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Vibration theory

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Vibrations are a common occurrence in oil and gas activities that can affect operations, planning, facility design, and interpretation of results. Vibration is common in drillstrings, on platforms, wherever large engines are operating, in seismic operations, and many other aspects of oil and gas. Understanding vibration theory and the mathematics of vibrations are important to successful operations. A refresher on differential calculus can come in handy as well.

The fundamental theories of vibration are not new. Indeed, Saint-Venant[1] published his theory on the vibrations of rods in 1867, and Love[2] published an entire treatise on vibration theory in 1926. The mathematics of vibration theory involves infinite series, complex functions, and Fourier integral transforms, and its physics involves Newtonian mechanics and stress analyses. Until recently, except under relatively simple conditions, the complexity of such mathematics had restrained the application of vibration theory to solving simple common problems. Now, however, state-of-the-art computers can perform these complex calculations in a reasonable time frame, making possible a wave of new studies.

Vibration theory

A vibration is a fluctuating motion about an equilibrium state. There are two types of vibration: deterministic and random. A deterministic vibration is one that can be characterized precisely, whereas a random vibration only can be analyzed statistically. The vibration generated by a pumping unit is an example of a deterministic vibration, and an intermittent sticking problem within the same system is a random vibration.

In mechanical systems, deterministic vibrations are excitations that elicit a response from a system, as shown schematically in Fig. 1. In theory, as long as two of the three variables (excitation, system, and response) are known, the third one can be determined; however, the mathematics might be challenging. Most often, the response function is sought, so that the excitation function and the system must be known.

Vibration systems can be linear or nonlinear, and discrete or continuous (Fig. 2). In all cases, a vibration system can be in one, two, or three mutually orthogonal dimensions. A linear system is a system in which proportionality (Eq. 1) and superposition (Eq. 2) are true, that is, in which:




When proportionality and superposition are not true, then the system is nonlinear.

A discrete system is one having a finite number of independent coordinates that can describe a system response. These independent coordinates are known as degrees of freedom (DOFs). If the motion of mass, either translational or rotational, of a vibrating system is a function of only one independent coordinate, then the system has one DOF. If two or more independent coordinates are required to describe one or both types of motion, then the system has two or more DOFs. If a system is continuous (an infinite set of independent coordinates is needed to describe the system response), it has an infinite number of DOFs. Because material structures all have a continuous nature, all systems have an infinite number of DOFs. Most systems have dominant DOFs; some even have a single dominant DOF. Such systems therefore can be characterized as discrete systems, which makes the mathematics more tractable.

If a system has a single DOF or set of DOFs in only one direction, it is a 1D system. If there are two mutually orthogonal directions for the DOF, it is a 2D system; and if there are three mutually orthogonal directions for the DOF, it is a 3D system.

As Fig. 3 shows, the excitation function can be periodic or transient, and absent or present. A periodic vibration is one that can be characterized mathematically as an indefinite repetition. A transient vibration is of finite length and is composed of waves that have a definite beginning and that eventually die out. These waves can be of extremely short duration or last for some time.

A standing wave is a vibration whose wave profile appears to be standing still, though actually the particles that make up the material are oscillating about an equilibrium position. Because of the geometry and boundary conditions of the material through which they are traveling, the waves and the reflected waves cancel and reinforce themselves over the same location in the material, which makes the wave profile appear not to be moving. The point at which no motion is occurring is a nodal point, or node. The point of maximum amplitude is the antinode.

In reality, all waves are transient in some way. If a wave is repeated over a longer time than it takes for a single wave to propagate through a material, then this series of waves can be called a vibration. All vibrations are transient, as well. If the vibration lasts longer than the time under analysis, then it can be characterized as infinite in length.

When the excitation is present and is actively affecting the system within the analysis time frame, the response is called a forced vibration. The response of a system with an absent excitation function—one that is not present within the analysis time frame—is called a free vibration. As such, the system can be responding to the removal of an excitation function. For example, if the response of a mass and spring system is sought after the system has been pulled down and released, the original excitation function (the pulling force) is considered absent because the analysis is being performed after the release.

Wave propagation

The method by which a vibration travels through a system is known as wave propagation. When an external force is impressed on a real-world elastic body, the body does not react instantly over its entire length. The point immediately under the external force reacts first, and then the section just under that point reacts to the previous section’s reaction, and so on. This series of reactions is called wave propagation because the reactions propagate through the body over a period of time at a specific velocity. If the rate of change of the external force is slow enough, static equilibrium analysis can model the reactions adequately for most engineering applications. This is called rigid-body analysis. If the external force changes rapidly, however, wave-propagation analysis is necessary to model the reactions effectively.

There are many types of elastic waves. Some listed in this section are longitudinal, lateral, and bending waves.[3] Some of these are shown in Fig. 4.

In longitudinal waves (also variously called compression/tension, axial, dilatational, and irrotational waves), the particles that make up the elastic medium are forced directly toward and away from each other, and the direction of the particles’ motion is parallel to that of the wave motion. In most steels, longitudinal waves travel at ≈16,800 ft/sec. Longitudinal waves are not dispersive. This means that all the wave components that make up a longitudinal wave travel at the same velocity and, hence, do not separate (disperse).

In lateral waves (also known variously as shear, torsional, transverse, equivoluminal, and distortional waves), the particles slip beside each other, and move perpendicular to the direction of the wave motion. Because slipping uses more energy, lateral waves are slower. In steel, for example, they travel at 10,400 ft/sec. A rapidly changing torsional force on a section of pipe will cause a lateral wave to propagate from the point of application to all other parts of the pipe. It propagates as an angular twist. Lateral waves are nondispersive and have a similar solution method as the longitudinal waves; however, shear or transverse waves are dispersive (i.e., the wave components that make up a shear wave travel at different velocities). Their wave components will disperse and "smear" the initial wave profile. This complicates the analysis significantly.

Bending waves (flexural waves) travel as a bend in a bar or plate and have longitudinal and lateral components. Rotary shears and moments of inertia complicate bending-wave analysis.

Wave-propagation studies in petroleum engineering areas generally have been confined to longitudinal, torsional, and lateral waves. Of these, longitudinal waves generally are easiest to model and are considered in this section. A compression wave is a stress wave in which the propagated stress is in compression. Likewise, a tension wave is a stress wave in which the propagated stress is in tension.

Wave reflection from various geometric boundaries

Wave propagation is the movement of a distinct group of waves through some material in response to an external force.

A key point in wave-propagation studies is how waves interact with geometric discontinuities. What happens as a wave meets a fixed or free boundary condition? Also, what happens to a wave as it encounters a geometrical area change or a change in material properties?

There are two limiting boundary conditions for wave propagation: a fixed (pinned) end (zero displacement) and a free end (zero stress). A fixed end is a boundary condition in which there is zero displacement. According to wave theory, during a wave encounter with a fixed end, the stress at the fixed end doubles during the passage of the wave. A reflection of a stress wave will simply bounce back with the same sign. A compression wave will reflect as a compression wave and a tension wave will reflect as a tension wave. At a fixed end, because the displacement is zero, the particle velocity will be zero. The wave particle velocity amplitude is inverted during a reflection from a fixed end.

A free end is defined as a boundary condition free to move. The stress at the free end is always zero. The effects on stress and particle velocity caused by a free end are opposite of the effects on stress on a fixed end. A compression wave encountering a free end reflects as a tension wave, and a tension wave reflects as a compression wave. The wave particle velocity values double during an encounter with a free end and reflect with the same sign.

As a wave encounters a change in cross-sectional area, some of the wave is reflected and some is transmitted (refracted). The amplitudes and sign of the waves depend on the relative change in cross-sectional area. The equation that describes the effect on the incident force, Fi, of a cross-sectional-area, density, or modulus-of-elasticity change for the transmitted wave is


and for the reflected wave is


If an incident wave encounters a junction where the relative change in cross-sectional area is greater than 1 (a smaller area to a larger area), most of the wave will transmit through the junction. Some of the wave will reflect from the junction and will keep the same sign. For example, a compression wave will transmit through the junction and keep going as a somewhat-diminished compression wave. The part of the wave that is reflected is still a compression wave, but its amplitude is less than that of the wave that transmitted though the junction.

On the other hand, if an incident wave encounters a junction where the relative change in cross-sectional area is less than 1 (a larger area to a smaller area), most of the wave will reflect off the junction, but some of it will transmit through the junction and will keep the same sign. For example, a compression wave will transmit through the junction and keep going as a diminished compression wave. The reflected part of the wave is a tension wave whose absolute amplitude is greater than that of the compression wave that is transmitted through the junction.

As with most drillstrings, there are many geometric discontinuities (changes in cross-sectional area) that will cause part of the wave to refract and part to reflect. For example, drill collars to heavyweight drillpipe to drillpipe all are geometric discontinuities. Sometimes, too, there are material discontinuities—changes in material density or modulus of elasticity—that cause refractions and reflections. A third possible type of discontinuity is when there are different endpoints. For example, if the pipe is stuck, one end can be modeled as stuck. If the pipe is hanging freely, such as with casing running, then the end is free.

More detailed information on wave propagation can be found in several sources.[4][5][6][7][8][9]

Wave behavior

Wave velocity depends primarily on density and modulus of elasticity but also is affected by damping and frequency. For example, hitting one end of a long steel rod with a hammer will generate a longitudinal wave that compresses the particles of the steel. The wave’s length is set by the length of time that the hammer is in contact with the end of the rod, whereas its magnitude is set by the force of the hammer blow. As the wave moves along the rod, the steel within the length of the wave is compressed. After the wave passes, the steel returns to its unstressed state, though not necessarily in the same location as before the wave passage.

As another example, twisting (shearing) a steel rod will generate a shear wave. A shear wave moves along the rod more slowly than the longitudinal wave does. Similarly to the longitudinal wave discussed above, its length is set by the duration of the twisting action, whereas its magnitude is set by the torque from the twisting action.

Waves act independently, but the stresses they create can be additive. For example, two equal compression waves that are generated simultaneously by hammer blows at each end of a long steel rod will meet in the center of the rod, pass through one another, and then each continue along the rod as if the other never existed (independence). While the waves are passing each other, however, the compression in the steel will be twice (additive) that of either wave.

Natural frequencies and resonance

Everything has a natural frequency, a frequency at which it would vibrate were it given the energy to vibrate and left alone. For instance, the human body has a natural frequency of ≈5 cycles/sec. All drill and rod strings have a natural frequency that depends on the material properties and geometry. The material properties determine the wave velocity, and the geometry determines how waves are reflected and refracted.

During wave propagation, the wave eventually reaches an end of the material. Some of the wave will reflect back to its source. If the reflection reaches the source at the same time a new wave is generated, the two waves will combine and be synchronized in phase. Later, if those two waves’ reflections return to the source at the same time the next new wave is generated, all three waves will combine. This will continue for as long as waves are generated under these conditions, and the resultant wave will increase in amplitude, theoretically to infinity. This is called resonance. The frequency at which resonance occurs is the natural frequency or an integer multiple of that frequency (called a harmonic). If this wave reinforcement is allowed to continue, the system eventually will either self-destruct or fatigue to failure.

A continuous system contains an infinite number of natural frequencies, whereas a discrete single-degree-of-freedom (SDOF) system (e.g., a point mass on a massless spring) has only one natural frequency. If two point masses are connected using two springs, then there are two natural frequencies in this 2DOF system. In general, the number of DOFs in a system determines the number of natural frequencies it has, which means that any discrete system will have a finite number of natural frequencies; however, in reality, there is an infinite number of natural frequencies because all systems are continuous. Some frequencies will have higher amplitudes than others. Such continuous systems with discrete higher-amplitude responses can be modeled with a discrete methodology.


Resonance energy does not reach an infinite value because of damping, the dissipation of energy over time or distance.[10] Without damping, or friction, the energy from vibrations would build until there is more energy than the structure can sustain, which can cause structural failure.

A wave propagating into a system adds energy to a system, whereas damping removes it. Generally, the dissipated energy from the vibration is converted to heat, and if damping does not take enough energy out of a system, the system can self-destruct from energy overload. The amount of energy in a system at a given time is reflected in the system’s stress/strain level. The more stresses/strains in the system, the higher the energy level. Once the stresses reach a value greater than the yield strength of the system, yield failure is imminent. If the stresses are greater than the ultimate strength of the material, failure is immediate.

In the borehole, three distinctive types of damping occur: viscous, Coulomb, and hysteretic. Viscous damping occurs when the damping force generated is proportional to the velocity of the particles. Coulomb damping (also called dry friction) is the force generated by the movement of materials past one another, and it usually is proportional to the force normal to the materials’ surfaces. The dynamic and static coefficients of friction are the proportionality constants. Hysteretic damping is the friction force generated by the relative motion of the internal planes of a material as a wave causes particle motion. Although this is true of all materials, some materials are viscoelastic (i.e., they show a much larger hysteretic effect than do others).

Viscous damping

As noted above, viscous damping occurs when the damping force is proportional to the velocity of the particles. Viscous damping is shown by:


One way that viscous damping arises in jarring analysis is from the interaction of a solid and liquid at their interface, such as where the steel contacts the liquid mud along the sides of a drillstring.

One method for determining the damping involves noting the decrement of acceleration over one vibration cycle.[11] An impulse is impressed on the drillstring to produce a wave. While the wave is decaying, the acceleration is measured and recorded multiple times at one location on the string and at the same phase (i.e., crest to crest). The time between recordings also is noted. These values are used in Eq. 6 to compute the damping coefficient (c). Unfortunately, though, this method gives the total damping and does not distinguish between viscous and Coulomb damping.


Coulomb damping is the friction that occurs when two dry surfaces slide over each other, and its force is a constant value that is independent of particle velocity and displacement, but dependent on the friction factor (μ) and the force normal to the friction surface. This value is:


The Coulomb damping force always is of the opposite sign from that of the particle velocity, so that the damping force reverses when the particle velocity changes signs. This discontinuity makes it a nonlinear damping force, shown as:


Nonlinearity makes a closed-form solution to an equation of motion difficult.

Hysteretic damping

Hysteretic damping also is called structural damping because it arises from internal friction within a structure. A wave moves through a material because the atomic structure is reacting to an applied force. As the atoms of the structure move, energy is lost through the interaction of these atoms with their neighboring atoms. Hysteretic damping is the energy lost when atoms move relative to each other.

If a material had a perfectly linear stress/strain relationship, hysteretic damping would not occur. In reality, though, there is no such thing as a perfectly linear stress/strain curve. Two curves develop on the stress/strain diagram while a material is stressed and relieved. The center area between these two curves represents the energy lost to internal friction. (This hysteresis loop is the reason for the name of this damping type.) This variation can be small, but the amount of energy dissipated can be large because high-frequency vibrations can cause this loop to be repeated many, many times over a given time period.[12]

The hysteretic-damping value is highly dependent on a number of factors. One factor is the condition of the material (i.e., chemical composition, inhomogeneities, and property changes caused by thermal and stress histories). Another is the state of internal stress from initial and subsequent thermal and stress histories. Also, the type and variation of stress—axial, torsional, shear, and/or bending—affect the hysteretic-damping value.

A way of looking at hysteretic-damping force is to set it proportional to the particle velocity divided by the wave frequency. This is shown in Eq. 9.[13]


Equivalent springs

Many systems can be modeled as multiple springs. Such springs can be combined into a single, equivalent spring (Fig. 5). For parallel springs, the sum of the spring constants is equal to the equivalent spring constant (Eq. 10). For series springs, the reciprocal of the sum of the reciprocals of the spring constants is equal to the equivalent spring constant (Eq. 11). A linear spring oscillates in a single translational direction. A torsional spring oscillates with an angular twist (Eq. 12).

Boundary and initial conditions

The boundary conditions (how the ends of a system are attached) and initial condition (condition of the system at the start in time) are extremely important in vibration and wave propagation analysis. The specific solution of any ODE or PDE requires a set of boundary and/or initial conditions. Usually, a displacement (boundary condition) and an initial velocity (initial condition) are specified.

In wave propagation, the boundary conditions also dictate wave behavior. For example, a compression wave is reflected from a free end as a tension wave and from a fixed end as a compression wave. If two rods are connected at their ends and are of different geometry or material, then a fraction of the energy of the wave is reflected and the remaining portion of the energy is refracted at their connection. Other types of boundaries direct the system response by limiting the DOF. This includes:

  • Boundary conditions of pinned, revolute, translational, translational and rotational
  • Forcing function
  • Mass spring and/or damper, and a semi-infinite connection

In addition, changes in material properties will affect the various constants and will cause wave-propagation reflections and refractions at the boundary between the properties. Fig. 6 shows some typical boundary conditions.

Mechanical vibration analysis

There are three components to mechanical vibration analysis:

  • Determine the geometric compatibilities
  • Determine the constitutive (material properties) equations
  • Determine the equilibrium condition

The geometric compatibilities are the displacement constraints and connections. They also include the continuous properties, which state that the system does not separate into individual pieces. (If it does, that is another problem altogether.) The constitutive equations represent the material properties, which include mass, damping, and spring coefficients. These constitutive equations include stress/strain relationships and Hooke’s law (Eqs. 13a and 13b):


or, in another form,


The coefficient of Δl in Eq. 13b often is called the spring constant or stiffness constant.

The equilibrium condition is based on both static and dynamic conditions. A static equilibrium states that the sum of the forces acting on an object is equal to zero:


A dynamic equilibrium is based on Newton’s second law and is the basis of many vibration analysis methods. The sum of the forces acting on an object is equal to its mass times the acceleration of the object. Other dynamic-equilibrium analysis includes virtual work methods and energy-balance methods (Hamilton’s principle).

Newton’s second law for a translational system is


and for torsional systems is


Newton’s second law can be rewritten in a form known as D’Alembert’s principle:


in which mẌ is treated as a force and is called an inertial force.

Some basic equations of vibration analysis are shown in Table 1.


Ac = cross-sectional area, L2, in.2
c = axial damping coefficient, mL/t, lbf-ft/sec
E = modulus of elasticity, m/Lt2, psia
E(t) = excitation function
Ft = transmitted force, mL/t2, lbf
Fd = damping force, mL/t2, lbf
Ff = friction force, mL/t2, lbf
Fh = hysteretic force, mL/t2, lbf
Fi = incident force, mL/t2, lbf
Fn = normal force, mL/t2, lbf
Fr = reflected force, mL/t2, lbf
h = hysteretic factor, dimensionless
I = second moment of inertia, L4, in.4
L = total length, L, ft
m = mass, m, lbm
r = generic constant
R(t) = response function
t = time, seconds
vs = sonic velocity, L/t, ft/sec
= second derivative with respect to time of displacement (acceleration), L/t 2 , ft/sec 2
Δl = change in length, L, in.
α = amplitude, various
ε = strain, L/L, in./in.
Ӫ = second derivative with respect to time of twist (acceleration) rad/sec2
μ = friction factor, dimensionless
ρ = density, m/L3, lbm/in.3
σ = stress, m/Lt2, psia
ω = frequency, 1/t, Hz


  1. Barré de Saint-Venant, A.J.C. 1867. Sur le choc longitudinal de deux barres élastiques. Journal de Mathématiques Pures et Appliquées 2 (12): 237.
  2. Love, A.E.H. 1926. A Treatise on the Mathematical Theory of Elasticity, fourth edition. New York: Dover Publications.
  3. Meyers, M.A. 1994. Dynamic Behavior of Materials. New York: John Wiley & Sons.
  4. Hudson, J.A. 1980. The Excitation and Propagation of Elastic Waves. Cambridge, UK: Cambridge University Press.
  5. Mal, A. and Singh, S.J. 1991. Deformation of Elastic Solids. Englewood Cliffs, New Jersey: Prentice-Hall.
  6. Sharman, R.V. 1963. Vibrations and Waves. London: Butterworth.
  7. Chin, W.C. 1994. Wave Propagation in Petroleum Engineering: Modern Applications to Drillstring Vibrations, Measurement-While-Drilling, Swab-Surge, and Geophysics. Houston, Texas: Gulf Publishing.
  8. Tolstoy, I. 1973. Wave Propagation. New York: McGraw-Hill Book Co.
  9. Doyle, J.F. 1989. Wave Propagation in Structures: An FFT-Based Spectral Analysis Methodology. New York: Springer-Verlag.
  10. Harris, C. and Crede, C. ed. 1988. Shock and Vibration Handbook, third edition. New York: McGraw-Hill Book Co.
  11. Dareing, D.W. and Livesay, B.J. 1968. Longitudinal and Angular Drill-String Vibrations With Damping. Journal of Engineering for Industry 90B (4): 671-679.
  12. Kolski, H. 1963. Stress Waves in Solids. New York: Dover Publications.
  13. Doyle, J.F. 1991. Static And Dynamic Analysis Of Structures With an Emphasis on Mechanics and Computer Matrix Methods. Dordrecht, The Netherlands: Kluwer Academic Publishers.

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See also

Basic vibration analysis

Vibrating systems with multiple sources of motion

Continuous vibration systems