Suppression of the stationary maglev vehicle–bridge coupled resonance using a tuned mass damper

Abstract

The coupled resonance that occurs between a maglev vehicle and an elevated bridge is a unique problem for an electromagnetic suspension maglev system when the vehicle is suspended above the bridge without moving. This problem causes the bridge to vibrate in a large amplitude, it significantly degrades the ride comfort of the maglev train, and it needs to be solved before the commercial application of the system. In this paper, the principle of the stationary vehicle–bridge coupled resonance is investigated, and it is shown that the levitation system is non-passive and it causes self-excited vibration when the first resonance frequency of the bridge approaches the critical frequency of the levitation system. To eliminate the coupled resonance, an approach using a tuned mass damper (TMD) to render passive the levitation control system is presented, and it shows that a TMD with appropriate parameters can stabilize the coupled system. A procedure for seeking out a group of suboptimal parameters of the TMD is also proposed. This procedure ensures that the designed result has a satisfactory performance as well as a reasonable stability margin. The location where the TMD should be mounted to achieve the best performance is discussed. The numerical simulation shows that the designed TMD can completely suppress the vehicle–bridge coupled resonance.

Keywords

Coupled maglev resonance TMD vehicle-bridge

1. Introduction

Compared with conventional railway systems, the maglev system has the advantages of low noise and high speed due to the absence of mechanical contact between the vehicle and the guideway. The outstanding climbing ability (7%–10% slope) and small turning radius (50 m–70 m) make the low speed electromagnetic suspension (EMS) maglev system suitable for application to an urban transportation system (Yan, 2008). However, using controlled electromagnetic forces to neutralize its weight, the dynamic behavior of the EMS maglev system becomes rather complicated when the vehicle is passing through an elevated bridge (or girder) at a speed lower than 10 km/h. A typical problem is the coupled resonance (which also appears as self-excited vibration or coupled vibration in other literature) that occurs between a maglev vehicle and a bridge.

Some examples exist of problems caused by the maglev vehicle-guideway self-excited vibration when the vehicle is moving at a very low speed. She et al. (2008) reported that the TR-08 high speed maglev vehicle on the Shanghai maglev demonstration line experienced violent coupled vibration while the vehicle was crossing a turnoff at a very low speed. Their research indicated that the flexibility and the first resonance frequency of the turnoff were the main factors that contributed to the coupled vibration. In some cases, the vehicle–bridge coupled resonance can cause control failures to the levitation control system, resulting in contact between the vehicle and the track. Albert et al. (2008) reported that the AMT (American Maglev Technologies) system achieved successful levitation on a guideway mounted to a concrete foundation in Florida, but encountered difficulties in achieving stable levitation when the vehicle was moved to an elevated guideway in the Old Dominion University campus. It was believed that the flexibility of the 90-feet (27.45 m) long, essentially simply supported elevated girders was the main reason that led to the difficulties of achieving a stable levitation. A similar problem has also been encountered in the low speed maglev vehicle CMS-03 A, on the Tangshan maglev test line. Figure 1 shows the vertical acceleration in the midpoint of a 14.5 -m long, elevated bridge when the CMS-03 A vehicle was passing over it at a very low speed. It can be seen that the amplitude of the bridge vibration grows with time. In some cases, the amplitude of the vibration will grow to be sufficiently large that some of the electromagnets in the vehicle will collide with the steel track, leading to a levitation failure.

However, as of today, only a small amount of previous work aiming to solve the resonance problem has been done. By applying the center manifold reduction and the method of multiple time scales, Wang et al. (2008a, b) theoretically investigated the stability of the coupled system in the presence of time delays in the feedback signals, and found that time delay in the feedback path is an important factor which might lead to the coupled resonance. Yet time delay is not the only thing that causes the coupled resonance problem. Shi et al. (2004) pointed out that the natural frequency of the controller and the first resonance frequency of the bridge being too close to each other might lead to coupled resonance. This conclusion seems reasonable because experiments have shown that the resonance frequencies of the bridge play an important role in the coupled resonance problem. However, more work still needs to be done in exploring an appropriate method to suppress the coupled resonance and this is the focus of the current paper.

The fact that the amplitude of the vibration grows with time indicates that the vibration energy of the bridge grows with time when the resonance occurs. Since there is no other way to transmit energy to the bridge except from the electromagnetic levitation system, it can be concluded that the vibration energy comes from the levitation system. In other words, the levitation system is non-passive (that is, the energy supplied to the system is greater than the energy dissipated by passive damping). The passivity analysis of the system is always ultilized to investigate the stability of the system, and it has been widely applied in controller design for elastic systems. Kelkar and Joshi (2004) introduced a passivity-based controller design methodology which is based on the concept of using compensation methods to render the system passive. Since the negative feedback connection of any passive and strictly passive system is asymptotically stable, the designed controller can stabilize the coupled system. By a demonstration of its applications to several dynamic systems, Kelkar and Joshi (2004) pointed out that the most important advantage of the passivity-based approach is that it has inherent robustness to unmodeled dynamics and parametric uncertainties, and it is uniquely suited for application to elastic systems. Shimizu et al. (2004) investigated the magnetic levitation system with two electromagnets located above a flexible beam. The system was decomposed into two subsystems − the electrical subsystem and the mechanical subsystem, and the controllers for each subsystem were designed separately to ensure that they were passive. It was shown that the passivity-based controller design strategy could suppress all elastic vibration modes and it didn’t need any model reduction. Similar work has also been done by Namerikawa and Kawano (2006), in which a passivity-based controller was designed to stabilize the iron ball suspension system. The experiments showed that the passivity-based controller achieved better performance than the conventional PID controller. The work listed above indicates that the controller designed using the passivity-based approach has significant robustness in the presence of unmodeled dynamics and parametric uncertainties.

In this paper, the passivity-based approach is also adopted for developing a method to stabilize the maglev vehicle–bridge coupled system. However, unlike the passivity analysis listed above, the passivity of the magnetic levitation system is investigated here based on “virtual excitation”; that is, the passivity of the levitation system is investigated by energy flow analysis on the assumption that the bridge is vibrating over a wide frequency range.

The tuned mass damper (TMD) is a passive device that has been widely used for reducing the structural vibration of flexible structures. A typical application of the TMD is in the reduction of vibration in bridges with moving loads. For example, Lin et al. (2005), and Shi and Cai (2008) discussed the application of the TMD in the suppression of bridge vibration excited by trains or vehicles, and they concluded that the TMD was effective in suppressing the vibration of the bridge. However, application of the TMD to a maglev elevated bridge to suppress the stationary maglev vehicle–bridge coupled vibration has not yet been well studied. In this paper, in contrast to the well known situation where the TMD is used for decreasing the vibration level of a flexible structure, the TMD is used as a feedforward compensator for the levitation control system, and it can be regarded as an application of one of the pacification methods that have been discussed by Kelkar and Joshi (2004).

The optimization of TMD parameters has been extensively studied by many researchers; for example, Den Hartog (1956) and Harris and Crede (1976). However, the design procedures proposed in the above publications cannot be directly applied to the suppression of the stationary maglev vehicle–bridge coupled resonance, as they cannot guarantee that the combined system is always passive. In this paper, a new procedure to determine the parameters of the TMD is proposed. The most significant difference from previous work is that this design procedure is independent of the bridge parameters.

The remainder of this paper is organized as follows: First, the passivity of the magnetic levitation system is investigated in section 2. Based on this investigation, the principle of the stationary maglev vehicle–bridge coupled resonance is revealed, and the condition to ensure the stability of the coupled system is presented. In section 3, the characteristics of the TMD are analyzed, and in section 4, a procedure to optimize the parameters of the TMD is presented. As an example, the application of the TMD to a simply supported bridge is demonstrated in section 5, followed by conclusions in section 6.

2. Passivity analysis of the magnetic levitation system

Figure 2 shows a cross-section of the maglev vehicle and the elevated guideway. The electromagnetic force, which supports the weight of the cabin, is buffered by an air spring to increase the ride comfort of the vehicle. A fundamental unit of the levitation system (referred to as a levitation module) consists of two electromagnets in series and an independent levitation controller. Since adjacent levitation modules are mechanically decoupled, the influence of one levitation module on other levitation modules can be neglected. Therefore, in this study, only one suspension unit is considered, as shown schematically in Figure 3.

In Figure 3, m₁ represents the mass of the electromagnet and the part that connects the electromagnet and the air spring; k_s and c_s are, respectively, the equivalent stiffness and damping of the secondary suspension system (air spring); and m₂ denotes the equivalent payload supported by the electromagnet. The structure of the bridge is temporarily of no particular concern here, since the passivity analysis of the levitation system is independent of the bridge.

Consider the structure of the electromagnet shown in Figure 2. Suppose the number of turns of the coil is N, the area of the magnetic pole is A, the space permeability is μ₀, and the gap between the magnetic pole and the lower surface of the steel track is δ(t). Then the magnetic flux, ϕ(t), through the iron core of the electromagnet is given by

φ (t) = \frac{μ_{0} ANi (t)}{2 δ (t)}

(1)

where i(t) is the total current through the electromagnets. Since a levitation unit consists of two electromagnets in series, the voltage balance equation for each electromagnet can be given by:

\frac{u (t)}{2} = i (t) R + N \frac{d φ (t)}{dt}

(2)

where R is the resistance of a single electromagnet and u(t) is the voltage across the electromagnet. Substituting (1) into (2), yields

\frac{u (t)}{2} = i (t) R + \frac{μ_{0} {AN}^{2}}{2 δ (t)} \overset{\cdot}{i} (t) - \frac{μ_{0} {AN}^{2} i (t)}{2 δ^{2} (t)} \overset{\cdot}{δ} (t) .

(3)

It has been shown previously by Wang et al. (2008a) that the total magnetic force F_m generated by the two electromagnets can be described as

F_{m} = \frac{μ_{0} {AN}^{2}}{2} (\frac{i (t)}{δ (t)})^{2} .

(4)

In Figure 3, the motion of the electromagnet and the cabin can be expressed respectively as follows

\begin{matrix} m_{1} {\overset{··}{y}}_{1} (t) + c_{s} [{\overset{\cdot}{y}}_{1} (t) - {\overset{\cdot}{y}}_{2} (t)] \\ + k_{s} [y_{1} (t) - y_{2} (t)] = - F_{m} (t) + m_{1} g, \end{matrix}

(5)

m_{2} {\overset{··}{y}}_{2} (t) + c_{s} [{\overset{\cdot}{y}}_{2} (t) - {\overset{\cdot}{y}}_{1} (t)] + k_{s} [y_{2} (t) - y_{1} (t)] = m_{2} g .

(6)

It is well known that a magnetic levitation system is inherently unstable, and to keep the levitation gap at a desired value, external control of the magnetic force is required. Here, the state feedback control scheme is applied, which gives

u (t) = {\bar{k}}_{1} [δ (t) - z_{0}] + {\bar{k}}_{2} {\overset{\cdot}{y}}_{1} (t) + {\bar{k}}_{3} i (t) + u_{0},

(7)

where z₀ is the desired levitation gap, which is a constant; δ = y₁–y₀;

{\bar{k}}_{1}

{\bar{k}}_{2}

and

{\bar{k}}_{3}

are the state feedback coefficients, and u₀ is the bias voltage which satisfies

\frac{μ_{0} {AN}^{2}}{2} (\frac{u_{0}}{2 {Rz}_{0}})^{2} = (m_{1} + m_{2}) g,

(8)

where g is the acceleration due to gravity. To investigate the passivity of the levitation system, it is supposed that the bridge is vibrating with a very small amplitude around the equilibrium point. Using equations (3)−(7), the electromagnetic force, F_m, under harmonic excitation of the bridge can be obtained. Since the amplitude of the excitation is very small, it is reasonable to investigate the response of the linearized model around the equilibrium point instead of the nonlinear model. The linearized model of the closed-loop system can be described as

{\begin{matrix} Δ u (t) = {\bar{k}}_{1} [Δ y_{1} (t) - Δ y_{0} (t)] + {\bar{k}}_{2} {\overset{\cdot}{y}}_{1} (t) + {\bar{k}}_{3} Δ i (t) \\ L_{0} \overset{\cdot}{i} (t) = Δ u (t) / 2 - Δ i (t) R + F_{i} [{\overset{\cdot}{y}}_{1} (t) - {\overset{\cdot}{y}}_{0} (t)] \\ Δ F_{m} (t) = 2 [F_{i} Δ i (t) - F_{z} [Δ y_{1} (t) - Δ y_{0} (t)]] \\ m_{1} {\overset{\cdot\cdot}{y}}_{1} (t) + c_{s} [{\overset{\cdot}{y}}_{1} (t) - {\overset{\cdot}{y}}_{2} (t)] + k_{s} [Δ y_{1} (t) - Δ y_{2} (t)] = - Δ F_{m} (t) \\ m_{2} {\overset{\cdot\cdot}{y}}_{2} (t) + c_{s} [{\overset{\cdot}{y}}_{2} (t) - {\overset{\cdot}{y}}_{1} (t)] + k_{s} [Δ y_{2} (t) - Δ y_{1} (t)] = 0 \end{matrix}

(9)

where

L_{0} = \frac{μ_{0} {AN}^{2}}{2 z_{0}}, F_{i} = \frac{μ_{0} {AN}^{2} i_{0}}{2 z_{0}^{2}}, F_{z} = \frac{μ_{0} {AN}^{2} i_{0}^{2}}{2 z_{0}^{3}},

and i₀ is the steady state current through the electromagnet. Here, the prefix “Δ” for the variables in (9) is to indicate that the variables following it are the perturbations around the static equilibrium. For simplicity, the prefix “Δ” is omitted in the rest of this paper.

Defining $v_{0} = {\overset{\cdot}{y}}_{0}$ allows the expression for the levitation force to be obtained from equations (9) in the Laplace form:

\frac{F_{m} (s)}{v_{0} (s)} = - \frac{s [m_{1} m_{2} s^{2} + c_{s} (m_{1} + m_{2}) s + k_{s} (m_{1} + m_{2})] C_{1}}{DEN (s)},

(10)

where

C_{1} = F_{i} {\bar{k}}_{1} + F_{z} {\bar{k}}_{3} - 2 {RF}_{z}, \bar{R} = R - \frac{{\bar{k}}_{3}}{2},

\begin{matrix} DEN (s) = m_{1} m_{2} L_{0} s^{5} + [m_{1} m_{2} \bar{R} + c_{s} (m_{1} + m_{2}) L_{0}] s^{4} \\ + [c_{s} (m_{1} + m_{2}) \bar{R} + k_{s} (m_{1} + m_{2}) L_{0} + m_{2} F_{i} {\bar{k}}_{2}] s^{3} \\ + [k_{s} (m_{1} + m_{2}) \bar{R} + m_{2} C_{1} + c_{s} F_{i} {\bar{k}}_{2}] s^{2} \\ + (c_{s} C_{1} + k_{s} F_{i} {\bar{k}}_{2}) s + k_{s} C_{1} . \end{matrix}

The excitation force that the bridge acts on the levitation module, F₀, is equal in strength to the electromagnetic force, F_m, but is opposite in direction. Therefore, the relationship between the excitation force and the excitation velocity can be obtained as:

\begin{matrix} G_{fv} (s) = \frac{F_{0} (s)}{v_{0} (s)} \\ = \frac{s [m_{1} m_{2} s^{2} + c_{s} (m_{1} + m_{2}) s + k_{s} (m_{1} + m_{2})] C_{1}}{DEN (s)} \end{matrix} .

(11)

Suppose that the parameters of the levitation system are chosen as in Table 1. Simulations show that choosing ${\bar{k}}_{1}$ = 200000, ${\bar{k}}_{2}$ = 2500 and ${\bar{k}}_{3}$ = −50 will ensure that the closed-loop response is satisfactory. Under the chosen parameters, the response of F₀ with respect to the frequency of the excitation is plotted in Figure 4. Kelkar and Joshi (2004) have shown that for the finite-dimensional, linear, time-invariant (LTI) system, G(s), the passivity is equivalent to the transfer function matrix of the system being positive and real. For a single-input, single-output (SISO) system, being positive and real is equivalent to Re[G(jω)] ≥ 0, or equivalently, the phase of G(jω) remains within ±90°. However, from Figure 4 it can be seen that the levitation system is non-passive since the phase response of the system exceeds ±90°.

Table 1.

Parameters of the levitation system

Variable	Value	Unit
m ₁	600	kg
m ₂	860	kg
k _s	60000	N/m
c _s	6000	Ns/m
z ₀	0.008	m
μ ₀	4π × 10⁻⁷	H/m
A	0.02352	m²
N	320	—
R	0.56	Ω

Suppose the excitation velocity of the bridge is given by

v_{0} (t) = V \cos (ω t + φ),

(12)

then the excitation force can be obtained following equation (11):

F_{0} (t) = F_{A} \cos (ω t + φ + θ),

(13)

where

F_{A} = | G_{fv} (j ω) | \cdot V

θ = \land G_{fv} (j ω)

. The average power generated by the excitation force, F₀, is

\begin{matrix} P_{0} (ω) = \frac{1}{T} \int_{0}^{T} F_{A} \cos (ω τ + φ + θ) V \cos (ω τ + φ) d τ \\ = \frac{F_{A} V}{2} \cos θ = \frac{1}{2} V^{2} | G_{fv} (j ω) | \cos θ, \end{matrix}

(14)

from which it can be seen that when

θ \in (- \frac{π}{2}, \frac{π}{2})

, the power of the excitation force, F₀, is positive, and the energy flows from the bridge to the levitation system. Otherwise, the power of the excitation force will be negative, and the energy flows from the levitation system to the bridge. Referring to Figure 4, it can be concluded that when the excitation frequency ω > 2πf_c (where f_c is the critical frequency of the levitation system), the energy flows from the levitation system to the bridge. In this case, if the energy flow into the bridge is greater than the energy dissipated by the bridge damping mechanisms, the amplitude of the vibration will grow with time.

Equation (14) can be rewritten as

P_{0} (ω) = \frac{1}{2} Re [G_{fv} (j ω)] V^{2} .

(15)

Substituting equation (11) into equation (15), yields

P_{0} (ω) = \frac{{NUM}_{2} (ω)}{{DEN}_{2} (ω)} \frac{V^{2}}{2},

(16)

where

\begin{matrix} {NUM}_{2} (ω) = α_{2} β_{5} ω^{6} + (α_{0} β_{5} + α_{2} β_{3} - α_{1} β_{4}) ω^{4} \\ + (α_{1} β_{2} - α_{0} β_{3} - α_{2} β_{1}) ω^{2} + (α_{0} β_{1} - α_{1} β_{0}), \end{matrix}

(17)

\begin{matrix} {DEN}_{2} (ω) = β_{5}^{2} ω^{10} + (β_{4}^{2} - 2 β_{3} β_{5}) ω^{8} \\ + (2 β_{1} β_{5} - 2 β_{2} β_{4} + β_{3}^{2}) ω^{6} \\ + (β_{2}^{2} + 2 β_{0} β_{4} - 2 β_{1} β_{3}) ω^{4} \\ + (β_{1}^{2} - 2 β_{0} β_{2}) ω^{2} + β_{0}^{2}, \end{matrix}

(18)

and

\begin{matrix} α_{0} = k_{s} (m_{1} + m_{2}) C_{1}, α_{1} = c_{s} (m_{1} + m_{2}) C_{1}, \\ α_{2} = m_{1} m_{2} C_{1}, \\ β_{0} = k_{s} C_{1}, β_{1} = c_{s} C_{1} + k_{s} F_{i} {\bar{k}}_{2}, \\ β_{2} = k_{s} (m_{1} + m_{2}) \bar{R} + m_{2} C_{1} + c_{s} F_{i} {\bar{k}}_{2}, \\ β_{3} = c_{s} (m_{1} + m_{2}) \bar{R} + k_{s} (m_{1} + m_{2}) L_{0} + m_{2} F_{i} {\bar{k}}_{2}, \\ β_{4} = m_{1} m_{2} \bar{R} + c_{s} (m_{1} + m_{2}) L_{0}, \\ β_{5} = m_{1} m_{2} L_{0} . \end{matrix}

In the remainder of this paper, V is set equal to 1 m/s, unless otherwise specified. According to equation (16), the P₀ vs. ω curve can be plotted, as shown in Figure 5. It can be seen that the excitation power, P₀, becomes negative when the excitation frequency is higher than f_c, which suggests that energy transfers to the bridge and the amplitude of the vibration may increase with time. This leads to an interesting conclusion: since higher order modes of the bridge whose resonance frequencies are higher than f_c can always be found, coupled resonance will always occur. This conclusion is valid if the damping of the bridge is ignored but is unlikely to be valid if the damping of the bridge is taken into account. Suppose that the equivalent damping of the k-th vibration mode is c_k, then the damping force is given by F_c(t) = c_kv₀(t). Referring to equation (12), the average power dissipated by the damping can be written as

\begin{matrix} P_{c} = \frac{1}{T} \int_{0}^{T} F_{c} (τ) v_{0} (τ) d τ = \frac{1}{T} \int_{0}^{T} c_{k} V^{2} \cos^{2} (ω τ + φ) d τ \\ = \frac{c_{k} V^{2}}{2} . \end{matrix}

(19)

Taking the power dissipated by the damping of the bridge into account, the total excitation power is

P_{A} = P_{0} + P_{c} .

(20)

Figure 6 shows the synthesized excitation power curve when the damping of the bridge is taken into account, from which it can be seen that in the presence of the damping, only a limited band of the curve is below zero. Therefore, for the bridge whose resonance frequencies are out of this band, it is unlikely for the coupled system to stimulate the coupled resonance. In addition, it can be concluded that, if there exists at least one mode of the bridge which satisfies the following conditions, the vehicle–bridge coupled resonance may occur:

(1) ω_{n} > 2 π f_{c};

(21a)

(2) Re [G_{fv} (j ω_{n})] + c_{0} < 0 .

(21b)

Here, ω_n denotes the n-th resonance frequency of the bridge.

3. Passification of the levitation system using a TMD

Figure 6 indicates that it is beneficial to the passivity of the coupled system if an extra damper can be added to the bridge to absorb the vibratory energy. This can be regarded as an approach of feedforward passification using a skyhook damper, which has been discussed by Huyanan and Sims (2007). The feedforward passification scheme is shown schematically in Figure 7, where G_c(s) is the transfer function of the skyhook damper.

Unfortunately, it is not easy to install a skyhook damper to an elevated bridge because a support to fix the other end of the damper is not always available. However, a TMD can always be applied in such a case. Figure 8 shows the schematic of a TMD applied to a maglev bridge. The dynamic equation of the TMD shown in Figure 8 can be written as

m_{a} {\overset{··}{y}}_{a} + c_{a} ({\overset{\cdot}{y}}_{a} - {\overset{\cdot}{y}}_{0}) + k_{a} (y_{a 0} + y_{a} - y_{0}) = m_{a} g,

(22)

where y₀ is the displacement of the bridge at the position where the TMD is installed, y_a₀ is the static displacement of the spring caused by the gravity of m_a, and

y_{a 0} = m_{a} g / k_{a}

. Let F_a denote the excitation force that the bridge exerts on the TMD, that is,

F_{a} = m_{a} {\overset{··}{y}}_{a}

. Considering that

v_{0} = {\overset{\cdot}{y}}_{0}

, the relationship between F_a and v₀ can be deduced from equation (22) in Laplace form as

G_{a} (s) = \frac{F_{a} (s)}{v_{0} (s)} = \frac{m_{a} s (c_{a} s + k_{a})}{m_{a} s^{2} + c_{a} s + k_{a}} .

(23)

Letting s = jω, it can be shown that at frequency ω, the excitation power transmitted to the TMD can be written as

P_{a} (ω) = \frac{1}{2} Re [G_{a} (j ω)] V^{2} = \frac{1}{2} \frac{m_{a}^{2} c_{a} ω^{4}}{(k_{a} - m_{a} ω^{2})^{2} + c_{a}^{2} ω^{2}} V^{2},

(24)

where V is the amplitude of v₀. Equation (24) suggests that P_a(ω) is always positive provided that c_a > 0, which also indicates that the TMD is passive. Taking the TMD into account, the total excitation power provided by the excitation source becomes

P_{F 0} (ω) = P_{0} (ω) + P_{a} (ω) = \frac{V^{2}}{2} {Re [G_{fv} (j ω)] + Re [G_{a} (j ω)]} .

(25)

Since P_a(ω) is always positive, it is possible to compensate the total excitation power P_F₀ to be positive by proper design of the TMD. According to equation (23), and setting s = jω, the frequency at which the imaginary part is zero can be obtained as:

ω = 0, or ω = \sqrt{\frac{k_{a}^{2}}{k_{a} m_{a} - c_{a}^{2}}} .

Therefore, the resonance frequency of the TMD is

ω_{a} = \sqrt{\frac{k_{a}^{2}}{k_{a} m_{a} - c_{a}^{2}}} .

(26)

According to equation (24), when the excitation frequency equals ω_a, the power absorbed by the TMD is

P_{a 0} = \frac{k_{a} m_{a}}{2 c_{a}} V^{2} .

(27)

On the other hand, from equation (24), the maximum value of P_a is

P_{am} = \frac{2 k_{a}^{2} m_{a}^{2}}{c_{a} (4 k_{a} m_{a} - c_{a}^{2})} V^{2},

(28)

and the corresponding frequency is

ω_{am} = \frac{\sqrt{2} k_{a}}{\sqrt{2 k_{a} m_{a} - c_{a}^{2}}} .

(29)

The expression for P_am is more complex than that for P_a. However, if the damping ratio of the TMD is small, namely, $c_{a} << \sqrt{k_{a} m_{a}}$ , then ω_a ≈ ω_am, and P_a₀ ≈ P_am. In most cases, the designed TMD is under-damped, so P_a₀ can be used as a good approximation of the maximum power absorption of the TMD.

The bandwidth is another important parameter of the TMD. Here, the bandwidth of a TMD is defined as the frequency range in which the power absorption of the TMD is greater than 0.5 P_am. When the damping ratio of the TMD is small, the bandwidth can be simply approximated as:

{BW}_{a} \approx \frac{c_{a}}{m_{a}} .

(30)

It shows that the bandwidth of the TMD is proportional to c_a; whereas equation (27) indicates that the maximum power absorption of the TMD is inversely proportional to c_a. Therefore, a compromise which concerns both the maximum power absorption and the bandwidth is required during the design of the TMD.

4. Optimization of the TMD parameters

A series of methodologies for the purpose of optimizing TMD parameters has been discussed by Den Hartog (1956), Harris and Crede (1976), and Huyanan and Sims (2007). However, these methodologies are not proposed for direct use here, as their common purpose is to design a TMD that can decrease the vibration level of a flexible structure, without guaranteeing that the designed result is capable of eliminating the coupled resonance. As for the maglev system, the TMD applied here is for the purpose of rendering the levitation system passive rather than reducing the vibration of the bridge.

Generally, the resonance frequency of the secondary suspension system of a maglev vehicle is less than 2 Hz, while the frequency of the vehicle–bridge coupled resonance is higher than 10 Hz; therefore, the impact of the secondary suspension system can usually be neglected in the analysis of the coupled resonance problem. Disregarding the secondary suspension system, equation (11) can be rewritten as

G_{fv}^{'} (s) = \frac{m_{1} C_{1} s}{m_{1} L_{0} s^{3} + m_{1} \bar{R} s^{2} + F_{i} {\bar{k}}_{2} s + C_{1}} .

(31)

Figure 9 shows a comparison of G′ _fv (s) and G_fv(s), from which it can be seen that they have only a little difference at frequencies higher than 10 Hz; thus the critical frequency and maximum negative power absorption of the levitation system can be calculated using equation (31) instead of equation (11). From equation (31), it can be shown that without the secondary suspension system, the power transmitting from the excitation source (bridge) to the levitation system can be written as:

P_{0}^{'} (ω) = \frac{m_{1} ω^{2} C_{1} (F_{i} {\bar{k}}_{2} - m_{1} L_{0} ω^{2})}{(C_{1} - m_{1} \bar{R} ω^{2})^{2} + ω^{2} (F_{i} {\bar{k}}_{2} - m_{1} L_{0} ω^{2})^{2}} \frac{V^{2}}{2},

(32)

from which it can be found that the frequency corresponding to P₀′(ω) = 0 is

ω = 0

, or

ω = \sqrt{F_{i} {\bar{k}}_{2} / m_{1} L_{0}}

, thus the critical frequency of the levitation system without the secondary suspension system is:

f_{c}^{'} = \frac{1}{2 π} \sqrt{\frac{F_{i} {\bar{k}}_{2}}{m_{1} L_{0}}} .

(33)

Using equation (32), it is possible to obtain the maximum negative power absorption of the levitation system, P_sm, and the corresponding frequency, f_sm (as shown in Figure 5). Unfortunately, it is not easy to get a closed-form solution. A large number of simulations show that f_sm can be fitted by a polynomial:

\begin{matrix} f_{sm} = 6.54 - 1.79 \times 10^{- 5} {\bar{k}}_{1} + 5.37 \times 10^{- 3} {\bar{k}}_{2} - 2.69 \\ \times 10^{- 7} {\bar{k}}_{2}^{2} - 0.148 \times {\bar{k}}_{3} + 3.87 \times 10^{- 4} {\bar{k}}_{3}^{2} . \end{matrix}

(34)

To make the system passive, the selection of the TMD parameters should ensure that the total excitation power, P_F₀, is identically greater than zero. Referring to equations (24), (25), and (32), this condition can be rewritten as:

\begin{matrix} \frac{m_{a}^{2} c_{a} ω^{4}}{(k_{a} - m_{a} ω^{2})^{2} + c_{a}^{2} ω^{2}} \\ + \frac{m_{1} ω^{2} C_{1} (F_{i} {\bar{k}}_{2} - m_{1} L_{0} ω^{2})}{(C_{1} - m_{1} \bar{R} ω^{2})^{2} + ω^{2} (F_{i} {\bar{k}}_{2} - m_{1} L_{0} ω^{2})^{2}} > 0 . \end{matrix}

(35)

This is the only requirement for the TMD parameters. Since there are three parameters for a single TMD, the design result is not unique. Taking the model uncertainties of the levitation system into account, it is preferable to find a suboptimal solution for the TMD parameters which provides a reasonable margin of the passivity. This procedure can be outlined as follows:

Use equation (33) to calculate the critical frequency of the levitation system, f_c. Then obtain f_sm and P_sm by using equations (34) and (32), respectively.

Set the resonance frequency of the TMD to be f_sm; namely, ω_a = 2πf_sm; then set the bandwidth of the TMD to be BW_a = 2π (f_sm − f_c). The expression of ω_a and BW_a can be obtained by using equations (26) and (30), respectively.

Set P_am = 4P_sm.

The procedure listed above can be generalized as follow:

{\begin{matrix} \frac{c_{a}}{m_{a}} \geq {BW}_{a}, (36 a) \\ \frac{k_{a} m_{a}}{2 c_{a}} \geq P_{am}, (36 b) \\ \frac{k_{a}}{m_{a}} = ω_{a}^{2}, (36 c) \end{matrix}

which yields

{\begin{matrix} m_{a} \geq \frac{2 {BW}_{a} P_{am}}{ω_{a}^{2}} (37 a) \\ k_{a} = ω_{a}^{2} m_{a} (37 b) \\ c_{a} \geq \frac{2 {BW}_{a}^{2} P_{am}}{ω_{a}^{2}} (37 c) \end{matrix}

It should be noted that the design process does not make use of any information about the bridge. This is because the TMD designed here is aimed to make the levitation system passive, and the design result is applicable to any kind of structure provided that the TMD and the electromagnet are at the same position on the structure. Where the TMD and the electromagnet are at different positions on a bridge, some special issues must be taken into account. This will be demonstrated in an example in the next section.

5. Numerical simulation

Suppose the parameters of the levitation systems are: m₁ = 800 kg, m₂ = 700 kg; and the control parameters are chosen as: ${\bar{k}}_{1} = 200000$ , ${\bar{k}}_{2} = 1500$ and ${\bar{k}}_{3} = - 50$ . Other parameters are the same as in Table 1. According to step (1) of the design process, the critical frequency of the levitation system can be calculated using equation (33), which gives f_c = 12.17 Hz. Using equations (34) and (32), the following values can be obtained: f_sm = 18.8 Hz, P_sm = 1930 W. Therefore, the bandwidth of the TMD is BW_a = 2π (f_sm − f_c) = 41.5 rad/s, and the maximum absorption power of the TMD should be P_am = 4P_sm = 7720 W. According to equation (37), the parameters of the TMD can be determined: m_a = 43.3 kg, k_a = 6.02 × 10⁵N/m, c_a = 1690 Ns/m.

Figure 10 shows a comparison of the excitation force in the frequency domain with and without the designed TMD. It can be seen that the system with the designed TMD is strictly passified since the phase curve is limited to being within ±90°.

Figure 11 shows a comparison of the total excitation power, P_F₀, with and without the TMD. It can be seen that the excitation power of the compensated system is positive at all frequencies. This also indicates that the compensated system should be stable.

To validate the effectiveness of the design, a simply supported single-span concrete bridge, which has been widely used in the low speed maglev routes, is considered, as shown in Figure 8. Suppose the span length of the bridge is L, the bending stiffness of the bridge is EI, and the displacement of the bridge is y(x,t). The differential equation for the vibration of the bridge has the following form:

EI \frac{\partial^{4} y}{\partial x^{4}} + ρ \frac{\partial^{2} y}{\partial t^{2}} = F (x, t),

(38)

where ρ is the mass per unit length of the bridge, and F(x,t) is the external force acting on the bridge. Suppose that the TMD is installed in the middle of the bridge, and that the electromagnet is also at the same location. Compared with the length of the bridge, the length of the electromagnet can be neglected; thus

F (x, t) = (F_{s} + F_{a}) δ (x - \frac{L}{2}),

(39)

where δ(·) is the Dirac function, F_s is the levitation force, which equals F_m in equations (9), and F_a is the component that the TMD acting on the bridge, which can be deduced from equation (22). Using modal analysis, the displacement of the bridge can be synthesized in the form

y (x, t) = \sum_{n = 1}^{\infty} q_{n} (t) φ_{n} (x) .

(40)

Here, ϕ _n (x) and q_n(t) are the modal shape and the generalized displacement of the n-th order resonance mode of the bridge, respectively. For a simply supported beam, ϕ _n (x) = sin(nπx/L). Substituting equations (39) and (40) into equation (38), multiplying both sides of the resultant equation by ϕ _n (x), and integrating both sides from 0 to L, yields

{\overset{··}{q}}_{n} (t) + ω_{n}^{2} q_{n} (t) = \frac{2 (F_{s} + F_{a})}{ρ L} \sin \frac{n π}{2},

(41)

where

ω_{n} = (\frac{n π}{L})^{2} \sqrt{\frac{EI}{ρ}} .

(42)

For the fundamental mode of the beam, the vertical displacement in the center of the bridge can be given by:

{\overset{··}{q}}_{1} (t) + ω_{1}^{2} q_{1} (t) = \frac{2 (F_{s} + F_{a})}{ρ L} .

(43)

This is equivalent to a mass-spring system having a natural frequency of ω₁, and mass of ρL/2. In the simulation, it is assumed that the first resonance frequency of the bridge is 16.7 Hz, and ρL = 1.8 × 10⁴kg. Figure 12 shows the simulation result, from which it can be seen that without the TMD, the coupled resonance occurs; while after the TMD is attached to the bridge, the coupled resonance vanishes.

As mentioned earlier, the simulation is based on the hypothesis that the TMD and the electromagnet are at the same location on the bridge; however, the vehicle can be at any location on a bridge, and the TMD is not always at the same position as the electromagnet. Fortunately, as will be shown below, it is not a necessary condition to satisfy this hypothesis. Suppose that the electromagnet is at a on the bridge, while the TMD is at b, (a, b∈[0, L]), then

F (x, t) = F_{s} δ (x - a) + F_{a} δ (x - b) .

(44)

Using equation (40), the vertical velocity of the bridge can be obtained. Moreover, referring to equations (32) and (24), the following equations can be derived:

P_{s} (ω) = \frac{1}{2} Re [G_{fv} (j ω)] V_{a}^{2} = \frac{1}{2} Re [G_{fv} (j ω)] (\sum_{n = 1}^{\infty} {\overset{\cdot}{q}}_{n} φ_{n} (a))^{2},

(45)

P_{a} (ω) = \frac{1}{2} Re [G_{a} (j ω)] V_{b}^{2} = \frac{1}{2} Re [G_{a} (j ω)] (\sum_{n = 1}^{\infty} {\overset{\cdot}{q}}_{n} φ_{n} (b))^{2} .

(46)

Therefore, the total power transmitted from the bridge to the levitation and TMD system is

\begin{matrix} P_{F 0} (ω) = P_{s} (ω) + P_{a} (ω) \\ = \frac{1}{2} \sum_{n = 1}^{\infty} {\overset{\cdot}{q}}_{n}^{2} {Re [G_{fv} (j ω)] φ_{n}^{2} (a) + Re [G_{a} (j ω)] φ_{n}^{2} (b)} . \end{matrix}

(47)

For the fundamental vibration mode of the bridge, n = 1, it gives

P_{F 0} (ω) = \frac{1}{2} {\overset{\cdot}{q}}_{1}^{2} {Re [G_{fv} (j ω)] φ_{1}^{2} (a) + Re [G_{a} (j ω)] φ_{1}^{2} (b)} .

(48)

Comparing equation (48) with equation (25), it can be found that provided $φ_{1}^{2} (b) \geq φ_{1}^{2} (a)$ , P_F₀(ω) will always be positive if equation (25) is positive for ω > 0. For the simply supported beam, mounting the TMD at the middle of the span will make $φ_{1}^{2} (b)$ to reach its maximum value. In this case, for any a∈[0, L], $φ_{1}^{2} (b) \geq φ_{1}^{2} (a)$ holds, and P_F₀(ω) > 0. In other words, wherever the electromagnet is, the system is always passive, and the fundamental vibration mode of the bridge is no longer able to cause the coupled resonance. This conclusion also holds when n = 3, 5, 7 … However, when n is even, say, n = 2, the conclusion will not hold, since the second order mode shape of the simply supported beam, which is sin(2xπ/L), equals zero when x = L/2. In this case, it is desirable to put the TMD at L/4 or 3L/4 of the bridge to make sure that $φ_{n}^{2} (b)$ is always larger than $φ_{n}^{2} (a)$ . This suggests that for all the vibration modes of the bridge, and for the electromagnet to be at any location on the bridge, two TMDs may be required for the purpose of passifying the system − one should be installed at the middle of the span, and the other should be installed at L/4 of the span. Alternatively, a compromise should be made if only one TMD is available. For example, putting the TMD at 3L/8 of the span is acceptable when the TMD parameters are chosen such that equation (47) is identically greater than zero. In practice, the coupled resonance is mostly caused by the fundamental mode of the bridge; therefore, a single TMD can solve this problem by mounting it to the middle of the span.

It should be mentioned that the preceding conclusion is not restricted to a simply supported beam. Provided that the TMD is mounted at a location where the absolute value of the modal shape reaches its maximum value such that $φ_{1}^{2} (b) \geq φ_{1}^{2} (a)$ , the system can also be made passive.

Compared with the TMD designed using the general design approach (see Den Hartog (1956)), the mass of the TMD designed in the previous example is quite light. This is because the TMD used here is to passify just one levitation module. However, when multiple levitation modules are taken into account, the designed TMD would be heavier. Suppose that there are a total of N levitation modules on the bridge, and their locations are represented by x_k, (k = 1, 2 , … , N). For the fundamental mode of the bridge, the total power transmitted from the bridge to the levitation modules are:

P_{s} (ω) = \frac{1}{2} \sum_{k = 1}^{N} Re [G_{k} (j ω)] V_{k}^{2} = \frac{1}{2} {\overset{\cdot}{q}}_{1}^{2} \sum_{k = 1}^{N} {Re [G_{k} (j ω)] φ_{1}^{2} (x_{k})},

(49)

where G_k(jω) is the transfer function of the k-th levitation module, and they have the same form as equation (11). Therefore, the total power dissipated by the TMD and the levitation modules is:

\begin{matrix} P_{F 0} (ω) = P_{s} (ω) + P_{a} (ω) \\ = \frac{1}{2} {\overset{\cdot}{q}}_{1}^{2} {\sum_{k = 1}^{N} {Re [G_{k} (j ω)] φ_{1}^{2} (x_{k})} + Re [G_{a} (j ω)] φ_{1}^{2} (b)}, \end{matrix}

(50)

which shows that, to ensure P_F₀ > 0, the power dissipation of the TMD should be larger than the power provided by the weighted sum of the levitation modules. As a rough approximation, the TMD parameters can be chosen to be N times the values designed for the single levitation module. For example, if the bridge takes at most 10 levitation modules (the number of levitation modules a bridge can take depends on the length of the span and the length of a levitation module), according to the design result obtained in the previous example, the TMD parameters should be amplified by 10 times, which gives m_a = 433 kg, k_a = 6.02 × 10⁶N/m, c_a = 1.69 × 10⁴ Ns/m. This result is conservative since most of the weights for the levitation modules,

φ_{1}^{2} (x_{k})

, are less than

φ_{1}^{2} (b)

; however, it provides a good margin of the passivity.

6. Conclusions

In this paper, the principle of the stationary maglev vehicle–bridge coupled resonance problem has been investigated and the conditions for the resonance to occur have also been presented. It has been shown that the non-passive nature of the levitation system is the reason that leads to the maglev vehicle–bridge coupled resonance.

To suppress the coupled resonance, a scheme of installing a TMD onto the bridge as a feedforward compensator was introduced. It was shown that that a TMD with appropriate parameters can effectively render the system to be strictly passive. A procedure to search for a suboptimal solution was also provided. The design process is independent of the structure of the bridge.

As an example, an application of the design process was demonstrated, and it showed the designed TMD was capable of suppressing the stationary coupled resonance caused by the fundamental mode of the bridge. More general cases were also discussed, including when the electromagnet is at a different location to the TMD, and the effect of higher order modes of the bridge. The more general case of multiple levitation modules on the bridge has also been discussed. The design procedure for the TMD parameters presented in this paper provides a general methodology for TMD design in an EMS maglev system.

Footnotes

Acknowledgements

The authors would like to acknowledge the support they received from the School of Mechanical Engineering, the University of Adelaide, and from their funding bodies.

Funding

This work was supported by National Key Technology R&D Program (grant number 2006BAG02B05-04) and the National Nature and Science Foundation of China (grant number 60404003).

References

Albert TE, Oleszczuk G and Hanasoge AM (2008) Stable levitation control of magnetically suspended vehicles with structural flexibility. In Proceedings of the 2008 American Control Conference, Washington DC, June 11–13, pp. 4035–4040.

Den Hartog

(1956) Mechanical Vibrations, 4th edn. New York: McGraw-Hill.

Harris

Crede

(1976) Shock and Vibration Handbook, 2nd edn. New York: McGraw-Hill.

Huyanan

Sims

(2007) Vibration control strategies for proof-mass actuators. Journal of Vibration and Control 13(12): 1785–1806.

Kelkar

Joshi

(2004) Control of elastic systems via passivity-based methods. Journal of Vibration and Control 10: 1699–1735.

Lin

Wang

Chen

(2005) Train-induced vibration control of high-speed railway bridges equipped with multiple tuned mass dampers. Journal of Bridge Engineering July/August: 398–414.

Namerikawa T and Kawano H (2006) A passivity-based approach to wide area stabilization of magnetic suspension systems. In Proceedings of the 2006 American Control Conference, Minneapolis, MN, June 14–16, pp. 4576–4581.

She LH, Wang H, Zou DS, et al. (2008) Hopf bifurcation of maglev system with coupled elastic guideway. In Proceedings of the Twentieth International Conference on Magnetically Levitated Systems and Linear Drives, San Diego, CA, December 15–18.

Shi

She

Chang

(2004) The bifurcation analysis of the EMS maglev vehicle-coupled-guideway system. Acta Mechanica Sinica 36(05): 634–640.

10.

Shi

Cai

(2008) Suppression of vehicle-induced bridge vibration using tuned mass damper. Journal of Vibration and Control 14(7): 1037–1054.

11.

Shimizu T, Sasaki M, and Wajima K (2004) Passivity based control of a magnetic levitation system with two electromagnets for a flexible beam. In Proceedings of AMC 2004, Kawasaki, Japan, pp. 129–134.

12.

Wang

Zhang

(2008a) Non-resonant response, bifurcation and oscillation suppression of a non-autonomous system with delayed position feedback control. Nonlinear Dynamics 51: 447–464.

13.

Wang

Zhang

(2008b) Sub-resonant response of a nonautonomous maglev system with delayed acceleration feedback control. IEEE Transactions on Magnetics 44(10): 2338–2350.

14.

Yan

(2008) Development and application of the maglev transportation system. IEEE Transactions on Applied Superconductivity 18(2): 92–99.