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Vibrating systems with multiple sources of motion

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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

Vol1 page 0026 eq 001.PNG....................(1)

and for mass 2 is

Vol1 page 0026 eq 002.PNG....................(2)

  • Fig. 1 – 2DOF free and undamped CLO system.

    Fig. 1 – 2DOF free and undamped CLO system.

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

Vol1 page 0026 eq 003.PNG....................(3)

and

Vol1 page 0026 eq 004.PNG....................(4)

for which the second differentiation is

Vol1 page 0027 eq 001.PNG....................(5)

and

Vol1 page 0027 eq 002.PNG....................(6)

Substituting back into the equations of motion and collecting terms, the result is:

Vol1 page 0027 eq 003.PNG....................(7)

and

Vol1 page 0027 eq 004.PNG....................(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:

Vol1 page 0027 eq 005.PNG....................(9)

Using linear algebra, the determinate is:

Vol1 page 0027 eq 006.PNG....................(10)

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

Vol1 page 0027 eq 007.PNG....................(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:

Vol1 page 0027 eq 008.PNG....................(12)

and

Vol1 page 0027 eq 009.PNG....................(13)

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

Vol1 page 0028 eq 001.PNG....................(14)

therefore,

Vol1 page 0028 eq 002.PNG....................(15)

The first natural frequency motions then are:

Vol1 page 0028 eq 003.PNG....................(16)

and

Vol1 page 0028 eq 004.PNG....................(17)

The mode shape for the second natural frequency is:

Vol1 page 0028 eq 005.PNG....................(18)

therefore,

Vol1 page 0028 eq 006.PNG....................(19)

The second natural frequency motions then are:

Vol1 page 0028 eq 007.PNG....................(20)

and

Vol1 page 0028 eq 008.PNG....................(21)

The complete solution then is both displacements added into one equation:

Vol1 page 0028 eq 009.PNG....................(22)

and

Vol1 page 0028 eq 010.PNG....................(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.

  • Example 1 – 2DOF free and undamped CLO system.

    Example 1 – 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

Vol1 page 0030 eq 001.PNG....................(24)

and for mass 2 as

Vol1 page 0030 eq 002.PNG....................(25)

This can be written in matrix form as

Vol1 page 0030 eq 003.PNG....................(26)

or as

Vol1 page 0030 eq 004.PNG....................(27)

where M = the mass matrix, C = the damping matrix, K = the stiffness matrix, Ẍ = the acceleration vector, Ẋ = the velocity vector, and X = the displacement vector, which are given as:

Vol1 page 0030 eq 005.PNG....................(28)

Vol1 page 0030 eq 006.PNG....................(29)

Vol1 page 0030 eq 007.PNG....................(30)

Vol1 page 0030 eq 008.PNG....................(31)

Vol1 page 0030 eq 009.PNG....................(32)

and

Vol1 page 0030 eq 010.PNG....................(33)

If the solution is assumed to be of the form

Vol1 page 0030 eq 011.PNG....................(34)

then

Vol1 page 0031 eq 001.PNG....................(35)

and

Vol1 page 0031 eq 002.PNG....................(36)

Substituting back into the equation of motion, the result in matrix form is:

Vol1 page 0031 eq 003.PNG....................(37a)

which can be rewritten as

Vol1 page 0031 eq 004.PNG....................(37b)

  • Fig. 2 – 2DOF free and damped CLO system.

    Fig. 2 – 2DOF free and damped CLO system.

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

Vol1 page 0031 eq 005.PNG....................(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.

  • Example 2 – 2DOF free and damped CLO system.

    Example 2 – 2DOF free and damped 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.[1][2][3][4][5][6][7][8][9]

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

References

  1. ↑ Achenbach, J.D. 1973. Wave Propagation in Elastic Solids, seventh edition. Amsterdam: North-Holland Publishing Co.
  2. ↑ Clough, R.W. and Penzien, J. 1975. Dynamics of Structures. New York: McGraw-Hill Book Co.
  3. ↑ Den Hartog, J.P. 1934. Mechanical Vibrations. New York: Dover Publications.
  4. ↑ Meirovitch, L. 1986. Elements of Vibration Analysis, second edition. New York: McGraw-Hill Book Co.
  5. ↑ Nashif, A.D., Jones, D.I.G., and Henderson, J.P. 1985. Vibration Damping. New York: John Wiley & Sons.
  6. ↑ Seto, W. 1964. Mechanical Vibrations. New York: McGraw-Hill Book Co.
  7. ↑ Shabana, A.A. 1991. Theory of Vibration, Volume I: An Introduction. New York: Springer-Verlag.
  8. ↑ Shabana, A.A. 1991. Theory of Vibration, Volume II: Discrete and Continuous Systems. New York: Springer-Verlag.
  9. ↑ Elmore, W.C. and Heald, M.A. 1969. Physics of Waves. New York: Dover Publications.

Noteworthy papers in OnePetro

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External links

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See also

Vibration theory

Basic vibration analysis

Continuous vibration systems

Differential calculus refresher

PEH:Mathematics_of_Vibrating_Systems

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