**2. Rigid-body bounce vibration mode and elastic natural vibration mode**

The ballasted tracks are structurally prone to deteriorate over time. They absolutely require periodic maintenance work and repair. Recently, many attempts were made to improve the ballasted track structure. For example, many types of elastic and/or viscoelastic structural members, such as rail pads, under-sleeper pads and under-ballast mats, were attempted to reduce ballast degradation [1–3]. In one experimental investigation, the sleeper's vibration characteristics, including the dynamic effects of sleeper/ballast interactions, were investigated through a modal analysis to predict the railway track's dynamic response [4]. Dynamic wheel/ rail interactions, which significantly contribute to impact vibration and noise, were also investigated for rail and wheel surface defects in field measurements and numerical simulations [5–8]. When considering the future of the railway management, it is impossible to disregard the necessity for frequent maintenance that is dependent upon manual aid. Therefore, the need exists to improve maintenance methods for the ballasted track based on findings from empirical and numerical investigations of the dynamic response characteristics and deteriora-

Running trains cause dynamic loads mainly through two mechanisms [9, 10]. One is the dynamic load from passing axle loads as a train passes. The related frequency characteristics, which depend on the number of axles passing per unit of time, are limited to low frequencies of only several Hertz to approximately 30 Hz. The other mechanism is the impact load that is generated dynamically by the rolling contact mechanism between the wheels and rails. The ballast layer transmits this sharp pulse-shaped impact load superimposed on the low-frequency loads from passing axles. This waveform, when transformed into a frequency domain, exhibits numerous vibration components with broadband characteristics that extend from low frequencies to several kilohertz. That is to say, dynamic response measurements of the ballasted track require high-precision measurements of vibration components ranging broadly from several Hertz to several kilohertz. Outputs of sensor would be degraded by noise of tens of millivolts deriving from the inductive currents of high-voltage overhead cables in conventional field measurements, which necessitated the use of a low-pass filter to alleviate that interference. Ensuring the measurement accuracy of high-frequency vibration components exceeding 50–100 Hz was impossible under such conditions. For that reason, no comprehensive discussion of these components has been reported in the literature to date.

This chapter gives accurate field measurements of the dynamic responses of the ballasted track with a train passage at a sampling frequency of 10 kHz without low-pass filters, using special sensing sleepers and sensing stones produced by the authors [11, 12]. Using the measurement effects, the analysis can be done with a focal point on propagation characteristics of dynamic loads inside the ballast layer and vertical natural modes of the ballast layer. Moreover, the author conducted a free fall-weight impact loading test on a full-scale mock-up of a ballasted track to ascertain the response motions of the ballast layer, with high rapidity and with high accuracy, in the high-frequency region immediately after the impact load was used [13]. Furthermore, using a direct solver MUMPS [14–17] corresponding to large-scale parallel computing of a distributed memory type, according to finite element transient response analysis FrontISTR [18] and Advance/FrontSTR [19] based on a fine ballast aggregate model, both the elastic natural vibration mode and the rigid-body bounce natural vibration mode of

the granular ballasted layer are simulated numerically.

tion factors of the ballast layer.

104 New Trends in Structural Engineering

In general, a dynamic load propagates as an elastic wave through the interior of an object, consequently inducing the natural vibration modes specific to the object, which can be applied to the ballasted track composed of the ballast aggregate. Although the ballast layer is a discontinuous structure, it presumably has natural vibration modes that are specific to the ballasted track. **Figure 1** shows some characteristics of the principal natural vibratory motions in the vertical direction of the ballast layer [20]. One characteristic is the rigid-body bounce natural vibration mode. Another is the elastic shrinkage natural vibration mode. The ballasted track structure represents a single degree of freedom system that includes sleepers, rails and other components which constitute the track structure mass, along with the ballast layer and roadbed which constitute the spring rigidity component. In the bounce natural vibration mode, this single degree of freedom system moves vertically and rigidly under a train's dynamic load. The ballast layer acts as an elastic one-directional spring in the vertical direction. Rigid-body natural vibration modes of six kinds exist: They are translational and rotational along each of the three axes. In both the dynamic loads applied to the ballast layer and in the responses of the ballast layer, the translational vibration components in the vertical direction are predominant.

Therefore, this research specifically examines the translational bounce behaviour in the vertical direction. According to the dynamics, the natural frequency of the first-order rigid-body bounce mode *f* 1 is given theoretically as *f* <sup>1</sup> = √ \_\_\_\_ *k*/*m*/(2*π*), where *k* and *m*, respectively, denote the ballast stiffness and overburden mass of the track structure. This mode reportedly exists at approx. 100 Hz [21].

The elastic vibration mode is the motion by which the whole ballast layer shrinks and stretches vertically as an elastic body. This natural vibration mode is considered not to occur in a normal-state ballast layer but to occur when the ballast layer is under high confining pressure generated by the train's weight applied to the layer. To date, no report of the relevant literature has described a study conducted to capture this mode, that is, ballast motion in the frequency domain related to this mode. Moreover, on a real track, natural vibration that

**Figure 1.** Principal vertical natural vibration motions of the ballast layer. (a) Rigid-body bounce mode, (b) elastic shrinkage mode.

entails bending deformation and torsional deformation of the members occurs. According to the physical theory related to standing waves on the railway track, the natural frequency of the elastic first-order vibration mode *f* 1 is given theoretically as *f* <sup>1</sup> = √ \_\_\_\_\_ (*E*/*ρ*)/(4*L*), where *E*, *ρ* and *L*, respectively, denote the ballast layer's Young's modulus, density and thickness.

## **3. Vibration tests by the use of a full-scale mock-up**

The author built a full-scale mock-up of the ballasted track structure and investigated natural vibration characteristics of the ballasted track by performing an experimental modal analysis based on impulse excitation tests.

**Figure 2** shows the profile/plane of the mock-up and the sensor positions [22–24]. To build the full-scale mock-up, the author employed new ballast using hard andesite, which complies with the same standard for the real track. They were compacted sufficiently. Type 3 prestressed concrete (PC) monoblock sleepers were used for the mock-up. They are used widely for the metre-gauge (1067 mm-wide) railway lines that are conventionally used for Japan Railway Companies. For the prevention of interference of vibration effects of the concrete frame, urethane foam panels were sandwiched between the mock-up and the outer concrete frame. The author installed acceleration sensors on the mock-up and the concrete frame, vibrated the mock-up and ensured that the vibrations were sufficiently isolated to assess the vibration-insulating properties. The author conducted tests by hitting the end of the sleeper laterally, longitudinally and vertically with an impulse hammer to make it vibrate. Test records included measurements of the acceleration responses: 22 sleeper locations and several ballast layer locations. From those data the accelerance was calculated: the transfer functions of the acceleration responses to the excitation force in the frequency domain. Then

experimental modal analysis was conducted considering the location relations of the measuring points. As a result, for the ballasted track, the author was able to identify the natural vibration frequencies and their modal shapes between the low-frequency domain and 1 kHz. **Figure 3** presents the natural vibration frequencies and the specific modal shapes of the ballasted track that were acquired from the test results [22–24]. Although there are rigidbody vibration modes of six types as described above, the figure shows only the rigid-body translational mode in the vertical direction. It also shows that the vertical and translational rigid-body vibration is generated at 98 Hz, which agrees well with earlier reported research results. Furthermore, six types of dominant natural vibration mode entailing the bending and torsional deformations of sleepers are identified as shown in the same figure. Nevertheless, the author was unable to capture any vertical elastic natural vibration mode of the ballast layer in this test. That is true probably because the elastic vibration modes of the whole ballast layer would occur only when the ballast layer continuity is sufficiently satisfied according to

Vertical Natural Vibration Modes of Ballasted Railway Track

http://dx.doi.org/10.5772/intechopen.79738

107

Dynamic responses were measured on an actual ballasted track of a main conventional railway line in Japan to identify the dominant natural vibration modes of the ballasted track. The track structure at the measurement site, consisting of continuous welded rail weighing 60 kg/m and type 3 PC sleepers, was designed based on the Japanese standard [25], which allows a running speed higher than 130 km/h. The measurement site was located on a solid embankment in a straight section. For spacing between the sleepers, 41–42 sleepers are positioned over a

the train loads on the ballast layer.

**4. Field measurements and spectral analysis**

**Figure 3.** Mode shapes of natural vibration acquired from test results.

**Figure 2.** Overview of full-scale vibration test.

**Figure 3.** Mode shapes of natural vibration acquired from test results.

entails bending deformation and torsional deformation of the members occurs. According to the physical theory related to standing waves on the railway track, the natural frequency of

The author built a full-scale mock-up of the ballasted track structure and investigated natural vibration characteristics of the ballasted track by performing an experimental modal analysis

**Figure 2** shows the profile/plane of the mock-up and the sensor positions [22–24]. To build the full-scale mock-up, the author employed new ballast using hard andesite, which complies with the same standard for the real track. They were compacted sufficiently. Type 3 prestressed concrete (PC) monoblock sleepers were used for the mock-up. They are used widely for the metre-gauge (1067 mm-wide) railway lines that are conventionally used for Japan Railway Companies. For the prevention of interference of vibration effects of the concrete frame, urethane foam panels were sandwiched between the mock-up and the outer concrete frame. The author installed acceleration sensors on the mock-up and the concrete frame, vibrated the mock-up and ensured that the vibrations were sufficiently isolated to assess the vibration-insulating properties. The author conducted tests by hitting the end of the sleeper laterally, longitudinally and vertically with an impulse hammer to make it vibrate. Test records included measurements of the acceleration responses: 22 sleeper locations and several ballast layer locations. From those data the accelerance was calculated: the transfer functions of the acceleration responses to the excitation force in the frequency domain. Then

is given theoretically as *f*

<sup>1</sup> = √

\_\_\_\_\_

(*E*/*ρ*)/(4*L*), where *E*, *ρ* and

1

**3. Vibration tests by the use of a full-scale mock-up**

*L*, respectively, denote the ballast layer's Young's modulus, density and thickness.

the elastic first-order vibration mode *f*

106 New Trends in Structural Engineering

based on impulse excitation tests.

**Figure 2.** Overview of full-scale vibration test.

experimental modal analysis was conducted considering the location relations of the measuring points. As a result, for the ballasted track, the author was able to identify the natural vibration frequencies and their modal shapes between the low-frequency domain and 1 kHz.

**Figure 3** presents the natural vibration frequencies and the specific modal shapes of the ballasted track that were acquired from the test results [22–24]. Although there are rigidbody vibration modes of six types as described above, the figure shows only the rigid-body translational mode in the vertical direction. It also shows that the vertical and translational rigid-body vibration is generated at 98 Hz, which agrees well with earlier reported research results. Furthermore, six types of dominant natural vibration mode entailing the bending and torsional deformations of sleepers are identified as shown in the same figure. Nevertheless, the author was unable to capture any vertical elastic natural vibration mode of the ballast layer in this test. That is true probably because the elastic vibration modes of the whole ballast layer would occur only when the ballast layer continuity is sufficiently satisfied according to the train loads on the ballast layer.
