The sine run, also called wave propagation, occurs at the wheel-rail systems with conically profiled, so tapering outwardly, rigidly coupled wheels. The purpose of the taper is the cornering without noise and wear caused by wheel flange contact: In curves, the offset to the outside wheel rolls with a larger scale on the rail from as the offset from the track center wheel, which is opposite to the outer remains, so that the axis turns the steering wheel in the curve. Transition curves allow a proportional development of offset and track curvature. Deviations from the ideal line will be more than offset by this principle of guidance, so that a rolling movement is generated.
This movement is sinusoidal at low amplitudes and has a constant wavelength, so increasing the speed with the frequency. It is operated at high speed fueled by dynamic forces (English hunting oscillation ) to the zig -zag run with the stop of the wheel flanges on the rails, so measures must be taken for damping to avoid excessive wear and loss of comfort.
Neither case of cylindrical wheel profile, typical of trams, even at individual wheels, such as the Talgo, the sine overflow occurs. The former take quite loud noises around tight corners, if the wheel flange lubrication is not sufficient, while the wheels are aligned in the bisector curves between the cars at the Talgo.
In a wheel with a conical wheel profile, which is off-center on two parallel rails, the radii of the two wheels on the points of contact with the rails are of different sizes. Since the two wheels are rigidly connected by the axle and rotate at the same speed, performs the wheel with the larger radius at the point of contact for a longer way forward as the wheel with the smaller radius. Therefore, a too far right standing wheel steers to the left, and vice versa. This causes a yaw movement which can be represented in a non-linear differential equation. In the limit of small amplitude is calculated as the solution to a sine curve. The formula was published in 1883 by John Bell.
In reality, the profiles are not exactly conical due to wear or are the same as a closure profile in order to avoid a sharp change of the running behavior due to wear. This combined with the above neglected finite curvature of the rail section to a non-linear dependency of the rolling radius of the lateral offset. If the contact points do not jump when changing the offset, this dependence can be approximated linearly for small deflections, see Equivalent conicity, so the above derivation remains valid.
The friction between wheel and rail, especially with rigid coupling of two wheelsets in a bogie, dampens the vibration and stabilizes a smooth run, while with increasing speed, the excitation of the oscillation is stronger. Outweighs the loss, returns a set of wheels, which has fallen into an off-center position by an error in the track position or by a curve, after a few periods back to the center position back. Outweighs the excitation, as the amplitude increases, until it is limited by increased friction at the flanges. The high speed large restoring forces disrupt passengers and load vehicle and track superstructure.
A Radsatzpaar in a bogie reached an uninterrupted run to 180 km / h; for higher speeds the wheel profile can be selected flatter ( see Equivalent conicity ) and the yaw motion of the truck against the car body are additionally damped (eg rotary damper on the bogie MD 36 or high-manganese plates on the bogie MD 522).
For an exactly sinusoidal run it only comes in a simplified model; the actual path is also influenced by the fact that a normal wheel usually does not exactly conical wheel profiles and optimized for the cone and flange system rail profile shape does not run exactly at all points, as well as unregulated forces, the on bogie or carbody be initiated. However, it can adjust the running of wheelsets technically so accurate that only wobbling movements occur in the millimeter range. For a better running performance and lower wear is achieved than with wheel sets with cylindrical wheel profile.
The maximum permissible speed of conventional rail vehicles is significantly influenced by the precise matching of wheel and rail profile and ensuring the continuous monitoring and improvement.
When the track curves with many narrow radii, the advantages of a conical wheel profile with sinusoidal running hardly come to fruition. Therefore, vehicles are often used with an easier to manufacture cylindrical wheel profile for trams, funicular railways and subways.