The Pfluger Column

Pfluger’s column is a pinned column under a uniform tangential follower force. Walker (1972) in­cluded damping and non-uniform follower force. Here, we include guided end in addition to pinned end. Stability of Pfluger’s column is governed by the differential equation

— ^2w + w"" + pf(x)w" = 0, x e[0, 1] (1) and the boundary conditions:

w = w" = 0; for a pinned end,

w = w’" = 0; for a guided end.

w(x) is the lateral deflection, ^ is a frequency parameter, p > 0 is a load parameter, and f(x) is a non-negative bounded function related to the distribution of the follower force. The operator fd2 is nonself-adjoint with respect to the boundary conditions. The operator —^2 + d4 + pf(x)d2, however, is self-adjoint in a generalized sense (Leipholz, 1974a) with respect to the operator d2 under the boundary conditions in Equation (2). The generalized self-adjointness implies

j (—&2u + u"" + pfu")v" dx = j (—&2v + v"" + pfv")u" dx. (3)

u(x) and v(x) are any two admissible functions satisfying all the boundary conditions for w(x).

We will show that the Pfluger column is a conservative system by symmetrizing it. We will also show that Leipholz’s generalized self-adjointness can be reduced to the classical self-adjointness.

Let us differentiate Equation (1) with respect to x and denote w’ by y(x). Then Equation (1) becomes

-^2y + y”” + p[f(x)y’]’= 0.

The follower force now assumes the appearance of a unidirectional loading. Whether or not the operator in Equation (4) is self-adjoint also depends on the boundary conditions for y(x).

At a pinned end of the column, w" = 0 implies y’ = 0. From Equation (1), w = w" = 0 implies w"" = 0, which in turn implies y= 0. As can be seen, a pinned end for w becomes a guided end for y. At a guided end of the column, w’ = 0 implies y = 0, and w’" = 0 implies y" = 0. It is seen that a guided end for w turns out to be a pinned end for y.

The system in Equation (4) is equivalent to the system in Equation (1), because Equation (4) and the associated boundary conditions are derived from Equation (1) via the transformation of variable: y = w’. Hence the eigensolutions of Equation (1) are also eigensolutions of Equation (4).

The operator in Equation (4) is self-adjoint. It can be shown in the usual manner that for admiss­ible y and z, the following inner products hold:

C[-U2y + y"" + p(fy’ )’]z dx = /Vn2z + z"" + p(fz’)’]y dx. (5)

J0 J0

The expression in Equation (5) can also be derived from the generalized self-adjointness in Equa­tion (3) by integration by parts to obtain

and then by denoting u’ by y and v’ by z. We have thus shown that the operator in Equation (4) is self-adjoint. So the Pfluger column can only have divergent type of instability.

/0V)2d*

y{x) fo f(x)(y’)2dx

The buckling load pcr has a Rayleigh quotient:

or in terms of w:

The foregoing derivation shows that the Pfluger column can be symmetrized into a self-adjoint system. Stability of the Pfluger column can be studied via its equivalent self-adjoint system. The relationship between the Pfluger column and its equivalent self-adjoint system is as follows.

Systems

Loading

End Conditions

Pfluger’s column

Tangential follower force

P-P P-G

G-P G-G

Equivalent self-adjoint

Unidirectional force

G-G G-P

P-G P-P

P stands for a pinned end, and G stands for a guided end. The equivalent self-adjoint system may be viewed as a continuum counterpart of the symmetrized discrete system. A G-G column has a rigid body translational mode. We will ignore it, as we are interested in the flexural modes only.

Note the static equation governing the bending moment in the Pfluger column, M"+f(x)M = 0, is also self-adjoint in the classical sense.

As an example, to study the stability of a G-G column under a uniform tangential follower force, we can instead study a P-P column under its uniform self-weight. While the former is a nonself­adjoint problem, the latter is a well-known classical self-adjoint problem. The critical weight of the self-adjoint problem is 18.57, so is the buckling load of the nonself-adjoint problem.

It is interesting to note that for a divergent nonself-adjoint discrete system Ax – XBx = 0, where A is symmetric, positive definite, B is not symmetric, and X is real, there exists a lower bound self-adjoint system Ay – oBTA-1 By = 0 such that o1 < (X1)2 and

Likewise, for Pfluger’s column, a lower bound self-adjoint system exists:

where K(x, f) = G(x, f)f(x)f(f) is a symmetric, positive definite kernel, such that /01[u"(x)]2dx

u(x)/0Ч1 K(x, f)u"(x)u"(f ) dx df ‘S. Greenhill’s Shaft

Greenhill’s shaft is a pin-ended bar in torsion. The system is not self-adjoint except when в, the angle between the applied torque vector and the tangent to the end of the bar, is equal to 1/2.

Leipholz (1974a) studied a pure tangential torque (в = 0) and showed that the Greenhill shaft is a conservative system of the second kind. Walker (1973) considered the case в = 1/2 and included an axial compression and a damping force. He developed a Lyapunov functional for stability study. Here we examine a bar in a viscoelastic medium of low density under a pure tangential torque and a pure axial torque (в = 1), respectively. We will symmetrize the systems and improve the stability boundary obtained by Walker.

The linearized system (Bolotin, 1963) is described by the differential equation

where

x(y – 1) for 0 < x < y

y(x – 1) for y < x < 1 is Green’s influence function. Let M = w(x) eiLx/2. The expression in Equation (11) becomes

(r2 + cr + k) f G(x, y)M(y) dy + M" + pM – iLM’ = 0,

0

w(0) = w(1) = 0,

and

H(x, y) = G(x, y) e lL(x y)/2 is a symmetric kernel. Greenhill’s shaft is thus symmetrized.
For a given k and L, the critical load has a Rayleigh quotient

„ l2 , Jo(w/)2dx ~kJoJo я(х’ y)u!(x)w(y) dx dy

Per — ~ . + lilt

4 w(x)

which describes the free vibration of a string. So pcr = -(L2/4) + n2 is the necessary and sufficient condition for stability when k = 0.

An estimate of pcr in Equation (17) can be obtained by assuming w(x) = sin(nx), resulting in

When L = 0, Equation (21) yields pcr = n2 + (k/n2). On the other hand, when L = ±2n, pcr = k/8((1/3) + (3/n2)). It is seen that k > 0 does increase the stability boundary, directly confirming Walker’s conjecture about an increase in the upper bound on L2 + 4p.

The case of a pure axial torque (в = 0) is considered next. Integrate Equation (11) twice with respect to x to obtain

The constants of integration a1 and a2 are found to be zero, in view of the boundary conditions z = 0 and z" – iLz’ = 0. Consequently, Equation (22) becomes

The dependent variable needs to satisfy only the boundary conditions z(0) = z(1) = 0.

Equation (23) is the same as Equation (14). The symmetrized system in Equation (16) and the stability boundary in Equation (20) also holds true for Greenhill’s shaft under an axial torque. Therefore, the shaft under an axial torque and that under a tangential torque behave in a similar manner and both have the same equivalent symmetrized system.