Dynamics of Railway VehiclesTractive & Braking MotionParasitic MotionRollingPitchingHunting, yawing, nosingBouncingShuttlingLurchingSwayingTwisting

Parameters of InterestForces (Y, Q, Hy)Accelerations ( ,, )Displacements (y, z)In longitudinal (x), lateral (y) and vertical (z) directions

Categorisation of parameters according to their nature of occurrence:

Static: Caused by effects of gravity and static frictionQuasi-static: Caused by cant deficiency or excess, wind forces, curving action on points and crossingsDynamic: Caused due to irregularities in track geometry and stiffness, discontinuities like rail joints and crossings, wheelset hunting, wheel flats and other vehicle irregularitiesFrequency ranges for vertical forces:Sprung masses: 0-20 HzUnsprung masses: 20-125 HzWelds, corrugations, rail joints, wheel flats: 0-2000 Hz

Klingels formula for lateral sinusoidal motion of wheel on rail: y = y0.sin2(x/L), where y0 is amplitude of lateral displacement, x is longitudinal displacement and L is the wavelength given by the formula:L = 2(rs/2) where, r is the radius and s is the distance between the left and right rail-wheel contact point in the central position and is the conicity of wheel tread.The frequency (f = v/L, where v is speed) of the above motion should be away from the natural frequency of the vehicle.

Wheelset HuntingNo forces in unconstrained modeIn constrained mode (2y0 > flange clearance) forces develop, wavelength reduces and frequency increases.At critical speed, resonance occurs.With increasing lateral and longitudinal stiffness from 0 (unconstrained wheelset) to (rigidly held wheelset), critical speed increases from 0 to a maximum and again drops sharply to 0.This indicates importance of selecting longitudinal and lateral stiffness, clearance and damping correctly.In this diagram, longitudinal and transverse stiffness have been considered as same (KL = Kt).

Stiffness (KL = Kt)Crtical speed

Mass-Spring-Damper System

Free Undamped System:y = ymax .cosnt wheren = 2f = (k/m) is the natural frequency of the systemk is spring stiffness and m is massSystem oscillates at natural frequencyIncrease in spring stiffness increases frequencyIncrease in mass reduces frequency

Free Viscous Damped Systemy = ymax .cos(ndt).e-(r/2m).t wherend = n (1-r/rc) is the natural frequency in damped conditionr is damping coefficient, rc = 2(km) is critical damping coefficient and r/rc is damping ratioSystem oscillates at natural frequency with eponentially reducing amplitudeAt critical damping (r = rc), nd becomes 0 and response becomes pure exponentially decaying.For practical applications, damping should be well below critical level.

FreeForcedUndampedDampedUndampedDamped

Viscous vs. Friction dampingViscous damping force is proportional to rate of change of displacement.Friction damping force is constant.Viscous damping reduces oscillations exponentially.Friction damping reduces oscillations linearly, as indicated by the following:y = [ymax (F/k).(2 /).t].cos(ndt) where F is the constant damping force.

Forced undamped Systemy = ymax .cosit where ymax = yi(max)/[1 (i/n)2]yi and i are amplitude and frequency of input disturbance.System oscillates at input frequencyAmplitude increases with input frequency, approaching when i = nThis is known as resonance. The speed at which it occurs is called critical speed.Resonance can be avoided by increasing n by using harder springs.More practically, resonance should occur at low or non-dwelling speeds.

Forced damped System

y = ymax .cos(it ) where ymax = yimax/[4(r/rc)2.(i/n)2 + ( 1 (i/n)2]Where is the phase difference given by ri/(k mi2)Resonant frequency will now be = n [1 - 2(r/rc)2] at (i/n) a little less than 1.System oscillates at input frequency but with a phase difference.Highest amplitude at a given damping ratio occurs at slightly lower than natural frequency.Increasing the damping ratio greatly controls the amplitude in the resonance zone.

Spring CharacteristicsSteel SpringsLaminated: bulky, low resilience, excessive self damping, no lateral actionHelical: more efficient than laminated, little self damping, limited lateral actionVolute: Non linear, otherwise similar to helical springsBelleville: Good resilience, low self damping, versatile for selecting different combination of stiffness and deflectionRing: Best resilience feature, very high self damping, suitable for couplers and buffersRubber SpringsHigh natural hysteresis, cuts out high frequency vibrations and noiseGenerally better resilience than steelNo sliding surfaces, resulting in less wearSupportable load depends on loaded vs free area. More layers of steel bonding will increase loadability but increase stiffness.V shaped (Chevron) springs take lateral and longitudinal loads also.Conical springs provide non-linear characteristicsNatural tendency to creep or flow or to become unstable.Elastic memory causes temporary or permanent change of properties if the loading pattern remains unchanged for too long.Joule Effect causes about 0.4% change in the initial deflection per 0C change in temperature.

Air SpringsGenerally of bellow type or rubber diaphragm typeStiffness inversely proportional to air volumeVolume and spring height maintained constant by levelling valve.Stiffness increases with air pressure.Frequency is nearly independent of load at higher pressuresCuts out higher frequencies and noise very effectively.Any degree of lateral and longitudinal stiffness can be designed.Much higher load to tare ratio can be obtained.Height can be maintained within close limits.

Design considerations for vertical ridingPrimary & secondary critical frequencies should be well separated.Both should be away from natural frequency of the car body.Coupling of bogie tilting oscillations with other modes should be avoided.Centre of gravity of bogie, tractive link between bogie & body and instantaneous centre of rotation should lie in the plane of the axle centre line.For lateral ridingFor lower lateral force, vertical distance between axle centre and C.G. of the body should be more.Heavy equipment should be mounted close to the C.G. along the longitudinal neutral axis of the body.For swing link type bolsters, long swing links should be used.Divergent swing links offer greater lateral stiffness.Convergent swing links produce lateral oscillations of lower frequency.

Separation of Primary & Secondary Suspension FrequenciesMagnification Factor

Limiting values of important parameters

Lateral forceLateral force transmitted by the axle boxes to the bogie frame is the same as total force transmitted by the vehicle to the track. This is known as H force.Lateral force between wheel and rail flange is known as Y force. It is the sum of lateral component of normal force and lateral frictional and creep forces on rail-wheel contact pointThe net of left & right Y forces (Y) and H force produce wheel set acceleration.Prudhommes formula: Limiting value of Hy: ( + P/3) where P is static axle load in kN. Value of is generally 10 for timber sleepers.Y (and Q) force can be measured only with a measuring wheel.Hy force is measured by load cell placed between axle box cover and axle face or between axle box and bogie frame. The bogie design should be such that no other bogie component should transmit the force from the axle. Alternatively, such a bogie component (such as an axle guide arm) can itself be strain gauged to measure Hy. For Hy force, UIC prescribes value of as 0.9 for coaches and locos, 0.8 for empty wagons and 0.75 for loaded wagons. For Y force, is 1 for coaches and 0.85 for wagons. This indicates that Y force is lower than Hy force.IR prescribes value of as 0.85 for all rolling stock for both Y and Hy force.Duration of the lateral force exceeding the limiting value is an important consideration.

Derailment Coefficient (Y/Q)Nadals formula: Y/Q < (tan )/(1 - tan) where Y and Q are the instantaneous values of lateral and vertical forces at rail-wheel contact point, is the wheel flange angle with the horizontal plane and is coeff of friction between wheel and rail flange.As increases and decreases, lateral stability improves.Above formula assumes single point rail-wheel contact. Actually, flange contact point is a little ahead of tread contact point.Actually, is different for flange and tread contact points.This formula assumes no resultant lateral acceleration at contact point.Duration of the ratio exceeding the limit value is an important consideration.UIC prescribes limiting value of 0.8 for Y/Q for a sliding mean over 2 meter and sliding interval of 0.5 meter.IR prescribes a limiting value of 1 lasting over 1/20 secs for both Y/Q and Hy/Q. When a measuring wheel is not used, measurement of Q is simplified by correcting the static wheel load by dynamic deflection of primary springs.AAR prescribes limiting values for a single wheel, a wheelset and all wheels on one side of a bogie.With a conventional measuring wheel using slip rings on the axle face to transfer the strain gauge signals, Hy cannot be measured simultaneously.

Ride Quality and IndexRide quality relates to vehicle and Ride Index relates to passenger comfort.Human response to vibrations depend on the amplitude, frequency and duration.Generally, frequencies in 0.5 to 5 Hz range cause highest discomfort and fatigue.Frequencies above 20 Hz are barely perceptible to human body.Lateral vibrations cause more discomfort than vertical ones.Sperling postulated ride quality as the product of displacement (y or z), acceleration (d2y/dt2 or d2z/dt2) and impulse (d3y/dt3 or d3z/dt3).This translates to RI = 0.896 [(3/f).F(f)]0.1F(f) is a weighting factor related to frequency.For Ride Quality, F(f) is taken as one. Mostly applicable to wagons.For passenger carrying stock, F(f) in lateral mode is higher than in vertical mode.F(f) below 0.5 Hz is taken as 0. Above 5 Hz, it reduces sharply.UIC prescribes assessment of ride quality by rms values of acceleration.

Ride Index for Passenger Stock

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