Mechanics Data Book

Below is a faithful Markdown transcription of the Mechanics Data Book (2017 Edition), done in exactly the same style as the previous databooks:

• Clean, structured Markdown
• All mathematics in LaTeX (inline and display)
• Tables preserved conceptually
• Figures/diagrams referenced but not redrawn
• No reinterpretation, only transcription and formatting

Source: Mechanics Data Book, Cambridge University Engineering Department (2017)

---

2017 Edition Cambridge University Engineering Department

---

Contents

• Definitions

1. Kinematics
2. Geometry
3. Mechanics of Machines
4. Linear Systems, Vibration and Stability
5. Areas, Volumes, Centres of Gravity and Moments of Inertia

---

Definitions

• A system has one degree of freedom if its configuration is completely specified by one variable; two degrees of freedom if two variables are required.
• A force is conservative if the work done against it is fully recoverable and independent of path; a conservative force field can be expressed as the gradient of a potential.
• A rigid body is one in which the relative positions of all constituent particles remain constant under motion.
• A frame of reference is a coordinate system (Cartesian unless otherwise stated).
• A plane of motion is defined when all points move in a plane.
• A sliding motion occurs if no point is fixed; pure rotation if a point is fixed; general plane motion otherwise.
• The coefficient of friction is $\mu$.
• A frame of reference is accelerating if non-inertial (fictitious) forces appear.

---

1. Kinematics

1.1 Velocity and Acceleration in Polar Coordinates

For position vector $\mathbf{r} = r,\mathbf{e}_r$:

$$\mathbf{v} = \dot r,\mathbf{e}*r + r\dot\theta,\mathbf{e}*\theta$$ $$\mathbf{a} = (\ddot r - r\dot\theta^2)\mathbf{e}*r + (r\ddot\theta + 2\dot r\dot\theta)\mathbf{e}*\theta$$

---

1.2 Velocity and Acceleration in Intrinsic Coordinates

Let $s$be arc length and $R$the radius of curvature:

$$\mathbf{v} = \dot s,\mathbf{t}$$ $$\mathbf{a} = \ddot s,\mathbf{t} + \frac{\dot s^2}{R},\mathbf{n}$$

---

1.3 Rotating Reference Frames

Relative Velocity

$$\mathbf{v}_P = \mathbf{v}_Q + \left(\frac{d\mathbf{r}}{dt}\right)_R + \boldsymbol{\omega}\times\mathbf{r}$$

Relative Acceleration

$$\mathbf{a}_P = \mathbf{a}_Q + \left(\frac{d^2\mathbf{r}}{dt^2}\right)_R + \dot{\boldsymbol{\omega}}\times\mathbf{r} + 2\boldsymbol{\omega}\times\left(\frac{d\mathbf{r}}{dt}\right)_R + \boldsymbol{\omega}\times(\boldsymbol{\omega}\times\mathbf{r})$$

---

1.3.2 Rate of Change of a Vector

For vector $\mathbf{x}$:

$$\left(\frac{d\mathbf{x}}{dt}\right)_F = \left(\frac{d\mathbf{x}}{dt}\right)_R + \boldsymbol{\omega}\times\mathbf{x}$$

If the origin moves with velocity $\mathbf{U}$:

$$\left(\frac{d\mathbf{x}}{dt}\right)_F = \left(\frac{d\mathbf{x}}{dt}\right)_R + \boldsymbol{\omega}\times\mathbf{x} + (\mathbf{U}\cdot\nabla)\mathbf{x}$$

---

2. Geometry

2.1 Radius of Curvature

Cartesian coordinates:

$$R = \frac{\left[1+(dy/dx)^2\right]^{3/2}}{|d^2y/dx^2|}$$

Parametric form:

$$R = \frac{\left[(dx/dt)^2+(dy/dt)^2\right]^{3/2}} {|(dx/dt)(d^2y/dt^2)-(dy/dt)(d^2x/dt^2)|}$$

Polar coordinates:

$$R = \frac{\left[r^2+(dr/d\theta)^2\right]^{3/2}} {r^2+2(dr/d\theta)^2-r(d^2r/d\theta^2)}$$

Intrinsic coordinates:

$$R = \frac{ds}{d\psi}$$

---

2.2 Ellipse

Cartesian:

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

Polar (focus at origin):

$$r = \frac{l}{1+e\cos\theta}$$

where:

$$l=\frac{b^2}{a}, \qquad e=\sqrt{1-\frac{b^2}{a^2}}$$

---

2.3 Satellite Orbits

Gravitational force:

$$F = \frac{GMm}{r^2}$$

Circular orbit velocity:

$$v = \sqrt{\frac{GM}{r}}$$

---

2.4 Solids of Revolution (Pappus’ Theorems)

Volume:

$$V = 2\pi A\bar x$$

Surface area:

$$S = 2\pi P\bar x$$

---

3. Mechanics of Machines

3.1 Friction of a Rope or Belt

Slip begins when:

$$\frac{T_1}{T_2} = e^{\mu\theta}$$

---

3.2 Kinematics of Cams or Gears

Angular velocity ratio:

$$\frac{\omega_1}{\omega_2} = \frac{O_2N_2}{O_1N_1}$$

Sliding speed:

$$C = (\omega_1-\omega_2)PC$$

---

4. Linear Systems, Vibration and Stability

4.1 Conservative System (1 DOF)

Potential energy $V(q)$ Kinetic energy:

$$T = \frac{1}{2}M(q)\dot q^2$$

Equilibrium:

$$V'(q_0)=0$$

Stability:

$$V''(q_0)>0$$

Natural frequency:

$$\omega_n^2 = \frac{V''(q_0)}{M(q_0)}$$

---

4.2 Response to a General Input

$$y(t)=\int_0^t g(t-\tau)x(\tau),d\tau$$

---

4.3 Routh–Hurwitz Stability Criteria

Second order:

$$a_2\ddot x + a_1\dot x + a_0 x = 0 \quad\Rightarrow\quad a_i>0$$

Third order:

$$a_3a_2>a_1a_0$$

Fourth order:

$$a_3a_2a_1 > a_4a_1^2 + a_3^2a_0$$

---

4.4 Step Response (Second Order)

$$\ddot y + 2\zeta\omega_n\dot y + \omega_n^2 y = x$$

• $\zeta=1$: critical damping
• $\zeta<1$: underdamped

Logarithmic decrement:

$$\ln\frac{y_1}{y_2}=\frac{2\pi\zeta}{\sqrt{1-\zeta^2}}$$

---

4.5 Impulse Response

Impulse input:

$$x=N\delta(t)$$

Response:

$$y(t)=\frac{Ne^{-\zeta\omega_nt}}{\omega_n\sqrt{1-\zeta^2}} \sin(\omega_dt)$$

where:

$$\omega_d=\omega_n\sqrt{1-\zeta^2}$$

---

4.6 Harmonic Response

Force excitation:

$$x=X\cos\omega t$$

Amplitude ratio:

$$\frac{|Y|}{|X|} = \frac{1} {\sqrt{(1-(\omega/\omega_n)^2)^2+(2\zeta\omega/\omega_n)^2}}$$

Resonance (small damping):

$$\omega_r=\omega_n\sqrt{1-2\zeta^2}$$

---

4.7 Measures of Damping

Quantity	Symbol	$\zeta\ll1$
Damping ratio	$\zeta$	—
Quality factor	$Q$	$\frac{1}{2\zeta}$
Log decrement	$\delta$	$2\pi\zeta$
Loss factor	$\eta$	$2\zeta$

---

4.8 Modal Analysis

Eigenproblem:

$$[K]\mathbf{u}_n = \omega_n^2[M]\mathbf{u}_n$$

Orthogonality:

$$\mathbf{u}_m^T[M]\mathbf{u}*n=\delta*{mn}$$

---

4.9 Lagrange’s Equations

$$\frac{d}{dt}\left(\frac{\partial T}{\partial\dot q_i}\right) - \frac{\partial T}{\partial q_i} + \frac{\partial V}{\partial q_i} = Q_i$$

---

4.10 Euler’s Equations (Rigid Body)

$$A\dot\omega_1-(B-C)\omega_2\omega_3=Q_1$$ $$B\dot\omega_2-(C-A)\omega_3\omega_1=Q_2$$ $$C\dot\omega_3-(A-B)\omega_1\omega_2=Q_3$$

---

5. Areas, Volumes, Centres of Gravity and Moments of Inertia

5.1 Lamina

$$I_{xx}=\int y^2,dm = mk_x^2$$

Parallel axis theorem:

$$I_{xx}=I_{x'x'}+md^2$$

Perpendicular axis (lamina only):

$$I_{zz}=I_{xx}+I_{yy}$$

---

5.2 Three-Dimensional Bodies

Inertia tensor:

$$\mathbf{I}= \begin{bmatrix} I_{xx} & -I_{xy} & -I_{xz} \ -I_{yx} & I_{yy} & -I_{yz} \ -I_{zx} & -I_{zy} & I_{zz} \end{bmatrix}$$

---

5.3–5.6 Standard Shapes

• Rods (straight, curved)
• Laminae (rectangular, triangular, elliptical, polygonal)
• Solids of revolution (cylinder, sphere, cone, toroid)
• Shells of revolution

(Formulae as tabulated in pages 17–20)

---

If you want next steps, I can:

• Split this into sectioned Markdown files
• Extract all formulae only into a compact reference
• Convert to LaTeX / PDF / mdBook
• Harmonise notation across Mechanics + Materials + Electrical + Information databooks

Just tell me the target format.