIC THEORY OF GASES Page Introduction The constituent partices of the atter ike atos, oecues or ions are in continuous otion. In soids, the partices are very cose and osciate about their ean positions.
More informationLecture 24: Spinodal Decomposition: Part 3: kinetics of the
Leture 4: Spinodal Deoposition: Part 3: kinetis of the oposition flutuation Today s topis Diffusion kinetis of spinodal deoposition in ters of the onentration (oposition) flutuation as a funtion of tie:
More informationUNITI DRIVE CHARACTERISTICS
Eectrica Drives: UNII DRIVE CHARACERISICS Motion contro is required in arge nuber of industria and doestic appications ike transportation systes, roing is, paper achines, textie is, achine toos, fans,
More informationHOW TO CALCULATE PRESSURE ANYWHERE IN A PUMP SYSTEM? Jacques Chaurette p. eng. www.lightmypump.com April 2003
HOW TO CALCULATE PRESSURE ANYWHERE IN A PUMP SYSTEM? Jaques Chaurette p. en. www.lihtmypump.om April 003 Synopsis Calulatin the total head of the pump is not the only task of the pump system desiner. Often
More informationA novel active mass damper for vibration control of bridges
IABMAS 08, International Conferene on Bridge Maintenane, Safety and Management, 37 July 008, Seoul, Korea A novel ative mass damper for vibration ontrol of bridges U. Starossek & J. Sheller Strutural
More informationSoftware Piracy: A Strategic Analysis and Policy Instruments
Software Piray: A Strategi Anaysis and Poiy Instruents Dyuti S. BANERJEE a Otober 00 Abstrat We eaine te governent s roe in restriting oeria iray in a software arket. Wefare aiization ay or ay not resut
More information11  KINETIC THEORY OF GASES Page 1. The constituent particles of the matter like atoms, molecules or ions are in continuous motion.
 KIETIC THEORY OF GASES Page Introduction The constituent partices of the atter ike atos, oecues or ions are in continuous otion. In soids, the partices are very cose and osciate about their ean positions.
More informationModule 7: AM, FM, and the spectrum analyzer.
Module 7: AM, FM, and the spetru analyzer. 7.0 Introdution Eletroagneti signals ay be used to transit inforation very quikly, over great distanes. Two oon ethods by whih inforation is enoded on radio signals,
More informationSHAFTS: TORSION LOADING AND DEFORMATION
ECURE hird Edition SHAFS: ORSION OADING AND DEFORMAION A. J. Clark Shool of Engineering Department of Civil and Environmental Engineering 6 Chapter 3.13.5 by Dr. Ibrahim A. Assakkaf SPRING 2003 ENES 220
More informationBest Execution in Mortgage Secondary Markets
Best Exeution in ortgage Seondary arkets hungjui Wang and Stan Uryav 2 ESEAH EPOT #20053 isk anageent and Finanial Engineering Lab Departent of Industrial and Systes Engineering University of Florida,
More informationAssigning Tasks in a 24Hour Software Development Model
Assigning Tasks in a Hour Software Deveopent Mode Pankaj Jaote, Gourav Jain Departent of Coputer Science & Engineering Indian Institute of Technoogy, Kanpur, INDIA 006 Eai:{jaote, gauravj}@iitk.ac.in
More informationMirror plane (of molecule) 2. the Coulomb integrals for all the carbon atoms are assumed to be identical. . = 0 : if atoms i and j are nonbonded.
6 Hükel Theory This theory was originally introdued to permit qualitative study of the πeletron systems in planar, onjugated hydroarbon moleules (i.e. in "flat" hydroarbon moleules whih possess a mirror
More informationTrigonometry & Pythagoras Theorem
Trigonometry & Pythagoras Theorem Mathematis Skills Guide This is one of a series of guides designed to help you inrease your onfidene in handling Mathematis. This guide ontains oth theory and exerises
More informationPhys101 Lectures 14, 15, 16 Momentum and Collisions
Phs0 Lectures 4, 5, 6 Moentu and ollisions Ke points: Moentu and ipulse ondition for conservation of oentu and wh How to solve collision probles entre of ass Ref: 9,,3,4,5,6,7,8,9. Page Moentu is a vector:
More informationHEAT CONDUCTION. q A q T
HEAT CONDUCTION When a temperature gradient eist in a material, heat flows from the high temperature region to the low temperature region. The heat transfer mehanism is referred to as ondution and the
More informationChapter 14 Oscillations
Chapter 4 Oscillations Conceptual Probles rue or false: (a) For a siple haronic oscillator, the period is proportional to the square of the aplitude. (b) For a siple haronic oscillator, the frequency does
More informationPhysics 211: Lab Oscillations. Simple Harmonic Motion.
Physics 11: Lab Oscillations. Siple Haronic Motion. Reading Assignent: Chapter 15 Introduction: As we learned in class, physical systes will undergo an oscillatory otion, when displaced fro a stable equilibriu.
More informationThe Virtual Spring Mass System
The Virtual Spring Mass Syste J. S. Freudenberg EECS 6 Ebedded Control Systes Huan Coputer Interaction A force feedbac syste, such as the haptic heel used in the EECS 6 lab, is capable of exhibiting a
More informationCENTRIFUGAL PUMP  1
CENTRIFUGAL UM CENTRIFUGAL UM Objeties. At onstant pump speed, determine the harateristi ure (pressure hane s. flow rate) of a entrifual pump.. At onstant pump speed, determine the effiieny as a funtion
More informationDoppler effect, moving sources/receivers
Goals: Lecture 29 Chapter 20 Work with a ew iportant characteristics o sound waves. (e.g., Doppler eect) Chapter 21 Recognize standing waves are the superposition o two traveling waves o sae requency Study
More informationPhysics 100A Homework 11 Chapter 11 (part 1) The force passes through the point A, so there is no arm and the torque is zero.
Physics A Homework  Chapter (part ) Finding Torque A orce F o magnitude F making an ange with the x axis is appied to a partice ocated aong axis o rotation A, at Cartesian coordinates (,) in the igure.
More informationOpen Source Development with a Commercial Complementary Product or Service 1. Ernan Haruvy, Suresh Sethi, Jing Zhou. University of Texas at Dallas
Open Soure Developent with a Coerial Copleentary rodut or Servie Ernan aruvy, Suresh Sethi, Jing Zhou University of Texas at Dallas Abstrat Opening the soure ode to a software produt often iplies that
More informationLab M3: The Physical Pendulum
M3.1 Lab M3: The Physical Pendulum Another pendulum lab? Not really. This lab introduces anular motion. For the ordinary pendulum, we use Newton's second law, F = ma to describe the motion. For the physical
More information10 UNSTEADY FLOW IN OPEN CHANNELS
0 UNTEY FLOW IN OEN CHNNEL 0. Introdution Unsteady flow in open hannels differs from that in losed onduits in that the eistene of a free surfae allows the flow rosssetion to freely hange, a fator whih
More informationA Primer on Dimensions and Units
1 Dienion v Unit A Prier on Dienion and Unit Glen Thornrot Mehanial Enineerin Departent Cal Poly State Univerity, San Lui Obipo Nearly every enineerin proble you will enounter will involve dienion: the
More informationExercise 4 INVESTIGATION OF THE ONEDEGREEOFFREEDOM SYSTEM
Eercise 4 IVESTIGATIO OF THE OEDEGREEOFFREEDOM SYSTEM 1. Ai of the eercise Identification of paraeters of the euation describing a onedegreeof freedo (1 DOF) atheatical odel of the real vibrating
More information1.3 Complex Numbers; Quadratic Equations in the Complex Number System*
04 CHAPTER Equations and Inequalities Explaining Conepts: Disussion and Writing 7. Whih of the following pairs of equations are equivalent? Explain. x 2 9; x 3 (b) x 29; x 3 () x  2x  22 x  2 2 ; x
More informationPHYSICS 151 Notes for Online Lecture 2.2
PHYSICS 151 otes for Online Lecture. A freebod diagra is a wa to represent all of the forces that act on a bod. A freebod diagra akes solving ewton s second law for a given situation easier, because
More informationNeuralNetwork SecurityBoundary Constrained Optimal Power Flow
IEEE TRANSACTIONS ON OWER SYSTEMS 1 NeuraNetwork SeurityBoundary Constrained Optima ower Fow Vitor J. GutierrezMartinez, Caudio A. Cañizares, Feow, IEEE, Caudio R. FuerteEsquive, Member, IEEE, Aejandro
More informationExplanatory Examples on Indian Seismic Code IS 1893 (Part I)
Doument No. :: IITKGSDMAEQ1V.0 inal Report :: A  Earthquake Codes IITKGSDMA Projet on Building Codes Explanatory Examples on Indian Seismi Code IS 1893 (Part I) by Dr. Sudhir K Jain Department of
More informationPlane Trusses. Section 7: Flexibility Method  Trusses. A plane truss is defined as a twodimensional
lane Trusses A plane truss is defined as a twodiensional fraework of straight prisatic ebers connected at their ends by frictionless hinged joints, and subjected to loads and reactions that act only at
More informationRelativity in the Global Positioning System
Relativity in the Global Positioning System Neil Ashby Department of Physis,UCB 390 University of Colorado, Boulder, CO 8030900390 NIST Affiliate Email: ashby@boulder.nist.gov July 0, 006 AAPT workshop
More information( C) CLASS 10. TEMPERATURE AND ATOMS
CLASS 10. EMPERAURE AND AOMS 10.1. INRODUCION Boyle s understanding of the pressurevolue relationship for gases occurred in the late 1600 s. he relationships between volue and teperature, and between
More informationSAT Math MustKnow Facts & Formulas
SAT Mat MustKnow Facts & Formuas Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Rationas: fractions, tat is, anyting expressabe as a ratio of integers Reas: integers pus rationas
More information28 Oscillations: The Simple Pendulum, Energy in Simple Harmonic Motion
Chapter 8 Oscillations: The Simple Pendulum, Enery in Simple Harmonic Motion 8 Oscillations: The Simple Pendulum, Enery in Simple Harmonic Motion Startin with the pendulum bob at its hihest position on
More informationarxiv:astroph/0304006v2 10 Jun 2003 Theory Group, MS 50A5101 Lawrence Berkeley National Laboratory One Cyclotron Road Berkeley, CA 94720 USA
LBNL52402 Marh 2003 On the Speed of Gravity and the v/ Corretions to the Shapiro Time Delay Stuart Samuel 1 arxiv:astroph/0304006v2 10 Jun 2003 Theory Group, MS 50A5101 Lawrene Berkeley National Laboratory
More informationChannel Assignment Strategies for Cellular Phone Systems
Channel Assignment Strategies for Cellular Phone Systems Wei Liu Yiping Han Hang Yu Zhejiang University Hangzhou, P. R. China Contat: wliu5@ie.uhk.edu.hk 000 Mathematial Contest in Modeling (MCM) Meritorious
More informationchapter > Make the Connection Factoring CHAPTER 4 OUTLINE Chapter 4 :: Pretest 374
CHAPTER hapter 4 > Make the Connetion 4 INTRODUCTION Developing seret odes is big business beause of the widespread use of omputers and the Internet. Corporations all over the world sell enryption systems
More informationANSYS Tutorial. Modal Analysis
ANSYS Tutorial Slides to accompany lectures in VibroAcoustic Desin in Mechanical Systems 2012 by D. W. Herrin Department of Mechanical Enineerin Lexinton, KY 405060503 Tel: 8592180609 dherrin@enr.uky.edu
More informationIsaac Newton. Translated into English by
THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY (BOOK 1, SECTION 1) By Isaa Newton Translated into English by Andrew Motte Edited by David R. Wilkins 2002 NOTE ON THE TEXT Setion I in Book I of Isaa
More informationSimple Harmonic Motion MC Review KEY
Siple Haronic Motion MC Review EY. A block attache to an ieal sprin uneroes siple haronic otion. The acceleration of the block has its axiu anitue at the point where: a. the spee is the axiu. b. the potential
More informationPY1002: Special Relativity
PY100: Speial Relativity Notes by Chris Blair These notes over the Junior Freshman ourse given by Dr. Barklie in Mihaelmas Term 006. Contents 1 Galilean Transformations 1.1 Referene Frames....................................
More information17. Shaft Design. Introduction. Torsion of circular shafts. Torsion of circular shafts. Standard diameters of shafts
Objetives 17. Shaft Design Compute fores ating on shafts from gears, pulleys, and sprokets. ind bending moments from gears, pulleys, or sprokets that are transmitting loads to or from other devies. Determine
More information( 1 ) Obtain the equation of the circle passing through the points ( 5,  8 ), (  2, 9 ) and ( 2, 1 ).
PROBLEMS 03 CIRCLE Page ( ) Obtain the equation of the irle passing through the points ( 5 8 ) ( 9 ) and ( ). [ Ans: x y 6x 48y 85 = 0 ] ( ) Find the equation of the irumsribed irle of the triangle formed
More informationWorking together to archive the UK Web. Helen HockxYu Head of Web Archiving, British Library
Working together to arhive the UK Web Heen HokxYu Head of Web Arhiving, British Library The UK Web Domain 4 th TLD after.om,.de and.net Over 10 miion.uk registered domain UK organisations aso use non.uk
More informationAnalysis of a Fork/Join Synchronization Station with Inputs from Coxian Servers in a Closed Queuing Network
Analysis of a Fork/Join Synhronization Station with Inputs fro Coxian Serers in a Closed ueuing Network Ananth Krishnaurthy epartent of eision Sienes and Engineering Systes Rensselaer Polytehni Institute
More informationFundamentals of Chemical Reactor Theory
UNIVERSITY OF CALIFORNIA, LOS ANGELES Civil & Environmental Engineering Department Fundamentals of Chemial Reator Theory Mihael K. Stenstrom Professor Diego Rosso Teahing Assistant Los Angeles, 3 Introdution
More informationMagnetic Materials and Magnetic Circuit Analysis
Chapter 7. Magneti Materials and Magneti Ciruit Analysis Topis to over: 1) Core Losses 2) Ciruit Model of Magneti Cores 3) A Simple Magneti Ciruit 4) Magneti Ciruital Laws 5) Ciruit Model of Permanent
More informationEngine turning moment diagram:
Chapter 3 Flywheel Application of slidercrank echanis can be found in reciprocating (stea) engines in the power plant i.e. internal cobustion engines, generators to centrifugal pups, etc. Output is nonunifor
More informationCHAPTER J DESIGN OF CONNECTIONS
J1 CHAPTER J DESIGN OF CONNECTIONS INTRODUCTION Chapter J of the addresses the design and heking of onnetions. The hapter s primary fous is the design of welded and bolted onnetions. Design requirements
More informationCIS570 Lecture 4 Introduction to Dataflow Analysis 3
Introdution to Dataflow Analysis Last Time Control flow analysis BT disussion Today Introdue iterative dataflow analysis Liveness analysis Introdue other useful onepts CIS570 Leture 4 Introdution to
More informationSpecial Relativity and Linear Algebra
peial Relativity and Linear Algebra Corey Adams May 7, Introdution Before Einstein s publiation in 95 of his theory of speial relativity, the mathematial manipulations that were a produt of his theory
More informationLecture L263D Rigid Body Dynamics: The Inertia Tensor
J. Peraire, S. Widnall 16.07 Dynaics Fall 008 Lecture L63D Rigid Body Dynaics: The Inertia Tensor Version.1 In this lecture, we will derive an expression for the angular oentu of a 3D rigid body. We shall
More information6. Fasteners and Fastening methods. Introduction. Fastener types. Fastener application. Screw thread terminology. Screw thread terminology (Fig. 6.
6. Fasteners and Fastening methods Ojetives Desrie many types of fastening systems and their uses. Understand priniples of stress area, pith diameters, and thread types and forms. Understand different
More informationLecture L9  Linear Impulse and Momentum. Collisions
J. Peraire, S. Widnall 16.07 Dynaics Fall 009 Version.0 Lecture L9  Linear Ipulse and Moentu. Collisions In this lecture, we will consider the equations that result fro integrating Newton s second law,
More informationThe Impact of Targeting Technology on Advertising Markets and Media Competition
From the SeectedWorks of Joshua S Gans December 2009 The Impact of Taretin Technooy on Advertisin Markets and Media Competition Contact Author Start Your Own SeectedWorks Notify Me of New Work Avaiabe
More informationExample: Suppose that we deposit $1000 in a bank account offering 3% interest, compounded monthly. How will our money grow?
Finance 111 Finance We have to work with oney every day. While balancing your checkbook or calculating your onthly expenditures on espresso requires only arithetic, when we start saving, planning for retireent,
More informationcos t sin t sin t cos t
Exerise 7 Suppose that t 0 0andthat os t sin t At sin t os t Compute Bt t As ds,andshowthata and B ommute 0 Exerise 8 Suppose A is the oeffiient matrix of the ompanion equation Y AY assoiated with the
More informationProperties of Pure Substances
ure Substance roperties o ure Substances A substance that has a ixed cheical coposition throuhout is called a pure substance such as water, air, and nitroen. A pure substance does not hae to be o a sinle
More informationSINGLE PHASE FULL WAVE AC VOLTAGE CONTROLLER (AC REGULATOR)
Deceber 9, INGE PHAE FU WAE AC OTAGE CONTROER (AC REGUATOR ingle phase full wave ac voltage controller circuit using two CRs or a single triac is generally used in ost of the ac control applications. The
More informationVersion 001 test 1 review tubman (IBII201516) 1
Version 001 test 1 review tuban (IBII01516) 1 This printout should have 44 questions. Multiplechoice questions ay continue on the next colun or page find all choices before answering. Crossbow Experient
More informationMathematics 1c: Solutions, Homework Set 6 Due: Monday, May 17 at 10am.
Mathematis : Solutions, Homework Set 6 Due: Monday, May 7 at am.. ( Points) Setion 6., Eerise 6 Let D be the parallelogram with verties (, 3), (, ), (, ) and (, ) and D be the retangle D [, ] [, ]. Find
More informationDSPI DSPI DSPI DSPI
DSPI DSPI DSPI DSPI Digital Signal Proessing I (879) Fall Semester, 005 IIR FILER DESIG EXAMPLE hese notes summarize the design proedure for IIR filters as disussed in lass on ovember. Introdution:
More informationModulation Principles
EGR 544 Communiation Theory 5. Charaterization of Communiation Signas and Systems Z. Aiyaziiogu Eetria and Computer Engineering Department Ca Poy Pomona Moduation Prinipes Amost a ommuniation systems transmit
More informationTERM INSURANCE CALCULATION ILLUSTRATED. This is the U.S. Social Security Life Table, based on year 2007.
This is the U.S. Socia Security Life Tabe, based on year 2007. This is avaiabe at http://www.ssa.gov/oact/stats/tabe4c6.htm. The ife eperiences of maes and femaes are different, and we usuay do separate
More informationVectors & Newton's Laws I
Physics 6 Vectors & Newton's Laws I Introduction In this laboratory you will eplore a few aspects of Newton s Laws ug a force table in Part I and in Part II, force sensors and DataStudio. By establishing
More informationRevista Brasileira de Ensino de Fsica, vol. 21, no. 4, Dezembro, 1999 469. Surface Charges and Electric Field in a TwoWire
Revista Brasileira de Ensino de Fsia, vol., no. 4, Dezembro, 999 469 Surfae Charges and Eletri Field in a TwoWire Resistive Transmission Line A. K. T.Assis and A. J. Mania Instituto de Fsia Gleb Wataghin'
More informationThe Mathematics of Pumping Water
The Matheatics of Puping Water AECOM Design Build Civil, Mechanical Engineering INTRODUCTION Please observe the conversion of units in calculations throughout this exeplar. In any puping syste, the role
More informationEE 201 ELECTRIC CIRCUITS LECTURE 25. Natural, Forced and Complete Responses of First Order Circuits
EE 201 EECTRIC CIUITS ECTURE 25 The material overed in this leture will be as follows: Natural, Fored and Complete Responses of First Order Ciruits Step Response of First Order Ciruits Step Response of
More informationIntroduction to Unit Conversion: the SI
The Matheatics 11 Copetency Test Introduction to Unit Conversion: the SI In this the next docuent in this series is presented illustrated an effective reliable approach to carryin out unit conversions
More informationModule 4. Analysis of Statically Indeterminate Structures by the Direct Stiffness Method. Version 2 CE IIT, Kharagpur
Mode 4 Anaysis of Staticay Indeterinate Strctres by the Direct Stiffness Method Version CE IIT, Kharagr esson 4 The Direct Stiffness Method: Trss Anaysis Version CE IIT, Kharagr Instrctiona Objecties After
More informationChapter 13 Simple Harmonic Motion
We are to adit no ore causes of natural things than such as are both true and sufficient to explain their appearances. Isaac Newton 13.1 Introduction to Periodic Motion Periodic otion is any otion that
More informationRetirement Option Election Form with Partial Lump Sum Payment
Offie of the New York State Comptroller New York State and Loal Retirement System Employees Retirement System Polie and Fire Retirement System 110 State Street, Albany, New York 122440001 Retirement Option
More informationUnit  6 Vibrations of Two Degree of Freedom Systems
Unit  6 Vibrations of Two Degree of Freedom Systems Dr. T. Jagadish. Professor for Post Graduation, Department of Mechanical Engineering, Bangalore Institute of Technology, Bangalore Introduction A two
More information4.3 The Graph of a Rational Function
4.3 The Graph of a Rational Function Section 4.3 Notes Page EXAMPLE: Find the intercepts, asyptotes, and graph of + y =. 9 First we will find the intercept by setting the top equal to zero: + = 0 so =
More informationProgramming Basics  FORTRAN 77 http://www.physics.nau.edu/~bowman/phy520/f77tutor/tutorial_77.html
CWCS Workshop May 2005 Programming Basis  FORTRAN 77 http://www.physis.nau.edu/~bowman/phy520/f77tutor/tutorial_77.html Program Organization A FORTRAN program is just a sequene of lines of plain text.
More informationAC VOLTAGE CONTROLLER CIRCUITS (RMS VOLTAGE CONTROLLERS)
AC TAGE CNTRER CRCUT (RM TAGE CNTRER) AC voltage controllers (ac line voltage controllers) are eployed to vary the RM value of the alternating voltage applied to a load circuit by introducing Thyristors
More informationProjectile Motion THEORY. r s = s r. t + 1 r. a t 2 (1)
Projectile Motion The purpose of this lab is to study the properties of projectile motion. From the motion of a steel ball projected horizontally, the initial velocity of the ball can be determined from
More informationComay s Paradox: Do Magnetic Charges Conserve Energy?
Comay s Paradox: Do Magneti Charges Conserve Energy? 1 Problem Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ 08544 (June 1, 2015; updated July 16, 2015) The interation energy
More informationFrom (2) follows, if z0 = 0, then z = vt, thus a2 =?va (2.3) Then 2:3 beomes z0 = z (z? vt) (2.4) t0 = bt + b2z Consider the onsequenes of (3). A ligh
Chapter 2 Lorentz Transformations 2. Elementary Considerations We assume we have two oordinate systems S and S0 with oordinates x; y; z; t and x0; y0; z0; t0, respetively. Physial events an be measured
More informationEffects of InterCoaching Spacing on Aerodynamic Noise Generation Inside Highspeed Trains
Effets of InterCoahing Spaing on Aerodynami Noise Generation Inside Highspeed Trains 1 J. Ryu, 1 J. Park*, 2 C. Choi, 1 S. Song Hanyang University, Seoul, South Korea 1 ; Korea Railroad Researh Institute,
More informationLesson 44: Acceleration, Velocity, and Period in SHM
Lesson 44: Acceleration, Velocity, and Period in SHM Since there is a restoring force acting on objects in SHM it akes sense that the object will accelerate. In Physics 20 you are only required to explain
More informationImpact Simulation of Extreme Wind Generated Missiles on Radioactive Waste Storage Facilities
Impat Simulation of Extreme Wind Generated issiles on Radioative Waste Storage Failities G. Barbella Sogin S.p.A. Via Torino 6 00184 Rome (Italy), barbella@sogin.it Abstrat: The strutural design of temporary
More informationA tutorial on inverting 3 by 3 matrices with cross products
A tutorial on inverting by atries with ross produts By Cedrik Collob Abstrat. his tutorial introdues the idea of inverting a by atrix and alulating its erinant with ross produts hopefully in a siple anner
More informationChapter 1: Introduction
Chapter 1: Introdution 1.1 Pratial olumn base details in steel strutures 1.1.1 Pratial olumn base details Every struture must transfer vertial and lateral loads to the supports. In some ases, beams or
More information) ( )( ) ( ) ( )( ) ( ) ( ) (1)
OPEN CHANNEL FLOW Open hannel flow is haraterized by a surfae in ontat with a gas phase, allowing the fluid to take on shapes and undergo behavior that is impossible in a pipe or other filled onduit. Examples
More informationRole of the Reference Frame in Angular Photon Distribution at ElectronPositron Annihilation
Journal of Modern Physis,, 5, 353358 Published Online April in SiRes. http://www.sirp.org/journal/jmp http://dx.doi.org/.36/jmp..565 Role of the Referene Frame in Angular Photon Distribution at EletronPositron
More informationCalculation Method for evaluating Solar Assisted Heat Pump Systems in SAP 2009. 15 July 2013
Calculation Method for evaluating Solar Assisted Heat Pup Systes in SAP 2009 15 July 2013 Page 1 of 17 1 Introduction This docuent describes how Solar Assisted Heat Pup Systes are recognised in the National
More informationON THE ELECTRODYNAMICS OF MOVING BODIES
ON THE ELECTRODYNAMICS OF MOVING BODIES By A. EINSTEIN June 30, 905 It is known that Maxwell s eletrodynamis as usually understood at the present time when applied to moing bodies, leads to asymmetries
More informationON THE ELECTRODYNAMICS OF MOVING BODIES
ON THE ELECTRODYNAMICS OF MOVING BODIES By A. EINSTEIN June 30, 905 It is known that Maxwell s eletrodynamis as usually understood at the present time when applied to moing bodies, leads to asymmetries
More informationPREDICTION OF MILKLINE FILL AND TRANSITION FROM STRATIFIED TO SLUG FLOW
PREDICTION OF MILKLINE FILL AND TRANSITION FROM STRATIFIED TO SLUG FLOW ABSTRACT: by Douglas J. Reineann, Ph.D. Assistant Professor of Agricultural Engineering and Graee A. Mein, Ph.D. Visiting Professor
More informationModule 6 : Lecture 1 DIMENSIONAL ANALYSIS (Part I)
Overview Modue 6 : Lecture DIMENSIONAL ANALYSIS (Part I) Many practica fow probes of different nature can be soved by using equations and anaytica procedures, as discussed in the previous odues. However,
More informationFinance 360 Problem Set #6 Solutions
Finance 360 Probem Set #6 Soutions 1) Suppose that you are the manager of an opera house. You have a constant margina cost of production equa to $50 (i.e. each additiona person in the theatre raises your
More informationINCOME TAX WITHHOLDING GUIDE FOR EMPLOYERS
Virginia Department of Taxation INCOME TAX WITHHOLDING GUIDE FOR EMPLOYERS www.tax.virginia.gov 2614086 Rev. 07/14 * Table of Contents Introdution... 1 Important... 1 Where to Get Assistane... 1 Online
More informationAUDITING COST OVERRUN CLAIMS *
AUDITING COST OVERRUN CLAIMS * David PérezCastrillo # University of Copenhagen & Universitat Autònoma de Barelona Niolas Riedinger ENSAE, Paris Abstrat: We onsider a ostreimbursement or a ostsharing
More information