Critical speeds and vibration

Every shaft solve also computes the shaft's natural vibrations: the lowest bending critical speeds with their mode shapes, the forward and backward whirl of the turning shaft, the response to unbalance, and the torsional and axial modes. Use them to keep the running speed clear of resonances, and to see how stiff supports, light parts and short overhangs move the first critical speed up.

The results appear below the charts in the centre of the shaft workspace. The governing critical speed is also a key result in the results column and can be checked against a minimum.

Inputs

The vibration results use the shaft model you already built. A few inputs matter only here:

A rigid support makes the critical speeds an upper bound. With Couple linked bearings, each linked bearing takes its tangent stiffness at the coupled state, in each plane, radially and in tilt, with the housing in series. A ball bearing with clearance is two to three times stiffer for small vibrations than its load over its displacement suggests.

Results

Bending modes

The first three bending critical speeds of the non-rotating shaft, each with its plane (xy or xz) and its mode shape drawn swinging along the shaft between its envelope. The supports are marked under the shape. When the supports are the same in both planes the xy plane's modes are listed; when they differ, the lowest three of both planes, each with its plane.

In the 3D model view, the shape picker shows any of these modes on the shaft instead of the deflection under the loads.

Campbell diagram

When the shaft turns, the polar inertia of the shaft and its parts stiffens forward whirl (the shaft's centre orbits with the rotation) and softens backward whirl (against it). The Campbell diagram plots each mode's natural frequency, forward and backward, over the shaft speed, with the Running speed line. Where a mode crosses that line, its speed is critical; the forward crossings are marked 1F, 2F and so on.

Below the chart, each critical speed is listed as forward, backward and standing (the non-rotating value).

On supports that are stiffer one way than the other, both bending planes are solved together, coupled by the gyroscopic moments. The whirl is then elliptic, and it is called forward when it turns mostly with the shaft.

Governing critical speed and the check

The 1st critical speed key result and the Min 1st critical speed check use:

• forward whirl, turning, when any part has polar inertia: the first synchronous whirl critical speed that unbalance excites, the one the rotor meets as it runs up. On supports that differ between the planes, any whirl with a forward share of at least 5 % counts, elliptic backward whirl included.
• bending, non-rotating otherwise: the first non-rotating bending critical speed.

The check's message says which one it used.

Unbalance response

When a load has a Part unbalance, the Unbalance response chart shows the radius of the shaft's orbit (µm) at each support and each unbalanced load, from standstill to 1.3 times the highest critical speed. Below it, Orbits at … rpm draws the orbits seen along the shaft at the speed you pick with the slider (by default the first peak), and each point lists its peak and the speed of the peak. The speeds are refined round every peak until it is resolved to 10⁻⁴, so they are not evenly spaced near a resonance and the peak does not depend on the grid.

The response turns with the shaft (synchronous whirl), so the parts' polar inertia acts on it. On round supports the orbit is a forward circle, which the polar inertia stiffens. Supports or housings stiffer one way give an elliptic orbit, and its larger radius is shown: the two bending planes are then solved together, coupled by the gyroscopic moments, as for the critical speeds. The structure and supports are damped by the damping ratio; the negative springs of a rotor's magnetic pull are not.

Torsional and axial modes

• Torsional modes: the shaft twisting, free at both ends, with the shaft's polar inertia and the parts' (gears, pulleys and rotors from their diameters, or a load's polar inertia). The rigid rotation is left out.
• Axial modes: the shaft stretching along its axis with the parts' masses, held at the locating support by the linked bearing's axial tangent stiffness at its load. When no bearing gives one, the support is taken as axially rigid and a note says so.

Frequencies are given in rpm (Hz × 60) so that they compare directly with the running speed.

Examples

Simple shaft

The simple template (Ø40 × 400 mm, rigid supports, no part mass) gives:

Giving the load L1 a Part mass of 8 kg lowers the first bending critical speed to 15 736 rpm. With a Part unbalance of 100 g·mm on it as well, the largest orbit radius at L1 in the response is 257.7 µm at 15 774 rpm, just above the critical speed (the forward whirl, stiffened by the shaft's own polar inertia); at the rigid supports it is zero.

Electric motor

The electric motor template is a 15 kW motor shaft whose rotor has polar inertia and a magnetic pull that acts as a negative spring. It gives:

Because the rotor has polar inertia, the check uses the forward whirl value, which is higher than the non-rotating one.

Raising a critical speed

• Shorten the span between the supports, or the overhang of an overhung part.
• Increase the diameter: stiffness grows faster than mass.
• Make parts lighter, or move heavy parts towards a support.
• Use stiffer supports: a bearing with higher stiffness, preload, or a stiffer housing.
• Switch on Couple linked bearings to see the real support stiffness instead of rigid supports, which overestimate the critical speeds.

Method and limits

• Timoshenko beam elements with the consistent translational and rotary mass, supports rigid or with their stiffness. Mode shapes by inverse iteration.
• The bending critical speeds listed first are non-rotating. Gyroscopic stiffening enters only the whirl results, the Campbell diagram, the unbalance response and the governing critical speed when a part has polar inertia.
• Supports and bearings are linear springs at one operating point. The speed dependence of bearing stiffness is not included, and neither are oil film bearings, seals or fluid forces.
• Damping is structural, one ratio for the shaft, supports and housings. There is no stability (logarithmic decrement) analysis.
• The coupled-plane whirl on supports that differ between the planes is solved in the lowest 24 non-rotating modes of each plane.
• Torsional modes are for the shaft alone, free at both ends: the inertia of a motor or a driven machine on the far side of a coupling is not included unless you model it as a part.
• The critical speeds are checked against ROSS 3.0, an independent open-source rotordynamics program, in the verification suite.

From the command line

axm new motor.axm.json --template electric-motor
axm solve motor.axm.json --detail --json
axm load update motor.axm.json L1 --unbalance 100

The JSON holds criticalSpeeds, governingCriticalSpeed, modes, whirl (with campbell), unbalanceResponse, torsionalModes and axialModes.

Related

• Deflection, critical speeds and whirl
• Shaft
• Bearings
• Motors
• Housings
• Units
• Tutorial: a simple shaft