Mechanics Data Book
Cambridge University Engineering Department
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Source: Mechanics Data Book, Cambridge University Engineering Department (2017)
2017 Edition Cambridge University Engineering Department
Contents
- Definitions
- Kinematics
- Geometry
- Mechanics of Machines
- Linear Systems, Vibration and Stability
- 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)
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