Presentation on theme: "Lecture #22 FORCE SYSTEM ASSOCIATED WITH SPUR GEARS Course Name : DESIGN OF MACHINE ELEMENTS Course Number: MET 214."— Presentation transcript:
Lecture #22 FORCE SYSTEM ASSOCIATED WITH SPUR GEARS Course Name : DESIGN OF MACHINE ELEMENTS Course Number: MET 214
To design a power transmission system involving gears, it is necessary to relate the torque and/or horse power applied to a shaft to the forces acting on the teeth of the gears connected to the shaft. To analyze the force system existing with the use of gears, a brief overview will be presented concerning how the point of contact existing between the teeth of mating gears evolves as the gears rotate. Understanding how the point of contact existing between the teeth of mating gears evolves as the gears rotate will enable the forces acting on the gears to be related to the torque and/or horse power applied to the shafts connected to the gears. After the geometry of the force system has been quantified to the extent necessary to relate horse power and/or torque to forces transmitted between gears, a brief overview will be presented concerning the profile that gear teeth can use to transmit the forces in the manner described.
Identifying the requirements for a gear tooth from the context of the force system existing between meshing gears prior to investigating the tooth profile facilitates understanding of why certain profiles are preferred for use as gear teeth. To investigate how the forces are typically transmitted between a pair of gears, consider the figures shown below. Each figure is provided with a caption describing certain features exhibited by the gears as the pinion is rotated.
Based upon the figures provided on the previous slide, as gear #1 (pinion) rotates tooth A1 which is attached to gear 1 will subsequently pass through the positions previously occupied in figure 1 by tooth B1 and then tooth C1. As is evident from figures 1-3 as gear #1 rotates, the point of contact existing between tooth A1 and tooth A2 on the gear varies. As is further evident from the figure, if all of the points of contact generated between tooth A1 and tooth A2 are connected, a line results that is referred to as the line of action or the path of contact. As will be shown, having all points of contact existing between the teeth of meshing gears lie on the same line enables the gears to retain properties of the friction wheel alternative. As will be shown, the profile chosen for the gear teeth is fundamental to enabling the points of contact existing between the teeth of meshing gears to evolve along a line. The forces transmitted between a pair of gears is transmitted between the teeth of the gears at the points of contact. As evident from the figures shown above, more than one pair of teeth may be in contact at any particular orientation of the gears. In the figures shown above, 3 pairs of teeth are in contact at each orientation shown. Having more than one pair of teeth in contact distributes the load to be transferred via the gears relieving the demands placed on a single tooth.
If the forces transmitted between the teeth of meshing gears are transmitted along the line of action at every point of contact, then regardless of the angular position of the gears, the forces transmitted between the gears maintain a fixed orientation in space. Maintaining a fixed orientation in space for the forces to be transmitted between gears enables the power transmitted between the gears to be independent of the angular position of the gears. This is a very desirable characteristic for gears to possess. To quantify the relationships existing among forces transmitted between gears with gear geometry including the diameter of the pitch circle, it is essential to define the orientation of the line of action relative to the geometry associated with meshing gears. To form a reference for specifying the orientation of the line of action, draw a line connecting the centers of the gears. The line connecting the gear centers will be referred to as the line of centers (see figure below). Next, draw a line perpendicular to the line of centers through the pitch point existing with the friction wheel implementation. The pitch point is the point of contact existing between the pitch circles associated with the gears. The pitch circles are indicative of the diameter that a friction wheel implementation must have in order to produce the same velocity and/or torque ratios that the gears will produce.
In light of the above relationships, considered just the base circles of a pair of gears in mesh and connect the two base circles by wrapping a thread around a portion of each base circle as shown in the figure below. Although the illustration provides additional features, the additional features will be ignored for the moment so the focus can be directed toward the motion existing along the line of action. As noted previously, the line of action is tangent to the base circles as shown.
If the tooth profile is selected properly, the line of action will also pass through the pitch point. Consequently the forces transmitted along the line of action can be slid along the line of action until either the tip or the tail of the force vector transmitted between the gear teeth coincides with the pitch point. The principle of transmissibility permits the forces to be relocated in this manner. The ability to translate the force vector along the line of action to the pitch point for any orientation of the gear due to all points of contact lying on the line of action enables the force system associated with the gears to be conveniently related to the power transmitted between the gears. The constructions involving the line of centers, the pitch point, the line perpendicular to the line of centers through the pitch point and the line of action are shown in a new figure which also includes a pair of teeth to emphasize that the forces transmitted between gears are due to the teeth contacting one another in a very prescribed manner to be discussed in more detail shortly. The illustration provided below locates the driver on the bottom of the gear pair which is different than the previous illustrations. Note also that the driver is rotating in a clockwise direction.
In order to properly resolve the forces acting along the line of action into components so bending stresses, direction of resisting torques,shaft size, deflection, etc, can be calculated properly, it is essential to realize that the above diagram illustrates the forces applied by the driver to the driven gear.
As will be discussed in more detail, the force components applied to the driver by the driven gear are in the direction opposite of the force components shown on the driven gear of the previous diagram. The force components applied to the driver gear by the driven gear represents the reaction of the driven gear in response of the actions of the driver gear acting on the driven gear. In order to reinforce the relevance of the sign for the pressure angle, note that the direction of rotation of the driver is clockwise. In addition, note as the gears rotate and the tip of the arrow representing the direction of rotation of the driver converges toward the pitch point ( pitch point not explicitly shown on the diagram) the arrow representing the direction of rotation of the driver will merge with the line of action indicating the direction of the force the teeth of the driver gear will impart to the teeth of the driven gear. Since the emphasis of the drawing is in the context of the driver gear, the forces transmitted from the teeth of the driver to the teeth of the driven gear will be along the line of action in the direction of the arrow representing the direction of rotation of the driver when merging with the line of action. To emphasize these consideration of the force system existing with a gear pair, a diagram showing more detail of the situation is provided below.
Although the direction of rotation of the driver can be used with a properly orientated line of action to establish the direction of the force and/or the force components acting on the driven gear as explained on the previous slide, a simpler technique involves the following. Note that if the pitch circles of the gears were envisioned as a rolling mill, then a piece of paper would flow though the rolling mill in only one direction.
Recall the following relationship existing between speed, torque, power. where power watts torque N-m angular velocity rads/sec Alternatively, where power, H.P. torque in-lbs angular velocity in rpm Using geometry associated with the pitch circles: where tangential force component of force w n being transmitted between gears radius of pitch circle
Substituting for the expressions of torque in terms of power levels to the following equations where w t = lbs Power can also be expressed in terms of wt and the pitch line velocity vt. where power in watts force N velocity m/s or in terms of customized units wherevelocity in ft/min force in lbs power in HP
The above expression can be rearranged to solve for w t in terms of customized units Accordingly, to relate horsepower and/or torque applied to a shaft to the forces acting on gears, calculate F t from HP or torque using the formulas provided above. After F t is calculated, determine F r and F n provided previously. After all force components are known, shear and moment diagrams, torque diagrams and other analysis tools can be employed (angle of twist, deflection) to assist in designing a shaft for use with gear systems. To reinforce the geometry involved in analyzing the force system associated with the use of gears, consider the diagram shown below.
Note: Fr serves to separate the shafts connected to the gears and for this reason Fr is sometimes referred to as the separating force. It should be noted, a distinction must be made as always, between the forces and/or torques acting on the gear due to the pinion as opposed to the reaction forces acting on the pinion due to the gear. The reaction forces acting on the pinion are equal and opposite to the acting on the gear due to the pinion. The determination of the driver and the driven gear are made of an analogous basis provided for pulleys. Accordingly, the procedure for determining the driver and driven gear of a pair will not be presented. The subsequent lecture will be devoted to working problems to reinforce the material presented herein.