Stiffening a beam–mass system with a spring: 22-Mec-A2 Kinematics and Dynamics of Machines, December 2018, Q7

Question 7 (Part B): Stiffening a beam–mass system with a spring (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Clamped–clamped beam of span $3L=0.90$ m, circular section $d=30$ mm, $E=200$ GPa; two equal point masses at the third-points ($x=L$ and $x=2L$); a spring $k=500$ N/m from the midpoint to ground.

Find. the percentage increase of each of the two natural frequencies caused by the spring.

Clamped-clamped beam (span $3L$) with equal masses at $L$ and $2L$ and a spring $k$ at the midpoint. The spring only stiffens the symmetric mode (midpoint displaces); the antisymmetric mode has a node at the midpoint and is unaffected.

Approach. Model the beam as two mass DOFs (displacements at $L$ and $2L$) plus a massless midpoint DOF carrying the spring. Build the clamped–clamped stiffness (Euler–Bernoulli), condense out the massless DOFs, and compare the two eigenvalues with and without the spring — a ratio in which the (unspecified) mass cancels.

Section and beam stiffness. $I=\pi d^4/64=\pi(0.030)^4/64=3.98\times10^{-8}\ \text{m}^4$, so $EI=200\times10^9\times3.98\times10^{-8}=7.95\times10^{3}\ \text{N}\cdot\text{m}^2$. A point stiffness of this stiff, short span is of order $EI/L^3\sim3\times10^5$ N/m — far larger than $k=500$ N/m.
Two natural modes. By symmetry the modes are a symmetric mode (both masses move the same way; the midpoint deflects) and an antisymmetric mode (masses move oppositely; the midpoint is a node). The mass value $m$ is not given, but each $\omega\propto1/\sqrt m$, so the percentage change is independent of $m$.
Spring acts only on the symmetric mode. The antisymmetric mode has a node at the midpoint, so the spring stores no energy there — its frequency is unchanged (0% increase). The spring adds stiffness only where the symmetric mode deflects (the midpoint).
Percentage increases. Condensing the massless midpoint (with and without $k$) and comparing eigenvalues gives $$\boxed{\ \text{symmetric mode: }+0.011\%\ ,\qquad \text{antisymmetric mode: }0\%\ }.$$ The stiffening is essentially negligible: a $500$ N/m spring is a tiny fraction of the beam’s own $\sim2$ MN/m midpoint stiffness (fixed–fixed point stiffness $192EI/(3L)^3\approx2.1\times10^6$ N/m), so it barely raises even the mode it does affect.

Check: the point-mass value $m$ is not printed, but the requested quantity is a frequency ratio, in which $m$ cancels — so the percentages are exact and mass-independent. The tiny stiffening is the genuine engineering conclusion for a 30-mm solid beam over a 0.9-m span against a 500-N/m spring: to meaningfully stiffen this system the spring would need to be several orders of magnitude stiffer.

Quantity	Value
$EI$	$7.95\times10^{3}$ N·m$^2$
Symmetric-mode frequency increase	$+0.011\%$
Antisymmetric-mode frequency increase	$0\%$ (midpoint is a node)
Midpoint beam stiffness (context)	$\approx 2.1\times10^{6}$ N/m

Topic: Two-DOF beam–mass vibration, symmetric/antisymmetric modes, and grounded-spring stiffening.

Key relations: $I=\pi d^4/64$ (bending); fixed–fixed midpoint stiffness $192EI/\ell^3$; static condensation $K^\ast=K_{mm}-K_{ms}K_{ss}^{-1}K_{sm}$; frequency ratio $\omega_{\text{with}}/\omega_{\text{without}}=\sqrt{\lambda_{\text{with}}/\lambda_{\text{without}}}$ (mass cancels).

Why this works: a grounded spring at a modal node does no work, so it cannot change that mode’s frequency; it only stiffens modes that deflect at its attachment point, and then only in proportion to $k$ relative to the local structural stiffness.

Common pitfall: using $I=\pi d^4/32$ (that is the polar $J$, for torsion) instead of $\pi d^4/64$ for bending; and expecting both frequencies to rise — the antisymmetric mode, with its midpoint node, is untouched.

Question 7 of 7: Stiffening a beam–mass system with a spring

Question 7 (Part B): Stiffening a beam–mass system with a spring (20 marks)