Date on Master's Thesis/Doctoral Dissertation


Document Type

Master's Thesis

Degree Name



Mechanical Engineering

Degree Program

Mechanical Engineering, MS

Committee Chair

Murphy, Kevin

Committee Co-Chair (if applicable)

Alphenaar, Bruce

Committee Member

Alphenaar, Bruce

Committee Member

Richards, Christopher

Author's Keywords

passive; automatic; balancing; collisions; train formation; train destruction


Many applications are inherently rotational in nature. These applications range from industrial products to consumer products. As seen in a traditional frequency response plot, the motion of a rotating imbalance does not approach zero as the forcing frequency is increased. Unlike a traditional forced mass spring oscillator, the motion of the rotating imbalance approaches some non-zero value. To account for this residual motion, some systems utilize a balancing device to reduce this motion. These balancing devices can be passive or active, depending on the design considerations. This paper will focus on the traditional, passive ball-type balancer due to its simplicity and extensive use in application. This paper derives the equations of motion for a vertically oriented ball-type balancing system. Due to the high non-linearity of these equations, a fourth order Runge-Kutta numerical integration method is used. The ball balancer equations of motion contain the proper physics needed for full operation such that the ball balancer can translate horizontally, vertically and rotate angularly in the presence of gravity. Acceleration terms are included such that a wide range of operating conditions can be tested. Additionally, n number of balls are present, which are affected by rolling friction and viscous fluid drag. Unlike many numerical models published in the past, the ball-to-ball interactions are not neglected within this model. These interactions include collisions, and train formations and separations. An application of the method presented by (Henon 1982) is utilized where the equations of motion are altered such that an exact integration step can be solved. This is based on the need for a displacement step (collision) or a force step (separation). Although the model presented can accommodate n number of balls, only a maximum ball count of two is considered. It is shown how the behavior of the balls affect the motion response of the ball balancer at rotational velocities above the translational resonance of the system. It is seen that a critical transition is reached; the operating point at which the ball balancer becomes effective at offsetting an eccentric mass. It is also seen that ball balancer displacement decreases until a point of saturation, after which ball balancer displacement increases. Also for the two ball case, it is shown that the spatial characteristics of the balls do affect steady state motion. The angle that separates two contacting balls alters the center of gravity of the train of balls such that the balancing capacity of the system is reduced. Although this effect is shown to be small for a two ball case, the balancing capacity is further reduced as the angle between two contacting balls becomes larger.