So called G-force, it is actually not a force. G-force is an acceleration rate expressed in units of G. G is the "acceleration of gravity", equals to 9.81m/s2 at the sea level. G-force equals acceleration divided by G. Acceleration is the rate of velocity change (δv) in a given period of time (δt). In vibration motion, acceleration is decided by frequency, amplitude and the time phase of the object. In physics and mathematics, The Simple Harmonic Vibration Model is often used to analyst and calculate velocity and acceleration.

Because G-force is decided by the two most important parameters of whole body vibration plates, the frequency and the amplitude, manufacturers of whole body vibration plates tend to use G-force as an important vibration intensity indicator. However, manufacturers may calculate G-Force using different formulas and different criteria.

At Vibration Therapeutic ®, we derive acceleration calculation from the Simple Harmonic Motion Equation:

Velocity v = ωDcos(ωt)

Acceleration a = -ω2Dsin(ωt)

D is the displacement from the equilibrium point, ω=2πf is the angular frequency, and t is the time.

Therefore, a=-(2πf)2Dsin(2πft)

In vibration motion, object moves in alternate directions. It achieves the max velocity at the equilibrium points and decreases to zero at the peak points. In the meantime, acceleration changes from zero at the equilibrium points and to the max value at the peak points, alternating between negative value and positive value in each cycle. Such a dynamic is expressed by the factor sin(2πft) in the equation.

In a single vibration circle (Fig to the right), the object moves in four phases, starting from 1) equilibrium position to maximum displacement, 2) then changing the direction from maximum displacement back to equilibrium position, 3) continuing travel to the max maximum displacement on the opposite end, and 4) changing direction again and back to the original equilibrium position. Velocity increases and decreases in the 4 phases. In the phases that object moves away from the equilibrium position, the velocity decreases from max to zero. While in the phases that object moves towards the equilibrium position, the velocity increases from zero to the maximum value.

In phase 1 & 3, velocity decreases from the maximum value to zero, acceleration is in negative range.

In phase 2 & 4, velocity increases from zero to maximum value, acceleration is in positive range.

The total time of single circle is 1/f. The time of a single phase is ¼ of 1/f. When the object is at the peak position, D = A (Amplitude).

Therefore, at the peak position, the object achieve the maximum positive and negative acceleration rate. The dynamic factor in the equation sin(2πft)=sin(π/2) =1. Therefore,

apeak = ±(2πf)2A

When the object moves close to a peak position, apeak = -(2πf)2A; when the object starts to move away from a peak position, apeak = (2πf)2A

The acceleration rate in vibration movement is in constant change on differentiation level, between zero and the maximum values (both positive and negative) in each phase. It is biased to use the peak acceleration rate in of a phase to evaluate the impact of vibration acceleration to human body, At Vibration Therapeutic, we introduce two acceleration concepts for the evaluation of the vibration acceleration:

1) Peak Acceleration Rate

Peak Acceleration Rate is the max acceleration rate of the object in a phase.

apeak = ±(2πf)2A

2) Phase Acceleration Rate

Phase Acceleration Rate is defined as the AVERAGE between zero and the Peak Acceleration Rate of the phase. For whole body vibration machine, we consider Phase Acceleration Rate better represents vibration acceleration than Peak Acceleration Rate.

Phase 1 & 3, aphase = -2π2f2A

Phase 2 & 4, aphase = 2π2f2A

It is IMPORTANT to note that, for vibration machines, the Amplitude quoted by most manufacturers is actually the Peak-to-Peak Amplitude. The Amplitude in the formula above is the Amplitude as defined in physics, which is half of the Peak-to-Peak Amplitude.

For most whole body vibration machines, the major movement is on vertical direction. Gravity need to be considered in the calculation of acceleration. Therefore, the acceleration values for each phase are:

Phase 1, aphase = -2π2f2A + G

Phase 2, aphase = 2π2f2A + G

Phase 3, aphase = -2π2f2A + G

Phase 4, aphase = 2π2f2A + G

These formula calculate the acceleration of the vibration PLATFORM of the machine. However, in vibration exercise, the user's body is not affixed on the vibration platform. When the vibration platform accelerates or decelerates faster than the acceleration of gravity, user's body may not fully stick on the platform. The body can be in projectile motion or free-fall motion.

In phase 1, the platform moves up in decelerated speed, at the point the absolute acceleration rate is greater than G, user's body is in the projectile motion, and the accelerating rate is -G. Although User's body and platform may not arrive at their respective peak positions in the same time, we know in phase 1, the body's acceleration is between -G and 0.

In phase 2, the platform moves down in accelerating speed, at the point the acceleration rate is greater than G, user's body is in free-fall motion, the acceleration rate is G. Although User's body and platform may not fall from their respective peak positions in the same time, we know in phase 2, the body acceleration is between 0 and G.

In phase 3, the platform continue moves down in decelerating speed, user's body continues with free fall and then catch up with the platform at a certain point. The body acceleration rate is between G and -2π2f2A + G.

In phase 4, the platform moves up in accelerating speed, user's body stays on the platform in the entire phase. The body acceleration rate is 2π2f2A + G

Our goal is to evaluate the impact of the highest Phase Acceleration Rate on user's body, which is Phase 4 Acceleration Rate.

Phase 4 Acceleration Rate = 2π2f2A + G

Phase 4 Acceleration Rate converted to G-force (divided by G):

Phase 4 G-force =2π2f2A/G + 1

Also in Phase 4, for reference, the maximum G-force can be calculated as

Phase 4 Peak G-force = (2πf)2A/G +1

Phase 4 G-force is the acceleration values we use to evaluate acceleration of the vertical movement of whole body vibration machine.

For Model VT003F vibration plate , given the highest frequency 40Hz, and the highest (peak-to-peak) amplitude 3mm (0.003m), the acceleration rates are calculate as

For reference, Phase 4 Peak G-force = 10.64

For Model VT020-3 vibration plate , given the highest frequency 15Hz, and the highest (peak-to-peak) amplitude 10mm (0.003m), the acceleration rates are calculate as

For reference, Phase 4 Peak G-force = 5.52

For Model VT017 vibration plate , given the highest frequency 10Hz, and the highest (peak-to-peak) amplitude 3mm (0.009m), the acceleration rates are calculate as

For reference, Phase 4 Peak G-force = 2.82

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