33703d5a-fad4-4e6a-9ca2-238751b741ba 73 3 5 auto

&[DATE] &[TIME] - &[FILENAME]

KEY & other Notes

Pronumeral conventions R -> Radius ~ m L -> Length ~ m θ -> Angle between assemblies ~ deg β -> Angle from straight of Universal Joints ~ rad, deg N -> Ratio Φ -> Diameter ~ m Ψ -> Fluctuation rate γ -> Index position ~ rad s -> angular displacement ~ Deg or radians ω -> Rotational speed angular velocity ~ rev/min, rad/sec, rev/sec a -> angular acceleration ~ rev/min^2, rad/sec^2, rev/sec^2 m -> mass ~ kg J -> Mass Moment of Inertia ~ kg.m^2 τ -> Torque ~ Nm

Section - Step titles

User defigned Variables

Measured geometry & fixed values

Calculated output

Questionable - TBC

Explanation

Referance

Check calculation

Subscript conventions

NB No part of these calculations consider the balance or imbalance of the Cardan Shafts. Voith supply Cardan shafts balanced to G... to BS..

Calculation refignments Get spelling etc correct. Get OEM transmision ratios. Get OEM Mass moments of imertia of the Cardan shafts. Resolve reactions on transmission frame Get a traction system diagrm in Improve explanations. Improve measurement accuracy

Voith

All measurements were taken alone with normal hand tools. Say ±5% They can easily be improved.

Geometry of 2nd Cardan Shaft (between FD1 and Transmission)

Cardan Joint Angles in the HORIZONTAL Plane

Angle of Bogie to Car

R.track 200 (m * :

90 <-> ∞

90 m is the specified min track Raduis

Radius of track CL curve

L.BogP 5512 * 1 + 14 / + (0.0254 * m * :

Pitch between bogies.

L.BogP 16.7957 m *

θ.Bog L.BogP 2 / (R.track / asin :

Check this measure again

Assuming No superelevation

θ.Bog deg 2.4065

Angle of Bogie to Car

Distance between Final Drive 1 Cardan joint and car centre line

L.Fd1Uj_BogCr 1.22 m * :

L.Fd1Uj_BogCr 1.22 m * :

Final Drive 1 Uni Joint to bogie centre of rotation

L.Fd1Uj_CarCl L.Fd1Uj_BogCr θ.Bog sin * :

Distance between FD1 cardin joint & CL of car

L.Fd1Uj_CarCl 0.0512 m *

L.Fd1Uj_CarCl 0.0512 m *

Angle at transmission end Cardan joint

L.TranUj_BogCr 2.51 m * :

L.TranUj_BogCr 2.51 m * :

Trans Cardan Joint to bogie centre of rotation

β.UjTranH L.Fd1Uj_CarCl L.TranUj_BogCr L.Fd1Uj_BogCr - / atan :

β.UjTranH deg 2.2741

Check Cardin shaft is on Cl of car. No it's not. FDs are 725 & 680 from brake disks. Assume centreline for simplicity. Trans Output???

Angle at trans end Cardin joint

Cardan shaft extension due to direct offset (Pythagoras Only)

Δ.LCs_Simp L.TranUj_BogCr L.Fd1Uj_BogCr - (2^L.Fd1Uj_CarCl 2^+ sqrt L.TranUj_BogCr L.Fd1Uj_BogCr - (- :

Δ.LCs_Simp L.TranUj_BogCr L.Fd1Uj_BogCr - (2^L.Fd1Uj_CarCl 2^+ sqrt L.TranUj_BogCr L.Fd1Uj_BogCr - (- :

Δ.LCs_Simp 0.001 m *

Δ.LCs_Simp 0.001 m *

Cardan shaft extension due to direct offset

There is additional expansion within the Joints at the cross The Shaft is not realy on the car centreline.

Angle at Final Drive 1 end Cardan joint

β.UjFd1H θ.Bog β.UjTranH + :

β.UjFd1H deg 4.6806

Angle at FD1 end Cardan joint

Cardan Joint Angles in the Vertical Plane

The Cardin flange faces are both close to vertical and the slope of the Cardan shaft measures approx 6 degrees rising towards the Final drive. Vertical curves not considered.

Spring travel is ~60mm. Swing arm length is 460mm. Torsion bar Leangth is 800mm. Torsion bar is parallel to and 300mm from swing arm. Air bag travel is ~150mm

β.UjTranV 6 deg * :

β.UjFd1V 6 deg * :

Vertical offet angle of Trans Cardan Joints

Assume constant 6 deg for simplicity.

Cardan Joint Angles in combined Planes

Horiz -> XOc Vert -> XOa combined -> XOP

β.UjTranΣ β.UjTranH tan (2^β.UjTranV tan (2^+ sqrt atan :

β.UjFd1Σ β.UjFd1H tan (2^β.UjFd1V tan (2^+ sqrt atan :

2nd Cardan shaft Cardan Joint Angles in combined planes

β.UjTranΣ deg 6.4106

β.UjFd1Σ deg 7.5888

Final drive Input rotation speed (average) with train speed constant.

I originally ran my calculations from the transmission to the final drive 1; then realised that it is better to run the calculations from the final drive to the transmission because in the time frames being considered the train’s inertia has a greater influence than the transmission. The calculations now run from the final drive to the transmission.

ν.Train 60 (km hr / * :

ν.Train m sec / 16.6667

Ν.Fd 3.2628 :

Final Drive ratio

Φ.wheel 0.940 m * :

Wheel diameter 940 - 865mm Φ

ω.Fd1 ν.Train Ν.Fd * Φ.wheel 0.5 * / :

SMath handles cycles, radians, revs and Hz in a peculiar way. Hence peculiar math to left and note to the right.

ω.Fd1 rev sec / 18.4146

ω.Fd1 rad sec / 115.7021

Final drive flanges are assumed to be paralel so Cardan Joint speed fluctuations will cancel out. Therefore Angular velocity of the input and output shafts of both the Final drives will be equal at all times.

ω.Fd1 rev min / 1104.8739

P.TE 380 kW * :

Rated output of traction Engine

ω.TE 2100 rev min / * :

Crankshaft speed of traction Engine

τ.TE P.TE : (

P.TE ω.TE / N m * 1727.968

Output tourque of traction Engine

2nd Cardan Shaft (between FD1 and Transmission) with FD1 considered as constant angular velocity.

Fluctuation Rates of Cardan Joints of 2nd Cardan shaft.

Ψ.UjFd1 1 β.UjFd1Σ cos / β.UjFd1Σ cos - :

from Elbe document

Ψ.UjFd1 0.0176

Fluctuation Rate for Cardan joint at final drive

Ψ.UjTran 1 β.UjTranΣ cos / β.UjTranΣ cos - :

Ψ.UjTran 0.0125

Fluctuation Rate for Cardan joint at Transmission

Ψ.Tot Ψ.UjFd1 Ψ.UjTran - :

Ψ.Tot 0.005

Fluctuation Rate for whole Cardan shaft

Cardan Lay Shaft Speed by Fluctuation Rate method

ω.Cls_max1 ω.Fd1 ω.Fd1 Ψ.UjFd1 2 / * + :

ω.Cls_min1 ω.Fd1 ω.Fd1 Ψ.UjFd1 2 / * - :

Cardan Lay Shaft Ideal Max & min Speeds by Fluctuation Rate method. Assuming train speed is contstant and zero backlash.

ω.Cls_max1 rev min / 1114.5939

ω.Cls_max1 ω.Cls_min1 / 1.0178

ω.Cls_min1 rev min / 1095.1539

ω.Cls_max1 ω.Cls_min1 - rev min / 19.44

Transmission end Cardan Joint Speed by Fluctuation Rate method

ω.Tran_max1 ω.Fd1 ω.Fd1 Ψ.Tot 2 / * + :

ω.Tran_min1 ω.Fd1 ω.Fd1 Ψ.Tot 2 / * - :

ω.Tran_max1 rev min / 1107.6637

Cardin Shaft Ideal Max & Min Speeds at Trans Joint Flange by Fluctuation Rate method. Assuming train speed is constant and zero backlash.

ω.Tran_max1 ω.Tran_min1 / 1.0051

ω.Tran_min1 rev min / 1102.0842

ω.Tran_max1 ω.Tran_min1 - rev min / 5.5795

2nd Cardan Shaft Input /output Speed by unequal angle Cardan Shaft method

ω.Tran_max2 β.UjTranΣ cos β.UjFd1Σ cos / ω.Fd1 * :

from Elbe document

ω.Tran_min2 β.UjFd1Σ cos β.UjTranΣ cos / ω.Fd1 * :

ω.Tran_max2 ω.Tran_min2 / 1.0051

ω.Tran_max2 rev min / 1107.6671

Cardan Shaft Ideal Max & Min output Speed by unequal angle Cardan Shaft method. Assuming Transmission output is constant at 2100rpm.

Close enuf to agreement

Agreement by 2 methods

ω.Tran_min2 rev min / 1102.0878

Reminder

Angular displacement of lay shaft in relation to Fd1 flange

R.track 200 m *

γ.1 s.2_1 γ.1 sin β.UjFd1Σ cos γ.1 cos * atan :

β.UjFd1Σ deg 7.5888

Angular displacement of transmission flange in relation to lay shaft

β.UjTranΣ deg 6.4106

γ.1 s.3_2 γ.1 sin β.UjTranΣ sec γ.1 cos * atan :

Angular transmission of transmision flange in relation to Fd1 flange

γ.1 s.3_1 γ.1 sin β.UjTranΣ sec γ.1 cos * atan sin β.UjFd1Σ cos γ.1 sin β.UjTranΣ sec γ.1 cos * atan cos * atan :

γ.1 s.forced γ.1 γ.1 sin β.UjTranΣ sec γ.1 cos * atan sin β.UjFd1Σ cos γ.1 sin β.UjTranΣ sec γ.1 cos * atan cos * atan (- (:

x s.2_1 x s.3_2 x s.3_1 x s.forced 4 1 sys

β.Fd1sam 75 deg * :

β.Transam 65 deg * :

γ.1 s.2_1sam γ.1 sin β.Fd1sam cos γ.1 cos * atan :

γ.1 s.3_1sam γ.1 sin β.Transam sec γ.1 cos * atan sin β.Fd1sam cos γ.1 sin β.Transam sec γ.1 cos * atan cos * atan :

γ.1 s.3_0sam γ.1 γ.1 sin β.Transam sec γ.1 cos * atan sin β.Fd1sam cos γ.1 sin β.Transam sec γ.1 cos * atan cos * atan (- :

γ.1 s.0_0sam γ.1 :

γ.1 s.3_2sam γ.1 sin β.Transam sec γ.1 cos * atan :

x s.0_0sam x s.2_1sam x s.3_2sam x s.3_0sam x s.3_1sam 5 1 sys

This sample plot shows the angular displacement effect at a scale made obvious by extreme angularity. It is not linked to the calcs.

γ.maxs 2.35 :

s.forced γ.maxs γ.maxs sin β.UjTranΣ sec γ.maxs cos * atan sin β.UjFd1Σ cos γ.maxs sin β.UjTranΣ sec γ.maxs cos * atan cos * atan (- (2 * :

On curved track this Angular displacement (less backlash (Nill)) must be accomidated by the trasmision frame and it's mounts.

s.forced deg 0.1447

L.TranF 3 m * :

Transmision frame width

Amp.CP L.TranF 2 / s.forced sin * :

Amplidude of the Control panel movement assuming all the displacement is taken in the trans frame mounts.

Amp.CP mm 3.7869

Angular Velocity of the Cardan lay shaft assuming constant velocity at the Final drive Flange.

γ.1 ω.Cls ω.Fd1 β.UjFd1Σ cos * 1 β.UjFd1Σ sin (2^γ.1 cos (2^* - / :

Derivative of angular displacement to give angular velocity. (check against fluctuation)

γ 0 rad * π 2 / 21 sys :

γ deg 09021 sys

R.track 200 m *

γ.1 ω.Tran2 ω.Fd1 β.UjTranΣ cos * 1 β.UjTranΣ sin (2^γ.1 π 2 / + cos (2^* - / :

ω.Cls ω.Fd1 β.UjFd1Σ cos * 1 β.UjFd1Σ sin (2^γ cos (2^* - / :

γ.1 ω.Tran ω.Fd1 β.UjFd1Σ cos * 1 β.UjFd1Σ sin (2^γ.1 cos (2^* - / β.UjTranΣ cos * 1 β.UjTranΣ sin (2^γ.1 π 2 / + cos (2^* - / :

γ.maxω 0 :

ω.Tranmax ω.Fd1 β.UjFd1Σ cos * 1 β.UjFd1Σ sin (2^γ.maxω cos (2^* - / β.UjTranΣ cos * 1 β.UjTranΣ sin (2^γ.maxω π 2 / + cos (2^* - / :

Angular Velocity of the Transmission output Flange assuming constant velocity at the Final drive Flange

x ω.Cls ω.Fd1 - x ω.Tran2 ω.Fd1 - x ω.Tran ω.Fd1 - 3 1 sys

ω.Tranmin ω.Fd1 β.UjFd1Σ cos * 1 β.UjFd1Σ sin (2^γ.minω cos (2^* - / β.UjTranΣ cos * 1 β.UjTranΣ sin (2^γ.minω π 2 / + cos (2^* - / :

γ.minω π 2 / :

This green plot flattens close to 0 as the track radius approaches ∞. (Assuming Cardan Shaft is on centre line of car) On straight track (with just a 6deg vertical offset the transmission Ujoint will null out the angular displacement of the Fd1 U joint; leaving the inertia due to the angular acceleration of the lay shaft to excite the vibration.

ω.Tranmax rev min / 1107.6671

ω.Tran_max1 rev min / 1107.6637

ω.Tran_max1 rev min / 1107.6637

ω.Tranmin rev min / 1102.0878

ω.Tran_min1 rev min / 1102.0842

ω.Tran_min1 rev min / 1102.0842

ω.Tran_max2 rev min / 1107.6671

ω.Tran_max2 rev min / 1107.6671

For power to be constant as the angular velocity fluctuates so does the torque; but the other way. The effect of this would only need to be investigated deeper during the design phase.

ω.Tran_min2 rev min / 1102.0878

ω.Tran_min2 rev min / 1102.0878

Close enough to agreement by 3 different methods

Delta Angular velocity of the Cardin Lay Shaft due to Final drive end joint.

Delta Angular velocity of the Transmission shaft due to Transmission end joint.

Super positioning of Delta Angular Transmission shaft velocity due to the combined effect of both joints.

Angular acceleration of the lay shafts of the Cardan shafts

a.1 0 (rev sec 2^/ * :

Because the final drive is assumed to be rotating at a constant rate

γ.1 a a.1 β.UjTranΣ cos * 1 β.UjTranΣ sin (2^γ.1 cos (2^* - / ω.Fd1 2^β.UjTranΣ cos * β.UjTranΣ sin 2^* β.UjTranΣ * 2 γ.1 * sin * 1 β.UjTranΣ sin (2^γ.1 cos (2^* - (2^/ - :

x a

Derivative of Angular displacement to give angular acceleration. 2nd for acceleration

ω.Fd1 β.UjFd1Σ cos * 1 β.UjFd1Σ sin (2^γ cos (2^* - / (γ diff rev min / 0 39.5702 - 000 38.2019 - 0081 sys

ω.Fd1 β.UjFd1Σ cos * 1 β.UjFd1Σ sin (2^γ.1 cos (2^* - / (γ 2 diff rev min 2^/ 0

Not working??

γ 0 1.571 21 sys

ω.Fd1 β.UjFd1Σ cos * 1 β.UjFd1Σ sin (2^γ cos (2^* - / (γ diff 0 4.144 1 s / * - 0 6.694 1015 -^* 1 s / * - 041 s / * - 0 6.463 1015 -^* 1 s / * - 81 sys

recast unitless x ff ω.Fd1 Hz / β.UjFd1Σ cos * 1 β.UjFd1Σ sin (2^x cos (2^* - / :

x plot_ff x ff π 4 / ff 21 sys :

1rst & 2nd  derivatives x D1_D2 x ff x diff x ff x 2 diff 21 sys :

x plot_ff x D1_D2

0 ff π 4 / ff 21 mat 116.724 115.698 21 mat

γ.mina 0.7747 rad * :

γ.maxa 0.7747 rad * - :

a.Cls2max a.1 β.UjTranΣ cos * 1 β.UjTranΣ sin (2^γ.maxa cos (2^* - / ω.Fd1 2^β.UjTranΣ cos * β.UjTranΣ sin 2^* β.UjTranΣ * 2 γ.maxa * sin * 1 β.UjTranΣ sin (2^γ.maxa cos (2^* - (2^/ - :

a.Cls2max rad sec 2^/ 18.7901

a.Cls2max rev sec 2^/ 2.9905

Angular acceleration of the lay shaft of the 2nd Cardan shaft

a.Cls2min a.1 β.UjTranΣ cos * 1 β.UjTranΣ sin (2^γ.mina cos (2^* - / ω.Fd1 2^β.UjTranΣ cos * β.UjTranΣ sin 2^* β.UjTranΣ * 2 γ.mina * sin * 1 β.UjTranΣ sin (2^γ.mina cos (2^* - (2^/ - :

a.Cls2min rad sec 2^/ 18.7901 -

a.Cls2min rev sec 2^/ 2.9905 -

β.Uj1 4 deg * :

Vertical offet angle of Cardan Shaft #1's Joints

a.Cls1max a.1 β.Uj1 cos * 1 β.Uj1 sin (2^γ.maxa cos (2^* - / ω.TE 2^β.Uj1 cos * β.Uj1 sin 2^* β.Uj1 * 2 γ.maxa * sin * 1 β.Uj1 sin (2^γ.maxa cos (2^* - (2^/ - :

a.Cls1max rev sec 2^/ 2.6207

β.Uj3 4 deg * :

Horizontal offet angle of Cardan Shaft 3's Joints

a.Cls3max a.1 β.Uj3 cos * 1 β.Uj3 sin (2^γ.maxa cos (2^* - / ω.Fd1 2^β.Uj3 cos * β.Uj3 sin 2^* β.Uj3 * 2 γ.maxa * sin * 1 β.Uj3 sin (2^γ.maxa cos (2^* - (2^/ - :

a.Cls3max rev sec 2^/ 0.7255

Mass Moment of Inertia of the Cardan shafts

Cardan shaft #1 Te - Tran 400 between centres.

Cardan shaft #2 Tran - Fd1 1280 between centres. Voith equp # 93139

Cardan shaft #3 Fd1 - FD2 1130 between centres.

m.Cls1 6324 * - (kg * :

m.Cls2 9024 * - (kg * :

m.Cls3 8024 * - (kg * :

m.Cls1 55 kg *

m.Cls2 82 kg *

m.Cls3 72 kg *

Mass of cardan lay shafts

R.Cls1 0.07 m * :

R.Cls2 0.06 m * :

R.Cls3 0.063 m * :

Radius of gyration cardan lay shafts

J.Cls1 m.Cls1 R.Cls1 2^* :

J.Cls2 m.Cls2 R.Cls2 2^* :

J.Cls3 m.Cls3 R.Cls3 2^* :

Have Asked Voith for the Mass moment of inertia of the lay shafts.

J.Cls1 0.2695 kg * m 2^*

J.Cls2 0.2952 kg * m 2^*

J.Cls3 0.2858 kg * m 2^*

Mass Moment of Inertia of the Cardan lay shafts.

τ.Cls1max a.Cls1max J.Cls1 * :

τ.Cls2 a.Cls2max J.Cls2 * :

τ.Cls3 a.Cls3max J.Cls3 * :

Inertial Torque caused by the angular aceleration of the lay shafts

τ.Cls1max N m * 4.4378

τ.Cls3 N m * 1.3026

τ.Cls2 N m * 5.5468

τ.Cls2Σ3 τ.Cls2 τ.Cls3 + τ.Cls2 τ.Cls3 - 21 sys :

τ.Cls2Σ3 N m / 6.8494 m 2^* 4.2443 m 2^* 21 sys

τ.Cls2 τ.Cls3 + τ.Cls2 τ.Cls3 - / 1.6138

Cardan Shafts 2 & 3 can be indexed so their inertial torques sum to reinforce each other or subtract to partially negate each other. Indexing then at 90 deg to each other will reduce the excitation energy applied to the transmission frame.

τ.Cls ω.Cls * W #

τ.Cls2Σ3 N m / 6.8494 m 2^* 4.2443 m 2^* 21 sys

1 ω.Fd1 4 * / ms 2.1607

Every

a torque of

is applied to the output shaft of the transmission

in alternating directions. This is over and above the torque driving the train along the track and as the result of the vertical offset of the 2nd Cardan shaft will occour when ever the car is moving.

In addition to the above inertial torque there is a small angular displacement when the car is going around a curve. This angular displacement is syncronised with the inertial torque and it's magnitude is only limeted by deflection or slip.

EXPLAIN ... @ every a torque of is applied to the output shaft of the transmission in alternating directions. This is over and above the torque driving the train along the track and as the result of the vertical offset of the 2nd Cardan shaft will occour when ever the car is moving. In addition to the above inertial torque there is a small angular displacement when the car is going around a curve. This angular displacement is syncronised with the inertial torque and it's magnitude is only limeted by deflection or slip.

1 ω.Fd1 4 * / ms 2.1607

τ.Cls2Σ3 N m / 6.8494 m 2^* 4.2443 m 2^* 21 sys

....................................................................................... 1 ω.Fd1 4 * / 2.1607 ms @ = @ every ωFD1, a torue of τ.Cls2Σ3 6.8494 m 2 ^ * 4.2443 m 2 ^ * 2 1 sys N m / @ = EXPLAIN 13 1 line

is applied to the output shaft of the transmission in alternating directions. This is over and above the torque driving the train along the track and as the result of the vertical offset of the 2nd Cardan shaft will occour when ever the car is moving. In addition to the above inertial torque there is a small angular displacement when the car is going around a curve. This angular displacement is syncronised with the inertial torque and it's magnitude is only limeted by deflection or slip.

Reactions of the transmission frame.

Reaction to the traction engine tourque

Reaction to the torque output to the 2nd Cardan shaft or x(-1) when the car is traveling the other way.

Scenario Summary Page

This page alone will not be meaningful to many, however, after reading the above, a few versions of this Scenario summary will highlight the relationship of the different factors.

R.track 200 m *

β.UjFd1Σ deg 7.5888

β.UjTranΣ deg 6.4106

These figures are just repeaters. They need to be changed above the place where they are used for calculations

ν.Train km hr / 60

ω.Fd1 rev min / 1104.8739

Average

ω.Tran_max1 ω.Tran_min1 - rev min / 5.5795

Fluctuation in the Angular Velocity of the lay shaft.

a.Cls2max rev sec 2^/ 2.9905

Angular acceleration of the lay shaft

τ.Cls2 N m * 5.5468

Torque due to the acceleration of the 2nd Cardan Lay shaft

τ.Cls2 τ.Cls3 + N m * 6.8494

Different Torques due to indexing the 2nd and 3rd Cardin shafts to reenforce or diminish the

τ.Cls2 τ.Cls3 - N m * 4.2443

τ.Cls2 τ.Cls3 + τ.Cls2 τ.Cls3 - / 1.6138

1 ω.Fd1 2 * / ms 4.3214

Period

Amplidude of the Control panel movement assuming all the displacement is taken in the trans frame mounts.

s.forced deg 0.1447

Amp.CP mm 3.7869