2. Free Vibration of Single-Degree-of-Freedom Systems

Introduction and Context

A mechanical system vibrates freely when it is displaced from equilibrium and then released without continued external excitation. The subsequent motion is governed by the system’s inertia, stiffness, and energy-dissipation mechanisms. Free vibration is therefore a fundamental way of understanding the intrinsic dynamic properties of a structure or machine. Its most important characteristics are natural frequency, mode of motion, decay rate, and the distribution of energy between kinetic and potential forms.

The single-degree-of-freedom (SDOF) model is the simplest useful representation of a vibrating system. Although real machines and structures often possess many degrees of freedom, an SDOF model can frequently describe a dominant mode or a localized component with sufficient accuracy. It also provides the theoretical basis for modal analysis, vibration testing, isolation, rotating machinery diagnostics, and control-system design.

An SDOF system is represented by a generalized coordinate x(t), which uniquely specifies its configuration. The standard translational model consists of a mass m, a spring of stiffness k, and, where appropriate, a damper with coefficient c. For rotational systems, the corresponding quantities are mass moment of inertia J, torsional stiffness k_t, and rotational damping coefficient c_t.

Learning Objectives

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

1. Formulate the governing equation of an undamped and damped SDOF system.

2. Determine natural frequency, damped natural frequency, damping ratio, and critical damping.

3. Obtain free-vibration responses from arbitrary initial displacement and velocity.

4. Interpret logarithmic decrement and estimate damping from measured vibration data.

5. Explain energy dissipation in viscous and Coulomb-damped systems.

6. Replace complex elastic or inertial arrangements by equivalent SDOF parameters.

7. Diagnose common modeling and calculation errors in free-vibration problems.

8. Apply free-vibration concepts to engineering design, testing, and maintenance.

The SDOF Idealization

For a translational system, Newton’s second law gives the equilibrium of forces in the direction of motion. With displacement x measured from the static equilibrium position, the general free-vibration equation is

m x¨ + c x˙ + kx = 0.

Here, m is the effective mass, c is the viscous damping coefficient, and k is the stiffness. Measuring displacement from static equilibrium eliminates the constant gravitational force from the equation. If displacement is measured from an unloaded reference instead, gravity must be included explicitly.

The idealized elements have distinct physical roles. The mass stores kinetic energy, the spring stores elastic potential energy, and the damper removes mechanical energy by converting it into heat or another nonrecoverable form. The quality of a model depends on whether these equivalent properties accurately represent the actual mode or component being studied.

For a rotational SDOF system,

J θ¨ + c_t θ˙ + k_t θ = 0,

where θ is angular displacement. The mathematical treatment is identical after replacing m, c, and k by J, c_t, and k_t.

Undamped Free Vibration

When damping is negligible, the equation becomes

m x¨ + kx = 0.

Assuming a solution of the form x = Xe^(st), the characteristic equation is

ms² + k = 0.

Thus,

s = ±iω_n,

where

ω_n = √(k/m)

is the undamped natural circular frequency in rad/s. The natural frequency in cycles per second is

f_n = ω_n/(2π).

The general displacement response is

x(t) = A cos(ω_n t) + B sin(ω_n t).

The constants A and B are determined by the initial displacement x(0) = x_0 and initial velocity x˙(0) = v_0. Since x(0) = A and x˙(0) = ω_n B,

x(t) = x_0 cos(ω_n t) + (v_0/ω_n) sin(ω_n t).

An equivalent amplitude-phase form is

x(t) = X cos(ω_n t − φ),

where

X = √[x_0² + (v_0/ω_n)²]

and

φ = tan⁻¹[(v_0/ω_n)/x_0],

with the quadrant selected consistently with the signs of x_0 and v_0.

The undamped response is periodic and does not decay. Its period is T_n = 2π/ω_n. The motion continues indefinitely because the ideal model contains no mechanism for energy loss. This assumption is useful for establishing natural frequency, but it is never exact in a physical machine.

Energy Interpretation

For an undamped system, the total mechanical energy is constant:

E = 1/2 m x˙² + 1/2 kx².

At maximum displacement, velocity is zero and all energy is elastic. At the equilibrium position, displacement is zero and all energy is kinetic. During the motion, energy transfers continuously between these two forms. This energy viewpoint provides a useful check on analytical and numerical solutions.

Viscously Damped Free Vibration

With viscous damping, the equation is

m x¨ + c x˙ + kx = 0.

The characteristic equation is

ms² + cs + k = 0.

It is convenient to define the undamped natural frequency and critical damping coefficient as

ω_n = √(k/m), c_c = 2mω_n = 2√(km).

The damping ratio is

ζ = c/c_c = c/(2mω_n).

The roots can then be written as

s = −ζω_n ± ω_n√(ζ² − 1).

The nature of the free response depends on ζ.

Underdamped Response: 0 < ζ < 1

Most lightly damped mechanical systems fall in this category. Define the damped natural frequency as

ω_d = ω_n√(1 − ζ²).

The response is

x(t) = e^(−ζω_n t)[A cos(ω_d t) + B sin(ω_d t)].

Using x(0) = x_0 and x˙(0) = v_0 gives

x(t) = e^(−ζω_n t) {x_0 cos(ω_d t) + [(v_0 + ζω_n x_0)/ω_d] sin(ω_d t)}.

The oscillation occurs at ω_d, while its envelope decays exponentially as e^(−ζω_n t). The decay constant ζω_n is also called the attenuation coefficient. The damped period is T_d = 2π/ω_d, which is slightly greater than the undamped period.

A phase-amplitude representation is

x(t) = X e^(−ζω_n t) cos(ω_d t − φ).

Unlike the undamped case, the amplitude is not constant. Mechanical energy also decreases exponentially in the underdamped model; because energy is proportional to amplitude squared, its envelope is proportional to e^(−2ζω_n t).

Critically Damped Response: ζ = 1

At critical damping, the two characteristic roots coincide. The response is

x(t) = (A + Bt)e^(−ω_n t).

For initial displacement x_0 and velocity v_0,

x(t) = [x_0 + (v_0 + ω_n x_0)t]e^(−ω_n t).

Critical damping is the boundary between oscillatory and nonoscillatory motion. It returns the system to equilibrium without overshoot in the ideal model and is often desirable in instruments, door closers, and suspension mechanisms when rapid settling is important.

Overdamped Response: ζ > 1

The roots are real and negative:

s_1 = −ω_n[ζ − √(ζ² − 1)],

s_2 = −ω_n[ζ + √(ζ² − 1)].

The response is

x(t) = C_1e^(s_1t) + C_2e^(s_2t).

There is no oscillation. The slower root controls the long-time response, so excessive damping can make the final return to equilibrium slower than critical damping.

Undamped Limit

If c = 0, then ζ = 0 and ω_d = ω_n. The damped solution reduces smoothly to the undamped solution. This limiting behavior is an important consistency check when deriving equations or writing computational programs.

Logarithmic Decrement and Experimental Damping Estimation

For an underdamped free response, successive peaks have amplitudes that decrease according to the exponential envelope. If x_r and x_(r+1) are successive peaks measured on the same side of equilibrium, the logarithmic decrement is

δ = ln(x_r/x_(r+1)).

For n cycles,

δ = (1/n) ln(x_r/x_(r+n)).

For viscous damping,

δ = ζω_nT_d = 2πζ/√(1 − ζ²).

Therefore,

ζ = δ/√(4π² + δ²).

For light damping, δ is small and ζ ≈ δ/(2π). This approximation is convenient but should not be used when damping is moderate or high.

In practice, logarithmic decrement is obtained from a free-decay test. A displacement, acceleration, or velocity transducer records the response after an impulse or an initial displacement. Peaks are identified, the ratio of amplitudes is calculated, and several cycles are often averaged to reduce measurement noise. Care is required when the measured signal contains offset, nonlinear damping, multiple frequencies, or sensor saturation.

Energy Dissipation in Damped Systems

For viscous damping, the instantaneous rate of energy loss is

P_d = c x˙².

Since this quantity is nonnegative, the total mechanical energy decreases according to

dE/dt = −c x˙².

For a free underdamped response, the energy associated with the exponential envelope decreases approximately as e^(−2ζω_n t). The loss is greatest when velocity is high, particularly near the equilibrium position.

The energy dissipated per cycle may also be related to the area enclosed by the force-displacement hysteresis loop. This provides a bridge between time-domain free-decay measurements and frequency-domain damping characterization.

Coulomb or Dry-Friction Damping

Coulomb damping arises from sliding friction and is approximately independent of velocity while motion continues. If the friction force has magnitude F_c and opposes velocity, the equation is

m x¨ + kx + F_c sgn(x˙) = 0.

This equation is nonlinear because the force changes sign whenever velocity changes sign. During each half-cycle, the friction force is constant in magnitude, so the equation can be solved piecewise.

For a half-cycle in which velocity is positive, the equation is

m x¨ + kx = −F_c.

The equilibrium about which this segment oscillates is shifted by −F_c/k. When velocity becomes negative, the shifted equilibrium changes to +F_c/k. Consequently, the peak amplitude decreases by a constant amount per half-cycle. Under ideal Coulomb damping, the reduction in successive peak amplitudes on the same side is approximately

ΔX = 4F_c/k

per complete cycle. The precise relationship depends on the convention used for defining peak amplitudes, but the key feature is linear, rather than exponential, amplitude decay.

Coulomb damping is especially relevant in joints, guides, brakes, seals, and mechanisms with sliding contact. It differs fundamentally from viscous damping: viscous force depends on velocity and vanishes at zero velocity, whereas dry friction changes sign and remains finite during sliding. Near rest, sticking may occur, and the system may stop before reaching the ideal mathematical equilibrium.

Equivalent Parameters in Engineering Models

Real systems rarely consist of one visibly isolated mass and one spring. An SDOF representation is obtained by calculating equivalent mass and stiffness with respect to a selected coordinate.

Equivalent Stiffness

Springs in series satisfy

1/k_eq = 1/k_1 + 1/k_2 + ⋯,

while parallel springs satisfy

k_eq = k_1 + k_2 + ⋯.

For a uniform axial member, k = EA/L. For a torsional shaft, k_t = GJ_p/L, where G is the shear modulus and J_p is the polar second moment of area. Beam bending stiffness depends on boundary conditions and loading position; it must be derived from the force-deflection relation or flexibility coefficient.

Equivalent Mass

If several masses move with different velocities, kinetic energy provides a systematic method. Let the generalized coordinate be q and let the velocity of mass i be v_i = a_i q˙. Then

T = 1/2 Σ m_i v_i² = 1/2 m_eq q˙²,

so that

m_eq = Σ m_i a_i².

Rotational inertia may be reflected into a translational coordinate through the kinematic relation v = rω. For a rigid body moving in translation and rotation, both contributions must be included.

Equivalent Damping

Similarly, if damper i has relative velocity b_i q˙,

P_d = Σ c_i(b_i q˙)² = c_eq q˙²,

and therefore

c_eq = Σ c_i b_i².

These energy-based expressions prevent a common error: adding masses or damping coefficients directly when their motions are not identical.

Initial-Condition Problems: A Reliable Solution Procedure

A practical solution should proceed in a disciplined sequence:

1. Select the generalized coordinate and define its positive direction.

2. Draw the free-body diagram and identify the equilibrium reference.

3. Derive the equation of motion using force balance, energy, or an equivalent method.

4. Compute ω_n, c_c, ζ, and, if applicable, ω_d.

5. Select the correct response form from the damping regime.

6. Apply x(0) = x_0 and x˙(0) = v_0 explicitly.

7. Check units, limiting cases, initial conditions, and long-time behavior.

For example, consider m = 10 kg, k = 4000 N/m, c = 20 N·s/m, x_0 = 0.01 m, and v_0 = 0. The undamped natural frequency is ω_n = √400 = 20 rad/s. The critical damping coefficient is c_c = 2(10)(20) = 400 N·s/m, so ζ = 0.05. The damped frequency is ω_d = 20√(1 − 0.05²) ≈ 19.975 rad/s. The response is therefore

x(t) = e^(−t) [0.01 cos(19.975t) + (0.05)(0.01)/√(1 − 0.05²) sin(19.975t)],

with displacement in metres and time in seconds. The decay rate is ζω_n = 1 s⁻¹. The system oscillates many times while its amplitude gradually decreases.

Applied Case Study: Free-Decay Testing of a Machine Support

Background

An illustrative industrial case concerns a motor mounted on a steel support frame. Excessive vibration was observed after the motor was switched off, suggesting that the support had a lightly damped natural mode near the operating speed range.

Situation and Problem

A technician displaced the motor-support assembly slightly and released it. A laser displacement sensor recorded successive positive peaks of 8.0 mm, 6.9 mm, 5.95 mm, and 5.14 mm. The time between the first and fourth peaks was 0.30 s. The effective mass, obtained from a finite-element model and experimental calibration, was 120 kg.

The engineering questions were: What are the damping ratio and natural frequency? Is the support likely to settle rapidly after shutdown? Which design change should be considered first?

Analysis

Using the first and fourth peaks, three cycles are involved. The logarithmic decrement is

δ = (1/3) ln(8.0/5.14) ≈ 0.147.

Hence,

ζ = 0.147/√(4π² + 0.147²) ≈ 0.023.

The measured damped period is T_d = 0.30/3 = 0.10 s, giving

ω_d = 2π/T_d ≈ 62.8 rad/s.

Since ω_d = ω_n√(1 − ζ²), the undamped natural frequency is approximately 62.8 rad/s, or about 10 Hz. The system is lightly damped, so its oscillation persists for several seconds after release.

Approach and Solution

The first design option was to increase structural stiffness, which would raise the natural frequency and potentially move the mode away from the motor’s excitation range. The second option was to add a viscous damper, which would increase ζ without substantially changing the natural frequency. Because the measured problem was prolonged decay rather than a direct resonance shift, a properly selected damper was the more targeted intervention.

Outcome

After installation of a damper, a repeat free-decay test showed a substantially larger logarithmic decrement. The support reached an acceptable vibration level in fewer cycles. The test also revealed that the damper had to be attached at a location with adequate relative motion; placing it near a nodal region would have produced little energy dissipation.

Learning Points

The case illustrates that free-decay testing can identify damping without requiring a continuous forcing experiment. It also demonstrates the distinction between changing stiffness, which primarily changes frequency, and changing damping, which primarily changes decay rate. Finally, equivalent parameters depend on the chosen mode and damper location, not merely on component ratings.

Professional Applications, Limitations, and Best Practices

Free-vibration analysis is used in suspension design, instrument calibration, structural health monitoring, rotor-support assessment, vehicle ride analysis, seismic isolation, and the validation of finite-element models. In experimental mechanics, the free-decay curve is often one of the quickest ways to estimate modal frequency and damping.

Several limitations must be recognized. The SDOF assumption fails when closely spaced modes contribute significantly, when the response contains strong nonlinearities, or when the component is coupled to a flexible foundation. Viscous damping is a convenient equivalent model, but real damping may arise from friction, material hysteresis, joints, fluid drag, or acoustic radiation. Damping may also depend on amplitude and frequency.

Common mistakes include confusing angular frequency with frequency in hertz, using ω_n rather than ω_d in a damped oscillation, applying a viscous-damping formula to Coulomb damping, neglecting the static-equilibrium reference, and combining equivalent masses without accounting for velocity ratios. Initial velocity is also frequently omitted incorrectly. A robust solution always substitutes t = 0 into the final expression and verifies both displacement and velocity.

Modern vibration systems commonly use digital accelerometers, laser vibrometers, data acquisition systems, and automated curve-fitting algorithms. Digital signal processing can estimate peaks, decrement, frequency, and uncertainty, but filtering choices and sensor bandwidth must be documented. Automated identification is useful only when the measured signal is genuinely dominated by one mode; otherwise, a multi-degree-of-freedom or operational modal-analysis method is required. Professional practice therefore combines analytical modeling, measurement validation, and transparent reporting of assumptions.

Chapter Summary

An SDOF free-vibration system is governed by inertia, stiffness, and damping. Without damping, the response is periodic at the natural frequency ω_n = √(k/m), and total mechanical energy remains constant. With viscous damping, the damping ratio ζ determines whether the system is underdamped, critically damped, or overdamped. Underdamped motion oscillates at ω_d = ω_n√(1 − ζ²) while its amplitude decays exponentially.

Logarithmic decrement provides a practical method for estimating damping from successive free-decay peaks. Viscous damping dissipates energy at a rate proportional to velocity squared, whereas Coulomb damping produces approximately linear amplitude reduction per cycle and requires piecewise analysis. Equivalent mass, stiffness, and damping should be derived using kinematics and energy, especially when components move differently. Free-vibration theory is valuable for design, testing, troubleshooting, and model validation, but its assumptions must be checked against real mechanisms and measured data.

Key Terms

Natural frequency: The intrinsic oscillation frequency of an undamped system.

Circular frequency: Frequency expressed in radians per second.

Damped natural frequency: Oscillation frequency of an underdamped system.

Damping coefficient: Constant relating viscous damping force to velocity.

Damping ratio: Ratio of actual viscous damping to critical damping.

Critical damping: The minimum damping that prevents oscillation in an ideal linear system.

Underdamping: A condition in which the system oscillates while amplitude decays.

Overdamping: A nonoscillatory condition with damping greater than critical.

Static equilibrium: Position at which constant forces are balanced.

Logarithmic decrement: Natural logarithm of the ratio of successive free-decay amplitudes.

Coulomb damping: Energy dissipation caused by approximately constant sliding friction.

Viscous damping: Damping force proportional to instantaneous velocity.

Equivalent mass: Generalized inertial parameter referred to a selected coordinate.

Equivalent stiffness: Generalized elastic parameter referred to a selected coordinate.

Attenuation coefficient: Exponential decay rate ζω_n of an underdamped response.

Free vibration: Motion occurring after initial disturbance without continuing external excitation.

Generalized coordinate: A coordinate that uniquely describes the modeled configuration.

Energy dissipation: Irreversible removal of mechanical energy from a vibrating system.

Review Questions

Short Answer Questions

1. Define an SDOF system and identify the physical meaning of its mass, stiffness, and damping elements.

2. Distinguish between natural frequency and damped natural frequency.

3. What is critical damping, and why is it important in engineering design?

4. Define logarithmic decrement and state its use in vibration testing.

5. Why must displacement often be measured from the static equilibrium position?

Long Answer Questions

1. Derive the free-vibration response of an undamped SDOF system for arbitrary initial displacement and velocity.

2. Derive the underdamped viscous free-vibration response and explain the significance of its exponential envelope.

3. Discuss the three damping regimes and compare their displacement responses and engineering uses.

4. Explain energy dissipation in viscous damping and contrast it with Coulomb damping.

5. Develop expressions for equivalent mass, stiffness, and damping using generalized-coordinate and energy arguments.

Analytical / Application Questions

1. A 25 kg machine component is attached to a spring of stiffness 10,000 N/m and a damper of 50 N·s/m. Determine ω_n, ζ, ω_d, and the response when x(0) = 5 mm and x˙(0) = 0.

2. A free-decay test gives amplitudes of 12.0, 10.2, 8.7, and 7.4 mm at successive peaks. Estimate the logarithmic decrement and damping ratio using the first and fourth peaks.

3. A mechanism contains two masses with velocities v_1 = q˙ and v_2 = 0.5q˙, and two dampers with relative velocities q˙ and 2q˙. Derive its equivalent mass and damping in terms of the component values, and explain why direct addition of all parameters would be incorrect.

22. Multiple Choice Questions with Answers

Question 1

For an undamped SDOF system with mass m and stiffness k, the natural circular frequency is

A. k/m

B. √(k/m)

C. √(m/k)

D. 2π√(k/m)

Correct Answer: B. √(k/m)

Question 2

An underdamped system has a damping ratio

A. ζ = 0

B. 0 < ζ < 1

C. ζ = 1

D. ζ > 1

Correct Answer: B. 0 < ζ < 1

Question 3

If the initial displacement is x0 and the initial velocity is zero, the undamped response is

A. x0 sin(ωnt)

B. x0 cos(ωnt)

C. x0e^(−ωnt)

D. x0t cos(ωnt)

Correct Answer: B. x0 cos(ωnt)

Question 4

For viscous damping, the critical damping coefficient is

A. mωn

B. 2mωn

C. kωn

D. 2kωn

Correct Answer: B. 2mωn

Question 5

The damped natural frequency of an underdamped system is

A. ωn/√(1 − ζ²)

B. ωn(1 − ζ²)

C. ωn√(1 − ζ²)

D. ζωn

Correct Answer: C. ωn√(1 − ζ²)

Question 6

For a viscously damped free response, the amplitude envelope is proportional to

A. e^(+ζωnt)

B. e^(−ζωnt)

C. sin(ζωnt)

D. t²e^(−ζωnt)

Correct Answer: B. e^(−ζωnt)

Question 7

Logarithmic decrement is primarily used to estimate

A. spring stiffness from static deflection

B. mass from kinetic energy

C. damping from successive free-decay peaks

D. friction coefficient from temperature

Correct Answer: C. damping from successive free-decay peaks

Question 8

Compared with viscous damping, ideal Coulomb damping generally produces

A. exponential amplitude decay

B. constant amplitude

C. approximately linear amplitude reduction per cycle

D. increasing amplitude with time

Correct Answer: C. approximately linear amplitude reduction per cycle

Question 9

If two masses move with velocities q˙ and 0.5q˙, their equivalent mass with respect to q is

A. m1 + m2

B. m1 + 0.5m2

C. m1 + 0.25m2

D. 0.5m1 + m2

Correct Answer: C. m1 + 0.25m2

Question 10

A system with ζ > 1 will ideally

A. oscillate at ωn

B. oscillate at ωd

C. return to equilibrium without oscillation

D. have constant mechanical energy

Correct Answer: C. return to equilibrium without oscillation