Vibration-damping rubber is commonly used as a vibration isolation measure when installing motors, pumps, and other rotating objects. When a rotating object is moving, it generates centrifugal force (centripetal force). If there is an imbalance, a fluctuating force caused by the imbalance occurs, which can lead to vibration. Balancing the rotating object is performed to suppress vibration, which is why balancing is done when changing car tires. So, what magnitude of force is generated?
Let's try to recall what we learned in physics class.
When an object is attached to a string and spun, the centrifugal force and centripetal force balance each other, causing the object to move in a circular motion at a constant speed. The centrifugal force acts outward, perpendicular to the circumference, the centripetal force acts in the opposite direction towards the center of the circle, and the object's velocity is in the circumferential direction.
If the radius is r and the angular velocity is ω, then the velocity v is given by
v＝rω ・・・（1）
The acceleration 'a' is in the same direction as the force and perpendicular to the velocity.
a＝rω^2 ・・・（2）
The force F is given by the mass of the object, where m is the mass of the object.
F＝ma＝mrω^2 ・・・（3）
If this circular motion is on a vertical plane, then a force projected onto the vertical axis acts vertically, and the force F, neglecting the initial phase,
P＝ma＝mrω^2sinωt ・・・（4）
It is expressed as follows.
If the motor's revolutions per minute are R, then the relationship between frequency f and ω is:
f=R/60 、 ω＝2πf ・・・(5)
It can be calculated as follows:
An object can be thought of as having unbalanced mass and a point mass.
If the balance is poor, this force increases as the rotational speed increases, resulting in greater vibration. You can experience this firsthand when aligning shafts during coupling.
If you look closely at equation (4), you will see mr, which represents torque. In complex machines, various axes are combined, so reducing the imbalance of each axis helps to suppress vibration and also eliminates excess torque, thus improving efficiency.
Now, when we analyze the vibration acceleration of the motor stand using an FFT analyzer, the vibration acceleration caused by the rotational speed R can be measured as the acceleration component of frequency f, which is obtained by equation (5). By focusing on the rotational speed of each axis in the analysis, we can find clues to investigate which axis is causing the vibration and thus the source of the vibration.
<Example 1>
When a motor was mounted on a base supported by spring k, the static deflection was 10 cm. The motor has an unbalanced force. To reduce this force to 1/35 or less, what rotation speed (revolutions per minute) should the motor be operated at?
Equation (5) from the previous issue
Natural frequency fo = 5/√δ, where δ = static deflection (cm)
twist
fo＝5/√10＝1.58 （Hz）
Equation (6) from the previous issue
Transmissibility τ=｜1÷{1-(ω/ωo)^2}｜
twist
τ = 1/35 = 1/{f/fo)^2 - 1}, where f > fo
Therefore
f＝9.48Hz
Convert to revolutions per minute
R＝9.48×60＝568.8 （r/min）
The motor will be operated at a rotational speed of 569 r/min.
<Example 2>
The vibrating screen is vibrated using an unbalanced vibration exciter.
What is the excitation force when the vibration exciter rotates at 750 r/min, the total mass of the rotating body is 1 ton, and the eccentricity is 50 mm?
From equation (5)
f＝750÷60＝12.5 Hz
From equation (3)
P=mrω^2=1000×0.05×(2π×12.5)^2=approximately 308×10^3(N)
An excitation force of approximately 308 kN is generated.
References: Practical Mechanical Vibration Engineering, by Masaharu Kunieda, Rikogakusha
(Excerpt from the email newsletter issued on September 26, 2005)
