Vibration testing engineers should identify the potential effects of shaker force to protect the overall system. For instance, if a shaker is designed to vibrate vertically, any cross-axis vibration on the shaker table can damage components, such as flexures. Force moments are one such effect that must be identified, particularly in the aerospace industry.
Force Moments
A moment is a measurement of an object’s tendency to rotate about a specific point or axis in response to a specific force. The axis around which the object rotates determines whether the measurement is torque or an overturning moment.
Torque is a rotational force that causes an object to rotate about the longitudinal axis (twisting or torsional moment). An overturning moment is a rotational force that causes an object to tip or bend around one or more transverse axes (bending or tipping moment).
Both measurements are calculated by multiplying the linear force by the perpendicular lever arm distance.
- Rotations over the longitudinal axis: torque
- Rotations over the transverse axis: overturning moment
Torque and overturning moments are vector quantities, meaning they have magnitude and direction. A physical convention called the “right-hand rule” can be used to determine the direction/rotation of the measurement.
Calculating Overturning Moments
Notably, vibration testing engineers calculate overturning moments during force-limited vibration testing (FLVT).
For this test configuration, force gauges are mounted between an upper and lower force plate, and the device under test (DUT) is attached to the upper plate (Figure 1). If the DUT bends back and forth in response to the linear shaker force, the force plates will subsequently apply force to the sensors on the z-axis.
Figure 1. A force sensor between aluminum force plates.
The following example demonstrates how mounting the force sensors on the x- and y-axes simplifies overturning moment calculations.
Example Calculation
To calculate an overturning moment about the x-axis, the engineer must multiply the force applied in the z-axis by the force sensors’ distance (in the y direction) from the center.
For example, if the DUT is attached to a ring plate, the overturning moment about the x-axis is calculated by summing all the y-axis coordinates multiplied by their respective z-axis force (Figure 2).
Figure 2. An example ring structure for component mounting during FLVT. Many ring structures have more than four force sensors.
In the example, the z-axis forces (y-coordinates) are 130 mm from the x-axis; the two other z-axis forces are on the x-axis (0 mm). To calculate the overturning moment about the x-axis:
(1) 
Addressing Overturning Moments
An unbalanced payload can lead to an unequal distribution of force at its attachment points. A common cause of unbalance is an offset of the payload’s center of mass from the plane of motion.
Vibration forces tend to cause an unbalanced payload to rotate. The structural interface should resist this rotation so the payload only moves in the plane of motion. The reaction torque that the system must supply to restrain the payload is the “overturning moment.”
Figure 3. An aluminum cube with an additional 178 g on the left upper rail.
Example
It does not take a large off-center mass to significantly change the overturning moment. In a simple experiment, we mounted a 178 g mass on the upper left rail of a 1,774 g cube (Figure 3-4).
Ten percent of the total mass was 106 mm above and 106 mm to the left of the center of mass. With the additional mass, the overall center of mass moved 8.6 mm up and 10.28 mm to the left.
Figure 4. An aluminum cube 212.7 mm in each dimension with a total mass of 1,774 g.
Without the additional mass, the cube had an average 1.351 Nm overturning moment around the x-axis. With the additional mass, this overturning moment increased to 3.493 Nm (Figure 5).
Figure 5. Comparison of overturning moment RMS. The overturning moment with the additional mass (green trace) is 2.58x greater than without (pink trace).
Overturning Moment Formulas in VibrationVIEW
VibrationVIEW calculates overturning moments in the time domain with virtual channels. Virtual channels perform linear mathematical operations on raw time signals to combine physical input data.
In the following example, the sum of the z-axis forces is virtual channel 33. The engineer will apply a notch to this channel (force) during FLVT. The overturning moment about the x-axis is calculated in virtual channel 36 according to the previous coordinate system calculation.
Figure 6. An example virtual channel configuration.
Phase Difference
VR10500 Controller and Virtual Channels
Any phase difference between channels within a single VR10500 controller, between multiple VR10500s, or between a VR10500 and a virtual channel is minimal. At 2 kHz, Vibration Research expects less than a 0.040-degree difference within a unit.
VR10500 Controller Phase Uncertainty Values
Specified at k=2 and valid for the ±10V/±1V/0-20V ranges
| Same Unit | Hardware Clocking | Internal Ethernet | Network Switch | |
| Uncertainty at 2 kHz (degrees) | ±0.040 | ±0.048 | ±0.052 | ±0.057 |
| Uncertainty at 20 kHz (degrees) | ±0.40 | ±0.48 | ±0.52 | ±0.57 |
Channel Comparison
With hardware clocking enabled, we conducted a simple test to compare channel performance within the same controller, between controllers, and between a controller and a virtual channel. Channels within the same controller had phase differences of 0.026 degrees, and those between different units had a phase difference of 0.03469 degrees. The difference between a virtual channel and a controller channel was about the same as between different units: 0.03433 degrees.
Figure 7. VR10500 controller phase uncertainty values measured within the same unit (Ch 3, Ch 16); between units (Ch 18); and between a virtual channel and a unit (Ch 36).