22-Mec-A2 Kinematics and Dynamics of Machines · Undated paper
Question 7 of 7: Two-DOF torsional vibration of a geared shaft
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper: National Exams — 16-Mec-A2 Kinematics and Dynamics of Machines · 3 hours, open book · Rubric: answer Q1 (compulsory) plus any four of Q2–Q7. Marks: 20 each. All seven questions are solved here.
Reference texts (subject). R. L. Norton, Design of Machinery, 6th ed. (mobility Ch. 2, synthesis Ch. 3, velocity/acceleration Ch. 6–7, cams Ch. 8, epicyclic trains §9.6–9.9, balancing Ch. 13); Uicker, Pennock & Shigley, Theory of Machines and Mechanisms, 5th ed.; C. E. Wilson & J. P. Sadler, Kinematics and Dynamics of Machinery, 3rd ed.; S. S. Rao, Mechanical Vibrations, 6th ed. (Part B multi-DOF torsional Ch. 6, impact Ch. 1).
Note — figures printed without dimensions. Every figure on this paper is printed without dimensions, and several carry inconsistent numbers (e.g. Q4 shows two conflicting tooth-count sets, and a ring $N_3=20$ cannot mesh a sun $N_1=32$; Q5 prints $AG_3=BG_3=82.5$ mm with $AB=179$ mm, a geometrically impossible triangle). The methods below are exact and complete; where a numerical answer depends on figure geometry (Q2, Q5) or on internally inconsistent data (Q4), the adopted reading is stated explicitly and the value is flagged as representative. Scale-free results (mobility, Grashof class, angular-velocity ratios, transmission-angle location, mode shapes, non-dimensional natural frequencies) are exact regardless of the figure dimensions.
Question 7: Two-DOF torsional vibration of a geared shaft (20 marks)
Find. Equations of motion; the two natural frequencies and mode shapes in terms of $G,J,I,L$; and how to shift $\omega_2$ by $\pm15\%$.
Approach. Take the two gear rotations $\theta_1,\theta_2$ as generalized coordinates; write Newton’s torsional equations with $k=GJ/L$; solve the eigenvalue problem of $\mathbf K$ relative to $\mathbf M=I\,\mathbf 1$.
Fig. 7. Fixed-free torsional two-DOF model: two equal disks $I$ on a massless shaft of two equal $GJ/L$ segments.
Equations of motion. Segment stiffness $k=GJ/L$. Torque balance on each gear:
$$I\ddot\theta_1+k\theta_1+k(\theta_1-\theta_2)=0,\qquad I\ddot\theta_2+k(\theta_2-\theta_1)=0.$$
In matrix form $\mathbf M\ddot{\boldsymbol\theta}+\mathbf K\boldsymbol\theta=\mathbf 0$ with
$$\mathbf M=I\begin{bmatrix}1&0\\0&1\end{bmatrix},\qquad \mathbf K=k\begin{bmatrix}2&-1\\-1&1\end{bmatrix}.$$
Frequency equation. With $\lambda=\omega^2I/k$, $\det(\mathbf K-\omega^2\mathbf M)=0$ gives
$$(2-\lambda)(1-\lambda)-1=\lambda^2-3\lambda+1=0\ \Rightarrow\ \lambda=\frac{3\pm\sqrt5}{2}=0.382,\ 2.618.$$
Natural frequencies. $\omega=\sqrt{\lambda\,k/I}=\sqrt{\lambda}\,\sqrt{GJ/(IL)}$:
$$\boxed{\omega_1=0.618\sqrt{\frac{GJ}{IL}}},\qquad \boxed{\omega_2=1.618\sqrt{\frac{GJ}{IL}}}.$$
(The factors are $1/\varphi$ and $\varphi$, the golden ratio.)
Mode shapes. From the first row, $\theta_2/\theta_1=2-\lambda$:
$$\text{Mode 1 }(\lambda=0.382):\ \{\theta_1,\theta_2\}=\{1,\,1.618\}\ \text{(in phase)};$$
$$\text{Mode 2 }(\lambda=2.618):\ \{\theta_1,\theta_2\}=\{1,\,-0.618\}\ \text{(out of phase; node in segment 2)}.$$
Shifting $\omega_2$ by $\pm15\%$. Since $\omega_2\propto\sqrt{k/I}=\sqrt{GJ/(IL)}$ and $J=\pi d^4/32$, so $\omega_2\propto d^2/\sqrt{L\,I}$. To raise $\omega_2$ by 15% ($\omega_2^2\times1.32$): increase shaft diameter by $1.32^{1/4}=1.072$ (a 7.2% larger dia), or shorten the segment to $0.756L$. To lower $\omega_2$ by 15% ($\omega_2^2\times0.72$): reduce the diameter to $0.922d$, lengthen the shaft to $1.32L$, or increase each gear inertia to $1.38I$. Any one of these hits the $\pm15\%$ target; changing the diameter is usually most compact.
