Sources and References

Primary textbooks — W.L. Cleghorn & N. Dechev, Mechanics of Machines, 2nd ed., Oxford University Press, 2015; Daniel J. Inman, Engineering Vibration, 5th ed., Pearson, 2022.

Online resources — MIT OCW 2.003SC Engineering Dynamics (Vandiver et al.); MIT OCW 2.14 Analysis and Design of Feedback Control Systems; S.S. Rao, Mechanical Vibrations, 6th ed., Pearson, 2017; W.T. Thomson & M.D. Dahleh, Theory of Vibration with Applications, 5th ed., Prentice Hall, 1998; B. Tongue, Principles of Vibration, 2nd ed., Oxford University Press, 2002; Cambridge Engineering Part IB Mechanics notes; NPTEL Lecture Series on Theory of Machines (IIT Kharagpur).

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Chapter 1: Mechanisms — Kinematics and Degrees of Freedom

1.1 Machines, Mechanisms, and Kinematic Chains

A machine is a device that transforms energy and transmits or modifies motion to do useful work. A mechanism is the kinematic skeleton of a machine — the arrangement of rigid bodies (links) connected by joints that constrains and guides relative motion. Understanding the distinction is the first step toward systematic analysis.

Every mechanism is built from two primitive concepts: links and kinematic pairs. A link is a rigid body forming one member of a mechanism. A kinematic pair (joint) is a constraint between two links that permits certain relative motions while preventing others. When links and joints are assembled into a closed network they form a kinematic chain; when one link of the chain is fixed to a frame of reference the result is a mechanism.

1.1.1 Classification of Links

Links are classified by the number of joints they carry:

• Binary link: two joints
• Ternary link: three joints
• Quaternary link: four joints

The fixed link (ground) is called the frame or fixed link. The link that receives input motion is the crank or driver. The link that delivers output is the follower. The link connecting driver to follower (when neither is grounded) is the coupler or connecting rod.

1.1.2 Classification of Kinematic Pairs

Kinematic pairs are classified by the number of degrees of freedom they permit and by the type of surface contact.

For the planar mechanisms studied in this course every revolute joint removes two DOF (prevents two translations in the plane, leaving one rotation) and every prismatic joint also removes two DOF (prevents one translation perpendicular to the slide direction and one rotation, leaving one translation). This accounting leads directly to Grübler’s equation.

1.2 Degrees of Freedom and Grübler’s Equation

Before any kinematic or kinetic analysis can be performed, one must know how many independent inputs are needed to uniquely specify the configuration of the mechanism. This count is the mobility or degrees of freedom of the mechanism.

1.2.1 Constraint Counting in the Plane

A single rigid body moving freely in the plane possesses 3 DOF (two translations and one rotation). An assembly of \( n \) links (excluding the fixed frame, which contributes no DOF) therefore has \( 3n \) DOF before any joints are introduced.

Each lower kinematic pair in the plane removes 2 DOF (it imposes 2 scalar constraints). Each higher pair removes 1 DOF. Let:

• \( L \) = total number of links including the fixed frame
• \( n = L - 1 \) = number of moving links
• \( J_1 \) = number of lower pairs (full joints: revolute or prismatic)
• \( J_2 \) = number of higher pairs (half joints: rolling contact, gear teeth)

1.3 Mechanism Inversion

Inversions of the four-bar chain yield the four-bar linkage, the crank-rocker, the double-crank, and the double-rocker, depending on the relative link lengths. Inversions of the slider-crank chain yield the slider-crank (engine form), the Scotch yoke mechanism, oscillating-cylinder engines, and the fixed-block crank mechanism. Recognizing inversions allows the analyst to apply one set of kinematic equations to many superficially different devices.

1.4 Grashof’s Criterion

Not every four-bar linkage allows continuous rotation of all links. Grashof’s criterion determines which link can make a full rotation.

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Chapter 2: Analytical Kinematic Analysis of Planar Mechanisms

2.1 Vector Loop Equations

The most systematic approach to kinematic analysis of planar mechanisms is the vector loop method. Each link is represented as a vector; the closed-loop condition (the vector sum around any closed loop must be zero) provides algebraic equations relating link positions, which can then be differentiated to obtain velocity and acceleration equations.

2.1.1 Position Analysis — The Loop Closure Equation

For a planar mechanism, represent link \( i \) of length \( r_i \) directed at angle \( \theta_i \) from the positive \( x \)-axis as the complex vector \( \mathbf{R}_i = r_i e^{i\theta_i} = r_i(\cos\theta_i + i\sin\theta_i) \). For a single closed loop:

\[ \mathbf{R}_2 + \mathbf{R}_3 - \mathbf{R}_4 - \mathbf{R}_1 = 0 \]

Separating real and imaginary parts gives two scalar equations:

\[ r_2 \cos\theta_2 + r_3 \cos\theta_3 - r_4 \cos\theta_4 - r_1 \cos\theta_1 = 0 \]\[ r_2 \sin\theta_2 + r_3 \sin\theta_3 - r_4 \sin\theta_4 - r_1 \sin\theta_1 = 0 \]

With \( \theta_1 = 0 \) (frame along \( x \)-axis) and \( \theta_2 \) as the known input, these two nonlinear equations in \( \theta_3 \) and \( \theta_4 \) can be solved analytically (Freudenstein equation) or numerically.

2.2 Velocity Analysis by Differentiation

The angular velocities \( \dot{\theta}_3 = \omega_3 \) and \( \dot{\theta}_4 = \omega_4 \) found here completely describe how fast each link rotates. The velocity of any point on the coupler can then be found by vector addition.

2.3 Acceleration Analysis by Differentiation

2.4 Kinematic Analysis of the Slider-Crank Mechanism

The slider-crank is analytically tractable in closed form. Let the crank (link 2) have length \( r \) and angle \( \theta \), the connecting rod (link 3) have length \( l \) and angle \( \phi \), and let \( x \) be the piston displacement from the crank center. The offset (eccentricity) is zero for a centered slider-crank.

2.4.1 Position

The closure equation in the \( x \)-direction gives:

\[ x = r\cos\theta + l\cos\phi \]

The closure in \( y \):

\[ 0 = r\sin\theta + l\sin\phi \implies \sin\phi = -\frac{r}{l}\sin\theta \]

Defining the crank ratio \( \lambda = r/l \):

\[ \cos\phi = \sqrt{1 - \lambda^2\sin^2\theta} \]

2.4.2 Velocity

Differentiating with respect to time:

\[ \dot{x} = -r\omega\sin\theta - l\dot{\phi}\sin\phi, \qquad 0 = r\omega\cos\theta + l\dot{\phi}\cos\phi \]

From the second equation: \( \dot{\phi} = -\frac{r\omega\cos\theta}{l\cos\phi} \). Substituting:

\[ \dot{x} = -r\omega\sin\theta + \frac{r^2\omega\cos\theta\sin\phi}{\cos\phi \cdot l} \]

For most engine analyses the approximation \( \lambda \ll 1 \) (long connecting rod) is used, giving:

\[ \dot{x} \approx -r\omega\left(\sin\theta + \frac{\lambda}{2}\sin 2\theta\right) \]

2.4.3 Acceleration

\[ \ddot{x} \approx -r\omega^2\left(\cos\theta + \lambda\cos 2\theta\right) \]

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Chapter 3: Kinetic Analysis, Force Transmission, and Balancing

3.1 Force Analysis of Linkages

Kinetic (dynamic) analysis determines the forces and moments acting at each joint and the inertial loads on each moving link. This information is essential for component sizing, bearing selection, and power calculation.

The analysis proceeds in two stages. First, kinematics is solved (Chapters 1–2) to find positions, velocities, and accelerations. Then Newton’s second law is applied to each moving link treated as a free body.

3.1.1 Equations of Motion for a Rigid Link

For a planar rigid link \( k \) with mass \( m_k \) and moment of inertia \( I_{G_k} \) about its own centre of mass \( G_k \):

\[ \sum \mathbf{F}_k = m_k \mathbf{a}_{G_k} \]\[ \sum M_{G_k} = I_{G_k} \alpha_k \]

These are six scalar equations per moving link (three in 3D, three in planar form: two force balances + one moment balance). With \( n \) moving links there are \( 3n \) equations and \( 3n \) unknown joint forces (each full joint has two force components). For a mechanism with one DOF and one unknown (input torque/force), the system is determinate.

3.2 Force Analysis of the Four-Bar Mechanism

3.3 Dynamic Force Analysis

Dynamic force analysis accounts for inertia effects (\( m\mathbf{a} \) and \( I\alpha \) terms). For a crank-slider mechanism, the piston’s inertia force \( F_i = m_p \ddot{x} \) alternates in direction with each stroke, creating unbalanced forces transmitted to the engine frame.

3.3.1 Inertia Forces and D’Alembert’s Principle

D’Alembert’s principle reformulates dynamics as a statics problem by introducing an inertia force \( -m_k \mathbf{a}_{G_k} \) and an inertia torque \( -I_{G_k}\alpha_k \) as fictitious applied loads. The link is then in “dynamic equilibrium” under real plus inertia loads, and all static analysis methods apply.

3.4 Balancing of Rotating Machinery

3.4.1 Static Balancing

A rotating shaft carrying multiple masses is statically balanced if the resultant centrifugal force is zero — equivalently, the centre of mass lies on the axis of rotation.

3.4.2 Dynamic Balancing

Static balance is necessary but not sufficient to eliminate bearing forces. Even when the net force is zero, a couple may exist if the masses are in different planes along the shaft. Dynamic balance requires both force and moment balance.

3.4.3 Partial Balancing of the Slider-Crank

For a single-cylinder engine the reciprocating piston cannot be fully balanced by a rotating counterweight. Adding a counterweight of mass \( m_b \) at radius \( r_b \) on the crank, oriented to oppose the piston, can cancel the primary (first-harmonic) force. However, this introduces a new lateral (transverse) force of the same magnitude. The engineer must choose the fraction of primary balance: typically 50% is used for single-cylinder engines as a compromise.

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Chapter 4: Gears and Gear Trains

4.1 Fundamentals of Gear Geometry

Gears transmit motion and torque between shafts by the meshing of teeth. Unlike friction drives (which slip under overload), toothed gears enforce a positive velocity ratio provided tooth geometry obeys the fundamental law of gearing.

4.1.1 The Fundamental Law of Toothed Gearing

4.1.2 Involute Tooth Geometry

The involute of a circle is the curve traced by the end of a taut string unwinding from the circle. Involute tooth profiles automatically satisfy the fundamental law because the common normal at any contact point is the common tangent to the two base circles, which passes through a fixed pitch point.

Key involute gear parameters:

• Module \( m = d/N \): pitch diameter divided by number of teeth (SI). All meshing gears must have the same module.
• Pitch circle diameter \( d = mN \)
• Pressure angle \( \phi \): angle between the common normal and the pitch circle tangent, standardized at \( 20° \) (formerly \( 14.5° \))
• Base circle diameter \( d_b = d\cos\phi \)
• Addendum \( a = m \) (tooth height above pitch circle)
• Dedendum \( b = 1.25m \) (tooth depth below pitch circle)
• Center distance for meshing gears: \( C = \frac{m(N_1 + N_2)}{2} \)
• Gear ratio: \( i = \omega_1/\omega_2 = N_2/N_1 = d_2/d_1 \)

4.2 Types of Gears

4.2.1 Spur Gears

Spur gears have teeth parallel to the rotation axis. They transmit motion between parallel shafts. Spur gears are the simplest and most common. They produce no axial thrust but are noisier than helical gears due to abrupt engagement of the full tooth width.

4.2.2 Helical Gears

Helical gears have teeth inclined at a helix angle \( \psi \) to the rotation axis. Engagement begins at one end of the tooth and progresses smoothly across the face — resulting in quieter, smoother operation. However, helical gears generate an axial (thrust) force component \( F_a = F_t \tan\psi \) that must be absorbed by thrust bearings. Crossed helical gears can connect non-parallel, non-intersecting shafts.

4.2.3 Bevel Gears

Bevel gears transmit motion between intersecting shafts (usually at \( 90° \)). Teeth are formed on the surface of a cone. Straight bevel gears are analogous to spur gears; spiral bevel gears provide smoother engagement analogous to helical gears. The pitch surfaces are cones meeting at the apex.

4.2.4 Worm Gears

A worm gear pair consists of a worm (essentially a screw) meshing with a worm wheel. It transmits motion between non-parallel, non-intersecting (crossed) shafts, typically at \( 90° \). Very large speed reductions (\( 10:1 \) to \( 500:1 \)) are achievable in a single stage. The lead angle of the worm determines whether the drive is reversible (backdrivable) or self-locking. Self-locking occurs when the lead angle is less than the friction angle.

4.3 Gear Trains

A gear train is a series of meshing gears. It amplifies torque, reduces speed (or vice versa), and can reverse the direction of rotation.

4.3.1 Simple Gear Trains

In a simple gear train each shaft carries exactly one gear. The overall velocity ratio is the product of successive gear ratios:

\[ \frac{\omega_{\text{out}}}{\omega_{\text{in}}} = \pm\frac{N_1}{N_2} \cdot \frac{N_3}{N_4} \cdots \]

Intermediate gears between driver and driven gear are called idler gears; they change rotation direction but do not affect the speed ratio magnitude.

4.3.2 Compound Gear Trains

In a compound gear train, at least one shaft carries more than one gear. Multiple reductions can be achieved in a compact package. The velocity ratio:

\[ \frac{\omega_{\text{out}}}{\omega_{\text{in}}} = \pm \prod_{k} \frac{N_{\text{driver},k}}{N_{\text{driven},k}} \]

4.4 Epicyclic (Planetary) Gear Trains

An epicyclic or planetary gear train differs from ordinary (simple or compound) trains in that at least one gear’s axis is not fixed to the frame — it revolves around another gear’s axis. This gives planetary trains exceptional versatility: very high ratios in compact packages, multiple operating modes by holding different members, and the ability to add or subtract speeds (differentials).

4.4.1 Components of a Planetary Train

A basic planetary set consists of:

• Sun gear (S): central gear, axis fixed to frame
• Planet gears (P): mesh with sun and ring, carried on planet carrier
• Ring gear (R, annular gear): internal gear, axis fixed to frame
• Planet carrier (C, arm): the link carrying the planet gear axes

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Chapter 5: Free Vibration of Single-Degree-of-Freedom Systems

5.1 Motivation and Overview

Mechanical vibration is the oscillatory motion of a system about an equilibrium position. Vibrations arise whenever stored energy alternates between potential and kinetic forms. While controlled vibration is useful (musical instruments, vibrating conveyors, ultrasonic machining), uncontrolled vibration causes fatigue, noise, structural failure, and reduced precision. This chapter establishes the mathematical framework for the simplest vibrating system: a single-DOF mass-spring-damper.

5.2 Equation of Motion

5.2.1 Newton’s Law Approach

Consider a mass \( m \) on a spring of stiffness \( k \) with a viscous dashpot of coefficient \( c \). Let \( x(t) \) be displacement from the static equilibrium position. The equation of motion (Newton’s 2nd law, with static deflection terms cancelling):

\[ m\ddot{x} + c\dot{x} + kx = 0 \]

5.2.2 Energy Method Derivation

5.3 Damped Free Vibration

Dividing the equation of motion \( m\ddot{x} + c\dot{x} + kx = 0 \) by \( m \) and introducing standard parameters:

5.3.1 Underdamped Response (\( \zeta < 1 \))

The characteristic equation \( s^2 + 2\zeta\omega_n s + \omega_n^2 = 0 \) has roots:

\[ s_{1,2} = -\zeta\omega_n \pm i\omega_n\sqrt{1-\zeta^2} = -\sigma \pm i\omega_d \]

where \( \sigma = \zeta\omega_n \) is the decay rate and \( \omega_d = \omega_n\sqrt{1-\zeta^2} \) is the damped natural frequency. The general solution:

\[ x(t) = e^{-\zeta\omega_n t}\left(A\cos\omega_d t + B\sin\omega_d t\right) = Ce^{-\zeta\omega_n t}\cos(\omega_d t - \phi) \]

with initial conditions \( x(0) = x_0 \) and \( \dot{x}(0) = v_0 \):

\[ A = x_0, \qquad B = \frac{v_0 + \zeta\omega_n x_0}{\omega_d} \]

The amplitude decays as an exponential envelope \( Ce^{-\zeta\omega_n t} \) while the response oscillates at frequency \( \omega_d \).

5.3.2 Critically Damped Response (\( \zeta = 1 \))

Repeated roots \( s_{1,2} = -\omega_n \) yield:

\[ x(t) = (A + Bt)e^{-\omega_n t} \]

with \( A = x_0 \), \( B = v_0 + \omega_n x_0 \). This is the minimum damping for which no oscillation occurs; it corresponds to the fastest return to rest without overshoot and is the design target for instrument needles and door closers.

5.3.3 Overdamped Response (\( \zeta > 1 \))

Two distinct real roots \( s_{1,2} = \omega_n\left(-\zeta \pm \sqrt{\zeta^2 - 1}\right) \):

\[ x(t) = Ae^{s_1 t} + Be^{s_2 t} \]

The system returns to equilibrium without oscillating, but more slowly than the critically damped case.

5.4 Logarithmic Decrement

The logarithmic decrement \( \delta \) is the standard experimental method for measuring damping ratio from a free-vibration decay record.

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Chapter 6: Forced Harmonic Response and Vibration Isolation

6.1 Undamped Forced Vibration

Apply a harmonic force \( F(t) = F_0\cos\omega t \) to an undamped SDOF system:

\[ m\ddot{x} + kx = F_0\cos\omega t \]

The particular solution (steady-state response):

\[ x_p(t) = \frac{F_0/k}{1 - (\omega/\omega_n)^2}\cos\omega t = X\cos\omega t \]

The amplitude \( X = \frac{F_0/k}{1 - r^2} \) where \( r = \omega/\omega_n \) is the frequency ratio. The static deflection \( X_{st} = F_0/k \).

6.2 Damped Forced Vibration — Frequency Response

6.3 Rotating Unbalance Excitation

Many vibration problems arise from rotating machinery with mass imbalance — eccentrically mounted rotors, unbalanced fans, crank mechanisms. A mass \( m_e \) (unbalanced mass) rotating at eccentricity \( e \) generates a centrifugal force:

\[ F(t) = m_e e \omega^2 \cos\omega t \]

For a machine of total mass \( M \) (including \( m_e \)):

\[ M\ddot{x} + c\dot{x} + kx = m_e e \omega^2 \cos\omega t \]

The steady-state amplitude:

\[ MX = \frac{m_e e r^2}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}} \]

or in dimensionless form with \( r = \omega/\omega_n \):

\[ \frac{MX}{m_e e} = \frac{r^2}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}} \]

6.4 Base Excitation

When the support (base) of an SDOF system moves harmonically — \( y(t) = Y\sin\omega t \) — the relative displacement \( z = x - y \) and the absolute displacement \( x \) of the mass are both of interest.

The equation of motion for the mass:

\[ m\ddot{x} + c(\dot{x} - \dot{y}) + k(x - y) = 0 \]

Or equivalently, in terms of absolute displacement \( x \):

\[ m\ddot{x} + c\dot{x} + kx = kY\sin\omega t + c\omega Y\cos\omega t \]

The steady-state amplitude of the mass:

\[ \frac{X}{Y} = \sqrt{\frac{1 + (2\zeta r)^2}{(1-r^2)^2 + (2\zeta r)^2}} \]

This ratio \( X/Y \) is the displacement transmissibility.

6.5 Transmissibility and Vibration Isolation

Key isolation design principles:

1. Transmissibility \( T_r < 1 \) requires \( r > \sqrt{2} \) (for any \( \zeta \)).
2. In the isolation region (\( r > \sqrt{2} \)), less damping gives better isolation — high damping actually degrades isolation performance above \( \sqrt{2}\omega_n \).
3. However, during startup (passing through \( r = 1 \)), low damping allows large resonant amplitude. Practical isolators balance these competing demands.
4. Natural rubber, elastomeric mounts, and air springs are common isolator materials providing both stiffness and damping.

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Chapter 7: Multi-Degree-of-Freedom Systems and Modal Analysis

7.1 Equations of Motion for Two-DOF Systems

Real structures have infinitely many DOF, but reduced-order models with two (or a few) DOF capture essential dynamics and introduce the key concepts of natural frequencies, mode shapes, and modal analysis that extend to full-scale finite element models.

7.1.1 Derivation of the 2-DOF Equations

Consider two masses \( m_1 \) and \( m_2 \) connected in series by springs \( k_1 \), \( k_2 \), \( k_3 \) (with \( k_1 \) to ground, \( k_2 \) between masses, \( k_3 \) from \( m_2 \) to ground or absent). Applying Newton’s second law to each mass:

\[ m_1\ddot{x}_1 + (k_1 + k_2)x_1 - k_2 x_2 = F_1(t) \]\[ m_2\ddot{x}_2 - k_2 x_1 + (k_2 + k_3)x_2 = F_2(t) \]

7.1.2 Matrix Form of the Equations of Motion

7.2 The Eigenvalue Problem — Natural Frequencies and Mode Shapes

7.2.1 Setting Up the Eigenvalue Problem

For free vibration (\( \mathbf{F} = \mathbf{0} \)), assume harmonic motion \( \mathbf{x}(t) = \boldsymbol{\Phi} e^{i\omega t} \):

\[ \left(-\omega^2\left[ M \right] + \left[ K \right]\right)\boldsymbol{\Phi} = \mathbf{0} \]

This has a non-trivial solution only when:

\[ \det\left(\left[ K \right] - \omega^2\left[ M \right]\right) = 0 \]

This is the characteristic equation. Its \( N \) roots \( \omega_1^2 \leq \omega_2^2 \leq \cdots \leq \omega_N^2 \) are the squared natural frequencies.

7.3 Orthogonality of Mode Shapes

The orthogonality of mode shapes is the theoretical foundation of modal analysis. It guarantees that the transformation to modal coordinates decouples the equations of motion.

7.4 Modal Matrix and Coordinate Transformation

Collect the mode shapes as columns of the modal matrix \( \left[ \Phi \right] = \left[\boldsymbol{\phi}^{(1)} \, \boldsymbol{\phi}^{(2)} \, \cdots \, \boldsymbol{\phi}^{(N)}\right] \).

Define the modal coordinate vector \( \mathbf{q}(t) \) through the transformation:

\[ \mathbf{x}(t) = \left[ \Phi \right]\mathbf{q}(t) = \sum_{r=1}^{N} \boldsymbol{\phi}^{(r)} q_r(t) \]

Substitute into \( [M]\ddot{\mathbf{x}} + [K]\mathbf{x} = \mathbf{F}(t) \) and premultiply by \( [\Phi]^T \):

\[ [\Phi]^T[M][\Phi]\ddot{\mathbf{q}} + [\Phi]^T[K][\Phi]\mathbf{q} = [\Phi]^T\mathbf{F} \]

By orthogonality, \( [\Phi]^T[M][\Phi] = [m_r] \) (diagonal matrix of modal masses) and \( [\Phi]^T[K][\Phi] = [k_r] = [\omega_r^2 m_r] \) (diagonal matrix of modal stiffnesses). The system decouples into \( N \) independent SDOF equations:

\[ m_r \ddot{q}_r + \omega_r^2 m_r q_r = f_r(t), \quad r = 1, 2, \ldots, N \]

where the modal force is \( f_r(t) = \left(\boldsymbol{\phi}^{(r)}\right)^T \mathbf{F}(t) \).

Each modal equation is an independent SDOF system that can be solved by the methods of Chapters 5 and 6. The physical response is then reconstructed by superposition: \( \mathbf{x}(t) = \sum_r \boldsymbol{\phi}^{(r)} q_r(t) \).

7.5 Free Response via Modal Superposition

For the undamped free-vibration problem (\( \mathbf{F} = \mathbf{0} \)), each modal equation gives:

\[ q_r(t) = A_r \cos\omega_r t + B_r \sin\omega_r t \]

The physical response:

\[ \mathbf{x}(t) = \sum_{r=1}^{N} \boldsymbol{\phi}^{(r)}\left(A_r \cos\omega_r t + B_r \sin\omega_r t\right) \]

The constants \( A_r \) and \( B_r \) are found from the initial conditions via the orthogonality relations:

\[ A_r = \frac{\left(\boldsymbol{\phi}^{(r)}\right)^T [M] \mathbf{x}(0)}{m_r}, \qquad B_r = \frac{\left(\boldsymbol{\phi}^{(r)}\right)^T [M] \dot{\mathbf{x}}(0)}{m_r \omega_r} \]

7.6 Forced Response via Modal Superposition

For forced vibration, each decoupled modal equation:

\[ m_r \ddot{q}_r + m_r\omega_r^2 q_r = f_r(t) \]

is solved by the methods of Chapter 6 (for harmonic \( \mathbf{F}(t) = \hat{\mathbf{F}} e^{i\omega t} \), each modal equation has a harmonic particular solution). Including modal damping \( \zeta_r \) (added phenomenologically):

\[ \ddot{q}_r + 2\zeta_r\omega_r\dot{q}_r + \omega_r^2 q_r = \frac{f_r(t)}{m_r} \]

For harmonic excitation \( \mathbf{F}(t) = \hat{\mathbf{F}}\sin\omega t \):

\[ q_r(t) = \frac{\hat{f}_r/m_r}{\sqrt{(\omega_r^2 - \omega^2)^2 + (2\zeta_r\omega_r\omega)^2}}\sin(\omega t - \phi_r) \]

where \( \hat{f}_r = (\boldsymbol{\phi}^{(r)})^T \hat{\mathbf{F}} \) and \( \phi_r = \arctan\frac{2\zeta_r\omega_r\omega}{\omega_r^2 - \omega^2} \).

The total physical response:

\[ \mathbf{x}(t) = \sum_{r=1}^{N} \boldsymbol{\phi}^{(r)} q_r(t) \]

This is the modal superposition method — the workhorse of structural dynamics and finite element analysis. For a structure with \( N \) DOF under broadband excitation, only the first few modes typically contribute significantly, allowing massive computational savings by truncating the modal sum.

7.7 Modal Analysis: Physical Interpretation and Engineering Significance

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Summary and Interconnections

This course presents two interlocking perspectives on machine dynamics. Part 1 establishes the geometric and kinematic foundations: how mechanisms are built (Grübler, Grashof), how they move (loop closure, differentiation), what forces they transmit (Newton/D’Alembert), and how gears achieve speed transformation (involute geometry, planetary trains). Part 2 translates the resulting forces and motions into the vibrational language of natural frequencies, damping, and resonance — the universal framework for predicting and controlling dynamic response in all mechanical systems.

The thread connecting the two parts is the concept of unbalanced periodic forces. Part 1 computes the forces; Part 2 predicts the response to those forces. Together they equip the mechanical engineer to design machines that run smoothly, reliably, and safely throughout their operating speed range.

Key Formulas Reference

Part 1 — Mechanisms

\[ M = 3(L-1) - 2J_1 - J_2 \quad \text{(Grübler)} \]\[ s + l \leq p + q \quad \text{(Grashof criterion)} \]\[ \frac{\omega_S - \omega_C}{\omega_R - \omega_C} = -\frac{N_R}{N_S} \quad \text{(Planetary train)} \]

Part 2 — Vibrations

\[ \omega_n = \sqrt{\frac{k}{m}}, \quad \zeta = \frac{c}{2\sqrt{km}}, \quad \omega_d = \omega_n\sqrt{1-\zeta^2} \]\[ M(r,\zeta) = \frac{1}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}, \quad r = \frac{\omega}{\omega_n} \]\[ \delta = \frac{2\pi\zeta}{\sqrt{1-\zeta^2}}, \quad \left[K\right]\boldsymbol{\phi}^{(r)} = \omega_r^2\left[M\right]\boldsymbol{\phi}^{(r)} \]\[ (\boldsymbol{\phi}^{(s)})^T[M]\boldsymbol{\phi}^{(r)} = 0, \quad (\boldsymbol{\phi}^{(s)})^T[K]\boldsymbol{\phi}^{(r)} = 0 \quad (r \neq s) \]