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Research Papers: Mechanisms and Robotics

# Improving Machine Drive Dynamics: A Structured Design Approach Toward Balancing

[+] Author and Article Information
B. Demeulenaere, R. S. Berkof

Mechanical Engineering Department, Katholieke Universiteit Leuven, Leuven B-3001, BelgiumMechanical Engineering Department, Stevens Institute of Technology, Hoboken, NJ 07030

Reuleaux (4) defined a mechanism as follows: “A closed kinematic chain of which one link is … made stationary, is called a mechanism. … In general, therefore, a … closed kinematic chain can be formed into a mechanism in as many ways as it has links. … the kinematic chain is compound or simple, and consists of kinematic pairs of elements.”

VDI, the German association of engineers.

The VDI guideline (3) recommends that measures be taken to reduce the input torque, shaking force, and∕or shaking moment if the ratio $amax∕g$ of the maximum acceleration of a mechanism’s output link to the gravity acceleration $g$ exceeds 1. Such an output link could be, for instance, the follower of a cam-follower mechanism or the rocker of a crank-rocker four-bar.

Fundamental concepts of shaking force and shaking moment balancing are provided in the work of Berkof and Lowen (17-18).

“The best model of a cat, is a cat—preferably the same cat,” attributed to N. Wiener (1894–1964), US mathematician.

While torque-current models for dynamic operating conditions are well known for dc permanent-magnet servo motors, they are not for induction motors. For the latter case, two such models are derived and experimentally validated in Ref. 30.

This is a translation, provided in the bilingual German-English VDI guideline (3), of the German Eigenbewegung, similar to the translation eigenvalue of Eigenwert.

Antibacklash gears are two gears back to back on the same shaft that can be slightly rotated at assembly (or by springs) with respect to one another, so as to take up the backlash (42).

These precision points lie on the path generated by a point on the coupler of a reference four-bar, described by Berkof and Lowen (17). The dynamic forces in the newly designed four-bar, passing through the precision points, are compared to the dynamic forces in this reference four-bar

Arakawa et al. (54) and VDI guideline (3) also proposed the use of pneumatic or hydraulic springs, of which the spring constant can be adjusted to the actual (average) drive speed, which is a possible, though expensive, option

Kirchof (11) pointed out that a cam-follower mechanism, of which the follower works against a spring, is probably the oldest balancing mechanism, as it was used as such in the knitting machine devised by William Cotton of Loughborough, Leicestershire (England) in 1864. This machine had heavy lead cams mounted on a camshaft of about $20m$. The followers were assured to have contact with the cams by heavy springs. However, in order to turn the camshaft by hand (which was required every now and then), Cotton provided balancing cams, also working against a spring. These cams were designed in a rather heuristic way (11), such that the statical input torque (that is, the input torque to overcome the spring forces) was zero. This was, however, not the only advantage. It was experimentally observed that the dynamic performance also substantially improved, since the fluctuation of the driving torque substantially decreased at the nominal speed. Consequently, a smaller motor could be used.

That is, the perpendicular distance between the line of action of the slider and the crank pivot.

The mechanical advantage of a mechanism is the ratio of the output force over the input force.

Norton (33) provided an excellent overview of the more classical input torque balancing methods in the area of multicylinder combustion engines.

J. Mech. Des 130(8), 082302 (Jul 10, 2008) (9 pages) doi:10.1115/1.2920473 History: Received July 06, 2007; Revised January 11, 2008; Published July 10, 2008

## Abstract

Linkage and cam-follower mechanism design are often based on the simplifying assumption of constant drive speed. Usually, however, the mutual interaction between the mechanism and its actuator gives rise to a fluctuating drive speed, possibly resulting in a variety of dynamic problems. This paper provides machine designers and researchers with a structured design approach toward solving such problems. As such, it constitutes both an extension and a complement to the 1979 input torque balancing survey by Berkof.

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## Figures

Figure 1

Ensemble of four cam-based centrifugal pendula, proposed by Beschkine in the 1939 French patent (6). The curves Si denote internal cam profiles, which impose some prescribed motion to the pendula Mi. Revolute joints connect the pendula to the driving shaft.

Figure 2

Two cam-based input torque balancing mechanisms proposed by Artobolewski in the 1958 Maschinenbautechnik paper (7): a cam-follower mechanism with a translating roller follower (left) and its kinematic inversion (right).

Figure 3

(a) torsional pendulum vibration absorber, where k denotes the torsional spring stiffness and J the flywheel inertia; (b) centrifugal pendulum vibration absorber (CPVA), where the rotor o1o2 imposes the rotation of the pendulum link o2o3 in the centrifugal field. The Z-axis is parallel to the drive shaft.

Figure 4

Simple toggle. As the angle α approaches 90deg, the links come into toggle and the mechanical advantage approaches infinity. However, frictional effects reduce the forces to much less than infinity although still quite high.

Figure 5

Experimental validation of the cam-based centrifugal pendulum, a cam-based input torque balancing mechanism displayed in Fig. 1, while designed and experimentally validated by Demeulenaere (65): one period of measured torque (solid line) and theoretical torque (dashed line) Mc (N m) for Ω=200,250,300,413.5rpm. While the mechanism functions almost as predicted at Ω=200,250rpm, the response is heavily dominated (peak amplitude of 2280Nm, that is, three times the predicted amplitude) by a resonance phenomenon at Ω=413.5rpm. This resonance phenomenon is due to the addition of a significant amount of rotational inertia to an axis that is not perfectly stiff.

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