The basic Whipple-Carvallo bicycle model for the study of stability takes into account only geometric and mass properties. Analytical bicycle models of increasing complexity are now available, they consider frame compliance, tire properties, and rider posture. From the point of view of the designer, it is important to know if geometric and mass properties affect the stability of an actual bicycle as they affect the stability of a simple bicycle model. This paper addresses this problem in a numeric way by evaluating stability indices from the real parts of the eigenvalues of the bicycle's modes (i.e., weave, capsize, wobble) in a range of forward speeds typical of city bicycles. The sensitivity indices and correlation coefficients between the main geometric and mass properties of the bicycle and the stability indices are calculated by means of bicycle models of increasing complexity. Results show that the simpler models correctly predict the effect of most of geometric and mass properties on the stability of the single modes of the bicycle. Nevertheless, when the global stability indices of the bicycle are considered, often the simpler models fail their prediction. This phenomenon takes place because with the basic model some design parameters have opposite effects on the stability of weave and capsize, but, when tire sliding is included, the capsize mode is always stable and low speed stability is chiefly determined by weave stability.

The stability of bicycles has drawn the attention of scientists since the development of the first modern bicycles. The first dynamic models for understanding the bicycle dynamics were written independently by two scientists at the end of the nineteenth century: Whipple (1899) and Carvallo (1899). The Whipple–Carvallo bicycle model (WCBM) (Meijaard et al., 2007) consists in the linearized equations of motion of the bicycle and the rider. This model makes it possible to conduct open loop analysis with the rider hands-off the handlebar. Many authors have analyzed bicycle stability using the WCBM (Limebeer and Sharp, 2006; Meijaard et al., 2007; Schwab et al., 2007; Sharp, 2008) by solving the eigenvalue problem in order to analyze the modes of vibration of the system.

Starting from the WCBM many linearized bicycle models of increasing complexity have been developed to study bicycle stability. Some researchers have extended the WCBM in order to include compliance of the front assembly (CFA). This compliance commonly includes the effects of the frame head tube, the fork, and the wheel (Doria and Roa, 2017; Doria et al., 2017; Klinger et al., 2014; Limebeer and Sharp, 2006; Plöchl et al., 2012; Sharp, 2008). These models add an additional velocity degree of freedom to the WCBM in order to take into account the lateral velocity of the front-assembly due to compliance. Other authors have extended the model including tire mechanics (Doria et al., 2013; Doria and Roa, 2017; Klinger et al., 2014; Sharp, 2008; Souh, 2015). When tire lateral slip is considered at least four degrees of freedom (DOFs) are needed to model the bicycle.

Even if bicycle tires may exhibit a non-linear behavior (Doria et al., 2013) and the shock absorbers that nowadays equip many bicycles have non-linear characteristics (Cossalter et al., 2010), very few authors have carried out stability analysis taking into account non-linear properties (Bulsink et al., 2015), because stability analysis with non-linear models requires cumbersome time-domain simulations and specific identification methods for extracting the properties of the modes of vibration from time-domain data.

The rider with his/her mass, stiffness and damping characteristics has a large effect on bicycle dynamics, even if the control actions (Kooijman and Schwab, 2013) are neglected and a completely passive behavior is assumed. For this reason, some researchers have integrated the bicycle model with rider models composed of rigid bodies (the limbs) connected by means of joints (the articulations) and by lumped stiffness and damping elements (Schwab et al., 2012; Doria and Tognazzo, 2014). These models make it possible to simulate the passive response of the rider both in the hands-off and in the hands-on configuration.

The extensions of the WCBM improve the quality and range of reliability of the stability analysis. For instance, when the front-assembly compliance and tire mechanics are included, the wobble mode appears as an additional mode of vibration and bicycle stability at relatively high speed is better predicted. Additionally, when the hands-on condition is analyzed, the weave, capsize and wobble modes are changed with respect to the hands-off condition, in particular the wobble mode becomes more damped due to the rider's arms influence (Klinger et al., 2014; Roa et al., 2018). Nonetheless, it is useful to determine the limits and potentialities of each model.

The purpose of this paper is to compare different models of increasing complexity in terms of their capability of predicting bicycle stability. Since bicycle stability depends on many parameters, only the effect of geometrical properties is analysed, tire properties, stiffness properties, and rider body properties are kept constant. The possible geometric configurations are explored numerically with a design of experiment (DOE) approach based on the space filling method proposed by Sobol (1967). This method compared with a random method assures a lower uncertainty for the same number of sample points (Saltelli et al., 2008) which are associated with the computational effort. The stability of each bicycle configuration is evaluated by means of numerical indices that are calculated from the eigenvalues obtained by means of the models of increasing complexity.

The simplest model considered in this research is the WCBM, that was checked
and reviewed in Meijaard et al. (2007). This model has two velocity degrees
of freedom (DOFs): the steer rotation of the handlebar around the steering
axis

Bicycle models.

The second model considered in this research is the improvement of the WCBM
recently proposed in Doria et al. (2017), see Fig. 1b. This model accounts
for front assembly compliance by introducing a revolute joint, a rotational
spring, and a rotational damper (not shown in the figure); the revolute
joint defines the deformation axis of the front frame that makes possible
the lateral displacement of the front wheel. Therefore, the number of DOFs
increases to three, and the new variable

Tire and fork parameters.

The most complex bicycle model considered in this research is the one
developed in Klinger et al. (2014). In this model lateral slips of front
and rear tires are allowed. A linear model of tire forces and torques is
adopted, since they depend in a linear way on side-slip and camber angles.
The DOFs are five:

In all the previous models the rider with the hands-off the handlebar is simulated assuming the center of mass and inertia properties as in Meijaard et al. (2007). The model developed in Klinger et al. (2014) makes it possible to simulate bicycle dynamics considering the rider with the hands-on the handlebar as well. In order to avoid introducing new DOFs, the rider's body is connected to the handlebars by means of arms equipped with joints located at the shoulders, elbows and wrists, according to the approach suggested in Schwab et al. (2012). In the hands-on model, a bent-forward posture is assumed, and the center of mass and inertia properties are modified accordingly (Moore et al., 2009).

Stability of bicycles and powered-two-wheeled vehicles usually is analyzed by plotting the real and imaginary parts of the eigenvalues against forward speed (Meijaard et al., 2007). This research focuses on the stability of city bicycles and the nominal geometric and mass properties of the reference bicycle are set equal to the ones of the benchmark bicycle defined in (Meijaard et al., 2007). Figures 2–5 represent the eigenvalue plots of the bicycle with the reference parameters of Tables 1 and 2 that are obtained carrying out stability analysis by means of the four models here considered.

Eigenvalues against speed using the WCBM and the nominal parameters in Tables 1 and 2.

Eigenvalues against speed using the WCBM with CFA and the nominal parameters in Tables 1 and 2.

Eigenvalues against speed using the complete model and the nominal parameters in Tables 1 and 2, hands-off.

Eigenvalues against speed using the complete model and the nominal parameters in Tables 1 and 2, hands-on.

In the field of bicycle dynamics very few stability indices have been
defined. With reference to stability analysis carried out by means of the
WCBM, which predicts only the weave and capsize modes, weave speed

The analysis of the eigenvalues plots becomes more complex when more
detailed models able to predict the wobble mode are used to study stability.
In this case, it is possible to introduce another modal stability index,
which is the wobble speed (

In order to improve the stability analysis, the concept of stability area (Doria and Roa, 2017) is used in the framework of this research.

Dealing with single mode stability, the stability area index is the area
formed by the curve of the real part of an eigenvalue and the speed axis
when this real part is negative:

Global stability can be studied considering the self-stability area, which
is the intersection of the stability areas of the modes. If two modes are
considered:

Nominal values and ranges of geometric and mass properties.

The aim of this research is to analyze the effect of the geometrical
parameters of the bicycle on the stability indices defined in the previous
section. This analysis is performed numerically by exploring the effect of
the various parameters by means of large series of simulations to assess the
sensitivity of the stability indices to the design parameters. The
sensitivity is analyzed combining two approaches. First, a variance-based
sensitivity method is used to explore the possible high-order interactions
between the design parameters that can affect the stability indices. Second,
a correlation analysis is used to investigate the main trend of the
variation in the stability indices associated with the change in each design
parameter across the whole domain. In both cases, eight design parameters
are considered:

The analysis of the interactions of the different design parameters and their effect on the stability indices is performed through the variance-based method presented in Sobol (2001); Saltelli et al. (2008) and numerically implemented by Cannavo (2012). In the framework of this approach, two factors are said to interact when their contribution to the variance of the output cannot be expressed as the sum of their single contributions. In this paper, the importance of the interactions between variables on the stability indices is studied. To this aim, the contribution to the total variance of the single parameters (i.e., first-order interactions), and the contribution to the total variance of the pairs of parameters (i.e., second-order interactions) are calculated. When these contributions are calculated, the remaining variance is associated to higher-order interactions (i.e., third-order or higher).

For a design parameter

Figure 6 presents the contribution of first, second, and higher order
interactions, to the variance of the weave mode indices calculated by means
of the various bicycle models. The sum of the sensitivities of the weave
speed index (

Sensitivity of the weave mode to first, second, and higher orders of interaction. Models of increasing complexity from bottom to top.

Sensitivity of the capsize mode to first, second, and higher orders of interaction. Models of increasing complexity from bottom to top.

Figure 7 summarizes the results of the sensitivity analysis for the capsize
mode indices, showing the contribution of the orders of interaction.
First-order interactions explain 97.8 % or more of the total variance of
the capsize speed index (

Sensitivity of the wobble mode to first, second, and higher orders of interaction. Models of increasing complexity from bottom to top.

Figure 8 shows the sensitivity of the wobble mode indices to the different
orders of interaction. The sum of the sensitivities explained by the
first-order interactions decreases for wobble, compared with those obtained
for the other two modes. For the wobble speed index (

The analysis of the interactions of the design parameters with the global
stability indices is also performed by means of the variance-based method.
Figure 9 presents the contribution of first, second, and higher order
interactions to the variance of Ssr

Sensitivity of the stability indices taking into account weave and capsize to first, second, and higher orders of interaction. Models of increasing complexity from bottom to top.

Sensitivity of the stability indices taking into account the three modes to first, second, and higher orders of interaction. Models of increasing complexity from bottom to top.

Figure 10 shows the sensitivity of Ssr

The correlations associated to the single modes of vibration are studied one
at a time considering both the speed index and the area index. A
space-filling computational experiment is used, implementing a quasi-Monte
Carlo exploration of the domain based on a Sobol low discrepancy sequence
(Sobol, 1967). 40 000 points are evaluated, and the results are used for the
computation of the correlation coefficients.

Figures 11, 12, and 13 summarize the results of the correlation analysis. The correlation coefficients are shown in solid red when they represent positive correlations, and in dotted blue when they represent negative correlations. The circles that represent each correlation coefficient have an area that is proportional to the magnitude of the index.

Correlation coefficients for the indices of the weave mode.
Models of increasing complexity from bottom to top. Speed range: 0 to 10 m s

Correlation coefficients for the indices of the capsize mode.
Models of increasing complexity from bottom to top. Speed range: 0 to 10 m s

Correlation coefficients for the indices of the wobble mode.
Models of increasing complexity from bottom to top. Speed range: 0 to 10 m s

Figure 11 presents the correlation coefficients related with the weave mode.
The basic WCBM shows that

Therefore, as far as the weave mode is concerned, it can be stated that the predictions on weave stability made by the simplest models are confirmed when more complex models are adopted, there are only some variations in the relative importance of the various parameters.

All the previous results dealt with the hands-off condition, the full model makes it possible to study the hands-on condition as well. Positioning the rider's hands on the handlebar has a very small influence on the stability of the weave mode considering both the weave speed index and the weave area index.

Figure 12 makes it possible to analyze the effect of geometric parameters on
the stability of the capsize mode. The basic WCBM shows that

Finally, the effect of geometric parameters on wobble stability is analyzed
by means of the correlation coefficients that are shown in Fig. 13. The 2
DOF WCBM does not simulate the wobble mode and the simplest model predicting
wobble is the WCBM with CFA. With this model, the parameter with the
strongest positive correlation coefficient with

Correlation coefficients for the stability indices taking into
account weave and capsize. Models of increasing complexity from bottom to
top. Speed range: 0 to 10 m s

Correlation coefficients for the stability indices that consider
the three modes (i.e., capsize, weave, and wobble). Models of increasing
complexity from bottom to top. Speed range: 0 to 10 m s

The correlation between the geometric parameters and the indices that define
the global stability of the bicycle can be considered the most important
result from the practical point of view. The key question is if the
predictions made by the simpler models hold true even when more realistic
bicycle models are considered. The WCBM can predict only weave and capsize
stability, therefore indices Ssr

The simplest model that can be used for calculating the self-stability indices taking into account three modes (weave, capsize, and wobble) is the WCBM with front compliance, see Fig. 15.

Correlation analysis shows that in this case parameter

The comparison between the stability indices calculated with 2 and 3 modes
shows that some parameters change their effect on stability: with 3 modes
the increase in

When the basic WCBM is extended by introducing front assembly compliance,
the most important effect is the appearance of a high frequency wobble mode,
which may become unstable at the highest speeds considered in the framework
of this research (i.e., 10 m s

The introduction of tire slip in the model has an important effect on
stability, because not only the wobble mode appears, but also the capsize
mode becomes stable over the whole range of speeds. Therefore, even if the
influence of the various geometric and mass properties on the stability of
the single modes (weave and capsize) is very similar to the one predicted by
the WCBM, the effect on global stability is often different. This phenomenon
takes place because some parameters have opposite effects on the stability
of weave and capsize and in the simple WCBM their effect on global stability
derives from a combination of the effects they have on the single modes.
Conversely, in the presence of tire slip, the capsize mode is always stable
and no longer relevant, thus the combined effects do not take place, and
global stability is dominated by the weave and wobble modes. Only parameters

A different posture of the rider with hands-on the handlebar does not
strongly change the effect of mass and geometric parameters on the stability
indices. The most important effect is that with hands-on

According to the presented results, it is possible to state that the simple WCBM gives useful hints for understanding the physical phenomena determining bicycle stability, especially at low speed. The simpler models give more accurate information about the stability of the single modes than about the global stability of the bicycle; therefore they can be useful when the stability of a single mode is the main concern, because the other modes can be stabilized by the rider, or become unstable at very high speed. Generally speaking, a full bicycle model is strongly recommended for studying the global stability properties of actual bicycles.

All the data used in this manuscript can be obtained by request from the corresponding author.

AD coordinated the research activity and analyzed the results. SR developed the numerical codes and carried out the simulations. LM developed the correlation analysis and analyzed the results.

The authors declare that they have no conflict of interest.

This research has been partially supported by the Colombian Administrative Department of Science, Technology, and Innovation (Colciencias) (grant no. Doctorate Formation Program 617 of 2013).

This paper was edited by Anders Eriksson and reviewed by James Sadauckas and one anonymous referee.