Transmission Rack And Pinion

[Thora Buckner]

Arack and pinion is a type of linear actuator that comprises a circular gear (the pinion) engaging a linear gear (the rack). Together, they convert between rotational motion and linear motion. Rotating the pinion causes the rack to be driven in a line. Conversely, moving the rack linearly will cause the pinion to rotate. A rack-and-pinion drive can use both straight and helical gears. Though some suggest helical gears are quieter in operation, no hard evidence supports this theory. Helical racks, while being more affordable, have proven to increase side torque on the datums, increasing operating temperature leading to premature wear. Straight racks require a lower driving force and offer increased torque and speed per fraction of gear ratio which allows lower operating temperature and lessens viscal friction and energy use. The maximum force that can be transmitted in a rack-and-pinion mechanism is determined by the torque on the pinion and its size, or, conversely, by the force on the rack and the size of the pinion.

The rack carries the full load of the actuator directly and so the driving pinion is usually small, so that the gear ratio reduces the torque required. This force, thus torque, may still be substantial and so it is common for there to be a reduction gear immediately before this by either a gear or worm gear reduction. Rack gears have a higher ratio, thus require a greater driving torque, than screw actuators.

A rack and pinion is commonly found in the steering mechanism of cars or other wheeled, steered vehicles. Rack and pinion provides less mechanical advantage than other mechanisms such as recirculating ball, but less backlash and greater feedback, or steering "feel". The mechanism may be power-assisted, usually by hydraulic or electrical means.

The use of a variable rack (still using a normal pinion) was invented by Arthur Ernest Bishop[2] in the 1970s, so as to improve vehicle response and steering "feel", especially at high speeds. He also created a low cost press forging process to manufacture the racks, eliminating the need to machine the gear teeth.

Rack railways are mountain railways that use a rack built into the center of the track and a pinion on their locomotives. This allows them to work on steep gradients, up to 45 degrees, as opposed to conventional railways which rely on friction alone for locomotion. Additionally, the rack and pinion addition provides these trains with controlled brakes and reduces the effects of snow or ice on the rails.

A rack and pinion with two racks and one pinion is used in actuators. An example is pneumatic rack-and-pinion actuators that can be used to control valves in pipeline transport. The actuators in the picture on the right are used to control the valves of large water pipeline. In the top actuator, a gray control signal line can be seen connecting to a solenoid valve (the small black box attached to the back of the top actuator), which is used as the pilot for the actuator. The solenoid valve controls the air pressure coming from the input air line (the small green tube). The output air from the solenoid valve is fed to the chamber in the middle of the actuator, increasing the pressure. The pressure in the actuator's chamber pushes the pistons away. While the pistons are moving apart from each other, the attached racks are also moved along the pistons in the opposite directions of the two racks. The two racks are meshed to a pinion at the direct opposite teeth of the pinion. When the two racks move, the pinion is turned, causing the attached main valve of the water pipe to turn.[3][4]

Rack-and-pinion drives are the preferred option in the machine tool sector when long ranges of motion and high loads are involved. However, their shortcomings particularly include deficiencies in the achievable positioning and path accuracy. The backlash as one of the main issues is well described in the literature and numerous solutions to reduce its negative effects exist. In contrast, there is a lack of literature regarding the scientific and systematic analysis of the transmission errors in rack-and-pinion drives. In this paper, the displacements originating in the drive train of a system with industrial components are measured under different operating conditions. The observed transmission errors are thoroughly analyzed in no-load operation and their sources are discussed. Subsequent investigations show significant load-dependent alterations of the transmission errors and direction-dependent characteristics, the causes of which are explained. It is shown, that transmission errors negatively affect the path accuracy of position controlled drives, which is amplified by excitation of the machine structure in certain operating conditions. To address this issue, different error compensation concepts are presented.

Modern manufacturing equipment is expected to deliver high production quality coupled with high dynamic performance. Both properties are largely determined by the installed drive systems. In addition to the achievable feed forces, they also define the accuracy and the static and dynamic rigidity. Rack-and-pinion drives (RPDs) are the preferred choice for applications with long travel distances and high loads [1]. The stiffness of these drive systems is independent of the travel distance and since only stationary rack elements are added to increase the axis length, whereas the inertia moved by the drive remains unchanged, arbitrarily long travels can be realized without inhibiting the dynamics [2]. This high scalability in combination with economical implementation make RPDs particularly suitable for heavy machinery [3]. Nevertheless, they also have some specific downsides. Of particular relevance for machine tools is the inferior positioning and path accuracy compared to other drive types [2]. A major issue limiting the positioning accuracy is backlash, which has a negative effect on both static and dynamic performance. However, the literature offers various approaches to eliminate its negative effects to a great extent by utilizing mechanical [3] or electrical [4] preload or designated model-based control strategies [5], that can also factor in elasticity [6].

In consequence, other sources of error become more apparent. One issue that mainly influences the dynamic accuracy and causes deviations during trajectory tracking is the transmission error (TE) of RPDs. TEs are defined as the deviations of position and velocity that occur between the input and output of gearings [7]. In the case of RPDs, such errors are present both in the meshing of the rack and pinion and in other gearing in the drive train. The result are periodic position differences between the drive motor and the table that can lead to vibration excitation and unsatisfactory surface finish of the workpieces. TEs are comprehensively described for gearwheel pairings [8], including vibration research [9], their minimization through design improvements [10] and implications for planetary gears [11]. For rack-and-pinion gearings literature regarding simulative studies utilizing FEM tools [12] as well as analytical models [13] exist. In addition, studies concerning automotive steering systems [14] and variable compression combustion engines [15] are available. However, in the context of position-controlled feed drives with RPD, there is hardly any literature to date that provides experimental data concerning the impact of TEs on the path accuracy and vibration excitation of the drive mechanics. As a consequence, approaches for their compensation are uncommon. This paper is intended to contribute to adress this gap.

Therefore in Sect. 2 a setup with industrial components for experimental investigation is presented. Section 3 analyzes the TEs of this RPD in detail under different operating conditions. Following this, the effects on the path accuracy of the controlled drive are examined in Sect. 4, involving the elaboration of potential excitations of the machine structure induced by the drive train in Sect. 4.1. Subsequently concepts to compensate for the deviations from a control engineering perspective are presented in Sect. 5.

To investigate the transmission uniformity of RPDs, an exemplary system that is comparable to common industrial implementations is used as a test bench. Figure 1 outlines the schematic structure of such a system with RPD and Fig. 2 shows a picture of the corresponding test bench utilized in this paper.

The servomotor actuates the pinion through a high-precision two-stage planetary gearbox. The linear motion is subsequently provided through the meshing of pinion and rack. To enhance smoothness of movement, gearing with a helix with an incline \(\beta\) of 19.5283\(^\circ \) is used. The drive is guided by two guide rails and the mass \(m_T\) of the table with the drive system is 420 \(\hbox kg\). The components used are listed in Table 1 and relevant parameters can be taken from Table 2.

A specific feature of the test bench is a linear direct drive (LDD) mounted in the center of the table in parallel to the feed drive. This allows to apply forces to the RPD and thereby simulate loads.

The central control unit is a Siemens CU320-2 with corresponding inverters for motor and LDD. The position control of the drive is done by a cascade control. Figure 3 illustrates such a configuration, as it is used in the setup under consideration. The structure consists of three interconnected control loops [16]. While the innermost current control loop has negligible influence on the system characteristics, the velocity and position control loops are decisive for the properties of the drive [17]. The velocity controller regulates the rotational velocity of the drive motor \(\dot\theta _M\) via the desired motor torque \(T_s\). The position controller, obtaining the target position \(x_s\), provides the desired table velocity \(v_s\). In case of an indirect position control the control loop is closed through feedback of the motor position \(x_M\). While this configuration has the benefit of reduced cost due to the lack of need for additional measuring systems, the missing feedback of the table position means that the position control can only react inadequately to disturbances affecting the table, such as friction and process forces. In the area of machine tools with high demands on positioning accuracy, a linear measuring system is therefore commonly added to allow for a direct position control of the table position \(x_T\) [2]. In the setup under consideration, an absolute position measuring system integrated in the guide rails is used. For improved trajectory tracking the position controller is supplemented with a velocity feed-forward controller [16]. The controllers are parameterized for the experiments according to the common setting rules used in the machine tool sector. The velocity controller with the proportional gain \(K_p\) and the time constant \(T_n\) is set according to the principle of the symmetrical optimum and is computed at 8 \(\hbox kHz\). The position controller operates at 1 \(\hbox kHz\) and the gain \(K_v\) is chosen in such a way that an overshoot-free positioning is achieved (aperiodic loop, damping ratio \(\zeta = 1\)) [17]. The corresponding parameters are listed in Table 2.

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