How to Increase Gear Tension on a Bicycle UPDATED

Empathize the speed of a bicycle, by analyzing forces and computing ability

This commodity shows how to calculate the power needed for a bicycle to move at a given speed. Equations are explained from kickoff principles, considering the forces acting on a wheel. Typical values are given for all the required parameters. The approach also allows you to summate speed if you know ability, although it's a bit more difficult that way around. The focus in this article is on the relationship between power and steady state speed, although this approach likewise provides the upper bound for analysis of acceleration.

Physics of bicycle speed

Applying bones physics to cycling, the equation F=ma (force = mass x acceleration), may be expressed as

F_P – F_R = 1000 a

where F_P is the propulsive forcefulness and F_R is the sum of all resistance forces.

Assuming a constant speed has been reached, acceleration is zero and the above equation simply becomes F_P = F_R . Multiplying this by velocity, v, converts the propulsive force into the propulsive power P_P .

P_P = five F_R

P_P may exist replaced by the input power P_in generated by the cyclist and the efficiency of the transmission arrangement η. This gives a general equation relating the stead state speed of a wheel to the power produced by the passenger:

P_in η = v F_R

The total resistance to movement is really made up of several separate resistance forces:

F_m , resistance due to gravity (when riding on a gradient)
F_A , air resistance to frontwards move
F_R , rolling resistance
F_RB , bike bearing resistance
F_B , crash-land resistance
F_W , air resistance to the rotation of the wheels

These resistance forces are sometimes represented differently, but the bones forces remain the same.

Resistance Due to Gravity

Free body diagram showing slope resistance forces for bicycle

Gravity acts on the combined mass of bicycle and passenger, causing a downward force, weight. If the basis is not horizontal, then a component of this forcefulness volition act to resist forrard motion. If the slope is specified as an bending, then the gravity force, mg, is multiplied by the cosine of the angle. Typically, the slope will exist specified as a gradient, G, and then that the resistance due to gravity is given past

F_k = m g sin( arctan(M) )

where thou is the combined mass of bike and passenger, g is the acceleration due to gravity (9.81 m/due south) and G is the gradient of the slope. The steepest roads have gradients of about xxx%, but these are very rare. A 12% gradient is normally considered to be very steep. You lot tin can easily check gradients on your normal routes using the free Google Earth Pro app. This video shows yous how, using the Tiptop Profile feature.

F_g is the same at whatsoever speed, even when stationary, information technology'due south simply the force pulling yous down the slope of the hill.

Air Resistance to Forrard Motion

Air resistance, or aerodynamic drag, is given by the standard equation:

F_A = ½ C_d A ρ v_A ⁱⁱ

where C_d is the elevate coefficient, A is the frontal expanse, ρ is the density of the air, 5_A is the air velocity in the direction the cycle is travelling. A_f is sometimes used to make information technology articulate that the drag coefficient is based on the frontal surface area, this is because aerofoil sections are oft specified using different measures of surface area.

Since changes to the shape of a bike or the position of the rider will change both the drag coefficient and the frontal area, it is mostly more user-friendly to consider C_dA as a single parameter, the drag area. This can be institute by testing. For example, in a wind tunnel, the strength F_A and wind velocity 5 are measured. Measurements are also taken of the air force per unit area, temperature and humidity, these are used to calculate the air density ρ. The drag area is then calculated by rearranging the standard air resistance equation:

C_d A = ii F_A / ρ 5_A ²

Information technology is mostly not necessary to accurately measure the frontal area and determine C_d independently. The elevate area is sufficient to calculate the elevate force at whatever speed and in any environment.

It is important to annotation that C_d actually varies with velocity, although it is generally adequate to care for it as a constant over a small range of velocities. As the air velocity increases, the airflow changes in significant ways such as becoming turbulent and laminar flow separating. This can lead to quite sudden changes in C_d . The critical parameter which affects the elevate expanse, for a given shape, is the Reynolds number, which depends on both velocity and the size of the object. The chief parts of a bicycle and cyclist are approximately cylindrical with diameters ranging from 1 mm (spokes) up to about 300 mm (body of cyclist), at typical cycling speeds this results in Reynolds numbers in the approximate range of 10³to x⁵. Over this range, the coefficient of elevate for a cylinder is approximately abiding. Notwithstanding, at higher speeds the coefficient changes significantly. Additionally, for aerofoil sections, the drag coefficient varies significantly over this range. For more streamlined bicycles there may, therefore, exist value in evaluating the drag area as a function of velocity.

Current of air complicates things in 2 ways. Firstly, a head wind means that the air velocity v_A is not the aforementioned as the bikes velocity v_Chiliad . Secondly, a cantankerous current of air changes the drag area of the bicycle since the air flows over the cycle and rider in a different direction. Air current may accept any management relative to the path of the bicycle. The showtime step in considering these issues is to separate the air current velocity into two components. The headwind component is tangential to the bicycle'southward path and has velocity v_WTan and the crosswind component is normal to the cycle's path and has velocity v_WNor . It is then possible to calculate the air velocity from the bike's velocity and the headwind component:

v_A = v_G + v_WTan

The yaw angle is the angle betwixt the wheel's path and the direction of airflow over the bicycle. Information technology is calculated:

Yaw Bending = arctan(5_WNor / 5_A )

When measuring drag in a wind tunnel, it should be measured at different yaw angles. Information technology is then possible to interpolate betwixt the measured angles to make up one's mind C_d A for any yaw bending.

Sometimes, a further simplification is made in which C_d A is combined with the cistron ½ and the density of air to requite a single aerodynamic elevate factor yard_A , given by

k_A = ½ C_d A ρ

This may be convenient but it can also mistakenly create the impression that air density is relatively constant. This is not the instance. Under extreme conditions air density could exist as high as 1.5 kg/1000³ (below body of water level, with dry air at -xxx C) or as low as 0.nine kg/yard³ (at 1,000m meridian with a temperature of 50 C and 100 pct humidity). On the highest route in the earth the density could even get as low as 0.seven kg/yardⁱⁱⁱ. Even nether normal conditions the density can vary between nearly 1.1 and 1.iii kg/thousand³.

Air Resistance to Wheel Rotation

For air current tunnel tests, a bicycle should exist mounted on rollers so that its wheels are rotating at the ground speed. This is because the motion of the wheels changes the period of air over the bike and rider. The force measured in the wind tunnel is, however, only the force of the air pushing backwards on the bike and rider. The air's resistance to the rotation of the wheels, acquired mostly by the spokes passing through the air, is non measured. This resistance has its own drag area which may exist measured by rotating the wheels in the air. The air resistance to wheel rotation is given by:

F_West = ½ C_d A ρ v_M ²

Note that ground velocity should probably be used here. It is clear that if a tail air current exactly matches the speed of the bicycle, then that the air velocity is nothing, at that place is still a resistance to the rotation of the wheels. This demonstrates that the use of air velocity, as in some sources, is not right. Nevertheless, there may well be some event caused by the air velocity which would not be detected by rotating the wheel in still air. This issue has not be fully explored in the literature.

It should also be noted that if a coast-downwards, or similar, examination is used to determine the drag area, then this will produce a combined drag area, for the resistance to both forward motion of the bicycle and rotation of the wheels. In this case, information technology is hard to deal with the yaw angle furnishings fully.

Rolling Resistance

Every bit a wheel rolls over the footing, both the wheel and the basis deform slightly. Some of this deformation is rubberband and so the free energy is recovered. The inelastic deformation generates heat and dissipates power. A solid steel wheel on a steel rail has very little inelastic deformation and therefore the rolling resistance is very low. The air within a pneumatic tire also has good energy recovery. However, some energy is still lost by friction within the tire casing. When riding over soft ground significant energy is also dissipated by the ground itself.

Small inertial loads, equally the wheel passes the small-scale bumps of a crude surface, may too be included equally part of rolling resistance, although strictly this is bump resistance, dealt with in the next section.

Reliable analytical models of rolling resistance do not exist. Instead, rolling resistance is measured for a given tire, road surface and inflation force per unit area. The rolling resistance is normally modelled as a single dimensionless coefficient of rolling resistance C_R which is the ratio of the normal to tangential force on the tires. The normal force is the combined weight of the bike and passenger pushing downwards against the road. On a perfectly horizontal route this is simple mg. The tangential forcefulness is the rolling resistance. On a hill with gradient M the rolling resistance is given past:

F_R =C_R m k cos( arctan(G) )

Crash-land Resistance

The small bumps, which are continuously encountered due to the roughness of the road surface, are normally included in the rolling resistance. Large polish bumps may not cause any resistance. The wheel may roll smoothly up ane side converting kinetic energy into potential free energy. When the bicycle rolls down the other side the potential energy is converted back into kinetic energy with most consummate energy recovery. Larger and steeper bumps may cause a significant resistance which requires consideration divide to rolling resistance. For instance, curbs and potholes on the road, or rocks and roots off-road.

When a bike encounters a square edged bump, the angle of force depends on the size of the bump and the size of the bike. If a square edges crash-land is taller than the cycle radius, then the wheel may be stopped completely. This frequently happens with the pocket-sized wheels of roller skates and skateboards. If interruption allows the wheel to motility backwards, relative to the vehicle, before it moves upwardly, then even in such an farthermost instance it may be possible for the bike to roll over the bump.

The lower the mass which must exist lifted over a crash-land, the lower the bump resistance. For this reason, pneumatic tires are excellent at minimizing bump resistance since only a small portion of the tire casing, weighting a few grams, is lifted over almost bumps. For bumps besides large to exist absorbed past pneumatic tires, suspension ways that merely the cycle and break arm must exist lifted over the bump, non the whole vehicle.

There is no belittling or experimental way of determining the general crash-land resistance. A meaning claiming is that the resistance greatly depends on the response of the passenger.

Bike Bearing Resistance

Wheel bearing resistance is typically included with rolling resistance, although it is a divide resistance which behaves differently. It has been shown that bearings produce a torque which varies linearly with both normal forcefulness and rotational velocity Mai, Poland, and Jenkins (1991). However, since bearing resistance is relatively small, treating it as a component of rolling resistance appears to produce negligible errors. In the most accurate tests to date, in that location was no velocity dependent component of begetting resistance included, with the bearing resistance included within the rolling resistance.

Experimental Validation of Bicycle Power Equations

This arroyo was kickoff demonstrated to predict the power of road cyclists by approximately 3% (Martin et al, 1998). These tests used wind tunnel tests to decide C_dA for a range of yaw angles and independent tests to determine the air resistance to wheel rotation, but reference values were used for rolling and cycle bearing resistances. Considerably higher accuracy has been accomplished by Fitton & Symons (2018) by because kinetic energy changes and accurately modelling changes in rolling resistance with cornering.

Group Resistances Together

Resistances may be grouped together for a number of reasons. Sometimes this is done to but to nowadays a simpler model. For instance bearing resistance is very modest and is oftentimes included with rolling resistance. Ane very good reason for grouping resistances is when a coast-downwardly examination is performed. A toll downwardly test involves recording the time information technology takes for a wheel to decelerate from some known speed, without pedaling, braking or turning. Deceleration is caused by the resistance forces, allowing them to be determined. Deceleration can be attributed to individual resistance factors by considering how they vary with velocity. The test rail gradient is measured and the calculated resistance due to gravity is removed kickoff. The abiding force, which does non vary with velocity is then attributed to rolling resistance, although bearing resistance and possibly bump resistance will also exist included. The remaining strength, which volition vary with the square of velocity, is attributed to air resistance. However, this is not the same air resistance as would be measured in a wind tunnel since it includes both the air resistance to forward motility and the air resistance to the rotation of the wheels. A more than general mode of expressing these forces is equally the zeroth club forces and second club forces, referring to the power of velocity with which they vary. In theory, it would also exist possible to identify outset order forces, which may include the velocity dependent component of bearing friction. In practice, declension-downward tests are not accurate plenty to find such effects. Therefore, coast-down tests determine two resistance parameters.

Calculating the Power Required to Achieve a Given Speed

The start footstep in computing power is to determine resistance coefficients. In the simplest example, this would mean a single drag area and a single rolling resistance coefficient. The total power required to overcome each resistance force would so exist calculated and the sum of these would be the full power required.

Calculating Speed for Known Ability

It is possible to calculate the bike velocity, for a given input power, by rearranging the equation for power. To exercise this, the resistance forces, which are dependent on velocity, must commencement be substituted into the power equation. Resistance forces which are not dependent on velocity may proceed to be represented equally private parameters. When the resulting equation is rearranged to make velocity the subject field, it becomes very complex. This means that it is of little intuitive value and it is oft more than useful to simply view a graph of ability against velocity. However, for computational analysis it can brand sense to apply the equation in this form. An case is given beneath: