This Bicycle Mechanics Analysis was developed by collaboration between jbriggs444, haruspex, and chestermiller. We all concur on its contents.
Moment Balance on Rear Wheel Assembly:
$$(T_U-T_L)R_{RS}-F_{SR}R_W=I_Rα\tag{1}$$
where
##T_U## = Tension in upper part of the chain
##T_L## = Tension in lower part of the chain
##R_{RS}## = Radius of rear sprocket
##F_{SR}## = Static friction force exerted by ground on rear wheel in
forward direction
##R_W## = Radius of wheels
##I_R## =Moment of inertia of rear wheel assembly
##α## = Angular acceleration of wheels
Moment of Inertia of Rear Wheel
$$I_R=m_RR_W^2\tag{2}$$
where
##m_R## = mass of rear wheel assembly
Eqn. 2 makes the reasonable approximation that all the mass of the wheel assembly is concentrated at the rim.
Kinematic Equation Between Angular Acceleration of Wheels and Forward Acceleration of Bike
$$α=a/R_W\tag{3}$$
where a is the acceleration of the bike and rider.
Combining Eqns. 1-3 for Rear Wheel Assembly Moment Balance
$$(T_U-T_L)R_{RS}-F_{SR}R_W=m_RR_Wa\tag{4}$$
Moment Balance on Pedal Assembly (neglecting rotational inertia of pedal sprocket assembly)
$$τ_P=(T_U-T_L)R_{PS}\tag{5}$$
where
##τ_P=## = Torque applied by rider to pedal sprocket
##R_{PS}## = Radius of pedal sprocket
Moment Balance on Front Wheel Assembly
$$F_{SF}R_W=m_FR_Wa\tag{6}$$
where
##F_{SF}## = Static friction force exerted by ground on front wheel in
backward direction
##m_F## = mass of front wheel assembly
Overall Force Balance on Bike and Rider
$$F_{SR}-F_{SF}=Ma\tag{7}$$
where
##M## = combined mass of bike and rider
Solution of Equations and Results
Equations 4-7 constitute a complete mathematical statement of our bicycle mechanics model. Our objective now will to be to use these equations to derive relationships for the:
- acceleration of the bike expressed as a function of the pedal torque applied by the rider
- static friction force on the rear wheel expressed as a function of the pedal torque applied by the rider
- static friction force on the rear wheel expressed as a function of the tension difference between the upper part of the chain and the lower part of the chain
If we combine Eqns. 4 and 5 to eliminate the tension difference between upper and lower portions of the chain, we obtain:
$$τ_P\frac{R_{RS}}{R_{PS}}-F_{SR}R_W=m_RR_Wa\tag{8}$$
Combining this with Eqns. 6 and 7 yields:
$$(M+m_R+m_F)R_Wa=τ_P\frac{R_{RS}}{R_{PS}}\tag{9}$$
Note from Eqn. 9 that including the rotational inertia of the wheel assemblies in the analysis is equivalent to increasing the total mass of the bike and rider (which already includes the mass of the wheel assemblies) by the mass of the wheel assemblies a second time. If we solve equation for the acceleration a, we obtain:
$$a=\frac{τ_P}{R_W(M+m_R+m_F)}\frac{R_{RS}}{R_{PS}}\tag{10}$$
If we substitute Eqn. 10 into Eqn. 8, we obtain the following relationship for the static frictional force on the rear wheel as a function of the torque applied by the rider on the pedal:
$$F_{SR}=\frac{τ_P}{R_W}\frac{R_{RS}}{R_{PS}}\frac{(M+m_F)}{(M+m_R+m_F)}\tag{11}$$
We can also express the static frictional force in terms of the tension difference between the upper part of the chain and the lower part of the chain (for whatever that's worth) by solving Eqn. 5 for ##τ_P## and substituting the result into Eqn. 11 to obtain:
$$F_{SR}=\frac{(T_U-T_L)R_{RS}}{R_W}\frac{(M+m_F)}{(M+m_R+m_F)}\tag{12}$$
The mass ratio term in Eqn. 12 captures the effect haruspex was referring to in his responses when he indicated that the frictional torque "lags" behind the propulsive torque.
This completes our analysis.
jbriggs444, haruspex, chestermiller