In quality inspection and performance evaluation of integrated robot joints, second-harmonic (2×) frequency testing has become an indispensable key step. Although it may look highly specialized, this test is directly linked to core performance indicators such as joint accuracy, stiffness, and reliability. This document explains—covering technical principles, engineering practice, and quality control—why second-harmonic testing is necessary and why it is increasingly important in modern robot manufacturing.

Core Structure and Dynamic Characteristics of Robot Joints

Typical Structure of an Integrated Joint Module

A typical transmission chain of an integrated joint is:

Motor -> Harmonic Drive / RV Reducer -> Output Shaft

Among these, the reducer is the core transmission component, and its performance largely determines the overall joint behavior.

Nonlinear Characteristics of Harmonic Drives

Harmonic drives are widely used in collaborative robot arm joints due to advantages such as high reduction ratio, near-zero backlash, and compact size. Their working principle is based on the elastic deformation of the flexspline:

- The wave generator rotates and drives the flexspline into an elliptical deformation.

- During deformation, the flexspline meshes with the circular spline to transmit motion.

- The circular spline is fixed, and the flexspline outputs reduced-speed rotation.

Because this transmission relies on elastic deformation, it inherently exhibits nonlinear stiffness. When the wave generator rotates at angular frequency ω, the flexspline experiences two meshing-and-disengaging events per revolution, which introduces a 2ω frequency component in output torque and vibration response—this is the physical origin of the second-harmonic phenomenon.

The Physical Essence of the Second-Harmonic Phenomenon

Mechanism of Frequency-Domain Features

During operation of a harmonic drive, the stress distribution in the flexspline changes periodically. Using the wave generator speed as the fundamental frequency f1, within one rotation cycle the flexspline undergoes:

- First meshing zone: the long-axis direction fully meshes with the circular spline.

- Transition zone: meshing depth gradually decreases.

- Second meshing zone: the short-axis direction forms another meshing zone.

- Transition zone: meshing depth decreases again.

This structural feature—one rotation, two meshing events—causes torque ripple, stiffness variation, and vibration response of the output shaft to exhibit a clear second-harmonic characteristic (2f1). From a Fourier-analysis perspective, this is a typical parametric excitation system, where the system stiffness itself varies periodically over time.

Spectral Identification in Vibration Signals

By measuring joint vibration with accelerometers or laser vibrometers, a frequency-domain response spectrum can be obtained. A typical vibration spectrum of a harmonic-drive joint includes:

- Fundamental (1×): corresponds to motor speed or input shaft frequency.

- Second harmonic (2×): characteristic frequency of the harmonic drive, usually the most prominent amplitude.

- Third harmonic (3×) and above: higher-order harmonic components with smaller amplitudes.

The magnitude of the 2× amplitude directly reflects:

- Meshing quality between the flexspline and circular spline.

- Machining accuracy of the wave generator.

- Appropriateness of bearing preload.

- Coaxiality errors introduced during assembly.

Engineering Significance of Second-Harmonic Testing

Stiffness Evaluation

Joint stiffness is a key parameter affecting robot positioning accuracy and dynamic response. The torsional stiffness of a harmonic drive is not a constant; it is time-varying with the meshing state. A larger second-harmonic amplitude indicates more severe stiffness fluctuation, leading to:

- Reduced positioning accuracy: additional position error under load changes.

- Worse trajectory tracking: oscillations during high-speed motion.

- Lower control stability: control algorithms struggle to compensate for time-varying stiffness.

Monitoring second-harmonic features allows indirect evaluation of the joint’s equivalent torsional stiffness and its fluctuation range, providing a basis for control-system design.

Assembly Quality Inspection

Second-harmonic amplitude is highly sensitive to assembly accuracy. The following assembly defects can cause abnormal second-harmonic behavior:

- Wave generator eccentricity: amplitude increases significantly (uneven flexspline deformation, unbalanced meshing forces).

- Improper bearing preload: frequency spread, more sidebands (introduces extra clearance or over-constraint).

- Flexspline mounted with tilt: 2× splits into double peaks (asymmetric meshing zone).

- Poor coaxiality between circular spline and flexspline: coupling between 2× and rotational frequency (misalignment of geometric axes).

Early Fault Warning

During service life, reducers undergo degradation such as wear and fatigue. Evolution of second-harmonic characteristics can serve as a condition-monitoring indicator:

- Gradually increasing 2× amplitude: flexspline fatigue crack growth, stiffness reduction.

- Shift in 2× frequency: bearing wear causes speed instability.

- New sidebands appear: localized damage such as pitting or spalling on tooth surfaces.

Compared with traditional periodic teardown inspections, online monitoring based on second-harmonic features enables predictive maintenance, avoiding unplanned downtime due to sudden failures.

Test Methods and Standards

Test System Configuration

A complete second-harmonic testing system typically includes:

- Excitation device: servo motor drives the joint at constant speed or variable speed.

- Sensor suite: triaxial accelerometers (mounted on joint housing), torque sensor (measures output torque ripple), encoder (phase reference signal).

- Data acquisition and analysis: high-sampling-rate DAQ (>= 10 kHz), FFT spectrum analysis, order tracking analysis (for variable-speed conditions).

Typical Test Procedure

Step 1: No-load running test

- Run at 30%, 60%, and 100% of rated speed.

- Record vibration spectra at each speed.

- Extract 1× and 2× amplitudes and compute their ratio.

Step 2: Load test

- Apply 50% and 100% of rated torque.

- Compare changes in second-harmonic characteristics under different loads.

- Evaluate load-dependent stiffness and damping behavior.

Step 3: Sweep test

- Sweep from low speed to high speed at a uniform rate.

- Plot a Campbell diagram to identify resonance points.

- Check whether the second harmonic couples with structural natural frequencies.

Relevant Standards

Although there is currently no standalone standard specifically for second-harmonic testing, the following standards provide frameworks for vibration and dynamic testing:

- ISO 10218-1:2011

- GB/T 30819-2014

- ISO 9283:1998

- ISO 14738:2002

Many robot manufacturers also establish internal second-harmonic test procedures within their quality systems and use it as a standard item for joint outgoing inspection.

Conclusion

Second-harmonic testing is an important means to understand and evaluate the performance of collaborative robot arm joints. It reveals inherent dynamic characteristics of harmonic drives and provides quantitative evidence for quality control, fault diagnosis, and performance optimization.

From the physical perspective, the second-harmonic phenomenon originates from the flexspline’s one rotation, two meshing events structure, representing an intrinsic response of a parametric excitation system. From the engineering perspective, the second-harmonic amplitude is directly associated with key indicators including stiffness fluctuation, assembly precision, and wear condition.

As robotics advances toward higher precision and higher reliability, second-harmonic testing will inevitably move from a laboratory method to a production-line standard, becoming an important safeguard for robot quality. For engineers engaged in robot design, manufacturing, and maintenance, a deep understanding of the principles and methods behind second-harmonic testing will help improve product competitiveness and drive technological progress in the industry.