The modal equations in classical mechanics

0045-794919053.00 + 0.00 Pergamon Ress pk

Compters & Strwfwes Vol. 37, No. 4, pp. 617-619, 1990 Printed in Great Britain.

TECHNICAL NOTE THE MODAL

EQUATIONS

IN CLASSICAL

MECHANICS

A. E. ANUTA JR Civil Engineering Department, Arizona State University, Tempe, AZ 85287, U.S.A. (Received 10 October 1989) AImtract-Tbe philosophical development of the modal equations, and several mathematical extensions from them, are discussed in relation to the various areas of classical mechanics. Some obvious similarities occur between all of these areas because of their common physical and mathematical foundations.

The constitutive form of the modal equations is developed from Rayleigh’s quotient by minimizing it with respect to the generalized coordinates, xi,

For a discrete, undamped, linear elastic system in free vibration, the modal equations are developed by an application of two fundamental principles: the conservation of energy, and the minimization of the energy function. The principle of conservation of energy is TCV=C,

*dV_p&) 8x1 ax,-’

T+V=O.

(2)

,...,

n,

(3)

where j is the imaginary number which is necessary to account mathematically for the fact that the velocity and displacement vectors are always out of phase with each other. Thus, frequency is a linear operator which mathematically describes a physical observation. The proportionality between the system velocities and displacements [eqn (3)] can be used to transform the system kinetic energy function into another function of proportional value, ?, determined solely by the system displacements: -o@(X) Substituting quotient’ [l]:

= T(f).

eiik = mijku - kern,,

i

(4)

(5)

wbere wR is ‘Rayleigh‘s frequency’; this equation is the system energy function, CAS 37/G-Q

(6)

(74

and the corresponding modal equations are [2]

eqn (4) into cqn (2) gives ‘Rayleigh’s

+g,

n.

Mathematically, a function is minimized by equating to zero all of its partial derivatives taken with respect to the independent variables, and then solving the resulting equation set to find the stationary values of the variables. Thus, the physical observation of the proportionality between the velocity and displacement vectors both generates the frequencies which are the dependent variables of the system energy function, and reduces the independent variables to the problem displacements only. For linear systems, the solution to the resulting equation set is a complete set of orthogonal displacements, each related to a corresponding frequency of vibration. Mass and stiffness are physically observable quantities. For discrete vibrating systems, the mass and stiffness coefficients, m and k respectively, are determined by measurements, which are actually experimental comparisons to defined physical standards. From these coefficients, the kinetic and potential energy functions obtain tigebraic values. Then, applying the ~nstitutive form of the modal equations to the kinetic and potential energy functions generates the modal coefficients, e:

For an undamped, linear elastic system, the potential energy is a function solely of the system displacements, V(x), whereas the kinetic energy is a function solely of the system velocities, T(k). A crucial observation in the development of vibration theory is that for linear systems in free vibration, the magnitude of the velocity vector equals the frequency of oscillation, o, times the co~~nding ~~itude of the displacement vector: i=l,2

,...,

(1)

where T is the system kinetic energy function, V is its potential energy, and C is a constant. Kinetic energy is a ~sitiv~~nite q~ntity, wbereas potential energy is determinate to within an additive constant depending upon the selection of an arbitrary reference datum. No generality is lost by selecting the reference datum so that

_$=jwx,,

i=l,2

i

ie#,x,x,x,=O,

i=l,2

,...,

n.

(7b)

The solutions to eqns (5) and (7b) are the complete set of mode shape vectors and their corresponding frequencies. These mode shapes and frequencies can subsequently be used to construct the equations of motion for the vibrating system in a manner similar to the conventional solution to the same problem. The equations of motion for a vibration problem are functions of both displacement and time. The time coordinate does not appear directly in the modal equations. However, in the development of the modal equations Rayleigh’s frequency, which is a time function, is the dependent variable about which the minimization is performed. The rows of the modal equation set are therefore representations of the system displacements at various, but equal, times. If T represents a temporal index and S a spatial

617

618

Technical Note

one, the modal coefficients can be interpreted as e,. This interpretation of the dimensions of the modal equation indices is significant in some of the associations between vibration theory and other areas of classical mechanics. Up to this point the modal equations have simply served as an alternate mathematical path leading from the mass and stiffness coefficients of the physical system to the equations of motion describing its behavior in space and time. For the conventional solution to the vibration problem, the continuity of the mathematical development is interrupted at this point and no further direct extension is possible. There are several reasons for this; one sufficient reason is that the dynamical equations of the conventional solution are quadratic in nature, as opposed to the cubic form of the modal equations. Because the dynamical equations are quadratic, they become identically zero under the required triple differentiation of the compatibility conditions. However, the cubic form of the modal equations reuslts in a non-trivial application of the compatibility conditions, and this is a window to further continuous mathematical developments relating vibration theory to other areas of mechanics [3]. The successful satisfaction of the compatibility conditions by the modal equations guarantees that the function surface they generate is continuous and therefore solvable. Unfortunately, the solution is not necessarily easy, because the modal equations are always singular and therefore possess multiple solutions. However, the true solution to these equations can be obtained through an occult principle, wherein the conflicting requirements of the equations themselves, their spatial and temporal constraints, and any auxiliary equations, are all brought into a conjunctive alignment [4]. The solution is tedious and can only be accomplished with the aid of a digital computer in a converging iteration scheme, but successful solutions are possible. Compatibility is a mathematical concept originally applied to the elasticity problem to guarantee that the function surface generated by the system displacements remains smooth and continuous. Although the physical interpretation of the mathematical terms changes when compatibility is applied to the vibration problem, the conclusions are still valid. The most obvious extension of the modal equation concepts beyond vibration theory is to the subject of statics, through the similarity between the minimization of the system energy function and Castigliano’s second theorem. Castigliano’s second theorem can be stated as: ‘A linear elastic structure in static equilibrium with deflections due to external forces will always assume a configuration of a minimum distortion energy’ [5]. Specifically, the displacements of the structure are determined by minimizing its potential energy function; there is no kinetic energy function. The most significant difference between the two subject areas is that the solution to the vibration problem is a function of both space and time, whereas the statics problem solution is a function of space only. There is also an obvious relationship between the subjects of statics and elasticity, which now brings the compatibility conditions into the association. The concepts of compatibility and the minimizing of the system energy function then connect the modal equations to the subject of elasticity through the distortion energy method for determining the displacements of an elastic body. The most obvious differences between the two areas are the discrete form of the modal equations and the continuous nature of elastic bodies. As momentum is the derivative of the kinetic energy function with respect to the generalized velocities, the principle of conservation of momentum links vibration theory and the subject of dynamics for systems without a potential energy function. The proportionality between the displacement and velocity vectors [eqn (3)] results in an

equality of the differentiation operation of dynamics and the minimization used to obtain the modal equations. Returning once again to the modal equations, to follow a second mathematical path, they are invariant under relativistic transformations [6]. That is, if two identical vibrating systems are attached to two Cartesian coordinate systems, one at rest and the other moving away at a uniform velocity, the modal equations for both systems are identical. Thus, their mode shapes are always equal. Because of the ‘red-shift’ of light, the frequencies of vibration between the two systems are not equal. As the invariance of the mode shapes and the red-shift of the frequencies at relativistic velocities agree with accepted physical observations, there now exists a link between vibration theory and the area of relativity. Plan&s law describes the emission of light in terms of its vibrational frequencies 171.Actuallv. Planck’s law contains a paradox for two reasons: (1) A frk;luency of vibration is a result of its corresponding mode shape and not vice versa; it is possible to devise more than one mode shape resulting in any given frequency when substituted into Rayleigh’s quotient, but the inverse is not true. (2) The mode shapes are invariant under relativistic transformations but frequencies are not. However, it is not practical to measure mode shapes on an atomic scale, so that, to agree with obtainable physical observations, Planck’s law is necessarily stated in terms of the emission frequencies. As Plan&s law is the beginning point for quantum mechanics, vibration theory is now linked to this subject by direct mathematical development and the orthogonal set of solutions they both generate [8]. Returning again to the modal equations, to follow another mathematical path, the dimensionality of the constitutive equations [eqns (6)] can be interpreted as [9] 1 E

- 0, -

(8)

where S is the space dimension and T is the time dimension, To satisfy eqn (8), the space-time product increases without limit. This conclusion concerning the behavior of the physical dimensions of the modal equations is a direct consequence of the principles of conservation of energy and minimization of the energy function. This conclusion also coincides with the ‘big-bang’ theory of the creation of the universe, which attempts to describe mathematically the physical observations of astronomy. That is, beginning with the assumption that all of the universe mass/energy is concentrated at a single point in the ‘cosmic-egg’, the system becomes unstable and expands explosively. The mass/energy diffuses in time throughout space in a universe which is at present expanding. Thus, the basic philosophical assumptions used to develop the modal equations can also be interpreted so as to link vibration theory and astrophysics. A fundamental assumption of the problems in both statics and elasticity is that all external forces are slowly applied so as not to disturb the system dynamically, or that these forces are applied a suEiciently long time in the past so as to allow any dynamical disturbances to decay through some unspecified energy dissipation or transfer mechanism. This assumption then justifies neglecting or omitting the kinetic energy function, and produces a mathematical solution independent of the time dimension. The subjects of statics and elasticity then form one boundary in the field of classical mechanics, representing systems with only a potential energy function. A visualization of the region of classical mechanics can be given on a set of phase-plane coordinates, consisting of the system generalized displacements and velocities; the boundary for this case is along the positive displacement axis. The principle of conservation of momentum is developed by a minimization of the system kinetic energy function with respect to the generalized velocities. Momentum, as a

619

Technic :a1 Note mathematical operator, is certainly applicable to general problems in dynamics. However, when it is considered in terms of the minimization of the system energy function, momentum is simply one term of the reduced or minimized equation; there is no justification for isolating it from the other terms. Thus, the principle of conservation of momentum, interpreted as a minimization principle, forms another mathematical boundary in classical mechanics representing systems with only a kinetic energy function. The boundary for this case is along the positive velocity axis. Vibration theory and classical quantum mechanics occupy the region in the phase-plane between the two boundaries, representing systems with both a kinetic and a potential energy function. The region for this case is the entire positive quadrant inside the two boundaries. Certainly the existence of real physical systems which have both a kinetic and a potential energy function is verifiable through physical observation. Whether or not systems devoid of one energy function or the other actually exist, or if they represent purely mathematical concepts at the limiting conditions, is not clearly established. The immediate concern here is only for the consequences of these various conditions upon the solutions which result from them. For discrete systems with n degrees of freedom, if the problem is deterministic, for systems with only one energy function, the solution will consist of only one set of n generalized displacements;t or, for systems with both energy functions, because there are two independent functions being minimized simultaneously, the solution will consist of n independent sets of n generalized displacements each. The material of the preceding paragraph is well established mathematically. What is new and unique is the overall unifying effect of the modal equation methods upon the entire area of classical mechanics. The principles of

t There are some interesting problems in elasticity involving ‘multiply connected bodies’ with only a potential energy function, which do have more than one mathematical solution for their displacements.

conservation of energy and minimization of the energy function govern all the areas of classical mechanics, both on the boundaries and within the area of the phase-plane description. The principle of minimization of the energy function can be stated as: ‘Any bounded physical system having either, or both, a kinetic and potential energy function, when disturbed, will always rearrange its configuration in space and time so as to minimize its energy function.’ This definition encompasses Castigliano’s second theorem, the distortion energy method, and the principle of conservation of momentum. It also represents a generalization of Hamilton’s least action principle [lo].

REFERENCES 1. J. Strutt (Baron Rayleigh), The Theory of Sound, 2nd Edn, Vol. I, pp. 88-89. Dover, New York (1945). 2. A. Anuta Jr, The modal equations. Cornput. Slruct. 18, 955 (1984). 3. A. Anuta Jr, The compatibility of the modal equations. Compur. Struct. 33, 1229-1232 (1989). 4. A. Anuta Jr, The mode shape reversal. Compur. Struct. 33, 103-116 (1989). 5. A. Castigliano, Thkorie de I’Equilibre a’es Systmes Elastiques et ses A&cations. Ch. I. A. Neero. - , Turin (1879): -6. A. Anuta Jr, The modal equations in relativity. Comput. Struct. 34, 679480 (1990). 7. M. Planck, Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum. Versuch Dt. Phys. Gesell. 2,237 (1900); Uber das Gesetz der Energieverteilung im Normalspektrum. Ann. Phys. 4, 553 (1901). 8. P. Dirac, The physical interpretation of the quantum

dynamics. Proc. R. Sot. 113, 621 (1927). 9. A. Anuta Jr, The dimensionality of the modal equations. Compul. Strucl. 35. 619-620 11990). 10. w. Hamilton