### Executive Summary — Bezerra Report

The purpose of this experiment is to compare the advantages of the stepper bike in terms of power and torque to a conventional bike. To offer a proper comparison, the time frame, applied force and pedal rate are equal in each case. Each design concept is modeled in SolidWorks and analyzed using COSMOS Motion. The motion study analyzes one full pedal cycle for each design. Hand calculations are then used to go beyond the motion study results.

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**Assumptions**

- One pedal cycle on the conventional bike is 180 degrees driving and 180 degrees non‐driving (returning to starting position). One pedal cycle on the stepper bike is 44 degrees driving and 44 degrees non‐driving (returning to starting position).
- The pedal rate for each design is equal. It is established that the conventional bike is being operated at 30 rpm. This means one full pedal cycle takes 2 seconds. To equate the pedal velocity in each analysis, the pedal velocity of the conventional bike was found. Due to different drive arm lengths, it was not adequate to relate pedal velocity in terms of angular displacement. Instead the velocity was determined in terms of inches along the radius of the drive arm per second. For a conventional bike operating at 30 rpm (input axle rotation rate of 180 degrees per second), the pedal velocity was found to be 23.56 in/sec (see attached “Pedal Rate Calculations”). This pedal velocity, when translated to the stepper design, results in a cycle rate of 1.30 seconds per cycle, or an input axle rotation of 46.15 rpm.
- The input axle of the conventional design is directly connected to the transmission. The input axle of the stepper design is connected an output sprocket through a belt connection. This utilizes a gear ratio between the drive arm (pin to pin distance is 6 inches) on the input axle and the output sprocket (pitch diameter of 2.33 inches) which is connected to the transmission.
- There is a 45 pound force applied to each pedal during the driving portion of the cycle. In the COSMOS Motion analysis there is a 0.04 second transition between applying a 45 pound force on the down‐stroke and a 0 pound force on the upstroke. This is implemented in the study through a step function outlined in the “Motion Study Setup” section.
- Each design was analyzed based on one full pedal cycle. The attached “Power Output Comparison” chart compares the two designs on a per unit‐of‐time basis.

Motion Study Setup

This motion study has 3 basic components. Gravity is a global force applied to the system at the standard value of 32.2 ft/s2. Next are the movements and forces. COSMOS Motion uses the term “motor” when defining a movement, whether linear or rotational. In this case, the motors create the rotational motion of the pedals based on the motor criteria. In the software, the motors are completely independent of the forces. The motors simply establish the motion profile of the given component. This motion exists with or without a force. Then a force is applied to a component controlled by that motion profile. A force alone does not create motion. It is the combination of the motors and forces that generates results. Below is the starting position of each study. The arrows about the input axle represent the motors and their direction. The arrows shown on the pedals represent the downward forces. The forces always remain in the downward direction but change in value from 45 lbs on the downward stroke to 0 lbs on the upward stroke.

Each design has different motor requirements. The conventional design is very simple. The drive arms rotate about the drive axle, as shown above, at a rate of 30 rpm in the forward direction. The stepper design uses a step expression instead of a constant velocity to define the motor and, in turn, define the motion. A displacement step expression follows the format: STEP(TIME,T1,A1,T2,A2,). “STEP” establishes that it is a step expression. “TIME” establishes that the T1 and T2 variables are in terms of real time. A1 is the initial angle and T1 is the initial time. A2 is the final angle and T2 is the final time. COSMOS then interpolates the expression to get from A1 to A2 in the time T2‐T1. Two STEP expressions are used to obtain the up and down motion. The following are the expressions used for each motor.

**Left STEP(TIME,0,0,0.65,44)+STEP(TIME,0.65,0,1.30,‐44) CCW Direction**

**Right STEP(TIME,0,0,0.65,44)+STEP(TIME,0.65,0,1.30,‐44) CW Direction**

In the study, the force on each pedal cannot remain a constant 45 lbs because of the 0 lb force on the pedal during the upward stroke. However, the force on each pedal must be applied in time with the motion of the pedal in order to keep a constant applied force of 45 lbs throughout the pedal cycle.. To do this, a step function is used. A step function has 4 variables: F1, T1, F2, and T2. The force F1 is applied from time T=0 until time T1. Then at time T2, force F2 is applied throughout the remainder of the study. The force between time T1 and T2 is linearly interpolated. Below is the value of each value for each motor for each design. The 0.04 second gap between T1 and T2 represents the transition of force between the left and right pedal halfway through the cycle.

Design | Pedal | F1 (lbs) | T1 (sec) | F2 (lbs) | T2 (sec) |
---|---|---|---|---|---|

Stepper | Right | 45 | 0.63 | 0 | 0.67 |

Stepper | Left | 0 | 0.63 | 45 | 0.67 |

Conventional | Right | 0 | 0.98 | 45 | 1.02 |

Conventional | Left | 45 | 0.98 | 0 | 1.02 |

### Results

The motion study determined the torque created by the force on the pedal and the power consumption in doing so. The attached “Power Output Comparison” table shows side by side results from the study. Also attached are the charts and data produced by COSMOS Motion for each study.

The first set of data on the comparison table is the power input. This is the power necessary to obtain the motion profile with the applied forces as previously outlined. The attached “Power Input” graphs show the power used over the course of one cycle. With the applied force and pedal velocity kept constant between the two designs, it is expected that the human energy necessary to operate each design would be equal. The “Power Output Comparison” table shows that the average human input energy for each design is equal (within an acceptable error), thus confirming this hypothesis.

The next set of data is derived from the torque generated by the force on the pedal (refer to the above images for the identification of the components mentioned in the table). The amount of torque generated throughout the one pedal cycle is shown on the “Torque at Axle” graphs. As the table shows, the stepper bike has a must higher force factor. This value is the average torque over one pedal cycle divided by the maximum torque available from the drive arm. The stepper design has a higher force factor due its small angular displacement. The conventional design is less efficient in this sense due to its circular motion and large angular displacement. While the stepper bike has a higher torque, it has a much lower speed at the input axle compared to the conventional bike. This is shown in the equation below. The input human power is held constant and is directly related to the product of the torque and velocity. The conventional design has a higher rpm but less torque and the stepper design has a lower rpm but more torque.

However, the two input axles are not comparable due to the nature of the stepper design. The appropriate comparison is between the output axle of the stepper design (as shown in the above image) and the input axle (which is also the output axle) of the conventional design. To make the comparison, the gear ratio effect of the stepper bike’s belt drive must be considered. The belt attaches to two pins on the drive arm spaced 6 inches apart. The belt then connects to a sprocket with a pitch diameter of 2.33 inches. This system is geared for speed. Therefore, the output axle has a higher speed but less torque than the input axle. Even with this torque reduction, the stepper design has more torque at the output axle than the conventional design does at its input (and output) axle (340 lbf‐in compared to 213 lbf‐in). This means the stepper design can use the same human power to generate 60% more torque than the conventional design. With this excess torque, the stepper bike can use a transmission to increase the overall bike velocity with the same human power.

### Conclusion

The numbers confirm that when the stepper bike and conventional bike are operated with the same amount of human input power, the stepper bike will outperform the conventional bike by producing 60% more torque. The drive arm on the stepper design undergoes a smaller angular displacement to maximize the output efficiency and create an excess of torque. Through a belt or chain system and transmission, this excess of torque can be converted to speed by operating the bike at a lower gear setting. The stepper bike is the superior design.

**Pedal Rate Calculations [for conventional bicycle]**

[Values marked with *****, indicate predetermined input]

Calculation | Value | Units | Description |
---|---|---|---|

A | 30.00* | rpm | rate of pedal travel |

B = A/60 | 2.00 | sec/cycle | cycle rate of one pedal |

C | 360.00 | deg/cycle | of pedal |

D = C/B | 180.00 | deg/sec | of pedal |

E | 7.50* | in | radius of drive arm |

F = (2*π*E)*(D/360) | 23.56 | in/sec | pedal travel |

**Pedal Rate Calculations [for BezErra Stepper bicycle]**

[Values marked with *, indicate predetermined input]

Calculation | Value | Units | Description |
---|---|---|---|

F = F | 23.56 | in/sec | pedal travel |

G | 20.00* | in | radius of drive arm |

H = [F/(2*π*G)]*360 | 67.50 | deg/sec | at input |

I | 88.00* | deg/cycle | |

J = I/H | 1.30 | sec/cycle | cycle rate of one pedal |

K | 6.00* | in | input “sprocket” diameter |

L = (π*K)*H/360 | 3.53 | in/sec | cord travel rate of input and output sprocket |

M | 2.33* | in | output sprocket diameter (McMaster 2299K19) |

N = L/(π*M)*360 | 173.82 | deg/sec | of output sprocket travel |

O = J*N | 226.61 | deg/cycle | of output sprocket travel |

P = (I/J)/360*60 | 11.25 | rpm | of input axle |

Q = P*(K/M) | 28.97 | rpm | of output sprocket |

**Note:** The above calculations assume a seamless transition between the driving and non‐driving pedal [for both the conventional and the BezErra Stepper bicycles].

### Power Output Comparison [Power]

Description | Conventional Value | Conventional Unit | Stepper Value | Stepper Unit | Calculation |
---|---|---|---|---|---|

max human power applied at input axle | 121 | watts | 182.66 | watts | A |

average human power applied at input axle | 75.57 | watts | 116.95 | watts | B |

cycle time period | 2 | secs | 1.3 | secs | C |

average human input energy | 151.14 | watt‐secs | 152.04 | watt‐secs | D = B*C |

### Power Output Comparison [Torque]

Description | Conventional Value | Conventional Unit | Stepper Value | Stepper Unit | Calculation |
---|---|---|---|---|---|

applied force | 45 | lbs | 45 | lbs | E |

max torque at input axle | 340.89 | lbf‐in | 912.34 | lbf‐in | F |

average torque at input axle | 212.9 | lbf‐in | 875.56 | lbf‐in | G |

force factor | 0.6245 | 0.9597 | H = G/F | ||

input axle speed (see Pedal Rate Calculation tables for conventional and Stepper Bicycles) | 30 | rpm | 11.25 | rpm | I |

input “gear” diameter | 1 | 6 | J | ||

output “gear” diameter | 1 | 2.33 | K | ||

gear ratio (speed) | 1 | 2.5751 | L = J/K | ||

gear ratio (torque) | 1 | 0.3883 | M = K/J | ||

output axle speed | 30 | rpm | 28.97 | rpm | N = I*K |

output axle torque (average) | 212.9 | lbf‐in | 340.01 | lbf‐in | O = G*M |

torque advantage | N/A | 1.597 | P = Os/Oc | ||

transmission ability utilizing torque advantage | 1 | 1.597 | Q = P | ||

speed of wheel with transmission ability | 30 | rpm | 46.266 | rpm | R = Q*N |

wheel size | 28 | in | 28 | in | S |

speed of bike | 2638.9 | in/min | 4069.8 | in/min | T = (S*π)*R |

speed of bike | 2.499 | mph | 3.854 | mph | U = T/12/5280*60 |

speed advantage | N/A | 154.22 % | V = Us/Uc*100 |

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