Cross-axis Apparent Mass Research Articles

Finite element modeling and parameter identification of the seated human body exposed to vertical vibration

Developing a biomechanical model which connected with the actual anatomy of the human body is helpful to understand the human response to vibration. A finite element model of the seated human body with 175 cm in stature and 68.6 kg in weight, which consists of seven segments, six joints and soft tissue, was established to reflect apparent mass based on the Hybrid III dummy model. By comparing the body segment mass percentages with previous data, the rationality of mass distribution in this model was verified. The biomechanical parameters play a crucial role in biodynamic modeling, while the joint and soft tissue parameters are difficult to choose due to the wide range of anthropometric parameters. In this study, the root-mean-square error between the calculated and the measured apparent mass was taken as objective function, and the effect of fifteen human parameters on the objective function was analyzed through sensitivity analysis. Then seven parameters with a considerable influence on the objective function were selected as design variables, and four approximate models were established for parameter optimization. Soft tissues and joint parameters of the model were determined by parameter identification, and the finite element model that can reflect vertical in-line and fore-and-aft cross-axis apparent mass of the human body without backrest was developed. The seated human model presented in this paper can also reflect the transmissibility from seat to the first thoracic spine and the main modes of the human body below 10 Hz, which is conducive to express the human response to vibration.

Mechanism associated with the effect of backrest inclination on biodynamic responses of the human body sitting on a rigid seat exposed to vertical vibration

Drivers and passengers in vehicles usually sit with their backs supported with a backrest. Sitting with a backrest in vehicles can affect the biodynamic responses of the human body and sitting comfort. Understanding the effect of back support on the biodynamic responses can assist in improving vehicle ride comfort. This study was designed to identify the mechanism associated with the effect of backrest inclination on the apparent masses measured at the seat pan and the backrest during vertical whole-body vibration based on experiment and modelling. The apparent masses of 14 subjects sitting on a rigid seat with various backrest inclination angles of 0º, 10º, 20º and 30º during exposure to vertical vibration at 0.5 ms−2 r.m.s. between 0.5 and 20 Hz were measured at the seat pan and the backrest. A biodynamic model previously developed by the authors was updated to include the interaction between the back of the human body and the seat backrest to represent the measured responses for each subject . The effect of backrest inclination on the biodynamic response and on the model parameters were identified. It was found that as the backrest inclination increased, the resonance frequency in the vertical in-line apparent mass measured at the seat pan increased, and the resonance frequency in the fore-and-aft cross-axis apparent mass measured at the seat pan decreased, while the resonance frequency in the normal apparent mass measured at the backrest was not changed. Statistical analysis found that with increasing backrest inclination, the vertical stiffness of the soft tissues at the ischial tuberosities, middle thighs and front thighs, as well as the stiffness at the lumbar joint tended to decrease, while the fore-and-aft stiffness and the damping of the soft tissues at the ischial tuberosities tended to increase. Analysis with the model of the median apparent masses of the 14 subjects disclosed the vibration modes of the body containing the fore-and-aft sliding motion of the pelvis and thighs, the vertical motion of the body on the soft tissue beneath the ischial tuberosities and thighs, and the bending motion of the spine. With the vertical backrest, the fore-and-aft and vertical motions of the pelvis and thighs were coupled together. As the backrest inclination increased, this coupling decreased and the coupling between the vertical motion of the pelvis and the bending of the spine increased.

Principal component analysis of the cross-axis apparent mass nonlinearity during whole-body vibration

During whole-body vibration (WBV), dynamic forces measured at the excitation-subject interface in directions other than the excitation axis, i.e. cross-axis response, are analysed using the principal component analysis (PCA) and virtual coherence techniques. The study applied these operations to the inline and cross-axis forces measured with twelve semi-supine human subjects exposed to longitudinal horizontal nominally random vibration between 0.25 and 20 Hz at root mean square acceleration levels of 0.125 ms−2 and 1.0 ms−2. The source identification is realised by a reversed path, aiming to identify relative contributions and correlations between the forces in response to a single axis excitation. The inline longitudinal and the cross-axis vertical forces were found to be correlated to each other from a low (e.g. 1–3 Hz) to a medium frequency range (e.g. 10–15 Hz). Above this range, where the forces were much reduced, the two forces tended to be independent in their contribution to the overall response. The singular vectors and virtual coherences were able to establish the degree of correlation in each of the frequency band identified. A signal processing framework is then proposed to take into account cross-axis responses for human vibration.

Nonlinearity in the localised apparent masses of the seated human body exposed to vertical vibration

Abstract The apparent mass of the seated human body exposed to vertical whole-body vibration has shown nonlinear characteristics with the principal resonance frequency decreasing with increasing vibration magnitude. Experimental studies have been designed to investigate the cause of nonlinearity but there seems no consistent conclusion on the role of body parts in the nonlinearity of the apparent mass. Biodynamic modelling provides insight into effect of each model parameter on the dynamics of the system so that the contribution of individual body parts on the nonlinearity can be investigated. In this paper, a multi-body dynamic model of seated human body that can represent the dynamic forces over the local area of contact beneath the ischial tuberosities and thighs was developed. Model parameters were determined for 14 individuals who were exposed to vertical whole-body vibration at 0.25, 0.5, and 1.0 m/s2 r.m.s. in a previous study. With the model parameters determined for 14 individuals and 3 vibration magnitudes, statistical analysis was performed to confirm the effect of vibration magnitude on model parameters. The model developed in this study contained six rigid bodies representing upper-torso, lower-torso, pelvis, thighs, legs, and viscera of the body, interconnected by linear rotational springs and dampers. The geometry parameters of the model were initially derived from a finite-element model of spine and were later scaled to each of the 14 individuals. It was found as vibration magnitude increased, the vertical stiffness beneath the ischial tuberosities, the middle thighs, and the front thighs, the stiffness of viscera, the stiffness of the lower thoracic joint, and the vertical damping beneath the ischial tuberosities decreased significantly. For the vertical apparent mass, reducing the vertical stiffness beneath the ischial tuberosities and thighs and the stiffness of viscera reduced the resonance frequency, while reducing the other stiffness parameters did not see the similar effect. For the fore-and-aft cross-axis apparent mass, reducing the fore-and-aft stiffness beneath the ischial tuberosities and thighs reduced the resonance frequency, while reducing the other parameters of stiffness did not show the similar effect. It was concluded that the soft tissues beneath the ischial tuberosities and thighs and the viscera are the parts principally causing the nonlinearity observed in the apparent masses of the human body.

Identification of biomechanical nonlinearity in whole-body vibration using a reverse path multi-input-single-output method

The study implements a classic signal analysis technique, typically applied to structural dynamics, to examine the nonlinear characteristics seen in the apparent mass of a recumbent person during whole-body horizontal random vibration. The nonlinearity in the present context refers to the amount of ‘output’ that is not correlated or coherent to the ‘input’, usually indicated by values of the coherence function that are less than unity. The analysis is based on the longitudinal horizontal inline and vertical cross-axis apparent mass of twelve human subjects exposed to 0.25–20 Hz random acceleration vibration at 0.125 and 1.0 ms-2 r.m.s. The conditioned reverse path frequency response functions (FRF) reveal that the uncorrelated ‘linear’ relationship between physical input (acceleration) and outputs (inline and cross-axis forces) has much greater variation around the primary resonance frequency between 0.5 and 5 Hz. By reversing the input and outputs of the physical system, it is possible to assemble additional mathematical inputs from the physical output forces and mathematical constructs (e.g. square root of inline force). Depending on the specific construct, this can improve the summed multiple coherence at frequencies where the response magnitude is low. In the present case this is between 6 and 20 Hz. The statistical measures of the response force time histories of each of the twelve subjects indicate that there are potential anatomical ‘end-stops’ for the sprung mass in the inline axis. No previous study has applied this reverse path multi-input-single-output approach to human vibration kinematic and kinetic data before. The implementation demonstrated in the present study will allow new and existing data to be examined using this different analytical tool.

Dynamic forces over the interface between a seated human body and a rigid seat during vertical whole-body vibration

Biodynamic responses of the seated human body are usually measured and modelled assuming a single point of vibration excitation. With vertical vibration excitation, this study investigated how forces are distributed over the body-seat interface. Vertical and fore-and-aft forces were measured beneath the ischial tuberosities, middle thighs, and front thighs of 14 subjects sitting on a rigid flat seat in three postures with different thigh contact while exposed to random vertical vibration at three magnitudes. Measures of apparent mass were calculated from transfer functions between the vertical acceleration of the seat and the vertical or fore-and-aft forces measured at the three locations, and the sum of these forces. When sitting normally or sitting with a high footrest, vertical forces at the ischial tuberosities dominated the vertical apparent mass. With feet unsupported to give increased thigh contact, vertical forces at the front thighs were dominant around 8Hz. Around 3–7Hz, fore-and-aft forces at the middle thighs dominated the fore-and-aft cross-axis apparent mass. Around 8–10Hz, fore-and-aft forces were dominant at the ischial tuberosities with feet supported but at the front thighs with feet unsupported. All apparent masses were nonlinear: as the vibration magnitude increased the resonance frequencies decreased. With feet unsupported, the nonlinearity in the apparent mass was greater at the front thighs than at the ischial tuberosities. It is concluded that when the thighs are supported on a seat it is not appropriate to assume the body has a single point of vibration excitation.

Finite element modelling of human-seat interactions: vertical in-line and fore-and-aft cross-axis apparent mass when sitting on a rigid seat without backrest and exposed to vertical vibration

Biodynamic models representing distributed human-seat interactions can assist seat design. This study sought to develop a finite element (FE) model representing the soft tissues of the body supported by seating and the vertical in-line apparent mass and the fore-and-aft cross-axis apparent mass of the seated human body during vertical vibration excitation. The model was developed with rigid parts representing the torso segments, skeletal structures (pelvis and femurs) and deformable parts representing the soft tissues of the buttocks and the thighs. The model had three vibration modes at frequencies less than 15 Hz and provided reasonable vertical in-line apparent mass and fore-and-aft cross-axis apparent mass. The model can be developed to represent dynamic interactions between the body and a seat over a seat surface (e.g. dynamic pressure distributions and variations in seat transmissibility over the seat surface).

Biodynamic Response of the Seated Human Body to Single-axis and Dual-axis Vibration: Effect of Backrest and Non-linearity

The biodynamic responses to the human body give an understanding of why human responses to vibration (changes in health, comfort, and performance) vary with the frequency and direction of vibration. Studies have shown that biodynamic responses also vary with the magnitude of vibration and that the backrests of seats influence the transmission of vibration to the seated human body. There has been little study of the nonlinearity in the biodynamic responses of the body to dual-axis excitation and no study of the influence of backrests during dual-axis excitation. This study investigated the apparent mass and cross-axis apparent mass of the human body exposed to random vibration (0.2 to 20 Hz) in all 15 possible combinations of four magnitudes (0, 0.25, 0.5 and 1.0 ms(-2) r.m.s.) of fore-and-aft vibration and the same four magnitudes of vertical vibration. Nonlinearity was evident, with the body softening with increasing magnitude of vibration when using a fixed magnitude of vibration in one direction and varying the magnitude of vibration in the other direction. The fore-and-aft apparent mass on the seat was greater without a backrest at the lower frequencies but greater with a backrest at the higher frequencies. The vertical apparent mass on the seat was decreased by the backrest at low frequencies. Cross-axis coupling was evident, with excitation in one axis producing a response in the other axis. It is concluded that the nonlinearity of the body evident during single-axis and multi-axis vibration, and the influence of backrests, should be taken into account when determining frequency weightings for predicting human responses to vibration and when optimising the dynamics of seating to minimise exposure to vibration.

An analytic model of the in-line and cross-axis apparent mass of the seated human body exposed to vertical vibration with and without a backrest

During vertical excitation of the seated human body there are vertical and fore-and-aft forces at the seat that are influenced by contact with a backrest, so it is desirable to take into account the effect of a backrest when developing models of the seated human body. Initially, a seven degree-of-freedom multi-body dynamic model was developed for the human body sitting with an upright posture unsupported by a backrest and exposed to vertical vibration. The model was optimized to fit the vertical apparent mass and the fore-and-aft cross-axis apparent mass measured on a seat. The model was then extended by the addition of vertical and fore-and-aft reaction forces to the upper lumbar spine to model the interaction between the human body and a backrest. By minimizing the least square error between experimental data and the analytical solution of the apparent masses on the seat and at the back, the human body model was able to represent both the vertical apparent mass and the fore-and-aft cross-axis apparent mass on the seat and at the back. Parameter sensitivity studies showed that the vertical apparent mass and the fore-and-aft cross-axis apparent mass on the seat and the backrest were all highly sensitive to the axial stiffness of the tissue beneath the pelvis. Pitch motion of the upper-body contributed to the vertical apparent mass and the fore-and-aft cross-axis apparent mass on the seat. The apparent mass at the back was more sensitive to the stiffness and damping of the lower back than the properties of the upper back.

A model of the vertical apparent mass and the fore-and-aft cross-axis apparent mass of the human body during vertical whole-body vibration

The apparent mass of the human body reflects gross movements caused by whole-body vibration and can be used to predict the influence of body dynamics on seat transmissibility. With vertical excitation, various models fit the measured vertical apparent mass of the human body, but experiments also show high fore-and-aft forces on the seat (the fore-and-aft cross-axis apparent mass) that have not influenced current models. This paper defines a model that predicts the vertical apparent mass and the fore-and-aft cross-axis apparent mass of the seated human body during vertical excitation. A three degree-of-freedom model with vertical, fore-and-aft and rotational (i.e. pitch) degrees of freedom has been developed with twelve model parameters (representing inertia, stiffness, damping, and geometry) optimised to the measured vertical apparent mass and the measured fore-and-aft cross-axis apparent mass of the body. The model provides close fits to the moduli and phases for both median data and the responses of 12 individual subjects. The optimum model parameters found by fitting to the median apparent mass of 12 subjects were similar to the medians of the same parameters found by fitting to the individual apparent masses of the same 12 subjects. The model suggests the seated human body undergoes fore-and-aft motion on a seat when exposed to vertical excitation, with the primary resonance frequency of the apparent mass arising from vertical motion of the body. According to the model, changes in the vertical, fore-and-aft, or rotational degree of freedom have an effect on the resonance in the fore-and-aft cross-axis apparent mass.

Modelling resonances of the standing body exposed to vertical whole-body vibration: Effects of posture

Lumped parameter mathematical models representing anatomical parts of the human body have been developed to represent body motions associated with resonances of the vertical apparent mass and the fore-and-aft cross-axis apparent mass of the human body standing in five different postures: ‘upright’, ‘lordotic’, ‘anterior lean’, ‘knees bent’, and ‘knees more bent’. The inertial and geometric parameters of the models were determined from published anthropometric data. Stiffness and damping parameters were obtained by comparing model responses with experimental data obtained previously. The principal resonance of the vertical apparent mass, and the first peak in the fore-and-aft cross-axis apparent mass, of the standing body in an upright posture (at 5–6 Hz) corresponded to vertical motion of the viscera in phase with the vertical motion of the entire body due to deformation of the tissues at the soles of the feet, with pitch motion of the pelvis out of phase with pitch motion of the upper body above the pelvis. Upward motion of the body was in phase with the forward pitch motion of the pelvis. Changing the posture of the upper body had minor effects on the mode associated with the principal resonances of the apparent mass and cross-axis apparent mass, but the mode changed significantly with bending of the legs. In legs-bent postures, the principal resonance (at about 3 Hz) was attributed to bending of the legs coupled with pitch motion of the pelvis in phase with pitch motion of the upper body. In this mode, extension of the legs was in phase with the forward pitch motion of the upper body and the upward vertical motion of the viscera.

Comparison of the apparent masses and cross-axis apparent masses of seated humans exposed to single- and dual-axis whole-body vibration

Humans are exposed to whole-body vibration in many types of environment. In almost all cases, the vibration to which the human is exposed comprises multi-axis vibration, such that vibration occurs in all directions simultaneously. Despite the complex nature of vibration to which humans are exposed in the workplace, almost all laboratory studies investigating the biomechanical response of the person have been completed using single-axis simulators. This paper presents a study whereby 15 male subjects were exposed to single-axis whole-body vibration in the x-, y- and z-directions and dual-axis vibration in the xy-, xz-, and yz-directions using a 6 degree-of-freedom vibration simulator. All vibration magnitudes were 0.4 ms −2 rms in each axis. Acceleration and force was measured in the x-, y-, and z-direction during all trials. Subjects sat in two postures (‘back-on’ and ‘back-off’) on a flat rigid seat. Apparent masses measured using single-axis and dual-axis vibration stimuli showed comparable results; similarly, cross-axis apparent masses (i.e. the ratio of the force in one direction to the acceleration in another direction) were almost identical for the single- and dual-axis vibration stimuli. All results were in agreement with data previously published using single-axis vibration. In most cases, the peaks in the apparent mass and the cross-axis apparent mass occurred at a slightly lower frequency for the dual-axis vibration than for the single-axis vibration. It is hypothesised that this change is due to a nonlinear effect, analogous to that which occurs with increasing vibration magnitude for single-axis vibration.

Apparent mass and cross-axis apparent mass of standing subjects during exposure to vertical whole-body vibration

The effects of posture and vibration magnitude on the vertical apparent mass and the fore-and-aft cross-axis apparent mass of the standing human body during exposure to vertical vibration have been investigated. Twelve male subjects were exposed to random vertical vibration over the frequency range 2.0–20 Hz at three vibration magnitudes: 0.125, 0.25 and 0.5 m s −2 rms. Subjects stood in five different postures: upright, lordotic, anterior lean, knees bent and knees more bent. The vertical acceleration at the floor and the forces in the vertical and fore-and-aft directions at the floor were used to obtain the apparent mass and the cross-axis apparent mass. The resonance frequency of the apparent mass was significantly reduced with knees bent and knees more bent postures, but there were only minor effects on the resonance frequency by changing the position of the upper body. Considerable cross-axis apparent mass, up to about 30% of the static mass of subjects, was found. The cross-axis apparent mass was influenced by all postural changes used in the study. In all postures the resonance frequencies of the apparent mass and the cross-axis apparent mass tended to decrease with increasing vibration magnitude. This nonlinear characteristic tended to be less clear in some postures in which subjects increased muscle tension.

Effect of Backrest and Torso Twist on the Apparent Mass of the Seated Body Exposed to Vertical Vibration

Occupational exposure to whole-body vibration is often combined with a requirement to perform twisting actions. This paper reports a study where the effect of twisting on the biomechanical response of the seated person was investigated. Twelve male subjects were exposed to vertical random whole-body vibration at 0.4 m/s2 r.m.s. Each subject sat in four different postures: 'back-on', 'back-off', 'twist' (where subjects were required to twist the torso by 90 degrees) and 'move' (where subjects were required to performing a moving task with extended arms). Similar apparent masses were measured for the 'back-on', 'back-off' and 'twist' conditions, where a peak occurred at about 6 Hz. For the 'move' condition, the peak in the apparent mass was attenuated indicating a different biomechanical response in this posture. The 6 Hz peak in fore-and-aft cross-axis apparent mass was eliminated in the 'move' condition. It is suggested that the change in biomechanical response is due to either the extended arms acting as a passive vibration absorber or that the twisting action interferes with the usual acceleration-muscle feedback system. Further work will be required to test these hypotheses.

Effect of seat surface angle on forces at the seat surface during whole-body vertical vibration

Twelve male subjects have been exposed to whole-body vertical random vibration so as to investigate the effect of seat surface angle, vibration magnitude and contact with a backrest on the ‘vertical apparent mass’ (calculated from forces normal to the seat surface and vertical acceleration) and ‘fore-and-aft cross-axis apparent mass’ (calculated from forces parallel to the seat surface and vertical acceleration). At each of four seat surface angles (0°, 5°, 10°, and 15°), the subjects were exposed to four vibration magnitudes (0.125, 0.25, 0.625, and 1.25 m s −2 rms) in the frequency range 0.25–15 Hz. The ‘vertical apparent mass’ and ‘fore-and-aft cross-axis apparent mass’ on the seat surface suggested resonances in the vicinity of 5 and 4 Hz, respectively. At all seat angles, both with and without a backrest, the resonance frequency in the ‘vertical apparent mass’ was greater than the resonance frequency in the ‘fore-and-aft cross-axis apparent mass’. Within subjects, the two resonance frequencies were not correlated in any condition. Seat angles up to 15° had a negligible effect on the ‘vertical apparent mass’ but a considerable effect on the ‘fore-and-aft cross-axis apparent mass’ on the seat surface, where ‘cross-axis apparent mass’ increased with increasing seat angle. At all seat angles, increasing the vibration magnitude decreased the resonance frequency in both directions. The least significant decrease in resonance frequency with increasing vibration magnitude occurred in the ‘fore-and-aft cross-axis apparent mass’ at the maximum seat angle of 15°. At low frequencies, the backrest reduced the forces in both directions, with the reduction greatest in the ‘fore-and-aft’ direction. The ‘fore-and-aft cross-axis apparent mass’ at resonance was correlated with subject mass and subject stature.