Vertical Apparent Mass Research Articles

Low-frequency vibration analysis of human body in semi-supine posture exposed to vertical excitation

In the new space vehicle architecture, the astronauts in semi-supine posture would be exposed to elevated vibration level. The present study is focused to develop a multi-body model of human in a semi-supine posture to analyze their biodynamic responses due to vertical excitation. The study proposes a linear 15-DOF model of a semi-supine human in the sagittal plane. The human body model is represented by five body segments as head-neck, torso-arm, pelvis, thigh, and legs. Each segment is interconnected with the translational and rotational spring-damper elements to simulate the relative motion. The tissues in contact with the rigid vibrating support are modeled using translational spring-damper elements. The model parameters were identified using optimization scheme by minimizing the least square error between normalized magnitude and phase of predicted and target vertical apparent mass in the frequency domain. The parameter sensitivity analysis shows that vertical stiffness and damping of underneath tissue of torso has substantial influence on peak modulus of apparent mass and corresponding resonance frequency. The modal analysis reveals that the principal resonance frequency for human body in semi-supine posture is greater as compared to seated and standing posture.

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.

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.

Vertical and dual-axis vibration of the seated human body: Nonlinearity, cross-axis coupling, and associations between resonances in transmissibility and apparent mass

The vertical apparent mass of the human body exhibits nonlinearity, with the principal resonance frequency reducing as the vibration magnitude increases. Measures of the transmission of vibration to the spine and the pelvis have suggested complex modes are responsible for the dominant resonance during vertical excitation, but the modes present with dual-axis excitation have not been investigated. This study was designed to examine how the apparent mass and transmissibility of the human body depend on the magnitude of vertical excitation and the addition of fore-and-aft excitation, and the relation between the apparent mass and the transmissibility of the body. The movement of the body (over the first, fifth and twelfth thoracic vertebrae, the third lumbar vertebra, and the pelvis) in the fore-and-aft and vertical directions (and in pitch at the pelvis) was measured in 12 male subjects sitting with their hands on their laps during random vertical vibration excitation (over the range 0.25–20Hz) at three vibration magnitudes (0.25, 0.5 and 1.0ms−2 rms). At the highest magnitude of vertical excitation (1.0ms−2 rms) the effect of adding fore-aft vibration (at 0.25, 0.5, and 1.0ms−2 rms) was investigated. The forces in the vertical and fore-and-aft directions on the seat surface were also measured so as to calculate apparent masses. Resonances in the apparent mass and transmissibility to the spine and pelvis in the fore-and-aft and vertical directions, and pitch transmissibility to the pelvis, shifted to lower frequencies as the magnitude of vertical excitation increased and as the magnitude of the additional fore-and-aft excitation increased. The nonlinear resonant behaviour of the apparent mass and transmissibility during dual-axis vibration excitation suggests coupling between the principal mode associated with vertical excitation and the cross-axis influence of fore-and-aft excitation. The transmissibility measures are consistent with complex modes contributing to motion of the body at the principal resonance: pitch motions of the upper thoracic and lumbar spine, and vertical and fore-aft motion of the pelvis and spine. The mode varies with the magnitude of vertical and fore-and-aft excitation.

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.

6축 힘측정판을 이용한 수직방향 전신진동에 대한 겉보기질량 및 겉보기편심질량에 대한 고찰

When whole-body is exposed to vertical vibration, asymmetry shape of human body affects the response on the translational(fore-aft, lateral, vertical) and rotational(roll, pitch, yaw) motion. While the translational motion has been studied with various titles, it has been rare to study the rotational motion of human body exposed to vertical excitation because of lack of experimental equipment. This study was performed by using a 6-axis force plate installing strain gage type sensors for the rotational response. Sixteen male subjects were exposed to vertical vibration on rigid seat in order to investigate apparent mass of three translational motion and apparent eccentric mass of three rotational motion. Random signal was generated to make excitation vibration which was on an effective frequency range of 3~40 Hz, and magnitude of 0.224 m/<TEX>$s^2$</TEX> r.m.s. The frequency range and magnitude used was selected for the vibration of passenger vehicle on idling condition. As the result, cross-axis apparent masses of fore-and-aft and lateral direction were not significant showing 20 % and 3 % of vertical apparent mass relatively. And apparent eccentric mass of pitch motion was dominant when compared to that of roll and yaw motion, which is reasoned by asymmetry direction of human body sitting on a seat.

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.

Apparent mass of the human body in the vertical direction: Inter-subject variability

The biodynamic responses of the seated human body to whole-body vibration vary considerably between people, but the reasons for the variability are not well understood. This study was designed to determine how the physical characteristics of people affect their apparent mass and whether inter-subject variability is influenced by the magnitude of vibration and the support of a seat backrest. The vertical apparent masses of 80 seated adults (41 males and 39 females aged 18–65) were measured at frequencies between 0.6 and 20 Hz with four backrest conditions (no backrest, upright rigid backrest, reclined rigid backrest, reclined foam backrest) and with three magnitudes of random vibration (0.5, 1.0 and 1.5 m s -2 rms). Relationships between subject physical characteristics (age, gender, weight, and anthropometry) and subject apparent mass were investigated with multiple regression models. The strongest predictor of the modulus of the vertical apparent mass at 0.6 Hz, at resonance, and at 12 Hz was body weight, with other factors having only a marginal effect. After correction for other variables, the principal resonance frequency was most consistently associated with age and body mass index. As age increased from 18 to 65 years, the resonance frequency increased by up to 1.7 Hz, and when the body mass index was increased from 18 to 34 kg m −2 the resonance frequency decreased by up to 1.7 Hz. These changes were greater than the 0.9-Hz increase in resonance frequency between sitting without a backrest and sitting with a reclined rigid backrest, and greater than the 1.0-Hz reduction in resonance frequency when the magnitude of vibration increased from 0.5 to 1.5 m s −2 rms. It is concluded that the effects of age, body mass index, posture, vibration magnitude, and weight should be taken into account when defining the vertical apparent mass of the seated human body.

Biodynamic Response and Spinal Load Estimation of Seated Body in Vibration Using Finite Element Modeling

Trunk biomechanical models play an indispensable role in predicting muscle forces and spinal loads under whole-body vibration (WBV) exposures. Earlier measurements on the force-motion biodynamic response (impedance, apparent mass) at the body-seat interface and vibration transmissibility (seat to head) have led to the development of different mechanical models. Such models could simulate the overall passive response and serve as an important tool for vehicle seat design. They cannot, however, evaluate physiological parameters of interest under the WBV. On the contrary, anatomical models simulating human's physiological characteristics can predict activities in muscles and their dynamic effects on the spine. In this study, a kinematics-driven nonlinear finite element model of the spine, in which the kinematics data are prescribed, is used to analyse the trunk response in seated WBV. Predictions of the active model (i.e., with varying muscle forces) as compared with the passive model (i.e., with no muscle forces) compared satisfactorily with measurements on vertical apparent mass and seat-to-head transmissibility biodynamic responses. Results demonstrated the crucial role of muscle forces in the dynamic response of the trunk. Muscle forces, while maintaining trunk equilibrium, substantially increased the compression and shear forces on the spine and, hence, the risk of tissue injury.

A Variable Parameter Single Degree-of-freedom Model for Predicting the Effects of Sitting Posture and Vibration Magnitude on the Vertical Apparent Mass of the Human Body

Models of the vertical apparent mass of the human body are mostly restricted to a sitting posture unsupported by a backrest and ignore the variations in apparent mass associated with changes in posture and changes in the magnitude of vibration. Using findings from experimental research, this study fitted a single degree-of-freedom lumped parameter model to the measured vertical apparent mass of the body measured with a range of sitting postures and vibration magnitudes. The resulting model reflects the effects of reclining a rigid backrest or reclining a foam backrest (from 0 to 30 degrees), the effects of moving the hands from the lap to a steering wheel, the effects of moving the horizontal position of the feet, and the effects of vibration magnitude (from 0.125 to 1.6 ms(-2) r.m.s.). The error between the modelled and the measured apparent mass was minimised, for both the apparent masses of individual subjects and the median apparent masses of groups of 12 subjects, for each sitting posture and each vibration magnitude. Trends in model parameters, the damping ratios, and the damped natural frequencies were identified as a function of the model variables and show the effects of posture and vibration magnitude on body dynamics. For example, contact with a rigid backrest increased the derived damped natural frequency of the principal resonance as a result of reduced moving mass and increased stiffness. When the rigid backrest was reclined from 0 to 30º, the damping decreased and the resonance frequency increased as a result of reduced moving mass. It is concluded that, by appropriate variations in model parameters, a single degree-of-freedom model can provide a useful fit to the vertical apparent mass of the human body over a wide range of postures and vibration magnitudes. When measuring or modelling seat transmissibility, it may be difficult to justify an apparent mass model with more than a single degree-of-freedom if it does not reflect the large influences of vibration magnitude, body posture, and individual variability.

Biodynamic Responses of the Seated Human Body to Single-axis and Dual-axis Vibration

Occupational exposures to vibration always involve multi-axis vibration. Since human responses to vibration are highly nonlinear and cross-coupled, it is to be expected that excitation in one axis will alter response to vibration in another axis. The purpose of this study was to investigate nonlinearity in the apparent masses of subjects seated without a backrest and exposed to single-axis and dual-axis vertical and fore-and-aft excitation. The driving point apparent masses and cross-axis apparent masses in the two translational directions were measured with twelve subjects exposed to random vibration (0.2 to 20 Hz) in all 15 possible combinations of four vibration magnitudes (0, 0.25, 0.5, or 1.0 ms(-2) r.m.s.) in the fore-and-aft and vertical directions. With single-axis excitation (either fore-and-aft or vertical), the median in-line apparent mass exhibited a nonlinear characteristic in which the body softened with increasing magnitude of vibration. With dual-axis excitation, at all magnitudes of vertical excitation the resonance frequency in the vertical apparent mass reduced as the magnitude of fore-and-aft vibration increased, and at all except the greatest magnitude of fore-and-aft excitation the resonance frequency in the fore-and-aft apparent mass reduced as the magnitude of vertical vibration increased. The coherency between the fore-and-aft acceleration and the fore-and-aft force was lowered by the addition of vertical excitation, and the coherency between the vertical acceleration and the vertical force was lowered by the addition of fore-and-aft excitation. The nonlinearity evident in both in-line apparent masses was also evident in the cross-axis apparent masses. It is concluded that with dual-axis excitation the fore-and-aft and vertical response of the seated human body is nonlinear, with resonance frequencies decreasing with increasing magnitude of vibration. Consequently, vibration in one axis (either fore-and-aft or vertical) affects the apparent mass of the body measured in the other axis (either vertical or fore-and-aft).

Apparent mass of the human body in the vertical direction: Effect of a footrest and a steering wheel

The apparent mass of the seated human body influences the vibration transmitted through a car seat. The apparent mass of the body is known to be influenced by sitting posture but the influence of the position of the hands and the feet is not well understood. This study was designed to quantify the influence of steering wheel location and the position of a footrest on the vertical apparent mass of the human body. The influences of the forces applied by the hands to a steering wheel and by the feet to a footrest were also investigated. Twelve subjects were exposed to whole-body vertical random vibration (1.0 m s−2 rms over the frequency range 0.13–40.0 Hz) while supported by a rigid seat with a backrest reclined to 15°. The apparent mass of the body was measured with five horizontal positions and three vertical positions of a steering wheel and also with hands in the lap, and with five horizontal positions of a footrest. The influence of five forward forces (0, 50, 100, 150, 200 N) applied separately to the ‘steering wheel’ and the footrest were also investigated as well as a ‘no backrest’ condition. With their hands in their laps, subjects exhibited a resonance around 6.7 Hz, compared to 4.8 Hz when sitting upright with no backrest. In the same posture holding a steering wheel, the mass supported on the seat surface decreased and there was an additional resonance at 4 Hz. Moving the steering wheel away from the body reduced the apparent mass at the primary resonance frequency and increased the apparent mass around the 4 Hz resonance. As the feet moved forward, the mass supported on the seat surface decreased, indicating that the backrest and footrest supported a greater proportion of the subject weight. Applying force to either the steering wheel or the footrest reduced the apparent mass at resonance and decreased the mass supported on the seat surface. It is concluded that the positions and contact conditions of the hands and the feet affect the biodynamic response of the body in a car driving posture. As the biodynamic response influences the vibration transmitted through seats, these factors should be considered in dynamic models of vehicle seating.

Apparent mass of the human body in the vertical direction: Effect of seat backrest

The transmission of vibration through a seat depends on various characteristics of the seat and the dynamic response of the human body. The dynamic response of the body can be represented by its apparent mass, but the effect of the seat on the apparent mass of the body is not well understood. This study was designed to quantify the effect of foam and rigid backrests on the vertical apparent mass of the human measured at the seat surface supporting the body. The apparent masses of 12 subjects were measured during exposure to random vertical vibration (1.0 ms −2 rms from 0.125 to 40 Hz) in a seat with a rigid backrest, in the same rigid seat with three thicknesses of foam backrest (50, 100 and 150 mm), and in the same seat with no backrest. The backrests were inclined at various angles: 0°, 5°, 10°, 15°, 20°, 25° and 30° for the rigid and 100 mm foam backrests, and 0°, 10°, 20° and 30° for the 50 and 150 mm foam backrests. With all vertical backrests (i.e., 0° inclination), there were resonances in the apparent mass of the body around 5 and 10 Hz. With no backrest, the apparent mass was increased at frequencies less than the resonance frequency but decreased at frequencies between 8 and 20 Hz, relative to the apparent mass with the vertical rigid and foam backrests. With the rigid backrest, the primary resonance frequencies in the apparent mass increased with increasing backrest inclination. With the foam backrests, the resonance frequencies decreased with increasing backrest inclination. At frequencies less than the primary resonance, the apparent mass decreased with increasing backrest inclination, particularly with the rigid backrest. Between 8 and 15 Hz, the apparent mass decreased with increasing inclination, most notably with the foam backrests. At inclinations less than 30°, there was little effect of foam thickness on the apparent mass, but at 30° an increase in the thickness of the foam decreased the frequency of the first resonances. Since contact with backrests and the characteristics of backrests influence the vertical apparent mass of the seated body, backrests should be expected to influence the transmission of vertical vibration through a supporting seat cushion.

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.

Relationship between measured apparent mass and seat-to-head transmissibility responses of seated occupants exposed to vertical vibration

The “to-the-body” and “through-the-body” biodynamic response functions of the seated human body exposed to vertical vibration are measured and analyzed in an attempt to identify relationships between the apparent mass and seat-to-head transmissibility measures. The experiments involved 12 male subjects exposed to three magnitudes of whole-body vertical random vibration (0.25,0.5,1.0 m/s 2 rms acceleration) in the 0.5–15 Hz frequency range, and seated with three back support conditions (none, vertical and inclined), and two different hands positions (hands in lap and hands on the steering wheel). The vertical apparent mass and seat-to-head transmissibility responses were acquired during the experiments, where the head acceleration was measured using a light and adjustable helmet-strap mounted accelerometer. The results showed that both the measured responses show good agreements in the primary resonances, irrespective of the back support condition, while considerable differences between the normalized apparent mass and seat-to-head transmissibility could be seen in the secondary resonance range for the two back supported postures. The seat-to-head transmissibility responses are further shown to be relatively sensitive to back supported postures compared with that of apparent mass responses. Relatively stronger effects of hands position were observed on the seat-to-head transmissibility responses compared with the apparent mass responses under back supported conditions. From the results, it is further concluded that seat-to-head transmissibility emphasizes the biodynamic response in the vicinity of the secondary resonance compared to the apparent mass. The seat-to-head transmissibility measure is thus considered to be more appropriate for describing seated body response to higher frequency vibration.

Nonlinear dual-axis biodynamic response of the semi-supine human body during vertical whole-body vibration

Nonlinear biodynamic responses are evident in many studies of the apparent masses of sitting and standing subjects in static postures that require muscle activity for postural control. In the present study, 12 male subjects adopted a relaxed semi-supine posture assumed to involve less muscle activity than during static sitting and standing. The supine subjects were exposed to two types of vertical vibration (in the x-axis of the semi-supine body): (i) continuous random vibration (0.25–20 Hz) at five magnitudes (0.125, 0.25, 0.5, 0.75, and 1.0 m s −2 rms); (ii) intermittent random vibration (0.25–20 Hz) alternately at 0.25 and 1.0 m s −2 rms. With continuous random vibration, the dominant primary resonance frequency in the median normalised apparent mass decreased from 10.35 to 7.32 Hz as the vibration magnitude increased from 0.125 to 1.0 m s −2 rms. This nonlinear response was apparent in both the vertical ( x-axis) apparent mass and in the horizontal ( z-axis) cross-axis apparent mass. As the vibration magnitude increased from 0.25 to 1.0 m s −2 rms, the median resonance frequency of the apparent mass with intermittent random vibration decreased from 9.28 to 8.06 Hz whereas, over the same range of magnitudes with continuous random vibration, the resonance frequency decreased from 9.62 to 7.81 Hz. The median change in the resonance frequency (between 0.25 and 1.0 m s −2 rms) was 1.37 Hz with the intermittent random vibration and 1.71 with the continuous random vibration. With the intermittent vibration, the resonance frequency was higher at the high magnitude and lower at the low magnitude than with continuous vibration of the same magnitudes. The response was typical of thixotropy that may be a primary cause of the nonlinear biodynamic responses to whole-body vibration.

Apparent mass of small children: Modelling

Mass-spring-damper models are widely available for quantifying the whole-body vibration characteristics of primates, human adolescents and human adults, but no models have previously been developed for small children. In this study a single degree of freedom, linear, mass-spring-damper with base support model was determined from the seated vertical apparent mass modulus function of each of eight small children of less than 18 kg in mass. A differential evolution optimisation algorithm was used in conjunction with a mean squared error measure and penalty functions to identify the optimal child model parameter values. The eight child models were characterised by a mean moving mass m 1 of 8.5 kg, a mean body stiffness k 1 of 21,131 N/m and a mean damping coefficient c 1 of 329 Ns/m. Comparison to the parameter values of similar models reported in the literature for Rhesus monkeys, Baboons, large children and adults suggests that the values obtained in the current study for small children are intermediate between the smaller primates and the larger humans. A regression analysis of the model parameters was performed as a function of subject mass for a data set consisting of the eight child models, 12 similar models for primates, and 60 similar models for large children and adults. The moving mass m 1 of the group of models grew with a power exponent of approximately unity, the body stiffness k 1 grew with a power exponent of approximately +1/2, the damping coefficient c 1 grew with a power exponent of approximately +3/4 and the dimensionless damping ratio was independent of subject mass. The natural frequency of the models grew with a power exponent of approximately −1/4.

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 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.