Theoretical Mechanics B - APS

1 Statics

1.1 Force

• modulus
• direction
• effect point

constraint

• Soft body constraint : belt
• Smooth contact surface : gear meshing
• Hinge
• Chain constraint
• Fixed end constraint
• bearing

1.2 Concurrent force system

all force start at one point

$$\begin{cases}\sum F_x=0\\sum F_y=0\\sum F_z=0\end{cases}$$

1.3 Torque & Couple

Torque

• point

$\mathbf{M_o(F)}=\mathbf r\times \mathbf F$

• axis : parallel part = 0

  $$M_z(\mathbf F)=(\mathbf r \times\mathbf F)\cdot \mathbf k$$

• sum : vector sum

Couple

two force same length, reverse direction, the effective line parallel, the combination of the two torque

$$\mathbf M=\mathbf r_{BA}\times F$$

put any where

$$\begin{cases}\sum M_x=0\\sum M_y=0\\sum M_z=0\end{cases}$$

1.4 Any force system

rigid body

1.4.1 Force translation

1.4.2 Simplify

$$\mathbf M=\sum \mathbf M_o(\mathbf F_i)\\mathbf F_R=\sum \mathbf F_i$$

• $\mathbf F_R$: major vector, free of position
• $\mathbf M$ : major torque

result

• $\mathbf F_R=0, \mathbf M\neq 0$ : change with position
• $\mathbf F_R\neq0, \mathbf M=0$
• $\mathbf F_R\neq0, \mathbf M\neq0$
  • perpendicular : sum force to another O
  • parallel : spiral Force
    • same direction : dextral
    • reverse : sinistral
  • combination : to another point of spiral force

1.4.3 Distributed load

1.4.4 Center of gravity

1.4.5 Force system balance

1.5 Application

1.5.1 Truss

• Node : 2 unknown can resolved

  

  • zero chain

    

• Cutting section : 3 unknown can resolved

  

1.5.2 Friction

• static friction : $F_{max}=f_sF_N$

  • $f_s$ : coefficient of friction

• kinetic friction : $F_k=f_kF_N$

  • $f_k<f_s$

• angle of friction : $\tan\theta=f_s$

  • auto lock

• Overturn or slide

  

• Roll resistance : $M_f=\delta F_N$

  

  • pure roll $F_s<F_{max}, M=M_f$

2 Kinematics

2.1 Particle

• Vector

• Rectangular coordinates

• Natural coordinates

2.1.1 Velocity composition

• Implicated motion
• Relative motion

$$\overrightarrow{v_a}=\overrightarrow{v_e}+\overrightarrow{v_r}$$

2.1.2 Acceleration composition

• Coriolis acceleration

$$\overrightarrow{a_a}=\overrightarrow{a_e}+\overrightarrow{a_r}+\overrightarrow{a_c}$$

• $\overrightarrow{a_c}=2\overrightarrow{w_e}\times\overrightarrow{v_r}$

2.2 Rigid Body

• translation
• rotation

• Plane movement

  • velocity

    • speed synthesis
    • speed projection
    • instantaneous center

    

  • acceleration : $\overrightarrow{a_B}=\overrightarrow{a_A}+\overrightarrow{a_{BAn}}+\overrightarrow{a_{BAt}}$

3 Dynamics

3.1 Particle motion differential equation

Newton law

• Inertial system
• Non-inertial system

3.2 Centroid motion theorem

two inference

3.3 Momentum theorem

• momentum p
• impulse I

3.4 Angular momentum principle

• Moment of inertia : $J$
• Parallel axis theorem
• Radius of gyration : $\rho$

• angular momentum
  • unit : kgm2/s

Rigid body fixed axis rotation differential equation

$$J_z\alpha=\sum M^e_{zi}$$

Rigid body plane motion differential equation

$$\begin{cases}ma_{cx}=m\ddot{x_c}=\sum F_{xi}\ma_{cy}=m\ddot{y_c}=\sum F_{yi}\J_c\alpha=J_c\ddot{\varphi}=\sum M_{ci}\end{cases}$$

3.5 Kinetic energy theorem

3.5.1 Work for different force

• Weight : $W=\int(-G)dz=G(z_1-z_2)$
• Spring : $W=\dfrac{k}{2}(\delta^2_1-\delta^2_2)$
• Gravity : $W=Gm_0m(\dfrac{1}{r_2}-\dfrac{1}{r_1})$
• Rotation : $W=\int\limits^{\varphi_2}_{\varphi_2}md\varphi$

3.5.2 Kinetic energy

• Translation : $T=\dfrac{1}{2}mv^2$
• Fixed axis rotation : $T=\dfrac{1}{2}J_zw^2$
• Combination : $T=\dfrac{1}{2}mv_c^2+\dfrac{1}{2}J_cw^2$

3.5.3 Kinetic energy theorem

$$T_2-T_1=\sum W_i$$

3.5.4 Potential Energy and Mechanical energy conservation

• Potential E
  • weight: $V=mg(z-z_0)$
  • spring : $V=\frac{1}{2}k\delta^2$
  • gravity : $V=-\dfrac{Gm_0m}{r}$
• Mechanical energy conservation : $T_1+V_1=T_2+V_2$

3.6 Collision

• Recovery factor : $e=\dfrac{v_{2n}’-v_{1n}’}{v_{1n}-v_{2n}},0\le e\le 1$

• impulse and angular momentum

  $$mv_c’-mv_c=\sum \mathbf{I_i^e}\L_{c2}-L_{c1}=\sum M_c(\mathbf{I_i^e})$$

• Heart collision

  • e=1
  • e=0

  

• collision center : $h=\dfrac{J_z}{ma}$

  • a : rotation center to centroid

3.7 D’Alembert’s principle

add inertial force, make dynamic problem to statics problem

Inertial force system simplify

Major vector : $\mathbf{F_I}=-m\mathbf{a_c}$

Major torque : $\mathbf{M_O(F_I)}=-\dfrac{d\mathbf{L_O}}{dt}$

3.8 Virtual work principle

generalized coordinates : make sure the position$\rightarrow$dof

3.8.1 Virtual displacement

small movement $\delta r$

• geometry : instantaneous center

  

• analytic : get position function

  

3.8.2 Principle of virtual work

$$\delta W=\sum\limits^n_1F_i\delta r_i=0$$

3.8.3 Generalized force balance

lock other virtual displacement

$Q_j=\dfrac{\delta W_j}{\delta q_j}$

3.9 General kinetic equation & Lagrange equation

• General kinetic equation

  $$\sum\limits^n_1[(F_{ix}-m_i\ddot{x_i})\delta x_i+(F_{iy}-m_i\ddot{y_i})\delta y_i+(F_{iz}-m_i\ddot{z_i})\delta z_i]=0$$

  

• Lagrange equation

  $$\dfrac{d}{dt}\cdot\dfrac{\partial T}{\partial \dot{q_j}}-\dfrac{\partial T}{\partial q_j}=Q_j=-\dfrac{\partial V}{\partial q_j}$$

  $$\dfrac{d}{dt}\cdot\dfrac{\partial L}{\partial \dot{q_j}}-\dfrac{\partial L}{\partial q_j}=0, L=T-V$$