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Vibrating systems with multiple sources of motion

From an analytical perspective, systems with two or more sources of motion are modeled with multiple degrees of freedom (DOFs). At the basic level, systems with two and more DOFs are similar to single DOF systems. In a 2DOF system, two independent equations of motion are required to define a system (e.g., a double classical linear oscillator (CLO) or a double pendulum), but the sources need not be the same. For example, a system could have a translational and a rotational DOF. As long as the equations of motion are independent of one other, it is a 2DOF system. These equations must be solved simultaneously. (See a refresher on differential calculus if needed)

Free undamped 2DOF system

As in the SDOF system, an undamped (c1 and c2 = 0) system will be developed first. Fig. 1 shows such a system. As before, Newton’s second law can determine the equations of motion. In the SDOF system, a solution in the form of sine and cosine was used. For the first 2DOF system, another valid form of a solution, a sine with a phase angle, Φ, is used to show another solution form. The equation for mass 1 is ....................(1)

and for mass 2 is ....................(2)

These two equations of motion must be solved simultaneously because they are coupled through the displacement terms, x1 and x2. As before, a solution is assumed and substituted back into the equations of motion. The assumed solution is ....................(3)

and ....................(4)

for which the second differentiation is ....................(5)

and ....................(6)

Substituting back into the equations of motion and collecting terms, the result is: ....................(7)

and ....................(8)

The only way not to have a trivial solution (C1 and C2 = 0) to these equations is to have the determinate of the coefficients be zero: ....................(9)

Using linear algebra, the determinate is: ....................(10)

which is a quadratic equation in terms of the square of the natural frequencies, ω2. This solves to: ....................(11)

There are two roots to this equation, which means that there are two natural frequencies. The solution therefore must be in terms of the two frequencies. Substituting the first natural frequency back into the equation of motion solution gives the result of the first natural frequency: ....................(12)

and ....................(13)

The ratio of C11 to C21, known as the mode shape, for the first natural frequency is: ....................(14)

therefore, ....................(15)

The first natural frequency motions then are: ....................(16)

and ....................(17)

The mode shape for the second natural frequency is: ....................(18)

therefore, ....................(19)

The second natural frequency motions then are: ....................(20)

and ....................(21)

The complete solution then is both displacements added into one equation: ....................(22)

and ....................(23)

The values of C11, C12, C21, C22, Φ1, and Φ2 depend on the initial conditions and the mode shapes, λ1 and λ2.

Ex. 1 is a 2DOF free and undamped CLO system.

Free damped 2DOF system

Adding damping complicates the equations considerably, but the procedure remains the same. In this case, it is easier to use linear algebra. Using the same model as before (see Fig. 2), but adding viscous dampers (c1 and c2 ≠ 0), the equations of motion for the independent DOFs are determined for mass 1 as ....................(24)

and for mass 2 as ....................(25)

This can be written in matrix form as ....................(26)

or as ....................(27)

where M = the mass matrix, C = the damping matrix, K = the stiffness matrix, = the acceleration vector, = the velocity vector, and X = the displacement vector, which are given as: ....................(28) ....................(29) ....................(30) ....................(31) ....................(32)

and ....................(33)

If the solution is assumed to be of the form ....................(34)

then ....................(35)

and ....................(36)

Substituting back into the equation of motion, the result in matrix form is: ....................(37a)

which can be rewritten as ....................(37b)

Because time is always positive and a nontrivial solution is desired, the only way this equation is true is if the determinate of the coefficient of Ceωt is zero; that is, if ....................(38)

The determinate of Eq. 38 is a fourth-order polynomial in terms of a, which means that there are four roots. These roots can be:

1. Four real and negative roots.
2. Two sets of complex conjugates with negative real parts.
3. Two real and negative roots and one set of complex conjugates.

If number one is the case, then the result is an exponentially decaying motion without oscillation. It is similar to the overdamped case for an SDOF system. If number two is the case, then the motions will be exponentially decaying oscillations for both DOFs. This is similar to the underdamped case for an SDOF system. Finally, for case number three, either condition can occur.

Ex. 2 is a 2DOF free and underdamped CLO system.

More information on linear algebra can be found on the mathematics of fluid flow page.

Forced damped 2DOF system

Adding forcing complicates the equations considerably yet again. The procedure is the same, however, and it is a matter of keeping the mathematics straight. Many texts are available to delve more deeply into this subject. For more information, please refer to the literature.

Multiple-DOF systems

The previous discussion of 2DOF systems points out how to handle any DOF system. The last example used matrix notation to define the system for the solution process. Multiple-DOF systems are solved similarly with the primary difference being the degree of the defining matrices is greater, as is the degree of difficulty in solving the system. The matrix will have the same number of rows and columns as the degree of freedom. There are other methods (e.g., finite-element modeling) that can be used to tackle the complexity of multiple-DOF systems.

Nomenclature

 c = axial damping coefficient, mL/t, lbf-ft/sec C = constant of integration, various C = damping matrix, mL/t, lbf-ft/sec k = spring constant, m/t2, lbf/in. K = stiffness matrix, m/t2, lbf/in. m = mass, m, lbm M = mass matrix, m, lbm t = time, seconds x = displacement, L, in. ẋ = first derivative with respect to time of displacement (velocity), L/t, ft/sec ẍ = second derivative with respect to time of displacement (acceleration), L/t 2 , ft/sec 2 X = displacement vector Ẋ = velocity vector, L/t, ft/sec Ẍ = acceleration vector, L/t2, ft/sec2 λ = mode shape, dimensionless Φ = phase angle, rad ω = frequency, 1/t, Hz