Rotating Unbalance Vibration Simulator

Analyse the unbalance vibration of spring-supported rotating machinery such as motors, fans and pumps. Adjust the rotor's small offset mass, support stiffness, damping and running speed to see the natural frequency, frequency ratio, vibration amplitude and force transmitted to the foundation update in real time, and find a vibration-isolating design clear of resonance.

Parameters

Total machine mass M

Mass of the whole machine vibrating on the springs

Unbalance mass m

Equivalent mass offset from the rotor's balance

Eccentricity e

Offset of the unbalance mass from the spin axis

Support stiffness k

N/m

Total stiffness of the isolation springs

Damping ratio ζ

Damping level of the support (dimensionless)

Running speed N

rpm

Operating rotation speed of the rotor

Results

—

Natural freq. f_n (Hz)

—

Frequency ratio r

—

Vibration amplitude X (mm)

—

Unbalance force F (N)

—

Transmitted force F_tr (N)

—

Resonance status

—

Rotating-machine vibration — animation

A rotor spins inside a machine block supported by springs and a damper, and the unbalance mass (orange dot) produces a centrifugal force. The whole machine vibrates up and down with an amplitude proportional to X.

Vibration amplitude vs running speed (resonance curve)

Amplitude magnification vs frequency ratio r

Theory & Key Formulas

$$F = m\,e\,\omega^{2}, \qquad X = \frac{(m e/M)\;r^{2}}{\sqrt{(1-r^{2})^{2}+(2\zeta r)^{2}}}$$

Unbalance exciting force F and steady-state vibration amplitude X. m: unbalance mass, e: eccentricity, M: total machine mass, ω: angular speed, ζ: damping ratio. The exciting force grows with the square of ω.

$$\omega_n = \sqrt{\frac{k}{M}}, \qquad r = \frac{\omega}{\omega_n}$$

Natural angular frequency ωₙ and frequency ratio r. k: support stiffness. The response peaks at r ≈ 1 (resonance), and for r ≫ 1 the amplitude X approaches the constant value m·e/M.

$$F_{tr} = X\,k\,\sqrt{1+(2\zeta r)^{2}}$$

Force F_tr transmitted to the foundation through the springs and damper. It falls as r increases, which is favourable for vibration isolation.

What is rotating unbalance vibration?

🙋

People talk about rotor "unbalance" a lot. What exactly is wrong with it? A motor just looks like it is spinning steadily.

🎓

In short, it is a state where the centre of mass of the spinning rotor is slightly off the axis of rotation. Manufacturing scatter, machining tolerance, assembly clearance — no matter how precisely you build it, a small eccentricity always remains. As that offset mass keeps spinning, the centrifugal force F = m·e·ω² becomes a rotating exciting force that shakes the whole machine. Think of a washing machine spin cycle with the load bunched to one side — it bangs around. That is rotating unbalance vibration in action.

🙋

I see. But does the centrifugal force really get that large? The eccentricity is only a few millimetres.

🎓

That is the dangerous part of this problem. The exciting force grows with the square of the angular speed ω. Double the speed and the force is four times larger; triple it and it is nine times. Raise the "Running speed N" slider on the left — the unbalance force F will jump up fast. Just 0.5 kg of unbalance mass offset by 20 mm hammers the machine with over 150 N at 1200 rpm. An eccentricity you could ignore at low speed becomes critical at high speed — that is why people say "the faster, the tougher".

🙋

When the frequency ratio r gets close to 1, the amplitude X becomes enormous. Is that what "resonance" is?

🎓

Exactly. The springs supporting the machine have a natural frequency ωₙ = √(k/M), and when the excitation frequency from rotation matches it — that is, when r = ω/ωₙ reaches 1 — the amplitude shoots up. That is resonance. Look at the "resonance curve" chart below: there is a peak at a certain speed. The top of that peak is the resonance point. Notice too that lowering the damping ratio ζ makes the peak sharper and higher. With little damping, resonance produces an outrageous amplitude.

🙋

So to avoid resonance, do I just keep the speed below the resonance point?

🎓

That is one option, but many designs actually run "well above resonance" instead. In the region r ≫ 1, the amplitude X settles to the constant value m·e/M; raising the speed further no longer increases it. So large blowers and turbines deliberately place the resonance point below the operating speed and run above it. The catch is that every start-up and shutdown must pass through the resonance band r ≈ 1, so you need a design that races through it quickly and uses damping to limit the amplitude. That is called passing through a critical speed.

🙋

To reduce the vibration at its root, what is ultimately the best move?

🎓

The most effective move is to balance the rotor. Make the unbalance m·e itself small and the exciting force drops directly. In practice a balancing machine drives the residual unbalance down to a standard grade such as ISO 21940. Next come support tweaks: change the spring stiffness to shift the resonance point away from the operating range, and raise the damping ratio ζ with dampers or rubber mounts. It also matters to watch the transmitted force F_tr. Even with a small amplitude, a stiff spring sends a large force into the foundation and shakes surrounding structures. Balance, resonance avoidance, damping and isolation — use this tool to watch all four together while you design.

Frequently Asked Questions

When an unbalance mass m rotates at an eccentricity e, a centrifugal force F = m·e·ω² acts on that mass. Here ω is the angular speed (rad/s), proportional to the rpm. Because the centrifugal force scales with the square of ω, doubling the speed makes the exciting force four times larger. This is why unbalance problems become dramatically worse at high speed: a small eccentricity that is negligible at low speed becomes a critical vibration source in a high-speed machine.

When the angular speed ω of the rotating excitation coincides with the natural angular frequency ωₙ = √(k/M) of the spring support, the frequency ratio r = ω/ωₙ equals 1 and the vibration amplitude rises sharply. That is resonance. The smaller the damping, the sharper and higher the resonance peak. This tool flags 0.9 ≤ r ≤ 1.1 as a resonance band (dangerous). Passing quickly through this band during start-up and shutdown — running through a critical speed — is a key theme in real machine design.

There are three approaches. (1) Balance the rotor to reduce the unbalance m·e itself — the most fundamental fix, done in practice on a balancing machine to drive the residual unbalance down to a standard grade (e.g. ISO 21940). (2) Keep the operating speed well away from the resonance band; running well above the natural frequency (large r) lets the amplitude approach the constant value m·e/M. (3) Add damping; raising the damping ratio ζ with dampers or rubber mounts suppresses the amplitude, especially near resonance.

When the frequency ratio r is much greater than 1, the amplitude X approaches the constant value m·e/M. This means raising the speed further no longer increases the amplitude, which is the basis for running high-speed machines above resonance. However, start-up and shutdown always pass through the resonance band r ≈ 1, so the amplitude spike there must be controlled with damping. The force transmitted to the foundation also falls as r increases, so running above resonance is favourable for vibration isolation as well.

Real-World Applications

Blowers, pumps and electric motors: Factory blowers, air-handling fans, pumps of every kind and general-purpose motors are all rotating machines. Casting scatter or build-up on an impeller, and bearing wear, create unbalance that shows up as vibration in service. At the design stage, engineers check the relationship between natural frequency and operating speed as in this tool, choose a speed clear of the resonance band, and design the support with vibration-isolating rubber or springs.

Condition monitoring of rotating machinery: Vibration of running motors and pumps is monitored continuously with accelerometers, and a rise in unbalance-related vibration (the once-per-revolution component) is treated as a sign of trouble. Debris on blades, a bent shaft and bearing degradation all appear in the vibration spectrum as growing unbalance or exciting force. Having an intuitive feel for "force grows with the square of speed" from this tool helps when reading field vibration data.

Critical-speed design of large turbines and generators: The rotors of steam and gas turbines and large generators pass through several critical speeds (resonance points) on the way to rated speed. The design keeps each critical speed well away from the operating speed and limits the amplitude during passage with damping from dampers and bearings. The resonance curve in this tool is the most basic model of this "race quickly through the critical speed" idea.

Vibration isolation of appliances and precision equipment: Washing-machine spin drums, vacuum-cleaner motors and HDD spindles all have rotating-unbalance countermeasures built in. A washing machine cancels the offset with a balancer and lowers the force into the cabinet with rubber mounts. The concept of transmitted force F_tr is a key metric for whether vibration "stays inside the machine or spreads to the surroundings as noise and shaking".

Common Misconceptions and Pitfalls

A common misconception is that "vibration isolation succeeds as long as the amplitude is small". There is a reason this tool shows the transmitted force F_tr separately from the amplitude X. Stiffening the support springs reduces the machine's own amplitude, but a stiff spring transmits a large force to the foundation even for a small displacement. Conversely, a soft spring lets the machine sway a lot yet transmits only a small force to the foundation. "Not shaking the machine" and "not transmitting force to the foundation" are different goals, and isolation design must watch both. In general, the operating speed must be above √2 times the natural frequency before any isolation benefit (transmissibility below 1) is obtained.

Next, the assumption that "more damping is always better". Raising the damping ratio ζ does lower the amplitude peak at resonance. But in the region well above resonance (r ≫ 1), more damping actually increases the transmitted force F_tr, because the term 2ζr sits inside the square root of F_tr = X·k·√(1+(2ζr)²). Damping is effective for "getting safely through the resonance band", but on a machine whose steady operation is well above resonance, too much damping hurts isolation performance. The amount of damping should be decided as a balance between resonance passage and steady-state isolation.

Finally, the overconfidence that "balancing makes the vibration zero". Balancing reduces the unbalance m·e to a practically acceptable level — it cannot make it zero. A residual unbalance always remains, and in service the unbalance changes over time as debris collects on blades, bearings wear and the rotor distorts with heat. Balancing once does not guarantee permanent peace of mind; periodic vibration measurement and re-balancing are needed. Also, a bent shaft and misalignment are vibration causes distinct from unbalance and are not fixed by balancing. Correctly separating the causes of vibration is the first step toward a fix.

Rotating Unbalance Vibration Simulator

What is rotating unbalance vibration?

Frequently Asked Questions

Real-World Applications

Common Misconceptions and Pitfalls

How to Use

Worked Example

Practical Notes