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The relationship between vibration units mm, mms, and mms²: In the field of vibration analysis, mm (millimeters), mm/s (millimeters per second), and mm/s² (millimeters per second squared) are three key physical quantities used to describe vibration characteristics, corresponding respectively to vibration displacement, velocity, and acceleration. Although these three units appear independent, they are actually closely connected through the fundamental laws of vibration, and their conversion relationships are directly related to the frequency characteristics of vibration. The following provides a detailed explanation from aspects such as physical meaning, mathematical relationships, and practical application scenarios. I. Physical meaning: Understanding the three units from the perspective of motion’s essence. The essence of vibration is the back-and-forth movement of an object near its equilibrium position, and displacement, velocity, and acceleration are the three fundamental parameters that describe this movement: – Displacement (mm): refers to the maximum distance an object moves away from its equilibrium position during vibration, reflecting the “amplitude” of the vibration. For example, when a machine part vibrates, if the maximum distance that a certain point moves back and forth is 0.5 mm, it means that the amplitude of displacement for that point is 0.5 mm. Displacement is directly related to the spatial extent of vibration; excessive displacement can lead to collisions between components, wear, or structural deformation. - Speed (mm/s): Describes the rate of change of displacement over time, reflecting the \"intensity\" of vibration. The magnitude of velocity is related to the number of vibrations an object undergoes per unit time and the amplitude of its displacement; for example, when the amplitude of displacement remains constant, the higher the vibration frequency, the greater the velocity. In engineering, velocity is often used as a key indicator to evaluate the vibration intensity of rotating machinery such as motors and bearings, as it can effectively reflect the efficiency of vibration energy transfer. - Acceleration (mm/s²): Describes the rate of change of velocity over time, reflecting the \"impactiveness\" of vibration. Acceleration is proportional to the square of the vibration frequency, and it is particularly sensitive to high-frequency vibrations. For example, the impact vibration at the moment the device starts up, as well as the high-frequency impacts resulting from gear meshing, are often reflected through acceleration values. Excessive acceleration can lead to issues such as material fatigue and loose bolts. II. Mathematical relationships: Conversion formulas in simple harmonic vibration. In practical engineering, most vibrations can be approximated as simple harmonic vibrations (i.e., vibrations that follow sine or cosine patterns), and the relationships between displacement, velocity, and acceleration can be derived using mathematical derivatives. Let the displacement expression for harmonic vibration be: x(t) = A \ sin(ωt + φ), where A is the amplitude of displacement (in mm), ω is the angular frequency (in rad/s), t is time (in s), and φ is the initial phase (which does not affect the amplitude calculation and can be ignored). The relationship between the angular frequency and the actual frequency f (in Hz, i.e., cycles per second) is: ω = 2πf. 1. Conversion between displacement and velocity: Velocity is the first derivative of displacement with respect to time. By taking the derivative of the displacement expression, we obtain: v(t) = \frac{dx}{dt} = Aω \cos(ωt+ φ). The amplitude of velocity, denoted as V (in mm/s), is given by: V = A \cdot ω = A \cdot 2πf. In other words, the amplitude of velocity equals the amplitude of displacement multiplied by 2π times the frequency. For example, when the displacement amplitude A = 0.1 \, \text{mm} and the frequency f = 50 \, \text{Hz}, the velocity amplitude V = 0.1 \times 2 \times 3.14 \times 50 ≈ 31.4 \,\text{mm/s}. 2. Conversion between velocity and acceleration. Acceleration is the first derivative of velocity with respect to time (or the second derivative of displacement with respect to time). By taking the derivative of the velocity expression, we obtain: a(t) = \frac{dv}{dt} = -Aω² \sin(ωt+ φ). The amplitude of acceleration, denoted as a (in units of mm/s²), is given by: a = V \cdot ω = V \cdot 2πf. It can also be determined directly from displacement: a = A \cdot ω² = A \cdot (2πf)². In other words, the amplitude of acceleration equals the amplitude of velocity multiplied by 2π times the frequency, or it equals the amplitude of displacement multiplied by (2π times frequency) squared. For example, when the velocity amplitude is V = 31.4 \, \text{mm/s} and the frequency is f = 50 \, \text{Hz}, the acceleration amplitude is a = 31.4 \times 2 \times 3.14 \times 50 ≈ 9859.6\, \text{mm/s²} (which is approximately equal to 10g, where g is the acceleration due to gravity, and 1g≈9807 mm/s²). III. The influence of frequency on the conversion relationships: Unit sensitivity in different frequency bands. The conversion relationships between these three units are all related to the frequency f, which means that at different frequencies, the values of displacement, velocity, and acceleration for the same vibration can differ significantly. – Low-frequency vibrations (f < 10 Hz): At lower frequencies, the value of 2πf is small, and in such cases, displacement is a more sensitive indicator of vibration. For example, the low-frequency vibrations of buildings during earthquakes (around 1–5 Hz) are primarily caused by excessive displacement; therefore, in engineering, displacement (in mm) is commonly used to assess the safety of structures. - Intermediate-frequency vibration (10 Hz < f < 1000 Hz): At moderate frequencies, the velocity is proportional to the first power of the frequency; this value is appropriate and can effectively reflect the vibration energy (which is proportional to the square of the velocity). The vibration of rotating machinery in industry (such as fans and pumps) falls within this frequency range; therefore, international standards (such as ISO 10816) use velocity (mm/s) as the criterion to determine whether the vibration is acceptable or not. - High-frequency vibration (f > 1000 Hz): At higher frequencies, the value of (2πf)² increases sharply, making acceleration a more sensitive indicator of vibration. For example, in gear meshing and the high-frequency vibrations of rolling bearings (1000–10000 Hz), their impact characteristics are primarily reflected through acceleration (mm/s²); therefore, acceleration sensors are commonly used for monitoring in such situations. IV. Precautions for conversion in practical applications 1. Standardization of amplitude types: In vibration measurement, the amplitude may be expressed as \"peak value,\" \"RMS value,\" or \"peak-to-peak value.\" It is necessary to standardize these types before performing conversions. For example, the RMS value of a sine signal is 1/√2 of its peak value, while the peak-to-peak value is 2 times the peak value. If the root-mean-square displacement is 0.07 mm (corresponding to a peak value of 0.1 mm) and the frequency is 50 Hz, then the peak velocity is 31.4 mm/s, while the root-mean-square velocity is 31.4/√2 ≈ 22.2 mm/s. 2. Treatment of non-harmonic vibrations: Complex vibrations (such as shocks and random vibrations) need to be decomposed into multiple harmonic components using Fourier transformation; after converting each component using the aforementioned formulas, they are then combined together. It is not possible to calculate them using a single frequency. 3. Unit scale conversion: In engineering, it is sometimes necessary to convert units into more intuitive scales; for example, acceleration is often expressed in terms of “g” (1 g ≈ 9807 mm/s²), while velocity is usually expressed in “m/s” (1 m/s = 1000 mm/s). It is important to pay attention to the scale transformation when making such conversions. V. Conclusion: mm, mm/s, and mm/s² describe vibration from the three dimensions of displacement amplitude, motion velocity, and impact intensity, respectively. The core relationship is that in harmonic vibration, velocity is the product of displacement and angular frequency, while acceleration is the product of velocity and angular frequency (or the product of displacement and the square of angular frequency). Frequency is the key parameter that connects the three; different frequency ranges require appropriate units for analysis – displacement for low frequencies, velocity for medium frequencies, and acceleration for high frequencies. Understanding the conversion relationships among the three can help engineers assess more accurately the impact of vibrations on equipment and structures, providing a basis for fault diagnosis and safe design.