A Linear Conservative Gyroscopic System

The deflection of a conservative gyroscopic system being considered here satisfies the equation:

The Cauchy-Schwartz inequality can be used to show that Equation (30) is implied by

uT(C—1KC + K — C2)(CKC—1 + K — C2)u — 4uTCKC1Ku < 0. (31)

With the identity uTAu = uTASu, where As = A + AT/2, for any real matrix A, AS > 0 (or AS < 0) is sufficient for uTAu > 0 (or uTAu < 0). Therefore, a sufficient condition in terms of the operators for the system to be unstable is:

(A.1) 2(K — C2) + C—1KC + CKC—1 < 0, or

(A.3) (C-1KC + K – C2)(CKC-1 + K – C2) – 2(CKC-1 K + KC-1KC) < 0.

The matrices in the above expressions are symmetric. Their negative-definiteness can be verified by using the Sylvester criteria. The union of the regions defined by Equations (A.1), (A.2) and (A.3) is a domain of instability.

For the two-degree-of-freedom system considered by Walker, the system is unstable if: k1 + k2 + 16 < 0;

or

kk < 0;

or

(K1 + k2 + 16)2 – 4k1k2 < 0,

per (A.1), (A.2), and (A.3), respectively. The result is the same as the exact solution, because all the matrix products here are diagonal. The product of two 2 x 2 skew matrices is diagonal.

4.2. Sufficient Condition for Stability

The solution is stable (periodic) if all the following three conditions are satisfied:

uT(CKC-1 + K – C2)u > 0 (32)

or

uTCKC-1Ku> 0 (33)

or

An approximation will be made to simplify Equation (34). Let us require that Equation (34) be true for all real vectors, including w, which is the eigenvector in the eigenvalue problem

(CKC-1 + K – C2)w = Xw. (35)

Then Equation (34) becomes

wT[(CKC-1 + K – C2)2 – 4CKC-1K]w > 0.

Consequently, a sufficient condition for stability is:

(B.1) 2(K – C2) + C-1KC + CKC-1 > 0, and

(B.2) CKC-1 K + KC-1KC> 0, and

(B.3) (C-1KC + K – C2)2 + (CKC-1 + K – C2)2 – 4(CKC-1K + KC-1KC) > 0.

The intersection of the regions in (B.1), (B.2) and (B.3) is a domain of stability.

According to the present criterion, the same 2-DOF system will be stable if all the conditions k + k2 + 16 > 0; and

kk > 0; and

(k1 + k2 + 16)2 – 4k 1^2 > 0,

are satisfied. Again, the result is the same as the exact solution.

5. Conclusions

Walker, Leipholz, and Inman and Olsen have studied conservative systems of the second kind. Our objective here is to obtain an equivalent self-adjoint system. Existence of the symmetrized systems confirms certain conservative systems of the second kind behave as a true conservative system. In this way, study of stability can be carried out on the symmetrized system. In general, it is easier to study a self-adjoint problem than a nonself-adjoint problem. For the conservative system of the second kind, including the Pfluger column, we also presented a lower bound self-adjoint system. For a linear conservative gyroscopic system, we gave a zero parameter sufficient condition for instability and one for stability. The criteria depend only on the characteristics of the system. For a simple 2-DOF system, the present criteria yield the exact solutions.

References

Bolotin, V. V., 1963, Nonconservative Problems of the Theory of Elastic Stability, Pergamon Press. Inman, D. J., 1983, Dynamics of Asymmetric Nonconservative Systems, ASME, Journal of Applied Mechanics, 50(1), 199-203.

Inman, D. J. and Olsen, C. L., 1988, Dynamics of Symmetrizable Nonconservative Systems, ASME, Journal ofApplied Mechanics, 55, 206-212.

Leipholz, H. H.E., 1974a, On Conservative Elastic Systems of the First and Second Kind, Ingenieur- Archive, 43, 255-271.

Leipholz, H. H.E., 1974b, On a Generalization of the Concept of Self-Adjointness and of Rayleigh’s Quotient, Mechanics Research Communication, 1, 67-72.

Walker, J. A., 1972, Lyapunov Analysis of the Generalized Pfluger Problem, Journal of Applied Mechanics, Vol. 39, Trans. ASME, Vol. 94, Series E, pp. 935-938.

Walker, J. A., 1973, Stability of a Pin-Ended Bar in Torsion and Compression, ASME, Journal of Applied Mechanics, 40, 405-409.

Walker, J. A., 1991, Stability of Linear Conservative Gyroscopic Systems, Journal of Applied Mechanics, 58, 229-232.