Lubrication with oils and greases can be effective for reducing friction and wear. Bearings can be used for higher forces, speeds, and precision. Some of the standard types of bearings are described in the following list:
•
Plain bearings
•
Generally used in low speed machines.
•
The main bearing action comes from the lubricant.
•
Solid bearings
•
This looks like a section of tube that is placed in a hole, and the shaft rotates inside.
•
Typical materials are:
-
Bronze
-
Sintered bronze (with graphite)
-
Cast iron
-
Powdered metal impregnated with oil
•
Made for slowly rotating equipment.
•
Lubrication is required, and problems will arise when not properly maintained.
•
Available in standard sizes.
•
Split bearings
•
Used on large machines at low speeds.
•
The two halves of the bearings are adjusted in position using shims.
•
Typical materials include:
-
Bronze
-
Bronze with babbitt
-
Babbitt-lined metal
•
Oil grooves are used for lubrication.
•
Thrust bearings
•
Opposes axial thrusts of rotating shafts.
•
Uses shoes of a variety of shapes:
-
Flat
-
Kidney shaped
•
An oil wedge approach is used to support the bearing.
•
Rolling bearings
•
Advantages:
-
Low friction at all times
-
Compact
-
High accuracy
-
Low wear
-
Come in standard sizes
A shaft in a hole is a simple bearing, as shown in Figure 12.50. In this example the shaft is a bolt with a shoulder (no threads) in the holes. Washers are used to hold the surfaces together. Given the rotation, a lock washer or lock nut would be used. In the lower example a brass bushing is used to reduce friction and wear. These are a low cost design option for limited-life or limited-use products.
Figure 12.50. Simple bearings.
Balls or rollers are captured between a moving and a stationary ring or plate. The motion is purely based on rolling with negligible friction. In addition, the motion occurs over multiple balls or rollers, balancing out loads and variations in geometries. The ball bearings in Figure 12.51 are captured with two different raceways. Radial loads require horizontal channels and axial loads require a vertical rolling surface. Likewise, the roller bearings in Figure 12.52 use horizontal and vertical walls. The larger contact surface of the rollers makes it possible to use them for much larger loads. Both of these bearing types are commodity components available for various shaft and hole sizes, as well a wide range of axial and radial loads.
Figure 12.51. Ball bearings.
Figure 12.52. Roller bearings.
Thrust ball and roller bearings are used mainly for axial loads or compression (Figure 12.53). They can withstand small radial loads. These commodity bearings perform well at low speeds for balanced loads.
Figure 12.53. Thrust bearings.
Problems
12.55
What types of bearings are needed for an automobile axle?
12.56
How are bushings and bearings different?
12.57
Locate two bearings that will support a shaft turning at 2000 RPM with a load of 20 kN.
1.4 Fatigue affected by forces generated at the wheel–rail interface: the importance of dynamic loads
We now turn from the wheel and rail, components obviously and directly affected by the stresses generated at the wheel–rail contact, to components away from the vicinity of the contact but nevertheless affected by the conditions at the contact. It is worth pausing to mention the nature of the forces at the contact. At its simplest level, the contact patch between each wheel and the rail must support that proportion of the vertical static load, the weight, which passes through it. Because of symmetry, this is known as the axle load (the wheel load equals half the axle load). In addition, along the direction of the rail, forces due to the acceleration, braking and traction at steady speed must be sustained. When a train passes through a curve, the lateral loads needed to generate curved motion must be considered, together with the load redistribution from inner (lower) to outer (higher) rail. All these loads are relatively easy to quantify, but the situation is made much more complicated by the generation of dynamic loads.
In a useful review (Hill and Everitt, 1988), some of the historical gropings towards an understanding of this important effect were outlined. Their observations were so pertinent, they are worth quoting at some length:
Over the years a number of individuals have had the insight to perceive the importance of an appreciation of the service requirements. For example (Beaumont, 1879) observed in 1879 that:
When a train was running, the wheels were lifted up and down again on the very many irregularities of the line at a velocity which induced severe shocks. The velocity at which impact shocks were transmitted through the wheels to the axle was not simply that of gravity and that of the velocity of the train, but very many of the shocks were thus transmitted at the velocity of recoil of a loaded spring, which was probably as much as 1300 feet per second [400 m s −1].
Notwithstanding this observation, the railway axle soon entered folklore as something to be designed with the nominal stress under a fatigue limit (e.g. Anon., 1920).
Between the two world wars fatigue was studied almost exclusively as an endurance limit problem. The attitude is still prevalent today despite publications by people who have actually measured operating loads and strains. For example, a paper from the mid-1950s (Moreau and Peterson, 1955) … with remarkable insight on the field testing of diesel locomotive axles. They commented:
It is now possible to predict, with reasonable accuracy, what stresses will be induced in a specific axle design by a certain load and the relationship between the stress and the number of stress applications which will cause a failure is also fairly well established. … There is, however, very little information available about the loads an axle is actually exposed to in service. To determine whether a part of a structure of a machine is strong enough, the engineer must know the type of loading to which it will be exposed. If he does not, he has no other choice than to make a guess and see if it fails. It is too expensive to learn about weaknesses in axles from failures and it is also too expensive to make them so heavy that they are bound to be strong enough in spite of the designer’s ignorance about the loads. … To study axles under service conditions it is necessary to study axles in high speed passenger service and in slow freight service, on curves and on tangent track, on good track and on bad. The axles might be damaged under conditions which occur only occasionally. To make sure that no such conditions are overlooked, the behaviour of the axles must be observed over long periods.
Over 70 years after Beaumont, Moreau and Peterson found the service operating environment to be very different from the view held by the majority at that time. For example, in the course of their investigation they observed that about once every 1000 miles [1600 km] a stress of nearly four times the normal value occurred.
One of the major reasons for this state of affairs until about 1940 was the lack of suitable transducers to make the measurements. Until that time, with the exception of a brief period of use of magnetic strain gauges and carbon strip gauges, only mechanical means had been available. These devices were direct descendants of the method reported by the 1849 Commission into the Application of Iron to Railway Structures (Anon., 1849):
… and a pencil was fixed to the underside of one of the girders of the bridge, so that when the latter was deflected by the weight of the engine or train either placed at rest or passing over it, the pencil traced the extent of the deflection upon a drawing board attached to the scaffold.
It is worth discussing these dynamic loads in more detail. It is now recognised that the magnitude of the dynamic loads induced by the passage of a wheel over a discontinuity in the rail, for example, a gap, dip, or damage patch, is determined by, of course, the magnitude of the discontinuity and by the axle load in combination with the ‘unsprung mass’ of the vehicle, that is, the mass below the main suspension in ‘hard’ contact with the rail.
1.4.1 An illustration of the magnitude and effect of dynamic loads
An example calculated using a simple model from data supplied by the Japanese Central Railway Company is shown in Fig. 1.2. This figure illustrates the forces generated as a function of time by the passage of a train over a small (5 mm) dip in the rail head. Two trains are shown, an old type (Series 100) and its replacement (Series 300). The intention was to increase the speed of operation from 180 to 230 km/h. The form of the response from both trains at both speeds is similar, with the dynamic forces showing two clear peaks with time, the so-called P1 and P2 forces. The dynamic magnification increases with speed and lies in a range approximately 2.5 to 3 times greater than the static force. Clearly, these magnified forces have a significant effect on the fatigue of wheels, rails and axles. They are significant too in their effect on track maintenance. This is summarised in Fig. 1.3, which is a representation of the typical track maintenance costs as a function of speed for both types of train. The important characteristics of the new train are shown: a smaller wheel load (reduced from 7.5 to 5.7 tonnes) and a smaller unsprung mass (reduced from 2.3 to 1.7 tonnes), the reduction of which is a particularly sensitive way of reducing dynamic track forces. In the example shown, if the old train had been run at the required higher speed of 230 km/h, the track maintenance costs would have increased by some 20%. However, the new lighter train produces a saving of some 10% even at the higher speed. Obviously this is a somewhat simplified view of a complex situation which depends on many parameters. However, it serves to capture the essence of the dynamic load problem and illustrates the need for track and train designers to work in conjunction with each other. It serves too to illustrate the constraint of higher speeds and structural integrity. For high speeds it is necessary to drive down mass in critical components, thus making them more prone to fatigue.
1.2. Dynamic forces produced by the passage of trains over a rail head geometry defect (track force response for a 0.0025 rad, 5 mm rail dip).
1.3. Generic effect of dynamic forces on maintenance costs.
1.4.2 Bearings and axles
The life of bearings has been much improved by increasing cleanliness of steels. If care is taken to lubricate the bearings correctly and to prevent the entry of dirt, then satisfactory long lives can be easily obtained. There are reports that bearings have failed after dismantling in order to gain access to axles for crack detection examinations. The reason for the need to examine axles arises because of their safety-critical nature and the few, but persistent, number of axles which fail in service.
The fatigue failure of axles in the first railways was the catalyst that led to the identification of fatigue as a failure mechanism. Many investigations were prompted by the accident on the Paris to Versailles railway in 1842, Fig. 1.4, the first time a railway accident had caused major loss of life (Smith, 1990). In later decades, the pioneering experiments of the German engineer Wöhler led to the identification of the fatigue limit for steels. It is something of a surprise therefore that failures still occur. Although it might be assumed that the simplest solution would be to increase the size of axles to reduce stresses, the counter-argument outlined previously is that axles form part of the unsprung mass of the vehicle which must be minimised to reduce the generation of dynamic stresses. Particularly as operational speeds of trains have been increased, the pressure to reduce unsprung mass has become more pressing.
1.4. The Versailles accident of 1842, caused by a broken axle: the first railway accident leading to a large loss of life.
The need for thorough understanding of the service loads to which axles are subjected in service has already been noted. The loading is principally sinusoidal due to the bending couples produced by the upward wheel load reactions being offset from the downward supporting bearing loads. However, the wheel loads can be greatly magnified by dynamic effects, and the equally distributed static loads on each wheel on straight track can be redistributed by cornering and wheel nip at tight gauges as well as by breaking and accelerating forces. The condition of the track and the wheels is paramount in determining the levels of dynamic forces involved, which also increase with speed.
Over the years many experiments have been performed to measure stresses on axles in service. Until recently, the usual procedure was to use slip rings to carry strain gauge signals from the rotating axle into appropriate recording equipment. Because of the bulkiness of the equipment involved, records have been obtained over relatively short times and therefore distances. New developments in electronics have produced miniature equipment containing great recording and processing power. A programme now underway (Smith and Hillmansen, 2001) uses such equipment, which can be directly mounted on the axle and left unattended for periods of several months. A continuous load spectrum is recorded for later analysis by, for example, rain flow counting. More interestingly, strain is recorded over short time intervals of about five seconds, but only retained if a large strain event triggers a storage command, together with a location signal via a Global Positioning Satellite signal. It is hoped that the key very large strain excursions will be captured and identified in this way, in order to clarify why failures occur and how the severity of loading is related to track condition.
1.4.3 Inspection of axles and crack detection in axles
Although failures of axles are rare, typically two or three per year on the UK railway system, the consequences can be catastrophic. Therefore great effort and cost are expended in examining axles for cracks based on a philosophy of setting inspection intervals which leave some margin in the time it would take for the largest non-detectable crack to grow to failure in the time between inspections. However, because of the relatively large sizes of cracks which may be reliably detected (orders of several mm) and the runaway nature of fatigue crack growth, see Fig. 1.5, it is not easy to set economic yet practical inspection intervals. Non-destructive testing methods, ranging from ultrasonics, magnetic particles, dye-penetrants and eddy currents, are notoriously difficult to apply with complete confidence that they will be certain of identifying all cracks above the assumed sensitivity level. Added to this are the uncertainties arising from testing a huge number of axles, in order to identify the very small sample of the population that may be cracked. Detection sensitivities are usually based on what size of crack a particular method might detect in a test-piece known to contain a crack. This is a very different situation from detecting which particular axle out of say 10 000 might be cracked. There is a suspicion that crack detection of axles is inefficient and if better understanding of the nature of what must be an ‘extreme-value’ fatigue event could be utilised, great savings on inspection may be possible.
1.5. Crack length plotted as a function of number of cycles. The initial defect size is chosen to be 100 μm and axle failure is assumed to follow rapidly after the crack has grown to 30 mm. A crack length of 5 mm can be detected with a reasonable degree of certainty using NDT methods. This figure clearly illustrates that once a crack length of 5 mm is attained, the axle is near the end of its life.
1.4.4 Gearboxes, drive shafts, brakes, springs and suspension components
As we move further away from the wheel–rail interface, a whole range of components suffer from potential fatigue problems, and while load inputs arising from dynamic running loads are significant, their effect becomes more attenuated with distance from the origin at the interface. Cases of failures in all the components mentioned in this section heading have been reported in specific types of vehicle, but none could be said to be generic. In the past brakes generally operated by shoes acting directly on the tread of the wheel. There were counteracting effects: the wear produced by the action of the brake ‘dressed’ the wheel and rubbed out incipient fatigue damage. On the other hand, excessive braking tends to induce thermal damage at the wheel’s running surface. Although brakes of this type still operate on older vehicles and on most freight vehicles, newer designs incorporate disc brakes, on which thermal crazing leads to spalling or fracture of the disc.
Brake pipes and connections are often made from rubber and rubber compound materials, the fatigue evaluation of which formed the basis of a study by Hansaka et al. (1999). Rubber springs, in blocks or formed into air bags, are frequently used in modern suspensions (Luo et al., 2003), where they are subjected to fatigue deterioration. Most applications involving elastomers and other non-metallics require specific experimental testing programmes as the mechanical properties of such materials are generally not so well defined as those of metals and are sensitive to environmental deterioration and loading frequency effects.
1.4.5 Fatigue problems below the rail
The passage of a train over the rail and its supporting structure leads to potential problems in the rail fastening, the rail supports and the foundation of the track. The modern method of fastening rails to sleepers is by metal clips, which in certain circumstances have failed. Sleepers, made from a wide variety of materials – wood, concrete, steel or composites – seem remarkably free from fatigue problems. However, ballast, the principal material used to make the track foundation, does suffer from continuous deterioration, which in the broadest sense may be classified as fatigue. Ballast is nothing more than a compacted pile of stones through which loads are transmitted by the contacting vertices. Wear at these points of contact causes settlement of the track and is the principal reason for the extensive and expensive maintenance needed to preserve the geometry of the top surface of the rail. It is not therefore surprising that continuous slab track, more expensive than ballast to install but much cheaper in maintenance costs, has been used on many modern railways.
This area has been the subject of a recent review (Dahlberg, 2001). Some empirical models of settlement of ballast were discussed. If settlement is expressed as a function of loading, either number of wheel passes or tonnage, a common feature is an initial rapid settlement blending exponentially into a longer-term and much slower constant settlement rate.
Traffic monitoring technology has been developed for vehicle detection in recent decades. These technologies are often based on the side of the road detector (based on the visual sound detector, detector and based on laser detector, it is highly affected by the surrounding environment) and in the detector (induction coil, magnetometer, piezoelectric sensors, its installation is destructive and may reduce the service life of the pavement) [143,148].
Using Rayleigh scatter-based optical frequency domain reflection (OFDR) technology with a spatial resolution of 2.56 mm, Samim Mustafa et al. investigated the optimal embedding depth of DOFS for pavement monitoring by considering three embedding depths (10, 30, and 40 mm) and three moving loads (pedestrians, cars, and trucks) in a series of field tests (Fig. 21(a)). It is found that this technology can not only detect moving loads on the road, but also obtain a wide range of information, including the type of load, the speed of load movement, load weight, shaft, shaft spacing (vehicles) and traffic flow [149].
Fig. 21. Applications of DOFS in pavement engineering [12,142,149–152].
Zhao et al. proposed a vehicle classification system based on distributed optical vibration sensing (DOVS) technology to collect vehicle classification data on a large scale (Fig. 21(d)). Firstly, the system consists of embedding sensing fiber as a distributed sensor to collect traffic-induced vibration signals, and then extracts several features from the original signals to estimate axle configurations and identify vehicle categories. At the same time, the empirical mode decomposition (EMD) method was used to reconstruct the signal for feature extraction, and the extraction algorithm was used to obtain the axle configuration, driving speed and frequency domain characteristics of each vehicle. Finally, the multi-step classifier is used to classify vehicles into different categories [143,148]. This system can accurately classify the two-axle cars and passenger cars of heavy vehicles, but can hardly identify the three-axle of two-axle trucks. Besides that, this system can provide both the position and speed of the vehicle with high precision.
Traction characteristic curves show the relationship between tractive effort at wheel rim (vertical axle) and the running speed (horizontal axis), usually obtained by test. Traction characteristic curve varies with traction type; if the traction type is the same, traction characteristic curves of various locomotive-hauled train (EMU train) has few differences. Traction characteristic curves for each type of motor are decided by the manufacturer. The following section will analyze the features of classification with examples of electric locomotive, diesel locomotive, and high-speed EMUs.
(1) Electric Locomotive
1) Motor Types
Economic, environmental, and energy supply factors play increasingly vital roles in the viability of systems using straight electric propulsion. This electric propulsion is applied both to the electrification of existing railways (now diesel-powered) and in the construction of rapid railway lines, and transit and intercity rail lines for urban passenger service. Electric locomotives must be considered in terms of both their traction motor characteristics and the system of power supply and transmission to those motors.
The electric locomotive obtains electric power from overhead lines. For traction purpose, the electrified railway adopts a single-phase AC power supply system in China. The high-voltage alternating current from overhead lines, via a pantograph installed on the locomotive, enters the transformer set on the locomotive, and then the current is transformed into low voltage alternating current, and furthermore, with the rectification of converter, the AC current will be converted into DC current. In order to let the current flow to traction motors, gear engagement between traction motor and the locomotive traction wheel was applied to supply the required torque to produce the tractive force. Finally, through the adhesion effect between wheel and rail, the locomotive traction force is generated.
Motive power is usually classified by type of motor, including single-phase series-wound DC motor and single-phase series-wound AC motor.
Single-phase series-wound DC motor has had extensive use from early days for street railway and rapid transit trains. It is the conventional propulsive unit for diesel-electric locomotives. The advent of thyristor solid-state silicon controlled rectifier systems has made this motor available to modern heavy-duty electric locomotives. Plant costs are reduced through the direct use of commercial (60-cycle) frequencies. Smoother operation and greater reliability are possible with stepless electronic control of motor current and voltage.
The characteristics of single-phase series-wound AC motors are essentially similar to those of the DC type (including the added advantages of the direct use of AC current with no commutator to maintain). Its performance is best at high speeds, but it lacks the DC low-speed capabilities and is bulky and less compact, which is an important consideration in the design of modern motive power.
On China’s railways, traction motor on electric locomotive, currently adopts the DC series motor. This is because this kind of motor’s mechanical characteristic curve is similar to a hyperbola, which is suitable for locomotive traction.
2) Motor-Generator Systems
DC motors applied for the long-line freight locomotives have many shortcomings, including the arcing of high voltage across the commutator and the transmission of high DC voltage with high line loss, expensive transmission equipment, or both.
One way to overcome these disadvantages is the application of a motor-generator combination. A high-voltage AC transmission line current, with its lower costs, energizes a generator in the locomotive. The generator drives DC motors with relatively low voltage. The electric locomotive becomes essentially a constant-horsepower machine with a transmission line power factor close to unity and with performance independent of the transmission-line voltage. The flexibility and smoothness of DC motor controls are available in conjunction with the cheaper transmission of high-voltage alternating current. The Great Northern and the Virginian railways use this type of power. Motor-generator units are weighty, bulky, and require large cabs and heavy running gear. The power loss is rather high.
An SS4 modern electric locomotive is shown in Fig. 2.2.
Fig. 2.2. SS4 modern electric locomotive.
3) The Induction Motor (Poly Phase and Split Phase)
The poly-phase induction motor differs in its characteristics from the motors mentioned before. A poly-phase current is applied to the stationary or stator windings, setting up a rotating magnetic field in which the rotor coils are placed. A current is induced in the rotor, which has a tendency to resist the rotation of the field. Since the field cannot stop, the rotor is pulled along, creating a certain amount of torque.
When no load is applied, the rotor revolves at a theoretically synchronous speed, i.e., at the same speed as the rotating field. (Practically, on account of wind age and bearing friction, it cannot revolve at a true synchronous speed.) There is no induced voltage in the rotor, hence no torque producing current flows. The speed in revolutions per minute is equal to twice the frequency of the field in revolutions per minute divided by the number of poles.
When a load is applied to the rotor, its speed slips below that of the field (synchronization speed), and the rotor conductors begin cutting magnetic lines of force of the rotating field. A voltage is induced in the rotor coils that is proportional to the relative speeds of the rotor and rotating field. Thus torque may be said to vary with the slip from synchronous speed to the speed of maximum load. The insertion of resistance in the rotor circuit determines the point of maximum torque, but since the resistance is inefficient (consuming the current), it should be kept at a minimum and only a small range of slip permitted between no-load (synchronous) and full-load or maximum torque speed. The poly-phase induction motor thus becomes a fairly constant-speed machine, regardless of the applied load.
For starting speeds, efficiency is not of prime importance. Resistance, therefore, can safely be introduced into the rotor circuit. It is then cut out step by step, keeping maximum torque and tractive effort in line with the actual speed.
Three-phase motors and transmission systems are frequently used in Europe, but have proved to be too cumbersome and inflexible in terms of application in China. A three-phase line current requires two trolley wires, with the rails constituting the third side of the circuit. The Norfolk and Western Railroad in the USA made use of a so-called split-phase system. A single-phase alternating current taken from a single overhead wire was changed to a three-phase current through a converter in the locomotive and fed to a three-phase motor. Two speeds of 14 and 28 mph were available by connecting the stator windings in series and in parallel, so as to have a four-pole and an eight-pole machine. The effective number of poles could be varied from the control panel. Thus, with four poles effective, twice the speed was obtained than with eight poles. One of the chief advantages of the split-phase induction motor is its ability to regenerate power automatically on descending grades.
(2) Electric Tractive Effort
There is a definite relation between torque, current, voltage, and speed, available in a series of characteristic curves furnished for each type of motor by the manufacturer, as shown in Fig. 2.3. The manufacturer’s curves are often plotted on the basis of substation voltage. It is safer in estimation work to take the trolley voltage as 90% of the substation voltage. Thus only 90% of the torque from a manufacturer’s curve based on substation voltage should be used in the equation for tractive effort. Single-phase DC motor voltages in successive steps may be applied to the motors to give variations in speed control. Since every tap or point on the controller for an AC motor is a running position, it is desirable to have a series of curves showing AC motor performance under the several applied voltages. These curves are usually plotted with current as the abscissa and with torque (or tractive effort), speed, voltage, and horsepower shown as ordinates. Characteristic curves for varying speeds are shown in Fig. 2.3. Again, note that with silicon solid-state controlled rectifiers, the stepless electronic control of DC series-wound motors is available for smoother and more reliable operation.
Fig. 2.3. Characteristic curve of an SS3 locomotive.
Fig. 2.3 shows the traction characteristic curve of an SS3 electric locomotive. Power of electric locomotive is supplied by power plants, so for each locomotive there is no limitation in terms of the capacity of power supply. Therefore, traction of electric locomotive is mainly restricted by the power of traction motor and adhesion between wheel and rail.
The group of similar hyperbolic curves, which are numbered 1, 2,…,8-III, is the tractive force determined by the hauling power of the electric locomotive. It can be calculated after converting the electrical mechanical properties of the motor to wheel traction by the traction motor.
In the operation of the locomotive, according to the requirements of the operation, it is expected that the locomotive traction and speed can vary in a wide range. Only one curve, F = F (V), is not enough. The speed of a mainline train is regulated by changing the input voltage of the motor and magnetic flux (by weakening magnetic field). SS3 locomotives have eight regulation levels; therefore, there are eight curves F = F (V). On the eighth gear there are three weakening magnetic field levels, 8–I, 8–II, and 8–III, respectively called I, II, and III weakening magnetic fields.
In the absence of complete motor performance data, tractive effort may be estimated by the following approximation as given by the area in their manual (railroad engineering). For DC locomotives, tractive effort is inversely proportional to the cube of the speed and to the square roots of the horsepower:
For single-phase motors:
If the tractive effort for a given speed is known, the tractive effort of any other speed can be determined.
(3) Electric Locomotive Ratings
Adhesive traction: Torque and tractive effort increase with the current. Tractive effort is unlimited, theoretically, since the power supply from the central stations is, for practical purposes, unlimited but a limitation is the point at which the drivers begin to slip, as determined by the factor of adhesion. The hatched line in Fig. 2.3 is the tractive effort curve limited by the adhesive force between wheel and rail, and it is calculated according to Eqs. (2.1), (2.2).
Traction force is limited by permitted current of motor: a second practical limitation is enforced by the heating of the motors at full load and high current value. As the heating does not occur at once, it is permissible to utilize the overload possibilities of the motors for a limited period. At least two ratings are usually specified: the continuous and the hourly.
The continuous rating as prescribed by the Chinese Institute of Electrical Engineers is that maximum tractive effort which the locomotive can exert continuously without harm to the motors. It is not the maximum load the locomotive can handle, but it is the heaviest load that can be handled continuously with safety. The continuous rating is based on that continuous current and corresponding tractive effort which keeps motor temperature under 60°C (160°C with silicon windings), starting with a cool (25°C) motor.
The hourly rated tractive effort is based on that current and tractive effort which can be exerted for 1 h without harm to the motor; that is, the motor temperature is elevated by 60°C in 1 h, provided that the motors are cool when the overload is applied.
Much higher currents may be taken for shorter periods of time (such as 1 min), especially in the beginning and temperatures of 100°C may occasionally be reached without harm to the windings. When this operation occurs, measurements such as stops, coasting grades, or light pull below continuous current ratings must be applied immediately to permit the motors to cool.
The oblique line AB in Fig. 2.3 represents the continuous rating.
1)
Traction force under different speeds
“Chinese traction calculation procedures” (TCP) stipulate that tractive effort of electric locomotive is to be taken via a continuous rating system. For example, in the SS3 locomotive, if the tractive effort for a given speed is known, the tractive effort of any other speed can be determined as follows: in the acceleration process (accelerate from 0 to locomotive construction speed), the tractive force takes adhesive traction (CD), and motor traction (DA, BE section), respectively, and the tractive effort determined by continuous rating (AB).
2)
Calculated speed and calculated tractive force
Calculated speed is important for the calculation of the tractive mass of train, its corresponding tractive force is called calculated tractive effort. TCP take the speed and traction determined by the sustained current of the full-field traction characteristics at the highest step as the lowest calculated speed and maximum calculated traction. The corresponding speed and tractive force at point A in Fig. 2.3 are the lowest calculated speed (vjmin) and maximum calculated tractive force (Fjmax) of SS3 electric locomotive, respectively.
3)
Starting tractive effort
Starting tractive effort is the maximum tractive effort which is provided by locomotive under the starting status. A locomotive’s starting traction stipulated by TCP is determined by calculating or specially testing based on certain restrictions, and is called calculated starting traction effort (TEs). The calculated traction effort of most freight locomotives is restricted by adhesion conditions. Calculated traction of passenger locomotives is restricted by starting current. In the calculation of SS3 locomotives under starting condition, starting calculated traction effort (TEs) is equal to adhesion traction effort when speed is zero as. A few locomotives use limited current traction effort as calculated starting traction effort.
4)
Electric locomotive characteristics
The value parameters of frequently used Chinese electric locomotives are given in Table 2.1.
In Table 2.1, vcmin is the minimum calculated speed, Tcmax is the maximum tractive effort, Ts is the starting tractive force, P and Pμ is the locomotive axle load and the locomotive adhesion axle load, and Lloc is the length of the locomotive.
(4) Effects on Location
The foregoing characteristics are of great interest to the locating engineer. The hauling capacity of electric locomotives is based not upon the maximum grade encountered but upon the profile as a whole. A grade that would require pusher engines in steam service might be overcome by an electric locomotive making additional current demands if the grade can be negotiated before the motors become too warm. Thus length of grade is an important consideration. It should not be longer than which can be traversed in 1 h or less at the speed and loading for the locomotive’s hourly rating. It should also be followed by much easier gradients or preferably descending gradients to permit the cooling of the motors. A profile with many short, heavy grades permits a heavier tonnage per locomotive unit than one that contains a few very long grades. Electrification may well be considered as a possible alternative to grade reductions. Electrification should not be considered as an alternative to good, careful location design. An electric locomotive turning in a good performance on a poorly located line will give a far better performance on a line with properly designed gradients.
3.2.2 Rotary electromagnetic vibration energy harvesters
The rotary EM-VEH is defined by the rotary energy transducers, even if the original motion is linear displacement (which can translate into rotation). The core component of rotary EM-VEHs is usually in the form of rotary generators. Some of them are equipped with motion conversion mechanisms converting other types of relative motion to rotational motion and driving a rotary generator. The earliest vehicle kinetic energy harvesting scheme was to install the axle generator at the end of the axle to provide power for the devices on the unpowered freight wagons. Nagode et al. [107] proposed an axle generator with an open structure for quick and convenient installation. As shown in Fig. 9(a), three sets of friction wheelsets are arranged in the shell at 120° intervals. One of them transmits the motion of car axle and drives the generator by two sets of gears with an 8 times ratio. The other two wheelsets are free to maintain contact between the driving wheelset and the axle. Two halves of the device are hinged on the same axle and are tied by spring-loaded bolts during installation. In tests, the axle generator with a “Y” configuration of three phases connection presented a power of 290 W at the vehicle speed of 88.5 km/h (55 mph), which is adequate for numerous devices. However, different from VEHs, the energy source of the axle generator is the kinetic energy of the vehicle axle, which causes extra running resistance and traction power consumption.
Fig. 9. Rotary energy harvester proposed by Nagode et al. (a) axle generator [107], (b) rotary EM-VEH with nut-screw transmission [101].
Vibration energy in the vehicle suspension is a form of kinetic energy. For vehicles, it is waste energy, which is eventually converted into heat by shock absorbers or wedges. Reasonable uses of vibration will not affect the energy consumption of the train or even can improve the vehicle dynamics. Hence, Nagode et al. [101] continued to investigate energy harvesting from vehicle suspension vibration. In addition to the linear EM-VEH mentioned above, a rotary one fitted inside the suspension coil spring was proposed, as shown in Fig. 9(b). The ball screw transforms the linear motion of suspension in rotation, and the planetary gearbox is connected to the screw and amplifies the rotation speed of generator. A peak power of 77 W was obtained in the excitation of ±19 mm and 1 Hz, which is an order of magnitude higher than powers encountered with linear EM-VEH.
Nelson et al. [108,109] proposed a rotary EM-VEH mounted to and spanning two rail ties harvesting energy from rail vertical displacement, with the goal of generating 40 W for a grade crossing warning light system. The energy harvesting system was realized by the rack gear, pinion gear, clutch, gearbox and PMDC generator, as shown in Fig. 10(a). A maximum power output of 7.41 W was calculated for a loaded train traveling at 88.5 km/h (55 mph), theoretically. In field tests, the harvester generated a 0.22 W average power when the speed of the loaded trains is 18.5 km/h (11.5 mph), which is not enough for power requirements. The first-generation prototype cannot utilize the upward vibration of the track resulting in a low power generation efficiency of the generator. Therefore, the second generation prototype that can simultaneously scavenge power from upward and downward displacements of the track was developed [110], as shown in Fig. 10(b). The pinion gear is driven by engaging with a rack gear fixed to the subgrade. The deployment of the two one-way clutches realizes the mechanical rectification of the bidirectional rotation of the pinion gear. They are engaged or freewheeled when the corresponding input shaft of them rotates clockwise (CW) or counterclockwise (CCW). When the rail travels downward, the pinion gear rotates CW and the right clutch is engaged. The right gear is driven to rotate CW resulting in the CCW rotation of middle gear and DC generation. When the rail travels upward, the pinion gear rotates CCW. The left shaft rotates CW by the motion transmission of driven gear pair. The left clutch is engaged and the left gear rotates CW leading to the CCW rotation of middle gear and DC generation. Hence, regardless of the upward or downward track displacement, the generator always rotates in one direction. This prototype was equipped with a gearbox with doubled speed ratio (1:100) compared to the last generation and presented a better power generation capacity. In simulations, a maximum power over 300 W can be realized in the condition of a loaded train traveling at 96.5 km/h (60 mph). Even at the train speed of 18.5 km/h (11.5 mph) with load, an average power of 42 W can be generated.
Fig. 10. Track-side rotary EM-VEHs proposed by Nelson et al. (a–b) nut-screw based EM-VEHs realizing unidirectional [108,109] and bidirectional energy rail vibration energy harvesting [110], (c) the hydraulic power harvester [111], (d) the vibration amplification mechanism for unfavorable conditions [109,112].
In view of the constrained output power of previous devices in unloaded conditions or at low speeds, a hydraulic power harvester was proposed in Fig. 10(c) [111]. A hydraulic cylinder is mounted on the bottom of rail, retracting and extending during the train passing by, and brings pressurized fluid to the hydraulic motor. The DC generator is driven to rotate after amplifying the rotary speed of the hydraulic motor by the planetary gearbox. A poppet check valve in parallel with the hydraulic motor is necessary to permit the fluid to flow towards the bottom chamber of the cylinder during extension. The reservoir must be placed about 11 feet higher than the check valve to surpass the opening pressure of check value. An average power of 11.8 W was realized in the rail deflection of 3.8 mm at 0.37 Hz. However, when the deflection amplitude is less than 2.8 mm, the output of the hydraulic power is still constrained.
In Fig. 10(d), a cam-based device was designed combined with energy harvesters [109,112] to enhance power capacity of the harvester with mechanical transmission in unfavorable conditions. When a train passes by (from the bottom right to top left in Fig. 10(d)), roller 5 as well as roller 1 and 4 are driven to the right along the curved grooves, resulting in CW rotation of the follower 2 driven by roller 2. When a train passes by from a negative direction, follower 1 is driven CCW by a roller. The CW rotation of follower 2 or CCW rotation of follower 1 is converted to CW rotational of the main shaft via a pair of spur gears and a chain gear system. In addition, a return mechanism was developed to avoid two followers unable to reset. The second-generation prototype combined with the cam-based magnifying mechanism realized an average power of 50 W when a loaded train was traveling at 18.5 km/h (12 mph).
Zuo's group has carried out long-term research and improvement on the mechanical rotary EM-VEH to harvest energy from rail track deflections. They were the first to come up with a harvester based on twin pairs of rack-pinion transmission [113], as shown in Fig. 11(a). This configuration can effectively reduce the working frequency of each rack-pinion pair and prolong the service life. Two one-way clutches embedded gears 1 and 2 make two gears only rotate CCW. When racks move up and down, only one clutch is engaged and drives the corresponding gear CCW. Another clutch is disengaged and the corresponding gear is driven CCW by the middle gear between gears 1 and 2. The electrical generator and flywheel are driven CCW by shaft 3. Flywheel speed regulation enables the harvester to provide a more reliable and continuous output. In the excitation with 6.4 mm amplitude and 1 Hz frequency, the overall system realized an efficiency of 22.2% and 1 W average power output. Then, their second generation device with a prominent mechanical efficiency improvement was proposed, as shown in Fig. 11(b) [114]. With the same motion conversion principle, the previous three shafts with six mounted bearings and four meshing gears were simplified by a single shaft system with two mounted bearings and two meshing gears. The gearhead is added to amplify the low speed of pinion shaft and provide an optimal angular velocity for the efficient operation of generator. In tests, the harvester was loaded at a sinusoidal excitation with 3 mm amplitude and the frequency is from 2 Hz to 5 Hz. The mechanical efficiency as well as overall efficiency was improved by a factor of 2–3 compared with the last version. The best mechanical efficiency of 71.1% was achieved with 0.19 Ω external resistance at 2 Hz, and an overall best efficiency of 46.7% was obtained with 2 Ω external resistance at 3 Hz and 4 Hz.
Fig. 11. Track-side rotary EM-VEHs proposed by Zuo et al. (a) the first generation prototype [113], (b) the second generation prototype with mechanical efficiency improvement [114], (c) the third generation prototype with anchorless mounting [115], (d) the fourth generation prototype with novel MMR [116].
In the third version, the basic transmission mechanism of the second generation was followed, but the DC generator was replaced by the AC generator, as shown in Fig. 11(c) [115]. In addition, an anchorless mounting mechanism was developed. An adjustable threaded rod is used to preload the springs under the harvester. The other end of rod is fixed on the stationary base plate, which rests on the ballast. The mounting mechanism is easy to install quickly and will not damage the substructure of the track and avoid potential risks. The overall harvesting system was simplified as a spring-mass-damping system and was delved deeper in modelling. They also first conducted field tests to validate the proposed design in real conditions. The calculated average power of harvester with an equivalent of 16.7 Ω in the Y connection according to recorded data was about 6.9 W average and 54 W peak when a test train composed of 100 cars ran at 64 km/h (40 mph).
Pan in Zuo's group developed a compact ball-screw-based design with a novel mechanical motion rectifier (MMR) [116], as shown in Fig. 11(d). Compared with the rack-pinion configuration, the ball screw mechanism is considered more advantageous in excitation with small displacement due to reducing the backlash during the converting process. When the track as well as nut move up, the ball screw will spin CCW (from the top view). The upper bevel gear is engaged with an embedded clutch and drives the right bevel gear to spin CCW (from the side view). The generator is driven CCW after the acceleration of gearhead. If the track moves down, the lower bevel gear engaged with clutch will become a driven gear and rotate CW (from the top view), and the upper bevel gear will be the idle one. The core parts of MMR with three bevel gears and two one-way clutches ensure the unidirectional rotation of generator regardless of the track motion. Field test results showed an average power of 2.24 W when the two-unit rapid train runs at 30 km/h (19 mph). The four generations of trackside rotary EM-VEHs proposed by Zuo's group have been summarized in Fig. 12.
Fig. 12. Design comparison between four generations of track-side rotary EM-VEHs proposed by Zuo et al.
Another device with this novel MMR and rack-pinion transmission was proposed by Zuo's group as shown in Fig. 13 [117]. The previous two pairs of rack-pinions are simplified with one pair. The authors investigated the performance of this system in harvesting the vibration energy of suspension system. The harvester mounted to a loaded rail car with a total weight of around 23 tons obtained a peak phase power of 73.2 W and an average power of 1.3 W in 20 s at 30 km/h (19 mph) on field tests.
Fig. 13. Onboard rotary EM-VEHs proposed by Zuo et al. with novel MMR for suspension vibration energy harvesting [117].
Zhang's group first proposed a rotary EM-VEH with rack-pinon transmission in 2016, as shown in Fig. 14(a) [118]. The principle of MMR in the system is similar to that proposed by Nelson et al. in Fig. 10(b). The improvements are that: (1) the two one-way clutches of the novel device are embedded in gears. (2) the generator speed is amplified only by gears and a rack and no extra gearbox is added. In tests, an overall system efficiency of 55.5% was obtained. The design of the second-generation prototype was optimized based on the harvester in Fig. 11(a). Two pairs of rack-pinions are simplified to one pair and the same conversion effect is realized, as illustrated in Fig. 14(b) [119]. Another difference is that the prototype is not equipped with a flywheel. The peak voltage of 58 V (in excitation of 1 Hz, 0.25 mm amplitude) indicates the capacity of the harvester to supply power for some trackside equipment. Their latest research is trying to integrate kinetic energy harvester with railway Dowty Retarders (DRs), as shown in Fig. 14(c) [120]. The device accumulates the vibration energy during train deceleration and assists train braking simultaneously. The mechanical transmission part is composed of a special closed-loop shaped gear and a screw-nut pair. When the decelerating train acts on the device, the spring is compressed and the lead screw spins CCW. At this moment, bevel gear A is engaged with the first unilateral bearing, driving bevel gears C and D (two-way clutches embedded). Subsequently, the bevel gear B is driven to rotate CW although the second unilateral bearing is disengaged from the main shaft. The sleeve rotates CW due to the engagement of the third unilateral bearing and drives the generator coupling with it to rotate in the same direction. When the train starts, the return force of the spring drives the nut moving up, and the lead screw spins CW. The first unilateral bearing is disengaged now, and bevel gear A keeps stationary. The sleeve, as well as the generator, are directly driven CW by the main shaft due to the engagement of the second unilateral bearing. The third unilateral bearing is disengaged, thus the bevel gears B, C, D also keep stationary as the bevel gear A. The closed-loop gear set does not work during the recovery process. The proposed design ensures the constant rotation of the DC generator and improves the efficiency. An efficiency of 51.9% was obtained with the excitation of an amplitude of 7.5 mm and frequency of 2 Hz in experiments.
Fig. 14. Track-side rotary EM-VEHs proposed by Zhang et al. (a–b) the prototypes for rail vibration energy harvesting [118,119], (b) the prototype with closed-loop MMR for railcar braking deceleration [120].
Gao's group is committed to the onboard applications of EM-VEHs in railway vehicles. In recent research, a new compact EM-VEH with an inertial pendulum is proposed by Gao et al. [121]. It was designed to be mounted on the bogie side frame or on the car body for powering self-powered sensor nodes for freight rail transport. As shown in Fig. 15(a), the inertia pendulum and gear ring are fixed together by screws and share the same shaft. The inertia pendulum is driven by the inertia motion, and the shaft and gear ring follow. Then, the gear is driven by the engagement with the internal teeth of the gear ring, resulting in the rotation of the DC generator and electricity generation. The radius of the energy harvester can be adjusted to match the frequency of the vibration source. In addition, a DC-DC circuit with supercapacitors was designed for the function of energy conversion and management. The whole system realizes an efficiency of 40–65% above the startup voltage of 1 V. As a complete product, the innovations are as follows: (1) A compact structure integrates the energy harvester and a DC/DC circuit with supercapacitors. In field tests, the harvester fixed on the bogie side frame of C70E freight wagon generated a stable 4.7 V voltage and 50 mA current, which can drive some low-power sensors for onboard monitoring. (2) It can scavenge the vibration energy in multiple directions in the same plane. In addition, they [122] developed a novel vibration-to-rotation conversion mechanism for rotary EM-VEHs mounted on the shock absorbers of a metro via the interaction between a magnet array on cylinder and a linear vibrating magnet, as shown in Fig. 15(d). The linear vibrating magnet fixed on the vibration source drives the cylinder embedded in a ring gear to rotate through the tangential force between magnets. A maximum open voltage of over 20 V was achieved in a low-frequency hand-shaking test. However, this motion conversion mechanism is only applied to small-scale harvesters.
Fig. 15. Onboard small-scale rotary EM-VEHs proposed by Gao's et al. (a) structure configuration and (b) field tests for pendulum-resonance-based EM-VEH [121], (c) installation and (d) structure configuration for EM-VEH with magnet array [122].
Wang et al. [123] proposed an all-in-one rotary EM-VEH mounted to the axle of the train to harvest wheelset energy, as shown in Fig. 16. The circular array magnet is a rotor driven by a rotating shaft, meanwhile, the coils rotate synchronously due to the static friction between the bearing and shaft. With the increase of rotation amplitude, the gravity torque of the counterweight attached to the foundation drives the coil's oscillation, which achieves a desired relative displacement with the magnet and generates current in the coils. In the tests with a wheel speed ranging from 420 to 820 rpm, the harvester generated an average power of 32.21 mW, and a corresponding power density of 1982 W/m3 at a matched resistance of 150 Ω. It also exhibited the power supply capacity for low-power devices, such as LEDs, smartwatch, temp/humidity, etc. The reviewed rotary EM-VEHs in this section are summarized and listed in Table 3.
Fig. 16. Small-scale rotary EM-VEHs proposed by Wang's et al. (a) structure configuration and (b) application for harvesting wheelset energy [123].
Table 3. Summary of rotary electromagnetic energy harvester for railway industry.
Empty Cell
Year
Mechanism
Installation
Power [W]
Power density [W/m3]
Efficiency [%]
Simulated or field test excitation
Nagode et al.
2012
Axle generator
Car axle
290 (avg)
/
/
Car speed of 88.5 km/h in lab test
[101]
2010
Ball screw transmission
Suspension system
70 (max)
/
/
Amplitude: ±19 mm, 1 Hz in lab test
Nelson et al. [108,109]
2009
Rack-pinion transmission
Span two sleepers
3.9 (avg)
/
/
A deflection of 19 mm at train speed 12 km/h in lab test
[110]
2011
Rack-pinion transmission
Span two sleepers
306 (avg)
/
/
Loaded train with speed of 96.5 km/h in simulation
[111]
2013
Hydraulic power harvester
Bottom of the rail
11.8 (avg)
/
/
Maximum amplitude: 3.8 mm, 0.37 Hz in lab test
[109,112]
2013
Cam-based mechanism for harvester
Rail side
50 (avg)
/
/
A loaded train is traveling at 18.5 km/h with the magnifying mechanism and harvester [110] in simulation
Zuo et al. [113]
2012
Rack-pinion transmission with flywheel
Span two sleepers
1/1.4 (avg)
/
22.2/16.9
6.4 mm (1 Hz)/12.7 mm (0.5 Hz) in lab test
[114]
2013
Rack-pinion transmission with flywheel
Span two sleepers
11.4–47 (max)
/
45–47
Amplitude: 2 mm, 2–5 Hz in lab test
[115]
2018
Rack-pinion transmission with flywheel
Span two sleepers
6.9 (avg)
/
/
Field test with freight cars at a constant speed of 64 km/h
[116]
2019
Ball screw transmission
Span two sleepers
2.24 (avg)
/
/
Field test with rapid train at 30 km/h*
[117]
2019
Rack-pinion transmission
Suspension system
1.3 (avg)
380.15
68
Field test with loaded car of weight around 23 tons at 30 km/h
Zhang et al. [118]
2016
Rack-pinion transmission
Rail bottom
/
/
55.5
Amplitude: 6 mm, 2 Hz in lab test
[119]
2017
Rack-pinion transmission
Rail bottom
12.07 (avg)
4.23 × 103
/
Track vibration by vehicle-track model in simulation
[120]
2021
Ball screw transmission
Rail side
/
/
51.9
Amplitude: 7.5 mm, 2 Hz in lab test
Gao et al. [121]
2020
Electromagnetic harvester with inertial pendulum
Bogie side frame or on the car body
0.263 (avg)
951.80
40–65a
Field test with C70E freight wagon
[122]
2022
Electromagnetic harvester with magnet array
Shock absorber
0.05 (rms)
/
/
Train runs at 80 km/h in simulation
Wang et al. [123]
2021
Rotary electromagnetic harvester
Axle
0.032 (avg)
1.98 × 103
/
Wheel speed ranging from 420 to 820 rpm in lab test
a
It refers the efficiency of overall harvesting system including interface circuit.
Rolling bearings are usually designed for maximum life. If the correct bearing is selected for the application, the bearing must be properly installed, lubricated, and maintained to achieve the designed life. Of the estimated 10 billion bearings manufactured annually, it is estimated that about 0.5% or 50 million bearings are removed from service annually because of damage or failure [1]. Ref. [1] states that bearing damage or failure can be segregated into four categories: fatigue, inadequate lubrication, contamination due to seal failure, or other reasons. The four categories of bearing failures are graphically displayed in Fig. 7.1. Examples of inadequate lubrication include selecting the incorrect lubricant, using the wrong amount of lubricant, and employing the wrong lubrication interval. Other reasons include failures due to improper handling, improper mounting, inappropriate loadings, and improper fits. The determination of the cause of bearing failure can be difficult at times since one failure mode may initiate another. For example, water ingress through a seal can cause rust on the raceways and rolling elements of a grease lubricated tapered roller bearing utilized on railway car axles. Red rust, iron (III) oxides, is abrasive and can wear material from the raceways. The loss of material can alter the geometric stress concentration, increase the radial clearance, and a potential loss of preload. Additionally, wear debris in the grease will negatively affect the grease’s ability to lubricate and can cause damage to the elastomeric seals. Excessive damage will compromise the seal’s ability to retain the grease or exclude contaminates from entering the bearing cavity.
Fig. 7.1. Four categories of bearing failures: fatigue, inadequate lubrication, contamination from inadequate sealing, and other reasons.
In previous chapters, we have considered cases where the load is applied to the structure essentially only once, i.e. the permanent load as discussed in Chapter 2. In practice, however, many structures have to withstand many repeated applications of load, the variable loads also discussed in Chapter 2. The repeated application of a load is generally known as cyclic loading.
From the early days of the industrial revolution in the first half of the 19 th century, structures were subjected to repeated applications of load. Pistons in steam engines underwent cyclic loading owing to the steam cycle pushing the piston backwards and forwards. These pistons then generated rotational movement in, e.g., mine shaft hoists or railway axles. These hoists and axles were subjected to bending loading and, each time the axle rotated, the stress at a point on the surface ranged from zero to tension to zero to compression and back to zero again (Figure 6.1).
Figure 6.1. Stress cycling from tension (+) to compression (−) as a result of axle rotation.
Engineers of the time were well able to calculate these stresses and ensured that the stresses were well below the yield strength of the iron or steel, the established good practice of the day. Nevertheless, mine hoists and railway axles failed, and perhaps the most significant was the railway accident that occurred just outside Paris at Versailles in 1842. More than 50 people lost their lives and it was subsequently established that the cause of the accident was a broken locomotive axle. A sketch of the broken axle is shown in Figure 6.2. Although not widely recognised at the time, the failure exhibited two characteristics that are typical of failures under cyclic loading:
Figure 6.2. Fatigue failure in a train axle after the Versailles train crash (by Joseph Glynn, 1843).
a)
the fracture location was at a stress raiser, in this instance the notch where there was a small change in axle diameter; and
b)
the fracture surface was flat, relatively featureless and at right angles to the applied stress.
To 19th century engineers, these fractures were puzzling as it was clear that even cyclic loads well below the elastic limit were sufficient to cause failure. The notion developed that the repeated application of load somehow caused the iron or steel to become ‘tired’. It may be that this notion was a direct parallel with a worker becoming tired after a day's toil of repetitive tasks. As a result, failure under cyclic loading became known as fatigue failure, and application of repeated loads became known as fatigue loading.
We now know that the notion of the material becoming ‘tired’ is false; the fracture is caused by the initiation and steady growth of cracks, but the terminology ‘fatigue’ is now firmly entrenched.
The study of the behaviour of materials under cyclic loading started in the mid-19th century as a first step towards preventing fatigue failures. At this time, welding had not been developed as a fabrication route and the investigations were mainly concerned with unwelded, or plain, material. As an understanding of how plain materials fail by fatigue is essential, in this chapter the principles of fatigue are discussed in relation to the performance of unwelded material. Chapter 7 demonstrates the dramatic (and negative) influence that welding has on fatigue performance.
14.5 Case Study: A Strengthening Method for Railways Bridges in Japan
In Japan, rail transport is a major means of passenger transport between major cities and metropolitan areas, and steel bridges have been widely used in the Japan railway system. However, as many of those bridges have been used in service for tens or even more than one hundred years, many of them need to be strengthened integrally for the whole bridges or repaired locally for certain steel members. Similar to Japan, a survey conducted about the railway bridges in Europe (covering over 220,000 bridges owned by 17 different railways) indicated that more than 35% of the bridges are more than 100 years old (Olofsson et al., 2005, 2007). With the purpose of reducing the stress levels and extend fatigue service life of aged steel bridges, a rehabilitation method for short-span railway, connection between longitudinal and lateral girder in plate girder bridge, and steel columns was proposed. This research was performed by Maebashi Institute of Technology, Waseda University, Taiheiyo Materials Corporation, and Railway Technical Research Institute.
14.5.1 Strengthening Method Description
The proposed method aims to be used for strengthening short-span steel railway bridge superstructures and longitudinal-lateral beam connection in plate girder bridges. The philosophy of the proposed method is to change the steel section to be the composite section by integrating with new materials, which in turn reduces stress ranges and extend the residual fatigue life of the aged structures. New materials including rubber-latex, rapid hardening concrete, reinforcement, and glass fiber-reinforced polymer (GFRP) plates are used. In the first step, the old structural steel is cleaned and then rubber-latex mortar is sprayed on the surface of the structural steel for protecting the structural steel from corrosion, increasing the bond strength on the steel-concrete interface, and reducing the noise in the service stage. Concrete and mortar, including styrene-butadiene rubber- latex, show various abilities especially in adhesion bonding, waterproofing, and shock absorption and abrasion resistance. In the second step, reinforcing bars and GFRP plates should be installed. GFRP plates are very easy to carry due to their light-weight. In this method, the GFRP plates are used as formworks for concrete casting. Then after that, the light-weight rapid hardening concrete is poured to finish the strengthening. For the maintenance or repairing of the railway bridge, rapid construction and light weight are the critical points. In this strengthening method, the concrete casting can be scheduled in the night time and will not affect the public traffic. The real effects of the present methods were confirmed by following application examples.
14.5.2 Application in Strengthening Short-Span Aged Railway Superstructures
The image of this strengthening method in strengthening short-span aged railway superstructures is shown in Fig. 14.11. By using the present strengthening method, the total procedures generally can be finished within 2 or 3 days without affecting the railway transport. In order to confirm the real effects of this method, both laboratory test and field test were performed.
Fig. 14.11. Strengthening of aged short-span steel railway bridges.
The old steel railway bridge used for laboratory tests in this study had been used in service for almost 100 years, which was originally built in 1912 with two longitudinal steel girders. The specimen was 4380 mm in length and was simply supported at a span of 3870 mm. After strengthening, the concrete thickness was 200 mm with a width of 1530 mm. GFRP plates with the thickness of 5 mm were used as formworks and shear connection devices with the concrete. Reinforcing bars of D13 nominal diameter were used for both longitudinal and transverse reinforcing bars in the concrete slab. Size dimensions of the test specimen are shown in Fig. 14.12. The strengthening sequence of the aged steel railway bridge is shown in Fig. 14.13.
Fig. 14.12. Size dimensions of test specimen for laboratory test (RBL-1& RBL-2).
Fig. 14.13. Strengthening of the laboratory test specimen. (A) Before strengthening. (B) Rubber-latex mortar coating. (C) GFRP and reinforcement arrangement. (D) Concrete casting.
The test specimen was supported by a roller system at two ends, with a loading beam in the span center. In order to check the reliability of the measuring equipment and the stability of the test specimen, two levels preloading of 50 kN, 100 kN were applied before the experiment was carried out. The static loading test set-up is shown in Fig. 14.14.
Fig. 14.14. Laboratory test set-up.
The load-displacement curve obtained from the numerical analyses was compared with the experimental data as shown in Fig. 14.15. The displacement was taken from the vertical deflection at the bottom mid-point of the composite bridge, which was the span center point on bottom surface of the longitudinal beam. Before strengthening, a simple test was performed to generate the load-displacement curve of the original old steel bridge (applied load was only up to 33.8 kN to avoid damage or plasticity of the old steel). For the old steel bridge that has been used around 100 years, fatigue is a major concern. Considering that the design loads are fixed values for railway bridges, so how to increase the bridge stiffness in service stage is the heart of this matter. And approximately 43.7% rigidity increase was also confirmed by comparing the slopes of the load versus displacement curves from the test results, indicating that the deformation of the bridge and the stress levels of the steel members can be greatly reduced. Therefore, service life of the aged steel bridge can be extended.
The noise problem is another concern in strengthening aged steel railway bridges. In Japan, there are a large number of such old steel railway bridges, which cannot be replaced by intrinsically more quiet concrete structures due to the increase in weight, cost, and construction height. Therefore, noise reduction was a strategic event in strengthening the old steel railway bridges. In this strengthening method, rubber-latex and concrete members were integrated with the old steel girders not only with the purpose of increasing its stiffness and reducing the stress levels but also with the aim of reducing the noise levels. In order to evaluate the noise reduction effect of the old steel railway bridge after strengthening, the hammer test was performed for the old steel girder and the hybrid girder after maintenance. Vibration accelerations were recorded by using accelerometers arranged on the web in the span center, as shown in Fig. 14.16.
Fig. 14.16. Impact test before and after strengthening. (A) Before integration. (B) After integration.
Noise measurement set-up for the old steel railway bridge before and after integration was shown in Fig. 14.16. As for steel girders the structure-borne noise was remarkable when the frequencies from 125 to 2000 Hz, sound pressure levels within this range were recorded. Besides, the all-pass value (AP) was also calculated within this range. The one-third octave filters results shown in Fig. 14.17 indicates that the sound pressure level of 5–15 dB was reduced after strengthening, and about 15 dB sound level reduction can also be confirmed for AP values.
Fig. 14.17. Impact noise before and after strengthening (about 15 dB decrease).
To confirm the real effects of the present strengthening method, an aged steel bridge being used in service was strengthened and tested. The steel railway bridge had been used in service for around 61 years, which was originally built in 1952 with four longitudinal girders. The bridge was 3850 mm in length and was simply supported at a span of 3160 mm. Concrete with the thickness of 200 mm, GFRP plates with the thickness of 5 mm, and reinforcing bars of D13 nominal diameter were employed for strengthening the old railway bridge. In this bridge, only two connecting beams were used at the ends between the longitudinal beams. The field test specimens were denoted by FT-1 (before strengthening) and FT-2 (strengthening), respectively. The size dimensions of the specimen are shown in Fig. 14.18. The aged steel railway bridge before and after strengthening and the strain gauges used in the field test are shown in Fig. 14.19.
Fig. 14.18. Size dimensions of test specimen for field test (RBF-1).
Fig. 14.19. Field test specimen before and after strengthening. (A) Before strengthening. (B) After strengthening. (C) Strain gauges.
In the field tests, six strain gauges in total were used on the steel girder in the mid-span section to measure the flexural strain. Four of them were attached to the lower surface of the bottom flange in mid-span section (ch.1–ch.4), while the other two were used on lower surface of the top flange in the mid-span of two outer girders (ch.5–ch.6), as shown in Fig. 14.18. Train 10,000 series and 11,000 series in Japan Railways were employed as the running trainloads in the field test. The deadweight of train 11,000 series is slightly heavier than that of the 10,000 series train, but the with similar passenger capacity. In the field tests, the flexural strain under five running trains (three of them belong to train 10,000 series and the other two belong to train 11,000 series) was measured on the aged bridge before strengthening. After strengthening, six running trains including three trains of 10,000 series and three trains of 11,000 series were employed in the tests.
The average maximum stress (the measured strain multiplied by the Young's modulus of structural steel) of each main girder under different running trains was summarized in Table 14.1. It is found that in all cases the maximum stress of the strengthened bridges was always much smaller than that of the unstrengthened bridges. For the bridge under the train 10,000, the tensile stress reductions of 21.1%, 8.7%, 42.3%, and 21.4% were confirmed in the four main girders from ch.1 to ch.4, respectively. For the whole section, the maximum tensile stress was reduced by 23.3%. According to MLIT codes, the residual service life of the strengthened bridges can be extended to 2.2 times of the original railway bridge (MLIT, 2009). The compressive stress reduction of 25.3% and 12.9% was also confirmed for the two top flanges. When subjected to train 11,000 series, the average maximum stress reductions of 16.8%, 1.6%, 39.5%, 18.1%, 17.9%, and 4.6% were also confirmed in chapters 1–6123456, as shown in Table 14.1. The reduction of stress ranges under live load can result in the greatly extension of the residual fatigue service life of the aged steel railway bridges, and the effects of the present strengthening method for short-span railway bridge superstructures were confirmed.
Table 14.1. Stress results on Flanges of the Aged Steel Railway Bridges
Specimen
Type
Average Maximum Stress (N/mm2)
ch.1
ch.2
ch.3
ch.4
ch.5
ch.6
FT-1
10,000
15.2
11.5
14.9
11.2
− 15.4
− 9.3
11,000
17.3
12.8
17.2
12.7
− 17.3
− 10.8
FT-2
10,000
12.0
10.5
8.6
8.8
− 11.5
− 8.1
11,000
14.4
12.6
10.4
10.4
− 14.2
− 10.3
Stress reduction (%)
10,000
21.1
8.7
42.3
21.4
25.3
12.9
11,000
16.8
1.6
39.5
18.1
17.9
4.6
14.5.3 Application in Strengthening Short-Span Aged Railway Superstructures
In steel plate girder bridges, the damage (like crack or fracture) frequently occurs on the longitudinal and lateral beam connections due to fatigue or corrosion, as shown in Fig. 14.20. For steel railway bridges subjected to large train impact and vibration, fatigue is more important than corrosion. Particularly in welded connections, fatigue damages have been frequently reported. On this background, the strengthening method proposed in this study was used to improve the fatigue performance of the connections between longitudinal and lateral beams in steel plate girder bridges. Rubber-latex mortar, GFRP plates, rapid hardening concrete, and reinforcements are used to enhance the stiffness, load carrying capacity, as well as the durability of connections in old steel railway bridges. To confirm the effects of the present strengthening method in strengthening such connections, a specimen was designed according to the real connection in an aged railway bridge in Japan. Each of the specimens was 2.1 m in length and was simply supported at a span length of 2 m. Vertical stiffeners were welded at support sections to prevent buckling failure before flexural failure. The typical geometry and design details of test specimen before and after strengthening are shown in Fig. 14.21.
Fig. 14.20. Strengthening of longitudinal-transverse beam connection in aged steel railway bridges (Lin et al., 2014c). (A) Rubber-latex mortar coating. (B) Set-up of GFRP plates & reinforcements. (C) Concrete casting.
Fig. 14.21. Size dimensions of the steel joint before and after strengthening. (A) Side elevation of the connection before strengthening. (B) Side elevation of the connection after strengthening. (C) A-A. (D) B-B section. (E) C-C section.
Depending on the railroad car axle locations, the longitudinal-lateral beam connection can be under either positive bending or negative bending moments. For this reason, two steel connections were used for this study, one connection was subjected to positive bending moment (denotes as SC-P-1 for original steel connection and SC-P-1 for the strengthened connection), and the other one was designed for negative bending moment (denotes as SC-N-1 for original steel connection and SC-N-1 for the strengthened connection). Before strengthening, a static loading test was performed on the original connection joint, the loading was stopped when the strain reached to 800 μ (about 70% of the yield strain) to avoid damage or plasticity of the structural steel. Thereafter, the steel joint was strengthened and loading tests were performed again to confirm the real effects of the present strengthening method. The set-up of the loading tests is shown in Fig. 14.22.
Fig. 14.22. Set-up of the loading test. (A) Before strengthening. (B) After strengthening-positive bending. (C) After strengthening-negative bending.
With the purpose of measuring the strain distribution on the welded steel connections, four 1-axis strain gauges and two 3-axis strain gauges were used on the web of the steel connection. Since diagonal fatigue cracks are mainly observed in web corners of the longitudinal beam, the principal (maximum or minimum) strains at those places (PS-1 and PS-2 in Fig. 14.21) are of primary interest in the strain measurement. Under the bending moment, the PS-1 is in tension while the PS-2 is in compression, thus the maximum principal strain at PS-1 and the minimum principal strain at PS-2 are discussed later.
Figs. 14.23 and 14.24 illustrate the principal strain results of the steel connection before and after strengthening. For the connection subjected to positive bending moment (SC-P), taking the applied load of 8 kN as an example, the corresponding maximum principal strain at PS-1 reduced from 588 μ in SC-P-1 to 18 μ in SC-P-2, indicating 97% reduction of the maximum tensile strain. On the other hand, the minimum principal strain at PS-2 reduced from − 562 μ in SC-P-1 to − 61 μ in SC-P-2, thus 89% reduction can be confirmed.
Fig. 14.23. Principal strain in the connection subjected to positive bending moment. (A) Maximum strain at PS-1. (B) Minimum strain at PS-2.
Fig. 14.24. Principal strain in the connection subjected to negative bending moment. (A) PS-1. (B) PS-2.
When subjected to negative bending moment, the principal strain in the original and strengthened steel connection was shown in Fig. 14.24. Taking the applied of 7 kN as an example, the maximum principal strain at PS-1 reduced from 356 μ in SC-N-1 to 55 μ in SC-N-2, and the minimum principal strain at PS-2 reduced from − 504 μ in SC-N-1 to − 7 μ in SC-N-2. Therefore, approximately 85% and 99% reduction in the principal strain were confirmed at PS-1 and PS-2, respectively.
Considering the relationship between the stress range (or magnitude) and the stress cycles, the residual fatigue life of the steel connection can be greatly extended, thus the effects of the present method for strengthening such connections can be confirmed. In addition, this method is also applicable for repairing or strengthening of other structures because of the easy obtained construction materials such as GFRP, concrete, reinforcing bars, as well as the easy operational approach.
Adhesive traction: Torque and tractive effort increase with the current. Tractive effort is unlimited, theoretically, since the power supply from the central stations is, for practical purposes, unlimited but a limitation is the point at which the drivers begin to slip, as determined by the factor of adhesion. The hatched line in Fig. 2.3 is the tractive effort curve limited by the adhesive force between wheel and rail, and it is calculated according to Eqs. (2.1), (2.2).
Traction force is limited by permitted current of motor: a second practical limitation is enforced by the heating of the motors at full load and high current value. As the heating does not occur at once, it is permissible to utilize the overload possibilities of the motors for a limited period. At least two ratings are usually specified: the continuous and the hourly.
The continuous rating as prescribed by the Chinese Institute of Electrical Engineers is that maximum tractive effort which the locomotive can exert continuously without harm to the motors. It is not the maximum load the locomotive can handle, but it is the heaviest load that can be handled continuously with safety. The continuous rating is based on that continuous current and corresponding tractive effort which keeps motor temperature under 60°C (160°C with silicon windings), starting with a cool (25°C) motor.
The hourly rated tractive effort is based on that current and tractive effort which can be exerted for 1 h without harm to the motor; that is, the motor temperature is elevated by 60°C in 1 h, provided that the motors are cool when the overload is applied.
Much higher currents may be taken for shorter periods of time (such as 1 min), especially in the beginning and temperatures of 100°C may occasionally be reached without harm to the windings. When this operation occurs, measurements such as stops, coasting grades, or light pull below continuous current ratings must be applied immediately to permit the motors to cool.
The oblique line AB in Fig. 2.3 represents the continuous rating.
1)
Traction force under different speeds
“Chinese traction calculation procedures” (TCP) stipulate that tractive effort of electric locomotive is to be taken via a continuous rating system. For example, in the SS3 locomotive, if the tractive effort for a given speed is known, the tractive effort of any other speed can be determined as follows: in the acceleration process (accelerate from 0 to locomotive construction speed), the tractive force takes adhesive traction (CD), and motor traction (DA, BE section), respectively, and the tractive effort determined by continuous rating (AB).
2)
Calculated speed and calculated tractive force
Calculated speed is important for the calculation of the tractive mass of train, its corresponding tractive force is called calculated tractive effort. TCP take the speed and traction determined by the sustained current of the full-field traction characteristics at the highest step as the lowest calculated speed and maximum calculated traction. The corresponding speed and tractive force at point A in Fig. 2.3 are the lowest calculated speed (vjmin) and maximum calculated tractive force (Fjmax) of SS3 electric locomotive, respectively.
3)
Starting tractive effort
Starting tractive effort is the maximum tractive effort which is provided by locomotive under the starting status. A locomotive’s starting traction stipulated by TCP is determined by calculating or specially testing based on certain restrictions, and is called calculated starting traction effort (TEs). The calculated traction effort of most freight locomotives is restricted by adhesion conditions. Calculated traction of passenger locomotives is restricted by starting current. In the calculation of SS3 locomotives under starting condition, starting calculated traction effort (TEs) is equal to adhesion traction effort when speed is zero as. A few locomotives use limited current traction effort as calculated starting traction effort.
4)
Electric locomotive characteristics
The value parameters of frequently used Chinese electric locomotives are given in Table 2.1.
In Table 2.1, vcmin is the minimum calculated speed, Tcmax is the maximum tractive effort, Ts is the starting tractive force, P and Pμ is the locomotive axle load and the locomotive adhesion axle load, and Lloc is the length of the locomotive.
Parking maneuvers take place at very low or zero speed. The torque acting on the tire at such conditions may become very large. The influence of the finite tread width is essential as the response to spin is now predominant. We might employ the equations developed above but then we should take care of the integration of the spin velocity to properly limit the buildup of the yaw transient slip. Similar problems arose when considering the problem of braking to standstill or starting from standstill, cf. Section 8.6, Eqns (8.112, 8.113)Eqn 8.112Eqn 8.113.
To achieve a much better agreement with experimental evidence, a different approach will be followed in the present application. It may be noted that an important characteristic is actually still missing. For the brush based model, Eqn (9.70) was used. The equation governs the variation of the aligning torque Mz that arises when the nonrolling tire is steered and the steer angle ψ is increased from zero to and beyond the state of full sliding. Ultimately, the torque reaches the magnitude that would also arise when the rolling tire is subjected to a constant rate of turning dψ/dt (while the slip angle remains zero) at a forward speed Vx that decreases to zero. Then, the radius of turn R reduces to zero and thus the spin approaches infinity. Figure 9.21 illustrates the situation.
FIGURE 9.21. Approaching the maximum torque at standstill in two ways: 1. by decreasing the turn radius R to zero and 2. by increasing the steer angle −ψ while standing still.
The missing characteristic will be modeled by using a for-this-purpose adequate model that has been developed by Van der Jagt (2000). In his dissertation a model study was discussed that is especially aimed at the generation of a proper moment response to steering at very low or zero speed. First, the brush model was used to gain general insight into the phenomena that occur. Qualitatively good results have been obtained using this model, notably when a sinusoidal steer angle variation is imposed and the state of almost full sliding is attained periodically. For practical usage, a special type of model was developed of a nature completely different from the models used so far. Since this model appears to perform very well in the near zero speed range, we have tried to incorporate Van der Jagt's model in the existing model structure. For a gradual transition from the new type of model to the existing one, when the speed approaches and surpasses a low-speed threshold has been taken care of.
The principle of Van der Jagt's approach is that at a given rate of change of the steer input the growth rate of the tire angular deflection β decreases in proportion to a function of the remaining difference between the maximum achievable deflection and the current deflection. The torsional stiffness is assumed to be a constant and the resulting characteristic of the torque becomes similar to a first-order response function. The calculated moment gradually approaches its maximum value. When the direction of rotation of the wheel about the vertical axis is changed, the distance to the new, opposite, peak torque is large and, accordingly, the rate of reduction of the moment is large as well. It is this feature of the model that is attractive since a similar behavior has been found to occur with the actual tire subjected to an alternating left and right sequence of turning. The equations that govern the moment generation at standstill are as follows:
(9.121a)
(9.121b)
(9.121c)
For the parameter value co = 2, Figure 9.22 presents the calculated variation of the torque vs the steer angle compared with experimentally obtained results as reported by Van der Jagt. The nonrolling tire (size P205/65R15) is loaded to 4800 N on a flat plate and subsequently steered at a rate of + and −1 deg./s. The correspondence is quite good except perhaps for the initial phase where the wheel starts to be steered from the condition where Mz = 0. To improve the model performance Van der Jagt suggests using an exponent co, the value of which depends on the last extreme of the deflection angle β. For possible further refinements of the model, we refer to the original work.
When, instead of the new approach, the Magic Formula would be used with the integration limitation as suggested according to Eqns (8.112, 8.113)Eqn 8.112Eqn 8.113, a sharp peak would arise in the curve where the direction of turning is changed. As a result, the moment decreases at a much slower rate than shown by the test result.
The problem is now how to integrate the new model feature in the original model structure. The transient slip quantity , Eqn (9.86), may be recognized to be proportional to the deflection angle. As can be seen from Eqns (9.80, 9.82, 9.83, and 9.86) this quantity is obtained through integration of
(9.122)
In the new configuration, the integration is conducted at a gradually decreasing rate while approaching the maximum torque value. We have
(9.123)
(9.124)
At zero speed wVlow = 1. The moment is found with the linear function, cf. (9.104):
(9.125)
For the standing tire with speed Vx equal to zero, the response to alternating steer angle variations will follow a course similar to that of Figure 9.22.
FIGURE 9.22. Calculated and experimentally assessed variation of the moment vs steer angle for a nonrolling tire pressed against a flat plate at a load Fz = 4800 N.
It is now desired to change gradually to the original equations when the tire starts rolling. The transition is accomplished by adding up the following two components. The first one decreases in magnitude with increasing speed until it vanishes at Vx = Vlow while the second part increases from zero to its full value also at Vx = Vlow. For the gradual change, the following speed window is used:
(9.126)
With this quantity (already used in (9.123)), the first part that prevails at low speed becomes
(9.127a)
and the fraction obtained from the original (here simplified) Eqns (9.105 and 9.104)
(9.127b)
The resulting expression for the spin moment now reads
(9.128)
A similar method may be employed to improve the low-speed model for the side force responding to lateral motions of the contact patch (cf. Figure 9.29, point S) and for the fore- and-aft force to longitudinal motions of the same point S.
The adapted model will now be applied to the simulation of the motion of a rigid quarter car model with mass mqc that, while a sinusoidal steering input is applied, starts moving after 1.6 s with a linearly increasing speed. The lateral acceleration of the quarter car axle results from the action of the side force that begins to develop after the wheel has started to roll:
(9.129)
The lateral wheel slip velocity is now not only a result of the yaw angle at a forward speed of the vehicle but also due to the lateral velocity of the wheel axle . We have
(9.130)
which serves as an input into the Eqns (9.88 and 9.91). The additional parameter values have been appended in Table 9.2.
Figure 9.23 shows the courses of variation of various quantities vs time. Simultaneously, in Figure 9.24, the moment is plotted vs steer angle. Several phenomena occur that deserve to be noted. The steer angle has an amplitude that is large enough to attain a level of the moment close to its maximum. The moment starts to decrease in magnitude as soon as the steer angle passes its peak value. The moment changes sign before the steer angle does the same. After 1.6 s, the forward speed increases linearly with time and the side force starts to build up as a result of the slip angle that begins to develop. The car shows a lateral vibration in the low speed range as indicated by the fluctuations of the side force. Evidently, the quarter car vibrates against the lateral tire stiffness. The moment amplitude decreases as the spin diminishes in amplitude due to the increasing speed. The side force amplitude increases because of the larger lateral oscillations of the quarter car mass induced by the increasing speed of travel at the constant steer input pattern with time. The loops shown in Figure 9.24 give a nice impression of the transition from the situation at standstill to the condition at higher speeds. At standstill the moment varies in accordance with the diagram of Figure 9.22.
FIGURE 9.23. Simulation results of a parking maneuver (car leaving the parking lot while steering sinusoidally).
FIGURE 9.24. The steer torque plotted vs steer angle during the maneuver of Figure 9.23.
As mentioned before, to get a more accurate calculation of responses to lateral and circumferential wheel displacements at or near forward speed equal to zero, one might apply, instead of the abrupt integration limitation suggested earlier, Eqns (8.112) and (8.113)Eqn 8.112Eqn 8.113, the same structure of additional Eqns (9.123) and (9.124), and an adaptation such as achieved in Eqn (9.128).
