Mechanics, Oscillations and Resonance

Introduction

Mechanics connects forces and motion. In electrical engineering it supports the analysis of motors, generators, robotic actuators, rotating machinery, accelerometers, loudspeakers, relays and vibration-sensitive instruments. Electromechanical systems also exchange energy across electrical and mechanical domains, making work, power, momentum and resonance central design concepts.

A particle model represents an object by its mass concentrated at a point. A rigid-body model includes size and orientation while assuming that distances between material points remain constant. Real structures deform, but rigid-body mechanics is often accurate when deformation is small relative to overall motion. When flexibility is significant, vibration models approximate the deformation using springs, dampers and concentrated or distributed masses.

Learning Objectives

After studying this chapter, students should be able to:

• describe particle motion using position, velocity and acceleration;
• construct free-body diagrams and apply Newton’s laws;
• use work–energy and impulse–momentum methods;
• analyze fixed-axis and general planar rigid-body motion;
• calculate torque, angular momentum, rotational energy and rolling motion;
• derive and solve the simple harmonic oscillator equation;
• classify damped motion and interpret energy dissipation correctly;
• analyze steady-state forced vibration and resonance;
• evaluate vibration isolation for machines and instruments; and
• recognize modeling, safety and condition-monitoring considerations.

1. Particle Kinematics

1.1 Position, velocity and acceleration

Relative to an inertial reference frame, the position of a particle is the vector \mathbf{r}(t). Velocity and acceleration are

Position is measured in metres (m), velocity in m·s⁻¹ and acceleration in m·s⁻². In Cartesian coordinates,

For one-dimensional motion with constant acceleration a, integration gives

These equations are special cases requiring constant acceleration; they must not be applied directly when acceleration varies with time, position or velocity.

For curved motion, tangential–normal components are useful:

where v is speed, \rho is the local radius of curvature, \mathbf{e}_t is tangent to the path and \mathbf{e}_n points toward the center of curvature. A particle can therefore accelerate even at constant speed because its direction changes.

Worked Example 1: Time-dependent motion

A robotic carriage has position x(t)=0.20t³-0.60t²+2t m. At t=3.0 s,

The constant-acceleration equations are inappropriate because a depends on time.

2. Forces and Newtonian Dynamics

Newton’s second law for a particle of constant mass is

where \sum\mathbf{F} is the resultant external force in newtons (N), m is mass in kilograms and \mathbf{a} is acceleration. More generally,

where \mathbf{p} is linear momentum. The form \sum\mathbf{F}=m\mathbf{a} assumes constant mass in an inertial frame.

Common forces include weight m\mathbf{g}, contact normal force, tension, spring force and friction. For dry sliding friction, the idealized Coulomb model is

where \mu_k is the kinetic-friction coefficient and N is the normal force. Before sliding, static friction satisfies |F_s|\leq \mu_sN and takes the value required for equilibrium up to its limiting value.

2.1 Free-body diagrams

A free-body diagram isolates one body and shows only forces exerted on it by its surroundings. Coordinate directions should be selected before resolving force components.

Figure: Unambiguous free-body diagram of a pulled block. The applied-force angle is measured at the force vertex from the horizontal reference.

For the illustrated block, assuming no vertical acceleration,

while the horizontal equation during rightward sliding is

Worked Example 2: Pulled block

A 20 kg block is pulled by a 100 N force at 30° above horizontal. Let \mu_k=0.20 and g=9.81 m·s⁻².

The upward force component reduces the normal force and consequently reduces friction.

3. Work, Energy and Power

The work done by a force along a path from position 1 to position 2 is

Work is measured in joules, where 1\ \text{J}=1\ \text{N·m}. Taking the scalar product of m\mathbf{a}=\sum\mathbf{F} with displacement and integrating gives the particle work–energy theorem:

For conservative forces, potential energy V can be defined so that W_c=-\Delta V. Near Earth’s surface, gravitational potential energy is V_g=mgh, while an ideal linear spring has

where k is spring stiffness in N·m⁻¹ and x is deformation from the unstretched position. If only conservative forces do work,

With nonconservative work W_nc, such as friction or actuator input,

Power is the rate of doing work:

For a rotating shaft, P=\tau\omega, where torque \tau is in N·m and angular velocity \omega is in rad·s⁻¹. Although the radian is dimensionless, retaining “rad·s⁻¹” aids interpretation.

4. Impulse, Momentum and Impact

Integrating Newton’s second law over a time interval gives

where impulse \mathbf{J} has units N·s. For a system of particles, internal forces cancel in action–reaction pairs, so total linear momentum is conserved if the net external impulse is zero.

For direct central impact, the coefficient of restitution is

using a common positive axis and assuming body 1 initially approaches body 2. Here 0\leq e\leq1 for ordinary passive impacts. Momentum may be conserved while kinetic energy is not. A perfectly elastic impact has e=1; a perfectly plastic impact has e=0 and the bodies leave with a common velocity.

Impulse–momentum analysis is particularly useful for relay impacts, actuator end stops and short-duration machine loads for which detailed force histories are unavailable.

5. Rigid-Body Kinematics

A rigid body can translate, rotate or undergo both motions. In planar motion, all points move in parallel planes and the angular velocity is perpendicular to the plane.

5.1 Fixed-axis rotation

For rotation about a fixed axis,

For a point at radius r from the axis, the tangential speed is v=r\omega. Its tangential and normal acceleration magnitudes are

The normal acceleration points toward the rotation axis.

5.2 General planar rigid-body motion

General planar motion combines translation of a reference point with rotation about that point. For any two points A and B fixed in the same rigid body,

Here \mathbf{r}_B/A is the position of B relative to A, and both points must belong to the same rigid body. In planar motion, the last term equals -\omega²\mathbf{r}_B/A. It is the centripetal relative acceleration, not an optional correction.

An instantaneous center of zero velocity can simplify velocity analysis: at a given instant, the body’s velocities are consistent with rotation about that center. It generally cannot be used as a fixed center for acceleration analysis because the instantaneous center itself may accelerate.

Worked Example 3: Translating and rotating link

A 0.40 m link is instantaneously vertical. Point A has velocity \mathbf{v}_A=1.20\mathbf{i} m·s⁻¹ and zero acceleration. The link has \boldsymbol{\omega}=3\mathbf{k} rad·s⁻¹ and \boldsymbol{\alpha}=2\mathbf{k} rad·s⁻². With \mathbf{r}_B/A=0.40\mathbf{j} m,

Thus B is instantaneously at rest, although it need not have zero acceleration:

This example demonstrates why velocity and acceleration conclusions must be kept distinct.

6. Rigid-Body Dynamics

6.1 Mass moment of inertia

The mass moment of inertia about an axis is

with units kg·m². It depends on mass distribution and axis location. The parallel-axis theorem is

where I_G is the inertia about a parallel axis through the center of mass G, and d is the perpendicular distance between axes.

6.2 Newton–Euler equations for planar motion

General planar rigid-body dynamics is governed by

The first equation determines center-of-mass translation; the second determines rotation about the center of mass. These equations apply whether or not the body has a fixed axis.

For fixed-axis rotation about an inertially fixed point O, the special rotational equation is

It should not be applied without correction about an arbitrarily accelerating point. More generally, angular momentum about a fixed inertial point is

The kinetic energy of a planar rigid body is

This translational-plus-rotational form is valid for general planar motion. For rotation about a fixed point, it reduces to T=\tfrac12 I_O\omega².

6.3 Rolling without slipping

For a circular body rolling on a stationary surface without slipping,

where R is radius. These are constraint relations along the surface. The geometric contact point is instantaneously at rest, but the material point at contact generally has nonzero acceleration.

For a body rolling down an incline of angle \beta without slipping,

This result assumes a rigid body, sufficient static friction, negligible rolling resistance and no energy loss. For a uniform solid cylinder, I_G=\tfrac12mR², so a_G=\tfrac23g\sin\beta. Static friction supplies torque but does no work at an ideal stationary contact point.

7. Free Oscillation and Simple Harmonic Motion

An ideal undamped mass–spring oscillator obeys

Dividing by m gives

where \omega_n is the undamped natural angular frequency in rad·s⁻¹. The frequency in hertz and period are

The solution is

with constants determined by initial displacement and velocity. Acceleration satisfies \ddot{x}=-\omega_n²x, the defining relation for simple harmonic motion.

The total mechanical energy is constant:

Energy alternates between kinetic and elastic potential forms.

8. Viscous Damping

A linear viscously damped oscillator satisfies

where c is the damping coefficient in N·s·m⁻¹. Define

The damping ratio \zeta is dimensionless.

For the underdamped case,

The displacement amplitude envelope decays as e^{-\zeta\omega_nt}. However, the instantaneous mechanical energy does not generally equal a constant times e^{-2\zeta\omega_nt}. Its exact rate of change is obtained by multiplying the equation of motion by \dot{x}:

Because \dot{x}² oscillates, instantaneous energy decreases nonuniformly. The factor e^{-2\zeta\omega_nt} describes the energy envelope—for example, energies evaluated at successive displacement extrema—and is commonly used for cycle-averaged energy when the averaging assumptions are appropriate.

The logarithmic decrement between successive positive displacement peaks is

It is valid for an underdamped linear single-degree-of-freedom system with viscous damping.

Worked Example 4: Damped instrument mount

For m=2.0 kg, k=200 N·m⁻¹ and c=8.0 N·s·m⁻¹,

The amplitude envelope is reduced to 1% when

This is an envelope statement, not a claim that every instantaneous displacement equals 1% of its initial value at that time.

9. Forced Oscillation and Resonance

For a harmonically forced oscillator,

After the transient has decayed, the steady-state response is

where

and r=\omega/\omega_n is the frequency ratio. The phase should be selected in the correct quadrant: it approaches 0 at low frequency and π at high frequency.

The displacement magnification factor is

For \zeta<1/\sqrt{2}, the maximum displacement occurs at

Thus the displacement peak is slightly below \omega_n when damping is nonzero. Different response measures—displacement, velocity, acceleration and transmitted force—can peak at different frequencies. “The resonant frequency” must therefore be tied to a stated response quantity.

Figure: Conceptual displacement frequency response. Greater damping lowers and broadens the resonance peak.

At very low frequency, the response is approximately the static deflection F₀/k. Near resonance, energy supplied each cycle can exceed energy dissipated until a larger steady amplitude produces an energy balance. In an ideal undamped system driven exactly at its natural frequency, no finite steady-state amplitude exists; the amplitude grows with time under the linear model.

10. Base Excitation and Vibration Isolation

If the support moves as y=Y\cos\omega t, and the spring and damper act through relative displacement and velocity, the governing equation is

The absolute displacement transmissibility is

The same mathematical form describes force transmissibility for a harmonically forced machine under the standard linear isolator model. Isolation, defined by T<1, begins when r>\sqrt{2}. Damping reduces the resonant peak but can increase high-frequency transmissibility. Isolator selection is therefore a compromise among startup resonance, steady operating speed, displacement limits and shock response.

11. Applied Case: Isolation of a Motor–Generator Set

Illustrative case. A 120 kg motor–generator set operates at 1800 r·min⁻¹ and excites a precision measurement bench. Four identical mounts are to provide a frequency ratio of 3 at operating speed with damping ratio \zeta=0.10.

Background and problem

The rotational frequency is

To achieve r=3, the required natural frequency is

Analysis and solution

The total mount stiffness is

Each of four equal mounts requires approximately

The total viscous damping coefficient is

At operating speed,

Only about 14.5% of the harmonic force amplitude is transmitted under the ideal model. The static deflection is

Outcome and engineering judgment

The operating point is in the isolation region. Nevertheless, the machine passes through r\approx1 during startup and shutdown, where transmissibility is much higher. The designer should verify transient displacement, mount load ratings, shaft alignment, floor flexibility and forcing harmonics. Rapid controlled passage through resonance or variable damping may be appropriate.

Learning points

• Isolation performance depends on frequency ratio, not stiffness alone.
• Added damping suppresses startup resonance but may reduce high-frequency isolation.
• Mounts must be selected for both dynamic and static loads.
• A single-degree-of-freedom model should be validated against structural modes and measurements.

12. Electromechanical Interpretation and Contemporary Practice

Mechanical variables have useful electrical analogies. Under the force–voltage analogy, force corresponds to voltage, velocity to current, mass to inductance, damping to resistance and compliance 1/k to capacitance. Such analogies assist in modeling loudspeakers, electromagnetic actuators and piezoelectric systems, but sign conventions and the selected analogy must remain consistent.

Modern machines commonly use accelerometers, optical encoders and current sensors for condition monitoring. Spectral peaks may indicate imbalance, misalignment, looseness or bearing defects. Machine-learning systems can classify patterns, but they do not replace physical interpretation. Training data may omit rare failures, operating speed can shift spectral components, and sensor faults can create false alarms. A defensible workflow combines calibrated sensing, mechanics-based features, uncertainty estimates and expert review.

13. Modeling Limitations, Safety and Best Practice

Linear models assume small motion, constant stiffness, viscous damping and time-invariant parameters. Real systems may exhibit Coulomb friction, backlash, nonlinear stiffness, impacts, temperature-dependent properties or multiple coupled modes. Resonance predictions are especially sensitive to stiffness, boundary conditions and mass distribution.

Professional practice should therefore:

1. identify system boundaries and degrees of freedom explicitly;
2. draw complete free-body diagrams with coordinate directions;
3. distinguish static, transient and steady-state behavior;
4. verify units and limiting cases;
5. estimate parameter uncertainty and conduct sensitivity analysis;
6. validate models through measurement where consequences are significant;
7. include guards, shutdown thresholds and safe resonance-crossing procedures; and
8. apply relevant machinery-vibration standards, such as the applicable ISO 20816 series, together with manufacturer and regulatory requirements.

Vibration data may reveal production schedules or equipment condition. Networked monitoring systems should therefore use controlled access, secure data handling and documented alarm governance. Safety-critical decisions should not rely solely on opaque automated classifications.

Chapter Summary

Particle kinematics describes position, velocity and acceleration, while Newton’s laws relate motion to external forces. Work–energy methods are effective when displacement is central; impulse–momentum methods are advantageous for short-duration forces and impacts.

Rigid-body motion includes translation and rotation. General planar analysis requires relative velocity and acceleration relations, together with the Newton–Euler equations about the center of mass. Fixed-axis formulas are important special cases rather than universal rigid-body equations.

An undamped linear oscillator undergoes simple harmonic motion at its natural frequency. Viscous damping creates underdamped, critically damped or overdamped behavior. In underdamped motion, displacement has an exponentially decaying envelope. Instantaneous mechanical energy instead obeys the exact balance dE/dt=-c\dot{x}²; an exponential energy law describes an envelope or suitable cycle average, not the instantaneous energy universally.

Harmonic forcing produces frequency-dependent amplitude and phase. Resonance can amplify displacement and transmitted force, while suitable frequency ratios permit vibration isolation. Engineering design must also address resonance crossing, nonlinear behavior, uncertainty, sensing reliability and safety.

Key Terms

Acceleration: Time rate of change of velocity.

Angular momentum: Rotational momentum measured about a specified point or axis.

Coefficient of restitution: Ratio describing relative separation and approach speeds in impact.

Critical damping: Damping level producing the fastest ideal nonoscillatory return to equilibrium.

Damping ratio: Actual viscous damping divided by critical damping.

Free-body diagram: Diagram showing all external forces acting on an isolated body.

Frequency ratio: Forcing frequency divided by natural frequency.

Impulse: Time integral of force, equal to change in linear momentum.

Instantaneous center: Point about which planar rigid-body velocity is instantaneously equivalent to pure rotation.

Mass moment of inertia: Measure of resistance to angular acceleration based on mass distribution.

Natural frequency: Frequency at which an unforced system oscillates in a specified mode.

Power: Rate of work or energy transfer.

Resonance: Large frequency-response amplification when forcing is near a system’s characteristic frequency.

Rigid body: Ideal body whose internal point-to-point distances remain constant.

Simple harmonic motion: Oscillation in which acceleration is proportional and opposite to displacement.

Transmissibility: Ratio of transmitted response amplitude to input amplitude.

Viscous damping: Damping force proportional and opposite to relative velocity.

Work–energy theorem: Principle equating net work to change in kinetic energy.

Review Questions

Short Answer Questions

1. Distinguish between velocity and speed.
2. State the conditions under which the constant-acceleration equations are valid.
3. Why can an instantaneous center generally not be treated as a fixed center for acceleration analysis?
4. Define critical damping and damping ratio.
5. Explain why instantaneous energy in a viscously damped oscillator does not follow a simple exponential at every instant.

Long Answer Questions

1. Derive the particle work–energy theorem from Newton’s second law.
2. Explain general planar rigid-body motion and derive the relative acceleration relation between two body-fixed points.
3. Develop the Newton–Euler equations for a planar rigid body and distinguish them from fixed-axis rotation equations.
4. Derive the steady-state amplitude and phase relations for a harmonically forced damped oscillator.
5. Discuss the competing effects of damping in resonance suppression and high-frequency vibration isolation.

Analytical / Application Questions

1. A 25 kg machine is supported by a spring of stiffness 40 kN·m⁻¹ and a damper of 400 N·s·m⁻¹. Determine its natural frequency, damping ratio and damped natural frequency.
2. A uniform solid cylinder rolls without slipping down a 12° incline. Determine its center-of-mass acceleration and the minimum role static friction must play in producing rotation.
3. Design the total stiffness of an isolator for an 80 kg instrument operating at 25 Hz so that the operating frequency ratio is 2.5. Calculate its static deflection and discuss resonance during startup.

Multiple Choice Questions with Answers

Question 1

A particle has initial velocity 3 m·s⁻¹ and constant acceleration 2 m·s⁻². What displacement occurs in 4 s?

A. 20 m

B. 24 m

C. 28 m

D. 32 m

Correct Answer: C. 28 m

Question 2

A 20 kg block is pulled by a 100 N force at 30° above horizontal on a level surface, with no vertical acceleration. Using g=9.81 m·s⁻², what is the normal force?

A. 96.2 N

B. 146.2 N

C. 196.2 N

D. 246.2 N

Correct Answer: B. 146.2 N

Question 3

A 4 kg object falls through 5 m from rest without energy loss. What is its speed after the fall, using g=9.81 m·s⁻²?

A. 7.00 m·s⁻¹

B. 9.90 m·s⁻¹

C. 12.5 m·s⁻¹

D. 19.8 m·s⁻¹

Correct Answer: B. 9.90 m·s⁻¹

Question 4

A constant 10 N force acts on a 0.50 kg particle for 0.20 s. What change in velocity occurs in the force direction?

A. 1 m·s⁻¹

B. 2 m·s⁻¹

C. 4 m·s⁻¹

D. 10 m·s⁻¹

Correct Answer: C. 4 m·s⁻¹

Question 5

Which expression gives the acceleration of point B relative to point A on the same planar rigid body?

A. \mathbf{a}_B=\mathbf{a}_A+\boldsymbol{\alpha}\times\mathbf{r}_B/A+\boldsymbol{\omega}\times(\boldsymbol{\omega}\times\mathbf{r}_B/A)

B. \mathbf{a}_B=\mathbf{a}_A+\boldsymbol{\omega}\times\mathbf{r}_B/A

C. \mathbf{a}_B=\boldsymbol{\alpha}\times\mathbf{v}_A

D. \mathbf{a}_B=\mathbf{a}_A-\boldsymbol{\alpha}\times\mathbf{r}_B/A

Correct Answer: A. \mathbf{a}_B=\mathbf{a}_A+\boldsymbol{\alpha}\times\mathbf{r}_B/A+\boldsymbol{\omega}\times(\boldsymbol{\omega}\times\mathbf{r}_B/A)

Question 6

A uniform solid cylinder rolls without slipping down an incline of angle \beta. Its center-of-mass acceleration is:

A. g\sin\beta

B. \frac{1}{2}g\sin\beta

C. \frac{2}{3}g\sin\beta

D. \frac{3}{2}g\sin\beta

Correct Answer: C. \frac{2}{3}g\sin\beta

Question 7

For a freely vibrating viscously damped oscillator, which statement about instantaneous mechanical energy E is exact?

A. E remains constant

B. E=E₀e^{-2\zeta\omega_nt} at every instant

C. \frac{dE}{dt}=-c\dot{x}²

D. \frac{dE}{dt}=-kx²

Correct Answer: C. \frac{dE}{dt}=-c\dot{x}²

Question 8

For m=2.0 kg, k=200 N·m⁻¹ and c=8.0 N·s·m⁻¹, what is the damping ratio?

A. 0.10

B. 0.20

C. 0.40

D. 0.80

Correct Answer: B. 0.20

Question 9

For a harmonically forced oscillator with \zeta=0.20, at what frequency ratio does the displacement amplitude peak?

A. 0.800

B. 0.959

C. 1.000

D. 1.200

Correct Answer: B. 0.959

Question 10

For base or force transmissibility with r=3 and \zeta=0.10, what is the transmissibility approximately?

A. 0.015

B. 0.145

C. 0.600

D. 1.45

Correct Answer: B. 0.145