Theory of Machines and Mechanisms - APS
1. Mechanism structure
Machinery is the summary of machine and mechanism
1.1 Basic contept
- Link : move unit
- Kinematic pair : connect the link
- Kinematic chain : Link+ KinematicP
- closed & open
- planar & spatial
- Mechanism
- Fixed link
- Driven link
- Driving link
1.2 Pair
- DOF : f=6-s
- Constraint of kinematic pair : 1 - 5,to Class I - V pairs
- Pairing element
Classificatiton
- High pair : point, line
- Low pair : face
Closed
- form-closed
force-closed
Revolute pair
- Sliding pair
- Helical pair
- Spherical pair
Planar kinematic pair
- Spatial kinematic pair
1.3 Kinematic diagram of mechanism
1.4 DOF
$f=3n-(2p_l+p_h)$
- compound hinges
- passive dof : dont affect the mechanism 滚子
- redundant constraint 重复
1.5 Assur group
DOF=0
- Binary group : 2l+3pl
- Tenary group : 4l+6pl
Substitute higher pair mechanism by lower pair mechanism
2. Kinematic Analysis
2.1 Instantaneous center of velocity
- absolute
relative
RP : at this
- SP : $\infty$
Ph
- Pure rolling : at touch p
- Not : at common normal
Kennedy-Aronhold theorem : 3 points velocity
$P_{13}=P_{12}+P_{23}\\quad=P_{14}+P_{43}$
2.2 Vector graphic method
One link
- velocity vector polygon of mechanism
- acceleration vector polygon
- velocity image of link
- acceleration image of link
Use same p on 2 links
3. Balance
3.1 Rigid rotor
static balance - single plane balance
$F_I=mw^2r\\sum F=F_I+F_b=0$
- mass-radius product
dynamic balance - 2 plane balance
$\sum F=0\\sum M=0$
4. Mechainery
Starting period
$W_d=W_r+E$
Steady motion period
$W_d’=W_r’$
Stopping period
$E=-W_r’$
4.1 Equation of motion
4.2 Equivalent dynamic model
- equivalent moment of inertia
- equivalent moment
- equivalent link
- equivalent mass
equivalent force
4.3 Speed fluctuation
increment of work
decrement of work
Coefficient of non-uniformity of operating velocity of machinery
$\delta=\dfrac{w_{max}-w_{min}}{w_m}$
Flywheel
5. Linkage mechanism-4 bar linkages
5.1 Basic mode of 4 bar linkages
- crank : 360, revolute pair of revolving motion
- rocker : some area, revolute pair of swing motion
- Crank-rocker mechanism
- Double-crank mechanism
- parallel-c m
- antip-c m
- Double-rocker mechanism
5.2 Other mode
Slider-crank m
Offset sc m
Centric sc m
double sc m
scotch-yoke m
Guide-bar m
- Crank and rotating gb m
- Crank and swing gb m
Crank and swing slider m
5.3 Crank ?
- $L_{min}+L_{max}\le \sum L_{rest}$
- Lmin near the revolute p
5.4 Quick-return motion
- Crank angle between the 2 limit position
- Coefficient of travel speed variation
5.5 Pressure angle
- $\gamma$ : transmission angle
- dead point : $\gamma=0$ , +flywheel, 2 different
5.6 Design
- analytic method
Drawing
6.Cam mechanism
- Cam
- Follower
6.1 Classification
- Plate cam
Cylinder cam
Knife-edge follower
- Roller f
Flat-faced f
Force-drive cam m
- Positive cam m
6.2 Motion
- $r_0$ : base circle
- $h$ : actuating travel
$\delta_0$ : Motion angle of at
- $\delta _{01}$ : far angle of repose
- $\delta_{0}’$ : motion angle for return travel
- $\delta_{02}$ : near angle of repose
Constant velocity curve : rigid impulse, order 1
Constant acceleration and deceleration motion curve : soft impulse, order 2
Polynomial motion
Simple harmonic motion : soft impulse
Sine acceleration motion
6.3 Design
- Cam pitch curve
- cam contour
6.4 Pressure angle
$F=G/[\cos(\alpha+\varphi_1)-(l+2b/l)\sin(\alpha+\varphi_1)\tan\varphi_2]$
7.Gear mechanism
7.1 Classification
- parallel axis
- intersecting axis
interlaced shaft
spur
- helical
spiral bevel
External gear
- Internal gear
7.2 Fundamental Law of Gear-Tooth Action
Velocity Ratio: $i_{12}=\dfrac{w_1}{w_2}=\dfrac{O_2P}{O_1P}$
- Pitch piont
- Pitch circle
- Pitch line
7.3 Involute Curve
- generating line BK=arc AB
- B : instantaneous center
- BK is tangent to circle
- no involute curve in the base circle
involute function : $\theta_k=tan\alpha_k-a_k$
- $\theta$ : evolving angle
- $\alpha$ : pressure angle
Feature
- i constant
- force along the line of action啮合线
- separability
7.4 Spur Gear
- $r_a$ : addendum circle
- $r_f$ : dedendum circle
- $s$ : tooth thickness
- $e$ : space width
- $p=e+s$ : circular pitch
- reference circle
- $h_a$ : addendum
- $h_f$ : dedendum
- $h=h_a+h_f$ : tooth depth
7.4.1 Standard gear
- $z$ : number of teeth
- $m=\dfrac{p}{\pi}$ : module
- $d=mz$ : diameter of reference circle
- $\alpha=20^\circ$ : pressure angle,
- $r_b=rcos\alpha$
- $e=s=\pi m/2$
- $c=c^*m$ : Clearance
- $h_a^*=1$ : addendum factor, $h_a=h_a^*m$
- $c^*=0.25$ : clearance factor, $h_f=(h_a^*+c^*)m$
- $a=m(z_1+z_2)/2=r_1+r_2$ : $a\uparrow,\alpha’>\alpha,c’>c,a\downarrow$ can’t assemble
7.4.2 Correct Meshing
- $m_1=m_2$
- $\alpha_1=\alpha_2$
7.4.3 Contact ratio
$B_1B_2<N_1N_2$
$\varepsilon_\alpha=\dfrac{B_1B_2}{p_b}\ge[\varepsilon_\alpha]$
- 1.1-1.4
- $p_b=pcos\alpha$
- $\varepsilon_\alpha=[z_1(tan\alpha_{a1}-tan\alpha’)+z_2(tang\alpha_{a2}-tan\alpha’)]/2/\pi$
7.4.4 Generating method
Tooth cutting : number of teeth too less
$z_{min}=\dfrac{2h_a^*}{sin^2\alpha}$
7.4.5 Modifying method
To use less 17, then $s\neq e$,modified gear
also, too small and not enough strength
- $x$ : modification coefficient
- $s=(\frac{\pi}{2}+2xtan\alpha)m$
- $e=(\frac{\pi}{2}-2xtan\alpha)m$
- $h_f=(h_a^*+c^*-x)m$
- $h_a=(h_a^*+x)m$
Equal displacement gear
- small, positive
- bigger, negtive
7.5 Helical Gear
- $\beta$ : helix angle
- Normal plane
- $m_n=m_t\cos\beta$
- $p_n= \pi m_n=\pi m_t\cos\beta$
- $\tan\alpha_n=\tan\alpha_t\cos\beta$
- $d=zm_t=zm_n/\cos\beta$
- $a=m_n(z_1+z_2)/\cos\beta$
- $\tan\beta_b=\tan\beta\cos\alpha_t$
7.5.1 Correct meshing
- $\beta_1=\beta_2$
- $m_{n1}=m_{n2}$
- $\alpha_{n1}=\alpha_{n2}$
7.5.2 Contact Ratio
$\varepsilon_\gamma=\varepsilon_\alpha+\varepsilon_\beta$ :thumbsup:
- $\varepsilon_\alpha=[z_1(tan\alpha_{at1}-tan\alpha_t’)+z_2(tang\alpha_{at2}-tan\alpha_t’)]/(2\pi)$
- $\varepsilon_\beta=B\sin\beta/(\pi m_n)$
7.5.3 Virtual gear
$z_v=z/\cos^3\beta\Rightarrow\quad z_{min}=z_{vmin}\cos^3\beta$:thumbsup:
8. Gear Train
8.1 Classification
- fixed axis gt
- epicyclic gt
- sun gear
- planetary gear
- planetary carrier
- DOF=2 : differential gt
- DOF=1 : planetary gt
- compound planetary gt
8.2 Fixed axis
$i=\dfrac{\sum z_{DrivenGears}}{\sum z_{DriveGears}}$
- +,-
8.3 Epicyclic gt
+$w_H\Rightarrow$ inverted gt