Dynamic response of elastic beam to a moving pulse: finite element analysis of critical velocity

3.1. Features of beam deflection

As an example, Fig. 4(a) shows the deflection profiles of beam at two different times, where the velocity of moving pulse and the pulse duration are taken as 200 m/s and 0.1 s, respectively. It is found that the vibration of beam is significant in a particular region in front of the moving pulse. The vibration behind the moving pulse is not significant and the beam deflection is around 3 mm. At time 0.2 s, the amplitude of vibration is small and decays as the distance from the pulse front increases. At time 0.6 s, the vibration in a particular region in front of the moving pulse is relatively stable. With the increase of time, this stable region increases gradually.

Fig. 4a) Beam deflection and b) equivalent stress vs. position at different times. The symbols “pn” and “vn” denote the nth peak and the nth valley of beam deflection, respectively, ahead of the pulse front

a)

b)

Fig. 5a) Vibration of the node at position 300 m and b) amplitude of deflection vs. frequency of vibration of the node

a)

b)

With the transverse vibration of beam, flexural waves propagate along the beam. The wavelength of the flexural waves ahead of the pulse front is approximately equal to the spatial length of the pulse. The phase velocity of flexural wave depends on the main frequency of the flexural wave [12]. As an example, the vibration of the node at position 300 m away from the symmetry boundary was analyzed in Fig. 5. The node is still at first, and then it begins to vibrate when the flexural wave arrives, as shown in Fig. 5(a). After a particular time, denoted as $t_s$, the node vibrates in a relative stable manner until the moving pulse arrives. In this case, the time when the moving pulse reaches this node, $t_{front}$, is 1.5 s. We determined the main frequency of vibration between $t_s$ and $t_{front}$ with the method of Fast Fourier Transform (FFT). The resulted amplitude of deflection, $A_d$, vs. frequency of vibration, $f$, curve of the node is shown in Fig. 5(b), where the main frequency is 7.2 Hz. This frequency is in agree with the frequency of harmonic wave travelling at the same velocity as that of moving pulse in an elastic beam, determined by [12]:

where $a$ is defined as:

with $A$ and $I$ being the area and the moment of inertia of the cross-section, respectively. This implies that the phase velocity of flexural wave is equal to the velocity of moving pulse. As shown in Fig. 6, the main frequency of vibration at 300 m is coincided well with the frequency of harmonic wave travelling at the velocity of moving pulse for any velocity of moving pulse. For a particular elastic beam, the main frequency of the flexural wave is not influenced by the spatial length of the moving pulse and only dependent on the velocity of moving pulse.

Fig. 6Comparison between the main frequency of vibration at 300 m and the frequency of harmonic wave for different velocities of moving pulse

3.2. Features of maximal equivalent stress

We first take the pulse duration to be 0.1 s but vary the velocity of moving pulse to figure out some features of the distribution of maximal equivalent stress. For a particular velocity of moving pulse, there is two phases for the distribution of maximal equivalent stress. In phase I, the maximal equivalent stress increases as the distance to the symmetry boundary increases until it reaches a maximum value, see Fig. 7(a). In phase II, the maximal equivalent stress is much stable. From Fig. 7(a), it is found that the maximal equivalent stress is influenced by the velocity of moving pulse dramatically.

Fig. 7a) Maximal equivalent stress vs. position and b) the average value of maximal equivalent stress vs. the velocity of moving pulse

a)

b)

To evaluate the level of maximal equivalent stress in phase II, we evaluate the average value of maximal equivalent stress in this phase, denoted as ${\stackrel{-}{\sigma }}_{e,max}$. The beginning position of phase II is determined by minimizing the variance of ${\stackrel{-}{\sigma }}_{e,max}$. The average beginning position of phase II for the velocities of moving pulse considered in Fig. 7(a) is about 50 m. As the velocity of moving pulse increases, ${\stackrel{-}{\sigma }}_{e,max}$ increases first before reaching its maximum value and then decreases, as shown in Fig. 7(b). It is found that ${\stackrel{-}{\sigma }}_{e,max}$ achieves its maximum value at a certain velocity, which is termed as the critical velocity. In this case of $T=$ 0.1 s, the critical velocity is 221 m/s.

The critical velocity is dependent on the pulse duration. We calculated a series of ${\stackrel{-}{\sigma }}_{e,max}$ with varying pulse duration and velocity of moving pulse, and the results are shown in Fig. 8. It is found that the critical velocity decreases with increasing pulse duration, but ${\stackrel{-}{\sigma }}_{e,max}$ along the line of critical velocity increases with increasing pulse duration.

Fig. 8The variation of σ-e,max with the velocity of moving pulse and the pulse duration: a) three-dimensional diagram and b) a contour plot

a)

b)

3.3. Mechanism of stress fluctuation

To explore the mechanism of maximal equivalent stress fluctuation with position, we also take the pulse duration of 0.1 s as an example. The variation of ${\sigma }_{e,max}$ with position at phase II is shown in Fig. 9(a).

For each position, the time when ${\sigma }_{e,max}$ is reached, $t_{MES}$, and the time when the pulse front arrives, $t_{front}$, were analyzed. Three velocities of moving pulse, namely 180, 220 and 260 m/s, are considered, which correspond to velocities lower than, closing to and higher than the critical velocity, respectively. The value of ${\sigma }_{e,max}$ means the maximum value of equivalent stress that a specific position ever had. In this beam model, the maximal equivalent stress implies the maximum degree of bending which appears at the peak/valley of the beam deflection. The serial numbers of peaks and valleys are defined in Fig. 4(a). The symbols “p$n$” and “v$n$” denote the $n$th peak and the $n$th valley of beam deflection, respectively, ahead of the pulse front. Fig. 4(b) shows the equivalent stresses in the beam. According to the positions of ${\sigma }_{e,max}$ shown in Fig. 4(b) and the deflection profile of beam shown in Fig. 4, we can determine that ${\sigma }_{e,max}$ appears in p1 and p2 when $t=$0.2 s and 0.6 s, respectively. In this way, the time that ${\sigma }_{e,max}$ appears in peaks and valleys of beam deflection can be determined, as presented in Fig. 9.

It is interesting to find that the position of maximum degree of bending successively appears at p1, v1, p2, and so on, of the beam deflection. The duration of ${\sigma }_{e,max}$ appearing at p1 is longer than those at v1, p2, and so on. This is because that the bending degrees of the following p$n$ or v$n$ also increase before the position corresponding to p1 attains its maximal bending degree. In phase II, it is found that the positions where ${\sigma }_{e,max}$ attains minimum point are coincident with the positions where the corresponding beam deflection switches from the peak to the valley or vice versa. The travelling speeds of peaks or valleys are the reciprocal of the line slopes of $t_{MES}$ for each corresponding segment. It can be seen from Fig. 9 that the slopes of all parts of the curve are equal to the slope of the time when pulse front arrives versus position curve. Therefore, the travelling speed of p$n$ or v$n$ is equal to the velocity of moving pulse. This result agrees well with the analysis of main frequency of flexural wave in Section 3.1.

Fig. 9Maximal equivalent stress, σe,max, the time when σe,max is reached, tMES, and the time when the front of moving pulse arrives, tfront, at each position for different velocities of moving pulse: a) 180 m/s, b) 220 m/s and c) 260 m/s. The symbols “pn” and “vn” represent the nth peak and nth valley of beam deflection, respectively, when σe,max is reached

a)

b)

c)

The relative distance between the pulse front and p1, $D$, are pulse-velocity dependent. The variations of $D$ for the three typical velocities of moving pulse are shown in Fig. 10. At the beginning time, they are almost identical but then fluctuate around different levels. It can be seen that when the velocity of moving pulse is less than the critical velocity, the position corresponding to p1 is ahead of the pulse front. When the velocity of moving pulse is larger than the critical velocity, the position corresponding to p1 is behind the pulse front. If the velocity of moving pulse closes to the critical velocity, the position corresponding to p1 is also very close to the pulse front. These phenomena indicate that the influences of the interaction between the moving pulse and flexural wave in the beam are sensitive and depend on their relative velocity.

Fig. 10The relative distance between the pulse front and the first peak of beam deflection for different velocities of moving pulse: 180, 220 and 260 m/s. Positive distance represents the position corresponding to p1 is ahead of the pulse front

3.4. Dimensional analysis of critical velocity

According to Fig. 8, it is noticed that the critical velocity is strongly influenced by the pulse duration. To comprehensively understand the influence factors of the critical velocity, dimensional analysis is carried out in this section. There are seven parameters that may affect the critical velocity, including three material parameters (density $\rho$, Young’s modulus $E$ and Poisson’s ratio $\nu$), two structural parameters (area $A$ and moment of inertia $I$ of beam cross-section), and two load parameters (pulse duration $T$ and pulse amplitude $p_0$). Thus, the critical velocity $V_{cr}$ can be written in the form:

where $C_0$ is the speed of elastic wave, which is equal to ${\left(E/\rho \right)}^{1/2}\text{.}$ Among the seven governing parameters, if taking $E$, $A$ and $T$ as the three variables with independent dimensions, we can obtain four independent dimensionless parameters, namely $\rho A/\left(E{T}^2\right)$, $I/A^2$, $p_0/E$ and $\nu$. Thus, dimensional analysis yields:

this means that the dimensionless critical velocity depends on four dimensionless parameters.

In fact, the last two parameters in Eq. (4) have no or slight influence on the value of the dimensionless critical velocity. By changing $p_0$ but keeping the other six parameters in Eq. (3), the finite element simulations show that ${\stackrel{-}{\sigma }}_{e,max}$ is proportional to $p_0$ for a particular velocity and the value of $V_{cr}/C_0$ is identical for different $p_0/E$. This is because that the beam is elastic and the finite element model is linear. Changing $\nu$ but keeping the other six parameters in Eq. (3), we found that $V_{cr}/C_0$ is unchanged. In the beam theory, the influence of Poisson’s ratio should not be neglected only when the effect of shear stress need to be considered. The effect of shear stress can be ignored when the flexural wave length is 20 times larger than the gyration radius ($R_g$) of the beam cross-section [13]. The relationship between the flexural wave length and the gyration radius of the cross-section satisfies the above condition. Therefore, the effects of $p_0/E$ and $\nu$ on the dimensionless critical velocity can be neglected. According to these understandings, Eq. (4) can be further reduced to:

in the following study, the two dimensionless parameters $p_0/E$ and $\nu$ are set to be 4.286×10^-8 and 0.3, respectively.

Fig. 11Vcr/C0 vs. ρA/(ET2) for different I/A2 a) and b) Vcr/C0 vs. I/A2 for different ρA/(ET2)

a)

b)

The relationship between $V_{cr}/C_0$ and $\rho A/\left(E{T}^2\right)$ for three sets of values of $I/A^2$ is illustrated in Fig. 11(a). It can be found that $V_{cr}/C_0$ increases as $\rho A/\left(E{T}^2\right)$ increases for a particular value of $I/A^2$. This relation can be fitted by a power law:

where ${\alpha }_1$ and ${\beta }_1$ are fitting parameters. The fitting results of ${\alpha }_1$ and ${\beta }_1$ corresponding to the three sets of values of $I/A^2$ are also shown in Fig. 11(a). For different $I/A^2$, ${\beta }_1$ is almost constant, which is approximately equal to 0.25.

Similarly, the relationship between $V_{cr}/C_0$ and $I/A^2$ for three sets of values of $\rho A/\left(E{T}^2\right)$ is illustrated in Fig. 11(b). For a particular value of $\rho A/\left(E{T}^2\right)$, $V_{cr}/C_0$ increases with the increase of $I/A^2$. A power function is also selected to fit the relationship between $V_{cr}/C_0$ and $I/A^2$:

where ${\alpha }_2$ and ${\beta }_2$ are fitting parameters. From Fig. 11(b), it is found that ${\beta }_2$ is very close to 0.25 for different $\rho A/\left(E{T}^2\right)$.

According to these understandings, for simplicity, the fitting function of $V_{cr}/C_0$ is taken as:

where $\alpha$ is a fitting parameter. Fitting all above finite element results, we obtain $\alpha =$ 2.36. Thus, we have:

where $R_g$ is the gyration radius of the beam cross-section, which is equal to ${\left(I/A\right)}^{1/2}$. Therefore, the critical velocity is affected by the speed of elastic wave, the gyration radius and the pulse duration. These three parameters characterize the material, structural and load properties, respectively.

According to Ref. [1], the spatial interval between the positive and the negative peak of the air pressure pulse, $L_p$, is constant for a particular type of trains. In our finite element model, this spatial interval is equal to $VT/2$. When the velocity of moving pulse reaches the critical velocity, we have $L_p=V_{cr}T/2$. In this critical situation, substituting $T=2L_p/V_{cr}$ into Eq. (8) yields:

The critical velocity decreases as the spatial interval grows when the gyration radius and the speed of elastic wave are constant. The spatial interval $L_p$ is strongly dependent on the shape of the fore- and after-bodies of trains [1]. The designers of high-speed train may make the critical velocity meet the safety standard by modifying the shape of the fore- and after-bodies of trains.