Newton’s Laws of Motion
First Law — Inertia
A body remains at rest or in uniform straight-line motion unless acted upon by a net external force.
A body in motion travels in a straight line unless forced to change direction. This resistance to any change in motion is called inertia. The law defines what a force is: the only thing capable of altering a body’s state of motion.
\[\sum \vec{F} = 0 \implies \vec{v} = \text{constant}\]
Second Law — Force and Acceleration
The net force acting on a body equals its mass multiplied by its acceleration.
\[\boxed{F = ma}\]
Force and acceleration act in the same direction. The relationship is linear: doubling the net force doubles the acceleration; doubling the mass halves it for the same force.
| Force doubled |
Acceleration doubled |
| Mass doubled |
Acceleration halved |
| Both doubled |
Acceleration unchanged |
Third Law — Action and Reaction
For every force one body exerts on another, the second body exerts an equal and opposite force back on the first.
\[\vec{F}_{A \to B} = -\vec{F}_{B \to A}\]
Key points:
- The forces are equal in magnitude and opposite in direction.
- They act on different objects — they are never a cancelling pair.
- The reaction force has the same nature as the action (both contact, both gravitational, etc.).
Centrifugal and Centripetal Forces
A body in motion travels in a straight line unless forced to change direction (Newton’s First Law). When moving in a circle, its inertia causes it to resist that continuous change of direction — in a rotating reference frame, this resistance appears as a fictitious outward “centrifugal” force. The real inward force that continuously deflects the body onto the circular path is the centripetal force. Because circular motion involves a constant change of direction, velocity is always changing even at constant speed. For small angles, the magnitude of this change in velocity simplifies to \(v\theta\).
Static Balance
A system is statically balanced when the resultant of all centrifugal forces due to rotating masses is zero — i.e., the centre of mass lies on the axis of rotation. Static balance requires the vector sum of all \(mr\) terms to be zero:
\[\sum mr = 0\]
A single correction mass placed in the same plane can achieve static balance.
Dynamic Balance
Static balance alone is insufficient for systems with masses distributed across different axial planes. Dynamic balance requires two conditions to be satisfied simultaneously:
- The vector sum of all centrifugal forces equals zero (static condition).
- The vector sum of all moments of these forces about any reference plane equals zero.
\[\sum mr = 0 \qquad \text{and} \qquad \sum mr\ell = 0\]
This generally requires two correction masses placed in two chosen planes.
Balancing in a Single Plane
For coplanar rotating masses, a graphical or analytical approach uses the force polygon. Closing the \(mr\) vector polygon determines the required balancing mass and its angular position.
Balancing in Several Planes
For masses distributed along a shaft, the procedure is:
- Choose a reference plane and construct a couple polygon using \(mr\ell\) terms, where \(\ell\) is the axial distance from the reference plane.
- Closing the couple polygon gives the first correction mass (magnitude and angle).
- Construct the force polygon using \(mr\) terms to find the second correction mass.
Balancing of Reciprocating Machinery
A reciprocating engine generates forces that are fundamentally harder to balance than rotating masses. The piston moves in one plane only — adding an opposite mass in that plane creates a new out-of-balance force perpendicular to it. The problem cannot be solved by a single counterweight alone.
Primary and Secondary Forces
Treating the connecting rod as two lumped masses — one at the crank pin, one at the gudgeon pin — splits the system into a rotating part and a pure reciprocating part. The total reciprocating force is:
\[F = mr\omega^2\!\left(\cos\theta + \frac{r}{\ell}\cos 2\theta + \cdots\right)\]
| \(\cos\theta\) |
Primary force |
\(\omega\) (crank speed) |
| \(\dfrac{r}{\ell}\cos 2\theta\) |
Secondary force |
\(2\omega\) |
| Higher terms |
Negligible in practice |
— |
The secondary force arises from the obliquity of the connecting rod; higher harmonics are generally ignored. Written separately:
Primary force (first-order):
\[F_{\text{primary}} = mr\omega^2 \cos\theta\]
Secondary force (second-order):
\[F_{\text{secondary}} = \frac{mr\omega^2}{n}\cos 2\theta\]
where \(n = \ell / r\) is the ratio of connecting rod length \(\ell\) to crank radius \(r\).
Methods for Complete Balancing
Crankshaft Counterweight (Partial Balance)
Adding a mass opposite the crank pin cancels the rotating component entirely but only reduces the primary reciprocating force — it cannot eliminate it. This trades a vertical unbalance for a smaller horizontal one.
Multi-Cylinder In-Line Engines
By phasing cranks at equal angles, primary and secondary forces and couples can cancel across the engine. Balance requires:
\[\sum mr = 0 \quad \text{and} \quad \sum mrl = 0\]
for both primary and secondary components.
In multi-cylinder engine balancing there are four things to check:
F (Force): the vector sum of all \(mr\) terms. If non-zero, there is a net radial force shaking the engine.
C (Couple) — the vector sum of all \(mrℓ\) terms. If non-zero, there is a rocking moment twisting the engine about a transverse axis, even if the net force is zero.
| 2-cyl, 180° |
✓ |
✗ |
✓ |
✓ |
| 4-cyl, 180° |
✓ |
✓ |
✓ |
✗ |
| 6-cyl, 60° |
✓ |
✓ |
✓ |
✓ |
The six-cylinder in-line engine achieves complete balance for both primary and secondary effects.
Partial Balance
Complete balance is not always possible or practical — particularly for single-cylinder engines. In such cases a balance mass is added to reduce the dominant unbalanced force, and a residual force in one direction is accepted to eliminate the more harmful force in the perpendicular direction.
Critical Speed
Every rotating shaft is also an elastic beam with natural frequencies of lateral vibration. The critical speed is the rotational speed at which the excitation frequency from residual unbalance equals a natural frequency, producing resonance.
For a shaft carrying a disc, using the static deflection \(\delta_{st}\) under its own load:
\[\boxed{N_c = \frac{60}{2\pi}\sqrt{\frac{g}{\delta_{st}}}}\]
This is convenient because \(\delta_{st}\) can be directly measured or calculated from beam theory.
Behaviour Around Critical Speed
| Well below \(N_c\) |
Shaft runs stiffly; centre of mass ≈ geometric centre |
| Approaching \(N_c\) |
Amplitude grows rapidly — whirling begins |
| At \(N_c\) |
Resonance; amplitude theoretically unbounded (damping limits it) |
| Well above \(N_c\) |
Shaft self-centres; centre of mass migrates toward axis |
Many turbines and high-speed spindles operate supercritically — above the first critical speed — and simply pass through it during run-up.
Multiple Critical Speeds
A real shaft has infinitely many lateral modes, each giving a critical speed:
\[N_{c_n} \propto n^2 \quad \text{(uniform simply-supported shaft)}\]
The first critical speed is the most dangerous: it is encountered first and produces the largest deflection amplitude.
Summary of Equations
| Centrifugal force |
\(F = mr\omega^2\) |
| Static balance |
\(\sum mr = 0\) |
| Dynamic balance |
\(\sum mr = 0\) and \(\sum mr\ell = 0\) |
| Primary reciprocating force |
\(F_{\text{primary}} = mr\omega^2 \cos\theta\) |
| Secondary reciprocating force |
\(F_{\text{secondary}} = \dfrac{mr\omega^2}{n}\cos 2\theta\) |
| Critical speed |
\(N_c = \dfrac{60}{2\pi}\sqrt{\dfrac{g}{\delta_{st}}}\) |
Summary
Balancing is fundamentally a vector problem. Both graphical (polygon) and analytical methods are used throughout. Dynamic balancing always requires satisfying both force and moment equilibrium, building progressively from simple single-plane cases up to the complexity of multi-cylinder engine analysis. Critical speed analysis adds a further dimension: even a perfectly balanced shaft can experience destructive resonance if its operating speed coincides with a lateral natural frequency.