Dynamic Displacement Amplification Flashcards by Stan Broster

What causes vibration and issues on rail tracks?

Stress waves are induced by the track structural responses
Vibration source at the wheel-rail interface
Discontinuity on the track
Variable support

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How does soft ground give annoyance to people?

Soft ground gives whole body vibration and rattling

In Sweden, soft soils cause vibration issues

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How does hard ground give annoyance to people?

Hard ground gives structural borne noise

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What are the three types of waves, and what order are they in?

Compression wave comes first (e.g. what you hear when someone is talking)

Shear wave comes second (e.g. slinky)

Rayleigh wave comes last

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What is the most destructive wave, and roughly how much energy does it have?

Rayleigh wave - contains 67% of energy

The Rayleigh wave is about 95% of shear wave (for typical Poisson’s ratio)

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What does ‘v’ in the compression wave equation stand for?

Poisson’s ratio

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What is the shear wave equation also equal to?

= sq. [G/p]

Where G = shear modulus,
rho = density

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What is a reduction of response with distance?

Attenuation

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What are two example sensors for vibration measurement?

Geophones (velocity)

Accelerometers (acceleration)

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What are low frequencies (e.g. 0-40 Hz) likely to cause damage to?

Damage to substructure
Vibration and contact noise (buildings and vehicle)
Damage to vehicle (carriages, bogies, axles, wheels)

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What are high frequencies (e.g. 400-1500 Hz) likely to cause damage to?

Damage to rail (part of superstructure)
Radiated sound/noise to residents and passenger
Damage to vehicle (wheels only)

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Define critical speed

What does it result in?

Critical speed is when the train speed reaches the surface ground wave velocity of a homogeneous subgrade

Results in high ground displacement and surface wave propagation

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If there is no high track stiffness track structure at the surface (e.g. for slab track), when does critical speed occur?

The critical velocity in this scenario is given by the in-situ Rayleigh wave velocity

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If there is a high-stiffness structure near the track surface, what might happen?

The train speed’s critical velocity might increase above the Rayleigh-wave velocity

ie. towards a critical track velocity

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What needs to be done to run trains at high speed (higher than Rayleigh wave speed)?

Need to ground stabilise (increase the stiffness of the soil)

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What is peak-to-peak displacement?

The distance from peak displacement as the rail moves downwards to peak displacement as the rail moves upwards

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What is required for high-speed lines?

Very stiff formations (e.g. concrete slab-track)

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For the effect of undrained shear strength on the maximum permissible speed for a homogenous subgrade (e.g. clay soils), what ratio is used?

NB. ratio not on eqn sheet
NB. n = Rayleigh wave percentage

The value of K_c varies with the plasticity index (I_p), and the overconsolidation ratio (OCR)

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How is the plasticity index calculated?

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What can this equation be used to calculate?

Undrained shear strength (C_u) required for a permissible max. train speed for a given Rayleigh wave percentage (n)

NB. C_u is sometimes written as S_u

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What is the equation that relates undrained shear strength with depth?

What is the Rayleigh wave velocity ratio (M_RWV)?

What should the Rayleigh wave velocity ratio (M_RWV) be equal to?

0.6

What is the critical track velocity ratio (M_CTV)?

NB. using ballasted track with no embankments, V_CTV should be similar to V_R (from the Rayleigh wave velocity ratio)

Define ballast peak particle velocity (PPV) What is it dependent on?

PPV of the ballast is the **induced ballast velocity** due to train passage - it is the **rate at which the sleeper moves up and down** - it is dependent on the train speed (see graph)

What happens to PPV when the track reaches critical velocity?

A **gradient change** occurs (see graph) **Track dynamics** start to occur

At what point does PPV increase linearly/non-linearly with train speed?

At **sub critical velocity ratios** (of <0.5-0.6), PPV increases linearly with train speed **Past 0.6** the PPV increases in a non-linear way

What test is used to determine subgrade stiffness (E_V2)

The **plate load test** The plate load test is also used for calculation of vertical track stiffness In the UK, the relationship between E_V2 and E_max (small strain stiffness) is given in pic

What effect does higher baseplate stiffness (K_pla) and higher train speed have on the ballast?

The ballast experiences **higher vibration speed and acceleration** (g)

Explain how PPV (and overall track geometry) can be improved?

By using **pads** This can reduce high contact force (spreads it out) by **slowing stress wave**; slowing down the loading onto the ballast and improving track geometry

What happens to the ballast if high accelerations (e.g. 1.4g) are experienced?

The ballast starts to **unpack** (shakes, lifts up and then down) If PPA exceeds **0.7-0.8g** unmodified ballast starts to **destabalise**

Name the different parameters that PPV is related to

What happens if PPV exceeds 20 mm/s?

**Ballast decompression** starts - ballast migration will lead to increased maintenance

Which ground wave profile is above/below critical speed?

Top - above critical speed Bottom - below critical speed

What do these two models show?

a) **quarter-train** model b) **pseudo full-train** model

For this equation of motion for dynamic analysis, what are the different solution methods?

- **Frequency domain** solutions - Modal superposition solutions - Implicit time stepping algorithms - Explicit time stepping algorithms NB. don't need to know how to solve eqn of motion in exam

What does this ground displacement profile at Rayleigh-wave development (single moving load, homogenous soil) show?

**Pre-critical speed** There is a '**bowl of deformation**' when the axle is moving across slowly

What does this ground displacement profile at Rayleigh-wave development (single moving load, homogenous soil) show?

**Post-critical speed** At critical speed, a '**cone**' forms

The graphs show sleeper deflection before and passing critical speed at Ledsgard. Describe what is happening.

- The **response becomes out-of-phase** (it 'loses' one of the peaks) - There is very high **deformation** at/after critical speed - All the **frequencies collided** in this case study, giving very high displacements

For the Ledsgard case study, what factor determined whether the behaviour (for deflection*) was linear or non-linear?

The values of deviatoric strains

When plasticity comes into effect, what happens to soil properties?

They can reduce

For the Ledsgard soils, how do the dynamic soil properties (soil density, s-wave velocity, p-wave velocity, damping ratio) vary with depth under the rail?

For the equations for non-linear modelling, what is the typical value for P_atm?

In this critical velocity envelope/critical speed curve/interpolated normalised response curve for Ledsgard (in both the downward and upward directions), what is shown?

**Dynamic amplification envelope** - shows that there is an increase in **dynamic deflection** as the critical velocity is approached The downward deflections are normalised to the max. quasi-static downward deflection (likewise for upwards)

What does this critical speed curve for the Ledsgard displacement results show?

**Dynamic only** component Only the displacement results for the dynamic-only component of the upward and downward response - ie. the quasi-static deflections for both the upward and downward deflections have been subtracted from the peak values Graph shows how the dynamic component only is developed with train speed ratio (M_CTV) - The distance between the upward and downward points represents the peak dynamic component of the propagating ground waves with train speed ratio M_CTV

What does this train-track interaction drawing show?

The stiffness change from 'floating' to 'fixed' track

What does this train-track interaction drawing show?

The development of hanging sleepers in the transition zone

What does this train-track interaction drawing show?

It shows **embankment consolidation** This leads to a larger elevation difference

What could be introduced at a track transition point improve passenger experience? What are example parameters that should be minimised at transition points?

- Could have a **rigid base** (see drawing) - Wheel displacement - Wheel/rail interaction force - Train body vertical acceleration (g)

What can happen if a bridge is not at 90 degrees perpendicular to the rail?

This can cause a **cross-track fault**, which could lead to **flange climb** and derailment