Installation — business terrible  1 part
September 8th, 2015
Pfluger’s column is a pinned column under a uniform tangential follower force. Walker (1972) included damping and nonuniform 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 nonnegative bounded function related to the distribution of the follower force. The operator fd2 is nonselfadjoint with respect to the boundary conditions. The operator —^2 + d4 + pf(x)d2, however, is selfadjoint in a generalized sense (Leipholz, 1974a) with respect to the operator d2 under the boundary conditions in Equation (2). The generalized selfadjointness 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 selfadjointness can be reduced to the classical selfadjointness.
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 selfadjoint 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 selfadjoint. 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 selfadjointness 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 selfadjoint. 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 selfadjoint system. Stability of the Pfluger column can be studied via its equivalent selfadjoint system. The relationship between the Pfluger column and its equivalent selfadjoint system is as follows.
Systems 
Loading 
End Conditions 

Pfluger’s column 
Tangential follower force 
PP PG 
GP GG 
Equivalent selfadjoint 
Unidirectional force 
GG GP 
PG PP 
P stands for a pinned end, and G stands for a guided end. The equivalent selfadjoint system may be viewed as a continuum counterpart of the symmetrized discrete system. A GG 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 selfadjoint in the classical sense.
As an example, to study the stability of a GG column under a uniform tangential follower force, we can instead study a PP column under its uniform selfweight. While the former is a nonselfadjoint problem, the latter is a wellknown classical selfadjoint problem. The critical weight of the selfadjoint problem is 18.57, so is the buckling load of the nonselfadjoint problem.
It is interesting to note that for a divergent nonselfadjoint discrete system Ax – XBx = 0, where A is symmetric, positive definite, B is not symmetric, and X is real, there exists a lower bound selfadjoint system Ay – oBTA1 By = 0 such that o1 < (X1)2 and
Likewise, for Pfluger’s column, a lower bound selfadjoint 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 pinended bar in torsion. The system is not selfadjoint 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.