Dynamic Force Analysis of Machinery

Introduction

A machine transmits motion and power through connected bodies. In an ideal kinematic analysis, links are treated as weightless and inertia is ignored. Real machines, however, contain masses that accelerate, decelerate, rotate, and reciprocate. Their inertia produces forces and moments that influence bearing loads, frame vibration, shaft torque, noise, fatigue, and power requirements.

Dynamic force analysis determines the forces required to produce a prescribed motion and the reactions developed at joints and supports. The principal tools are Newton–Euler equations, which apply directly to real forces and moments, and D’Alembert’s principle, which introduces inertia forces and converts a dynamic problem into an equivalent static equilibrium problem.

This chapter develops these methods for rotating and reciprocating machinery. The treatment assumes rigid links, ideal pin or sliding joints, and known motion unless stated otherwise. Flexible-body vibration, lubrication forces, backlash, and nonlinear contact are outside the basic scope, although their effects are discussed later.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Explain the physical meaning of inertia forces and inertia couples.
2. Apply Newton–Euler and D’Alembert formulations to machine links.
3. Determine shaking forces and shaking moments in reciprocating mechanisms.
4. Calculate joint reactions and driving torque.
5. Estimate instantaneous power and mean power requirements.
6. Identify assumptions, limitations, and practical sources of error in dynamic analysis.

1. Fundamental Concepts

1.1 Force systems in machinery

A link in a machine may be subjected to:

• externally applied forces, such as gas pressure, spring force, friction, and gravity;
• joint forces from pins, sliders, bearings, and guides;
• forces caused by its translational acceleration;
• moments caused by its angular acceleration;
• externally applied couples, such as motor torque or resisting torque.

For a rigid body of mass m, the resultant force is related to the acceleration of its centre of mass G by

where ∑F is the vector sum of all external forces in newtons, m is mass in kilograms, and a_G is the acceleration of the centre of mass in metres per second squared.

The rotational equation about the centre of mass is

where ∑M_G is the resultant external moment about G in newton-metres, I_G is the mass moment of inertia about the relevant axis in kg·m², and α is angular acceleration in rad/s². These equations apply to plane motion of a rigid body when forces and moments are expressed consistently.

For general three-dimensional motion, the vector equation involves angular momentum and products of inertia. The present chapter focuses on planar mechanisms, which cover many engines, pumps, compressors, presses, and balancing systems.

1.2 Newton–Euler formulation

The Newton–Euler method draws a free-body diagram of each link and writes force and moment equations directly. For planar motion, the three scalar equations are

The method is especially transparent when joint forces are required. Its main discipline is correct free-body-diagram construction: every force must have a physical source, and action–reaction pairs must be assigned to different bodies.

1.3 D’Alembert’s principle

D’Alembert’s principle introduces an inertia force

acting through the centre of mass, and an inertia couple

The negative signs indicate that these fictitious quantities oppose acceleration. The dynamic equations can then be written in static-equilibrium form:

D’Alembert’s principle does not create an additional physical force. It is an analytical device that allows conventional equilibrium techniques to be used for an accelerating body.

1.4 Translational and rotational inertia

For pure translation, α = 0 and only the inertia force is required. For pure rotation about a fixed axis, the centre of mass may have centripetal acceleration while the body also experiences an inertia torque. For general plane motion, both terms are present.

The acceleration of a point P on a rigid link relative to G is

Here ω is angular velocity, α is angular acceleration, and r_{P/G} is the position vector from G to P. The second term is tangential acceleration and the third is normal or centripetal acceleration. In planar calculations, these vectors are commonly resolved into x and y components.

2. Dynamic Analysis Procedure

A reliable solution generally follows this sequence:

1. Specify the configuration and input angular velocity.
2. Perform kinematic analysis to determine positions, velocities, and accelerations.
3. Identify applied loads, masses, centres of mass, and mass moments of inertia.
4. Isolate each link with a free-body diagram.
5. Apply Newton–Euler equations, or add D’Alembert inertia forces and couples.
6. Solve joint reactions and the required driving force or torque.
7. Check force balance, units, signs, and limiting cases.
8. Repeat over the operating cycle if peak loads, shaking forces, or power variation are required.

The dynamic analysis is only as good as the preceding kinematic analysis. A small error in acceleration can produce a large error in inertia force, particularly at high speed because many acceleration terms vary with angular speed squared.

3. Reciprocating Mechanisms

3.1 Slider–crank mechanism

A slider–crank mechanism consists of a crank of radius r, a connecting rod of length l, and a slider. Let θ be the crank angle measured from the line of stroke and let ω be constant crank angular velocity. The slider displacement from inner dead centre can be expressed exactly by geometry. A commonly used approximate acceleration equation for a sufficiently long connecting rod is

The sign convention must be stated. In this expression, a_s is directed according to the selected positive slider coordinate. The first term is the primary acceleration and the second is the secondary acceleration. The approximation is most useful when l/r is relatively large; it becomes less accurate as the connecting rod becomes short.

The corresponding inertia force of a reciprocating mass m_r is

Thus, even with constant crank speed, the reciprocating inertia force changes continuously and contains primary and secondary components. The primary component has crank-frequency variation, while the secondary component has twice the crank frequency.

3.2 Gas or external force

In an engine, the force on the piston due to cylinder pressure is

where p is the gauge pressure acting on the piston in pascals and A is piston area in square metres. If atmospheric pressure, underside pressure, friction, or pumping effects are relevant, the net pressure force must be used rather than pA alone.

The net force transmitted through the connecting rod is affected by gas force, inertia force, gravity, and friction. For an ideal frictionless piston, the piston equation along the line of stroke is

where F_{pin,s} is the connecting-rod force component on the piston and m_p is piston mass. A positive or negative result indicates the actual direction relative to the assumed directions.

3.3 Connecting-rod obliquity and side thrust

The connecting rod is generally inclined to the line of stroke. If φ is its inclination, the rod force R has components

where R_s is the line-of-stroke component and R_n is the lateral component. The lateral component produces piston side thrust on the cylinder wall. Side thrust contributes to friction and wear and is one reason that rod length, piston offset, and crank geometry matter in engine design.

For an ideal slider without friction, the slider guide reaction balances the transverse component of the rod force. With friction, the reaction also contributes a force opposing sliding motion, and the required rod force must be found using an appropriate friction model.

3.4 Crank torque

The instantaneous torque about the crank centre is the moment of forces acting on the crank. If a tangential force F_t acts at crank radius r, then

More generally, if a force F acts at a perpendicular distance d from the crank centre,

The driving torque required at a specified instant must balance the torque produced by gas or external forces, inertia effects, friction, gravity, and any resisting load. It is incorrect to calculate torque from gas force alone when reciprocating masses are significant.

4. Rotating Machinery and Unbalance

4.1 Centrifugal force

A mass m rotating at radius r with angular velocity ω has radial acceleration rω². The required centripetal force is

The equal and opposite force transmitted to the shaft, bearings, or frame is commonly called the centrifugal or unbalance force. Its magnitude increases with the square of speed. Therefore, doubling speed increases an unbalance force by a factor of four, provided mass and radius remain unchanged.

For several rotating masses, complete balance in a plane requires

when the vector quantities are summed. For masses in different axial planes, both force and couple balance are needed. The couple-balance condition is

where a is the axial distance from a selected reference plane and vector signs represent angular positions.

4.2 Shaking force and shaking moment

A shaking force is the resultant unbalanced force transmitted by a mechanism to its frame. For a reciprocating mass, it is associated primarily with m_r a_s. For rotating masses, it is associated with the vector resultant of rotating unbalance forces.

A shaking moment is the resultant moment of these dynamic forces about a reference point. A machine may have zero resultant force but a nonzero shaking moment, or zero shaking moment but a nonzero resultant force. Both conditions must be examined in high-speed machinery.

Balancing strategies include adding counterweights, arranging cylinders with suitable phase angles, using opposed reciprocating masses, and selecting crank configurations that cancel primary or secondary components. Complete balancing may be impossible because a counterweight that balances reciprocating force can introduce an undesirable vertical or transverse force.

5. Worked Example: Slider–Crank Inertia Force

A slider–crank has crank radius r = 0.05 m, connecting-rod length l = 0.20 m, reciprocating mass m_r = 2.0 kg, and crank speed 1200 rev/min. Determine the approximate reciprocating inertia force at θ = 0° and θ = 90°, using the sign convention of the acceleration equation above.

First convert speed:

At θ = 0°,

Therefore,

At θ = 90°, cos θ = 0 and cos 2θ = cos 180° = −1:

Thus,

The signs show direction relative to the selected positive slider coordinate. The large difference between the two values demonstrates why inertia loading must be evaluated throughout the cycle rather than at one arbitrary position.

6. Joint Reactions in a Connecting Rod

Consider a connecting rod of mass m, centre-of-mass acceleration a_G, angular acceleration α, and mass moment of inertia I_G. Let the pin forces at the crank and piston ends be resolved into x and y components. If the forces on the rod are F_A and F_B, the equations are

Taking moments about G eliminates the unknown translational resultant through G:

The moment equations include the perpendicular distances from G to the force lines of action. Solving the three simultaneous equations gives the pin reactions. The force exerted by the rod on an adjacent link is equal and opposite to the force exerted by that link on the rod.

A common shortcut is to neglect connecting-rod mass and rotational inertia. This may be acceptable for preliminary analysis when the rod is light relative to the piston and crank assembly, but it is not valid automatically. At high speed, even a modest rod mass can create significant angular inertia torque.

7. Driving Torque and Power

Instantaneous mechanical power is

where P is power in watts, T is torque in newton-metres, and ω is angular velocity in rad/s. This relation applies to a rotating shaft and uses angular velocity in rad/s, not revolutions per minute.

If torque varies during a cycle, the mean power over one complete cycle is

For constant angular speed, this becomes

where T_mean is the time-average torque. The work over a cycle is the area under the torque–angle diagram:

Since torque multiplied by angular displacement has units of joules, the diagram provides a direct energy interpretation. A flywheel stores and releases kinetic energy to reduce speed fluctuation when input torque and resisting torque vary over a cycle. Its energy change is

for a flywheel of moment of inertia I when speed changes between ω_min and ω_max.

8. Applied Case: Dynamic Loading of a Single-Cylinder Compressor

Background

Consider an illustrative single-cylinder compressor operating at constant crank speed. The piston, pin, and a portion of the connecting rod are represented by an equivalent reciprocating mass. Cylinder pressure varies strongly during compression, while the crankshaft experiences a nearly constant external resistance.

Problem

The designer observes excessive bearing vibration and wants to determine whether the dominant cause is gas torque, reciprocating inertia, or rotating unbalance.

Analysis approach

The mechanism is analysed at small crank-angle increments. For each increment:

1. piston position, velocity, and acceleration are calculated;
2. cylinder pressure is obtained from measured pressure data;
3. gas force is calculated from net pressure multiplied by piston area;
4. reciprocating inertia force is calculated as −m_r a_s;
5. connecting-rod and crank-pin reactions are obtained from Newton–Euler equations;
6. crank torque is found from the moment of all forces about the shaft centre;
7. the resulting force and moment transmitted to the frame are resolved into harmonic components.

The results show a large primary shaking force at crank frequency and a smaller secondary force at twice crank frequency. The gas-pressure torque dominates during the compression portion of the cycle, whereas inertia torque becomes important near dead centres because acceleration is high. A small rotor eccentricity produces a force proportional to ω² and is therefore especially significant at the highest operating speed.

Solution and outcome

The engineering response is not simply to increase bearing size. The recommended sequence is to verify rotor balance, reduce unnecessary reciprocating mass, adjust counterweight magnitude and phase, and use a torsional and structural vibration assessment. A flywheel can smooth angular-speed fluctuation but cannot eliminate frame shaking forces. Mount stiffness and isolation may reduce transmitted vibration, but they do not remove the source forces.

Learning points

The case illustrates that dynamic loading is a superposition of several effects with different phase relationships. Correct diagnosis requires time-resolved force and torque data, not only average power. It also shows why a mechanism can deliver acceptable mean power while still suffering damaging peak bearing reactions.

9. Assumptions, Limitations, and Practical Accuracy

The basic rigid-body model usually assumes:

• rigid links and an undeformable frame;
• ideal pin and sliding joints;
• known mass properties;
• negligible clearance and backlash;
• prescribed crank speed or motion;
• friction neglected or represented by a simple known model;
• no external vibration or flexible support motion.

These assumptions can be unsuitable when operating speed approaches a structural natural frequency, when bearing clearance is large, when impacts occur, or when flexible shafts and rods significantly deform. In such cases, multibody dynamics, finite-element modelling, experimental modal analysis, and measured force validation may be required.

Sign conventions are a major source of error. A force calculated as negative is not wrong; it means its actual direction is opposite to the assumed direction. Every reported reaction should therefore include its magnitude, direction, and reference configuration.

Dimensional checks are essential. In F = ma, kg × m/s² = N. In T = rF, m × N = N·m. In P = Tω, N·m × rad/s = W because the radian is dimensionless. Speed must be converted from rev/min before use in acceleration and power equations.

10. Professional Practice and Contemporary Tools

Dynamic analysis is commonly implemented in spreadsheets, symbolic algebra systems, numerical scripts, and multibody simulation software. Software does not replace engineering judgment: the analyst must select coordinate systems, define joint constraints, enter correct mass properties, and validate predicted reactions against physical measurements.

Best practice includes:

• documenting all geometry, mass, inertia, and sign conventions;
• reporting both peak and mean loads;
• checking conservation of energy where appropriate;
• comparing simplified and detailed models;
• validating critical predictions using strain gauges, accelerometers, torque transducers, or pressure sensors;
• considering fatigue, noise, vibration, and safety factors rather than only static strength;
• retaining traceable assumptions and version-controlled computational models.

AI-assisted modelling can help generate parameter sweeps, identify unusual trends, or fit pressure data, but its outputs require independent verification. Safety-critical force predictions must remain auditable and physically interpretable.

Chapter Summary

Dynamic force analysis extends kinematic analysis by accounting for mass, acceleration, and angular acceleration. Newton–Euler equations relate external forces and moments to translational and rotational acceleration. D’Alembert’s principle introduces opposing inertia forces and couples so that an accelerating body can be treated through an equivalent equilibrium diagram.

In slider–crank mechanisms, reciprocating inertia force contains primary and secondary components. Connecting-rod obliquity produces side thrust, while pressure, inertia, friction, and gravity together determine joint reactions and crank torque. Rotating unbalance produces forces proportional to mass, eccentricity, and the square of angular speed. Complete balancing requires both force and couple conditions.

Driving torque varies over a cycle, and power is obtained from torque multiplied by angular velocity. A flywheel smooths speed variation by exchanging kinetic energy, but it does not eliminate shaking forces. Reliable analysis requires accurate kinematics, carefully drawn free-body diagrams, consistent units, explicit assumptions, and validation against measurements or more detailed models.

Key Terms

Inertia force: An analytical force equal to −m a, opposing translational acceleration.

Inertia couple: An analytical couple equal to −Iα, opposing angular acceleration.

Newton–Euler method: Direct application of translational and rotational equations of motion.

D’Alembert’s principle: Conversion of a dynamic problem into equivalent static equilibrium by adding inertia actions.

Centre of mass: Point at which the translational motion of a body’s mass may be represented.

Mass moment of inertia: Measure of resistance to angular acceleration.

Reciprocating mass: Mass that moves substantially along a repeated linear path.

Primary inertia force: Component varying at crank frequency.

Secondary inertia force: Component varying at twice crank frequency in a slider–crank approximation.

Shaking force: Unbalanced dynamic force transmitted to the machine frame.

Shaking moment: Unbalanced moment transmitted to the machine frame.

Obliquity angle: Inclination of a connecting rod from the line of stroke.

Joint reaction: Force exerted between connected machine members.

Unbalance: Unequal distribution of rotating mass about the rotation axis.

Driving torque: Torque required to maintain prescribed machine motion.

Resisting torque: Torque opposing the driving motion.

Mean power: Average power over a specified time or operating cycle.

Flywheel: Rotating energy-storage element used to reduce speed fluctuation.

Free-body diagram: Isolated-body representation showing all external actions.

Review Questions

Short Answer Questions

1. What is the physical meaning of an inertia force?
2. State the planar Newton–Euler equations for a rigid body.
3. Why does a reciprocating inertia force contain primary and secondary components?
4. What is meant by a shaking force?
5. Why must angular speed be expressed in rad/s when using P = Tω?

Long Answer Questions

1. Explain D’Alembert’s principle and compare it with the Newton–Euler method.
2. Derive the approximate acceleration expression for a slider–crank mechanism and discuss its limitations.
3. Explain how connecting-rod obliquity produces piston side thrust.
4. Describe the procedure for determining joint reactions in a moving connecting rod.
5. Explain the difference between force balancing and couple balancing in rotating machinery.

Analytical / Application Questions

1. A slider–crank has known r, l, m_r, and constant ω. Develop a procedure to calculate the inertia force at every crank angle and identify the angle producing the largest approximate value.
2. A rotating assembly has a measured vibration force that increases by a factor of approximately four when speed doubles. What physical mechanism does this suggest, and what additional measurement would help confirm the diagnosis?
3. For a compressor with measured cylinder pressure and known geometry, propose a method for constructing the instantaneous crank-torque diagram and estimating mean power and flywheel energy fluctuation.

Multiple Choice Questions with Answers

Question 1

Which equation represents the translational Newton equation for a rigid link?

A. ∑F = m a_G

B. ∑M = I_G α

C. P = Tω

D. F = m r ω²

Correct Answer: A. ∑F = m a_G

Question 2

In D’Alembert’s principle, the inertia force on a body is

A. m a_G in the direction of acceleration

B. −m a_G opposite to acceleration

C. I_G α through the centre of mass

D. zero for every rotating body

Correct Answer: B. −m a_G opposite to acceleration

Question 3

For a rotating mass, the magnitude of centrifugal unbalance force is

A. m r ω

B. m r/ω²

C. m r ω²

D. mω/r²

Correct Answer: C. m r ω²

Question 4

The secondary inertia component in the approximate slider–crank acceleration varies with

A. crank frequency

B. twice crank frequency

C. half crank frequency

D. no frequency

Correct Answer: B. twice crank frequency

Question 5

A 2 kg mass has acceleration 300 m/s². The magnitude of its inertia force is

A. 150 N

B. 298 N

C. 600 N

D. 900 N

Correct Answer: C. 600 N

Question 6

The correct relation for instantaneous shaft power is

A. P = T/ω

B. P = Tω

C. P = ω/T

D. P = T + ω

Correct Answer: B. P = Tω

Question 7

A force acting at perpendicular distance d from a crank centre produces torque

A. F/d

B. Fd

C. F + d

D. d/F

Correct Answer: B. Fd

Question 8

Complete balance of rotating masses in one plane requires the vector condition

A. ∑m = 0 only

B. ∑r = 0 only

C. ∑mr = 0

D. ∑m/r = 0

Correct Answer: C. ∑mr = 0

Question 9

Which component is not generally eliminated by a flywheel?

A. Angular-speed fluctuation

B. Stored kinetic-energy variation

C. Frame shaking force

D. Cycle-to-cycle torque variation at the shaft

Correct Answer: C. Frame shaking force

Question 10

If rotational speed doubles while rotating mass and radius remain constant, the centrifugal force becomes

A. half as large

B. twice as large

C. four times as large

D. unchanged

Correct Answer: C. four times as large