Dynamic Analysis of Linkages - Theory of Machines (TOM) - Mechanical Engineering

Introduction

When a number of bodies are connected so that the motion of one body produces a predictable motion in the others, the assembly is called a mechanism. A mechanism transfers motion and often transmits power. The individual parts which have relative motion with respect to other parts are called kinematic links or elements. Each link must be a resistant body so that it can transmit force and motion without excessive deformation.

Kinematic Link, Rigid and Resistant Bodies

• Kinematic Link or Element: Every part of a machine having some relative motion with respect to another part is a kinematic link or element. Links together with joints form mechanisms.
• Rigid Body: A body is said to be rigid when distances between any two of its points remain constant under applied loads (no deformation considered in kinematic analysis).
• Resistant Body: A body which deforms to some extent under external load but still resists and transmits force and motion. For example, belts are treated as rigid under tensile loading for kinematic purposes but behave as resistant bodies when deformation (stretching) is considered in more detailed analysis.

Types of Links

• Rigid Links: Deformations are negligible. Examples: crank, connecting rod, piston, cylinder.
• Flexible Links: Deformation occurs but within permissible limits. Examples: belt drives, rope drives.
• Fluid Links: Power is transmitted through fluid pressure; the fluid behaves as a link. Examples: hydraulic brake, hydraulic lift, hydraulic jack.

Constrained Motion

Constrained motion is the desired, predictable motion obtained from a mechanism when an input is applied. The constraints are provided by geometry and the joints between links.

• Completely constrained motion: Motion is constrained by the system itself so that the motion between elements is definite and independent of the direction of applied forces.
• Successfully constrained motion: Motion is constrained by external supports or surrounding conditions. The mechanism may permit more than one possible motion, but the surroundings restrict the output to a single desired motion. Example: a shaft that could both reciprocate and rotate but is prevented from reciprocation by bearings and loads, so it only rotates.
• Incompletely constrained motion (Unconstrained): For a given input more than one output is possible; the motion between elements of a pair may occur in more than one direction and depends on applied forces. Example: removing a guiding collar may allow a shaft to both rotate and translate.

Kinematic Pair

Two links are said to form a kinematic pair when they are in contact and their relative motion is constrained by the contact. Pairs are classified in several ways.

Classification according to type of relative motion

• Turning pair (Revolute pair / Pin joint): Relative motion is pure rotation. Example: circular shaft in a bearing; hinge joints.
• Sliding pair (Prismatic pair): Relative motion is pure translation. Example: rectangular slider in a guide.
• Rolling pair: Relative motion is pure rolling without slipping. Example: wheel rolling on a flat surface; balls in a ball bearing.
• Screw pair (Helical pair): Relative motion combines rotation and translation in a fixed ratio - obtained by mating helical threads. Example: lead screw and nut.
• Spherical pair (Ball-and-socket): Relative motion is rotation in three dimensions about a common centre. Example: vehicle mirror joint; ball-and-socket joint.

Classification according to type of contact

• Lower pair: Area or surface contact between links. Examples: shaft in bearing, slider-crank pairs.
• Higher pair: Line or point contact between links. Examples: cam and follower, gear teeth (approximately line contact), rolling contact between wheel and surface.
• Wrapping pair: One link wraps around another producing many contact points similar to higher pairs. Example: belt wrapped over a pulley.

Classification according to closure of contact

• Closed (Self-closed) pair: Links are held together mechanically to maintain contact (permanent contact). Most lower pairs are closed pairs.
• Open (Forced-closed) pair: Contact is maintained only by external forces; if the external force is removed, contact may be lost. Examples: some clutches, door closers.

Kinematic Chain

A kinematic chain is a series of links connected by kinematic pairs so that the first link is connected to the last link, forming a closed chain whose relative motions are constrained. If one link of a closed chain is fixed, the assembly becomes a linkage or mechanism.

Degrees of Freedom (Mobility)

• Definition: The minimum number of independent parameters required to completely specify the position of all links of a mechanism is the degrees of freedom (DOF) or mobility.
• Alternate view: Number of independent inputs required to determine the constrained motion of the mechanism.
• General relation: DOF = 6 - restraints for a rigid body in space; for planar mechanisms refer to the planar form of Kutzbach-Grübler criterion below.

Maximum number of motions possible of an object in 3D

Typical restraints and DOF for simple pairs (planar mechanisms)

• Turning (revolute) pair: Restraints = 5; DOF = 1 (one relative rotation).
• Lower pair (general): In planar lower pairs DOF = 1; spherical pair in space has DOF = 3.
• Higher pair: Often contributes DOF = 2 in planar contact (depends on contact constraint).

Degrees of Freedom of Planar Mechanisms - Kutzbach's Equation

For planar mechanisms, Kutzbach's equation (also called Kutzbach-Grübler criterion in literature) gives the mobility:

F = 3(l - 1) - 2j - h

where

• F = degrees of freedom (mobility) of the mechanism
• l = number of links (including the fixed frame)
• j = number of binary joints (lower pairs)
• h = number of higher pairs

If redundant motions exist (motions that do not affect the intended output), include the redundant count Fr:

F = 3(l - 1) - 2j - h - Fr

Redundant motion: Motion of a link that does not influence the primary outputs of the mechanism. Example: a link that rotates within a groove but does not change the overall kinematic behaviour.

Redundant motion

In the figure above, link 3 may rotate within a groove without affecting the mechanism's output; such motion is redundant.

Redundant link

Link 4 in the figure above is a redundant link if removing it does not change the mobility of the mechanism.

Linkage vs Structure

• Linkage / Mechanism: A kinematic chain in which at least one link is fixed and the motion of a movable link produces definite motions in other links; typically F ≥ 1.
• Structure / Frame: A mechanism whose degree of freedom is zero (F = 0); no relative motion is possible between links.
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Types of Chains

• Kinematic chain: Chain whose degree of freedom equals one (F = 1).
• Unconstrained chain: Chain whose degree of freedom is greater than one (F > 1).

Note: Degrees of freedom is the number of independent inputs required to obtain a constrained output.

Grübler's Equation (Special case)

When all joints are lower pairs (h = 0) and the mechanism is planar with a single degree of freedom (F = 1), Kutzbach's equation simplifies to the form commonly called Grübler's equation for planar 1-DOF linkages:

F = 3(l - 1) - 2j - h

For F = 1 and h = 0:

1 = 3(l - 1) - 2j

which reduces to:

3l - 2j - 4 = 0

This shows that in planar single-DOF mechanisms made entirely of lower pairs, the number of links l must be even (since 3l must be even). The smallest practical closed chain satisfying this is l = 4, giving the well-known four-bar linkage.

Four-bar Mechanism

• Definition: A simple closed chain consisting of four links connected by four turning (revolute) pairs.
• Practical condition: For the four-bar to be physically possible, the sum of the three shorter links must be greater than the longest link (triangle inequality in closed chain).

Four-Bar Mechanism

Inversions of Four-bar Mechanism

Different mechanisms are obtained by fixing different links of the four-bar; these are called inversions. For a four-bar (l = 4), the number of inversions is four (one for each link being grounded). Common inversions:

• Double crank mechanism - both adjacent links to the fixed link can fully rotate.
• Crank-rocker mechanism - one link oscillates (rocker), the other is a crank.
• Double rocker mechanism - both links oscillate.

Grashof's Law

Grashof's condition predicts whether a given four-bar linkage will allow continuous relative rotation between some links. Let s = length of shortest link, l = length of longest link, p and q = lengths of the other two links. Grashof's condition states:

(s + l) ≤ (p + q) gives at least one link capable of full rotation relative to the others.

• If (s + l) < (p + q), Grashof's condition is satisfied and at least one crank can rotate fully. Depending on which link is fixed, different inversions occur: if the shortest link is fixed → double crank; if the shortest is adjacent to the fixed link → crank-rocker; if the shortest is the coupler → double rocker.
• If (s + l) = (p + q), the mechanism is at the limit (change point) and may still permit full rotation depending on link arrangement.
• If (s + l) > (p + q), Grashof's condition is not satisfied and no link can make a full rotation - leads to double rocker behaviour.

Special cases with equal opposite links produce parallelogram or deltoid linkages depending on arrangement, giving characteristic motions (e.g., parallelogram linkage produces parallel displacement).

Transmission Angle

Transmission angle μ is defined as the angle between the output link and the coupler (the link connecting input and output). It is an important indicator of force transmission efficiency: values close to 90° are desirable. μ varies with input crank angle; it is maximum when crank angle θ = 180° and minimum when θ = 0° in the usual four-bar geometry.

Single Slider-Crank Mechanism

A single slider-crank mechanism consists of four links connected by three turning pairs and one sliding pair: crank, connecting rod (coupler), slider and frame. It is widely used to convert reciprocating motion to rotation and vice versa (example: engine piston-crank assembly).

Inversions of Single Slider-Crank

• Crank fixed: Whitworth quick-return mechanism; rotary internal combustion engine arrangements.
• Connecting rod fixed: Crank and slotted-lever quick-return mechanism; oscillating cylinder engines.
• Piston/slider fixed: Hand pump (bull engine/pendulum pump).

Crank and slotted lever quick return motion mechanism

The crank and slotted lever quick-return mechanism is an inversion where the connecting rod is fixed, producing unequal cutting and return stroke durations in shaping and slotting machines.

Let β = cutting stroke angle, α = return stroke angle. Then α + β = 360° and the quick return ratio (QRR) is defined as the ratio of cutting stroke time to return stroke time; QRR > 1 for a quick-return mechanism.

Double Slider-Crank Chain

A chain with four links connected by two turning pairs and two sliding pairs is called a double slider-crank chain. Different inversions are obtained by fixing different links:

Double Slider Crank Chain

• Slotted bar fixed: Elliptical trammels (generates approximate ellipse).
• One slider fixed: Scotch-yoke mechanism (direct conversion between rotation and reciprocation).
• Link connecting sliders fixed: Oldham's coupling (to accommodate parallel but offset shafts).

Mechanical Advantage of a Mechanism

The mechanical advantage (M.A.) is the ratio of output force (or torque) to input force (or torque) at any instant. For mechanisms where power is transmitted, instantaneous mechanical advantage equals the ratio of corresponding virtual displacements or the inverse ratio of corresponding velocity ratios.

Toggle Position

A mechanism is said to be in toggle position when the angle between the input link and the coupler becomes 180°. In this position the velocity of the output link is zero for an infinitesimal input, so the instantaneous mechanical advantage tends to infinite. Toggle positions are used in press or clamp mechanisms to produce large forces from small input motions, but they also produce singularities in the motion and require careful design consideration.

Dynamic Analysis of Linkages - Introduction

Dynamic analysis extends kinematic analysis by including the effects of mass (inertia) and external forces. In practice, it is essential for determining driving torque (or power) required, reaction forces in bearings, shaking forces transmitted to the frame, and for designing flywheels and balancing measures for smooth operation.

Basic concepts

• Inertia force: The product of mass and acceleration (m·a) acting opposite to the acceleration of the mass centre (D'Alembert's principle representation).
• Inertia torque: For a rotating link about its mass centre, inertia torque = I·α where I is mass moment of inertia about the considered axis and α is angular acceleration.
• Shaking force: Unbalanced inertial force transmitted to the frame, usually resulting from reciprocating masses (e.g., piston mass) and unbalanced rotating masses.
• Turning moment: Net torque on the crankshaft at any instant = driving torque (from pressure or prime mover) - resisting torque (from load) - inertia torques of rotating links and accessories.

D'Alembert's Principle (for dynamics of mechanisms)

D'Alembert converts a dynamic problem into a static-like problem by introducing inertial forces and moments. For each rigid body in the mechanism:

• Apply the real external forces and moments.
• Apply the inertial force equal to -m·a at the mass centre (opposite to acceleration).
• Apply the inertial moment equal to -I·α about the mass centre (opposite to angular acceleration).
• Enforce static equilibrium of forces and moments to obtain reaction forces, input torque, or other unknowns.

Dynamic force analysis of slider-crank mechanism (outline)

To determine the driving torque required by a slider-crank (common in reciprocating engines), follow these steps conceptually:

• Perform kinematic analysis to obtain positions, velocities and accelerations of piston, connecting rod and crank as functions of crank angle θ.
• Compute inertia force of the reciprocating mass (piston + small end of connecting rod, usually modelled as lumped mass): F_inertia = m_r · a_p, acting opposite to piston acceleration.
• Resolve inertia forces along the line of action to obtain equivalent forces/moments on the crank.
• Compute inertia torque of rotating links (crank, crankpin mass, flywheel), T_inertia = I_total · α_crank.
• Compute instantaneous resisting force at the piston (e.g., gas pressure × piston area) and resolve to a torque at crank: T_resist = F_resist × crank radius × transmission ratio (function of θ).
• Sum moments about crankshaft: T_drive = T_resist + T_inertia + T_friction + T_acceleration_terms. The required driving torque varies with crank angle and defines the turning moment diagram.

Mathematical relations for slider-crank kinematics (summary)

For a slider-crank with crank radius r, connecting rod length l, crank angle θ (measured from inner dead centre), piston displacement x (measured from inner dead centre):

x = r(1 - cos θ) + (l - √(l² - r² sin²θ))

Velocity of piston: v = dx/dt = (dx/dθ)·ω where ω is crank angular speed.

Acceleration of piston: a = d²x/dt² = (d²x/dθ²)·ω² + (dx/dθ)·α where α is crank angular acceleration (often α = 0 in steady-speed operation so a ≈ (d²x/dθ²)·ω²).

Inertia force and equivalent crank torque

For reciprocating mass m_r:

Inertia force F_in = m_r · a (directed opposite to piston acceleration).

Equivalent inertia torque on crank: T_in_eq = F_in · r · sin φ

where φ is the angle between the connecting rod and crank or appropriate lever geometry angle; sign conventions must be consistent when summing torques.

Turning moment diagram, mean torque and fluctuation of energy

The turning moment diagram is a plot of instantaneous crank torque versus crank angle. Areas above the mean torque line correspond to energy delivered to the crank; areas below correspond to energy removed (use during compression, cut-off). The mean torque corresponds to average power divided by angular speed. The fluctuation of energy is the difference between the maximum and minimum accumulated energy in a crank revolution and is countered by the flywheel.

Flywheel design (brief)

A flywheel stores kinetic energy to smooth out the fluctuation of energy and maintain near-uniform angular speed. For a given maximum permissible fluctuation of speed, the required moment of inertia I_f of the flywheel is found from energy fluctuation ΔE and allowable speed variation.

Typical design relation: ΔE = I_f · ω · Δω ≈ I_f · ω² · (Δω/ω) where Δω is the allowable variation in angular speed. From turning moment diagram calculate ΔE per revolution, then obtain I_f.

Shaking forces and balancing

• Shaking force: Caused by unbalanced reciprocating masses and unbalanced rotating masses. These produce alternating forces on the frame leading to vibrations.
• Primary balancing: Cancelling first-order (ω²) inertial forces by arranging counterweights so resultant centripetal and reciprocating force components are reduced. Complete balancing in single-cylinder in-line engines is not possible for all orders without additional balancing shafts.
• Secondary balancing: Addresses second-order (2ω) effects, important when connecting rod length is finite - piston acceleration contains both primary and secondary components due to connecting rod angularity.
• Partial balancing: Often used to reduce transmitted forces at design operating speeds when full balancing would create unacceptable rotating couples.

Example outline - Dynamic torque requirement for a slider-crank (conceptual)

Given: reciprocating mass m_r, crank radius r, connecting rod length l, engine speed N (rev/s), gas pressure distribution p(θ) on piston, friction torque estimate T_f.

Procedure (conceptual):

• Compute ω = 2πN.
• From geometrical relations obtain piston acceleration a(θ) = (d²x/dθ²)·ω².
• Compute inertia force F_in(θ) = m_r · a(θ) acting on piston.
• Compute equivalent inertia torque T_in(θ) about crank: resolve F_in through crank-rod geometry to crank radius.
• Compute gas force on piston F_g(θ) = p(θ)·A (A = piston area) and convert to torque T_g(θ) = F_g(θ)·r·(function of θ based on geometry).
• Compute instantaneous crank torque: T_crank(θ) = T_g(θ) - T_in(θ) - T_f (sign convention positive if driving).
• Plot turning moment diagram T_crank(θ) vs θ, find mean torque T_mean = (1/2π) ∫_0^{2π} T_crank(θ) dθ.
• Compute fluctuation of energy from areas between T_crank curve and mean torque line; use ΔE to size flywheel so that allowable speed variation is not exceeded.

Applications and design considerations

• Machine tools: Quick-return mechanisms and slider-crank arrangements are used in shapers, slotters and some presses.
• Internal combustion engines: Slider-crank for piston motion; dynamic analysis is essential for selecting crankshaft section, counterweights, and flywheel design to limit speed fluctuation and bearing loads.
• Pumps and compressors: Reciprocating mechanisms require dynamic balancing to reduce frame vibrations and increase life.
• Robotics and manipulators: Linkage dynamics affect actuator sizing, control, and trajectory planning.
• Balancing and vibration reduction: Correct dynamic design reduces noise, wear, and improves precision in machines.

Summary

This chapter summarises the core kinematic concepts required before dynamic analysis: types of links and pairs, kinematic chains, degrees of freedom using Kutzbach's equation, four-bar and slider-crank mechanisms and their inversions, Grashof's condition, transmission angle and toggle positions. Dynamic analysis then introduces inertia forces and moments using D'Alembert's principle, outlines the method to obtain instantaneous torques and forces (turning moment diagram), and shows how these results inform flywheel design, balancing and vibration control. These topics form the basis for designing and analysing practical mechanisms used in engines, machines and power-transmission systems.