Static and dynamic analyses of mountain bikes and their riders

Mountain biking is a globally popular sport, in which the rider uses a mountain bike to ride on off-road terrain. A mountain bike has either a front suspension system only or a full-suspension system to decrease the external vibration resulting from the terrain irregularities and to increase riding comfort. Despite the added comfort of full-suspension of mountain bikes, there are some disadvantages because the chain-suspension interaction and bobbing effect absorb some of the rider's pedalling power and lead to the reduction of pedalling efficiency. In this study, a technique for evaluating the pedalling efficiency of a bike rider in seated cycling by using engineering mechanics is developed. This method is also found to be useful for determining the correct crank angle for the beginning of the downstroke and that of the upstroke during each pedalling cycle. Next, five mathematical models of rider-bike systems are developed in Simulink and SimMechanics, including one hard-tail (HT) bike, and four full-suspension (FS) bikes [single pivot, four-bar-linkage horst link, four-bar-linkage faux bar, and virtual pivot point (VPP)]. In each of the five rider-bike systems, a PID controller is applied on the rider's elbow to prevent his upper body from falling down due to gravity. A pedalling controller is also developed in Simulink, which is based on the previous theory for evaluating the rider's pedalling efficiency written in Matlab. Another PID controller is used for the pedalling control by sensing the real-time moving speed and applying a suitable pedalling force to achieve a desired speed. The dynamic responses for each of the five rider-bike systems moving on a flat road surface (without bumps) and rough terrain (with bumps) are investigated. The values determined include the pedalling force, pedalling torque and power, forward velocity, contact forces of front and rear wheels, compressions of front suspension (front fork) and rear suspension (rear shock absorber), sprocket distance, chain tension force, and vertical accelerations of handlebar and seats. The numerical results reveal that, while moving on flat road surface, the pedalling efficiency of hard-tail bike is highest, and the bobbing effect of the VPP bike is most serious. However, while moving on rough terrain, the riding conditions for each of the four full-suspension bikes are more stable than the hard-tail bike